Unification of Finite Element Methods

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UNIFICATION OF FINITE ELEMENT METHODS

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NORTH-HOLLAND MATHEMATICS STUDIES

UNIFICATION OF FINITE ELEMENT METHODS

Edited by

H. KARDESTUNCER University of Connecticut Storrs Connecticut U.S.A.

1984

NORTH-HOLLAND-AMSTERDAM . NEW YORK. OXFORD

94

I S B N : 0 444 X 7.5100

Piihli,slier.s:

ELSE 'IER SCIENCE PUBLIS IERS R . V . P . O . Box IY91 1000 BZ Anisterdam The Netherlands Sole rli.strihritorsfbr tlie U .S . A . trtrrl Cirtiot!ii:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC 52 Vantlerbilt Avenue NewYork. N.Y. 10017 U.S.A.

Library of Congresi Cataloging in Publication Data

Main entry under t i t l e : Unification of f i n i t e element methods. (North-Holland mathematics studies) Bibliography: p 1. Finite element method. 2. Argyris, J. B. (John E.), 1916I. Kardestuncer, m e t t i n . 11. Series. TA347.F5U55 1984 620' .001'515353 84-6006 ISBN 0-444-87519-0 ( U . 6 . )

.

.

PRINTED IN T H E NETHERLANDS

7th

Dedicated to

Professor John H. Argyris for his pioneering and continuing contributions to the finite element methods

Alliance of Industry and Academe

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vii

PROFESSOR JOHN H. ARGYRIS A man who unifies engineering and mathematics with elegance My first encounter with Professor John H. Argyris’ work occurred during my graduate studies at MIT in the mid fifties. His elegant treatment of Bernoulli’s virtual work and energy principles mounted on Menabrea’s il minimo lavoro with Castigliano’s two theorems, St. Venant’s theories of torsion, Maxwell’s reciprocity principles, Lord Rayleigh’s variational principles, Muller-Breslau’s and Otto Mohr’s unit load ideas, etc. gave me the impression that this man belonged to the last century. Yet the methodology presented (stiffness and flexibility methods in structural analysis) was so new that it was unknown to my fellow students and did not even exist in the curriculum. A few years later, I learned that he was the holder of the prestigious Chair of Aeronautical Structures at the University of London where he was also Professor of Aerospace Sciences and at the same time was Director of the Institute of Statics and Dynamics and Director of the Computer Center at the University of Stuttgart. I began to wonder if perhaps there were more than one J.H. Argyris, and whose work was I studying? The more I studied his work and the more I learned of his accomplishments the more convinced I was that the man must be older than I thought; perhaps he was born a century before the last. However, when I finally met him in 1961, I was sure that he must be the grandson of the man whose work was so inspiring me and guiding my doctoral dissertation at the Sorbonne. A citizen of Great Britain, a resident of West Germany, John H. Argyris was born August 19, 191 6, in the Land of’the Gods. A child prodigy who graduated from the Technical University of Athens at the age of eighteen, he received the all-German Prize of Deutscher Stahlbauverband during his postgraduate studies in Munich when h e was only twenty years old. Many believe that it was not merely coincidental that Sir Isaac Newton was born o n Christmas day of 1642, the same day (with acceptable approximation based o n the theories presented in this volume) that another genius, Galileo Galilei, had died. I am curious to know what genius it was who died on August 19, 1916.

I had every intention here to write more about Professor Argyris and his work but the more I wrote the more I became convinced that my writing could in no way reflect the accomplishments of this great man. H i s life can not be told in an essay; his work can not be assessed in an article; his abundant energy can not be formulated as an energy functional. He is beyond and above all that most of us know of him. H. Kardestuncer

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MAIN DISTINCTIONS OF PROFESSOR JOHN H. ARGYRIS,D.Sc., Dr.h.c.mult. 1937

Dip1.-1ng.D.E.

Munich

1954

D.Sc. (Eng.)

University of London

1955

Fellow R.Ae.S.

Royal Aeronautical Society, London

1962

Honorary Associate Hon. A.C.G.I.

City Guilds Institute, London

1970

Laura h.c.dott.Ing.

University of Genoa, Special Distinction on the 100th Anniversary of the Faculty of Applied Mechanics and Ship Building

1971

George Taylor Prize

Royal Aeronautical Society, London

1971

Silver Medal

Royal Aeronautical Society, London

1972

Principal Editor

Computer Methods of Applied Mechanics and Engineering (Journal)

1972

dr.techn.h.c. and jus docendi

University of Norway, Trondheim

1973

Corresponding member

Academy of Sciences of Athens (Positive Sciences)

1974

Honorary Fellow

Groupe pour l'Avancement des MCthodes Numeriques de l'hgenieur (GAMNI), Paris

1975

von Karmin Medal

Highest Scientific Award, American Society of Civil Engineers, New York

1976

Honorary Fellow Hon.F.C.G.1.

City Guilds Institute, London

1979

Member A.S.C.E.

American Society of Civil Engineers, New York

Main Distinctions of Professor John H. Argyris

X

1979

Copernicus Medal

Highest Award in Natural Sciences Polish Academy of Sciences, Warsaw

1980

Gold Medal

of the Land Baden-Wurttemberg

1980

Honorary Professor

Northwest Polytechnical University, Xian, People’s Republic of China

1981

Timoshenko Medal

Highest Scientific Award, American Society of Mechanical Engineers, New York

1981

Life Member A.S.M.E.

American Society of Mechanical Engineers

1981

Member

The New York Academy of Sciences, New York

1982

I.B. Laskowitz Award with Gold Medal

Highest Astronautical Award of the New York Academy of Sciences

1983

Fellow of the AIAA

Highest Grade of Membership, American Institute of Aeronautics and Astronautics, New York

1983

Dr .Ing .E.h.

University of Hanover, Honorary Doctorate

1983

Honorary Professor

Technical University of Peking (Beijing)

1983

Honorary Life Member

New York Academy of Sciences, New York

1983

World Prize in Culture and Election as Personality of the Year 1984

Centro Studi e Ricerche delle Nazioni Accademia Italia, Salsomaggiore Terme

1984

Honorary Professor

Qinghua University, Beijing

1940

340 scientific publications

and continuing

xi

ACKN'OWLEDGMENTS The UFEM series could not take place without the generous help of the following friends, organizations, and societies. Their encouragement, support, and sharing of the ideals of the conference are sincerely appreciated and gratefully acknowledged.

Organizing Committee Members H. Clark, Hon. Chairman (UConn) H. Kardestuncer, Chairman (UConn) W.W. Bowley (UConn) J.J. Connor (MIT) H.A. Koenig (UConn) A. Phillips (Yale) R.J. Pryputniewicz (WPI)

H. Allik (BBN) W.W. Bowley (UConn) F. Camaretta (Sikorsky) A.D. Carlson (NUSC) M.K.V. Chari (General Electric) L. Collatz (Hamburg, Germany) J.H. Connor (MIT) A.C. Eringen (Princeton) S. Gordon (Electric Boat)

Session Chairmen H.A. Koenig (UConn) R. Lalkaka (United Nations) H. Mayer (Hamilton Standard) D.H. Norrie (Calgary) T. Onat (Yale) A. Phillips (Yale) T.H.H. Pian (MIT) J.A. Roulier (UConn)

Local Arrangements J.J. Farling (UConn, Conf. & Inst.) G.D. Smith (UConn) G.M. Wallace (UConn)

Analysis & Technology, Inc. AVCO Lycoming Corp. Bolt Beranek & Newman, Inc. Control Data Corp. Electric Boat General Electric Hamilton Standard Conf. & Inst. (UConn)

Advisory Board Members I. Babugka (Maryland) L. Collatz (Hamburg, Germany) A.C. Eringen (Princeton) R.H. Gallagher (Arizona) J.T. Oden (Texas) T.H.H. Pian (MIT) O.C. Zienkiewicz (Swansea, U.K.)

Participatmg Societies AIAA ASME CC-ASCE

Sponsoring Organizations Kaman Aerospace Corp. Naval Underwater Systems Center Northeast Utilities Perkin Elmer Pratt & Whitney Aircraft Sikorsky Aircraft UConn Foundation UConn Research Foundation

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LIST OF CONTRIBUTORS J.H. Argyris (l), Institut fur Statik und Dynamik der Luft- und Raumfahrtkonstruktionen, University of Stuttgart, Stuttgart, Fed. Rep. Germany. J.F. Abel (47), Department of Structural Engineering, Cornell University, Ithica, New York, U.S.A. S.N. Atlun’ (65), CACM, Georgia Institute o f Technology, Atlanta, Georgia, U.S.A. 1. Bubufku (97), Institute of Physical Science and Technology, University of Maryland, College Park, Maryland, U.S.A. K . 4 Buthe (123), Department o f Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. J. Bieluk (1 49), Department of Civil Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania, U.S. A. J.H. Bramble (1 67), Department of Mathematics, Cornell University, Ithica, New York, U.S.A. C A . Brebbiu (185), The Institute of Computational Mechanics, Ashurst Lodge, Southampton, England M.A. Celiu (303), Civil Engineering Department, Princeton University, Princeton, New Jersey, U.S.A. A. Chuudhury (1 23), Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. J. Sf. Doltsinis (l), Institut fur Statik und Dynamik der Luft- und Raumfahrtkonstruktionen, University of Stuttgart, Stuttgart, Fed. Rep. Germany. J. F. Hajjur (47), Department of Structural Engineering, Cornell University, Ithica, New York, U.S.A. T -Y. Hun (47), Department of Structural Engineering, Cornell University, Ithica, New York, U.S.A. A.R. Ingruffeu (47), Department of Structural Engineering, Cornell University, Ithica, New York, U.S.A. K. Izudpunah (97), Computational Mechanics Center, Washington University, St. Louis, Missouri, U.S.A. H. Kurdestuncer (207), Department of Civil Engineering, University o f Connecticut, Storrs, Connecticut, U.S.A. R. C MucCumy (149), Department of Civil Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania, U.S.A. D. S. Mulkus (235), Mathematics Department, Illinois Institute of Technology, Chicago, Illinois, U.S.A. A. Needleman (249), School of Engineering, Brown University, Providence, Rhode Island, U.S.A. 7: Nishioku (65), CACM, Georgia Institute of Technology, Altlanta, Georgia, U.S.A.

xiv

List

0.f

Contributors

A.K. Noor (275), NASA Langley Research Center, The George Washington University, Hampton, Virginia, U.S.A. E. T. Olsen (235), Mathematics Department, Illinois Institute of Technology, Chicago, Illinois, U.S.A. J.E. Pusciuk (1 67), Brookhaven National Laboratory, Upton, New York, U.S.A. R. Pemcchio (47), Department of Structural Engineering, Cornell University, Ithica, New York, U S A . A. Philpott (321), Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. G.F. Pinder (303), Civil Engineering Department, Princeton University, Princeton, New Jersey, U.S.A. R.J. Pryputniewicz (207), Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts, U.S.A. G. Strung (321), Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. B. Szubo (97), Computational Mechanics Center, Washington University, St. Louis, Missouri, U.S.A.

xv

PREFACE The 7th UFEM gathering, like its predecessors, advances further toward its goal of accomplishing “a unified method” in computational mechanics. No matter how powerful a methodology might be for a certain class of problems, it often presents shortcomings for others. Since engineering problems today are very complex and contain subregions with completely different physical and geometrical characteristics, certainly no single method is capable of handling the entirety of the problem. Consequently, the identification of various methodologies, each suitable for a particular subregion, and their unification have recently been in the minds of many researchers. The flow chart in Fig. 1 indicates three stages of such a unifica.tion: unified formulation, unified means, and unified methods.

Fig. 1. Flowchart for the Unification of Methods in Mechanics.

Preface

XVi

The components of the first stage of this unification are illustrated in Fig. 2. -_____ FORMULATIONS IN MECHANICS

t d---'-TENSORIAL E o u A ~ l o S

[

EMPIRICAL EOUATIONS

L_

, , /

/

, -

,

,

i

/ ' '

INTEGRAL EOUATIONS

_

-

-5-

,

,'

_

~

- .DIFFERENTIAL EOUATIONS ~

L

~--__

ANALYTICAL SOLUTIONS

-.-

Fig. 2. Unification of Formulations on Mechanics.

Many of the papers presented here address various stages of unification, and we believe that in the near future commercial or in-house codes will be developed to accomplish this task. The possibility of unifying various numerical methodologies using interactivegraphics has been investigated by John ABEL and his co-workers. Their work is fostering the unification concept with a unified means which interconnects analysis methods and design parameters. They are not only improving man-machine communication but communication between methodologies employed in different regions of the domain and stages of processing. ATLURI and NISHIOKA emphasize the unification (hybridization) of various methodologies (numerical, analytical, and experimental) in engineering for the solution of complex problems (e.g. crack propagation in 3-D domain with irregular geometry and material properties) for which none of the existing methodologies alone is sufficient. The authors have, in fact, been unifying these methodologies in their earlier work and they advocate the necessity of unification. The problems in this presentation, drawn from the field of fracture mechanics, demonstrate the use of more than one methodology (in time and space) for their solution. Undoubtedly, one can easily apply concepts presented in this paper to other problems. Intermethod compatibilities and error bounds, however, remain to be explored. In the opinion of the editor, the concepts presented here are firm enough ground to stand on when reaching for further goals in unification. Dealing primarily with problems for which energy functionals exist, BABUSKA and his co-workers present h-, p-, and h-p versions of the finite element methods.

Preface

xvii

They claim that error measures in stresses often do not follow monotonic behavior of the error measure in the energy norm. To overcome this difficulty, they introduce an extraction function and demonstrate the selection of such a function during adaptive post-processing. A numerical example accompanying the presentation uses the extraction technique. Contact problems, in particular between nonlinear deformable bodies subject to large deformation with sticking, sliding and separating, are among the most difficult problems in solid mechanics. BATHE and CHAUDHARY present a solution algorithm that they have developed for two-dimensional contact problems. They believe that alongside finite differences, finite elements, and surface integral techniques, there is still room for more reliable and effective algorithms to analyze general problems in this field. Numerical results for two problems - a pipe buried in soil and a traction of a rubber sheet embedded in a rigid channel - accompany the paper. BIELAK and MACCAMY unify variational finite element methods with the boundary integral equation method using the former in the interior of the domain and the latter at the exterior. They apply the methodology to a two dimensional electromagnetic interface problem: the interaction between air and a dielectric obstacle subject to two different sets of Maxwell’s equations. In t h s problem, a homogeneous differential equation defined over an infinite domain interfaces with a nonhomogeneous differential equation defined over a finite domain. After reviewing the fundamental principles b e h n d various approximate methods, BREBBIA embarks on the unification of finite elements and boundary elements. While acknowledging the power and potential of the former, he points out certain advantages of the latter and maintains that the complexity of the problems at hand necessitates combining (unifying) many methodologies. He refers to these as “the discrete element methods” and cites some recent attempts coinciding with the philosophy behind UFEM gatherings. KARDESTUNCER and PRYPUTNIEWICZ explore the possibility of unifying finite element modeling with laser experimentation in two different stages of the, procedure. The first part deals with evaluation of the stiffness and/or flexibility matrix coefficients for irregular (geometrically as well as physically) elements by experiment. The second part deals with determining the unknown values of the function by lasers. This, in turn, leads t o a reduction in the order of the stiffness matrix and to an increase in the accuracy of the results. Measurement techniques and numerical examples accompany the presentation. The main theme of the paper by MALKUS and OLSEN centers on the question of whether the NCR (Nagtegaal redundant constraint) element which fails to satisfy the LBB condition can be used for incompressible media. The NRC element is a quadrilateral macroelement with four triangles and has been used successfully by

xviii

Preface

others for problems involving inelastic deformation. Here, the authors discuss why NRC elements violate the LBB condition for convergence and how this condition can be removed so that the NRC element can be used for plane and axisymmetric incompressible flows, In order to demonstrate this, they present error estimates for the element when it is used in Stokesian flow. In his work, NEEDLEMAN applies finite element techniques to necking instabilities subject to classical and nonclassical constitutive relationships. The presentation is accompanied by a numerical analysis of tension tests using constitutive descriptions for polycrystalline metal. He presents remarkably good agreement between the FEM analysis and experimental results and claims that this is the result of incorporating into FEM modeling the constitutive relations for polycrystalline metals arising due to crystallographic texture. He also points out that localized shear stresses play a significant role in texture development. Two recent advances toward the unification of various methodologies in one physical problem are presented by A.K. NOOR. They are (i) a hybrid method based on the combination of the direct variational techniques with perturbation methods, and (ii) a two-stage direct variational technique. Advantages of both forms of unification are illustrated for nonlinear steady-state thermal and structural problems. The author also points out other combinations of various methodologies and research areas for more effective solution of nonlinear problems. Comparative numerical studies accompany the presentation. PHILPOTT and STRANG idealize the internal fiber of a human patella as a plane truss and then, using linear programming techniques, they try to optimize the system to accomplish minimum weight. After presenting the standard procedure for a fixed geometry problem, they develop an algorithm for problems with variable geometry, indeed an interesting and difficult task. Most truss problems consist of members with zero loads which in turn introduce degeneracy during the optimization procedure. The authors attempt in particular to deal with this difficulty of optimization. The alternating-direction collocation (ADC) method is presented by CELIA and PINDER with particular application to multi-dimensional transport equations. These authors enhance the ADC procedure by adding a small number of quintic elements along the principle direction of flow governed by the convection-dominated transport equation. A numerical example confined to a rectangular region and a flow chart for the enhanced ADC procedure accompany the presentation. All of these invited presentations have been written specifically to honor Professor Argyris with the understanding that they also follow as much as possible the spirit of the conference.

Preface

XiX

On behalf of the Organizers and Advisory Board members, I would like to thank our distinguished speakers, session chairmen, and participants for their kind cooperation. Some prepared their papers under a severe deadline, some traveled long distances, and some took time off from their demanding tasks to be here today. The result is most gratifying. The Editor

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xxi

CONTENTS Dedication to Professor John H. Argyris Main Distinctions of Professor John H. Argyris Acknowledgments List of Contributors Preface

Chapter 1

Chapter 2

Chapter 3 Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

On the Natural Approach to Flow Problems J.H. Argyris & J. St. Doltsinis (University of Stuttgart, Fed. Rep. Germany)

V

ix xi xiii xv

1

Interactive Computer Graphics for Finite Element, Boundary Element, & Finite Difference Methods J.F. Abel, A . R Ingraffea, R. Perucchio, T.-Y. Han, & J.F. Hajjar (Cornell University)

47

Hybrid Methods of Analysis S.N. Athri & T. Nishioka (GeorgiaInstitute of Technology)

65

The Postprocessing Technique in the Finite Element Method. The Theory & Experience I. Baburka (University ofMaryland), K. Izadpanah, & B. Szabo (Washington University)

97

On Finite Element Analysis of Large Deformation Frictional Contact Problems K . 4 Bathe & A . Chaudhary (MIT)

123

Mixed Variational Finite Element Methods for Interface Problems J. Bielak & R. C. MacCamy (Carnegie-Mellon University)

149

Preconditioned Iterative Methods for Nonselfadjoint or Indefinite Elliptic Boundary Value Problems J.H. Bramble (Cornell University) & J.E. Pasciak ( Brookhaven National Laboratory)

167

On the Unification of Finite Elements & Boundary Elements C.A. Brebbia (ICM, Southampton, England)

185

xxii

Chapter 9

Con ten t S

Unification of FEM with Laser Experimentation H. Kardestuncer (University of Connecticut) & R.J. Pryputniewicz (WorcesterPolytechnic Institute)

207

Chapter 10 Linear Crossed Triangles for Incompressible Media D.S.Malkus & E.T. Olsen (IIT, Chicago)

235

Chapter 11 The Numerical Analysis of Necking Instabilities A, Needleman (Brown University)

249

Chapter 12 Recent Advances in the Application of Variational Methods t o Nonlinear Problems A .K. Noor (NASA - Langley)

275

Chapter 13' Collocation Solution of the Transport Equation using a Locally Enhanced Alternating Direction Formulation M.A. Celia & G.F. Pinder (Princeton University)

303

Chapter 14 Numerical & Biological Shape Optimization A. Philpott & G. Strang (MIT)

32 1

Index

345

Unification of Finite Element Methods H. Kardestuncer (Editor) @ Elsevier Science Publishers B.V.(North-Holland), 1984

1

CHAPTER 1 ON THE NATURAL APPROACH TO FLOW PROBLEMS J. H. Argyris & J. St, Doltsinis

The paper surveys recent work on fluid dynamics performed a t the ISD, University of Stuttgart. I t i s i n particular directed to a natural description o f the flow phenomena and includes also a consideration o f thermally coupled problems. The derivation o f the relevant finite element equations when referred to natural quantities i s outlined and examples of application are given. For a discussion on the associated modern developments in numerical solution techniques the reader may consult 1281

.

1.

INTRODUCTION

The present paper surveys recent work on fluid dynamics performed a t the ISD, University of Stuttgart. The paper serves i n the main as a survey on modern developments i n finite element methods for fluid motion, and i s particularly devoted to a natural description o f the relevant phenomena. Its main attention i s focused on incompressible media. First draft of the theory has been presented a t a lecture given a t the Conference on Finite Elements in Water Resources i n Hanover i n 1982. In section 2, the natural terminology [l, 21 i s introduced and methodically applied to the formulation o f field quantities characteristic of fluid motion, such as the scalar pressure field and the vectorial velocity field. The condition of conservation of mass i s derived i n natural terms and natural measures for the stress and the rate of deformation are connected by the appropriate constitutive relations. Aiming a t the analysis of fluid motion coupled with thermal phenomena, the natural approach i s subsequently extended to the considemtion o f the temperature field and the heat flow 13

1.

Section 3 indicates the tmnsition to finite domains as a foundation for the development o f the finite element theory o f the flow problem. The streamline upwind/PetrovGalerkin formulation o f [ 4 1 may be used for the discretisation technique i n connection with either the strict fulfilment o f the incompressible statement or with the penaltyapproach to the condition o f incompressibility. In a subsequent step the finite element discretisation o f the thermally coupled fluid flow problem i s considered and the governing equations are established. For typographical brevity, we omit in t h i s paper a discussion of numerical integration schemes i n the time domain. Also the important task of an effective solution o f the

J.H. Argyris & J. St. Doltsinis

2

equations governing the flow problem i s not handled in the present contribution. For this purpose the reader i s referred to the presentation in1281

.

The theory presented in the paper i s applied in section 4 to the numerical analysis of some typical examples of viscous fluid motion. Thus, the convection dominated flow over a step i s considered for the two- and the three-dimensional case, and the solution o f thermally coupled flows i s demonstrated on the BBnard instability phenomenon i n a fluid between two planes of different temperatures. The interested reader may consult [281 for an analysis of cavity flows involving free and forced heat convection with a change from liquid to solid phase of the material.

2.

ON THE NATURAL APPROACH TO FLUID MOTION

2.1 Natural approach In the natural methodology of continuum mechanics, a l l considerations are established on or derived from an infinitesimal tetrahedron element which replaces the classical parallelopiped applied in the traditional cartesian point o f view. For comparison purposes both elements are shown in fig. 2.1 together with the associated coordinate systems. An elegclnt application of the tetrahedron element demands the use of supernumerary or homogeneous reference systems. One of these m y be defined by the directions of the six edges of the tetrahedron. The natural formulation of the mechanics and thermomechanics of solids [5, 3, 2 1 may be based on the Lagmngean approach in the sense that the tetrahedron constitutes then a moving and deforming material element. In our present considerations o f fluid motion, however, we prefer to adopt the Eulerian description L6, 71 in which the tetrahedron represents a fixed geometrical element in space. Before developing the natural concept we first review alternative representations of a vector i n three-dimensional space 13, 21 and illustrate then the argument on the twodimensional case depicted in fig. 3.2. Consider the vector r defined by cartesian en tries

In the natural terminology the vector r may, on the one hand, be composed from nonunique independent vectorial contributions taken along the tetrahedral edges

This forms the so-called component description of a vector. On the other hand, we may introduce as measures the unique orthogonal projections o f the vector f onto the natural coordinate axes,

This forms the total description o f a vector.

On the Natural Approach t o Flow Problems

3

Considering fig. 2.2, we observe that for a given component representation vector its artesian form r i s deduced by the transformation

re

of the

with the matrix

gNb =

[codq;)]

The total natural entries

4.=d)...,1;

I'=f,2, 3

(2.5)

of (2.3) are then obtained through the relation

where (2.4) has been used. n e symmetric matrix

establishes the direct connection between total and component definitions. I t i s evident that due to the redundancy of the natural quantities, the above transformations are not invertible

.

Finally, we note that the scalar product o f two arbitrary vectors Q and given in one of the equimlent forms

6 may be

as i s easily confirmed with the aid of (2.4) and (2.6)

2.2 Pressure field We proceed next to the description of a scalar field e.g. the pressure p i n the fluid which for arbitrary non-steady conditions i s a function of the time t and the position vector X We express this by

.

and examine the consequences of the different representations o f the vector W on the description of the scalar field. Positions may be defined by component coordinates X c dependence i n (2.9) the pressure gradient then reads

, cf.

(2.2). From the associated

where the chain rule confirms the transformation (2.6) between total natural and cartesian specification o f a gradient vector and justifies the total notation of (2.10).

J.H. Argyris & J. St. Doltsinis

4

Actually, &Cp) comprises the rates of change of p i n the natural directions (and hence the corresponding orthogonal projections of the gradient vector), Consider next the transformation rule (2.4) leading from the component to the artesian definition of the gradient vector,

Here the component vector

%,'PI

i s clearly

(2 12) 0

In fact, s f p ) merely comprises the formal deriwtives of p with respect to a nonunique dependence on X t and represents component contributions to the gradient vector. Applying next the chain rule to the gradient of (2.10) we obtain the relation

(2 13) which agrees with the transform tion of (2.6) between component no tura I and tota I na tural quaniities. In conclusion we list the inwriance of the expression

which furnishes the increment of the scalar p associated with a change of spatial location, and may be verified by the chain rule, or via an appropriate interpretation of (2.8).

2.3 Velocity field The extension of the above terminology to the description of a vector field i s stmightforword. Consider for instance the velocity field,

v = d t ,d

(2.15)

which i s i n general unsteady. The acceleration of a certain particle may be obtained by the so-called material differentiation of the velocity vector with respect to time as

(2.1 6) The f i r s t term of the expression in (2.16) represents the loca I deriw tive of the velocity with respect to time and i s to be evaluated a t a fixed location. The second term represents the contribution of convection and i s dependent on the gradient of the velocity field.

On the Natural Approach to Flow Problems

5

,

Disregarding for the time being a particular representation o f the velocity vector its gradient may be measured with respect to one of the different specifications of the location vector X Taking component natural coordinates Xc we obtain i n analogy to (2.10) the total natural gradient

.

In (2.17) the Cartesian gradient

may be related i n analogy to (2.1 1) to the component gradient of V

, (2.19)

which represents an extension of (2.12) and i s derived from a functional dependence of Y on Applying once more the chain rule to (2.17) we deduce

at,

(2.20) which relates directly the total to the component natural gradient and represents an extension of (2.13) to a vector field Y We also note the invariance of the expression

.

(2.21) which isanalogous to (2.14) and represents the convective acceleration term of (2.16). Here Y symbolises one o f the three differently defined representations of the velocity vector. The inwriance of (2.21) may be confirmed by the chain rule. We next turn our attention to the particle acceleration o f (2.16) and observe that i t may be represented by one of the alternative forms adopted for the description of the velocity vector y Thus, i n component natural terms we have

.

(2.22) with the total gradient matrix of V,

(2.23)

J. H. Argyris & J. St. Doltsinis

6

The cartesian form of the acceleration on the other hand may be expressed as (2.24) Here V comprises the artesian components of the velocity, cf. (2.1). Here the cartesian gradient matrix

(2.25) should not be confused with the expression o f (2.18), which i s not limited to a particular representation of the vector Y The total natural formulation of the acceleration i s given by

.

(2.26) with the associated component gradient matrix of

5

We observe that relations (2.4) and (2.6) between the alternative vector specifications also apply to the acceleration, as confirmed by (2.24) and (2.26).

2.4 Continuity condition We proceed to the natural formulation of the continuity condition. To this end, consider in fig. 2.3 the infinitesiml tetrahedron element defined by the lengths of the six edges

1 = r i d 14

8

18 1' i r j = rt?

Q = ~...J ,

(2.28)

.

with a volume c/ When determining the flow of mass through the element as induced by component natural velocities we stipulate the column matrix

containing the rate of change of a l l component velocities along the edges

9.

' s

Consider next a component natural velocity characterised by the intensity and assume for the time being an incompressible fluid with density f , Under these condithrough the centre of the face A' not tions mass permeates a t a rate f5'4' containing 1'' into the element and i s discharged a t a rate f ( ~ + dA''~ ~ c through the opposite face at a distance I"/J from the point of input. The balance of inputoutput of fluid m s s due to the component natural velocity ye" i s seen to be simply

On t h e Natural Approach to F l o ~Problcwis )

7

(2.30) where

(2.31)

denotes the rate o f change of Gd along 4.Generalising to a l l natural directions 4. and applying the column matrix eC of (2.29) we obtain the rate of mass supply by summation o f the individual contributions defined by (2.30) as arising for a l l component Hence the condition of conservation o f mass for an incomnatural velocities of U; pressible fluid i s given by

.

where

i=

1, 2,

3

(2.33)

i s the artesian counterpart of mC. Note also the summation vectors

(2 .34)

In the case of a compressible fluid, expression (2.30) for a typical rate of the component mass supply must account for a change of the density f along o( Consequently, the condition of conservation o f mass (2.31) i s modified into

.

Here the density gradients

(2 .36)

(2 e37) correspond to the definitions of (2.10) and (2.1 1 )

.

J.H. Arg-vris & J. St. Doltsinis

8

2.5 Rate of deformation and stress A specification of the behaviour of fluid flow demands the introduction of suitable stress and deformation measures. In classical continuum mechanics (see e.g. [83), the rate o f deformation i s defined by the symmetric part of the Cartesian gradient o f the instantaneous velocity field. With reference to (2.18) we may thus write

(2.38)

The associated instantaneous material spin i s then

and i s defined by the antisymmetric part. The rate of deformation of the material

i s correspondingly specified by the symmetric part of the Cartesian velocity gradient matrix. In what follows we refer to the column matrix

as the Cartesian rate of deformation. Natural measures of the rate o f deformation were originally defined by reference to the deformation of the fluid material instantaneously occupying the tetrahedron element [9, 71, They may be expressed i n terms of the natural definitions of the velocity gradient [lo] Thus, the tom1 natural rate of deformation i s given by

.

4

comprises the rates of extension of the material along the six The column matrix natural directions [5] I t may be related to the Cartesian definition of the rate of def o r m tion via

.

(2.43)

9

On tlie Natural Approach to Flow Probleins

c

.

where the detailed structure of the transformation matrix may be found in 191 The component natural rate of deformation i s defined i n analogy to the total one i n (2.42) as

(2.44)

The column matrix

6,may be related uniquely to the total natural rate of deformation

by

(2 .45) where the transformation matrix

i s also given in [ 91 and presumes that component velocities vary only along the direction of their action [ 101 In this case, o f (2.44) and & , o f (2.29) are identical.

.

4

For stresses we must adopt a corresponding definition to the rate o f deformation so that their scalar product satisfies the condition of invariance for the virtual rate of work. Thus the column matrix

=

[ G"

6 ' '

C3'

h

GIL

f i b z 3& G " ]

Q .47)

comprises the Cauchy stresses i n their Cartesian form and corresponds to the rate of deformation o f (2.41). The natural component stresses

6

correspond following 19, 5 1 to the total natural rate of deformation while the total natural stresses

dt

o f (2.42),

Q .49) correspond to the component natural rate o f deformtion of (2.44) (c.f. fig. 2.4). The invariance of the virtual rate of work may now be expressed as

(2 .50) Bearing i n mind (2.43) and (2.45) we easily confirm the relation

(2.51)

J.H. Argvris & J. St. Doltsinis

10

connecting the different representations of the stress state,

2 . 6 Constitutive relations for incompressible viscous fluids In formulating the stress-strain relations appertaining to the fluid motion, i t i s convenient to split the stress state into hydrostatic and deviatoric contributions. We may ignore here an account of the standard Cartesian approach (see e.g , I 1 11) and apply instead the natural approach to this subject as developed in [7, 91. Considering first total stresses we write

=t =

%If

+

(2 .52)

=tb

and obtain the hydrosbtic part of the t o b l stress in the form

Qiy

=

- L'

=

c, e,t

I 3

b,

(2.53)

where the matrix

€6

(2 .54)

performs the summation of the component stresses in each row and yields the total hydrostatic stress i n each of the natural directions. The deviatoric part of the total stress follows then from (2.52) as

in which relation (2.51) between total and component definitions i s used. Partitioning next the component stress as

we may derive the hydrostatic and the deviatoric part by application of (2.51) to the total quantities of (2.53) and (2.55), respectively. A decomposition of the total natural rate of deformation

st =

4"

f

Sib

(2 .57)

into volumetric parts

(2 .58) and deviatoric parts

11

On the Natural Approach to Flow Problems

(2.59) proceeds along the same argument. Also the component natural rate of deformation

may be partitioned analogously. Consider next an incompressible fluid, i.e. a fluid undergoing only isochoric deformations. In this case the volumetric rate o f deformation must vanish. This yields,

(2.61) which i s equivalent to (2.32). In the absence of viscous effects the incompressible fluid i s described as an ideal one for which deviatoric stresses are absent. Then the stress field derives simply from a static pressure p Consequently, the total stresses reduce to

.

and by

(2.51)the component stresses become

Gc

ccy= - pff-e,

(2.63)

-

In a viscous incompressible fluid on the other hand, a rate of deformation which i s exclusively deviatoric because of (2.61) leads to deviatoric total stresses of the form

-

or to deviatoric component stresses,

where

p

denotes the viscosity coefficient o f the fluid.

For the viscous case the stress i s ultimately obiuined by a superposition of a hydrostatic contribution arising from (2.62), (2.63)and the deviatoric contribution of (2.64), (2.65). Thus, the total natural stress reads

bt

= 2 / 4 4- p e ,

(2.66)

J.H. Argyris & J. St. Doltsinis

12 and the component one

(2.67)

We observe that the constitutive relations i n (2.66) and (2.67) are expressed i n terms of corresponding stress and rate o f deformation measures.

I f standard computational procedures are to be applied to the analysis of the isochoric motion of an incompressible fluid, one may use the so-called penalty approach. The isochoric condition (2.61) can then be relaxed and the pressure p i s related to the volumetric rate of deformation as follows

where (2 .69)

k

represents the penalty parameter. In (2.68), may be interpreted as a modulus of viscous compressibility and i s expressed in (2.69) i n an analogous manner to the wellknown elastic bulk modulus. The strictly incompressible constitutive relations (2.66), (2.67) m y now be modified accordingly. For instance, (2.67) assumes i n the penalty approach the form

(2.70) in which

5 4 -1 2

(2.71)

may be used as an alternative penalty parameter.

2.7 Fluid motion coupled with thermal phenomena In this subsection we consider fluid motion coupled to thermal phenomena. To this purpose we assume the following unsteady temperature field

r *

= T(~,x)

(2 .72)

where denotes the position vector. In extension of the argument in subsection 2.2 the time mte of the temperature of a particle may be expressed in the alternative forms

On the Natural Approach to Flow Problems

13

(2 .73)

The different formulations of the tempemture gmdient in (2.73) may be compared to the definitions i n (2.10), ( 2 . l l ) a n d (2.12), respectively. In the present case the invariance condition of (2.14) becomes

(2 .74)

The time rate of the temperature i s associated with a mte of heat stored i n the fluid material. The latter may be expressed per unit material volumeas

(2 .75) where C denotes the specific heat capacity of the fluid. In accordance with (2.73) the rate of heat stored i n the material may be composed in the Eulerian approach of two parts. Thus, the rate o f heat stored in a unit volume when fixed i n space reads

a7

PJ = P a t

(2 .76)

and i s associated with the tempemture rate obiained a t a fixed location. The contribution

(2 .77) i s the heat convection term due to the motion o f the fluid and may be presented in one of the alternative formulations, natural or artesian, as shown in (2.77). We proceed next to the specification of the heat supply to a unit volume o f space due to a dire:ted heat flow, i.e. conduction. Following subsection 2.1 the heat flow vector with Cartesian entries

p

,

J.H. Argyris & J. St. Doltsinis

14

(2 .78)

may alternatively be represented by the component natural contributions

(2 -79) or by the total natural quantities

(2.80) The reader is reminded of the interrelations between the alternative representations of the heat flow vector in accordance with (2.4) and (2.6). When determining the heat flow through an infinitesimal tetrahedron element shown in fig. 2.5, asarising from component natural heat fluxes [3] we have to introduce the column matrix

,

Consider now in fig. 2.5 a component natural heat flux characterised by the intensity and the outprogressing through the tetrahedron. Noting the input f:.(* put f9+0'4):Ad of the heat rate emerging a t a distance 1% from the point o f input we deduce for the rate of heat supply to the element as contributed by the component natural heat flux

$'

s',"

(2.82) where

(2.83)

:i

.

denotes the rate of change of Genemlising for a l l natuml directions along we apply the column m t r i x Ct of (2.81) and obtain the rate of heat supply by summation of a l l individual contributions as expressed by (2.82). Hence, we find

(2 .84)

On the Natural Approach to Flow Problems i s the Cartesian counterpart of the column matrix

where

15

4,

and reads

We conclude this subsection by presenting a natural counterpart to the Fourier's law relating the heat flow to the temperature gradient [3]. Starting with the Cartesian form

(2.86) wh re tion

2

den0

3s

the thermal conductivity o f the fluid, we LJduce the natum rela-

(2.87) by an appropriate application o f (2.6)and (2.4) or (2.11). We note that

(2 .88) symbolises the natural thermal conductivity matrix connecting via (2.87) the total natwith the component temperature gradient We ural heat flow vector ft also observe that the connection between $t and the total temperature gradient q ( T ) i s simply given by the thermal conductivity 2 o f the material as i n

;P,(T)

.

(2.86).

3.

DISCRETISATION BY FINITE ELEMENTS

3.1 Weak form o f the equations governing fluid and thermal flow Bearing in mind our prospective application o f the finite element technique to the flow problem we write i n the following the basic equations i n their weak form assuming a Thus, a weak form o f the momentum finite volume Y bounded by the surface balance may be expressed i n natural terms as

s .

3

-

8

the associated rate of deformawhere symbolises a virtual velocity field and tion, Also, f denotes the body force vector acting per u n i t volume and n a normal

J.H. Argyris & J. St. Doltsinis

16

operator yielding the surfoce tractions. Alternative formulations of (3.1) i n natural or in Cartesian terms are possible as outlined i n section 2. The virtual rate o f kinetic energy, for instance, on the left-hand side i n (3.1) i s given i n terms of one of the expressions

(3.2)

2

offered in (2.8) for the scalar product of two vectors. Clearly, the accelerution consists, in the Eulerian apprcach adopted here of a I y a I part and a convective part and may be specified in the component natural form &$ of (2.32), the Cartesian form of (2.24), or the total natural form of (2.26). We also observe that the component natural stress Cc in (3.1) obeys the constitutive laws of subsection 2.6. For a weak formulation of the isochoric condition we rely on expression (2.61) and write i n na tura I terms

where

represents the virtual pressure field.

We next turn our attention to the heat flow as occurring concurrently with the fluid motion. The heat balance of the volume in question may be expressed i n natural terms as

Y

I/

r/

where ? denotes a virtual temperature field. In (3.4), the first integral on the lefthand side i s due to the rate of heat stored i n the material, i n accordance with (2.75). I t i s specified through the local term of (2.76) and the convective term of (2.77). The second integral reproduces the rate of heat supply (2.84) by heat conduction. I t balances the stored heat expression with due consideration o f the rate of dissipation in the material as given by the right-hand side of (3.4). The second integral i n (3.4) associated with the heat flux may be transformed as follows (cf. [ 31)

where due to (2.8) and (2.87)

17

On the Natural Approach to Flow Problems Furthermore, the boundary condition

5

under the temperature 7 expresses the local heat exchange between the surface and the surrounding medium under a temperature Ts ; the associated heat transfer coefficient i s denoted by o( Thus (3.5) m y be brought into the final form

.

By substitution of (3.8) i n (3.4) one obtains the fundamental expression for the derivation of the relevant finite element relations,

3.2 Natural finite element equations for fluids

To set up a finite element formulation o f the flow problem consider first the weak momentum equation (3.1) i n conjunction with an approximate representation of the velocity field within each finite element expressed by

The column matrix

comprises the component natural contributions to the velocity vector a t any one o f the r) nodes of an element

CorrespondingI y, the matrix

contains the diagonal matrices, (3.13)

J.H. Argyris & J. St. Doltsinis

18

o f dimensions 6 x 6 which interpolate the velocities depend only on the total natural coordinates Kt

.

gj . Note also that the

W.'J

J

The local part of the acceleration within the element may now be established immediately via (3.9) as

(3.14)

Before entering into the derivation o f the convective part o f the acceleration we observe that the velocity field (3.9) may alternatively be described by

where

and

3

13

i s here the super row matrix of the component nodal velocities,

the column matrix,

Hence, the velocity gradient may be written as

(3.18)

(3 .19)

The convective term of the acceleration (cf.

(2.21)) may now be expressed as,

(3.20) i n which the velocity gradient i s represented by (3.18). The total natural velocity appearing in (3.20) obeys the interpolation rule of (3.9) i n the form

19

On the Natural Approach to Flow Problems with the column matrix (cf. (3.10))

Here nodes

6

comprises the field of total natural velocitiesat each of the 0 element

J

App Iying next expression (2.22), we obtain the acceleration by a summation o f the local part (3.14) and the convective part (3.20) in the form,

(3.24)

We now proceed to the mte of deformation within the finite element. To this purpose we consider the total natural rate of deformation Jt of (2.42) and rewrite i t in the form

>= d,.-.1 ' where the operator

=

d,

(3.25)

i s the (6 x 6) diagonal matrix

J

and,

(3.27)

Btc

Here 6,9 symbolises the a - t h column of the matrix i n (2.6), respectively i n (2.7). Application o f the interpolation rule (3.21) furnishes the total natural rate o f deformation within the element as,

J.H. Argyris & J. St. Doltsinis

20

(3.28)

Turning our attention to the virtual velocity field

Gt

introduced in (3.1) we set,

(3.29)

@

The definition of the column matrix i s i n line with that of i n (3.22). As to & i t s formation i s that of W N of (3.12) but may be based on different interThe associated virtual rate of deformation reads # "'J' polation functions 6;' then in analogy to (3.28)

.

(3.30)

In finite element theory, forces are assumed to be transmitted exclusively through the element nodes. Let the column matrix

comprise the component natural element contribution to the force vector a t each of the n element nodes,

In accordance with the invariance rule (2.8), the component natural representation of the nodal force vector pu' of (3.32) corresponds to the total natural definition of the nodal velocity vector b$j of (3.23). Disregarding for simplicity the volume forces on the right-hand side of (3.1) and expressing the surfoce integral through the nodal quantities the virtual work expression (3.1) assumes for a finite element of volume c/ the form

(3.33)

1/ Introducing the kinematic relations (3.29), natural forces a t the element nodes as

(3.30) i n (3.33) we obtain the component

21

On the Natural Approach t o Flow Problems

(3.34)

The acceleration term on the right-hand side of (3.34) may be transformed with the aid of (3.24) into,

(3.35) where

(3.36)

J corresponds to the Lagrangeun mass matrix while

(3.37)

accounts for the nonlinear convective contribution inherent to the present Eulerian approach. To specify the stress term on the right-hand side of (3.34) we call upon expression (2.67) for the component natural stresses and obtain

I/

J

Using the kinematics as prescribed in (3.28), the f i r s t integral on the right-hand side of (3.38) i s transformed into,

22

J.H. Argyris & J. St. Doltsinis

where

(3.40)

v represents the viscosity matrix of the element and reflects the deviatoric response of the isochoric fluid. With respect to the second integral on the right-hand side of (3.38) we introduce the approxima tion

P = V

(3.41)

to describe the pressure field within the element. In (3.41)

and contains the pivotal values of the pressure and 9C within the row matrix

p

i s the column matrix

the interpolation functions

Introducing (3.41) into (3.38) one obiuins,

(3.44)

(3.45)

J i s the hydrostatic element matrix. Using (3.44), (3.39) and (3.35), the component natural forces (3.34) of the element may ultimately be expressed as

23

On the Natural Approach t o Flow Problems

3.3 Transition to artesian definitions; discretised Navier-Stokes equations Before proceeding to the assembly of the element contributions (3.46) within the region considered, we transform (3.46) into a global artesian system o f reference. Denoting the respective artesian element nodal forces by

and the corresponding velocities by

j=

I,

..., Y

(3.48)

we m y apply relation (2.4) connecting natural and artesian definitions of vectors to obtain on the element level,

(3.49) and

(3.50) We note also that in

v=wv

(3.51)

the interpolation matrix W corresponds to the definition of ,&# i n (3.12) but with (cf. (3.13)) of dimension 3 x 3, in order to maintain consistency with the entries artesian definition. One may now substitute (3.50) i n (3.51) to express Y i n terms of V, Relating on the other hand f l to via (2.4) and expressing the latter through the interpolation (3.9) we deduce a second expression for Y Thus,

c3J'

.

6

.

and hence

(3.53) Applying next the transformation to the velocities (2.6) we obtain for the toiul elementa l velocities

24

J.H. Argyris & J. St. Doltsinis

Jt = r B J J

(3.54)

An analogous argument to that used i n (3.52) yields in the present case

5

(3.55)

and hence

(3.56) Substituting in (3.49) nishes the Cartesian forces

I

as

given i n (3.46), and

as defined in (3.54) fur-

(3.57) Using (3.56) and (3.53) as well as (3.36) and (3.50) we may verify that the first term in the second expression of (3.57) reduces to

v =

[/pGWdlTB',,J

2

Y

- [/pGCdJ1 r/ Note the expression for the elemental mass matrix vv)

=IpGcld/ Y

= WI

C;

(3.58)

(3.59)

25

On the Natural Approach to Flow Problems Consider next the second term on the right-hand side o f (3.58). Application of (3.37) for the natural convectivity matrix yields the Cartesian counterpart

4

in which use i s made of the relation (2.18) connecting and denotes the super row matrix of the cartesian nodal velocities.

$,

, Also,

f

Finally, the Cartesian viscosity and hydrostatic elemental matri:

d = r&;&j dN rs,, I

(3.61)

and

represent standard transform tions and do not require further elaboration.

zi

, the weightWe observe i n the above finite element idealisation that identity of the interpolation functions, reduces the discretisation proing functions, with wj cedure to that of Galerkin. In most structural applications, this method leads to symmetric matrices and the associated solutions are known to possess the property of best approximation. In convection dominated flow problems, however, we adopt a suggestion o f [41 and prefer to apply the streamline upwind/Petrov-Galerkin technique. In this case zj and W j are taken to be different. Bearing i n mind the aforementioned publication i n which a detailed description o f the method i s given we restrict our present account to an elaboration o f the alternative natural formulation. Following [ 4 ] , the weighting functions 6j are formed as

,

(3.63) where w j i s the standard interpolation function a t the j - t h element node and S j a perturbation defined by

J.H. Argyris & J. St. Doltsinis

26

which induces an upwinding i n the streamline direction. The scalar coefficient (G i s specified in [4] asa function of the velocityand the element dimensions. The natural expression for Sj i n (3.64) may be seen to simply rely on the invariance of alternative expressions of scalar products as shown in (2.8). In (3.64) d j i s assumed to be a function of the total natural coordinates X t The associated gradient $< follows then the definition of (2.12) with dj i n place of the pressure. In conclusion we note that the upwind technique introduces an additional dependence on the velocity into the finite element characteristics. As outlined i n [41 under certain conditions the upwind scheme affects merely the weighting of the acceleration term in (3.34) but not that of the stress term, In this case the element viscosity matrix i s symmetric.

.

Turning next our attention to the entire flow domain, the element contributions to the nodal forces as given by (3.57) may be summed up and yield the global relation

R

which represents the discretised form of the Navier-Stokes equations. In (3.65) ,denotes the column matrix of the nodal forces applied to the flow domain, y and 1, are the corresponding velocities and accelerations, and the column matrix P defines the pressure field in the entire flow domain. The matrices fl D h / , 0 and N may and be deduced by a straightforward assembly procedure from the matrices m , b l h of the individual elements.

,d

3.4 lsochoric condition. Exact analysis and approximate penalty formulation We now proceed to the discretisation o f the isochoric condition using the natural methodology and consider to this end the last expression i n (3.3). Introducing a relation analogous to that of (3.41) for the variation of and expressing as i n (3.28) we deduce for a finite element h e condition

dt

(3.66)

v

J

where the matrix

(3.67)

I/

Y

h,

cincides with the matrix and i n (3.45) for the case when GJ'= * l j TO obtain the artesian form of (3.66) we refer to (3.54) and deduce ILj = %j

.

21

On the Natural Approach t o Flow Problems

Hence the Cartesian counterpart of the natural matrix

f

is

(3.69)

The isochoric condition for the entire flow domain may now be symbolised by (cf. (3.68))

G'V

=

o

(3.70)

where the column matrix Y comprises the velocities a t the nodal points of the finite in (3.69). element mesh, and 6 i s composed by the individual element matrices

9

In the penalty approach the isochoric condition i s relaxed in accordance with (2.68). As a consequence the weak formulation in (3.3) i s correspondingly affected. Adopting the finite element approximation in (3.66) one obtains i n the penaltyopproach

(3.71)

The matrix

R"

may be seen to represent the integral expression,

(3.72)

Y Solution of (3.71) for the pressure yields

(3.73)

where use i s made of (3.54), (3.69) when forming the alternative Cartesian expression on the right-hand side of (3.73). Substitution of (3.73) in (3.57) determines a pure velocity formulation. Isolating the two last terms in the final expression in (3.57) we consequently have

The matrix

(3.75)

28

J. H. Argyris & J. St. Doltsinis

represents the elemental viscosity i n the penalty approach and i s a symmetric matrix i n an ordinary Galerkin approximation. The above procedure corresponds to the mixed finite element technique of [15] i n which velocity and pressure field are approximated independently, An alternative penalty formulation of the viscous incompressible problem may be obtained by substitution o f (2.68) i n (3.38). This leads to a pure velocity formulation i n (3.46) or (3.57) without the need o f a separate approximation for the pressure. On the other hand, this advantage involves necessarily a reduced integration [13, 141 Sumscheme for the volumetric part of theassociated viscosity matrix marisingthe discretised Navier-Stokes equations for the entire flow domain may be written in the penalty approach as

.

2

(3.76) where the relaxed isochoric constraint i n the viscosity matrix i s included in accordance with one or the other approximation technique.

-

4

in

3.5 Finite element equations for heat flow A s a final item we consider the finite element approximation of the heat balance in the fluid as governed by (3.4) and (3.8). To this end we write the tempemture field within the element as

(3.77) where the column matrix

comprises the temperatures a t the element nodes and the row matrix

the'interpolation functions. Analogously, we express the virtual tempemture field as

(3.80)

-

where the weighting functions ?j i n 2 ' may be constructed i n accordance with the streamline upwind/Petrov-Galerkin concept, as detailed i n (3.63) for Applying (3.77) we may obtain the local part o f the temperature rate as

aj

.

29

On the Natural Approach t o Flow Problems

-==r at

ri

(3.81)

Correspondingly the convective part becomes

(3.82)

where use i s made of (3.21) for

5 .

With the aid of (3.80), (3.81) and (3.82) the first integral in the heat balance of (3.4) may be transformed into

(3.83) The matrix

(3.84)

represents the heat capacity matrix of the element in a Lagrangean approach and must be supplemented in the present Eulerian presentotion by the convective contribution associated in (3.83) with the coefficient matrix

J.H. Argyris & J. St. Doltsinis

30

k

The second integral expression in (3.85) refers to a cartesian specification, the transition from the first natural expression being a consequence o f (2.74). The second integral on the left-hand side of (3.4) may be put as a consequence of (3.8) into the finite element form

J

s

being the element surface. Application of (3.80) and (3.77) yields the equiwlent expression

(3.87) The element conductivity matrix i s thus given by

I t s transcription into the artesian form m y be established by substitution o f (2.88) for and application of the gradient relations (2.20), (2.17) and (2.18). We find

Arc

Furthermore, we observe in (3.86) that

(3.90)

S

J

represents a prescribed heat rate through the element surface.

On the Natural Approach t o Flvw Problems

31

Concerning the rate o f dissipation defined by the integral on the right-hand side o f (3.4), one may write,

and

(3.92)

J

Y

6,

Here t t B $ and Q may be deduced from the mechanical account o f the flow problem i n subsection 3.2. Collecting the contributions (3.83), (3.86) and (3.91) into the overall heat balance of the element as expressed by (3.4) we obtain

(3.93) where a

a

a

f = P,

fd

(3.94)

i s a generalised heat rate. The finite element equations for the entire flow domain assume then the form

(3.95) *

a

r,a

are column matrices comprising quantities a t the nodes o f the i n which 7, finite element mesh and , k L are the relevant global matrices deduced by assembly of the respective element matrices.

c

,

J.H. Argyris & J. St. Doltsinis

32

4.

NUMERICAL EXAMPLES

In this section we present some examples illustrating the application o f the preceding theory on the solution o f pure and thermally coupled flow problems. Details o f the numerical solution methods, omitted i n this paper, may be studied i n 1281. There, the numerical aspects are discussed taking account o f the pertinent literature on the subject [16 251, which include recent developments, We should stress here that the streamline upwind/Petrov-Galerkin scheme i s applied to a l l our examples. The capabili t y of this method i s demonstmted i n what follows for convection dominated flow i n two and three dimensions. The solution o f thermally coupled flows i s illustrated on the BBnard type instability. Cavity flows with free and forced convection including a change of phase are treated i n [ 2 8 ] .

-

4.1 Flow over a step The transient incompressible flow over a step demonstrates the applicability o f the independent p - Y formulation and a two stage solution strategy as described i n [4, 281 Due to the high Reynolds number a turbulent flow field develops necessitating the use of upwind techniques. The geometry of the flow domain and the boundary conditions used in the calculation are sketched i n fig. 4.1 together with the material data o f the medium (air). A t the inlet a constant velocity profile i s prescribed which yields a Reynolds number o f 14950 based on the step height. A zero velocity component i n cross flow direction i s assumed a t the upper side and zero pressure a t the outlet of the channel, The flow region i s discretised by a mesh o f 1700 bilinear plane elements QUAP4 as indicated i n fig. 4.1. Starting from a quiescent i n i t i a l condition the development o f the turbulent flow i s investigated up to a total duration o f = 67.5 ms using 900 time increments. In fig. 4.2 the onset of turbulent flow i s shown i n the upper two streamline plots while the other plots depict the fully turbulent flow field. The disturbances i n the flow field near the outlet may be caused by the somewhat unrealistic pressure boundary condition. Also, the zero cross flow condition a t the upper side o f the flow region seems to be not well adjusted to the process. Despite a l l these shortcomings the long-time exposure of the flow over a step ( w t e r , visualised b y aluminium powder) shown i n fig. 4 . 3 and extracted from 1261 compares quite well with the streamlines a t the instant = 56.25 ms of the numerical investigation.

t

t

4.2 Flow i n a quadratic duct with a step The efficiency of the upwind scheme and its three-dimensional generalisation involving the two stage solution algorithm i s demonstmted i n this example. The flow region, the boundary conditions and the data o f the fictitious material are depicted i n fig. 4.4. A t the inlet cross-section a constant flow velocity i s assumed, the Reynolds number of 200 A zero pressure condition i s adopted a t the being based on the duct dimension H outlet. The flow domain i s discretised by 1368 linear volumetric elements HEXE8 as shown i n fig. 4.5. Calculations were performed i n 60 time steps from the i n i t i a l canditions to a steady state a t dimensionless times = 6. A t the final stage, projections

.

t

.

On the Natural Approach to Flow Problems

33

o f the nodal point velocity vectors onto the xy- and yz-planes are shown i n fig, 4.6. The following example i s concerned with the solution o f a coupled fluid/thermaI problem.

4.3 BBnard convection i n a rectangular box In a fluid heated from below buoyancy driven convection rolls w i l l develop above a critical value o f the Rayleigh number (cf. fig. 4.7)

This process i s analysed for water enclosed i n a rectangular box, disregarding any threedimensional effects. The lower and upper plate o f the box are held a t a constant temperature, but the vertical side walls are assumed to be subject to an adiabatic state. The fluid i s i n i t i a l l y set a t the same temperature as the upper plate and i s then heated from below. As soon as the critical Rayleigh number Ra = 1708 i s exceeded convection rolls begin to develop. To avoid the difficulties associated with the bifurcation phenomenon a t the critical Rayleigh number a perturbation i n temperature i s applied which determines the rotational sense o f the first vortex. The analysis i s continued unt i l stationary conditions are attained. The mechanical and thermal data o f the fluid (water) are quoted i n fig. 4.7 together with the discretisation by QUAP4 plane elements. The Rayleigh number i s ewluated to be 20250 which exceeds by far the critical value. This fact facilitates the generation = 450 s and involves 60 time o f an unstable process. The calculation extends over steps, varying between 2.5 s and 20 s. The small increments prove necessary i n the i n i t i a l process of the formation o f the convection rolls within the time i n t e r w l between 100 sand 150 s. The temperature perturbation applied for the initiation o f the convective flow i s removed after 150 s when a l l vortices are formed. Fig. 4.9 exhibits isotherms and streamlines a t different stages o f the process. The development o f the convection rolls and the steady state condition i s i n good agreement with experimental and analytical investigations 1271. The time between the initiation o f convection up to the f u l l y developed flow corresponds to the predictions. The series o f differential interferograms reproduced in fig. 4.8 shows the formation o f convection rolls i n silicone o i l under similar conditions.

t

The calculation o f the coupled fluid and thermal problems was performed by an iterative sequential solution o f the two individual problems (cf. 1 3 2 ) . The thermal equation, dominant i n the BInard convection phenomenon, w a s solved first followed by the solution o f the flow problem. A l l coupling quantities were taken into account, i.e. the convective terms i n the thermal problem and the buoyancy forces i n the flow problem, the latter being calculated using the Boussinesq approximation. The iterative solution o f the discretised equations leads to linear equation systems with non-symmetric coefficient matrices due to the convection terms, The equation system o f the thermal problem was solved using the QR-factorisation for the non-symmetric coefficient matrix.

J. H. Argyris & J. St. Doltsinis

34

For the flow problem the penalty approach w a s applied with the convection terms on the right-hand side so that standard solution methods were eligible. Upwinding w a s used in both problems with an upwind parameter of 0.258. Convergence below the limit € = 10'' in the heat rates and velocities respectively was required to terminate the iteration of the individual problems. The sequential solutions were continued until both the velocity and the temperature increments were reduced below the convergence limit of t = to-' between consecutive iterations.

REFERENCES

-

Argyris, J.H. et al., Finite element method the natural approach, Fenomech Comput. Meths. Appl. Mech. Engrg. 17/18 (1979)1-106.

'78,

Argyris, J.H., Doltsinis, J.St., Pimenta, P.M. and Wtistenberg, H., Thermomechanical response of solidsat high strains natural approach, Fenomech '81, Comput. Meths. Appl. Mech. Engrg. 32 (1982)3-57.

-

Argyris, J.H. and Doltsinis, J.St., On the natural formulation and analysis of large deformation coupled thermomechanical problems, Comput. Meths. Appl Mech. Engrg. 25 (1981)195-253.

.

Brooks, A .N. and Hughes, T.J. R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Fenomech '81, Comput. Meths. Appl. Mech. Engrg. 32 (1982)199-259. Argyris, J.H. and Doltsinis, J.St., On the large strain inelastic analysis in natural formulation Part I . Quasistatic problems, Comput. Meths. Appl. Mech. Engrg. 20 (1979)213-252. Part II. Dynamic problems, Comput. Meths. Appl. Mech. Engrg. 21 (1980)91-128.

-

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Argyris, J.H. e t al., Eulerian and Lagmngean techniques for elastic and inelastic deformation processes, TICOM 2nd Int. Conf., Austin, Texas, 1979. In: Compututional Methods in Nonlinear Mechanics (J .T. Oden, Editor), NorthHolland Publishing Company (1980)13-66. Argyris, J.H., Doltsinis, J.St. and Wtistenberg, H., Analysis of thermo-plastic forming processes natural approach, Computers and Structures, to appear.

-

Prager, W., Introduction to mechanics of continua, Ginn and Co.,

Boston

(1961).

Argyris, J, H,, Three-dimensional anisotropic and inhomogeneous elastic media, matrix analysis for small and large displacements, Ing.-Archiv 34 (1965)33-55. Argyris, J.H. and Doltsinis, J.St., A prime on superplasticity in natural formulation, Comput. Meths. Appl Mech. Engrg to appear.

.

.,

Hohenernser, K., Prager, W., Uber die Ansdtze der Mechanik isotroper Kontinw, ZAMM 12 (1932)21 6-226.

35

On the Natural Approach t o Flow Problems

P2J Argyris, J. H. and Mareczek, G. , Finite element analysis of slow incompressible viscous fluid motion, Ing. Archiv 43 (1974) 92-109.

-

31 Malkus, D.S. and Hughes, T.J.R.,

Mixed finite element methods reduced and selective integration technique: a unification of concepts, Comput. Meths. Appl. Mech. Engrg. 15 (1975) 63-81. Oden, J.T.,

RIP-methods for Stokesian flows, In: Finite Elements i n Fluids, Vol.

4 (R.H. Gallagher e t a l . , Editors), John Wileyand Sons Ltd., 1982. Taylor, R.L. and Zienkiewicz, O.C., Mixed finite element solution of fluid flow problems, In: Finite Elements in Fluids, Vol. 4 (R.H. Gallagher et al., Editors), John Wileyand Sons Ltd., 1982. Felippa, C.A. and Park, K.C., Direct time integration methods i n nonlinear structural dynamics, Comput. Meths. Appl, Mech. Engrg. 17/18 (1979) 277-313.

71 Glowinski, R., Dinh, Q.V. and Periaux, J., Domain decomposition methods for nonlinear problems in fluid dynamics, Fenomech ‘81, Comput. Meths. Appl. Mech

. Engrg., to appear.

Hestenes, M.R., Stiefel, E., Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Stand. 49 (1952) 409-436. Jennings, A . and Malik, G.M., The solution o f sparse linear equations by the conjugate gradient method, Int. J. Num. Meths. Engrg. 12 (1978) 141-158. Dennis, J.E. and More, J .J., Quasi-Newton methods - Motivation and theory, SlAM Review 19 (1977) 46-89. Matthies, H. and Strang, G., The solution o f nonlinear finite element equations, Int. J. Num. Meths. Engrg. 14 (1974) 1613-1626.

,

Thomasset, F. Implementation of finite element methods for Navier-Stokes equations, Springer New-York, 1981. Hughes, T.J.R., Winget, J., Levit, I. and Tesduyer, T.E., New alternating direction procedures in finite element analysis based upon EBE approximate fhctorimtion, Recent Developments i n Computer Methods for Nonlinear Solid and Structural Mechanics (eds. S.N. Atluri and N. Perrone), ASME Applied Mechanics Symposium Series, New York, 1983.

G.J., Methods of Numerical Mathematics, Springer-Verlag, PI Marchuk, York - Heidelberg - Berlin, 1975.

,

New

Hughes, T. J .R., Levit, I. and Winget, J. An element by element solution algorithm for problems of structural and solid mechanics, Comput. Meths. Appl. Mech , Engrg , 36 (1983) 241 -254.

36

J.H. Argyris & J. St. Doltsinis

[26]

Toni, I., Experimental investigation o f flow separation over a step, IUTAM Symposium Freiburg 1957, Grenzschichtforschung/Boundary layer research, H. Gartler ed., Springer-Verlag, 1958, 377-386.

[27]

Kirchartz, K.R. Oertel, H., Zeitabhtingige Zellularkonvektion, (1982), T 211 T 213.

[28]

Argyris, J., Doltsinis, J.St., Pimenta, P.M. and Wustenberg, H., Natural finite element techniques for fluid motion, Comput. Meths. Appl. Mech. Engrg., to appear,

-

,

ZAMM 62

On the Natural Approach t o Flow Problems

Paral lelopiped

Carterion directions (a)

Cartesian apprmch

Tetrahedrm

Natuml directions

(b)

Fig. 2.1

Natural apprmch

Cartesian and natural system of reference

37

w

00

4

t

Natural and Cartesian directions

(a)

.s

Reference system

2

Bl

Component definition

Cartesian definitions

Total definition

(Non unique cornpition of a vector)

(Unique decomposition of o vector)

(b)

Fig. 2.2

Alternative representations of vector

r

Natural and cartesian specifications of a vector for the two-dimensional case

On the Natural Approach to Flow Problems

Fig. 2.3

M o s s supply to a natural element due to a component velocity vector ycd

Fig. 2.4

Corresponding definitions of natural stresses and rates of deform tion

Fig. 2.5

Heat supply to a natural element due to a component heat flux vector

4:

39

P

0

-

v, = 10 m/s

vr = 0

yI

'vy= 0

-2s--l

Ov,

= "vv = 0

p=o

I

X

= 0360m H = a056 m

Material data (air1

S = 0.020 m

p = 1.293

L

Fig. 4.1

I(

= 17.3 10-6Pas kg/m3

Flow over a step. Description and finite element discretisation

,f H

I

On the Natural Approach to Flow Problems

Fig. 4.2

Fig. 4.3

Flow over a step. Streamlines during development o f turbulent flow

Visualisation of flow over a step by aluminium powder i n water [26]

41

J.H. Argyris & J. St. Doltsinis

42

v,.vy.v,.O

/t

v,. 1.0

T vy :v, :0

H

p.0

I

) .

L = 7.0 H = 2.5

s Fig. 4.4

i

g = 2.5

p

t

200.0

1.0

Flow in a quadratic duct with a step

A 1368 HEXEB -Elements

3306 Unknown velocities 1368 Unknown pressures

Fig. 4.5

Flow in a duct, Discretisation

On the Natural Approach t o Flow Problems

I

I

I

43

A E C b

I

--_____________

$+ .T. . . . . . . ......... . . . . . . . . .

. . . . . . . . . . . . . . . . . . I

9

I

.

.

*

,

.

.

-

a

I T

. I

. . . .

. .

.

. . . .

,

. . . . . . . . . I

, , , , , , .

,

, ,

,

I

I

,

,

,

,

,

,

1

I

,

,

. . I

,

, ,

I

, ,

* 1

,

,

. -

.

, .

.

_ - - . . - .

, - - - ..

*""<--. *

' d

Fig. 4.6

Flow in a duct. Projections of nodal point velocity vectors a t a stationary state

J. H. Argyris & J. St. Doltsinis

44

\\

\Perturbotion in T

v-r,+,j/r, = I 01 L H

:

p

012117 O01m

b =I ‘1 = 1,

Material data lwoterl

:

1793 I O - ~ Pa s

p = 9998 = 4217 A : 05683 c

K :

333 K

a = 0 46 10.’ g = 9 81

Fig. 4.7

Fig, 4.8

kglm3 Jlkg K Wlm K 1I K m/s’

BBnard convection in a rectangular box

Differential interferogmm of tmnsient convection of silicone oil 1271

Oir tlre ,Vutiirul Approuclr t o l ~ l o wProblenrs

Fig. 4.9

B&nard convection, Distribution of tempemture (left) and stream function (right) a t transient and quasi-sktionary conditi ons

45

This Page Intentionally Left Blank

Unification of Finite Element Methods H. Kardestuncer (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

47

CHAPTER 2 INTERACTIVE COMPUTER GRAPHICS FOR FINITE ELEMENT, BOUNDARY ELEMENT, & FINITE DIFFERENCE METHODS

J.F. Abel, A.R. Ingraffea, R. Perucclzio, T - Y. Han, & J.F. Hajjar

The t r e n d toward u b i q u i t y o f i n t e r a c t i v e g r a p h i c s i n computational e n g i n e e r i n g i s h e l p i n g t o develop an atmosphere conducive t o t h e combination and u n i f i c a t i o n o f v a r i o u s numerical a n a l y s i s methods. A number of f e a t u r e s o f i n t e r a c t i v e computer a i d e d design which f o s t e r u n i f i c a t i o n a r e discussed. Examples a r e drawn from t h r e e p r i n c i p a l aspects o f computer graphics i n e n g i n e e r i n g a n a l y s i s : p r e processing, postprocessing, and i n t e r a c t i v e a d a p t i ve a n a l y s i s . INTRODUCTION The purpose o f t h i s paper i s t o present and e x p l o r e some ideas about how t h e growing use of i n t e r a c t i v e computer g r a p h i c s i s b o t h a f f e c t i n g t h e u t i l i z a t i o n o f numerical methods by e n g i n e e r i n g a n a l y s t s and i n f l u e n c i n g t h e p o s s i b l e u n i f i c a t i o n o f these methods. Recent decades have seen a dramatic development and p r o l i f e r a t i o n o f numerical methods. T h i s a c t i v i t y has i n c l u d e d t h e s t r e n g t h e n i n g and d i v e r s i f i c a t i o n o f e s t a b l i s h e d methods such as f i n i t e d i f f e r e n c e s and f i n i t e elements, t h e r a p i d expans i o n o f r e d i s c o v e r e d techniques such as b o u n d a r y - i n t e g r a l equations, t h e i n c r e a s i n g r e c o g n i t i o n of u n i f y i n g f o r m u l a t i v e n o t i o n s such as v a r i a t i o n a l approaches , and t h e coinbi n a t i o n o f methods t o r e a l i z e d i v e r s e advantages such as t h e m i x i n g o f f i n i t e elements and boundary elements. The ideas presented here r e p r e s e n t some r e c e n t c o l l e c t i v e thoughts o f a research group a t C o r n e l l U n i v e r s i t y t h a t has, f o r n e a r l y a decade, been i n v o l v e d i n developments o f i n t e r a c t i v e computer graphics f o r t h e i r a p p l i c a t i o n s t o both computational mechanics research and computer-aided design. Because most o f t h i s research has been l i m i t e d t o s t r u c t u r a l e n g i n e e r i n g and mechanics and has i n v o l v e d p r i m a r i l y f i n i t e element and boundary element approaches, t h i s can h a r d l y be considered a g l o b a l view o f d i s c i p l i n e s and methods. Nevertheless, t h e emphasis i n t h i s paper on growing t r e n d s i n g r a p h i c s and t h e i r r e l a t i o n t o u n i f i c a t i o n o f a n a l y s i s methods can be considered a manifesto, a l b e i t modest, r e g a r d i n g d e s i r a b l e d i r e c t i o n s f o r f u t u r e developments i n i n t e r a c t i v e graphics. C l e a r l y i n t e r a c t i v e graphics i n i t s e l f i s d i s t i n c t from a n a l y s i s and, t h e r e f o r e , can be viewed from one p e r s p e c t i v e as having l i t t l e t o do w i t h u n i f i c a t i o n o f a n a l y s i s techniques. However, i t has been w i d e l y demons t r a t e d t h a t i n t e r a c t i v e g r a p h i c s can be h i g h l y successful i n b r e a k i n g down b a r r i e r s and d i f f i c u l t i e s i n t h e performance o f e n g i n e e r i n g a n a l y s i s . The enhanced access t o , and c o n t r o l o f , a n a l y s i s can i n t h e same way s i g n i f i c a n t l y h e l p remove o b s t a c l e s t o u n i f i c a t i o n , as w i l l be argued

J.F. Abel et al.

48

subsequently. The p r o v i s i o n o f a proper atmosphere f o r amalgamation i s no l e s s i m p o r t a n t than t h e r e c o g n i t i o n o f t h e p o t e n t i a l f o r u n i f i c a t i o n . U l t i m a t e l y , t h e obvious goal i s t h e p r o v i s i o n o f t h e most e f f e c t i v e and a p p r o p r i a t e a n a l y s i s methodologies f o r engineers t o c a r r y out design r e s p o n s i b l y and c r e a t i v e l y . The computational environment f o r a n a l y s i s i s changing r a p i d l y . Computeraided a n a l y s i s and design i n both p r a c t i c e and i n d u s t r y a r e being t i e d t o g e t h e r wit h computerized i n f o r m a t i o n r e p o s i t o r i e s and data f l o w c a p a b i l i t i e s , w h i l e a t t h e same t i m e i n c r e a s e d computational power i s being p r o vided t o t h e i n d i v i d u a l engineer. Three c h i e f c u r r e n t developments i n hardware and software are e n g i n e e r i n g w o r k s t a t i o n s which p r o v i d e i n t i m a t e access t o computing, n e t w o r k i n g which enables t h e s h a r i n g and r a p i d t r a n s f e r o f l a r g e volumes o f data among w o r k s t a t i o n s , and i n t e r a c t i v e g r a p h i c s c a p a b i l i t i e s i n t e g r a l t o t h e w o r k s t a t i o n s . The l a s t o f these i s t h e theme here. I n t h i s environment, w i t h i n t h e next decade, n e a r l y a l l computing w i l l be a s s o c i a t e d w i t h i n t e r a c t i v e graphics. Both l i t e r a l l y and f i g u r a t i v e l y , i n t e r a c t i v e computer graphics i s becoming t h e "window" t o t h e world o f analysis. I n t e r a c t i v e computer graphics was f i r s t a p p l i e d t o p r e p r o c e s s i n g and p o s t p r o c e s s i n g only, and these a r e s t i l l t h e phases o f a n a l y s i s where e n g i n e e r i n g p r o d u c t i v i t y i s most d r a m a t i c a l l y improved by i t s use. However, although a n a l y s i s o r p r o c e s s i n g i t s e l f has t r a d i t i o n a l l y been a batch p r o cedure, t h e continued i n c r e a s e o f cheap, d i s t r i b u t e d computational capab i l i t y and t h e growing demands o f d e s i g n e r l a n a l y s t s i n d i c a t e t h a t i n t h e near f u t u r e some i n t e r a c t i v e processing w i l l be e f f e c t i v e and d e s i r a b l e even f o r such c o m p u t a t i o n a l l y i n t e n s i v e procedures as f i n i t e element a n a l y s i s . This n o t i o n has given r i s e t o i n t e r a c t i v e - a d a p t i v e a n a l y s i s [l], which i s d e f i n e d as a n a l y s i s d u r i n g which t h e engineer can c o n t i n u o u s l y m o n i t o r what i s happening and can choose t o i n t e r v e n e a t any t i m e t o change system c h a r a c t e r i s t i c s , models, a n a l y s i s parameters, and a1 gor i t h m s . I t i s most s u i t e d t o n o n l i n e a r and t i m e - v a r y i n g analyses. The engineer i s a b l e t o move backward and forward i n t h e a n a l y s i s a t w i l l and t o invoke a v a r i e t y o f p a r a l l e l analyses a t any stage. Moreover, d i f f e r e n t types o f analyses can be s t r u n g t o g e t h e r i n a sequence. T h i s a n a l y s i s c o n t r o l c a p a b i l i t y c l e a r l y must r e l y on i n t e r a c t i v e computer graphics f o r i t s success. I n t h i s paper a l l t h r e e aspects o f i n t e r a c t i v e computer graphics i n e n g i n e e r i n g a n a l y s i s are used t o i l l u s t r a t e t h e n o t i o n s discussed. UNIFYING INFLUENCES OF INTERACTIVE GRAPHICS Experience w i t h t h e development o f i n t e r a c t i v e g r a p h i c a l s o f t w a r e programs f o r a n a l y s i s a p p l i c a t i o n s i n d i c a t e s t h a t , f o r w e l l designed i n t e r a c t i v e systems, programming i s a major task t h a t exceeds i n complexity and d i f f i c u l t y t h e programming o f much a n a l y s i s s o f t w a r e [2]. In addition t o the a r a t i o n a l u n d e r l y i n g database; m o d u l a r i t y t o f o s t e r usual requirements e x p a n d a b i l i t y ; c l e a r , s t r u c t u r a l programming; a system f o r a n t i c i p a t i n g , handling, and c o r r e c t i n g e r r o r s ; t r a n s p o r t a b i l i t y t o t h e maximum e x t e n t p o s s i b l e w h i l e m a i n t a i n i n g necessary performance and response; and t h e design f o r human f a c t o r s o r "user f r i e n d 1 i thorough documentation ness" places e x t r a demands on t h e program developer. Among these a r e completeness o f f u n c t i o n , f l e x i b i l i t y i n sequence, and a humanized command language and menus [31 141. The achievement o f these l a s t o b j e c t i v e s

--

--

-

Interactive Computer Graphics

49

tends t o r e s u l t i n a g e n e r a l i t y o f t h e i n t e r f a c e between t h e user and t h e computer t h a t i s conducive t o f l e x i b i l i t y , d i v e r s i f i c a t i o n , and u n i f ic a t i on o f a n a l y s i s methods. I n a d d i t i o n t o t h e fundamental design o f i n t e r a c t i v e software, a f a c t o r which i s a f f e c t i n g t h e way i n which p a r t i c u l a r a n a l y s i s methods f i t i n t o an o v e r a l l design process i s t h e i n c r e a s i n g use o f system models independent from a n a l y s i s models. I n t h e a p p l i c a t i o n o f , say, f i n i t e element methods, one was p r e v i o u s l y l i k e l y t o d e s c r i b e t o t h e computer t h e geometry, boundary c o n d i t i o n s , and l o a d i n g s o f t h e problem under

I n t e r a c t i v e System M o d e l l i n g ( P r e l i m i n a r y D e s i g n , Geometry)

I n t e r a c t i v e Preprocessing (Geometry, A t t r i b u t e s , A n a l y t i c a l M o d e l )

t

T r a n s l a t o r (s) Ana 1y s i s

A n a l y s i s M e t h o d (s) I n v e r s e T r a n s l a t o r (s)

t I I I n t e r a c t i v e Postprocessing

I n t e r a c t i v e Design/Re-Design (Geometry, A t t r i b u t e s )

F i g u r e 1.

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The computer-aided design process. I n t e r a c t i v e computer graphics i s t h e p r i n c i p a l medium f o r man-machine communication and c o n t r o l . The i n t e g r a t i o n o f t h e v a r i o u s stages shown i s p r e d i c t e d on computerized databases and data flow.

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c o n s i d e r a t i o n i n terms o f t h e elements, nodes, and c o n n e c t i v i t y o f t h e d e s c r i p t i o n t h a t was e n t i r e l y l i n k e d w i t h t h e a n a l y t i c a l d i s c r e t i z a t i o n . With t h e t r e n d toward computer-aided design, a n a l y s i s i s now u s u a l l y viewed as being i n t e g r a t e d i n t o an o v e r a l l design process such as d e p i c t e d i n general form i n F i g u r e 1. The l i n k s between t h e v a r i o u s steps shown a r e achieved through a computerized database and data flow. Here t h e geometric d e s c r i p t i o n o f t h e design problem i s c r e a t e d on t h e computer by a modeller or, i f a modeller i s n o t a v a i l a b l e , i t i s created as t h e f i r s t s t e p o f preprocessing. Then, i n preprocessing, a d d i t i o n a l a t t r i b u t e s o f t h e problem, such as m a t e r i a l p r o p e r t i e s , boundary c o n d i t i o n s , and l o a d i n g s , a r e a s s o c i a t e d w i t h t h e geometry. The r e s u l t i s t h e complete d e s c r i p t i o n o f an a n a l y s i s problem t h a t e x i s t s w i t h i n t h e computer p r i o r t o any meshing and, indeed, t h a t i s independent o f any p a r t i c u l a r method. I n t h e f u t u r e , t h e generation o f t h e a n a l y t i c a l model o r d i s c r e t i z a t i o n a p p r o p r i a t e f o r a p a r t i c u l a r method o f a n a l y s i s w i l l probably be automated [5]. T h i s w i l l be p a r t i c u l a r l y s u i t a b l e f o r a n a l y s i s procedures t h a t i n c l u d e s e l f - a d a p t i v e mesh a l g o r i t h m s t o achieve a s p e c i f i e d degree o f refinement o r accuracy i n each p o r t i o n o f t h e system. However, f o r a t l e a s t t h e next decade, it i s c l e a r t h a t t h e e n g i n e e r i n g a n a l y s t w i l l need t o continue t o e x e r c i s e s t r o n g i n f l u e n c e over t h e design o f t h e a n a l y t i c a l model because o f i t s d i r e c t e f f e c t on accuracy and r e l i a b i l i t y o f t h e a n a l y s i s . I n t e r a c t i v e mesh generators a r e i d e a l l y s u i t e d t o t h i s task [6]. Moreover, e f f e c t i v e i n t e r a c t i v e preprocessors p r o v i d e a f l e x i b i l i t y t h a t i s o n l y beginning t o be recognized. Not o n l y a r e computer-assisted mesh generators i n many cases a b l e t o be g e n e r a l i z e d f o r a v a r i e t y o f a n a l y t i c a l approaches, b u t t h e i n t e r a c t i v e t o o l s i n c o r p o r a t e d f o r , say, s u b s t r u c t u r i n g o r p a r t i t i o n i n g a l s o present n a t u r a l o p p o r t u n i t i e s f o r preprocessing f o r h y b r i d o r combined techniques. Another aspect o f i n t e r a c t i v e s o f t w a r e design t h a t s i g n i f i c a n t l y a l l e v i ates s p e c i f i c i t y t o a s i n g l e a n a l y s i s t y p e i s t h e humanization o f command languages. One d e t e c t a b l e t r e n d i s toward t h e use o f commands more n a t u r a l t o t h e engineer. For example, i n t h e s p e c i f i c a t i o n o f boundary c o n d i t i o n s f o r f i n i t e element a n a l y s i s , one may be asked t o s p e c i f y t h e t y p e o f c o n d i t i o n (e.g., "symmetry") r a t h e r t h a n a node-by-node l i s t o f r e s t r a i n t codes f o r each degree o f freedom. The c o r o l l a r y i s t h a t t h e preprocessing software o r t h e t r a n s l a t o r programs t h a t a r e t h e i n t e r f a c e t o t h e s p e c i f i c a n a l y s i s program ( F i g u r e 1) must i n c l u d e i n t e r n a l coding which a u t o m a t i c a l l y converts between a n a l y s i s - s p e c i f i c data and t h e engi n e e r ' s n a t u r a l vocabulary.

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Two f a c t o r s r e l a t i n g t o data o r g a n i z a t i o n f o r i n t e r a c t i v e systems potent i a l l y c o n t r i b u t e t o t h e d i v e r s i f i c a t i o n o f such software. The f i r s t o f these i s t h e o r g a n i z a t i o n o f data s t r u c t u r e s f o r e f f i c i e n t and r a p i d g r a p h i c a l d i s p l a y . This i n v o l v e s t h e e x p l o i t a t i o n o f geometric coherence which may d i f f e r from t h a t imposed by a p a r t i c u l a r a n a l y s i s method. For instance, i n d i s p l a y i n g a f i n i t e element mesh, one may n o t wish t o draw t h e mesh element by element because t h i s w i l l i n v o l v e t h e redrawing o f l i n e s f o r t h e edges o f adjacent elements. Instead, one would p r e f e r t o t a k e advantage o f t h e coherence o f mesh l i n e s by drawing each complete l i n e i n one step. The rearrangement o f element edge data i n t o complete mesh-line data i n t h i s f a s h i o n i s perhaps more reminiscent o f data o r g a n i z a t i o n s a p p r o p r i a t e t o h i g h e r o r d e r f i n i t e d i f f e r e n c e operators. Another example a r i s e s from t h e d e s i r e t o p r o v i d e d i s p l a y s i m p l i f i c a t i o n s f o r three-dimensional geometries. Here, one technique used t o p e r m i t c l e a r e r viewing o f three-dimensional f i n i t e element meshes i s t h e s e l e c t e d removal

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o f l i n e s o r t h e s e l e c t i v e d i s p l a y of some elements o r groups o f elements. One u s e f u l v e r s i o n of t h i s i s t h e d i s p l a y of o n l y t h e e x t e r i o r s u r f a c e s o f elements ['I] [&I]. The d i s t i n c t i o n here i n t h e data s t r u c t u r e between i n t e r i o r and e x t e r i o r element faces c r e a t e s an obvious 1 ink t o boundary element techniques. The second d a t a - r e l a t e d f a c t o r i s t h e o v e r a l l data t r a n s m i s s i o n scheme u n d e r l y i n g computer-aided design systems w i t h data f l o w s such as i n d i c a t e d s c h e m a t i c a l l y i n F i g u r e 1. The need not o n l y t o t r a n s m i t i n f o r m a t i o n from one stage o f design t o another b u t a l s o t o exchange data between d i f f e r e n t computer-aided design systems has g i v e n r i s e t o s t a n d a r d i z e d procedures, most n o t a b l y t h e I n i t i a l Graphics Exchange S p e c i f i c a t i o n o r I G E S [Sl. Although s t i l l under development and n o t y e t as e f f i c i e n t as d e s i r able, I G E S i s now being s u c c e s s f u l l y used t o t r a n s m i t geometrical data. Moreover, t h e s p e c i f i c a t i o n i s being expanded t o i n c l u d e a n a l y s i s i n f o r m a I G E S data f i l e s a r e r i g i d l y f o r t i o n , p a r t i c u l a r l y f i n i t e element data. matted b u t a r e so complete and f l e x i b l e i n t h e i r r e p r e s e n t a t i o n o f data t h a t t h e f i l e s can be considered e s s e n t i a l l y system independent. The f l e x i b i l i t y o f I G E S a r i s e s because i t s e n t i t i e s may be e i t h e r geometric o r nongeometric and because v a r i o u s e n t i t i t e s may be a s s o c i a t e d i n ways a p p r o p r i a t e t o any a p p l i c a t i o n . The i m p l i c a t i o n f o r a n a l y s i s o f t h e e x i s t e n c e of such schemes as IGES i s t h e n o t i o n t h a t e i t h e r problem d e s c r i p t i o n s o r a n a l y s i s models and r e s u l t s can be s t a n d a r d i z e d i n a comp l e t e , n e u t r a l form. T h i s f o s t e r s an exchange o f data t h a t p e r m i t s n o t o n l y comparisons b u t a l s o combinations o r h y b r i d i z a t i o n s o f a n a l y t i c a l

F i g u r e 2.

Assignment o f boundary c o n d i t i o n s d u r i n g f i n i t e element p r e p r o c e s s i n g f o r s h e l l s t r u c t u r e s [lo]. Note t h e n a t u r a l t e r m i n o l o g y f o r boundary c o n d i t i o n types (middle p o r t i o n o f menu a t r i g h t ).

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procedures. Some o f t h e ideas i n t r o d u c e d thus f a r are now i l l u s t r a t e d by d i s c u s s i o n s and examples o f i n t e r a c t i v e programs developed f o r research o r i n s t r u c t i o n a l purposes a t C o r n e l l . A l l o f t h e subsequent f i g u r e s are photographs taken from graphic d i s p l a y devices, and these have been reduced t o t h e degree t h a t some small t e x t i s not e a s i l y l e g i b l e . I n each o f these cases, t h e small t e x t c o n t a i n s l i t t l e p e r t i n e n t i n f o r m a t i o n and i s r e t a i n e d o n l y t o i l l u s t r a t e t h e n a t u r e o f t h e computer graphics d i s p l a y , I n a d d i t i o n , t h e photographs o f c o l o r d i s p l a y s are, by n e c e s s i t y , reproduced here i n black and white. PREPROCESSING

A s i n g l e glimpse a t an i n t e r a c t i v e preprocessor f o r s h e l l a n a l y s i s [ l o ] provides an example o f t h e humanization o f commands. I n F i g u r e 2, t h e menu page f o r t h e assignment o f boundary c o n d i t i o n s i s shown. The commands along t h e r i g h t edge are d i v i d e d i n t o groups. The upper group c o n s i s t s o f permanent menu items f o r t h e m a n i p u l a t i o n o f t h e main image, w h i l e t h e group below these are commands and m o d i f i e r s s p e c i f i c t o t h e immediate t a s k . Here t h e user has a c t i v a t e d t h e command ASSIGN w i t h t h e boundary c o n d i t i o n t y p e SYMMETRY and t h e m o d i f e r s e t t o LINE. (All active commands and m o d i f i e r s a r e i n d i c a t e d by a box drawn about them.) Simply by p o i n t i n g t o t h e m e r i d i o n a l c u t s o f t h e h y p e r b o l i c c o o l i n g tower s h e l l shown, t h e engineer assigns t h e symmetry c o n d i t i o n t o t h i s l i n e . By f a r t h e most s i g n i f i c a n t e f f o r t i n preprocessing i s t h e c r e a t i o n o f t h e

F i g u r e 3.

Complete three-dimensional f i n i t e element mesh f o r a p r o p e l l e r generated by l o f t i n g between s e c t i o n s [8].

Interactive Computer Graphics a n a l y t i c a l model o r mesh. Even w i t h mesh g e n e r a t i o n a l g o r i t h m s , some i n t e r v e n t i o n by t h e engineer i s u s u a l l y necessary o r d e s i r a b l e t o c o n t r o l t h e design of t h e d e s c r e t i z a t i o n . For example, i n t h e d i s c r e t e t r a n s f i n i t e mapping methods of mesh generation, t h e d e n s i t y and g r a d a t i o n o f t h e mesh i n a r e g i o n i s determined by t h e placement and spacing o f nodes around t h e boundary of t h e r e g i o n [7l, and t h i s t a s k i s b e s t performed by t h e analyst interactively. The techniques o f mesh c r e a t i o n t h a t have been found successful f o r i n t e r a c t i v e a p p l i c a t i o n s have a l s o proven t o be a p p l i c a b l e f o r a v a r i e t y of d i f f e r e n t a n a l y s i s types. A t C o r n e l l , t h e l o f t i n g s p e c i a l i z a t i o n of d i s c r e t e t r a n s f i n i t e mapping has been used f o r b o t h f i n i t e element [7] C8l and boundary element [ l l ] C121 preprocessing. F i g u r e 3 shows a complete three-dimensional f i n i t e element mesh f o r a p r o p e l l e r - l i k e s o l i d generated i n t e r a c t i v e l y i n t h i s fashion. The mesh f o r t h e hub has been c r e a t e d by l o f t i n g among p a r a l l e l p l a n a r s e c t i o n a l meshes w i t h d i f f e r e n t c i r c u l a r diameters, w h i l e t h e l o f t i n g s e c t i o n s f o r t h e blades a r e n o t p a r a l l e l and, a t t h e r o o t o f t h e blades, a r e nonplanar In s e c t i o n s i n t e r a c t i v e l y s e l e c t e d from t h e s u r f a c e o f t h e hub mesh [8]. F i g u r e 4, t h e mesh i n F i g u r e 3 i s s i m p l i f i e d f o r c l e a r e r v i e w i n g by d i s p l a y i n g o n l y t h e e x t e r i o r f a c e t s o f elements, and t h e r e s u l t i s c l e a r l y suggestive o f t h e a p p l i c a b i l i t y o f t h i s meshing procedure f o r boundary element preprocessing. An example o f a d i r e c t a p p l i c a t i o n o f t h i s meshing t e c h n i q u e f o r a boundary element problem i s shown i n F i g u r e s 5 and 6 [ll]

CW. The methods i n these examples c o u l d be s i m i l a r l y extended t o such a l t e r n a t i v e s as f i n i t e d i f f e r e n c e mesh generation. However, one o b s t a c l e remaini n g t o t h e complete g e n e r a l i z a t i o n o f i n t e r a c t i v e p r e p r o c e s s i n g techniques

F i g u r e 4.

The mesh i n F i g u r e 3 i s s i m p l i f i e d f o r c l e a r e r viewing by d i s p l a y i n g o n l y e x t e r i o r (boundary) f a c e t s o f elements.

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f o r a n a l y t i c a l model g e n e r a t i o n i s t h e nonorthogonal i t y o f t h e c o o r d i n a t e systems employed i n t h e mesh generation procedures. Many f i n i t e d i f f e r ence a l g o r i t h m s , f o r example, r e q u i r e a mesh which c o n s i s t s of i s o c o o r d i n a t e l i n e s i n an orthogonal c o o r d i n a t e system. The common f i n i t e e l e ment mesh g e n e r a t i o n techniques based on i s o p a r a m e t r i c o r t r a n s f i n i t e mappings do n o t a u t o m a t i c a l l y f u l f i l l t h i s requirement. Nevertheless, i t i s c l e a r t h a t t h e meshing methods can be extended w i t h minor adaptions t o l e s s r e s t r i c t i v e a n a l y s i s methods such as f i n i t e c e l l methods. I n many cases, boundary element a n a l y s i s r e q u i r e s n o t o n l y a d i s c r e t i z a t i o n of t h e s u r f a c e o f t h e domain b u t a l s o an i n t e r i o r mesh. The conside r a t i o n o f body e f f e c t s such as g r a v i t y loads and o f n o n l i n e a r i t i e s a r e two examples. I n a d d i t i o n , i f f u l l f i e l d o r s e l e c t e d i n t e r i o r r e s u l t s are required, t h e n i n t e r i o r p o i n t s must be chosen as l o c a t i o n s f o r t h e evaluat i o n o f responses. Although i n t e r i o r meshes f o r t h e boundary element method need n o t s a t i s f y t h e same s t r i c t t o p o l o g i c a l and c o n n e c t i v i t y requirements as a f i n i t e element mesh, t h e use o f an i n t e r i o r f i n i t e e l e ment d i s c r e t i z a t i o n i s acceptable i n most cases. The combination o f f i n i t e and boundary element preprocessing c a p a b i l i t i e s i s t h e r e f o r e f r u i t f u l . A modest two-dimensional example i s shown i n F i g u r e s 7 and 8 [13] [14]. Here t h e m o t i v a t i o n f o r use o f an i n t e r i o r mesh i s t h e i d e n t i f i c a t i o n o f b o t h i n t e r i o r p o i n t s and a n a t u r a l i n t e r p o l a t i o n scheme f o r t h e e v a l u a t i o n and v i s u a l i z a t i o n o f t h e f u l l s t r e s s f i e l d . F i r s t t h e boundary element mesh i n F i g u r e 7 i s c r e a t e d i n t e r a c t i v e l y by s e l e c t i n g t h e number, spacing, and g r a d a t i o n o f mesh p o i n t s a l o n g each edge. Then, i n F i g u r e 8, an automatic i n t e r i o r mesh generation scheme i s used t o f i n d r a p i d l y an i n t e r i o r mesh of sampling p o i n t s w i t h a n a t u r a l i n t e r p o l a t o r y b a s i s , i n t h i s case t r i a n g u l a r regions. This mesh i s determined s o l e l y by t h e

F i g u r e 5.

Planar s e c t i o n s f o r c r e a t i o n by l o f t i n g methods o f a boundary element mesh f o r t h e r o o t of a t u r b i n e blade [ll] [12].

Interactive Computer Graphics boundary nodes and by a s i n g l e i n t e r i o r d e n s i t y parameter. Although t h e r e s u l t i n g mesh c o u l d be a d j u s t e d i n t e r a c t i v e l y , t h e design c o n s i d e r a t i o n s c o n s t r a i n i n g t h e i n t e r i o r mesh a r e l e s s s t r i n g e n t t h a n those f o r a f i n i t e element a n a l y s i s . It i s s u f f i c i e n t t h a t t h e mesh i s f i n e i n t h e r e g i o n of expected s t r e s s c o n c e n t r a t i o n , and t h i s refinement i s achieved d i r e c t l y as a consequence o f t h e design o f t h e boundary element mesh w i t h s m a l l e r elements i n t h i s region.

POSTPROCESSING The combination of f i n i t e and boundary element c a p a b i l i t i e s c a r r i e s over i n t o postprocessing, p a r t i c u l a r l y f o r an example m o t i v a t e d by f u l l - f i e l d v i s u a l i z a t i o n such as t h e two-dimensional problem shown i n F i g u r e s 7 and 8. F i g u r e s 9 and 10 show how an i n t e r a c t i v e two-dimensional f i n i t e p o s t processor [151 i s used, w i t h o u t s i g n i f i c a n t m o d i f i c a t i o n , t o i n s p e c t t h e r e s u l t s o f t h e boundary element a n a l y s i s . The f i r s t o f these f i g u r e s shows t h e examination o f t h e v a l u e o f displacements a t nodes by p o i n t i n g t o a p a r t i c u l a r node; t h e deformed mesh can a l s o be d i s p l a y e d a t any s e l e c t e d m a g n i f i c a t i o n . F i g u r e 10 i s a b l a c k and w h i t e photo o f c o l o r s t r e s s contours throughout t h e domain. Here some of t h e s t r e s s ranges have been darkened i n t e r a c t i v e l y t o produce a c l e a r e r view o f how s t r e s s e s vary near t h e c e n t r a l hole. The general g r a p h i c a l problem i n p o s t p r o c e s s i n g i s t h e v i s u a l i z a t i o n o f how a s c a l a r , v e c t o r , o r t e n s o r response parameter v a r i e s over t h e geome t r y o f t h e system. P a r t i c u l a r l y f o r three-dimensional s o l i d s o r s u r faces, i t i s u s u a l l y p o s s i b l e t o d i s p l a y o n l y a s i n g l e parameter component over t h e s u r f a c e o f t h e body w h i l e u s i n g c o l o r a t i o n t o convey

F i g u r e 6.

The complete boundary element mesh r e s u l t i n g f r o m F i g u r e 5.

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simultaneously t h e t h r e e d i m e n s i o n a l i t y o f t h e geometry, F i g u r e 11 [15]. As l o n g as one has both an a p p r o p r i a t e way o f e v a l u a t i n g response parame t e r s a c c u r a t e l y and an i n h e r e n t o r assumed b a s i s f o r s p a t i a l i n t e r p o l a t i o n o f these parameters, an i n t e r a c t i v e g r a p h i c a l postprocessor need n o t be s p e c i f i c t o a p a r t i c u l a r method o f a n a l y s i s and may, i n f a c t , be used e f f e c t i v e l y even f o r combined methods,

As an i l l u s t r a t i o n , f o l l o w i n g a r e some o f t h e major design f e a t u r e s of a general -purpose i n t e r a c t i v e g r a p h i c a l postprocessor c u r r e n t l y under d e v e l opment a t C o r n e l l . The postprocessor i s being designed t o accommodate three-dimensional geometries w i t h speci a1 iz a t i ons a v a i l a b l e f o r both curved surfaces i n three-dimensional space and two-dimensonal problems. The postprocessor serves f o r f i n i t e element, f i n i t e d i f f e r e n c e , boundary element, and combined f i n i t e / b o u n d a r y element analyses. F l e x i b i l i t y f o r a v a r i e t y o f t e c h n i c a l analyses i s b u i l t i n , i n c l u d i n g s t r e s s a n a l y s i s , heat conduction, and compressible f l u i d f l o w , The b a s i c d i s p l a y c a p a b i l i t y c o n s i s t s o f c o l o r c o n t o u r i n g on t h e surfaces as shown i n F i g u r e 11. The user i s a b l e t o choose any viewing angle and may have s i n g l e o r m u l t i p l e views o f t h e system. Moreover, t h e user i s a b l e t o s e c t i o n , c u t , o r " u n f o l d " t h e geometry i n t e r a c t i v e l y i n a v a r i e t y o f c o o r d i n a t e systems t o o b t a i n views o f t h e i n t e r i o r d i s t r i b u t i o n s o f parameters. When a c u t i s made, t h e program a u t o m a t i c a l l y i n t e r p o l a t e s i n t e r i o r r e s u l t s t o o b t a i n t h e v a r i a t i o n o f t h e parameter over t h e exposed surface. When no i n t e r i o r r e s u l t s are a v a i l a b l e f o r a boundary element a n a l y s i s , t h e program can t r e a t postprocessing i n much t h e same manner as i t would handle r e s u l t d i s p l a y s on surfaces such as s h e l l and membrane s t r u c t u r e s . F i n a l l y ,

?I

B

F i g u r e 7.

I

I n t e r a c t i v e boundary element preprocessing f o r a twodimensional problem. C r e a t i o n o f a d i s c r e t i z e d o u t l i n e f o r one q u a r t e r o f a s t r e t c h e d p l a t e w i t h a c i r c u l a r h o l e [13].

In teructive Computer Graphics

51

a l t h o u g h t h e program has i t s own i n t e r n a l data o r g a n i z a t i o n designed f o r e f f i c i e n t i n t e r a c t i v e g r a p h i c a l m a n i p u l a t i o n , i t i s assumed t h a t data from a l l d i f f e r e n t types o f analyses a r e r e c e i v e d by t h e program i n a standardi z e d I G E S format. This represents a f i n i t e element p r e j u d i c e c o i n c i d i n g w i t h t h e i n i t i a l d i r e c t i o n s of t h e IGES t r e a t m e n t of a n a l y s i s data, b u t f o r t h e most p a r t i t seems f e a s i b l e t o expect t h a t t h e r e s u l t s from n e a r l y a l l a n a l y s i s types can be t r a n s l a t e d i n t o a f i n i t e - e l e m e n t - l i k e format.

INTEKACTIVE-ADAPTIVE ANALYSIS I n t e r a c t i v e - a d a p t i v e analyses has been d e f i n e d e a r l i e r as continuous g r a p h i c a l m o n i t o r i n g of t h e progress o f a n a l y s i s by t h e engineer w i t h t h e o p p o r t u n i t y t o i n t e r r u p t and t o change e i t h e r t h e design o r t h e a n a l y s i s [l]. I n a sense, t h i s i s a combination o f p r e p r o c e s s i n g and p o s t p r o c e s s i n g w i t h a d d i t i o n a l f a c i l i t i e s o f user c o n t r o l over a n a l y s i s . Obviously i f a n a l y s i s r e s u l t s a r e t o be monitored i n r e a l time, t h e computation must be very r a p i d o r t h e user w i l l be w a s t i n g h i s t i m e s i t t i n g before a very s l o w l y changing graphic d i s p l a y . C u r r e n t l y , t h i s l i m i t s t h e use o f i n t e r a c t i v e - a d a p t i v e techniques t o r e l a t i v e l y small problems, t o e f f i c i e n t s p e c i a l -purpose a n a l y s i s and design systems, t o t e a c h i n g a p p l i c a t i o n s , and t o research. However, t h e t r e n d s toward i n c r e a s i n g l y c o s t - e f f e c t i v e comp u t i n g and memory and i n c r e a s i n g l y r a p i d data t r a n s f e r i n d i c a t e t h a t t h e s e c o n s t r a i n t s w i l l be a l l e v i a t e d . An example of t h e use o f t h e approach t o m o n i t o r n o n l i n e a r s t r u c t u r a l behavior as i t i s being c a l c u l a t e d i s shown i n F i g u r e 12. Here f o u r d i f f e r e n t p o r t i o n s o f t h e d i s p l a y a r e used t o

F i g u r e 8.

For t h e problem o f F i g u r e 7, c r e a t i o n o f a mesh o f i n t e r i o r p o i n t s used t o o b t a i n f u l l - f i e l d i n f o r m a t i o n from a boundary element a n a l y s i s C131 C141.

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t r a c e a n a l y s i s progress. The main viewport shows t h e deformation behavior o f t h e s t r u c t u r e , w h i l e t h e upper l e f t and upper c e n t e r viewports i n c l u d e graphs o f response s e l e c t e d i n t e r a c t i v e l y by t h e user; l a s t l y , t h e upper r i g h t m o n i t o r summarizes a n a l y s i s i n f o r m a t i o n such as l o a d l e v e l , l o a d step, number o f i t e r a t i o n s , etc. The a n a l y s i s d i v e r s i f i c a t i o n o p p o r t u n i t i e s presented by i n t e r a c t i v e a d a p t i v e a n a l y s i s a r e several. F i r s t i s t h e a b i l i t y t o perform analyses which a i d i n e s t a b l i s h i n g a n a l y s i s parameters; f o r example, t h e a b i l i t y t o perform r a p i d n a t u r a l p e r i o d c a l c u l a t i o n s a s s i s t s i n d e t e r m i n i n g a p p r o p r i a t e t i m e increments f o r marching a l g o r i t h m s . Second, one r o l e f o r i n t e r a c t i ve-adapti ve approaches i s t o change a n a l y s i s s t r a t e g i e s a f t e r a n a l y s i s has begun. For example, i n F i g u r e 13 t h e a n a l y s t has, p a r t l y on t h e b a s i s o f an i n i t i a l a n a l y s i s , i d e n t i f i e d a group o f elements t o c o n s t i t u t e an e l a s t o p l a s t i c n o n l i n e a r s u b s t r u c t u r e , w h i l e t h e remaining elements a r e I n a d d i t i o n , a growing number o f grouped i n t o a l i n e a r s u b s t r u c t u r e [8]. s e l f - a d a p t i v e a l g o r i t h m s a r e becoming a v a i l a b l e f o r such aspects as f r a c t u r e propagation, s u b s t r u c t u r i n g , marching schemes, and mesh improvements. A1 though many of these s e l f -adapti ve a1 g o r i thms a r e s t a b l e , some have h e u r i s t i c bases and a r e best monitored t o ensure t h e continued v a l i d i t y o f computations. For example, t h e f i n i t e element mesh changes t h a t are necessary t o t r a c e an a r b i t r a r i l y p r o p a g a t i n g crack may l e a d t o meshes w i t h e x c e s s i v e l y d i s t o r t e d elements; such d i s t o r t i o n s can be c o r r e c t e d by t h e user under an i n t e r a c t i v e - a d a p t i v e approach. F i n a l l y , t h e implementat i o ? of i n t e r a c t i v e - a d a p t i v e procedures presents two o t h e r p o s s i b l e comput i n g strategies t h a t are p a r t i c u l a r l y s i g n i f i c a n t for nonlinear analysis.

7 i g u r e 9.

I n t e r a c t i v e boundary element p o s t p r o c e s s i n g f o r t h e displacements o f t h e problem i n Figures 7 and 8 u s i n g a twodimensional f i n i t e postprocessor [14].

Interactive Computer Graphics The f i r s t o f these i s p a r a l l e l a n a l y s i s i n which an a l t e r n a t i v e a n a l y s i s t y p e i s invoked t o diagnose system behavior. Examples i n c l u d e b u c k l i n g and f r e e - v i b r a t i o n t e s t s on n o n l i n e a r s t r u c t u r e s a t an a r b i t r a r y stage of l o a d i n g . The second i s s e q u e n t i a l a n a l y s i s where d i f f e r e n t analyses a r e performed t o i n f e r t h e e f f e c t of t h e o r d e r o f a p p l i c a t i o n o f i n f l u e n c e s . For example, a s t a t i c p r e l o a d i n g may precede a n o n l i n e a r s t r u c t u r a l dynamic a n a l y s i s . It i s apparent t h a t i n t e r a c t i v e - a d a p t i v e a n a l y s i s approaches h o l d s t r o n g promise f o r t h e e x p l o r a t i o n , development, and implementation o f u n i f i e d o r combined numerical methods. For instance, t h e same c a p a b i l i t y f o r adapt i v e s u b s t r u c t u r i n g i l l u s t r a t e d above c o u l d be used t o determine i n t e r a c t i v e l y t h e p a r t i t i o n i n g o f a geometric e n t i t y i n t o "married" f i n i t e element and boundary element zones. One o f t h e most e x c i t i n g uses o f i n t e r a c t i v e - a d a p t i v e a n a l y s i s procedures a t Cornel 1 t o date has, i n f a c t , been as a " t e s t bed" f o r research on numerical methods [l].

CONCLUSIONS The i n t e n t i o n o f t h i s paper has been t o demonstrate how i n t e r a c t i v e graphi c s i s a f f e c t i n g t h e environment i n which a n a l y s i s i s performed and t h e r e by i s i n f l u e n c i n g p o s s i b l e u n i f i c a t i o n o r h y b r i d i z a t i o n o f numerical methods. Although t h i s impact i s o n l y b e g i n n i n g t o be f e l t i n t h e comput a t i o n a l mechanics community, t h e r a p i d growth i n a v a i l a b i l i t y o f i t e r a c t i v e g r a p h i c a l c a p a b i l i t i e s w i l l d e f i n i t e l y a f f e c t such developments i n t h e near f u t u r e . It takes o n l y a small leap o f i m a g i n a t i o n t o see how i n t e r a c t i v e techniques a l r e a d y i n use f o r s p e c i f i c numerical methods such

F i g u r e 10.

F u l l - f i e l d s t r e s s v i s u a l i z a t i o n from a boundary element a n a l y s i s u s i n g t h e same f i n i t e element postprocessor as i n F i g u r e 9 [14] C151.

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60

as f i n i t e elements can be g e n e r a l i z e d f o r a v a r i e t y o f a n a l y t i c a l approaches. Future developments o f i n t e r a c t i v e g r a p h i c a l s o f t w a r e systems f o r e n g i n e e r i n g a n a l y s i s and design should be designed t o account f o r p o s s i b l e d i v e r s i f i c a t i o n o f a n a l y t i c a l procedures. One obvious recommendation i s t h a t s t r i c t s e p a r a t i o n be maintained between system models (geometry and a t t r i b u t e s ) and a n a l y t i c a l models (meshes o r o t h e r d i s c r e t i z a t i o n s ) . Under such c i rcumstances , developers o f i n t e r a c t i v e s o f t w a r e should be urged t o "go t h e e x t r a m i l e " t o p r o v i d e t h e a n a l y s i s f l e x i b i l i t y t h a t may be needed f o r e f f e c t i ve engi n e e r i ng design

.

ACKNOWLEDGMENTS P o r t i o n s o f t h e work used as examples i n t h i s paper have been sponsored by t h e N a t i o n a l Science Foundation, t h e N a t i o n a l Aeronautics and Space A d m i n i s t r a t i o n , and by v a r i o u s p r i v a t e companies; t h e w r i t e r s a r e g r a t e f u l f o r t h i s support. However, t h e views expressed here a r e t h e w r i t e r s ' and a r e n o t those o f any sponsor. The w r i t e r s would a l s o l i k e t o express t h e i r a p p r e c i a t i o n t o t h e i r present and former colleagues a t t h e Program o f Computer Graphics a t C o r n e l l U n i v e r s i t y , p a r t i c u l a r l y Donald P. Greenberg, D i r e c t o r o f t h e Program, and W i 11iam McGui re, a f a c u l t y c o - i n v e s t i g a t o r f o r much o f t h e research c i t e d .

F i g u r e 11.

Postprocessing u s i n g c o l o r contours on t h e surface o f a s o l i d i n c o n j u n c t i o n w i t h a d i f f u s e r e f l e c t i o n model t o convey t h r e e - d i mensi onal it y [ 151.

Interactive Computer Graphics REFERENCES and Abel , J . F., I n t e r a c t i v e - a d a p t i v e l a r g e displacement a n a l y s i s w i t h r e a l -time computer graphics, Computers and S t r u c t u r e s

c11 Gattass, M.,

16 (1983) 141-152. C2l Kame1 , H. A.,

Design and implementation o f i n t e r a c t i v e e n g i n e e r i n g software, i n : Abel, J. F., e t a l . (eds.), I n t e r d i s c i p l i n a r y F i n i t e Element A n a l y s i s ( C o l l e g e o f Engineering, C o r n e l l U n i v e r s i t y , I t h a c a , NY, 1981) 773-803.

c31 Pesquera, C. I., McGuire, W.,

and Abel, J. F., I n t e r a c t i v e g r a p h i c a l p r e p r o c e s s i n g o f three-dimensional framed s t r u c t u r e s , Computers and S t r u c t u r e s 17 (1983) 97-104.

c41 Abel, J. F.,

I n t e r a c t i v e computer qraphics i n aDDlied mechanics. i n : Proc. 9 t h U. S. Nat. Cong. App. Mechs’., (ASME,’ New York, 1983) .

97-104. c51 Yerry, M. A.,

and Shephard, M. S . , Authomatic three-dimensional mesh g e n e r a t i o n by t h e m o d i f i e d - o c t r e e technique, I n t l . J . Num. Meth. Engrg. ( t o appear).

C6l Shephard, M. S., and Abel, J. F.,

The i n t e g r a t i o n o f f i n i t e element methods i n t o CAD/CAM, i n : Kardestuncer, H. (ed.), F i n i t e Element Handbook (McGraw H i l l , New York, t o appear 1984).

F i g u r e 12.

Use o f m u l t i p l e d i s p l a y windows t o m o n i t o r n o n l i n e a r s t r u c t u r a l behavior as i t i s being c a l c u l a t e d d u r i n g i n t e r a c t i v e - a d a p t i v e a n a l y s i s C13.

61

J.F. A b e l e t al. Perucchio, R., I n g r a f f e a , A. R., and Abel, J . F., I n t e r a c t i v e computer g r a p h i c preprocessing f o r three-dimensional f i n i t e element a n a l y s i s , I n t . J . Num. Meths. Engrg. 18 (1982) 909-926. Han, T. Y., Adaptive s u b s t r u c t u r i n g and i n t e r a c t i v e graphics f o r three-dimensional f i n i t e element a n a l y s i s , Ph.D. Thesis, Department o f S t r u c t u r a l Engineering, Cornel 1 U n i v e r s i t y (1984). Smith, B. M., e t al., I n i t i a l graphics exchange s p e c i f i c a t i o n (IGES), Version 2.0 (NBSIR 82-2531 (AF), N a t i o n a l Bureau o f Standards, 1982). Chang, S . C., An i n t e g r a t e d f i n i t e element n o n l i n e a r s h e l l a n a l y s i s system w i t h i n t e r a c t i v e computer graphics, Ph.D. Thesis, Department o f S t r u c t u r a l Engineering, Cornel 1 U n i v e r s i t y (1981). Perucchio, R., and I n g r a f f e a , A. R., I n t e r a c t i v e computer g r a p h i c s p r e p r o c e s s i n g f o r three-dimensional boundary i n t e g r a l element a n a l y s i s , Computers and S t r u c t u r e s 16 (1983) 153-166. Perucchio, R., An i n t e g r a t e d boundary element a n a l y s i s system w i t h i n t e r a c t i v e computer graphics f o r three-dimensional l i n e a r - e l a s t i c f r a c t u r e mechanics, Ph.D. Thesis, Department o f S t r u c t u r a l Engineering, Cornel 1 Uni v e r s i t y ( 1984).

F i g u r e 13.

I n t e r a c t i ve-adapti ve substr.uctu r i ng d u r i n g e l a s t o p l a s t i c The a n a l y s t has s e l e c t e d those elements (shown a n a l y s i s [S]. d a r k e r ) which should c o n s t i t u t e a n o n l i n e a r s u b s t r u c t u r e d u r i n g t h e n e x t stage o f a n a l y s i s .

Interactive Computer Graphics [13] Han, T. Y., A general two-dimensional , i n t e r a c t i v e g r a p h i c a l f i n i t e / b o u n d a r y element proprocessor f o r a v i r t u a l stage environment, M.S. Thesis, Department o f S t r u c t u r a l Engineering, Cornel 1 U n i v e r s i t y (1981). [14] Han, T. Y.,

unpublished student p r o j e c t ( C o r n e l l U n i v e r s i t y , 1982).

[15] Schulman, M. A., The i n t e r a c t i v e d i s p l a y o f parameters on two- and t h r e e - dimensional surfaces, M.S. Thesis, Department o f A r c h i t e c t u r e , Cornel 1 U n i v e r s i t y (1981).

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Unification of Finite Element Methods H.Kardestuncer(Editor) 0 Elsevier Science PublishersB.V. (North-Holland),1984

65

CHAPTER 3 HYBRID METHODS OF ANALYSIS S.N. Atluri & T. Nishioka

Three different types of hybrid methods of analysis are (i) a hybrid-analytical-numerical method, (ii) discussed: a hybrid-experimental-numerical method, and (iii) a hybrid-numerical method, for treating complex engineering problems wherein either an analytical (closed form solution) method, or an experiment, or a single-type of numerical method, by itself, is insufficient; and thus a marriage of convenience between the methods is mandatory. INTRODUCTION

In his invitation to us to contribute to this volume dedicated to Professor J. H. Argyris, one of the greatest mechanicians of this century, the chairman of this conference, Professor Kardestuncer, suggested that we dwell on the theme of "hybrid methods of analysis". It often happens that one's name gets associated with a certain line or school of thought. And so it is that the first author wondered if Professor Kardestuncer suggested the title of this paper because of his (the first author's) happy association with another distinguished academician, Professor Theodore Pian, whose name has become more or less synomymous with "hybrid finite element methodsll, wherein the word 'hybrid' connotes "inter-f inite-element constraints", "Lagrange Further reflection convinced us that multipliers", and so forth. Professor Kardestuncer had something quite different in mind for us to on "hybrid methods" and we were led to ponder about write "compatibility" between "methods" and "inter-method" constraints, etc. Then we discovered, much to our pleasant surprise, that in fact we had been using "hybrid methods of analyses" in our research. Thus, the contents of the paper, which we hope would do justice to its

-

-

predetermined title, began to unfold. We present in the following, three such "hybrid methods of analyses". The first deals with a "hybrid-analytical-numerical method"

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S N . Atluri & T. Nishioka

of solving for the intensity of stress-state near the front of semi-elliptical or quarter-elliptical surface cracks in commonplace engineering structures. (We use the word 'analytical' in the context of closed form solutions.) The second deals with a "hybrid-experimental-numerical method" used in studies of dynamic fast crack propagation. The last deals with a "hybrid-numerical method" which employs two or more types of discretization of a continuum (space as well as time) problem.

I.

A brief discussion of each of these topics ensues.

METHOD FOR SURFACE FLAWS One of the most challenging problems of applied fracture mechanics of the past decade has been that of accurate and cost-effective evaluation of "stress-intensity factors" (or the strengths of the singularities of the stress and strain states) along the borders of embedded or surface flaws in complex structural geometries such as aircraft attachment lugs, nuclear reactor pressure-vessel-nozzle junctions, etc. The shapes of these flaws are often assumed, to a first approximation, as elliptical or part-elliptical. The most common approach has been to numerically model the crack-tip region with either "finite-elements" or their variants (such as "boundary-elementstt,etc). A variety of such techniques, with a majority of them being classifiable as singular finite element methods (wherein the basis functions for elements near the crack-front are augmented by those that would induce the known "inverse-square-root" singularity in strains near the crack-front), have been reported [ 1 , 2 ] in literature. Even though excellent results for the stress-intensity factors were obtained through these methods, since the crack-front region was explicitly modeled numerically (by some form of discrete elements), the number of "elements" and hence the number of algebraic equations were excessive, thus rendering these methods prohibitively expensive, if not altogether impractical, for routine application purposes. To circumvent the above problem, the authors have recently developed a methodology based on the Schwarz-Neuman alternating method. In the alternating technique, as applied to the problem of cracks in finite solids, two solutions are needed, generally. One of these solutions is for stresses in the uncracked finite body at the location of the considered crack, and the other solution is for the problem of an HYBRID-ANALYTICAL-NUMF,RICAL

infinite well as subject mentioned

body with a crack whose faces are subject to arbitrary normal as shear tractions. For cracks in complex-shaped finite bodies to complex external loads, the first solution previously would, in general, lead to a rather complex stress-field at the

Hybrid Methods of Analysis

67

location of the considered crack. The analytical results for the second solution, mentioned above, have so far been limited to only crack-face tractions that can be represented by polynomials of the cubic order or lower [3,4]. Recently Vijayakumar and Atluri [51 have overcome this limitation and derived a general solution for an embedded elliptical crack, subject to arbitrary-order polynomial type crack-face tractions (both normal as well as shear). This general solution was recast in a form more suitable for application, in a finite-element alternating technique, by Nishioka and Atluri [6]. The methodology developed in [5,6] is applicable, in general, for combined Modes I, 11, and 111 problems of arbitrary-shaped flawed structural components subject to arbitrary external loading. For typical three-dimensional surface-flaw problems, it was shown [6,7,8,9] that the "hybrid-analytical-numerical" procedure developed in [6] is least order of magnitude less expensive than the singular-finite-element procedures [1,2] developed in

one

the mid 1970's. We provide, in the following, some "hybrid-analytical-numerical'' method.

technical details of this

Analytical Solution for an Elliptical Crack in an Infinite Solid with Arbitrary Crack-Face Tractions In this section, only the Mode I problem is considered. The complete, general solution including the Modes I1 and 111 is given in Refs. [5,61. Suppose that x1 and x2 are Cartesian coordinates in the plane of the elliptical crack and x 3 is normal to the crack plane such 1.2

--

that

describes the border of the elliptical crack of aspect ratio (al/a2). The ellipsoidal coordinates 5, (a=1,2,3) are defined as the roots of the cubic equation:

(I.2)

Let form:

the normal traction along the crack surface be expressed in the

S.N. Atluri& T. Nishioka

68

'''' l

l

M

m

i=O j=O m=O n=O

(i,j) x2m-2n+i 2n+j X 2 *3,m-n,n 1

where A ' s are undetermined coefficients and the parameters i and j specify the symmetries of the load with respect to the axes of the ellipse, x1 and x2. I Using the well-known Trefftz's formulation as discussed in detail in [5], the problem is reduced to that of finding appropriate potential functions. A s shown in [ 5 J , the solution corresponding to the load expressed by (1.3) can be found in terms of the potential function:

where m

a 2k+i+j F2k-2R+i,2R+j

- a x12k-2R+iaX2R+j 2

2k+i+j+1 +(

3

ds

Jqo

(1.5)

and

and C's are also undetermined coefficients. The components of displacement ui and stress Gi, in terms of the derivatives of the potential functions (a comma followed by an index such as j implies partial differentiation w.r.t. x.) are given by: I u1 = (1-2v)f 3,l u2 u3

-

(1-2V)f

=

3,2

-2(1-v)f

+

+

3,3

x3f3,31

(I.7a)

x3f3,32

(I.7b)

+

x f 3 3,33

(I.7c)

and '11

=

"( f 3,11+2v3,22+x3f 3,311)

' 2 2 = 2!J(f 3,22"'

'12

=

3,ll+x3f 3,322)

2v(f3,12-2vf3,K!+X3f3, 312)

a33 = 2p(-f 3,33+x 3f 3,333)

(I.8a) (I.8b)

(I.8c)

(I.8d)

69

Hybrid Methods of Analysis

u31

= 2px

0

= 2px f

32

(I. 8 e )

f 3 3,331

(I.8 f )

3 3.332

where lI and L, are the shear modulus and Poisson's ratio. For later convenience, the stresses given by E q . (1.8) through E q s . (1.4-6) are expressed in a matrix form:

(c1

61 =

[PI

6x1

6xN Nxl

(1.9)

where [PI is the function of the coordinates (x1,x2,x3) and N is the total number of coefficients A OT C. Satisfying the boundary condition on the crack surface, the relation between the coefficients A and C can be summarized in a matrix form: = [B] {C) (1.10) Nxl NxN Nxl The detailed

complete expression of components of [B] is given in Ref.

[61

Expressions for the first, second, and third derivatives of the potential functions (1.4,.5) are given in [5,61. It is shown [5,6] that the evaluation of far-field displacements and stresses in the "infinite body" containing an elliptical crack hinges on the evaluation of a generic elliptical integral of the type:

I,

a k5Rlam1 uk+R+l 1

2

3

(I.11)

ds

JQO

wherein, 3; implies the jth order derivative with respect to xcl. To evaluate (1.11). we expand u k+R+l in terms of x2 and carry out term by c1 term differentiations. Then we get:

2p- 2q- kl x;q-2 r- R

1 . ( 2(p-q) p - 2 ~ )! (2 -2r) ! (2r)! ! (:-r) ! (r) ! (2p-2q-k1) ! (2q-2r-R1) ! X

(2r-m 1) !

(I. 1 2 )

S.N. Atluri & T. Nishioka

I0

where (.) denotes a multiplication, and m

(I.13)

Thus, one of the key algebraic steps in the successful application of the in conjunction with an analytical results originally derived in [ S ] , alternating method, is the evaluation of the elliptical integrals of the An efficient and accurate algorithm for generic type given in (1.13). accomplishing this key step has been presented in [ 6 ] .

I. 3

The "Hybrid-Analytical-Numerical"

"Finite-Element Alternating" Method As noted earlier, the present "hybrid-analytical-numerical" technique is based on two solutions: Solution Analytical solution for the problem of an elliptical crack subjected to arbitrary normal and shear loading on the surface of the crack, in an infinite solid, as presently discussed. Solution A general numerical technique such as the finite element method or boundary element method for the solution of stresses in the uncracked finite body with geometry and loading being identical to the cracked structure in question. Solution 1 is discussed earlier, while the finite element method is used in the present paper to generate Solution 2. The present finite element alternating method requires the following steps. (1) Solve the uncracked body under the given external loads by using finite element method. The uncracked body has the same geometry with the given problem except the crack. To save computation time in solving the finite element equations, a special solution technique is implemented. This wil be explained later. ( 2 ) Using the finite element solution, we compute the stresses at the location of the original crack. (3) Compare the residual stresses calculated in Step ( 2 ) with a In the present study one percent of the permissible stress magnitude. maximum external applied stress is used for the permissible stress magnitude.

(4) To satisfy the stress boundary condition, reverse the residual stresses. Then determine coefficients A of Eq. (1.3) by using the following least-squares method:

71

Hybrid Methods of Analysis

Ia

=/s'

-

,U :(

(a = 1,2,3) is a minimum

U$))2dS

(1.14)

C

R

where U3, is the reversed residual stresses calculated by the finite element solution, a(O)is defined by equation of the type (1.3), S c is the 3cl region of the crack, and I, are the functionals to be minimized. Rewriting Eq. ( 1 . 3 ) in a matrix form: (1.15) we obtain the relation and substituting Eq. (1.15) into Eq. (1.14), between the coefficients A and the residual stresses:

where (1.17)

(1.18)

(5)

Determine the

coefficients C

of

Eq. (1.4) in the potential

functions

by solving Eq. (1.10) ({C) = [B]-'{A}). (6) Calculate the stress intensity factors for the current iteration by substituting coefficients C in the following analytical expressions [5,6] specialized here for the "Mode I" problem:

&-(?)

2k-2R+i

sine 2R+j (T)

c3,k-R,1 (i,j)

(1.19)

1 2 where 8 i s the elliptic angle measured from x1 axis, and 2 1

2

A = a sin 8 (7)

+ a22cos28

Calculate the

(1.20)

residual stresses on external surfaces of the

S.N. Atluri & T. Nishioka

12

body due to the loads in Step ( 4 ) . To satisfy the stress boundary conditions, reverse the residual stresses and calculate equivalent nodal forces. These nodal forces {Q} can be expressed in terms of coefficients C:

@Im

=

- [G],{C3

(1.21)

and [GIrn=

Jf

“lT[nl

PIdS

(1.22)

m where m denotes the number for a finite element, [N] is the matrix of isoparametric element shape functions, [n] is the matrix of the normal direction cosines and [ P I is the basis function matrix for stresses and Although the matrix [ P I has the can be derived from Eq. (1.8). singularity of order l / G at the crack-front, the functions in [ P I decay very rapidly with the distance from the crack-front. Thus, the matrices [G,] are calculated only at the surface elements which satisfy the condition rmin < 5aly where rmin is the distance of the closest nodal point of each surface element from the center of the ellipse and a is 1 the semi-major axis of the ellipse. (8) Consider the nodal forces in Step (7) as external applied loads acting on the uncracked body. Repeat all steps in the iteration process until the residual stresses on the crack surface become negligible (Step 3). To obtain the final solution, add the stress intensity factors of all iterations. As seen from above, for the finite element alternating method, we need to solve the following type of finite element equations: (1.23) and

(1.24)

in which the superscript denotes the cycle of iteration, [K] is the global (assembled) stiffness matrix of the uncracked body and remains the same during the iteration process, and s_‘ is the nodal displacement vector for ith iteration. Qi is the nodal force vector for ith iteration and depends on the solution for the previous iteration qi-l - as expressed by Eq. ( 1 . 2 4 ) . An efficient equation solver OPTBLOK developed by Mondkar and Powell [ l o ] is used to save computational time in solving Eq. (1.23).

Hybrid Methods of Analysis

73

The solution algorithm is divided into three parts, i.e. (i) reduction of stiffness matrix, (ii) reduction of load vector, and (iii) back In OPTBLOK the reduction of stiffness matrix is done only substitution. once, although the reduction of load vector and back substitution may be repeated for any number of load cases. Thus, denoting CPU time for each part by T1, T2, and T3, respectively. The total CPU time T in solving Eq. (1.23) using OPTBLOK can be expressed by T = T

1

+ ( n + 1 (T

2

+ T 3)

= (T

1

+

T2 + T3)

+

n(T2 + T3)

(1.25)

where n is the tota number of iterations. Since TI is much larger than (T2 + T3), a great amount of computational time can be expected to be saved by comparing with the case in which Eq. (1.23) is solved for each To illustrate this situation, we iteration (T* = (n + l)(T1 + T2 + T3)). consider the example of a set of linear equations with the number of equations of 1960, and half band width of 200, wherein the CPU time for reduction of load vector and back substitution was about 5.6% of the total CPU time (T2 + T3 0.056T). Since for a typical problem, the present alternating method needs three iterations (n = 3), the additional cost in this case is only about 16.8%, which is considerably smaller than 300% in the case when Eq. (1.25) is solved for each iteration. Now, some comments concerning the solution of surface flaw problems in finite bodies, through the present "hybrid-analytical-numerical'' procedure are in order. Since the analytical solution for an elliptical crack in an infinite solid is implemented as Solution 1, it is necessary to define the residual stresses over the entire crack plane including the fictitious portion of the crack which lies outside of the finite body. Moreover, it is well known that the accuracy of the least squares fitting inside of the fitted region can be increased with the number of polynomial terms; however, the fitting curve may change drastically in the region outside of the fitting. For these reasons, in Ref. [ 6 1 numerical experimentation was carried out for arriving at an optimum pressure distribution on the crack surface extended into the fictitious region. For a semi-elliptical crack which lies in the region of -al < x l

<

2 al and 0 x2 5 a it was concluded that the fictitious pressure 2' which, for the region of -a2 5 x2 5 0, remains constant in the x2 direction but varies in the x1 direction, gives the best result among the several numerical experiments performed in Ref. [ 6 ] , even though the results for other types of assumed pressure in the fictitious region differed only slightly (t-2%). The procedure of the fictitious pressure distribution for a semi-elliptical surface crack was successfully used on

S.N. Atluri & T. Nishioka

74

the analyses of surface cracks, in finite thickness plates subject to remote tension as well as remote bending [6], and in pressure vessels [7,9]. In the present paper, taking acc0ur.t of the conclusion drawn in Ref. [6], the fictitious pressure distribution given below is employed for the analysis of a quarter-elliptical corner crack. For the first quadrant (x1,x2 2 0) (namely the actual surface crack), the residual stress can be calculated by the finite element method and is a function of the coordinates x1 and x2. For the other quadrants, the fictitious residual stress is defined as

R

u33

1

=

R a33(0,0)

for the second quadrant

(x <0,x2>O) 1-

for the third quadrant

(xl

[ U R~ ~ ( X ~ , O )for the fourth quadrant

,x2
(x >O,x
(I.26)

1.4

Results for Quarter-Elliptical Corner Cracks in Aircraft Attachment The geometry of the lug with two symmetric quarter-elliptical corner cracks is shown in Fig. 1. The lug material is 7075-76 Aluminum with Young's modulus E=71.71 GPa (10.4 x 106 psi) and Poisson's ratio U = 0.33. To simulate pin loading, the cosine bearing pressure defined cos )I acting on only a half of the boundary -n/2 5 $ I by SRR 2p (TRjt) r / 2 , as shown in Fig. 1 is considered. The analysis was performed for nine crack geometries as follows ~

ap/%

=

0.5, 1 . 2 , and 2.0

= 0.2, 0.5, and 0.8 %/t where a and ah denote crack lengths at the surfaces of the plate and P hole, respectively. Thus, a = a2 and ah = al for ap/ah = 0.5, and a = P P al and ah = a2 for ap/ah = 1.2 and 2.0. The typical finite element model used for the uncracked lug is shown in Fig. 2. This model consists of 140 twenty-noded isoparametric elements with 2250 degrees of freedom (before imposition of boundary condition). Due to the symmetry, a half of the lug was used in the analysis. The displacements were imposed as u 3 = 0 at x3 = -L, and u l = 0 at XI = -Ri. The matrices [GIm were calculated on the surface of x L = 0, t, R = Ro (x3 LO), and XI = Ro-Ri ( x 3 < 0) satisfying the earlier stated condition, rmin < 5a1. First, only stress analyses of the uncracked lug shown in Fig. 2 were performed to examine the effect of the lug length, changing L = 5Ri to 6Ri. The average value of normal stress a33 at the original crack

Hybrid Methods of Analysis location Thus,

differs

the

0.02% as t h e l u g l e n g t h changes from 5Ri t o 6Ri.

only

following

75

In addition, the

were done w i t h L = 5Ri.

analyses

magnitude

o f s h e a r stresses which p r o d u c e t h e Mode II and Mode 111 stress

intensity

factors

stresses stress

531

and

was

also

032

were,

Thus,

U33.

the

examined.

The

respectively,

Mode

average v a l u e of t h e s h e a r 0.5

and 0.1% of t h e normal

I s t r e s s i n t e n s i t y f a c t o r i s dominant and

o t h e r modes a r e n e g l i g i b l e i n t h i s c a s e .

To a

q u a n t i f y t h e e f f e c t s of a f i n i t e body, c r a c k a s p e c t r a t i o , e t c . ,

magnification

factor

( n o r m a l i z e d stress i n t e n s i t y f a c t o r ) Fi,

defined

by t h e f o l l o w i n g e q u a t i o n , i s used (1.27)

where ai

is

integral

o f second k i n d , k

The

a

r e f e r e n c e stress m a g n i t u d e , E(k) i s t h e c o m p l e t e e l l i p t i c 2 2 2 2 = (a, a 2 ) / a l and A i s d e f i n e d by Eq. ( 1 . 2 0 ) .

denominator

stress

exact

constant

of

reference

stress

elliptic

integral

ap/ah =

the

magnification

on

the

is

s i d e o f Eq. (1.27)

for

corresponds t o t h e

t h e e l l i p t i c a l crack subject t o the

crack surface, i n an i n f i n i t e solid.

The

is c h o s e n t o be: Ui = U = ( P / 2 R i t ) . The c o m p l e t e P of second k i n d , E ( k ) , i n Eq. (1.27) i s g i v e n by 1.2111

U i

and 1.4429 f o r a / a = 1.2. F i g u r e s 3a-c show P h f a c t o r s as a f u n c t i o n of t h e e l l i p t i c a l a n g l e f o r t h e

r a t i o s of a p / a h = 0.5,

aspect 3a-c

Ui

factor

0.5 and 2.0,

for

angle

t h e right-hand

intensity

pressure

-

1.2, and 2.0,

respectively.

The e l l i p t i c a l

measured from t h e h o l e s u r f a c e i n t h e s e c a s e s .

always

Figures

t h e m a g n i f i c a t i o n f a c t o r s a s a f u n c t i o n of t h e c r a c k l e n g t h a t

show

t h e p l a t e s u r f a c e , a p , f o r t h e c r a c k d e p t h o f a h / t = 0.2, 0.5, and 0.8. M a g n i f i c a t i o n f a c t o r s i n c r e a s e as t h e c r a c k l e n g t h a p d e c r e a s e s , due t o t h e f a c t t h a t t h e stress c o n c e n t r a t i o n e x i s t s around t h e p i n h o l e . The

CPU

time

for

all

the

above a n a l y s e s w a s a p p r o x i m a t e l y 1800

s e c o n d s u s i n g t h e CYBER 74. 11.

HYBRID-EXPERIMENTAL-NUMERICAL In

numerical general,

this

instance,

a

(H-E-N)

combined

METHODS OF ANALYSIS

u s e of t h e e x p e r i m e n t a l as w e l l as

in n o t p o s s i b l e from t h e u s e of e i t h e r a n e x p e r i m e n t o r a n u m e r i c a l techniques

a n a l y s i s alone.

is

made

to

ascertain

information

that

is,

Such h y b r i d t e c h n i q u e s a r e r a t h e r commonplace [ l l ] .

Here w e d i s c u s s one h y b r i d - e x p e r i m e n t a l - n u m e r i c a l technique i n p a r t i c u l a r , i n v o l v i n g dynamic c r a c k p r o p a g a t i o n and arrest. I n dynamic fracture mechanics, under e i t h e r q u a s i - s t a t i c or dynamic l o a d i n g conditions,

crack

p r o p a g a t i o n i s p o s t u l a t e d t o commence when t h e dynamic

S.N. Atluri & T. Nishioka

16

ap/ah

RI=1905mm

= 20

a h / ! = 05

R0=5715mm

L

= 5R,

t = 12 70mm

L-22860mm

F i g . 1.

Pin-Loaded, Corner-Cracked At t ach m e n t Lug

- ;;pjr

Fi g . 2 .

F i n i t e Element Model o f t h e Uncracked Lug

3.0;

3 h

-08

t

o dah=05

N

0 w% 2

0

&Is 4

-

s

-

@I0

HOLE SURFACE (0')

PLATE 1-

SURFACE (90')

-

Y" 00 0.0

I

'

o

n

'

30

I

60

(DEGREES) (a )

Fig. 3.

*

'

A

90

apph-2.0

L

0

30

60

(DEGREES) (C

1

Stress-Intensity Factor Solutions f o r 9 Different C r a c k G e o m e tr i e s in a n A t t ach m en t Lug

90

Hybrid Methods of Analysis

77

stress-intensity factor near the crack-tip exceeds a given material property, KId, the so-called initiation toughness. Thereafter, the crack-propagation is governed by the criterion, KIt = KID(v), where KIt is the time-dependent-stress-intensity factor near the tip of the propagating crack in the given problem, and KID is the material's dynamic fracture toughness which depends on the velocity, v, of the propagating crack-tip. One of the primary aims of experimentation in dynamic fracture is to determine the material's fracture toughness as a function of crack-velocity, KID(v). This involves measuring directly, the crack-tip stress data. To this end, dynamic photoelastic methods, or methods of However, these techniques for caustics, have been developed [ 1 2 ] . optically measuring the stress-state near the crack-tip are limited to transparent materials, such as photoelastic polymers and resins, and to plane-stress situations which cause thickness changes in the specimen near the crack-tip. However, such direct measurements are not possible for structural steels or other metallic materials under loading conditions that deviate from that of plane-stress. In such situations, a H-E-N technique (also referred to as a "generation calculation") is useful. Thus, for instance, in the dynamic crack-propagation experiment on a specimen, one may carefully measure the initial conditions on the specimen as well as the crack-length versus time history. These experimental data can then be used as input to a well-formulated numerical analysis program to determine the dynamic stress-intensity factor at the tip of the propagating crack. When such analyses are repeated on a number of specimens, one can accurately determine the dynamic fracture toughness property of the material as a function of crack-velocity. It may be worthwhile to note that for dynamic crack propagation in finite elastic bodies, such as the above-mentioned test specimen, the interaction with the crack-tip of stress waves reflected from the boundaries andlor emanated from other moving crack-tips plays an important role in determining the intensity of the dynamic singular stress-field at the considered crack-tip. Because of the analytical intractability of such elasto-dynamic crack problems, computational techniques are mandatory. This, coupled with the above described problems in experimentation, make the H-E-N methods indispensible in studies of dynamic crack propagation and arrest. Once the material property KID(V) is determined, it may be used, in conjunction with specified initial conditions and loading conditions on the cracked structure, to predict the history of crack propagation and arrest,if any, in the structure under question. Such a calculation is

S.N. Atluri & T. Nishioka

78

often called a "prediction calculation". In the following, we present briefly "generation" and "pred i ct i on" calculations

.

the

details o f both the

A

Synopsis of the Formulation of the "Generation" Problem Consider two instants of time tl and t2 = t l + At. Assuming, without loss of generality, that the crack propagation is in pure Mode I, let the crack lengths at tl and t2 be C 1 and 12 = C1 + AZ, respectively, Let the displacements, strains, and stresses at tl and t2 be, 1 2 2 2 The variables respectively, (ui, Eij, and as.) and (ui, Eij, and Oij). 1J are presumed known. It has been shown [ 1 3 ] that the at time t 1 variational principle governing the dynamic crack propagation between t l and t2 can be written as: 11.1

+ a..)6c2 1 + 1~

ij

p ( i i2

+ s .1) 6 u i2}

dV

(11.1)

In the above, V 2

is the domain, and sa2 the external boundary where -1 time-dependent tractions are prescribed, at time t2; Ti are the prescribed tractions at tl at sol ( z so*) as well as at AX+; ( )+ indicates the upper half of the crack-face, which only is considered in 1 1 2 the present Mode I problem. It is seen that the integrand (CT V ) (6ul) i) j in the last term of the rhs of Eq. (11.1) corresponds to the term of energy-release rate due to dynamic crack propagation. The Eq. (II.'l) may thus be viewed as a virtual energy-balance relation for dynamic crack-propagation, and hence the present numerical method based on Eq. (11.1) i s inherently energy-consistent. In Eq. ( I I . l ) , 1 a1.) are known, while (a2 c2 and w.2) are iJ ij' ij' i the variables. Now, Eq. (11.1) is used to develop a finite element Thus, the domain V 2 is discretized into a approximation at time t2. finite number of elements, with a domain V s immediately surrounding the crack-tip being treated as the so-called "singular element", and the

(ci,

Hybrid Methods of Analysis

79

domain V 2 - V, being mapped by the well-known, 8-noded, isoparametric In the singular-element V,, the basis functions for assumed elements. displacements are the crack-velocity dependent eigen-function solutions to the elasto-dynamic problem of crack-propagation in an infinite domain, as discussed in this paper. Note that at time t2, in the present Mode I problem, the crack-tip located at x = C + AX and hence the singular-element is centered at 1 1 x1 = C l + A Z . In developing the equations for the finite element mesh at 1 1 t2, it is seen from Eq. (11.1) that the variation of Oij and ui must be known in the finite element mesh at t2. However, oij, 1 uj, 1 and Gj' were is

solved for in the finite element mesh at tl. In the mesh at tl, the crack-tip was located at x1 - El, and hence the crack element was Thus, between tl and t2(tl + At) the crack element is centered at C1. translated by an amount A E . While the crack-element is translated, only the elements surrounding the moving crack-tip are distorted. Thus, the finite element meshes at times tl and t2 differ only in the location of the crack-tip (and hence the crack-element) and the shapes of the immediately surrounding isoparametric elements. Thus, the known data at and u in the mesh at tl is interpolated easily into corresponding j Further details of the above translating data in the mesh at t2. singularity-element method of simulating dynamic crack propagation in

Uij

arbitrary shaped finite bodies can be found in [13,141. We now remark briefly on the basis functions for assumed = 1,2) be fixed displacements used in the singular element. Let x,(a rectangular coordinates in the plane of the present two-dimensional elastic body, with the crack-tip moving along the xl axis and x2 is normal to the crack-axis. We introduce a coordinate system (6,x ) which 2 remains fixed w.r.t. the propagating crack-tip, such that 5 = x1 - vt, where v is, without loss of generality, the constant speed of crack-propagation. It equations, governing

can be shown [13,14] that the this problem, for the wave

elastodynamic potentials $

(dilatational) and $ (shear) are:

and a similar equation for $, except that Cd in Eq. (11.2) Is to be replaced by cs, where cd and cs are the dilatational and shear wave speeds respectively. The "steady-state" eigen-function solution to the homogeneous part of Eq. (11.2), namely, the solution which appears time-invariant to an observer moving with the crack-tip, and satisfies the prescribed traction conditions on the crack-face ( 5 < 0, x2 = LO) can

80

S.N. Atluri& T. Nishioka

be derived easily, as indicated in [13] and elsewhere. We use these eigen function solutions for an infinite body, as basis functions for assumed displacements within the "crack-tip-singularity-element". However, to satisfy the full Eq. (11.2), the undertermined coefficients, p . below, in the eigen function expansion are taken to be functions of J time. Thus, within the singular element

are the above described eigen-functions, and Bj are where u aj undetermined parameters, which are to be determined from the finite element equations for the cracked body. As seen from Eq. (11.3), the eigen functions u "1 depend on the crack-tip velocity. In the present numerical approach, the crack-tip velocity is assumed to be constant within each time-increment At, say v1 between t l and tl + At, and v2 between t2 and t2 + At, etc. Thus, t and t + At, the eigen-functions embedded in the between 1 1 singularity-element correspond to velocity v1 and those between t2 and t2

+

At correspond to velocity v2. Thus, the present finite element method is capable of handling non-uniform-velocity crack propagation.

The total velocities and accelerations of a material particle in the singular element, within each time step, corresponding to Eq. (11.3), can be written as: (11.4)

and (11.5) where ( ) , = a ( )/a<, and ( * ) implies a time derivative. 5 The salient features, pertinent to the studies reported in this paper, of the present method, the mathematical details of which are reported elsewhere [13,14], are as follows: ( a = 1,2) lead to the familiar (1/ (i) The eigen functions u a1 d ( r ) ) singularities in strains and stresses. Thus, the coefficient B,(t) is directly related (to within a scalar constant) to the dynamic stress intensity factor, KI(t). (ii) The compatibility of displacements, velocities, and accelerations of material particles at the boundary of surrounding elements with those of the surrounding (usual) isoparametric elements is

Hybrid Methods of Analysis

81

satisfied through a continuous least-squares approach. If the displacements, velocities, and accelerations of the nodes at the boundary of the singular-element, Vs, are q , 4 , and q respectively, the above least-squares technique leads to linear algebraic relations between the sets (4, 4 , q ) and (B, 8, and$) where are undetermined parameters in the eigen-function expansion, Eq. (11.3), in the singular-element. From these equations and the final finite element equations governing the nodal displacements, velocities, and accelerations of the cracked can be computed directly. Thus, the structure, the variables B, 8, dynamic stress-intensity factor, as well as its first two time derivatives, are computed directly in the present procedure. (iii) The "transient" finite element equations are integrated in time, using the well-known Newmark's @-method [ 13,141. (iv) Because of the use of the eigen functions in a moving coordinate system, as in Eq. (11.3), in the singular-element, there is the presence of an "apparent" damping matrix for the singular element. Further, for the same reason, this damping matrix as well as the stiffness matrix of the singular-element, are unsymmetric. However, the stiffness and mass matrices of the surrounding isoparametric elements are, of course, symmetric. Thus, the final finite element equation system will have a "small" degree of unsymmetry. This equation system is solved, in the present studies, using a simple iterative scheme. In the "generation" calculation, the crack velocity history, v(t) in (11.4,.5), is provided from the experiment. This history, as well as the initial conditions, are used as input to the finite element program to determine KI(t). 11.2

Details of the "Prediction" Calculation The problem here is to predict the time histories of crack-length

[I(t)], crack-velocity [E(t) Z v(t)], and possible crack-arrest, for a specified relationship of dynamic fracture toughness [ K I ~ ] versus crack-velocity [v]. Let the prediction problem be considered to have been solved up to time tl. In order to find the solution at t2( tl + At), the crack-velocity at t2, namely, v2 3 Z(t,) must be found. To this end, it is first noted that the dynamic stress-intensity factor can be written as :

K

I

=

Since, in

K (t,v) 1 the

present

(11.6)

procedure, the velocity of crack-propagation is

S.N. Atluri & T. Nishioka

82 assumed

t o be c o n s t a n t w i t h i n e a c h t i m e - s t e p ,

predict

the

at

velocity

+

[ti

(At)/2]

an approximate procedure t o

w i l l be sought.

T a y l o r s e r i e s e x p a n s i o n , i t i s s e e n from Eq. (11.6)

Using d o u b l e

that:

m

(11.7)

where

is

KIP

the

predicted

v a l u e of KI a t t l

+

(At/2).

One c a n , upon

expanding terms, w r i t e Eq. (11.7) a s :

where in

a(

) / a t , and R i s " r e s i d u e " of t h e T a y l o r e x p a n s i o n i n d i c a t e d (11.8). Note t h a t u s e i s made of t h e s a l i e n t f e a t u r e of t h e

(') =

Eq.

present

procedure, t h a t B l ( t )

analysis

Since

during K

predicted

dynamic

crack

arrest

dynamic-toughness

arr ID

Thus,

KI(t),

B1 b e i n g t h e c o e f f i c i e n t

propagation,

KI

KID

=

,

using the

Eq. (11.8) and t h e s p e c i f i e d K I D v s Z ( t ) r e l a t i o n , t h e

or

IP v e l o c i t y v ( z C) a t t h e t i m e [ t i

crack

3

(11.3).

of t h e f i r s t e i g e n - f u n c t i o n s a s i n Eq.

in

the

is

Kfgr,

present

+

( A t / 2 ) ] c a n be p r e d i c t e d .

is

crack-arrest

If the

p r e d i c t e d i f KIP

<

p r o c e d u r e , c r a c k arrest i s p r e d i c t e d as a

t e r m i n a l e v e n t , i f any, i n t h e p r o p a g a t i o n a n a l y s i s . Using system and,

of from

Pl(t2)]

the

above p r e d i c t e d c r a c k - v e l o c i t y v a l u e , t h e f i n i t e e l e m e n t

equations these,

at

the

time actual

(II.l),

are c o n s t r u c t e d ;

stres-intensity

factor KI(tz)[-

t 2 , b a s e d on Eq.

dynamic

Thus, t h e a c t u a l K I a t t l + ( A t / 2 ) i s computed, as,

i s computed.

c o r r e l a t i o n between t h e p r e d i c t e d KIP o f Eq. (11.8) and t h e a c t u a l K I

The

(11.9) c a n be s e e n t o depend on t h e r e s i d u e , "R", of Eq. (11.8). t h i s c o r r e l a t i o n , a f u r t h e r approximation i s i n t r o d u c e d i n t h e work t h a t t h e r e s i d u e R a t t l + ( A t / 2 ) can be approximated by i t s

of

Eq.

To

ensure

present known

value a t [ t l

procedure, w r i t t e n as:

the

-

( A t / 2 ) ] , i n t h e g e n e r i c sense.

generic

a l g o r i t h m u s e d t o f i n d KIP

Thus, i n t h e p r e s e n t

a t tl

+

( A t / 2 ) c a n be

Hybrid Methods of Analysis

-

bI(tl

-

%) -

Kp:

( - %)] tl

83

(11.10)

In all the presently reported computations, when Eq. (11.8) with R Z 0 was used, a maximum error of the order of 3% between K I ~and KI was noted. However, when Eq. (11.10) was used, this maximum error reduced to the order of 0.5%.

11.3

Example a H-E-N or "Generation" Analysis To demonstrate the "generation" type calculations, we first treat a wedge-loaded rectangular double cantilever beam specimen (WL-RDCB), the crack-propagation histories and dynamic stress-intensity factor histories in which were directly measured by Kalthoff et al. 1121. The relevant geometric data of the WL-RDCB specimen are indicated in Fig. 4 which also shows the finite element model wherein the moving-singularity-element is shown hatched, at the beginning of crack propagation. The material constants used in the present analysis are: E = 3380 MN/m2 and Poisson's 0.33. In the experiments of [ 1 2 ] , several test specimens, ratio, v wherein cracks were initiated from blunted notches with crack-propagation initiation stress-intensity factors K I ~larger than the fracture toughness KIc, were studied. Note that the actual loading mechanism in the experiment is closer, in numerical simulation, to loading the finite element model at point A in Fig. 4 , with the material to the left-hand side of line BA in Fig. 4 also considered to be participating in the motion. In the first attempt at the analysis, however, the loading was modeled to act a point B in Fig. 4 instead, and the material to the left of line AB was not modeled. In the remainder of the paper, the numerical model wherein load was applied at point A of Fig. 4 and the material to the left of line AB (Fig. 4 ) was also modeled, is often referred to as the "actual loading condition" and the other one as the "simplified loading condition" ,

-

respectively. In their report, Kalthoff et al. [ 1 2 ] identify the RDCB specimen as specimen 4 . For convenience, the same with KZq value of 2.32 identification is used in the presently reported numerical simulation.

S.N. Atluri & T. Nishioka

84 noted

As

e a r l i e r , t h e "generation" c a l c u l a t i o n uses a s input the measured c r a c k l e n g t h (and hence c r a c k - v e l o c i t y ) h i s t o r y .

experimentally The

output

of

the

stress-intensity

is

calculation

factor

at

the

directly

computed

dynamic

t h e t i p of t h e propagating c r a c k f o r v a r i o u s

time i n s t a n t s .

5 shows t h e c o n s i d e r e d c r a c k v e l o c i t y and l e n g t h h i s t o r y f o r 4 as r e p o r t e d i n [ 1 2 ] . F i g u r e 5 a l s o shows t h e p r e s e n t l y dynamic Kr a s a f u n c t i o n of time, a l o n g w i t h c o n p a r i s o n

Figure

RDCB

specimen

computed

experimental performed as

[12].

The

present

calculation

for

KI

was

( i ) d i r e c t c o m p u t a t i o n , s i n c e KI is

a l t e r n a t e ways:

mentioned e a r l i e r , (ii) from a c r a c k - t i p i n t e g r a l which g i v e s d i r e c t l y

the

crack-opening e n e r g y , and u s i n g t h e c r a c k - v e l o c i t y dependent r e l a t i o n

between energy three of

of

three

a s t h e undetermined p a r a m e t e r 8 1 i n t h e e l e m e n t b a s i s f u n c t i o n s

same

the

results

in

KI

and

the

from

a

global

r a t e , and (ifi) c a l c u l a t i n g f r a c t u r e

energy-release

energy balance r e l a t i o n .

It is seen t h a t a l l t h e

values agree excellently, thus pointing t o t h e inherent consistency

the

present

numerical

results

in

loading

condition".

Fig.

procedure.

were

5

on

based

It s h o u l d be p o i n t e d o u t t h a t t h e

u s i n g t h e forementioned " s i m p l i f i e d

s e e n from F i g . 5 , t h e p r e s e n t n u m e r i c a l r e s u l t s

As

e x h i b i t a pronounced peak as compared t o t h e e x p e r i m e n t a l r e s u l t s . Figure energy (F),

for

used.

It fact

variation

of

d i f f e r e n t energy q u a n t i t i e s :

(T); s t r a i n e n e r g y ( U ) ;

energy

4,

when

t h e "simplified loading condition" i s

noted

that

i n t h e p r e s e n t p r o c e d u r e , e a c h of t h e

specimen

should

be

U , and F i s c a l c u l a t e d s e p a r a t e l y and d i r e c t l y .

T,

+ T

U

that

mechanisms

accuracy

of

input

and f r a c t u r e e n e r g y

W,

dissipation the

shows

kinetic

RDCB

quantities the

6

(W);

Thus,

+ F i s e q u a l t o W a t a l l times (no o t h e r e n e r g y are

f o r h e r e ) i s a n i n h e r e n t check on

accounted

the calculation.

T h a t t h i s i s s o c a n b e s e e n from F i g .

6.

7

Figure

demonstrates

loading-conditions

the

effects

of

the

alternate

employed i n t h e f i n i t e e l e m e n t model of RDCB specimen.

I n b o t h the c a s e s , t h e model i s l o a d e d s o t h a t KI = 2.32 MN/m3i2. For 9 t h i s v a l u e of KIq, t h e d e f o r m a t i o n p r o f i l e s of t h e c r a c k f a c e when t h e l o a d i s a p p l i e d a t p o i n t s A and B , r e s p e c t i v e l y , are shown i n F i g . 7. It is

seen

from

Fig.

displacements)

mm)

at

7

that

points

for

the

same

value

A and B , r e s p e c t i v e l y ,

and 972.8N (and 0.74 mm).

of

are:

KIql

load

(and

970.7N (and 0.615

Thus, when t h e l o a d i s modeled t o a c t a t B

( t h e s o - c a l l e d " s i m p l i f i e d l o a d i n g case") t h e r e is more a p p a r e n t i n p u t of e n e r g y t o t h e specimen t h a n when t h e l o a d i s modeled t o a c t a t A ( t h e so-called

as

in

" a c t u a l l o a d i n g case").

Fig.

5

is

used,

but

When a n i d e n t i c a l c r a c k - l e n g t h h i s t o r y

with

t h e "actual loading condition",

the

Hybrid Methods of Analysis

85

L

e

'Moving

2 0

-1

Singular Element

e= 16mm h=IO mm H = 63.5 L=321mm 4 6 7 . 8 + 16mm Fig. 4.

F i n i t e Element Model o f Double C a n t i l e v e r Beam Specimen [ P r o p a g a t i n g S i n g u l a r Element Shown Hatched a t t i m e t = 01

p, ond K, tram G, [ C r a c k Openining Enetpy) Present. K, from G, I T o t o I Energy Bolance)

' Present

. 2.0.

*

Kallhoff

et 0 1

I' -

I

..*a

0 . 0

90

180

270

360

450

L [wsecl

F i g . 5.

Stress-Intensity Factor V a r i a t i o n a l w i t h Time f o r a P r o p a g a t i n g Crack [ S i m p l i f i e d Loading C o n d i t i o n ]

F i g . 6.

Energy V a r i a t i o n s D u r i n g Crack P r o p a g a t i o n [Simplif i e d Loading C o n d i t i o n ]

S.N. Atluri & T. Nishioka

86

computed dynamic k-factors are shown in Fig. 8. Comparing Figs. 5 and 8, it is seen that an apparently small modification in the load-condition modeling contributes to a substantial difference in the K-factor variation. It is seen that the results in Fig. 8, for the "actual loading case", agree remarkably well with the experimental results (considering the possible rate-sensitive behaviour of Araldite B as opposed to the present linear elastic modeling). The variation in energies W, U, T, F for the "actual loading case'' is shown in Fig. 9. Comparing Figs. 6 and 9, it is seen that W in the "simplified loading case" is higher than in the "actual", T is higher in the "simplified" in the "actual", and that the variations of U and F are than qualitatively similar in both the "loading cases". These results indicate clearly the need to numerically simulate the experimental boundary/initial conditions as closely as possible in order to obtain meaningful results from a H-E-N analysis of the type presented here. In closing it may also be pointed out that even though the above analysis is for a, transparent specimen (in which a direct measurement of the K-factor, shown in the above comparison with the numerical results from the present H-E-N analysis, was possible), similar H-E-N analysis on steel specimen were peformed by the authors [ 1 5 ] . HYBRID-NUMERICAL METHODS In the following, we present procedures whereby two (or more) seemingly different numerical methods may be employed simultaneously. Even though the two methods in question may be arbitrary, for convenience we consider the two currently popular ones: the finite element method (FEM) and the boundary element method (BEM). Such coupling methods were discussed elaborately in [ 1 6 ] . Here we present a synopsis. Let the region, V , occupied by the material be decomposed into two regions V 1 and V2, where V 1 is the BEM modeled region and V 2 is the FEM modeled region. Let p be the interface between V 1 and V2, as indicated in Fig. 10. The matching conditions are 111.

+

-

(a)

ui = ui at p

(b)

:t

= -t-

i

+ = Lim u (x), where ui i X '

-

-P

(111.1)

at p

-

u

i

=

Lim u (x) i X ' X

-

-P

Hybrid Methods of Analysis

*LczII.l A -B

I

'

Reaction f o r c e at A 970.7 N at 8 9 7 2 8 N

2(

Present

a

Input D a t a

'

Kalthoff et 01 (Experimental Results)

15

7

2

10

'E

....

$

Y

F i g . 7.

Crack S u r f a c e Deformat i o n P r o f i l e s : A Comparison f o r 2 Differe n t Loading C o n d i t i o n s

x-

'00

E 05

.

0 100 I00

200 ZOO

300

t [Ps=

Fig. 8.

Fig. 9.

Energy V a r i a t i o n s During Crack P r o p a g a t i o n [ A c t u a l Loading C o n d i t i o n ]

-r p

1

400 400

500 5 00

2

300

200

Y

>

100

S t r e s s - I n t e n s i t y F a c t o r Varia t i o n w i t h T i m e f o r a Propag a t i n g Crack [ A c t u a l Loading Condition]

88

S.N. Atluri & T. Nishioka

F i g . 10. Schematic R e p r e s e n t a t i o n of Domains Modeled by D i f f e r e n t D i s c r e t i z a t i o n Methods

Hybrid Methods of Analysis

with similar definitions for coupling are considered.

the

89

+

-

tractions ti and ti.

Three cases of

111.1 Coupling of FEM with Direct BIE Method

The notation is given in Fig. 10. technique yields the equations for V1:

Application of the Direct BIE

(111.2)

Application of the FEM to V1 yields the equation

K_s=_a III.l.A

Direct Coupling.

(111.3)

Equations (11.2,.3)

may be writen as follows:

(111.4)

where _q,* is the vector of nodal displacements at p for the BIE modeled is the vector nodal displacements at p for the FEM region and where CJ modeled region. The vector _a,* is that of nodal tractions at p for the BIE region, and (la is the vector of equivalent nodal forces for the FEM region. Two possibilities arise: (i) lump the tractions at BIE nodes, (ii) distribute the forces for the FEM region. + By satisfying the equations ga = q* and the condition t = -ti at p, -a i using either (i) or (ii), direct coupling is achieved as in a substracturing procedure. However, the assembled equations for V1 U V2 are unsymmetric. Thus, in the direct-coupling procedure, a significant advantage of the FEM (viz: symmetric banded matrices) is lost without appearing to gain much. III.1.B Coupling Through the Variational Method. The function ui(p), where p is a point in V1, generated from the solution of (111.2), using the Direct BIE method, satisfies the Navier equations exactly. Let the Let this be FEM interpolation for the displacement field at p be -UFO. written in the form

S.N. Atluri & T. Nishioka

90

where is the vector of FEM nodal displacements at p. Let the known 9-Fp and Su be not yet substituted into boundary values of ui and ti on S ui i (111.2). Then the solution u (p) in V1, that satisfies the inter-region i continuity condition:

at all points of p, and the relevant boundary conditions at S and S 4 u1 01 can be obtained from the stationary condition of the functional

The displacements

and tractions may be interpolated over S1 =

U p, as:

_u(Q)

=

!j*(Q)g*

sul

sul

(III.7a)

where *! and N* are functions defined appropriately over S1, Q is a point on S1, and q* and Q* are the master-vectors of nodal displacements and tractions over S1. From (111.2) it follows that

-

9*

= B-l

A9

*

(requiring the inversion of an unsymmetric, densely populated matrix). Equation (III.7b) then yields

L(Q)

=

k!

*,B-1,A q *

Suhstitution of (111.8) Into (111.6) gives:

(111.8)

Hybrid Methods of Analysis

"

91

I

I

(111.9)

It should be noted that ['plBIE

where q

'

'p(g*

gFp)

are, as yet, unknown.

-FP

87T

=

P

The stationary condition

(6cJ*) = 0

leads to algebraic equations for -q* in terms of -Fp' q On expressing -q* thus in terms of q the functional T can be expressed in the form -Fp P

[np IBIE

T = f !FP

T

[-KlBIE 4Fp

+

(111.10)

gBIE gFp

where -%IE is now symmetric. Thus, a symmetric equivalent stiffness matrix has been obtained for the BIE modeled region, which can be added to that of the FEM modeled region. Thus, a symmetric system matrix is obtained, at the expense, however, of inverting the unsymmetric matrix B_. The procedure given in Eq. (111.9) is general and can be used to link several BIE and FEM modeled regions. A simplification occurs if the condition u

= u

-Fo

-B

where served

onp

(111.11)

sB

is the B I E interpolation for 2. The integral over p in (111.6) the purpose of enforcing this condition. This integral now

reduces to

if (111.11) is satisfied a priori.

!,(Q)

= _N*(Q) g*

on

s1

Thus, for the BIE region

- P (111.12)

This

results in

some simplification to

the

algebra leading t o the

S.N. Atluri & T. Nishioka

92

equivalent stiffness matrix [,KIBIE defined in (111.10). However, the inversion of ,B still remains. Further simplifications arise if the BIE region is completely surrounded by FEM regions. In this case Sl = P and S = = 0. If u1 '51 the assumed displacement field at P for the BIE is identical to that for the FEM assumed displacement field at P, then an equivalent symmetric stiffness matrix for the BIE region can, a priori, be obtained as: *-1

TJr

4) ,N + ,N*

[(,M ,B

T

~ r - 1

(,M ,B ,A)

I

dp

(111.13)

The inversion of B still remains. 111.2 Coupling of FEM with Indirect Boundary Solution Method Consider the mixed boundary value problem for the BEM region V1:

u

i

=

-

u

at

i

(III.14a)

So

1 (III.14b) (111.144

s l = s uso.up u1

(I11.14d)

1

where Sl is the boundary of V1. single-layer potential:

The solution may be represented by a

(111.15) the unknown single where p is a point in V1, Q is a point on S1, S,(Q) layer potential, and U is the known Kelvin solution [ 1 6 ] . The stress ji field corresponding to (111.15) may be written as: /-

t (P) = j

- f Sj(P) +

where P is also on S1.

Si(Q)Tji(P,Q)

dSQ

In vector form, (III.15,.16)

(111.16)

may be written as: (111.17)

93

Hybrid Methods of Analysis

(111.18)

since

is continuous at the boundary [16], it follows that: (111.19)

Now, S ( Q ) may be interpolated over S1 as: =

_M(Q)s

(111.20)

where a is a vector of unknown parameters; or the boundary S1may be partitioned into elements S 1 , S2, ..,,SM; and the potenial 5 may be locally interpolated over each boundary element. The resulting interpolation could still be written in the form of (111.20) where, now, CL is a vector of nodal values of 5, On substituting (111.20) into (III.l7,.18,and . 1 9 ) , we obtain:

t ( P ) = “P)?

(111.23)

Since ~ ( p ) in (111.17) satisfies the Navier equations of elasticity identically, the one that satisfies the boundary conditions (111.14a,b,c) may be determined from the simplified potential energy functional [16], as the condition:

T

- b & y - yFP]. ds

(111.24)

is minimum

(111.25) where

q -FP

functions. varying (

are On T

~

nodal displacements at p, and M substituting (111.21,.22,.23, with ) ~respect ~ ~ to both and g

a

FP’

TP

are the interpolation

and . 2 5 ) into (111.24) and one obtains:

94

S.N. Atluri & T, Nishioka

(111.26)

where (_P* + P*T ) / 2 is the symmetric stiffness matrix of the region V 1 modeled by indirect boundary solution (IBS) method. Eqs. (111.26) may now be added to other eymmetric equations of the FEM modeled region V2. Thus, unlike the direct boundary integral method, no unsymmetric-matrix inversions arrive in the case of coupling of IBS method with FEM. However, a close examination of Eqs. (11.17,.18,.19, and $ 2 4 ) reveals that surface integrations must be performed twice. ACKNOWLEDGEMENTS The results presented herein were obtained during the course of investigations supported by the U.S. AFOSR under grant 81-0057C to Georgia Institute of Technology. The authors gratefully acknowledge this support as well as the encouragement of Dr. A. Amos. It is a pleasure to sincerely thank Ms. J. Webb for her assistance in the preparation of this manuscript. FOOTNOTES 1. Regents' Professor of Mechanics 2. Visiting Assistant Professor 3. Note, however, that in the presently considered symmetric 'mode I' problem only C J and ;~ a(') are nonzero. 33 4. This can be derived [16] from the usual potential energy functional, when the displacement field, in addition to satisfying the compatibility condition, also satisfies the equilibrium equations. REFERENCES Atluri, S. N., "Higher-Order, Special, and Singular Finite Elements", Chapter 4 in: State-of-the-Art Surveys 2 Finite Element Technology (Eds.: A. K. Noor and W. D. Pilkey), ASME, New York, NY, (19831, pp. 37-126. Atluri, S . N. and Kathiresan, K., "3-D Analyses of Surface Flaws in Thick-Walled Reactor Pressure Vessels Using - a Displacement-Hybrid Finite Element Method", Nuclear Engineering and Design, Vol. 51, No. 2, (1979), pp. 163-176. S., Eaetanya, A. N., and Shah, R. C., Kobayashi, A. "Stress-Intensity Factors for Elliptical Cracks" in: Prospects of Fracture Mechanics (Eds.: G. C. Sih, H. C. van Elst, and D. Brock), Noordhoff Int. Pub., (19751, pp. 525-544.

Sorensen, D. R. and Smith, F. W., "Semielliptical Surface Cracks Subjected to Shear Loading" in: Pressure Vessel Technology, Part I1 (Materials and Fabrication) Proceedings, Vol. 3. ICPVT, Tokyo,

Hybrid Methods of Analysis

95

ASME, NY, (1977), pp. 545-551. Vijayakumar, K. and Atluri, S . N., "An Embedded Elliptical Crack, in an Infinite Solid, Subject to Arbitrary Crack-Face Tractions", -Journal of Applied Mechanics, Vol. 48, (March 1981), pp. 88-96. Nishioka, T. and Atluri, S . N., "Analytical Solution for Embedded Elliptical Cracks, and Finite Element Alternating Method for Elliptical Surface Cracks, Subjected to Arbitrary Loadings", Engineering Fracture Mechanics, Vol. 17, No. 3, (19831, pp. 247-268. Nishioka, T. and Atluri, S . N., "Analysis of Surface Flaws in Pressure Vessels by a New 3-Dimensional Alternating Method" in: Aspects of Fracture Mechanics in Pressure Vessels and Piping, ASME PVP, Vol.58, (19821, pp. 17-35: Nishioka, T. and Atluri, S. N., "Integrity Analyses of Surface-Flawed Aircraft Attachment Lugs: A New, Inexpensive, 3-D Alternating Method," AIAA Paper No. 82-0742, 23rd SDM Conference, AIAA/ASCE/ASME/AHS, (10-12 May 1982), New Orleans, pp. 287-300. O'Donoghue, P., Nishioka, T., and Atluri, S . N., "Multiple Surface Cracks in Pressure Vessels", Engineering Fracture Mechanics (In Press), Georgia Tech Report (1983). Mondkar, D. P. and Powell, G. H., "Large Capacity Eqn. Solver for Structural Analysis", Computers & Structures, Vol. 4, (1974), pp. 699-728. Kobayashi, A. S . , "Hybrid Experimental-Numerical Stress Analysis", Experimental Mechanics, Vol. 23, No. 3, (19831, pp. 338-347. Kalthoff, J. F., Beinert, J., and Winkler, S . , "Measurements of Dynamic Stress Intensity Factors for Fast Running and Arresting Cracks in Double Cantilever Beam Specimens" in Fast Fracture and Crack Arrest (Eds.: G. T. Hahn, and M. F. Kanninen), ASTM STP 627, (1977), pp. 161-176.

--

Atluri, s. N., Nishioka, T., and Nakagaki, M., "Numerical Modeling of Dynamic and Nonlinear Crack Propagation in Finite Bodies by Moving Singular Elements'' in Nonlinear and Dynamic Fracture Mechanics (Eds.: N. Perrone and S . N. Atluri), ASME-AMD Vol. 35, (1979), pp. 37-67. Nishioka, T. and Atluri, S . N., "Numerical Modeling of Dynamic Crack Propagation in Finite Bodies, by Moving Singular Elements, Part 1 - Formulation, Part 11-Results", Journal of Applied Mechanics, Vo. 47, (1980), pp. 570-583. Nishioka, T. and Atluri, S . N., "Finite Element Simulation of Fast Fracture in Steel DCB Specimen", Engineering Fracture Mechanics, Vol. 16, No. 2, (1982), pp. 157-175. Atluri, S .

N. and Grannell, J. J., Boundary Element Methods (BEM) GIT-ESM-SA-78-16, Georgia

and Combination BEM-FEM, Report Institute of Technology, (19781, 84 pp.

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Unification of Finite Element Methods H. Kardestuncer (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

97

CHAPTER 4 THE POSTPROCESSING TECHNIQUE IN THE FINITE ELEMENT METHOD. THE THEORY & EXPERIENCE I. BabuKka, K. Izadpanah, & B. Szabo

The p a p e r a d d r e s s e s t h e h , p , and h-p versions of t h e f i n i t e e l e m e n t method i n c o n n e c t i o n w i t h a postprocessing technique f o r e x t r a c t i n g t h e values of a f u n c t i o n a l . T h i s t e c h n i q u e combines t h e f i n i t e e l e m e n t method w i t h t h e a n a l y t i c a l i d e a s of t h e t h e o r y of p a r t i a l d i f f e r e n t i a l e q u a t i o n s of e l l i p t i c type.

1.

INTRODUCTION

F i n i t e e l e m e n t c o m p u t a t i o n s i n s t r u c t u r a l mechanics u s u a l l y have two p u r p o s e s : ( 1 ) t o d e t e r m i n e t h e s t r e s s and d i s p l a c e m e n t f i e l d s and ( 2 ) t o d e t e r m i n e t h e v a l u e s of c e r t a i n f u n c t i o n a l s d e f i n e d on d i s p l a c e m e n t f i e l d s as, f o r example, t h e s t r e s s i n t e n s i t y f a c t o r s , stresses at s p e c i f i c p o i n t s , r e a c t i o n s , e t c . C o m p u t a t i o n s of t h e s e v a l u e s i n v o l v e t h e f i n i t e e l e m e n t s o l u t i o n . F o r example, t h e s t r e s s components are o f t e n computed a t t h e Gauss p o i n t s of t h e e l e m e n t s and t h e s t r e s s e s a t any o t h e r p o i n t s are t h e n computed by t h e i n t e r p o l a t i o n t e c h n i q u e , t h e s t r e s s i n t e n s i t y f a c t o r s is d e t e r m i n e d by t h e J - i n t e g r a l o r c u r v e f i t t i n g t e c h n i q u e , e t c . We s h a l l r e f e r t o t h e s e o p e r a t i o n s as postprocessing. U s u a l l y t h e v a l u e s of t h e s e f u n c t i o n a l s a r e needed t o be known w i t h h i g h e r a c c u r a c y and r e l i a b i l i t y t h a n t h e d i s p l a c e m e n t o r stress field itself.

I P a r t i a l l y s u p p o r t e d by t h e O f f i c e of Naval R e s e a r c h u n d e r g r a n t number N O 0 0 1 4-77-C-0623. 2 P a r t i a l l y s u p p o r t e d by t h e O f f i c e -of Naval R e s e a r c h u n d e r g r a n t number NO001 4-81 -K-0625.

98

1. Babuika et al.

Assuming t h a t we h a v e t h e f i n i t e e l e m e n t s o l u t i o n and w i s h t o determine c e r t a i n f u n c t i o n a l values t h e following questions arise: 1 ) What s h o u l d t h e r e l a t i o n s h i p be b e t w e e n t h e computat i o n a l e f f o r t s p e n t on t h e f i n i t e e l e m e n t s o l u t i o n and t h e e f f o r t s p e n t on p o s t p r o c e s s i n g : Is it b e t t e r t o u s e a v e r y s i m p l e and i n e x p e n s i v e p o s t p r o c e s s i n g t e c h n i q u e as f o r example d i r e c t e v a l u a t i o n of t h e s t r e s s e s from t h e f i n i t e e l e m e n t s o l u t i o n i n t h e d e s i r e d p o i n t s o r s h o u l d one s e l e c t a more O f c o u r s e we h a v e t o r e l a t e t h e a n s w e r t o expensive technique. t h e a c h i e v e d a c c u r a c y and t o t h e r e l i a b i l i t y and r o b u s t n e s s of t h e postprocessing procedures under consideration. 2 ) G i v e n a f i n i t e e l e m e n t s o l u t i o n , what is t h e l a r g e s t a c c u r a c y of t h e f u n c t i o n a l v a l u e s one c a n a c h i e v e by t h e postprocessing technique. I n o t h e r w o r d s , what is t h e maximal i n f o r m a t i o n c o n t a i n e d i n t h e f i n i t e e l e m e n t s o l u t i o n which c o u l d be u s e d f o r t h e e x t r a c t i o n of t h e d e s i r e d v a l u e . 3 ) How do t h e v a r i o u s v e r s i o n s of t h e f i n i t e e l e m e n t method, i . e . , t h e h - v e r s i o n , t h e p - v e r s i o n and t h e h-p v e r s i o n b e a r on t h e i m p o r t a n c e of p r o p e r s e l e c t i o n of t h e postprocessing techniques. T h e s e q u e s t i o n s a r e d i s c u s s e d i n some d e t a i l s . 2.

THE EXTENSION OPERATORS. THE h , p AND h-p T H E FINITE ELEMENT METHOD

VERSIONS OF

T h e r e a r e t h r e e v e r s i o n s of t h e f i n i t e e l e m e n t methods b a s e d on t h e common v a r i a t i o n a l ( e n e r g y ) p r i n c i p l e . They are c h a r a c t e r i z e d by t h e s y s t e m a t i c s e l e c t i o n ( e x t e n s i o n ) of t h e f i n i t e e l e m e n t s p a c e s l e a d i n g t o t h e c o n v e r g e n c e of t h e f i n i t e e l e m e n t s o l u t i o n s t o t h e e x a c t one. The h-version is t h e c l a s s i c a l and most commonly u s e d method of e x t e n s i o n : t h e p o l y n o m i a l d e g r e e of e l e m e n t s p is f i x e d and mesh r e f i n e m e n t is u s e d f o r c o n t r o l l i n g t h e e r r o r of a p p r o x i m a t i o n ( h r e f e r s t o t h e s i z e of t h e e l e m e n t ) . T y p i c a l l y t h e p o l y n o m i a l d e g r e e of e l e m e n t s is l o w , u s u a l l y p = 1 o r 2 . P r o p e r s e l e c t i o n of t h e mesh and i t s r e f i n e m e n t s t r o n g l y i n f l u e n c e s t h e e r r o r and i t s d e p e n d e n c e on t h e c o m p u t a t i o n a l effort. I n t h e p v e r s i o n t h e mesh is f i x e d and t h e p o l y n o m i a l d e g r e e of e l e m e n t s is i n c r e a s e d e i t h e r u n i f o r m l y o r s e l e c t i v e l y o v e r t h e mesh. The h-p version combines t h e h and p - v e r s i o n s , i . e . , e r r o r r e d u c t i o n is a c h i e v e d by a p r o p e r mesh r e f i n e m e n t and conc u r r e n t c h a n g e s i n t h e d i s t r i b u t i o n of t h e p o l y n o m i a l d e g r e e of elements. The p e r f o r m a n c e of t h e v a r i o u s e x t e n s i o n s o p e r a t o r s c a n b e compared f r o m v a r i o u s p o i n t s of v i e w , t h e most i m p o r t a n t of which a r e human and c o m p u t e r - r e s o u r c e s r e q u i r e m e n t i n r e l a t i o n

The Postprocessing Technique in the Finite Element Method

99

t o t h e d e s i r e d l e v e l of p r e c i s i o n . Such r e l a t i o n s h i p s a r e d i f f i c u l t t o q u a n t i f y and a r e s u b j e c t d u e t o v a r i o u s f a c t o r s , t h e r e f o r e t h e p e r f o r m a n c e o f t h e e x t e n s i o n o p e r a t o r s is u s u a l l y O f course r e l a t e d t o t h e number of d e g r e e s of f r e e d o m N . e v a l u a t i o n of an e x t e n s i o n o p e r a t o r would n o t b e m e a n i n g f u l w i t h o u t c o n s i d e r i n g t h e g o a l s of c o m p u t a t i o n . F o r e x a m p l e , i f o n l y s t r e s s i n t e n s i t y f a c t o r s a r e d e s i r e d , t h e n t h e a c c u r a c y of t h e computed d i s p l a c e m e n t s , r e a c t i o n s o r s t r e s s e s are n o t of importance. I n many c a s e s t h e c o m p u t a t i o n h a s m u l t i p l e g o a l s .

3.

rix/

THE MODEL PROBLEM

I n o r d e r t o i l l u s t r a t e t h e e s s e n t i a l p r o p e r t i e s of f i n i t e e l e m e n t s o l u t i o n and e x t r a c t i o n t e c h n i q u e s , we h a v e s e l e c t e d a model p r o b l e m which r e p r e s e n t s some o f k e y f e a t u r e s of a l a r g e c l a s s of e n g i n e e r i n g problems. S p e c i f i c a l l y l e t u s consider t h e plane s t r a i n p r o b l e m of twodimensional e l a s t i c i t y (homogeneous i s o t r o p i c m a t e r i a l ) w i t h E and u r e p r e s e n t i n g t h e modulus of e l a s t i c i t y and P o i s s o n k-l+l+ r a t i o r e s p e c t i v e l y (E > 0 , 0 < u < . 5 ) . The domain Figure 1 D, a square panel with a The model p r o b l e m c r a c k is shown i n F i g . 1 .

i

We s h a l l be c o n c e r n e d h e r e w i t h p r o b l e m s i n which o n l y t r a c t i o n s are p r e s c r i b e d a t t h e b o u n d a r y ( i . e . , f i r s t b o u n d a r y v a l u e p r o b l e m of e l a s t i c i t y ) .

We d e n o t e t h e d i s p l a c e m e n t v e c t o r f u n c t i o n by and t h e c o r r e s p o n d i n g s t r e s s t e n s o r by

_u = { u l , u 2 )

The s t r a i n e n e r g y f u n c t i o n a l is

+

( l - v ) ( - au2)2 a x2

+ 1-2v 2

au

(-Jaxl

+

-)a u 2 ax2

]dx1dx2

.

(3-1)

I. Babufka et al.

100

The s o l u t i o n u s a t i s f i e s t h e Navier-Lam; e q u a t i o n s . I t is p o s s i b l e t o e x - r e s s t h e s o l u t i o n t h r o u g h two holomorphic f u n c t i o n s $ ( z p , $ ( z ) u s i n g t h e t h e o r y of M u s k h e l i s h v i l i [ I ] .

where z

and and

z

=

= x1 +'(z).

x1

+

-

ix2,

ix2,

=

p

E 2vj, = 3-4u K

(3.7)

mean c o n j u g a t e v a l u e s t o

reap.

z

The components of t h e s t r e s s t e n s o r a r e e x p r e s s e d by KolosovMuskhe 1i s h v il i f o r m u l a e

and

Re + ' ( z )

i s t h e real p a r t of

$' (z)

.

The c o r r e s p o n d e n c e between t h e d i s p l a c e m e n t s (and t h e s t r e s s ) f i e l d and t h e f u n c t i o n s and $ is one t o one up t o t h e c o n s t a n t s y and y ' i n + and $ , r e s p e c t i v e l y , s a t i s f y i n g y 7' = 0 . the relation

+

-

I n our model problem we c o n s i d e r t h e f o l l o w i n g ( e x a c t ) s o l u t i o n

101

The Postprocessing Technique in the Finite Element Method

T(z)

where

=

m.

B ( z ) is a h o l o m o r p h i c f u n c t i o n on D. F u n c t i o n ze1/2 is t o on D . b e u n d e r s t o o d as t h e p r i n c i p a l b r a n c h of z-’/2 F u n c t i o n ~ ( z ) is u n i q u e l y d e f i n e d b y ( 3 . 9 ) and ( 7 . 7 ) ( 3 . 8 ) . The t r a c t i o n s on t h e b o u n d a r y o f D a r e d e f i n e d by ( 3 . 4 ) ( 3 . 5 ) . I t c a n b e r e a d i l y v e r i f i e d t h a t t h e two e d g e s of t h e crack a r e t r a c t i o n f r e e . We w i l l now d i s c u s s t h e f i n i t e e l e m e n t s o l u t i o n and t h e p o s t p r o c e s s i n g t e c h n i q u e i f t h e t r a c t i o n s a r e p r e s c r i b e d on t h e b o u n d a r y of D s o t h a t t h e e x a c t s o l u t i o n t o t h e p r o b l e m is g i v e n by ( 7 . 7 ) - ( 7 . 9 ) . S p e c i f i c a l y we now c o n s i d e r t h e c a s e E = 1 , v = 7. The s t r a i n e n e r g y of t h e exact s o l u t i o n is: W = 42.16491 240.

4.

THE FINITE ELEMENT SOLUTION

We h a v e s o l v e d t h e model p r o b l e m by t h e t h e f i n i t e e l e m e n t method.

The

h and p - v e r s i o n s p - v e r s i o n of t h e f i n i t e

6

A

The meshes f o r t h e

of

Figure 2 p-version,

A:

Mesh 1 , B: Mesh 2

e l e m e n t method was implemented i n t h e e x p e r i m e n t a l c o m p u t e r program COMET-X d e v e l o p e d a t t h e C e n t e r f o r C o m p u t a t i o n a l Mechanics of Washington U n i v e r s i t y i n S t . Louis [2]. The two meshes shown i n F i g . 2A,B were u s e d . The p o l y n o m i a l d e g r e e s w e r e t h e same f o r a l l elements and r a n g e d f r o m 1 t o 8 . The s h a p e f u n c t i o n s on t r a p e z o i d a l e l e m e n t s of mesh 2 w e r e c o n s t r u c t e d by b l e n d i n g f u n c t i o n t e c h n i q u e . The h - v e r s i o n s o l u t i o n was o b t a i n e d b y means of t h e c o m p u t e r p r o g r a m FEARS d e v e l o p e d a t t h e U n i v e r s i t y of Maryland [ 7 , 41.

102

I. BabuSka et al.

FEARS u s e s q u a d r i l a t e r a l e l e m e n t s of d e g r e e one. The program is a d a t i v e and r o d u c e s a s e q u e n c e of n e a r l y o p t i m a l meshes. The mesh from t h i s s e q u e n c e w i t h 319 S e e [3P [41 [51 f 6 1 “71 e l e m e n t s and number of d e g r e e s of freedom N = 617 is shown i n F i g . 3.

.

Figure 3 The mesh c o n s t r u c t e d by t h e a d a p t i v e program FEARS

5.

ERROR OF THE FINITE ELEMENT SOLUTION MEASlJRED I N ENERGY NORM

We d e n o t e t h e e x a c t s o l u t i o n by uo and t h e f i n i t e e l e m e n t s o l u t i o n by ypE. The e r r o r o f t h e f i n i t e element s o l u t i o n is d e n o t e d by

2,

-e

=

so - gFE.

We measure t h e magnitude of t h e e r r o r by t h e energy norm

n*IIE,

The Postprocessing Technique in the Finite Element Method

103

T h i s measure is e q u i v a l e n t t o m e a s u r i n g t h e e r r o r i n t h e s t r e s s components by i n t e g r a l s of i t s s q u a r e s ( t h e L2 norm). I n o u r c a s e when t r a c t i o n s are s p e c i f i e d at t h e boundary

and

(5.7) The e x t e n s i o n s o p e r a t o r s u n d e r c o n s i d e r a t i o n m o n o t o n i c a l l y i n c r e a s e t h e f i n i t e e l e m e n t s p a c e s e i t h e r by i n c r e a s i n g t h e d e g r e e of e l e m e n t s o r r e f i n i n g t h e mesh. Therefore t h e energy norm of t h e e r r o r m o n o t o n i c a l l y d e c r e a s e s . We c a n w r i t e

and e x p e c t t h a t f o r p r o p e r l y c h o s e n p t h e f u n c t i o n C ( N ) is n e a r l y c o n s t a n t e s p e c i a l l y f o r l a r g e r N. The number p > 0 i s t h e r a t e of c o n v e r g e n c e of t h e e r r o r measured i n t h e e n e r g y norm.

I t is p o s s i b l e t o e s t i m a t e t h e v a l u e of p . I n o u r c a s e t h e r a t e p is governed by t h e s t r e n g t h of t h e s i n g u l a r i t y of t h e s o l u t i o n . I t can be shown t h a t f o r t h e p - v e r s i o n [ 8 ] , [ 9 ]

w i t h E > 0 a r b i t r a r i l y small and C i n d e p e n d e n t of N. The h - v e r s i o n u s i n g t h e u n i f o r m mesh y i e l d s t h e e s t i m a t e

<

1I ell

CN- "4

w i t h t h e r a t e i n d e p e n d e n t of t h e d e g r e e of e l e m e n t s . o p t i m a l r e f i n e m e n t o f t h e mesh l e a d s t o t h e e s t i m a t e

The

(5.7) (FEARS u s e s p = 1 ) where t h e r a t e is i n d e p e n d e n t of t h e s t r e n g t h of t h e s i n g u l a r i t y .

The h-p v e r s i o n w i t h o p t i m a l mesh and the estimate IlellE -

p - d i s t r i b u t i o n leads t o

e <

Ce - y N

I. BaburIka et al.

104

where e = 113 i n d e p e n d e n t l y of t h e s t r e n g t h of t h e s i n g u l a r i t y and y > 0. The r e l a t i v e e r r o r i n t h e e n e r g y norm d e f i n e d as II ell

-

=

E,R

-

II 2 II

"0"E

h a s been l o t t e d i n F i g . 4 on l o g - l o g s c a l e f o r t h e (mesh 1 , 2 7 , f o r t h e h - v e r s i o n w i t h

p-version

NUMBER OF DEGREES OF FREEDOM

Figure 4 R e l a t i v e e r r o r i n t h e e n e r g y norm v s d e g r e e s of freedom ( 1 ) h - v e r s i o n , u n i f o r m mesh, ( 2 ) h - v e r s i o n , a d a p t i v e l y c o n s t r u c t e d mesh, ( 7 ) p - v e r s i o n Mesh 1 , ( 4 ) p - v e r s i o n Mesh 2 a d a p t i v e l y c o n s t r u c t e d mesh and f o r t h e h - v e r s i o n w i t h u n i f o r m mesh. The p o l y n o m i a l d e g r e e of e l e m e n t s is a l s o shown i n t h e f i g u r e , The shown s l o p e s a r e t h e t h e o r e t i c a l s l o p e s of t h e I t is seen t h a t t h e r a t e of c o n v e r g e n c e [u =I/*and 1/31. o b s e r v e d r a t e of c o n v e r g e n c e c l o s e y a g r e e s w i t h (5.5) ( 5 . 7 ) From ( 5 . 4 ) we can compute C ( N ) f o r t h e p - v e r s i o n . The r e s u l t s a r e given i n Table 1 . T a b l e s 2 and 3 show a n a l o g o u s r e s u l t s f o r t h e h - v e r s i o n . The comparison between T a b l e s 1-3 shows t h a t f o r 596 a c c u r a c y we need N = 1770 when u s i n g p - v e r s i o n Mesh 2 , N = 2290 f o r h - v e r s i o n w i t h a d a p t i v e l y r e f i n e d mesh and N = 146000 f o r hv e r s i o n w i t h u n i f o r m mesh.

105

The Postprocessing Technique in the Finite Element Method

P

N

1

35 95 135 239 347 479 675 81 5

2

3 4 5 6 7

8

II e II

32 -61 96 18.75% 15.89% 13.24% 1 1 .06% 9.47%

67 101

143 221 30 1 61 7

N

51 167 591

5.

2.010 1.816 1.997 2.059 2.061 2 * 079

8.27%

2.088

7.37%

2 099

TABLE 2 R e l a t i o n s h i p b e t w e e n It ell and E,R w i t h a d a p t i v e l y c o n s t r u c t e d mesh

N

C ( N) / II uoII

E,R

I en E,R

for the

h-version

[ u = l/2 ] C(N)/IIuOIE

32.918 26.78% 21.35% 16.79% 17.61% 9.63%

II ell

N

E,R

36.02% 27.07% 19.81%

2.035 2.665 2.562 2 501 2.366 2 394

C(N )

/ 1I uo 1I

- 967 974 .977

*

C O M P U T A T I O N OF THE STRESSES

The f i n i t e e l e m e n t method p r o v i d e s t h e s o l u t i o n which c o n v e r g e s t o t h e e x a c t s o l u t i o n i n t h e e n e r g y nor:?E We h a v e s e e n t h a t t h e e r r o r measured i n t h i s norm decreases monotoni c a l l y and v e r y o r d e r l y . We now examine t h e p o i n t w i s e e r r o r a n We d e n o t e t h e e r r o r i n s t r e s s e s f o r t h e h and p - v e r s i o n . t h e s t r e s s components as

106

1. BabuSku et al.

and t h e r e l a t i v e e r r o r by

,SFE1

and a r e r e s p e c t i v e l y t h e s t r e s s compolj lj n e n t s c o r r e s p o n d i n g t o t h e e x a c t and f i n i t e e l e m e n t s o l u t i o n . We w i l l compute t h e s t r e s s e s d i r e c t l y f r o m t h e d e r i v a t i v e s of ii and s t r e s s - s t r a i n law. P i g . 5 shows t h e r e l a t i v e e r r o r E e! i n T~~ at t h e p o i n t (.0,.1 ) computed by t h e p - v e r s i o n . 1J where

T .

DEGREE p

se

100

2

I

3

4 5 6 7 8 9

K W

i

W

a

I

10

25 50 I00 200 400 800 NUMBER OF DEGREES OF FREEDOM

DEGREE p 2 3 4 5 6 7 8

I

I00

se 10

b W

i

W K

I

10

25

50

I00

200

400

800

NUMBER OF DEGREESOF FREEDOM

Figure 5 The r e l a t i v e e r r o r o f e F j computed by t h e p - v e r s i o n R R R a ) Mesh 1 b ) Mesh 2. (1 ) e l l , ( 2 ) e 2 2 , ( 3 ) e 1 2

H S Z N

2

H S B N

The Postprocessing Technique in the Finite Element Method

T

107

d

Fi

ul

0 -4

a, 1

3

a

I. BabuSka et al.

108

F i g . 6 shows i s o m e t r i c drawings of t h e e r r o r i n T~~ for v a r i o u s p v a l u e s f o r meshes 1 and 2. The e r r o r v a l u e s were computed on a u n i f o r m g r i d w i t h t h e g r i d p o i n t s ( i h , j h ) h = .I, i , j = - 10, 10. A t points other than the grid points, t h e v a l u e s were computed by l i n e a r i n t e r p o l a t i o n .

I n t h e c a s e of t h e

h - v e r s i o n , t h e e r r o r is d i s c o n t i n u o u s at t h e boundary o f e v e r y e l e m e n t . T h e r e f o r e we compute t h e s t r e s s e s i n t h e c e n t e r of e v e r y e l e m e n t where i n c r e a s e d a c c u r a c y can be e x p e c t e d .

I n F i g . 7 we show t h e l e v e l - l i n e s of t h e e r r o r i n

(using t h e mesh shown i n F i g . 3) i n t h e u p p e r r i g h t q u a r t e r of t h e domain D . The l o c a l maxima and minima are shown a l s o i n t h e f i g u r e . The e r r o r is l a r g e i n t h e neighborhood of t h e t i p of t h e c r a c k . The l e v e l - l i n e s and t h e l o c a l maxima and minima depend on t h e used i n t e r p o l a t i o n t e c h n i q u e . We s e e i n c o n t r a s t t o t h e p - v e r s i o n t h a t t h e o s c i l l a t i n g b e h a v i o u r of t h e e r r o r i s not s o s t r o n g h e r e ; n e v e r t h e l e s s , it h a s t o be u n d e r l i n e d t h a t i f t h e s t r e s s e s w i l l be computed everywhere d i r e c t l y from displacements s t r o n g o s c i l l a t o r y behaviour w i l l appear i n every element. T~~

The c e n t e r of t h e e l e m e n t s a r e c h a n g i n g w i t h t h e mesh. To show t h e convergence of t h e s t r e s s e s , we s e l e c t e d f o r t h e T a b l e 4 t h e c e n t e r p o i n t s which a r e c l o s e s t t o t h e t i p of t h e c r a c k The t a b l e shows t h e e r r o r i n $ ( i n t h e f i r s t q u a r t e r of D ) . and t h e magnitude of t h e e x a c t v a l u e s of t h e s t r e s s .

I I

Figure 7 The l e v e l l i n e s of t h e e r r o r e of 22 -~

q u a r t e r of

D

computed by t h e

T~~

h-version

i n t h e upper

The Postprocessing Technique in the Finite Element Method

109

TABLE 4

The r e l a t i v e e r r o r of t h e s t r e s s e s i n t h e n e i g h b o r h o o d of t h e origin. Coordinates

1.7,

I

1 e22 R 1

I ey2 I

ITEII

[OI I IT12

No. of elements

N

16

51

-25

a25

75.54% 5.037

19.90% 7.751

20.7% 1.557

47

143

-125

.I 25

77.95%

10.84%

15.14%

x2

9

IT1'1 1

%

To d e p i c t t h e b e h a v i o u r i n a f i x e d p o i n t (.25, ,2 5 1 w e s e l e c t t h e c e n t e r p o i n t s c l o s e s t t o i t . T a b l e 5 shows t h e r e s u l t s . If we d e s i r e t o compute t h e s t r e s s components i n t h e n o d a l p o i n t ( . 2 5 , . 2 5 ) we h a v e 4 v a l u e s f o r d i s p o s i t i o n and a l s o t h e i r a v e r a g e . T a b l e 6 we shows t h e r e l a t i v e e r r o r s . The v a l u e i n t h e l i n e s 1 , 2 , 7 , 4 a r e computed from t h e e l e m e n t s o r d e r e d c o u n t e r c l o c k w i s e s t a r t i n g w i t h t h e u p p e r - r i g h t one. The l i n e A shows r e l a t i v e e r r o r of t h e a v e r a g e o f t h e s t r e s s v a l u e s computed i n t h e f o u r e l e m e n t s . I n c o n t r a s t t o t h e m o n o t o n i c and o r d e r l y b e h a v i o u r of t h e e r r o r m easured i n t h e e n e r g y norm, t h e a c c u r a c y i n t h e s t r e s s e s is poor and nonmonotonic , a l t h o u g h t h e s t r e s s e s a r e c o n v e r g i n g i n i n t e g r a l s e n s e ( i n t h e e n e r g y norm) m o n o t o n i c a l l y . In a d d i t i o n , t h e q u a l i t y of t h e computed s t r e s s components is v e r y different.

1. BabuJka et al.

110

TABLE 5

The r e l a t i v e c r r o r of t h e s t r e s s e s i n t h e neighborhood o f

(-25,-25).

I

I

I

I

I

I

R

I eE2 I

ley1 I

Coordinates

le121

I

I

I

I

51

16 64

167

256

591

*25

-25

-373

.375

35.54% 5.037

19.W 3.751

2Q.7044 1.553

8.01%

7.7% 7.062

16.06$

10.78$ 4.331

6.45s 1.794

4.113

c

;

z

E

1.268 L k

7.

-1875

-1875

10.2G$

5.817

*d

F:

POSTPROCESSING

We have s e e n t h a t s t r e s s e s computed d i r e c t l y from f i n i t e element s o l u t i o n s are n o t a c c u r a t e . N e v e r t h e l e s s , o f t e n t h e v a l u e s of t h e s t r e s s e s is t h e main aim o f t h e c o m p u t a t i o n . We w i l l show n?w t h a t by u t i l i z i n g t h e a n a l y t i c a l s t r u c t u r e o f t h e Navier-Lame e q u a t i o n s it is p o s s i b l e t o compute s t r e s s e s w i t h t h e a c c u r a c y comparable t o t h e a c c u r a c y of t h e e n e r g y of t h e f i n i t e element s o l u t i o n ( w h i c h is t h e s q u a r e of t h e e r r o r measured i n t h e e n e r g norm). We w i l l o u t l i n e t h e main i d e a . F o r more, s e e [81, [9y, [ l o ] . L e t ~0 = (xO,1,xo,2) C D radius p centered i n

-x0.

and d e n o t e by S ( x o , p ) t h e d i s c of Further, l e t D(xO,p) = D

-S(xo,p). See F i g . 8. The boundary of D ( x O , p ) is denoted by a D ( x O , p ) = aa U r where r is t h e boundary of t h e d i s k

S(xo,p)

.

w(x0,x)

5

-

We now d e f i n e t h e extraction ( d i s p l a c e m e n t ) f u n c t i o n (w , w ) which c o r r e s p o n d s t o t h e f u n c t i o n s $ , $ i n 1 2

The Postprocessing Technique in the Finite Element Method

t h e s e n s e of ( 3 . 2 ) ( 3 . 7 ) =

i(z)

and a r e d e f i n e d as follows

- Z O ) - ~ +; * ( Z )

A(z

111

(7.1)

TABLE 6

The r e l a t i v e e r r o r of t h e s t r e s s e s i n ( . 2 5 . 2 5 ) No. of elements

N

e;2

.034 .042 2.41 .026 3.09

A

43

147

1 2

3 4

4.09 1.46 4.41 4-41 6.07

5.47 2.68 .099 13-47 11.74

17-42 22.96 55 69 3.85 28.85

2 3 4

12.12 19.37 6.75 5.87 17.51

10.65 15.35 8.68 5.94 12.63

17 -36 17-36 5.47 68.41 60.22

A 1

8.1 1 8.10

10.17 6.62 5 -05 13-64 15.84

12.66 17.78 8.1 5 3-71 37 -48

2

3

4

A

64

167

64.41

13.01

2 3 4 1

61 7

72.22

11.16 35.86 .093 106.43 71 -49

A

319

77.95 4.35 3.38

7.57 2.31 2.01 17.42

1

221

1.60

4.97 6.60 1.76 7.14

10.99 4.79 12.21 17.27 9.71

A

106

%

1

where i * ( z ) and e * ( z ) are a r b i t r a r y h o l o z o r p h i c f u n c t i o n s on D ( n o t o n l y on D ( x O , p ) ) . Note t h a t 4 , $ a r e h o l o m o r p h i c on Q ( x o , p ) f o r any 0 < p . A l t h o u g h t h e domain

D(xO,p)

is d o u b l y c o n n e c t e d , t h e

I. Babufka et al.

112

d i s p l a c e m e n t f u n c t i o n w d e f i n e d by (7.1 ) t h r o u g h ( 7 . 3 ) by ('3.2) is a s i n g l e v a l u e d f u n c t i o n , and it is a n a d m i s s i b l e displacement function. Denote by

TLU],

TrW]

t h e stress t e n s o r s a s s o c i a t e d with t h e

dD

Fig. 8.

The domain

D(xo,p).

displacement functions and w. Denote t h e outward normal t o aQ(x0,p) by n. Then B e t t i ' s l a w can be w r i t t e n i n t h e form

T h i s e q u a t i o n c a n be r e w r i t t e n

The f u n c t i o n s 4 , 6 a s s o c i a t e d t o t h e s o l u t i o n w r i t t e n i n t h e neighborhood of zo:

2

c a n be

113

The Postprocessing Technique in the Finite Element Method

S(Z)

=

bo

$(Z)

=

+

bl(Z-ZO) S(z)

-

+

aD

(7.7)

-

(7 08)

Z0~'(Z).

Using ( 7 . 1 ) - ( 7 * 3 ) and ( 7 . 6 ) - ( 7 . 8 ) we g e t

j ( u , T I W 1 * n ) d s-

O((Z-Z,)~)

i n ( 7 . 5 ) and l e t t i n g

p

+

0

(~,T[~]*n)ds aD

By a p p l i c a t i o n of ( 3 . 4 ) - ( 3 . 6 ) T11

(x -0 )

~

~

T 1 2 ( -0 x

~ )

we get

=

2 Re(al

+ El -

b,)

( 7 -10)

=(

2Re(a, 2 ~

+) a1 +

bl)

(7.11)

=

I m bl

.

(7.12)

By p r o p e r s e l e c t i o n o f A , B we can o b t a i n t h a t t h e r i g h t hand s i d e of ( 7 . 9 ) be T . Note t h a t any c h o i c e of l,j* (7.1 ) and ( 7 . 2 ) does not change t h e r i g h t hand I n our problem when t h e t r a c t i o n s are p r e s c r i b e d a t a D , t h e f u n c t i o n g ( x ) = T [ u l * n i s g i v e n . ( 7 . 9 ) can t h e r e f o r e be w r i t t e n i n - t h e form

where F is ( f o r p r o p e r c h o i c e o f A , B ) t h e e x a c t v a l u e of t h e stress component a t x = xo. Of c o u r s e uo is n o t known b u t uFE i s . T h e r e f o r e we d e f i n e (7.14) By s u b t r a c t i n g (7 .1 3 )

f u n c t i o n a l FFE e x a c t l y ) is

(7.14), t h e e r r o r i n t h e extracted (provided t h a t i n t e g r a l s a r e evaluated

I. Babufka et al.

114

To t h i s e n d , l e t

L e t u s a n a l y z e now ( 7 . 1 5 ) .

1 =

(V1

,v2)

be

t h e ( e x a c t ) s o l u t i o n of t h e p r o b l e m when t r a c t i o n s T L w J * n a r e p r e s c r i b e d at aD. 1 f y b e c a u s e y i s s i n g u l a r a t x = xO, b u t 1 is n o t . E x i s t e n c e of v s a t i s f y the equilibrium condition.

is

guaranteed because T[w3*n We c a n w r i t e (7.16)

where W(u, v) is t h e u s u a l e n e r g y s c a l a r p r o d u c t a s s o c i a t e d w i t h W(u) d e f i n e d i n ( 3 . 1 ) . TJsing one of t h e b a s i c p r o p e r t y of t h e f i n i t e e l e m e n t method, namely

we o b t a i n from ( 7 . 1 5 ) ( 7 . 1 6 ) F

- FFE

2W(U0

=

-

UFE,

-V

-

z p ~ )

and h e n c e IF

-

FFEI

<

211~0

-

lFEIIEIIx

- 1~~11 E

(7.15)

So f a r we d i d n o t d i s c u s s t h ? c h o i c e of $,(z) and i , ( z ) , ( 7 . 1 8 ) shows t h a t $, and 5 , s h o u l d be s e l e c t e d so t h a t llx - v 1I is a t l e a s t of t h e o r d e r of iiu0 - upEli. -FE If 1: - XFEIIE :: CllU - ~ F ‘IEE we g e t IF - FFEl < CIIuo - u 112 < C(W(uo) - W(uFE)) and t h e r a t e of c o n v e r g e n c e FE E is t w i c e t h a t of t h e r a t e of t h e e r r o r measured i n t h e e n e r g y norm. Note t h a t i n e q u a l i t y ( 7 . 1 8 ) is u p p e r bound which neglects p o s s i b l e c a n c e l l a t i o n i n t h e energy i n t e g r a l . 8.

SELECTION OF THE EXTRACTION FUNCTION

When x0 i s n o t c l o s e t o t h e b o u n d a r y of D , t h e n we can s e l e c t i, = i, = 0. When x0 is c l o s e t o a D , then i, and i, s h o u l d be s e l e c t e d s o t h a t T L W 1 * n= 0 on t h a t p a r t of t h e b o u n d a r y which is c l o s e t o xO. O t h e r w i s e , we would n o t a c h i e v e t h a t lly - -vF E ~ ~ E w i l l be s m a l l . I n t h e f o l l o w i n g we o u t l i n e b r i e f l y t h e p r o c e d u r e f o r

s o t h a t T L W 1 * n= 0 on t h e c r a c k c o n s t r u c t i n g i, and 2, s u r f % c e s . T O s i m p l i f y t h e n o t a t i o n we w i l l w r i t e 0 i n s t e a d etc. of 0, D e f i n e an a u x i l i a r y f u n c t i o n

n(z)

on

D

115

The Postprocessing Technique in the Finite Element Method

U s i n g ( 3 . 4 ) and ( 3 . 5 ) t h e t r a c t i o n s on t h e c r a c k s u r f a c e c a n b e w r i t t e n as f o l l o w s

T ~ ~ ( z + )-

iT12(z+)

T ~ ~ ( z -- )

=

~(z,)

+ n(z-1

(8.2a)

=

@(z-) + n(z+)

(8.2b)

where z+ and zr e s p e c t i v e l y d e n o t e t h e u p p e r and l o w e r s u r f a c e of t h e c r a c k . TJsing (7.1)-(7.3) we g e t ~ ( z )=

-H(z-EO)-2+2~(z-zO)(z-Eo)-3-E(z-Eo)-2+~~(z)

(8.3)

Setting

we o b t a i n

where Q(z)

=

- A ( Z - Z ~ ) - ~+

I(Z-~~)-~ (8.7)

+ Note t h a t

2A(z

-

z,)(z

Q(z+) = Q ( z - ) .

Now we s e l e c t

6,

-

E0)-?

-

E(z

-

E0)-2

.

Similarly

so that

( 8 . 5 ) and ( 8 . 8 ) d e f i n e now @* and $+. By t h i s s e l e c t i o n we a c h i e v e t h a t T ~ * ( z + ) = T , ~ ( z + )= T ~ ~ ( z - )= T ~ ~ ( Z += )0. The r e l a t i o n (8.8) c a n be e a s i l y a c h i e v e d . For example, f o r

I, Babufka et al.

116

we g e t

which is one term i n ( 8 . 7 ) . C o n s e q u e n t l y we g e t t h e o t h e r t e r m s and combining them (8.8) is a c h i e v e d .

9. NUMERICAL PERFORMANCE OF THE E X T R A C T I O N TECHNIQUE We now p r e s e n t t h e r e s u l t s o f c o m p u t a t i o n a l e x p e r i m e n t s based on our model problzm a?d t h e e x t r a c t i o n f u n c t i o n d e s c r i b e d i n S e c t i o n 8 ( u s i n g +*, E + ) . F i g . 9. shows t h e r e s u l t s a n a l o g o u s t o t h o s e shown i n F i g . 5 b u t s t r e s s components -rid was computed by t h e e x t r a c t i o n technique. The s l o p e s h o n i n t h e f i g u r e shows t h e r a t e p = 1 (1.e. t h e r a t e of t h e c o n v e r g e n c e of t h e e n e r g y a n d n o t t h e e n e r g y norm). For comparison t h e e r r o r e R 1 2 f o r mesh 2 )

computed d i r e c t l y ( s e e F i g . 5b) is shown a l s o i n F i g . 9. 10 shows t h e i s o m e t r i c drawings ( i n t h e same s c a l e as i n

10

25

50

100

200 400

Fig.

800

NUMBER OF DEGREESOF FREEDOM Figure 9 The r e l a t i v e e r r o r of - r i j computed by p o s t p r o c e s s i n g of t h e R R p - v e r s i o n f o r Mesh 1 and Mesh 2. ( 1 ) eRl l (2) e22, ( 3 ) e I 2 )

H S B W

Z

H S B N

The Postprocessing Technique in the Finite Element Method

7:

a

by u

a

a, c, 3

computed

T 22

05

e22

The behavior of the error

postprocessing of the p-version.

117

I, Babufku et ul.

118

F i g . 6 ) of t h e e r r o r i n technique.

computed by t h e p o s t p r o c e s s i n g

722

T a b l e 7 shows t h e r e l a t i v e e r r o r eRi . i n t h e s t r e s s e s 7 i j at t h e poin: (.25,, . 2 5 ) computed by t h e 6 o s t p r o c e s s i n g t e c h n i q u e t a k i n g +* = 6~ = 0 ( b e c a u s e t h e p o i n t is n o t c l o s e t o t h e b o u n d a r y ) . T h i s d a t a s h o u l d be compared w i t h t h e r e s u l t s of T a b l e 5 and 6 . TABLE

7

The r e l a t i v e e r r o r e F j i n t h e s t r e s s e s ( . 2 5 , . 2 5 ) computed by p o s t p r o c e s s i n g .

No. a€ elements

R

T i j

at t h e p o i n t

N

el 1

R e22

51

22.51% 13.07% 5.92s 2.01%

12.7096 8.2%

22.51% 14.W 8.93%

12.7%

16 43 106 31 9

221 61 7

20.91% 11 .a% 4.6% 1 -47%

16 64 256

51 167 591

20.91% 12.51% 6.87%

143

eR 12

3.74% 1.31%

9.86%

6.2%

We s e e t h a t f o r a d a p t i v e meshes t h e e r r o r i s of o r d e r N-'

.

9

'$5

$g

'2j

E 8%

.ri

SZ and

f o r uniform meshes of o r d e r N1/2 S i m i l a r l y , a s i n t h e c a s e of t h e p - v e r s i o n we s e e an o r d e r l y c o n v e r g e n c e w i t h t h e r a t e as t h e s q u a r e of t h e e r r o r measured i n t h e e n e r g y norm ( a s t h e o r e t i c a l l y expected). 10.

CONCLUSIONS

The shown c o m p u t a t i o n s a r e c h a r a c t e r i s t i c i n t h e f o l l o w i n g way. The c o n v e r g e n c e i n t h e e n e r g y norm is monotonic and v e r y orderly. F o r t h e smooth s o l u t i o n t h e p - v e r s i o n is e s p e c i a l l y e f f e c t i v e . For unsmooth s o l u t i o n s t h e r e f i n e m e n t of t h e meshes i n t h e h - v e r s i o n is v e r y e s s e n t i a l . The convergence of s t r e s s e s i n a f i x e d p o i n t is v e r y " c h a o t i c , " t h e a c c u r a c y i n v a r i o u s components can be v e r y d i f f e r e n t . The r a t e of convergence of t h e p o s t p r o c e s s e d v a l u e s a r e as t h e s q u a r e of t h e e r r o r measured i n t h e e n e r g y norm. I n t h e case of t h e h - v e r s i o n , u n i f o r m ( o r p i e c e w i s e u n i f o r m ) meshes and smooth s o l u t i o n t h e superconvergence o c c u r s i n t h e c e n t e r of t h e e l e m e n t s . The r a t e is h2 l o g h i . e . , e f f e c t i v e l y as t h e s q u a r e of t h e e r r o r i n e n e r g y norm f h ) . T h e r e f o r e , t h e g a i n

The Postprocessing Technique in the Finite Element Method

119

f o r t h e e l e m e n t s of d e g r e e 1 is n o t i n t h e r a t e o f c o n v e r g e n c e of t h e p o s t p r o c e s s e d v a l u e b u t 1s i n t h e ( F o r p > 1 t h e g a i n of t h e p o s t p r o c e s s i n g a p p e a r s magnitude. also i n the rate.) The p o s t p r o c e s s i n g is e s p e c i a l l y i m p o r t a n t f o r t h e p - v e r s i o n , a l t h o u g h it is a l s o e s s e n t i a l f o r t h e h - v e r s i o n e s p e c i a l l y f o r unsmooth s o l u t i o n s and f o r g e n e r a l meshes.

11.

EFFECTIVITY OF THE POSTPROCESSING T E C H N I Q U E

I n t h e i n t r o d u c t i o n we raised a number o f q u e s t i o n s c o n c e r n i n g the postprocessing. We now b r . i e f l y a d d r e s s t h e s e q u e s t i o n i n t h e l i g h t of o u r r e s u l t s . D e t a i l e d a n a l y s i s w i l l be made i n a forthcoming paper.

I ) I t is c o s t e f f e c t i v e n o t t o s a v e c o m p u t a t i o n a l e f f o r t on a p o s t p r o c e s s i n g p r o c e d u r e e s p e c i a l l y when n o t an e x c e s s i v e number o f e x t r a c t i o n s is made. The c o s t of o b t a i n i n g r e l i a b l e and a c c u r a t e v a l u e s by p o s t p r o c e s s i n g is much s m a l l e r t h a n t o o b t a i n comparable a c c u r a c y by i n c r e a s i n g p i n t h e p - v e r s i o n o r r e f i n e t h e meshes i n t h e h - v e r s i o n . The p o s t p r o c e s s i n g u s u a l l y removes v e r y r e l i a b l y t h e " c h a o t i c " b e h a v i o u r of t h e e r r o r s i n s t r e s s e s . The e f f e c t i v i t y of t h e p o s t p r o c e s s i n g is c h a r a c t e r i z e d by h i g h e r r a t e of c o n v e r g e n c e t h a n i n t h e e n e r g y norm. 2 ) The r a t e of c o n v e r g e n c e as t h e s q u a r e o f t h e r a t e of t h e e r r o r i n t h e e n e r g y norm is t h e o r e t i c a l l y t h e maximal one which c a n be d i r e c t l y e x t r a c t e d . The p o s t p r o c e s s i n g t e c h n i q u e we o u t l i n e d l e a d s t o t h i s r a t e . 3 ) Developoment and i m p l e m e n t a t i o n of t h e p o s t p r o c e s s i n g t e c h n i q u e s i n f i n i t e e l e m e n t programs i s p r a c t i c a l l y n o t a v e r y s i m p l e t a s k . We m e n t i o n some a s p e c t s : a ) A number of e x t r a c t i o n f u n c t i o n s must be developed. Although many a n a l y t i c a l s o l u t i o n s of s p e c i a l problems a r e v e r y h e l p f u l f o r s u c h d e v e l o p m e n t , t h e g e n e r a l a p p r o a c h e s p e c i a l l y f o r nonhomogeneous material s t i l l needs f u r t h e r r e s e a r c h . b ) S p e c i a l c a r e must be e x c e r c i s e d i n t h e n u m e r i c a l e v a l u a t i o n of i n t e g r a l s b e c a u s e t h e e x t r a c t i o n f u n c t i o n c a n have s i n g u l a r c h a r a c t e r . c ) The p o s t p r o c e s s i n g t e c h n i q u e f o r n o n l i n e a r problems c o u l d be e s p e c i a l l y i m p o r t a n t b u t a d d i t i o n a l r e s e a r c h is n e c e s s a r y . REFERENCES [l]

M u s k h e l i s h v i l i , N. I . , Some b a s i c r o b l e m s of t h e m a t h e m a t i c a l t h e o r y of e l a s t i c i t y TP. N o o r d h o f f , Groningen, N e t h e r l a n d s , 1963).

[2]

Basu, P. K . ,

M.

P. ROSSOW, B.

S. S z a b o , T h e o r e t i c a l

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manual and u s e r ' s g u i d e f o r COMET-X ( C e n t e r f o r C o m p u t a t i o n a l Mehanics, Washington U n i v e r s i t y , S t . Louis).

[3]

M e s z t e n y i , C . , W.Szymczak, PEARS u s e r ' s manual f o r U N I V A C 1100 ( U n i v e r s i t y of Maryland, I n s t i t u t e f o r p h y s i c a l S c i e n c e and Technology Tech. Note BN-991, October 1 9 8 2 ) .

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G i g n a c , D. A , , I. Babugka, C . M e s z t e n y i , An i n t r o d u c t i o n t o t h e FEARS program, David W. T a y l o r Naval S h i p R e s e a r c h and Development C e n t e r R e p o r t DTNSRDC/CMLD8 3 / 0 4 , F e b r u a r y 1987.

[5]

Babus'ka, I . , A . M i l l e r , M . V o g e l i u s , A d a p t i v e methods and e r r o r e s t i m a t i o n f o r e l l i p t i c problems of s t r u c t u r a l m e c h a n i c s , U n i v e r s i t y of Maryland, I n s t i t u t e f o r P h y s i c a l S c i e n c e and Technology, Tech. Note BN-1009, J u n e 1983, t o a p p e a r i n t h e P r o c e e d i n g s of ARO Workshop on A d a p t i v e Methods f o r P a r t i a l D i f f e r e n t i a l E q u a t i o n s , SIAM, 1984.

[6]

BabuEika, I . , M . V o g e l i u s , Feedback and a d a p t i v e f i n i t e e l e m e n t s o l u t i o n i n o n e - d i m e n s i o n a l boundary v a l u e p r o b l e m s , U n i v e r s i t y of Maryland, I n s t i t u t e f o r P h y s i c a l S c i e n c e and Technology Tech. Note 1006, October 1983.

[7]

Babus'ka, I . , W. C. R h e i n b o l d t , R e l i a b l e e r r o r e s t i m a t i o n and mesh a d a p t a t i o n f o r f i n i t e e l e m e n t method; i n C o m p u t a t i o n a l methods i n n o n l i n e a r mechanics ( J . T . Oden, e d . , North-Holland P u b l . Co., Amsterdam, 1980, pp.

67-1 0 9 ) [8]

Babugka, I . , B. A. S z a b o , I . N. K a t e , The p - v e r s i o n of t h e f i n i t e e l e m e n t method, SIAM, J . Numer. Anal 18

( 1 981

51 5-545.

[g]

Babus'ka, I . , B. A. S z a b o , On t h e r a t e s of c o n v e r g e n c e o f t h e f i n i t e e l e m e n t method, I n t e r n a l J . Numer. Methods Engrg. 18 ( 1 9 8 2 ) 323-341.

[lo]

Babugka, I . , A . M i l l e r , The p o s t - p r o c e s s i n g i n t h e f i n i t e e l e m e n t method, Part 1 , C a l c u l a t i o n of d i s p l a c e m e n t s s t r e s s e s and o t h e r h i g h e r d e r i v a t i v e s of d i s p l a c e m e n t s , U n i v e r s i t y of Maryland, I n s t i t u t e f o r P h y s i c a l S c i e n c e and T e c h n o l o g y , Tech. Note BN-992, December 1982. To a p p e a r i n I n t e r n a l J . Numer. Methods Engrg. , 1984.

[ll]

Babugka, I . , A. M i l l e r , The p o s t - p r o c e s s i n g a p p r o a c h i n t h e f i n i t e e l e m e n t method, Part 2 , The c a l c u l a t i o n of s t r e s s i n t e n s i t y f a c t o r s , U n i v e r s i t y of Maryland, I n s t i t u t e f o r P h y s i c a l S c i e n c e and Technology, Tech. Note BN 993, December 1 9 8 2 , t o a p p e a r i n I n t e r n a l J . Nurner. Methods Engrg. , 1984.

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Babugka, I . , A . M i l l e r , The p o s t - p r o c e s s i n g a p p r o a c h i n t h e f i n i t e e l e m e n t method, Part 3 , A p o s t e r i o r i e r r o r e s t i m a t e s and a d a p t i v e mesh s e l e c t i o n , U n i v e r s i t y o f Maryland, I n s t i t u t e f o r P h y s i c a l S c i e n c e and T e c h n o l o g y , Tech. Note BN 1007, J u n e 1983, t o a p p e a r i n I n t e r n a l J. Numer Methods E n g r g . , 1984.

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Unification of Finite Element Methods H. Kardestuncer (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

123

CHAPTER 5 ON FINITE ELEMENT ANALYSIS OF LARGE DEFORMATION FRICTIONAL CONTACT PROBLEMS

K.-J. Bathe & A. Chaudhary

We c o n s i d e r t h e s o l u t i o n o f c o n t a c t problems i n v o l v i n g l a r g e deformations o f t h e c o n t a c t i n g bodies and s t i c k i n g o r s l i d i n g o n t h e c o n t a c t i n g s u r f a c e s . A f i n i t e element s o l u t i o n procedure i s d e s c r i b e d and t h e r e s u l t s o f some numerical s t u d i e s a r e presented. The o b j e c t i v e i n t h i s presentation i s t o give f u r t h e r i n s i g h t i n t o the s o l u t i o n procedure a l r e a d y presented i n an e a r l i e r paper [8].

1.

INTRODUCTION

Much progress has been made d u r i n g t h e l a s t decade i n t h e development o f computational techniques f o r n o n l i n e a r a n a l y s i s . These advancements have come about through s i g n i f i c a n t fundamental c o n t r i b u t i o n s i n d i s c r e t i z a t i o n t h e o r i e s and numerical a l g o r i t h m s , b u t i m p o r t a n t has a l s o been t h e crossf e r t i l i z a t i o n t h a t has taken p l a c e between t h e v a r i o u s approaches used f o r t h e numerical s o l u t i o n o f problems. For t h i s c r o s s - f e r t i l i z a t i o n t h e conferences on t h e u n i f i c a t i o n o f numerical methods - a t one o f which t h i s paper i s p r e s e n t e d - have p r o v i d e d an e x c e l l e n t forum. One area o f r e s e a r c h and development t h a t has o b t a i n e d much a t t e n t i o n by a n a l y s t s u s i n g v a r i o u s p o s s i b l e numerical approaches i s t h e a n a l y s i s These problems can b e most d i f f i c u l t t o s o l v e o f c o n t a c t problems [ l - 8 1 . and a l t h o u g h much research e f f o r t has been expended on t h e s o l u t i o n o f c o n t a c t problems u s i n g f i n i t e d i f f e r e n c e methods, f i n i t e element techniques, s u r f a c e i n t e g r a l methods, e t c . , t h e r e i s s t i l l much room f o r m r e r e l i a b l e and e f f e c t i v e a l g o r i t h m s t o analyse general c o n t a c t c o n d i t i o n s . The o b j e c t i v e i n t h i s p r e s e n t a t i o n i s t o discuss c e r t a i n aspects o f a c o n t a c t s o l u t i o n a l g o r i t h m t h a t we have developed and researched [81. We c o n s i d e r two-dimensional p l a n e s t r e s s , p l a n e s t r a i n o r a x i s y m m e t r i c cond i t i o n s . The c o n t a c t i n g b o d i e s can b e s u b j e c t e d t o l a r g e deformations w i t h s t i c k i n g , s l i d i n g and s e p a r a t i o n on t h e c o n t a c t i n g s u r f a c e s . I n t h e n e x t s e c t i o n we d e s c r i b e i n d e t a i l t h e c o n t a c t problem we cons i d e r , and i n S e c t i o n 3 we discuss o u r numerical s o l u t i o n procedure. The

( ? ) P r o f e s s o r o f Mechanical Engineering (*)Research A s s i s t a n t

K.-J.Bathe & A. Chaudhary

124

o b j e c t i v e i n t h i s s e c t i o n i s t o g i v e i n s i g h t i n t o o u r a l g o r i t h m . The gove r n i n g equations a r e d e r i v e d i n d e t a i l i n r e f . [8], which s h o u l d be cons u l t e d f o r a more f u l l account o f o u r s o l u t i o n technique. I n S e c t i o n 4 we then p r e s e n t t h e r e s u l t s o f v a r i o u s numerical experiments t o i l l u s t r a t e o u r observations on t h e a l g o r i t h m . These experiments show how t h e c o n t a c t s o l u t i o n procedure works, what t h e d i f f e r e n t assumptions are, and how t h e method can be a p p l i e d . I n t h e p r e s e n t a t i o n we c o n s i d e r s t a t i c a n a l y s i s , b u t t h e a l g o r i t h m i c steps used can a l s o be employed i n a dynamic s o l u t i o n .

2.

STATEMENT O F CONTACT PROBLEMS CONSIDERED

We can use o u r a l g o r i t h m f o r t h e a n a l y s i s o f a number o f f l e x i b l e bodies coming i n t o c o n t a c t w i t h each o t h e r o r w i t h r i g i d bodies. However, f o r ease o f p r e s e n t a t i o n o f t h e t h e o r y and o u r a l g o r i t h m we now c o n s i d e r two bodies, b o t h f l e x i b l e , t h a t p a r t l y come i n t o c o n t a c t . Figure 1 shows s c h e m a t i c a l l y t h e two bodies, which we c a l l t h e c o n t a c t o r and t h e t a r g e t . The bodies can undergo very l a r g e deformations, and can come i n t o cont a c t , w i t h o r w i t h o u t s l i d i n g and can separate again. However, we o n l y c o n s i d e r s t a t i c a n a l y s i s c o n d i t i o n s (hence t h e motions o f t h e bodies a r e "slow" s o t h a t t h e e f f e c t s o f i n e r t i a and damping f o r c e s can be n e g l e c t e d ) . The f o l l o w i n g equations govern t h e problem we consider. a)

where

T~~

The l i n e a r momentum e q u a t i o n f o r t h e t a r g e t and t h e c o n t a c t o r ,

i s the ( i , j ) t h

component o f t h e Cauchy s t r e s s t e n s o r and fiB i s

t h e i ' t h component o f t h e body f o r c e v e c t o r , b o t h a r e r e f e r r e d t o t h e curr e n t c o n f i g u r a t i o n s o f t h e bodies. Equation (1) must be s a t i s f i e d througho u t t h e m o t i o n o f t h e bodies s u b j e c t t o t h e a p p r o p r i a t e c o n s t i t u t i v e r e l a t i o n s t o e v a l u a t e t h e Cauchy s t r e s s e s . b ) The boundary c o n d i t i o n s correspond t o p r e s c r i b e d displacements on Sd, p r e s c r i b e d s u r f a c e forces on S f and a p r i o r i unknown c o n t a c t c o n d i t i o n s on t h e p o s s i b l e area o f c o n t a c t Sc.

We n o t e t h a t i n F i g . 1 t h e surfaces

Sd, S f and S c are d i s t i n c t from each o t h e r , and t h e i r sum c o n s t i t u t e s t h e complete s u r f a c e o f t h e t a r g e t and t h e c o n t a c t o r . Considering t h e s u r f a c e s S d and Sf on t h e bodies, we have t h e boundary condi ti ons , (2)

(3)

Large Deformation Frictional Contact Problems where uis

125

and fis denote t h e imposed displacements and a p p l i e d ,surface

f o r c e s , and t h e n . a r e t h e d i r e c t i o n cosines o f t h e normal t o t h e s u r f a c e . J We may n o t e t h a t we assume b o t h bodies t o be s u p p o r t e d a g a i n s t r i g i d body m t i o n s , hence t h e p r e s c r i b e d displacements on S d must be such as t o make t h e t a r g e t and t h e c o n t a c t o r , w i t h o u t c o n t a c t between them, s t a b l e s t r u c tures. Considering t h e surfaces S

C'

t h e displacements a r e f r e e from c o n s t r a i n t s

and no forces a r e developed on t h e s e s u r f a c e s as l o n g as t h e r e i s no cont a c t . However, c o n t a c t i s reached as soon as m a t e r i a l p a r t i c l e s o f t h e t a r g e t and c o n t a c t o r s u r f a c e s touch each o t h e r , and t h e n t h e f o l l o w i n g L e t tnCand tnTbe t h e c o n t a c t t r a c t i o n s ( f o r c e s /

considerations are v a l i d .

u n i t area) i n t h e d i r e c t i o n normal t o t h e c o n t a c t s u r f a c e s , w i t h tnC acting upon t h e c o n t a c t o r and tnTa c t i n g upon t h e t a r g e t , see F i g . 2 .

Contact i s

e s t a b l i s h e d as l o n g as t n C i s p o s i t i v e ( a c t i n g i n t o t h e body), and d u r i n g c o n t a c t we have T tnC

=

(4)

tn

Also, d e n o t i n g t h e d i f f e r e n t i a l displacement increments i n t h e d i r e c t i o n normal t o t h e c o n t a c t s u r f a c e

as dunC f o r t h e c o n t a c t o r and dunT f o r t h e

t a r g e t , t h e n d u r i n g c o n t a c t we have dun C

=

dunT

For t h e e v a l u a t i o n o f t h e t a n g e n t i a l t r a c t i o n s t h a t a c t o n t o t h e t a r g e t T and t h e c o n t a c t o r we use Coulomb's law o f f r i c t i o n . L e t t t Cand tt r e p r e s e n t t h e developed t a n g e n t i a1 t r a c t i o n s a l o n g t h e c o n t a c t s u r f a c e s , and l e t dutT-'

be t h e r e l a t i v e i n c r e m e n t a l displacement between the m a t e r i a l

p a r t i c l e s o f t h e c o n t a c t o r and t h e t a r g e t .

Itt

C

I

friction. i s It t

Then du;-'

i s zero as l o n g as

T < ~-r tnC(and hence I t t I < p t n T ) where 1-1 i s t h e c o e f f i c i e n t o f C

I

Further, t h e maximum t a n g e n t i a l t r a c t i o n t h a t can b e reached =

LI t n

T T (and a l s o Itt I = 1-1 tn ) , and when t h i s t a n g e n t i a l t r a c t i o n

i s developed we have IdutT-'1

1. 0 ,

hence r e l a t i v e t a n g e n t i a l m o t i o n between

t h e t a r g e t and c o n t a c t o r p a r t i c l e s i s t h e n p o s s i b l e . The " d i r e c t i o n " o f r e l a t i v e m o t i o n i s such t h a t t h e developed t a n g e n t i a l t r a c t i o n s oppose t h e motion. We s h o u l d n o t e t h a t a t t h e b e g i n n i n g o f t h e a n a l y s i s , t h e a c t u a l area o f c o n t a c t ( b e i n g a p a r t o f t h e p o s s i b l e area o f c o n t a c t ) i s unknown, and

K.-J. Bathe & A . Ckaudkary

126

T so a r e t h e c o n t a c t t r a c t i o n s tnC and t t C(and tn , t:).

The d e t e r m i n a t i o n

o f t h e a c t u a l area o f c o n t a c t and t h e corresponding c o n t a c t f o r c e s , w h i l e t h e t a r g e t and c o n t a c t o r a r e s u b j e c t e d t o small o r l a r g e deformations w i t h l i n e a r o r n o n l i n e a r c o n s t i t u t i v e b e h a v i o r , i s t h e key t a s k o f t h e s o l u t i o n procedure. 3.

SUMMARY OF CONTACT SOLUTION PROCEDURE

The s o l u t i o n procedure we have developed s o l v e s f o r t h e motion o f t h e c o n t a c t o r and t a r g e t bodies using t h e b a s i c c o n s i d e r a t i o n s sumnarized i n t h e p r e v i o u s s e c t i o n , Since t h e e q u i l i b r i u m , c o m p a t i b i l i t y and c o n s t i t u t i v e r e l a t i o n s must be s a t i s f i e d t h r o u g h o u t t h e - i n general - h i g h l y n o n l i n e a r response h i s t o r y , an i n c r e m e n t a l s o l u t i o n i s performed. I n t h i s s e c t i o n we aim t o d e s c r i b e t h e s o l u t i o n procedure p r e s e n t e d a l r e a d y i n r e f . [8] t o render more i n s i g h t i n t o t h e d e t a i l e d o p e r a t i o n s o f t h e a l g o r i t h m . 3.1

The incremental equations o f motion

As l o n g as t h e r e i s no c o n t a c t , t h e incremental s o l u t i o n i s performed as d e s c r i b e d i n r e f . [9, c h a p t e r 61. Namely, assuming t h a t t h e s o l u t i o n i s known f o r t h e c o n f i g u r a t i o n a t t i m e t, t h e i t e r a t i o n i s performed t o o b t a i n t h e s o l u t i o n corresponding t o t i m e t + A t . S i n c e t h e e f f e c t o f i n e r t i a forces i s neglected, t h e governing equations i n t h i s i t e r a t i o n are, u s i n g t h e f u l l Newton method and t h e n o t a t i o n o f r e f , [9], t+AtK(i-l)

-

t+At,(i) where ttAtK(i-l) -

(i)

AU =

=

t+AtR

t+Atu(i-1) -

-

-

t+AtF(i-1)

-

(i1

(7)

i s t h e t a n g e n t s t i f f n e s s m a t r i x ( + ) corresponding t o t i m e

t t A t and t h e end o f i t e r a t i o n (i-l), ttAtF(i-l) -

i s a nodal p o i n t f o r c e

v e c t o r corresponding t o t h e i n t e r n a l element stresses, t+At&

i s the vector

o f e x t e r n a l l y a p p l i e d loads and A!()’ i s t h e v e c t o r o f incremental nodal p o i n t displacements. Note t h a t a t t h e b e g i n n i n g o f t h e i t e r a t i o n , f o r i = l , we have t h e i n i t i a l c o n d i t i o n s

The c o n t r i b u t i o n s i n ttAtK(i-l) and t+AtF(i-l) a r e those o f t h e c o n t a c t o r and t h e t a r g e t . Since t h e r e i s no c o n t a c t as y e t , these c o n t r i b u t i o n s

(+)Note t h a t t h e t i m e s u p e r s c r i p t t t A t s i g n i f i e s h e r e t h e c o n f i g u r a t i o n (and l o a d ) a t t i m e t t A t and does n o t i m p l y a dynamic a n a l y s i s .

Large Deformation Frictional Contact Problems

127

a r e uncoupled, b u t t h e s o l u t i o n i s p o s s i b l e because both the target and t h e contactor bodies a r e properly supported ( a n assumption we s t a t e d i n Section 2 ) . As described i n d e t a i l i n r e f . [8], during each i t e r a t i o n using Eqs. ( 6 ) and (7), t h e algorithm checks using t h e c u r r e n t configurations of the contactor a n d t a r g e t bodies whether t h e contactor has "penetrated" t h e t a r g e t . I f a contactor s u r f a c e node i s within t h e t a r g e t , contact has been e s t a b l i s h e d and during such conditions t h e governing incremental equilibrium equations a r e

where t + A t-c~ ( i - l ) i s a contact s t i f f n e s s matrix, t + A t ~ ( i - l ) i s a vector o f -C

contact forces and t + A t-c~ ( i - ' ) i s a vector o f geometric o v e r l a p s , i . e . penetrations of t h e contactor nodes i n t o t h e t a r g e t . Figure 3 i l l u s t r a t e s schematically t h e meaning o f t h e vector ttAtA -c ( i - l ) . The Lagrange multip l i e r s A x ( ' ) can be i n t e r p r e t e d as increments i n the nodal p o i n t forces acting on t h e contact surfaces required t o prevent t h e overlap t+At, (i-1) -C . However, s i n c e E q . (8) has been derived by l i n e a r i z i n g about t h e s t a t e a t t h e end of i t e r a t i o n ( i - 1 ) , t h e s e contact force increments can be very approximate (due t o geometric and material n o n l i n e a r i t i e s and t h e f r i c t i o n a l r e s t r a i n t s ) a n d a r e not d i r e c t l y used t o c a l c u l a t e t h e contact forces. Instead we use E q . ( 8 ) t o impose t h e geometric c o n s t r a i n t s of no material overlap and evaluate t h e contact forces from t h e i n t e r n a l s t r e s s e s of t h e contacting bodies. t + A t R (i-1) Before di scuss i ng the eval u a t i on of the contact forces -C l e t us consider some f u r t h e r important d e t a i l s regarding E q . ( 8 ) by cons i d e r i n g t h e following t h r e e d i f f e r e n t cases. Y

Case of p e r f e c t s l i d i n g , u

=

0.0

In t h e case of no f r i c t i o n , our s o l u t i o n procedure e s t a b l i s h e s only one additional equation corresponding t o A A ( i ) f o r each contactor node t h a t has penetrated t h e t a r g e t body. This equation corresponds t o t h e displacement c o n s t r a i n t s of no material overlap normal t o t h e contact s u r f a c e , whereas tangent t o t h i s s u r f a c e t h e contactor and t a r g e t bodies can s l i d e f r e e l y on each o t h e r . Hence, t h e only contact t r a c t i o n s developed

K . 4 Bathe & A. Cliuudhary

128 r

r

are tnLand t n ' . Note t h a t a t convergence using Eq. (8) we must have

t t n tA ( i - 1 )

;

--c

-

(9)

and s i n c e no e x t e r n a l f o r c e s a r e a p p l i e d on t h e c o n t a c t surface ( t) t t A t R (i-1)

-

t+AtF(i-l)

-

-C

where t h e "approximately equal s i g n " i s used because convergence i s o n l y o b t a i n e d w i t h i n s p e c i f i c convergence t o l e r a n c e s . Also, s i n c e o n l y c o n t a c t forces normal t o t h e c o n t a c t s u r f a c e can be e s t a b l i s h e d , t h e components i n t h e v e c t o r t+AtR -c (i-')

must correspond t o (compressive) t r a c t i o n s a c t i n g

normal t o t h e c o n t a c t s u r f a c e . Case o f p e r f e c t s t i c k i n g , p =

m

I n t h e case o f s t i c k i n g t h e s o l u t i o n procedure e s t a b l i s h e s two equations corresponding t o AA(i)as soon as o v e r l a p i s detected, and e l i m i n a t e s t h e geometric o v e r l a p . Note t h a t as f o r t h e o t h e r cases t h e c o n t a c t o r node corresponding t o which t h e equations a r e e s t a b l i s h e d can come i n t o c o n t a c t anywhere along t h e c o n t a c t surface o f t h e t a r g e t . A t convergence, we have again t h a t Eqs. ( 9 ) and (10) a r e s a t i s f i e d . Case o f s t i c k i n g o r s l i d i n g , p > 0 We discussed i n S e c t i o n 2 t h a t when t h e f r i c t i o n c o e f f i c i e n t i s nonzero and small , t h e normal and t a n g e n t i a l t r a c t i o n s developed d u r i n g c o n t a c t determine whether s l i d i n g occurs. Consequently, t h e s o l u t i o n procedure assumes i n t h e f i r s t i t e r a t i o n from no c o n t a c t t o c o n t a c t s t i c k i n g c o n d i t i o n s and e s t a b l i s h e s corresponding t o Ax(') two equations. The s o l u t i o n i n t h i s i t e r a t i o n y i e l d s c o n t a c t tractions: c a l c u l a t e d assuming s t i c k i n g c o n d i t i o n s , t h a t a r e used t o e s t a b l i s h whether t h e r e a c t u a l l y a r e s t i c k i n g o r s l i d i n g c o n d i t i o n s . The updated c o n d i t i o n s on s t i c k i n g and s l i d i n g t o g e t h e r w i t h t h e corresponding c o n t a c t forces a r e employed i n t h e n e x t i t e r a t i o n t o e s t a b l i s h two ( f o r s t i c k i n g ) o r o n l y one ( f o r s l i d i n g ) c o n s t r a i n t equations corresponding t o each c o n t a c t o r node i n c o n t a c t . Note t h a t d u r i n g t h e s o l u t i o n and t h e i t e r a t i o n s , a node may a l s o change i t s s t a t e from s l i d i n g back t o s t i c k i n g .

A very i m p o r t a n t p a r t o f t h e sol u t i o n procedure i s hence t h e e v a l u a t i o n o f the c o n t a c t f o r c e s , w i t h t h e i r normal and t a n g e n t i a l components, and t h e d e c i s i o n o f whether s l i d i n g o r s t i c k i n g c o n d i t i o n s p r e v a i l . 3.2

E v a l u a t i o n o f c o n t a c t forces and s l i d i n g o r s t i c k i n g c o n d i t i o n s Using Eqs. ( 7 ) and ( 8 ) we c a l c u l a t e a t t h e b e g i n n i n g o f t h e ( i + l ) s t

( f ) E x t e r n a l l y a p p l i e d forces on t h e c o n t a c t s u r f a c e c o u l d a c t u a l l y be i n cluded i n Eq. (10)

129

Large Deformation Frictional Contact Problems i t e r a t i o n ttAtU(i), f o r c e v e c t o r tTAtF(i) -

from which we can d i r e c t l y e v a l u a t e t h e nodal p o i n t which i s e q u i v a l e n t ( i n t h e v i r t u a l work sense) t o

t h e c u r r e n t s t r e s s e s ttAt~kp(i).

Next, t h e f o l l o w i n g v e c t o r o f f o r c e s i s

evaluated

where we n o t e t h a t w i t h no c o n t a c t between t h e c o n t a c t o r and t h e t a r g e t t h e elements i n AE(i) must a l l be s m a l l a t convergence, i . e . , AR(i

+!?

as i + a. However, i f t h e r e i s c o n t a c t , t h e n t h e elements i n corresponding t o t h e t a r g e t and c o n t a c t o r nodes on t h e c o n t a c t s u r f a c e Sc ( t h a t a r e a f f e c t e d by t h e c o n t a c t ) must a t convergence i n mesh d i s c r e t i z a t i o n and i t e r a t i o n be equal t o t h e c o n t a c t f o r c e s . T h i s means t h a t these f o r c e s must s a t i s f y t h e f r i c t i o n c o n d i t i o n s summarized i n S e c t i o n 2 .

The compo-

nents i n AR(i) corresponding t o t h e nodes t h a t a r e n o t i n c o n t a c t must o f course sti7-1 be small (approach zero) a t convergence. The f i r s t s t e p i n e v a l u a t i n g whether t h e f o r c e s A&(i)

satisfy the

f r i c t i o n a l c o n d i t i o n s i s t o e v a l u a t e from A&(i) normal and t a n g e n t i a l t r a c t i o n s t h a t a c t o n t o t h e c o n t a c t o r . Since, c o n s i d e r i n g t h e c o n t a c t surface, t h e c o n d i t i o n s on t h e c o n t a c t o r a r e s t a t i c a l l y e q u i v a l e n t t o those on t h e t a r g e t , we c o n s i d e r o n l y t h e c o n t a c t o r . The d i s t r i b u t e d c o n t a c t t r a c t i o n s a r e those normal and t a n g e n t i a l f o r c e s ( p e r u n i t area) t h a t g i v e t h e c o n s i s t e n t nodal p o i n t f o r c e s s t o r e d i n A&(i).

L e t tnkand t t k be t h e

i n t e n s i t i e s o f t h e normal and t a n g e n t i a l t r a c t i o n s a t t h e c o n t a c t o r node k e v a l u a t e d u s i n g t h i s approach. F u r t h e r , l e t t h e t o t a l r e s u l t a n t normal k and tt and t a n g e n t i a l f o r c e s on segment j , e v a l u a t e d from t h e values t n f o r a l l nodes k i n c o n t a c t , be :T and, :T see F i g . 4. The s o l u t i o n p r o cedure uses t h e values T n j and T t j t o e v a l u a t e ( g l o b a l l y ) whether t h e f o r c e c o n t a c t c o n d i t i o n s a r e s a t i s f i e d and, i f necessary, updates t h e s e c o n d i t i o n s f o r the next i t e r a t i o n .

I f TnJ < 0, t h e segment i s r e l e a s e d f o r t h e n e x t i t e r a t i o n because t h e segt+AtR ( i ) ment cannot be i n t e n s i o n . To e v a l u a t e t h e c o n t a c t f o r c e v e c t o r -c , t h e t r a c t i o n s tt and tn a r e s e t t o z e r o o v e r t h e segment.

I]:fTI

5 1-1 T n j , t h e segment i s assumed t o s t i c k i n t h e n e x t i t e r a t i o n .

e v a l u a t e t h e c o n t a c t f o r c e v e c t o r ttAtR -c ( i ) ,

To

t h e t r a c t i o n s t t and tn f r o m

t h e p r e v i o u s i t e r a t i o n a r e employed.

I f IT:(>

,p:T

t h e segment i s assumed t o s l i d e i n t h e n e x t i t e r a t i o n .

e v a l u a t e t h e c o n t a c t f o r c e v e c t o r t+AtR -c (i),

To

t h e normal t r a c t i o n tn from

t h e p r e v i o u s i t e r a t i o n i s employed b u t t h e t a n g e n t i a l t r a c t i o n

130

K.-J. Bathe & A. Chaudhary

tt i s updated t o a c o n s t a n t value o f

s u r f a c e area o f t h e segment.

-

't

I ~j 1,

Aj

t

"n

where A . i s t h e

J

Using t h i s value o f tt, Coulonb's law o f

f r i c t i o n i s s a t i s f i e d g l o b a l l y o v e r t h e segment, b u t a c o n s t a n t value o f t a n g e n t i a l t r a c t i o n i s assumed t o a c t o v e r t h e segment. The t r a c t i o n s tt and tn o v e r each c o n t a c t segment thus o b t a i n e d a r e employed t o e v a l u a t e t h e nodal p o i n t c o n s i s t e n t c o n t a c t f o r c e s t + A t R-c ( i ) W i t h t h e s t a t e s o f t h e segments updated t o " r e l e a s e " , " s t i c k i n g " , " s l i d i n g " and t h e c a l c u l a t i o n o f ttAt&c(i)

.

or

completed, t h e s o l u t i o n procedure

e s t a b l i s h e s t h e s t a t e s o f t h e c o n t a c t o r nodes as summarized i n T a b l e 1 and t h e n (see S e c t i o n 3.1) e s t a b l i s h e s i n Eq. (8) two c o n t a c t equations f o r each node i n t h e s t i c k i n g c o n d i t i o n and one e q u a t i o n f o r each node i n t h e s l i d i n g c o n d i t i o n . We may n o t e t h a t by means o f t h e above c a l c u l a t i o n s t h e d i s t r i b u t e d f o r c e and f r i c t i o n a l e f f e c t s on t h e segments a r e concent r a t e d t o t h e nodes, c o n s i s t e n t w i t h usual f i n i t e element procedures. 3.3

Convergence o f t h e i t e r a t i v e scheme

-

To study t h e convergence o f t h e i t e r a t i o n s i t i s convenient t o c o n s i d e r t h e t h r e e d i f f e r e n t cases, p = 0.0, p =

and p > 0 b u t o f s m a l l value.

When y = 0.0 (case o f p e r f e c t s l i d i n g ) t h e e q u i l i b r i u m r e l a t i o n s i n Eq. (8) reduce t o those w i t h o u t c o n t a c t c o n d i t i o n s (see r e f . [9], c h a p t e r 6 ) supplemented w i t h t h e c o n s t r a i n t t h a t t h e c o n t a c t o r nodes cannot penet r a t e the t a r g e t but instead w i l l s l i d e without resistance over the t a r g e t segments. Hence, a t convergence, o n l y c o n t a c t f o r c e s t h a t a c t normal t o t h e c o n t a c t s u r f a c e a r e t r a n s m i t t e d . Note t h a t s i n c e t h e c o n t a c t o r nodes s l i d e o v e r t h e t a r g e t Segments, t h e t a r g e t nodes can be w i t h i n o r o u t s i d e t h e c o n t a c t o r . Hence, t h e f i n i t e element d i s c r e t i z a t i o n f o r t h e c o n t a c t o r and t h e t a r g e t s h o u l d be such t h a t t h e r e s u l t i n g m a t e r i a l o v e r l a p i s acceptable. These c o n s i d e r a t i o n s a r e a l s o i m p o r t a n t when p > 0 . The case o f p = -, i . e . , p e r f e c t s t i c k i n g , i s achieved by s i m p l y choosi n g p l a r g e enough s o t h a t t h e t a n g e n t i a l t r a c t i o n s on t h e c o n t a c t s u r f a c e a r e always l e s s than p t i m e s t h e normal t r a c t i o n s . The s o l u t i o n o b t a i n e d i n t h e i n c r e m e n t a l a n a l y s i s i s i n t h i s case path-dependent because t h e c o n t a c t o r nodes s t i c k throughout t h e a n a l y s i s t o t h e m a t e r i a l p o i n t s o f t h e t a r g e t segments w i t h which they f i r s t come i n t o c o n t a c t ( u n l e s s t e n s i o n r e l e a s e o c c u r s ) . Hence, a d i f f e r e n t sequence o f e x t e r n a l l o a d a p p l i c a t i o n w i t h t h e same f i n a l l o a d d i s t r i b u t i o n may l e a d t o s i g n i f i c a n t l y d i f f e r e n t r e s u l t s . However, convergence u s i n g Eqs. (8) t o (10) means t h a t a t each l o a d l e v e l t h e e q u i l i b r i u m , c o n s t i t u t i v e and c o m p a t i b i l i t y c o n d i t i o n s , w i t h i n t h e assumptions o f t h e f i n i t e element d i s c r e t i z a t i o n , a r e s a t i s f i e d . The most d i f f i c u l t types o f problems t o s o l v e a r e those f o r which p i s g r e a t e r t h a n zero b u t s m a l l , s o t h a t depending on t h e unknown normal and t a n g e n t i a l t r a c t i o n s along t h e c o n t a c t s u r f a c e , i n some areas s t i c k i n g and i n o t h e r areas s l i d i n g may o c c u r . Considering t h e e q u i l i b r i u m r e l a t i o n s

Large Deformation Frictional Contact Problems

131

i n Eq. (8) and t h e procedure f o r e v a l u a t i n g t h e c o n t a c t f o r c e s , we can make t h e f o l l o w i n g i m p o r t a n t o b s e r v a t i o n s r e g a r d i n g t h e convergence o f t h e iterations : O

O

Consider t h a t corresponding t o t h e c o n f i g u r a t i o n a t t i m e t, t h e cond i t i o n s f o r a l l c o n t a c t o r nodes a r e known. With t h e i n c r e a s e i n t h e e x t e r n a l l y a p p l i e d l o a d f r o m t i m e t t o t i m e t + A t some nodes reach t h e " s l i d i n g c o n d i t i o n " d u r i n g t h e i t e r a t i o n s , which r e s u l t s i n t o i n c r e m e n t a l displacements and a r e d i s t r i b u t i o n o f t h e i n t e r n a l element s t r e s s e s , u n t i l f i n a l l y t h e c o n t a c t t r a c t i o n s s a t i s f y Coulomb's law o f f r i c t i o n ( g l o b a l l y , f o r each o f t h e segments, see S e c t i o n 3 . 2 ) . T h i s means t h a t a t convergence o f t h e i t e r a t i o n s f o r t h e l o a d l e v e l t t A t , t h e c o n t a c t o r nodes a r e l a r g e l y i n t h e " s t i c k i n g c o n d i t i o n " a l t h o u g h d u r i n g t h e i t e r a t i o n s they may have s l i d . However, n o t e t h a t r e g a r d i n g t h e p h y s i c a l i n t e r p r e t a t i o n o f t h e s o l ut i o n r e s u l t s , a c o n t a c t o r node has been i n s l i d i n g f r o m t i m e t t o t i m e t t A t , whenever t h e c o n t a c t o r node i s n o t any more a t t h e same t a r g e t m a t e r i a l p a r t i c l e as i t was a t t i m e t. Hence, t h e f i n a l cond i t i o n o f s t i c k i n g f o r a c o n t a c t o r node a t t i m e t + A t does n o t a l o n e t e l l whether t h e node has or has n o t been s l i d i n g from t i m e t t o t i m e ttAt.

Considering t h e e v a l u a t i o n o f t h e t a n g e n t i a l t r a c t i o n s i n s l i d i n g we r e c a l l t h a t these t r a c t i o n s a r e c a l c u l a t e d by r e d u c i n g t h e developed t a n g e n t i a l t r a c t i o n s t o t h e magnitude compatible w i t h t h e normal t r a c t i o n s ( u s i n g Coulomb's law o f f r i c t i o n ) . Hence, t h e d i r e c t i o n o f r e l a t i v e t a n g e n t i a l s l i d i n g along t h e c o n t a c t s u r f a c e does n o t d i r e c t l y e n t e r i n t o t h e d e t e r m i n a t i o n o f t h e d i r e c t i o n o f t h e t a n g e n t i a l c o n t a c t t r a c t i o n s . However, o u r e x p e r i e n c e i s t h a t t h e c a l c u l a t e d t a n g e n t i a l t r a c t i o n s do oppose t h e motion p r o v i d e d t h e f i n i t e element r e p r e s e n t a t i o n i s f i n e enough and t h e i n c r e m e n t a l s o l u t i o n i s performed i n s m a l l enough s t e p s . We a r e purs u i n g f u r t h e r t h e o r e t i c a l and computational s t u d i e s o f t h i s o b s e r v a t i o n .

So f a r we considered o n l y how convergence i s reached. However, i n t h e i t e r a t i o n s , a c t u a l convergence c r i t e r i a a r e necessary t h a t measure when t o accept t h e c a l c u l a t e d s o l u t i o n . The convergence c r i t e r i a we have used measure t h e i n c r e m e n t a l energy and t h e change i n t h e c o n t a c t f o r c e s . Namely, r e f e r r i n g t o Eq. ( 8 ) , t h e s o l u t i o n i s accepted once t h e f o l l o w i n g r e l a t i o n i s satisfied,

where ETOL i s t h e energy convergence t o l e r a n c e , and once f o r a l l nodes i n c o n t a c t , r e f e r r i n g t o E q . ( l l ) , t h e components i n AJ(i) the contact forces s a t i s f y t h e r e l a t i o n

corresponding t o

K.-J. Bathe & A . Chaudhary

132

I In!

(i-1)

-

!a!

( i -2)

I12

where RCTOL i s t h e c o n t a c t f o r c e convergence t o l e r a n c e . used a r e ETOL = 0.001 and RCTOL = 0.01.

T y p i c a l values

We i l l u s t r a t e how t h e a l g o r i t h m proceeds i n s o l u t i o n s by means o f some numerical r e s u l t s g i v e n i n t h e n e x t s e c t i o n . 4.

NUMERICAL EXPERIMENTS

Various a n a l y s i s r e s u l t s o b t a i n e d w i t h o u r s o l u t i o n a l g o r i t h m and comparisons w i t h s o l u t i o n s p r e v i o u s l y r e p o r t e d have been presented i n r e f . [8]. The o b j e c t i v e i n t h i s s e c t i o n i s t o supplement t h e a n a l y s i s r e s u l t s o f r e f . [8] b y showing i n more d e t a i l how t h e s o l u t i o n i s o b t a i n e d and p r e s e n t i n g some r e s u l t s on t h e e f f e c t s o f mesh s e l e c t i o n and l o a d s t e p s i z e . We consider two problems a l r e a d y discussed i n r e f . [8]; namely, t h e a n a l y s i s o f a b u r i e d p i p e and t h e s o l u t i o n o f a r u b b e r sheet moving i n a r i g i d convergi ng channel. 4.1

A n a l y s i s o f a B u r i e d Pipe

F i g u r e 5 shows t h e b u r i e d p i p e considered. The o b j e c t i v e o f t h e a n a l y s i s i s t o p r e d i c t t h e t r a c t i o n s along t h e p i p e - s o i l i n t e r f a c e . Both, t h e p i p e and t h e s u r r o u n d i n g s o i l a r e considered l i n e a r e l a s t i c media. I n r e f . [8] we presented t h e s o l u t i o n t o t h e problem u s i n g t h e f i n i t e element i d e a l i z a t i o n o f F i g . 6, now c a l l e d mesh B . I n o r d e r t o study t h e e f f e c t o f d i s c r e t i z a t i o n on t h e s o l u t i o n r e s u l t s we now a l s o g i v e t h e s o l u t i o n t o t h e problem u s i n g t h e coarse mesh (mesh A) and t h e f i n e mesh (mesh C) shown i n F i g s . 7 and 8. Note t h a t mesh B i s o b t a i n e d by subd i v i d i n g each 8-node i s o p a r a m e t r i c element o f mesh A i n t o f o u r 8-node i s o p a r a m e t r i c elements, and mesh C i s o b t a i n e d i n t h e same manner from mesh B. F i g u r e 9 shows t h e computed t r a c t i o n s u s i n g t h e d i f f e r e n t meshes. The s o l u t i o n s have been o b t a i n e d i n a f o u r s t e p s o l u t i o n , i . e . by a p p l y i n g t h e t o t a l overburden pressure Po i n f o u r equal steps and u s i n g ETOL = 0.001 and RCTOL = 0.01. For comparison a l s o t h e s o l u t i o n s f o r zero f r i c t i o n and i n f i n i t e f r i c t i o n , o b t a i n e d u s i n g mesh B y a r e shown. Note t h a t t h e t r a c t i o n s tn and tt p l o t t e d i n F i g . 9 a r e t h e mean t r a c t i o n s o v e r a segment; hence, f o r t y p i c a l p o i n t s r e p r e s e n t i n g t h e t r a c t i o n s o n segment j we have t j = TA /: . and tnj = T:/A. (see Section t J J 3.2). F i g u r e 9 shows t h a t t h e d i f f e r e n c e s i n t h e c o n t a c t t r a c t i o n s c a l c u l a t e d u s i n g meshes B and C a r e reasonably s m a l l . Figures 10 and 11 show t h e t r a c t i o n d i s t r i b u t i o n s f o r each i t e r a t i o n u s i n g meshes B and C f o r a one s t e p s o l u t i o n . The f i g u r e s show t h e c a l c u l a t i o n o f t a n g e n t i a l t r a c t i o n s ( i t e r a t i v e l y updated) t o s a t i s f i y Coulomb's l a w

Large Deformation Frictional Contact Problems

133

o f f r i c t i o n g l o b a l l y o v e r each segment. A t convergence - see F i g s . l O ( e ) and l l ( g ) - t h e mean updated t a n g e n t i a l t r a c t i o n s o v e r a segment a r e e s s e n t i a l l y equal t o t h e mean t r a c t i o n s p r i o r t o updating. I n order t o study the e f f e c t o f using a d i f f e r e n t load incrementation, we show i n F i g . 12 t h e t r a c t i o n d i s t r i b u t i o n s c a l c u l a t e d when u s i n g t h e 4 equal l o a d increments t o reach t h e t o t a l overburden p r e s s u r e . We n o t e t h a t t h e s o l u t i o n o b t a i n e d t h i s way i s very c l o s e t o t h e s o l u t i o n c a l c u l a t e d when one l o a d increment i s used t o a p p l y t h e t o t a l overburden p r e s s u r e - see F i g s . l l ( g ) and 1 2 ( d ) . 4.2

A n a l y s i s o f a Rubber Sheet Moving i n a R i g i d Converging Channel

F i g u r e 13 shows t h e r u b b e r s h e e t considered. The r i g h t f a c e o f t h e sheet i s s u b j e c t e d t o a displacement h i s t o r y p u l l i n g i t i n t o t h e channel and t h e n pushing i t back t o i t s o r i g i n a l l o c a t i o n . The displacements a r e imposed s l o w l y s o t h a t i n e r t i a f o r c e s can b e n e g l e c t e d . T h i s problem was analyzed i n r e f . [8] u s i n g t h e mesh shown i n F i g . 14, here c a l l e d mesh B. We now a l s o g i v e t h e s o l u t i o n t o t h e problem u s i n g t h e c o a r s e r mesh shown i n F i g . 15, c a l l e d mesh A. F i g u r e 16 shows t h e p r e d i c t e d t a n g e n t i a l and normal t r a c t i o n s c a l c u l a t e d u s i n g meshes A and B . We n o t e t h e c l o s e correspondence between t h e r e s u l t s o b t a i n e d a l t h o u g h mesh A represents q u i t e a coarse i d e a l i z a t i o n o f t h e r u b b e r sheet. I n these s o l u t i o n s t h e convergence t o l e r a n c e s ETOL = 0.001 and RCTOL = 0.01 were used. I n t h e f i r s t s t e p a r a t h e r l a r g e number o f i t e r a t i o n s was necessary (19 f o r mesh A and 26 f o r mesh B ) , b u t from t h e second steD onwards an averaqe o f about 4 i t e r a t i o n s p e r s t e p f o r mesh A and 5 i t e k a t i o n s p e r s t e p for mesh B was used. 5.

CONCLUDING REMARKS

The o b . i e c t i v e i n t h s paDer was t o d e s c r i b e c e r t a i n i m p o r t a n t aspects o f o u r c o n t a c t s o l u t i o n a l g o r i t h m and t h u s supplement t h e d e s c r i p t i o n I n t h e paper we focussed on some p h y s i c a l and numerical g i v e n i n r e f . [8]. key aspects o f t h e s o l u t i o n prucedure, and we i l l u s t r a t e d o u r o b s e r v a t i o n s through t h e r e s u l t s - p r e s e n t e d i n d e t a i l - o f some numerical s o l u t i o n s . C o n s i d e r i n g t h i s work,we summarized i n t h e c o n c l u s i o n s o f r e f . [ 8 ] a number o f areas where f u r t h e r research would be very v a l u a b l e . ACKNOWLEDGEMENTS We a r e g r a t e f u l f o r t h e f i n a n c i a l s u p p o r t p r o v i d e d by t h e U.S. and t h e ADINA users group f o r t h i s work.

Army

134

K . 4 Bathe & A . Chaudhary Table 1 S t a t e o f Contactor node as Decided by S t a t e s o f A d j o i n i n g Segments

STATE

OF ADJOINING SEGMENTS

STATE OF NODE

one a d j o i n i n g segment

o t h e r a d j o i n i n g segment

sticking

sticking sliding tension release

sticking

sliding

sliding tension release

sliding

t e n s i o n r e 1ease

tension release

tension release

REFERENCES

[l] A r g y r i s , J.H., D o l t s i n i s , J., Pimenta, P.M. and Wustenberg, H., "Thermomechanical Response o f S o l i d s a t High S t r a i n s - N a t u r a l Approach", J. Computer Methods i n A p p l i e d Mechanics and Engineering, V01.32-34, 1982, pp. 3-57. [2] de Pater, A.D., and K a l k a r J.J. , "The Mechanics o f t h e Contact Between Deformable Bodies", D e l f t U n i v e r s i t y Press, 1975. H a l l q u i s t , J.O. , "A Numerical Treatment o f S l i d i n g I n t e r f a c e s and Impact", Computational Techniques f o r I n t e r f a c e Problems, AMD-Vol American S o c i e t y o f Mechanical Engineers, 1978.

. 30,

Hughes, T.J.R, T a y l o r , R.L., and Kanoknukulchai, W., "A F i n i t e Element Method f o r Large Displacement Contact and Impact Problems", i n Formulations and Computational A l g o r i t h m s i n F i n i t e Element A n a l y s i s , K.J. Bathe e t a l . eds., M.I.T. Press,1977. Campos, L.T., Oden, J.T., and K i k u c h i , N.,"A Numerical A n a l y s i s of a Class o f Contact Problems w i t h F r i c t i o n i n E l a s t o s t a t i c s " , Comp. Meth. i n Appl. Mech. and Eng., Vol. 34, pp. 821-845, 1982. Kalkar, J.J., A l l a e r t , H.J.C., and de Mul, J., "The Numerical C a l c u l a t i o n o f Contact Problem i n t h e Theory o f E l a s t i c i t y " , i n N o n l i n e a r F i n i t e Element A n a l y s i s i n S t r u c t u r a l Mechanics, W. Wunderlich e t a l , eds., S p r i n g e r Verlag, 1981. H a r t n e t t , M.J., "The A n a l y s i s o f Contact Stresses i n R o l l i n g Element Bearings", J. L u b r i c a t i o n Technology, ASME, Vol. 101, pp. 105-109, 1979.

Large Deformation Frictional Contuct Problems

135

[8] Bathe, K.J., and Chaudhary, A.B., " A S o l u t i o n Method f o r P l a n a r and Axisymmetric Contact Problems", I n t . J . Num. Meth. i n Ehgg., i n Press. [9] Bathe, K.J., " F i n i t e Element Procedures i n E n g i n e e r i n g A n a l y s i s " , P r e n t i c e - H a l l , 1982.

K.-J.Bathe & A. Chaudhary

136

PRESCRIBED FORCES ON S /

,

f

A PRIORI CONDITIONS ON CONTACT SURFAC:ES Sc UNKNOWN PRESCRIBED DISPLACEMENTS ON Sd

a)

Condition p r i o r t o contact

CONTACT REGION, NO A P R I O R I KNOWLEDGE OF REG I O N

b)

Condition a t contact

F i g u r e 1 Schematic r e p r e s e n t a t i o n o f two c o n t a c t i n g bodies

Large Deformation Frictional Contact Problems

Figure 2 Contact t r a c t i o n s on actual area o f contact

137

138

K.-J. Bathe & A . Chaudhary

CONTACTOR NODE k + l OVERLAP AT CONTACTOR NODES CONTACTOR NODE k

TARGET BODY

CONTACTOR BODY

(b)

Overlap a t contactor nodes

Figure 3 Schematic representation of overlap between two contacting bodies

CONTACTOR BODY

CONTACTOR SEGMENT j

T\i

' f l I O N

TANGENTIAL DISTRIBUTION TRACTION DISTRIBUTIDN

Figure 4 Normal and tangential t r a c t i o n s onto contractor body. Normal t r a c t i o n i s p o s i t i v e when acting inward t o the body, tangential t r a c t i o n i s p o s i t i v e when acting from node k t o node ( k t l )

Large Deformation Frictional Contact Problems

OVERBURDEN PRESSURE, Po

4

4

E=20.7x107

i

4

4

4

c

c

kPa

E - 1 8 . 4 ~ 1 0 ~k P a v=o .33

Figure 5

Pipe b u r i e d i n s o i l subjected t o t o t a l 100 kPa overburden pressure Po

(3.5m.0)

M.N.O.

FORMULATION

(3.5.3.5)

SEGMENT 3 NODE

I SOPARAMETRIC BEAM ELEMENTS

Figure 6

ELEMENTS

F i n i t e element i d e a l i z a t i o n o f b u r i e d p i p e i n s o i l ; mesh 6

139

K . J . Bathe & A. Chaudhary

140

,M.

Figure 7

Coarse mesh f i n i t e element i d e a l i z a t i o n o f b u r i e d p i p e i n s o i l ; mesh A

,M.N.O.

Figure 8

. . FORMULATION

N 0

FORMULATION

F i n e mesh f i n i t e element i d e a l i z a t i o n o f b u r i e d p i p e i n s o i l ; mesh C

Large Deformation Frictional Contact Problems

1.2

I

141

x MESH A MESH B

@

C

0 MESH

1.0

0.8

0.6

1

,u

=

0.4

/

u=O.25

\

0.2

/ V 0

I

I

15

30

I

/

45

K

MESH B

I

I

60

75

90

ANGLE, 0

Figure 9 Computed tractions at total load along pipe/soil interface in analysis of buried pipe; solution obtained using four equal size increments to total load for each mesh.

1.2 -

t

I T t R A T l O N NO. I

0.11

z

NORPAL 1MCllONI

= L

0.1

-

g%! E; 0.0

P

$ -0.1 23

ef Bl

-0.11

.

TAUGtNTlAL 1RlCTlWS BtiW UPMlllcl

1AHCtNllN

lRAClIM(S A i T t R UPMTIIG

K . 4 Bathe & A. Chaudhary

142

1.2 0.0

0.4 0.0

-0.4 -0.0

1.2

I -

0.0 0.4

-

0.0

-

--

-0.4

-

1.2

-

0.0

I

0.4

-

lTtRlllON NO. 3

30

-0.4

-

90

_____

ITERATION NO. 5 1

M 0.0

60

I

---

I

r -

90

60

___-_ I

Figure 10 Mean t r a c t i o n s , T'/A. and Tj/A.,for mesh B i n the t J n~ i t e r a t i o n s . One s t e p t o t o t a l l o a d and f i v e t e r a t i ons t o convergence.

Large Deformation Frictional Contact Problems

0.0

I-

-0.0

-o'l

I

0.1 0.0

.

[

_-0.1

60

--_---__

-

0.1

F i g . 11

143

144

K.J.Bathe & A. Chaudhary

0.8

0.4

-

1.2

-

M

0.0 -0.4

-

-0.8

-

0.4

L

60

---

90

-r-

I 30

0.0

0.4

iiiluiim m. 5

-1

___

KO

901

1lo

0.0

_.

1 6 0

90

r

Figure 11 Mean t r a c t ons, T j / A . and Tj/A.,for mesh C i n the t J n J i t e r a t i o n s One step t o t o t a l load and seven i t e r a t i o n s t o convergence.

Large Deformation Frictional Contact Problems

TRACTIONS A r l E R ilRST LOU0 INCREHfNI

c

N O R M 1 TRACTlOnS

-0.4

TUNGENT lUL TRACTlONS U i l E R UPOATING

TRACTlONS BEFORE UPDATlnC

t T R U C l l O n S AFTER SECONU LOAO INCREMENT

2.

0.0

-0 8 -0'4

1.2

t

-

-

TRACTIONS UiTER I H l R O LOUD INCREHENT

0.8 /

0.0 0.4

. .. -2-

-

-0.4

-

-0.8

-

L ! O _ -

160

'1

TRACTlONS AFTER FOURTH L M O I I C R E H E I I T

0.8

0 . 0 L -

30

Figure 12 Mean t r a c t i o n s , T i / A j and T i / A j , f o r mesh C a t convergence f o r each load s t e p . Total load applied i n four equal s i z e load increments.

145

146

K. J.Bathe & A. Chaudhary

.

PRESCRIBED DISPLACEMENT OVER ENTIRE FACE

RUBBER MATERIAL c c 2-

12

u1-0.15

I

-

MOONEY-RIVLIN MATERIAL MODEL C,=25.0 C2=7.0

x

(a)

2 urn

Problem considered

.

0

h

1.0

V l r YO

= X L I

8.0

16.0

24.0

32.0

TIME

(b)

Displacement h i s t o r y imposed on r i g h t f a c e o f sheet, At=0.5 Figure 13

Rubber sheet analyzed

,T. L . FORMULATION

(12'3) (12,1)

X-DISPLACEMENT PRESCRIBED OVER THE ENTIRE FACE (15.1.25)

y . CONTACTOR

(-3,-0.25)

Figure 14

TARGET SURFACE SURFACE

F i n i t e element mesh used i n a n a l y s i s o f rubber sheet; mesh

,T.L.

B

FORMULATION X-DISPLACEMENT PRESCRIBED OVER THE E N T I R E FACE

CONTACTOR SURFACE

Figure 15

TARGET SURFACE

F i n i t e element mesh used i n a n a l y s i s o f rubber sheets mesh A

Large Deformation Frictional Contact Problems

l4.Ol

-

MESH A

MESH 8

12.0

t

/

,

.. , ‘tn

TIME 14

6.0 4.0 4.0

tn TIME 8

2.0

-2.0

-4*0

tt TIME 14

t ( a ) A t times 8 and 14

-2.0

-

-4.0

(b)

A t times 18 and 24

Figure 16 Predicted tractions in analysis o f rubber sheet

147

This Page Intentionally Left Blank

Unification of Finite Element Methods H. Kardestuncer (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

149

CHAPTER 6

MIXED VARIATIONAL FINITE ELEMENT METHODS FOR INTERFACE PROBLEMS J. Bielak & R. C MacCarny

This paper presents a procedure for solving interface problems: that is, situations in which different partial differential equations are to be solved in adjacent regions. One of the regions is infinite in extent with homogeneous equations. The other is finite but the equations can be inhomogeneous. The method combines variational finite element methods inside with integral equation methods outside. A prototype situation, that of electromagnetic theory, is discussed.

1.

Introduction.

We present here variational formulations for a class of interface problems. These problems have the following form. Let R be a bounded region in the plane with boundary r and exterior n+. Let p be a smooth, positive function in a and define the second order elliptic operator L by, Lu = (PUxIx + (PUy)y. (1.1) Let q and y be functions defined on f1 and r respectively which are smooth but may be complex. Then given (possibly complex) functions f and g on r and a nonnegative constant Po we seek u (possibly complex) such that: ~ u + q u= o in LL; A U + P 02 u = o in n+ u

-

=

+

u +f,

-

+

yun = u n + g

-

on

r d

u satisfies a radiation condition in u is bounded as 1x1 a, for Po = 0." Here the plus and minus denote limits from call this problem (PP ) .

n+

(1.2)

if and

>

$,

n.

0

We

0

(*)

More generally one can allow A given.

u

Alog 1x1 as 151

-.

a,

150

J. Bielak & R.C. MacCamy

In the last section we indicate how such problems arise in the study of two dimensional electromagnetic fields for various choices of p, q, y and P O . In particular, we indicate that the problem with PO = 0 is of considerable importance. The problem P p o also arises in the study of two-dimensional elastodynamics as discussed in [l]. In all these applications one has, q = ~ 2 , P > o or q = ia2, a > 0 . (1.3) Our object is to give a variational formulation which satisfies two conditions: (1) One has to work only over L2 and r. (2) All boundary conditions are natural. The main problem is, of course, to account for the exterior region. We do this by exploiting the fact that the exterior equation has constant coefficients. Thus we can invoke the ideas of boundary integral equations for exterior problems. The necessary results are collected in section two. In section three we show how to use the results from the exterior problem to find problems which are equivalent to (P ) in which one satisfies (1.2), and has non-local boundary *O conditions connecting u and its normal derivative on r. These transformed problems are then given variational formulations. We obtain a whole family ( V P ) 6 of these variational problems depending on a parameter 6, 0 6 5 1. In sections four and five we discuss the numerical implementation of our variational problems with finite elements. Our methods are not completely new. The ideas bear some resemblance to the use of hybrid methods for interior Dirichlet problems [2]. They have some elements in common with [3] and [ 4 1 and the work in [5] is, in effect, the special case ( V P ) l for a simpler problem. Complete proofs of the results quoted here can be found in [l].

<

2.

The exterior problem.

We consider the equation, 2 Au + P 0u = 0, We put,

Po 1.

(2.1)

0.

where HA2) is the Hankel function of second kind and order is a fundamental solution for (2.1) and gpo zero. Thus g satisfies a railation condition for $0 > 0. Note, however, a. that go(x,y) becomes logarithmically infinite as 1x1 We use to define simple and double layers, S[cpl and 8[cpl with denz!%y cp:

-

15 1

Mixed Variational Methods for Interface Problems

For smooth curves r and functions cp the properties of 8 and B are well known. They satisfy (2.1) in C1 and in li' and the radiation conditions in ll'. We define integral operators S, N and D on r by, XE

r

(2.4)

The kernels Npo and Dpo are continuous while S p 0 has a logarithmic singularity when x =y. Moreover, one has the symmetry properties,

One has, then, the well known limit relations: S [ Y I * = S[VI,

+

an

( =)-

+

&[+I-

=

2 ; ~+ N[+]

=

i

1

~ + vD[Pl.

(2.6)

The layers can be used to obtain representations for solutions of (2.1). We have, for any solution of (2.1) in v If

v

v

=

-

r~iv-1 - S[V,I

n.

in

(2.7)

satisfies a radiation condition in =

+

+

8[vn] - n[vnl

in

ci+

if

The representations in n+ for PO complicated. The result is that if which is bounded at infinity then, v

=

+ -

8[vn]

O[V+]

+

C[v],

>

B, =

0

v

+

n+ we have, (2.8)

0.

is a little more is a solution of (2.1)

Svnds

r

n,

=

0,

(2.9)

where C[v] is a constant. From (2.7) and (2.613 we obtain, for a solution of (2.1) in i l l 1 (2.10) 2 u = D[u-] - S[Un]. Similarly, for PO > 0, and a solution of (2.1) in a+, satisfying the radiation condition, 1 + 2 u

=

+ -

S[U,]

+ I.

D[u

(2.11)

J. Bielak & R. C. MacCamy

152

Finally, for B o , = 0, and a solution of ( 2 . 1 ) bounded at infinity,

in

n+,

Equations (2.10) and (2.11) can be used to establish existence theorems for the solution of Neumann problems. If u,(uA) is specified on r then (2.10) ((2.11)) becomes an integral equation which can be solved for u-(u+).+ Then ( 2 . 7 ) ((2.8)) yields a solution of ( 2 . 1 ) in n ( n ) with the specified normal derivative. There is also a procedure to solve the Dirichlet problem which we will need. This procedure, developed in [61, is as follows. To find a solution of (2.1) in n (n+) with u- ( u + ) specified take , (2.13) u = 8[xl I with x satisfying, - + (2.14) S[Xl = u (u 1 . Then, by (2.612,

, i

= - 1

+ 3 X + "XI.

(2.15)

The method just given has to be modified for the exterior problem with Po = 0. The appropriate procedure is, u

=

S[x]

+

S[x] + C =,'u

C;

f x d s = 0."

(2.16)

r

Remark: There exists a countable infinity of values of Po, m r which the above integral equation methods will fail. We will assume that PO is not one of those values.

3.

The variational problems.

A s a first step in obtaining our variational procedures for the interface problems let us rephrase that problem using the results of section two. Suppose u is a solution of p p o l Po > 0 and put cp = yui. Then we will have u+ = ep-g by (1.2). We substitute this into (2.11) to ogtain.

;u+

(*)

If (Po)

+

D[u-] - S[cp]

=

-S[g].

u is allowed to have the behavior then one replaces 0 here by A .

(3.1)

u

N

A logl&l

in

Mixed Variational Methods for Interface Problems

153

But now we can use the transition conditions again to rewrite (3.1) as,

Thus denotep{%

is equivalent to the following problem which we Find u and cp such that,

PI.

1 We give a variational formulation of 6 . Multiply (3.3) by 7 , integrate by parts and use (3.3) to obtain, (3.4) Then multiply (3.313 by S(:U-+

r

7

and integrate over

Diu-1 - S[cp])Tds

=

r:

SFTds.

r

(3.5)

Our variational problem is then to find u and cp such that (3.4) and (3.5) hold for any (v,JI). For PO 7 0 the method has to be modified. Instead of (3.5) we obtain, S ( ~ U - +DLu-1 - S[Vl

r

+ C)Tds

=

SFTds

r

(3.5')

and we have to add the condition,

Here we have to find u, cp and and (3.6) hold for any (v,$ 1 .

(PO #

C

so that (3.41, (3.5')

Let us give a notation for the problem (3.41, (3.5) 0). Let u = (u,cp), V = (v, $ 1 and put

Then our variational problem is: find U such that forall V, G1 (U,V) = 51 (V) (VP)? Remark: In the applications (see the last section) the functions f and g 9re usually u;flu;fn for some function uo satisfying Au + pou = 0 1 in all spac'e. For such a function we have by (2.101, Z f - D[fI + S[gl = 0. It follows

J. Bielak & R.C. MacCamy

154

+

from ( 3 . 3 ) 3 that F = f = u 0 ’ We obtain a second variational problem by using ( 2 . 1 3 ) ( 2 . 1 5 ) . We use ( 2 . 1 3 ) in n+ with x to be determined. Then from ( 2 . 1 4 ) , ( 2 . 1 5 ) and the transition conditions to obtain, u n = + x + “XI + g, u = S[XI + f on r. (3.8) Then ( P p o ) is equivalent to the problem, denoted by u and x such that, 1 Lu + qu = 0 in Ci, yun = + “XI + g on

zx

-

P o I find

r

u = s[xl + f on r. We again give a variational formulation obtaining,

(3.9)

(3.11)

For

PO

=

0 we have to replace ( 3 . 1 1 ) by,

L,f(U--S[X] -C)Tds = i l f z d s ,

2r

Po >

r

SXds

r

=

(3.11’)

0.

We introduce a notation analogous to ( 3 . 7 ) . 0 , u = (u,x), v = (V,C)

We put, for

(3.12) L

Then ( 3 . 1 0 ) and ( 3 . 1 1 ) are, 0

G (U,V)

0

= 3; ( V )

( V P ) O.

We want to study the form of the variational problems a little more closely. For simplicity let us assume that y = p. In the applications this is usually true or else it can be achieved by changing variables. Let us write ( 3 . 7 ) and ( 3 . 8 ) in the obvious notations,

G0 ( U , V )

= G0 (

(u,X),(v,$))

= All(u,v)

0 (X,V)+ A21 0 (u,$ 1 + A 02 & X , $1 + A12

(3.13)

We want to demonstrate the symmetries here, using ( 2 . 5 ) . we have, by ( 2 . 5 ) 1 ,

First

Mixed Variational Methods for Interface Problems

(3.14)

4 2

Next ,

155

2.512

yields,

Clearly one has the choice of using (VP)l or (VP)O. The advantage of (VP)l is that it yields ui directly as part of the solution. Its disadvantage is that it makes the computation of the external field a little complicated. One must determine from the interior solution, compute u+ and - from theu-transition onditions and then do the two inteun grations in ( 2 . 8 ) . (VP)s yields the external field more readily with the single integration ( 2 . 1 2 ) but requires another integration, ( 2 . 1 5 ) I. to obtain u;. We observe that there is really a whole family of variational problems (VP)6 f 0 < 6 < 1. We simply multiply (VP) by 6 and (VP) by (1 - 6) and add. Then if we put Ir = (u,rP,x), Ir = (v,l,C) and (3.18)

we have the variational problems, find 6 6 G ( b , b ) = 3 (k)

Ir

such that for any b, (VP)a.

One can check that for

6 = 1 / 2 we have the symmetry relation, G 1 / 2 (b, -Ir) = G 1 ' 2 ( b , k ) . (3.19)

We return to this relation in the next sections. An analysis of the problems P1 and Po, as well as the variational problems (VP) and (VP)O, is presented in [ 1 1 , for the case $ 0 > 0. (The case $0 = 0 can be treated similarly.) We review the results briefly. There are some technical conditions. We have indicated in section two that a countable infinity of $,'s must be avoided. Further, if q in ( 1 . 2 ) 1 is real and positive in ri then it could happen that the problem Lu + qu = 0 in n, pun = 0 on r could have non-zero solutions. We assume that n is such that this cannot happen. Then the following facts have been established.

156

J. Bieluk & R.C. MucCumy

1. Suppose f f Hr(r) and g f Hr- (r) for some r 1/2. Then (6")has a unique (generalizeh.) solution (u,cp) with U f Hr+1/2(W u f ~ ~ ( r U; ) ,E H,- (r) If one computes 'u 'a%-'Li mo;:'( the transition conaitions then (2.8) yields a (classical) solution of (1.212 in n,' satisfying the radiation condition and with u f H, loc(n+). The combined function is a (generalized) solution of ( P F ~ ) . 2. Under the same conditions (Po) has a unique generalized solution ( u , ~ ) ,with same regularity; (2.13) yields a solution; of (1.2)2 with the same regularity and the combined function yields a (generalized) solution of (Pp0). 3. For f E , H ~ / ~ ( T and ) g E H- 2 ( r )( v P ) ~( ( ~ ~ 1 0have ) u f H1 (l4 and unique solutions ( u , ~ )( ( u , ~) ) wi[h V(X) E H-1/2(r). Results 1 and 2 are established by using known facts about boundary value problems in to reduce P1 ( P o ) to an equation of Riesz-Schauder type for ~ ( x on ) the space H-1/2(r). Then one can use the uniqueness of solutions of (P o ) to show the homogeneous Riesz-Schauder equations have onfy the trivial solution. In order to prove result 3 one has to establishcoercivity results of the form

.

-

i ) ~+~ I I C P I I ~ (r) The estimates (3.20) can where I I U I=I ~I I U I( t I be established by c nsidering'48e ad joint variational problems for (VP)l and (VP)8, respect'vely. It turns out that because of (3.17) the adjoint of (VP)I ((VP)O) is essentially (VP)O ((VP)l); hence one has a symmetric argument.

4.

Approximate variational problems.

In order to implement the variational problems numerically one introduces finite dimensional approximate spaces. We illustrate with (VP)l; the 0th rs are analogous. According to result 3 in section 3 , (VP)' has a solution (u,cp) f H1 (12) x H - 1 / 2 (r). We introduce families of subspaces, (4.1)

These are to be finite dimen ional and to depend on parameters hfl and hy. We put s h = S'fi x Shy. Then our approximate variational problem is: Find Uh = (uh ,cp h ) E Sh such that for any Vh = (vh , $ h E Sh , (AVP) h h h G ( U ,V ) = 3(V ) . (AVP)' is equivalent to sets of algebraic equations. Let

157

Mixed Variational Methods for Interface Problems

h

(W~,...,W

h

h

I

h

(41,...,QN ) be bases for

uh

hr.

and

S

N

Nhii =

z

h h

h

UiWiI q

=

i=l

kllii

L1

hr

LA Nh,. Then we have

and (AVP)

S

hr h h z rp.4. i=l 1 1

(4.2)

is equivalent to the algebraic equations h h h h +

~~~2~ = 2; &21ii

sh E

N

1R hL' I

(4 h )i

&222 =

+

a

(4.3)

(4.4) =

SFqi hds P

L

and the matrices are determined by

We shall say a little more about numerical implementations in the next section. Here we want to review some further theoretical results from [l]. The results requ're the fo lowing approximation properties of the spaces s'rl and SAT: (A.1) There exists a constant 1 , and an inteaer k such'that for any w E H 4 ( l l ) I 1
(A.2)

There exists a constant y 2 b E H&I (r),- 1 / 2

s ch th t for any ?i E S with

> 0 and a & I

kl

k1

>

>

1

1/2

there is a

r'$

The following results are established in [l]. Put h = hn + hr- Then if h is sufficientxy small: (1) Equations (4.5) have a unique solution. (2) Suppose U = {u,rp] is the solution of (VP)l and

J. Bielak & R.C.MacCamy

158

1

E H - 1 1 +€(TI with E < min(k,k 1 in (A.l) and 7A.2). $hen there exisrs a constant dent of h such that,

u E H1+~(12),

c, indepen-

Ashan example of the meaning of the above result one can take s l2 to consist of piecewise linear functions in R (k = 2) land Shy to consist of piecewise constant functions on L' (k = 1). Suppose then that the solution of (VP) has u E H2(n) and cp E H1/2(r) ( € = 1). Then take E = 1 in (4.6) and get (4.7)

Thus we obtain order h convergence in the natural n rm for (VP)l. One can also show that there is a constant c independent of the choice of h such that

P

The proofs of the above results proceed in several stages. One shows f'rst that the coercivity result{ (3.20) hold for any Uh E S' when V is restricted to V This is done by first using (3.20) to get a V E S which makes the inequalities valid and then using regul rity results to show that V can be approximated with a V' E Sh. These coercivity results on Sh enable one to establ'sh optimality; that is, to show that U is approximated by ,'U in the natural norm, as well as it is ossible to approximate U, in that norm, by elements of S Then one invokes (A.l) and (A.2). The L estimate (4.8) is obtained via the Aubin-Nitsche tricz.

.

R.

5.

Implementation of the numerical procedure.

For purposes of illustration we now discuss the actual implementation of the finite element method described in the preceding section for the case in which n is the unit circle and f and g are given by u$ and U6,nr respectively, where uo represents a harmonic incident plane wave field. The material in ,Li is homogeneous and uo is symmetric with respect to a diameter. The problem has been solved exactly in [ 7 1 . We divide the region R into circular sectors, with wedges around the origin, and consider a piecewise linear approximation for u in the polar coordinates r and 8 ; cp is taken to be a piecewise constant function on r. With this approximation the elements of the matrices 8 1 1 and Al2 and the load vector can be evaluated explicitly by direct integration. For WZ1 we integrate numerically with

sh

Mixed Variational Methods for Interface Problems

159

standard Gauss-Legendre formulas since the kernel in (2.4) that enters into the bilinear form Azl in ) Q ! : ( is continuous. Due to the logarithmic singularity of gPo which appears in the bilinear form A22 we use a modified Gauss-Legendre formula [ 8 1 that accommodates this singularity explicitly in evaluating the elements of &2. Nhi, X Nhn matrix with the elements indicated inA1h. iil;n and $21 are Nhn Nhy and Nhy Nhid, respectively, and $22 Nh The last three matrices will be full. will be NhT However, since tf;e diameters of the elements inside l2 and the lengths of the intervals on l' are about the same and of size h, then we see that Nh = N2 Thus, although the matrices $12, $21 and 822 fg11 their size is much smaller than that of $11 and an effective numerical procedure is still possible. Remarks: (i) Note that while $11 and $22 are symmetric matrices, $12 and f?21 are not generally the transpose of each other. Therefore, the system (4.3) is, in general, asymmetric. (ii) The form of equations (4.3) permits condensation. Suppose we are primarily concerned with the interior region. Then we may eliminate gh and consider the system,

.

.

sh

where BJ~ = -1/23?123322$321 and = -A12Aj$fh. The matrix B has nonzero elements only for nodes on t e boundary r. Fkus it represents the impedance of the exterior region n+, and constitutes, in effect, a discretized nonlocal absorbing represents the corresponding effective forcing boundary. function. Although 211 is in general an asymmetric matrix it turned out to be symmetric for the present problem. (iii) The condensation procedure requires that the matrix $22 be inverted; thus, it is not valid for values of Po for which the operator S[cpl in (2.4) cannot be inverted. Direct solution of the complete system (4.3)1, however, was possible for values of PO approaching these critical values. (iv) Other condensation schemes are clearly possible; see [ l ] for a scheme that is applicable if one is mainly concerned with the exterior region. (v) A symmetric discretized formulation of the general problem P is always possible by using the variational formulation VP1i2 in view of the symmetry relationship (3.19) provided one chooses real basis functions. The price we pay for this symmetry is that the system (4.3) is replaced by a set of similar structure of Nhn + 2Nhy equations instead of the Nhn + Nhy in (4.3). See [l] for details. After condensation, however, the corresponding system leads to equations of the form,

sh

wnich is similar to ( 5 . 1 ) and is clearly symmetric. the corresponding effective forcing function.

Lh

is

J. Bielak & R.C MacCamy

160

A comparison between the exact and approximate values of u at the center of the circle is shown in Table 1 for different values of Nhy (Nhn = N i ) for several combinations of the system parameters. The rresults tend to confirm our theoretical estimates that the convergence is of order h2 for the elements used.

6.

Two dimensional electromagnetic problems.

Many of the problems of electromagnetic problems can be idealized in the following way. One has a field everywhere in space, which we think of as filled with air. One introduces dielectric or metallic obstacles and seeks to determine both the fields induced in the obstacles and the distortion of the original field outside. This is an interface problem: one has different sets of Maxwell's equations in air and in the obstacles and transition conditions across the boundaries. The above problems are usually considered for the case of time-periodic fields of a single frequency and this is the case we consider. (By taking inverse Fourier transforms one can, in principle, solve time dependent problems from the periodic case.) The variables to be determined are the electric and magnetic fields t?. and 3. These satisfy Maxwell's equations. We write down theze equations when the material in question is either a dielectric or non-ferromagnetic metal in the time periodic case with frequency w. Further, we render the position variables x non-dimensional by dividing by a representative lengtf; a. The equations are: curl

2=

iwpa2,

curl

3

= KE,

where

K

-iwEa

=

for dielectrics,K

=

Ua

for metal.

(6.2)

Let us first do some scaling in the problem. Air is a dielectric with permeabilities p ~ €0. , We introduce dimensionless fields ;and 2 by writing, 8

H

=

hog,

$

=

iwgOahOg.

(6.31

Then (6.1) becomes ,

where k

2

= w l o€ a 2 =

P2

for dielectrics, k

=

icupoUa2

=

ia2

for metal.

161

Mixed Variational Methods for Interface Problems

The parameters k, a and $ are dimensionless. We allow for the possibility that the obstacles are inhomogeneous so that k , a and $ can depend on position. We can now describe the interface problem. Let n denote the obstacle regi n and d its exterior. Then in ll+ we have p = p o l k = = ui2a2C(OE0. Thus we have, curl

3

=

8,

curl

;=

P 20-E in

ri+

curl

3

=

c”)#,

curl

#

kg

fA

=

in

MO

.

r

The transition conditions across = an are that the tangential components of g and 2 are continuous, that is, if 2 is the normal to r:

-

n x g + = s x g ;

N

n x z + = n x g -

on

rl

(6.7)

n+ and

where the plus and minus denote limits from

n.

The problem is to be drive? by an incident field E 0 ,# 0 The differences 5 50- and satisfying (6.6)1 in all space. H -Zo are to satisfy a radiation condition. If we let E N and H represent the scattered fields in n+ then we sty11 have gquations (6.6) but (6.7) is replaced by,

-

n x g - = g x g+ + 2 x g 0;

n x g + = n x x - + n x g0 on

rl

N

with

3

and

2

(6.8)

satisfying a radiation condition.

We now specialize the geometry. We suppose that the obstacles consist of cylinders of uniform cross section n parallel to the z-axis. Then we limit ourselves to fields E, # (and which depend only on x and y, not z. N It can be shown that all such fields are combinations of fields of the following type: A 1 A 2 A Transverse magnetic (TM): 2 = E(x,y)k; 5 = H (x,y)i + H (x,y)j Transverse electric (TE):

5

=

H(x,j)t;

3

1

A

= E (X,j)l

h + E2 (x,j)].

(6.9)

Let us determine the structure of such fields. (TM) We have, A A p IA 2h H2 - H1 = kE. i+H J), E 1 - Ex] = -(H PO X Y Y

(*)

(6.10)

One can allow E o and go to have singularities in In fact, when = 0 one must have singularities in non-trivial problem. order to obtain

T2

n+.

162

J. Bielak & R.C. MacCamy

We introduce a function u

u

by the formulas, 2 u =

HIr

=

X

Po

(6.11)

PO

Then (6.1011 is satisfied if E = u if PO u ~ + )(-PO~ U (T

cc

and (6.lOI2 is satisfied )

Y Y

=

-ku.

(6.12)

Converse1 if u satisfies (6.12) and we put E = u and define Hy: H2 by (6.11) we have a solution of (6.10). Observe that onthe surface of our cylinder we have, for fields

TM

(6.13) where T and v are the unit tangent and normal to the boundary of n -in the x-y plane. ( T E ) We have, E:

-

E1 =

h!- h, Po

This time we define

u

uY = k E 1,

A A 10 20 H i - Hxj = k ( E i + E 1 ) . Y

(6.14)

by, LI

X

=

-kE

2,

H=u,

(6.15)

so that (6.16) Instead of (6.13) we have, (6.17) We can now obtain four different problems, all fitting the framework discussed in the preceding sections. We again let n+ denote the exterior of n in the x-y plane. The exterior region is air.

-

TM fields.

(I)

Dielectric cylinder

(11)

Dielectric cylinder - TE fields.

Mixed Variational Methods f o r Interface Problems

-

u

163

+ + uo/ +

u

=

(111) Metallic cylinder

uxx

+

u YY

-

u (IV)

=

=

-8,2 in

+ + uo; +

u

Metallic cylinder 3

-

u

=

+ + uo, +

u

Remarks: 1. For any materials except ferromagnetic ones there is only a small variation in p. Hence it is not a bad approximation to assume p/po = 1. 2. Although the theory can be carried through for any choice of the parameters there are really only two important cases. At low (say, 60 cycle) frequencies the parameter $ for a dielectric is very small while the parameter a for metals is O(1). At higher frequencies, say w = 0 ( 1 0 1 0 ) the parameter B is O ( 1 ) but the parameter a is very large. Thus the dielectric problems are both meaningful at higher frequencies. For the metallic cylinder problem at low frequencies one can, with small error, put $ - 0. This is what is usually done with a statement that ong 'neglects displacement current in air". This is the origin of our problem (PO). At higher frequencies the usual approximation is that the metal has "infinite conductivity" in which case one simply solves an exterior boundary value problem with equal to zero on the obstacle. Thus problems I11 and -tang e IV are really meaningful only if Po = 0.

Acknowledgement. This work was supported by the National Science Foundation under Grants CEE-8210859 (J.B.) and MCS-8219675 (R.C. MacC.).

T-able 1.

Relative Displacement at Origin

N

Re 1/2

.25n .50n n

2

.25n -50s n

hr

Im

Re

Exact

20

10

C

Im

Re

Im

Re

Im

0.8807 0.5327 0.1343

0.1421 0.0249 -0.2314

0.8802 0.5335 0.1367

0.1401 0.0225 -0.2301

0.8800 0.5334 0.1369

0.1398 0.0222 -0.2295

0.8800 0.5333 0.1368

0.1398 0.0222 -0.2293

1.8209 -0.5755 -2.3700

-0.0394 1.7429 -0.5224

1.8051 -0.5650 -1.8809

-0.0419 1.7012 -0.6694

1.8003 -0.5619 -1.7531

-0.0421 1.6856 -0.6850

1.7983 -0.5606 -1.7050

-0.0420 1.6785 -0.6859

Mixed Variational Methods for Interface Problems

165

References [l]

Bielak, J. and MacCamy, R.C., An exterior interface problem in two-dimensional elastodynamics, Quart. of Appl. Math. 41 ( 1 9 8 3 ) 1 4 3 - 1 6 0 .

[2]

Fix, G.J., Hybrid finite element methods, in: Noye, John (ed.), Numerical Simulation of Fluid Motion (North-Holland, Amsterdam, 1 9 7 8 ) .

[3]

MacCamy, R.C. and Marin, S.P., A finite element method for exterior interface problems, lnt. Jrnl. Math.and Math.Ana1. 3 ( 1 9 8 0 ) 3 1 1 - 3 5 0 .

[4] Aziz, A.K. and Kellogg, R.B., Finite element analysis of a scattering problem, Math. of Comp. 3 7 ( 1 9 8 1 ) 261-272. [5]

Johnson, C. and Nedelec, J.C., On the coupling of boundary integral and finite element methods, Math. of Comp. 3 5 ( 1 9 8 0 ) 1 0 6 3 - 1 0 7 9 .

[6]

Hsiao, G. and MacCamy, R.C., Solutions of boundary value problems by integral equations of the first kind, SIAM Review 1 5 ( 1 9 7 3 ) 6 8 7 - 7 0 5 .

[7]

Trifunac, M.D., Surface motion of a semi-cylindrical alluvial valley for incident plane SH waves, Bull. Seism. SOC. Am 6 1 ( 1 9 7 1 ) 1 7 5 5 - 1 7 7 0 .

[8]

Harris, C.G. and Evans, W.A.B., Extension of numerical quadrature formulae to cater for end point singular behavior over finite intervals, Int. J. Comp. Maths. 6 B (1977)

219-227.

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Unification of Finite Element Methods H. Kardestuncer (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

167

CHAPTER 7 PRECONDITIONED ITERATIVE METHODS FOR NONSELFADJOINT OR INDEFINITE ELLIPTIC BOUNDARY VALUE PROBLEMS J.H. Bramble & J.E. Pasciak

We c o n s i d e r a G a l e r k i n - F i n i t e Element a p p r o x i m a t i o n t o a general l i n e a r e l l i p t i c boundary v a l u e Droblem which may be n o n s e l f a d j o i n t o r i n d e f i n i t e .

\Je

show how t o p r e c o n d i t i o n t h e equations s o t h a t t h e r e s u l t i n g systems o f l ' i n e a r a l g e b r a i c equations l e a d t o i t e r a t i o n procedures whose i t e r a t i v e convergence r a t e s a r e independent o f t h e number o f unknowns i n t h e s o l u t i o n .

1.

INTRODUCTION. I n r e c e n t y e a r s , t h e a p p l i c a t i o n o f i t e r a t i v e methods t o

p r e c o n d i t i o n e d l i n e a r systems has been e x t r e m e l y s u c c e s s f u l i n a v a r i e t y of complex p h y s i c a l a p p l i c a t i o n s [3,16].

Many a r t i c l e s a r e a v a i l a b l e

i n t h e l i t e r a t u r e which r e p o r t on t h e f a v o r a b l e performance o f such methods [3,6,10,12]. The two aspects o f a r e s u l t i n g a l g o r i t h m c o n s i s t o f t h e p r e c o n d i t i o n e r and t h e u n d e r l y i n g i t e r a t i v e method [l ,8,12].

Various

i t e r a t i v e methods, t h e most p o o u l a r b e i n g t h e c o n j u g a t e g r a d i e n t (CG) and c e r t a i n normal forms o f t h e CG method, have been c o n s i d e r e d e x t e n s i v e l y b o t h f r o m a t h e o r e t i c a l and an experimental v i e w p o i n t (see references t h e r e i n ) .

[lo]

I t has been demonstrated t h a t , i n general

and t h e

,

i t e r a t i v e a1 g o r i thms w i t h t h e same t h e o r e t i c a l convergence r a t e s

J.H. Bramble & J.E. Pasciak

168

converge, i n p r a c t i c e , a t about t h e same r a t e ' .

The q u e s t i o n o f choosing

an a p p r o p r i a t e p r e c o n d i t i o n e r i s much more d i f f i c u l t .

The

p r e c o n d i t i o n e r must i n some way be s i m i l a r t o t h e i n v e r s e o f t h e system which i s b e i n g solved.

Consequently, t h e e v a l u a t i o n o f t h e

p r e c o n d i t i o n e r u s u a l l y r e q u i r e s t h e s o l u t i o n o f a system o f equations and

s o i f t h e method i s t o r e s u l t i n an improvement o f computational e f f i c i e n c y , t h e p r e c o n d i t i o n e r must have some p r o p e r t y which makes i t e a s i e r t o s o l v e than t h e o r i g i n a l system.

The i t e r a t i v e convergence

r a t e o f t h e a l g o r i t h m i s extremely s e n s i t i v e t o t h e choice o f Indeed, t h e c h o i c e o f a more a n n r o o r i a t e

preconditioner.

p r e c o n d i t i o n e r may reduce t h e number o f i t e r a t i o n s by an o r d e r o f magnitude o r more i n a g i v e n problem. I n t h i s paper we i l l u s t r a t e some techniques f o r a n a l y s i n g p r e c o n d i t i o n e d i t e r a t i v e methods f o r nonsymmetric problems.

We w i 11

discuss t h e problem o f choosing an a p p r o p r i a t e p r e c o n d i t i o n e r and study two d i f f e r e n t i t e r a t i v e a l g o r i t h m s .

T y p i c a l f i n i t e element

d i s c r e t i z a t i o n o f an e l l i p t i c boundary value DrOblem leads t o a m a t r i x problem (1.1) where

Mc = d. M

i s the " s t i f f n e s s " matrix associated w i t h the d i s c r e t i z a t i o n

and i s n o n s i n g u l a r and

d

and

Mi;' such t h a t

preconditioner (M1)-l

c

are vectors.

M1

i s symmetric p o s i t i v e d e f i n i t e ,

i s e a s i e r t o compute than

i n some sense"

(M)-'.

We seek a

(M)-',

and

(M1)-'

"approximates

System ( 1 . 1 ) can o f course be r e p l a c e d by t h e

e q u i v a l e n t system (1.2) The m a t r i x

Mt M i ' M i ' Mc = Mt M i '

M'

f

M

t

M i ' 'M;

M

M-i

d

.

i s symmetric p o s i t i v e d e f i n i t e and t h e

f i r s t a l g o r i t h m i s d e f i n e d by a o p l y i n g t h e conjugate g r a d i e n t method t o (1.2). (1.3)

A l t e r n a t i v e l y , (1.1) i s e q u i v a l e n t t o t h e problem -1

t

-1 d.

Mi;' Mt M i ' Mc = M1 M M1

The number o f i t e r a t i o n s t o reach a d e s i r e d accuracy may vary by a t most a f a c t o r of f i v e [6,10].

Nonselfadjoint or Indefinite Elliptic Boundary Value Problems

169

M" : M - l Mt MY;' M a l t h o u g h n o t u s u a l l y symmetric, i s a 1 symmetric o p e r a t o r w i t h r e s p e c t t o t h e i n n e r p r o d u c t defined by

The m a t r i x

I n e CG method can be a p p l i e d t o ( 1 . 3 ) i n t h e and leads t o A l g o r i t h m I 1 o f S e c t i o n 2.

,.>>

i n n e r product

Our a n a l y s i s suggests t h a t t h e

p r e c o n d i t i o n e d i t e r a t i v e method based on (1.3) i s more r o b u s t t h a n t h a t based on ( 1 . 2 ) s i n c e r e s u l t s f o r (1.2) r e q u i r e a d d i t i o n a l hypotheses.

In

f a c t , we have n o t been a b l e t o o b t a i n r e s u l t s f o r t h e scheme based on (1.2) unless t h e elements used i n t h e methods a r e o f " q u a s i - u n i f o r m " size. We s h a l l p r e s e n t two general theorems which can be used t o d e r i v e c e r t a i n d i s c r e t e s t a b i l i t y estimates.

Such e s t i m a t e s l e a d t o bounds on

t h e i t e r a t i v e convergence r a t e s o f a l g o r i t h m s f o r f i n d i n g t h e s o l u t i o n o f m a t r i x equations r e s u l t i n g from t h e f i n i t e element d i s c r e t i z a t i o n o f e l l i p t i c boundary v a l u e problems which may be nonsymmetric and/or indefinite.

We show how these general r e s u l t s can be a p p l i e d i n a

f i n i t e element a p p r o x i m a t i o n t o t h e Poincar;

problem.

Both s t r a t e g i e s

depend upon a p r i o r i s t a b i l i t y e s t i m a t e s f o r t h e continuous problem and use t h e a p p r o x i m a t i o n p r o p e r t i e s o f t h e d i s c r e t i z a t i o n t o d e r i v e t h e s t a b i l i t y e s t i m a t e f o r t h e matr-ix problems. The f i r s t theorem leads t o a s t r a t e g y which uses a p o s i t i v e d e f i n i t e symmetric problem as a p r e c o n d i t i o n e r f o r a more c o m p l i c a t e d nonsymmetric and/or i n d e f i n i t e problem.

The problem o f t h e e f f i c i e n t

s o l u t i o n o f p o s i t i v e d e f i n i t e problems, a l t h o u g h n o t c o m p l e t e l y s o l v e d , has been e x t e n s i v e l y researched.

For example, m a t r i c e s corresponding

t o p o s i t i v e d e f i n i t e symmetric problems o f t e n have c e r t a i n diagonal dominance p r o p e r t i e s which i m p l y t h a t v a r i o u s sparse m a t r i x packages [9,11]

can be used f o r t h e i r s o l u t i o n .

Also, t h e r e a r e " f a s t s o l v e r "

a l g o r i t h m s a v a i l a b l e f o r c e r t a i n e l l i p t i c oroblems on a v a r i e t y o f domains [5,14,15].

Our a n a l y t i c a l r e s u l t s guarantee t h a t t h e i t e r a t i v e

convergence r a t e f o r o u r a l g o r i t h m s i s independent o f t h e number o f unknowns i n t h e system.

Thus t h e c o s t o f convergence t o a g i v e n

accuracy grows l i n e a r l y w i t h t h e s i z e o f t h e problem. The f i r s t s t r a t e g y i s a p p l i c a b l e t o , f o r example, problems where the d i f f e r e n t i a l operator d e f i n i t e operator

L

A

can be decomposed i n t o a symmetric p o s i t i v e

and a compact ( b u t n o t s m a l l ) p e r t u r b a t i o n

B.

The

J. H. Bramble & J. E. Pasciak

170 operators

A, L, and B

a r e approximated by d i s c r e t e o p e r a t o r s

Ah,

and Bh d e r i v e d by f i n i t e elements. The d i s c r e t e approximation Lhy t o t h e s o l u t i o n u o f t h e o r i g i n a l problem i s d e f i n e d as t h e s o l u t i o n of (Lh

(1.4)

-+ B h ) U

= F.

Problem ( 1 . 4 ) can be r e p l a c e d by t h e e q u i v a l e n t problem Lhl (Lh t Bh)U = Lh’ F

(1.5)

,

We d e r i v e t h e a p p r o p r i a t e s t a b i l i t y e s t i m a t e s f o r ( 1 . 5 ) which guarantee t h a t t h e CG method a p p l i e d , w i t h r e s p e c t t o

<<*,*>>,

t o (1.3) converges

a t a r a t e independent o f t h e number o f unknowns i n t h e d i s c r e t i z a t i o n . a d d i t i o n , t h e s t a b i l i t y r e s u l t s y i e l d immediately e s t i m a t e s f o r t h e discretization error

In

u-U.

We g i ve a second theorem which , under a d d i t i o n a l hypotheses, provides another s t a b i l i t y estimate.

T h i s e s t i m a t e , under a f u r t h e r

r e s t r i c t i o n , can be used t o show t h a t t h e CG method a p p l i e d t o (1.2) converges t o t h e s o l u t i o n o f (1.2) a t a r a t e which i s independent o f t h e number o f unknowns i n t h e d i s c r e t i z a t i o n . An o u t l i n e o f t h e remainder o f t h e paper i s as f o l l o w s .

I n Section 4

we d e s c r i b e two conjugate g r a d i e n t a l g o r i t h m s f o r m a t r i x problems. S e c t i o n 3 g i v e s some p r e l i m i n a r i e s and n o t a t i o n t o be used i n t h e paper. I n S e c t i o n 4 we s t a t e t h e t y p e o f e s t i m a t e s needed t o guarantee r a p i d convergence f o r some it e r a t i ve methods f o r s o l v i ng nonsymmetri c and/or i n d e f i n i t e problems.

Two theorems used t o d e r i v e t h e s t a b i 1 ity estimates

a r e given i n S e c t i o n 5.

I n S e c t i o n 6 we apply t h e theorems t o a f i n i t e

element approximation o f a general e l l i p t i c boundary value problem. F i n a l l y i n S e c t i o n 7 we a p p l y a s t a b i l i t y e s t i m a t e t o bound t h e discretization error.

2.

CONJUGATE GRADIENT ALGORITHMS.

We d e s c r i b e t h e a l g o r i t h m s which r e s u l t from a p p l y i n g t h e conjugate g r a d i e n t method t o t h e p r e c o n d i t i o n e d sqstems (1.2) and ( 1 . 3 ) . I n e i t h e r t o the co o f (1.1) and t h e i t e r a t i v e a l g o r i t h m produces a sequence o f

case we assume t h a t we a r e g i v e n an i n i t i a l a p p r o x i m a t i o n solution

c

Nonselfadjoint or Indefinite Elliptic Boundary Value Problems

iterates

ci

for

residual error

i > 0.

d-Mc

We s t o p t h e i t e r a t i v e procedure when t h e

becomes s u f f i c i e n t l y s m a l l .

We n o t e t h a t a p p l y i n g

t h e c o n j u g a t e g r a d i e n t method t o p r e c o n d i t i o n e d systems as

i11 u s t r a t e d in t h e f o l 1owing a1 g o r i thms i s n o t novel however we in c l ude t h e d e t a i l s f o r completeness. A p p l y i n g t h e c o n j u g a t e g r a d i e n t method t o (1.2) g i v e s t h e f o l l o w i n g algorithm: M I = Mt Mi;' M i ' M

ALGORITHM I . (1)

Define

(2)

For

t -1 M-l ( d-Mco). ro = po = M M1 1

i> 0

define

ri o pi a i = (MI p i l o p i = c:

Cit1

1

+ a. p

i i

A p p l y i n g t h e c o n j u g a t e g r a d i e n t method i n t h e p r o d u c t t o (1.3) g i v e s t h e f o l l o w i n g a l g o r i t h m : M" = M i ' Mt MY1 M

ALGORITHM 11. (1)

Define

(2)

For

.

ro = po = M i ' Mt Mi1(d-Mco).

i> 0

define

ci+l

= ci

+ a. 1 pi

ri+l

= ri

- a i M"

t~~

<<-,*>>

inner

171

J.H. Bramble & J. E. Pasciak

172

3.

PRELIMINARIES AND NOTATION. Throughout t h i s paper we s h a l l be concerned w i t h s o l v i n g boundary

value problems on a bounded domain

.

r

boundary

n

contained i n

R2

with

To s t a t e o u r s t a b i l i t y estimates, we s h a l l make use o f

v a r i o u s spaces o f f u n c t i o n s d e f i n e d on

R

.

The space

L2(R)

i s the

c o l l e c t i o n o f square i n t e g r a b l e f u n c t i o n s on R ; t h a t is,a f u n c t i o n defined f o r (x,y) i n R i s i n L2(R) i f

The

LL(R)

f(x)

i n n e r p r o d u c t i s d e f i n e d by

(f,g)

: f(x,y)

g(x,y)dxdy

for

f, g E

2 L (Q).

R 1 We s h a l l a l s o use t h e Sobolev space H (n). Loosely, a f u n c t i o n af af 2 f, and - a r e a l l i n L (R). Thus f o r i n H1(R) i f aY 1 f u n c t i o n s i n H (n), we can d e f i n e t h e D i r i c h l e t f o r m by

We s h a l l a l s o denote t h e E

r

L

2

(r)

f g ds

f

is

i n n e r p r o d u c t by

. 2

r, t h e Sobolev space o f L ( a ) - f u n c t i o n s 2 rth o r d e r p a r t i a l d e r i v a t i v e s belong t o L ( Q ) w i l l be denoted by

For any p o s i t i v e i n t e g e r whose Hr( 0). We a l s o l e t values o f and

Ci

C

and

and

C Ci

Ci

for

i> 0

denote p o s i t i v e c o n s t a n t s .

may be d i f f e r e n t i n d i f f e r e n t places however

s h a l l always be independent o f t h e mesh parameter

h

The C

defining

173

Nonseljadjoint or Indefinite Elliptic Boundary Value Problems t h e a p p r o x i m a t i o n method.

Thus

C

and

Ci

w i l l always be independent

of t h e number o f unknowns i n t h e d i s c r e t i z a t i o n . To d e f i n e t h e a p p r o x i m a t i o n o f l a t e r s e c t i o n s we s h a l l need a c o l l e c t i o n o f f i n i t e element a p p r o x i m a t i o n subspaces { S h l , 0 < h( 1, 1 c o n t a i n e d i n H (R). T y p i c a l l y , f i n i t e element a p p r o x i m a t i o n subspaces

Q i n t o subregions o f s i z e h and t o be t h e s e t of f u n c t i o n s which a r e continuous on R and

a r e d e f i n e d b y p a r t i t i o n i n g t h e domain defining

Sh

piecewise p o l y n o m i a l when r e s t r i c t e d t o t h e subregions (see [4,7,17] for details). o f size on

h

and d e f i n e

Sh

t o be t h e f u n c t i o n s which a r e continuous

and l i n e a r on each o f t h e t r i a n g l e s .

52

R i n t o triangles

For example, one c o u l d p a r t i t i o n

be p a r t i t i o n e d i n t o r e c t a n g l e s and f u n c t i o n s which a r e continuous on

Sh

R could

Alternatively,

c o u l d be defined t o be t h e

R and b i l i n e a r on each o f t h e

rectangles.

4.

ESTIMATES FOR THE CONJUGATE. GRADIENT METHOD. Our a n a l y s i s o f i t e r a t i v e a l g o r i t h m s f o r p r e c o n d i t i o n e d systems i s

based on s t a b i l i t y e s t i m a t e s f o r t h e continuous o r n o n d i s c r e t e problem and t h e e r r o r e s t i m a t e s between t h e continuous s o l u t i o n s and t h e i r d i s c r e t e approximations.

To s t u d y t h e p r o p e r t i e s o f t h e s o l u t i o n s o f

boundary v a l u e problems i n p a r t i a l d i f f e r e n t i a l equations, i t i s natural t o consider operators i n t h e i r basis f r e e representations since complete s e t s o f b a s i s f u n c t i o n s a r e u s u a l l y t o o complex t o be o f much p r a c t i c a l value.

Consequently, i t i s n a t u r a l t o t h i n k o f t h e process

o f s o l v i n g f o r t h e d i s c r e t e s o l u t i o n o f t h e f i n i t e element equations as a b a s i s f r e e o p e r a t o r on t h e f i n i t e element subspace

Sh

of

H'(R)

r e p r e s e n t d i f f e r e n t i a l and s o l u t i o n o p e r a t o r s by t h e n o t a t i o n L,

or

T

denoted

.

We

A, B,

whereas t h e i r d i s c r e t e c o u n t e r p a r t s s h a l l be r e s p e c t i v e l y

Ah, Bh, Lh

and

Th.

The CG method can be a p p l i e d t o f i n d t h e s o l u t i o n

X

o f t h e problem

Lhx=Y where

L,,

i s a symmetric p o s i t i v e d e f i n i t e o p e r a t o r w i t h r e s p e c t t o some

inner product (cf.

L13J).

produces an a p p r o x i m a t i o n

The CG a l g o r i t h m r e q u i r e s an i n i t i a l guess X

n

to

X

after

n

i t e r a t i v e steps.

X,

It i s

and

114

J. H. Bramble & J. E. Pasciak

w e l l known t h a t

where

i s t h e c o n d i t i o n number f o r

y

r a t i o o f t h e l a r g e s t eigenvalue o f i f Lh

where

L

Lh h

and i s d e f i n e d t o be t h e

t o the smallest.

We n o t e t h a t

s a t i s f i e s the i n e q u a l i t y

(*,-),

number y

denotes t h e H - i n n e r product, then t h e c o n d i t i o n

i s bounded by

Thus estimates o f t h e t y p e (4.3)

Cl/Co.

i n c o n j u n c t i o n w i t h (4.2) l e a d t o convergence e s t i m a t e s f o r t h e CG method a p p l i e d t o (4.1). The problem o f f i n d i n g t h e f i n i t e element s o l u t i o n i n t h e examples X

of a

Sh.

We

o f l a t e r s e c t i o n s can be reduced t o s o l v i n g f o r t h e s o l u t i o n nonsingular operator equation (4.4)

AhX=Y Ah

where

i s a nonsymmetric and/or n o n p o s i t i v e o p e r a t o r on

s h a l l f i r s t p r e c o n d i t i o n t h e system, m u l t i p l y by t h e a d j o i n t and t h e n a p p l y t h e CG method i n t h e a p p r o p r i a t e i n n e r p r o d u c t . We assume t h a t we have a symmetric p o s i t i v e d e f i n i t e o p e r a t o r Th

d e f i n e d on

Sh

f o r a preconditioner.

The types o f p r e c o n d i t i o n e r s

f o r which we can g e t a n a l y t i c r e s u l t s w i l l be d e s c r i b e d i n l a t e r s e c t i o n s We note t h a t problem (4.4) can be r e p l a c e d by t h e problem o f finding X i n S satisfying

A; Th Th Ah X =

(4.5) where t o the

A;

i s the L

2

(a)

" LL(a)

At

Th Th

- adjoint

of

Ah.

The CG method w i t h r e s p e c t

i n n e r p r o d u c t can be used t o s o l v e (4.5).

The

convergence r a t e o f t h e r e s u l t i n g a l g o r i t h m i s bounded by (4.2) i n 2 t h e L (a) norm where Y i s bounded by Cl/Co f o r any Co and C1

satisfying

175

Nonselfadjoint or Indefinite Elliptic Boundary Value Problems

I n c e r t a i n a p p l i c a t i o n s , e s t i m a t e ( 4 . 6 ) can be used t o d e r i v e bounds on t h e i t e r a t i v e convergence r a t e o f A1 g o r i thm I. A l t e r n a t i v e l y , problem ( 4 . 4 ) i s a l s o e q u i v a l e n t t o t h e problem o f finding

i n Sh

X

satisfying

T A*T A X = T A*T Y . h h h h h h h

(4.7) The o p e r a t o r inner product

B

?

T A* T A h h h h

( T i ' W, V ) .

i s symmetric p o s i t i v e d e f i n i t e i n t h e

A p p l y i n g t h e CG method t o t h e s o l u t i o n o f

(4.7) i n t h i s i n n e r p r o d u c t g i v e s an a l g o r i t h m which converges a t a r a t e d e s c r i b e d by (4.2) where CO(Til W,W)

(4.8)

f o r any

y < C1/Co

f (Th AhW, AhW)

5

Cl(Thl

Co W,W)

and

C1

for all

satisfying W

E

Sh

.

I n a p p l i c a t i o n s , e s t i m a t e (4.8) i s used t o d e r i v e i t e r a t i v e convergence r a t e s f o r A1 g o r i thm II.

5.

STABILITY THEOREM.

In t h i s s e c t i o n we g i v e general r e s u l t s which can be used t o d e r i v e e s t i m a t e s o f t h e form (4.6) and ( 4 . 8 ) . L e t R be a continuous o p e r a t o r and Rh be i t s d i s c r e t e Theorem 1. approximation. Assume t h a t t h e f o l l o w i n g s t a b i l i t y and e r r o r e s t i m a t e s h o l d:

For any

E

> 0

there e x i s t s

CE

such t h a t

176

J.H. Bramble & J.E. Pasciak

Then t h e r e e x i s t s ho

0

such t h a t f o r

h < ho

(5.4) Remark 1.

E s t i m a t e ( 5 . 4 ) combined w i t h

guarantees a u n i f o r m ( i n d e p e n d e n t of

h ) i t e r a t i v e convergence r a t e f o r

t h e CG i t e r a t i o n f o r t h e s o l u t i o n o f (I+Rh) where

*

oroduct.

*

(I+Rn)U = F 1 H (R) i n n e r I + R h = ThAh and

denotes t h e a d j o i n t w i t h r e s D e c t t o t h e I n o u r f i n i t e element a p p l i c a t i o n s ,

Thus ( 5 . 4 ) and ( 5 . 5 ) w i l l i m p l y ( 4 . 8 ) f o r t h e p a r t i c u l a r examples o f the next section. Theorem 2.

Let

T1

and

T2

be c o n t i n u o u s o p e r a t o r s and

be t h e i r c o r r e s p o n d i n g d i s c r e t e a p p r o x i m a t i o n s . three estimates hold:

for

i = 1,2.

Then

TA

and

Th2

Assume t h a t t h e f o l l o w i n g

177

Nonselfadjoirit or Indefinite Elliptic Boundary Vulue Problems Remark 2.

Estimate

(5.8) i s an i n v e r s e p r o p e r t y f o r t h e o p e r a t o r

Th1

and i n a p p l i c a t i o n s i s d e r i v e d from t h e h y p o t h e s i s t h a t t h e mesh Estimate (5.9) coincides w i t h

elements a r e of "quasi uniform" s i z e .

.

Ah = (T;)-l

( 4 . 6 ) when Remark 3.

The proofs o f t h e above two theorems a r e s i m p l e and

consequently w i l l n o t be i n c l u d e d .

6.

THE P O I N C A R i PROBLEM. To i l l u s t r a t e o u r approach we c o n s i d e r a f i n i t e element

a p p r o x i m a t i o n o f t h e Poincare' problem i n t h i s s e c t i o n .

We c o n s i d e r t h e

f o l l o w i n g model problem: -Au t

au f KU ax

= f

in

R

(6.1) au -arl + a % au where

A

=

a2 ax

f

fYu=O

,a2 ~ ,2n

and

tangential d i r e c t i o n s along

r'.

on

r

a r e r e s p e c t i v e l y t h e normal and

T

For s i m p l i c i t y we have c o n s i d e r e d

c o n s t a n t c o e f f i c i e n t s i n d e f i n i n g t h e d i f f e r e n t i a l e q u a t i o n as w e l l as t h e boundary c o n d i t i o n .

Our r e s u l t s and i t e r a t i v e a1 g o r i thms e x t e n d t o

v a r i a b l e c o e f f i c i e n t problems w i t h o u t any c o m p l i c a t i o n s .

We a l s o assume

t h a t t h e s o l u t i o n o f (6.1) e x i s t s and i s unique. The f i n i t e element a p p r o x i m a t i o n t o (6.1) can t h e n be d e f i n e d by t h e G a l e r k i n technique.

M u l t i p l y i n g (6.1) by an a r b i t r a r y f u n c t i o n

i n t e g r a t i n g by p a r t s shows t h a t t h e s o l u t i o n

The f i n i t e element a p p r o x i m a t i o n function

U

in

Sh

U

which s a t i s f i e s

to

u

u

41 and

satisfies

i s t h e n d e f i n e d t o be t h e

J.H. Bramble & J.E. Pasciak

I78

Equation (6.3) can be used t o d e r i v e a system o f equations o f t h e form

(1.1) d e f i n i n g t h e d i s c r e t e s o l u t i o n U, i . e . , u s i n g a b a s i s f o r S h y (6.3) g i v e s N equations f o r t h e N unknowns d e f i n i n g U i n t h a t basis. To d e s c r i b e i t e r a t i v e methods f o r t h e s o l u t i o n o f (6.3) and/or t h e corresponding m a t r i x system, we s h a l l need t o use some o p e r a t o r notation.

F i r s t we consi der t h e Neumann problem w - A w = f

i n G.

aw

r

on

- = 0

au

Given a f u n c t i o n

f

D(w,e)

(a),t h e

f

to

w

o f (6.4) i s i n

i s s u f f i c i e n t l y smooth.

H2(G.)

We denote t h e

=

w. T i s a as t h e map which takes f t o T f 2 2 L (G.) i n t o H ( Q ) . The f i n i t e element approximation t o

+ (w,e)

W

in

Th f 5 W .

satisfying

Sh

= (f,e)

The d i s c r e t e s o l u t i o n o p e r a t o r takes

solution

T

(6.4) i s t h e f u n c t i o n (6.5)

2

r

i f as we s h a l l assume,

solution operator bounded map o f

L

in

Th

for all Th

e

E

Sh

.

can t h e n be defined as t h e map which

i s a map from

2

L (Q)

onto

Sh

and t h e

f o l l o w i n g convergence e s t i m a t e i s w e l l known ( c f . [ 2 ] ) :

I n a s i m i l a r manner, we can d e f i n e s o l u t i o n o p e r a t o r s f o r t h e f o l l o w i n g v a r i a t i o n a l problems:

and

We d e f i n e t h e s o l u t i o n o p e r a t o r s

R1z:

2

X and R w

I

$.

The corresponding

Nonselfadjoint or Indefinite Elliptic Boundary Value Problems

X

f i n i t e element approximations a r e g i v e n by t h e s o l u t i o n s

and

179

Y

in

satisfying

Sh

az

D(X,e)

+ (X,e)

= (5’8)

D(Y,e)

+ (Y,e)

=
+ (K-l)(z,e)

e

for all

E

Sh

and aw + y w y e >

for all

e

E

Sh

,

respectively. R

1 h

z :X

The d i s c r e t e s o l u t i o n o p e r a t o r s a r e t h e n d e f i n e d by 2 and Rh w Y and t h e f o l l o w i n g convergence e s t i m a t e s h o l d :

=

and

(6.8)

I n terms o f o p e r a t o r s , problem (6.1) i s e q u i v a l e n t t o ( I + R 1 + R2 ) u =- T A u = T f . The e x i s t e n c e and uniqueness p r o p e r t i e s o f s o l u t i o n s o f (6.1) can be used t o show t h a t f o r any

E

> 0

there i s a constant

CE

such t h a t

The d i s c r e t e e s t i m a t e

i s immediate f r o m t h e d e f i n i t i o n o f

i Rh

i n terms o f o p e r a t o r s as 2 ( I + Rh1 + Rh)U z T A U = Th f h h

.

Problem (6.3) can be s t a t e d

.

180

J.H. Bramble & J.E. Pasciak

A p p l y i n g Theorem 1 we g e t t h e f o l l o w i n g s t a b i l i t y e s t i m a t e :

The second i n e q u a l i t y i n (6.11) can be e a s i l y d e r i v e d from t h e d e f i n i t i o n s

Co

The constants size (6.12)

and

i n (6.10) a r e independent o f t h e mesh

C1

Now i t i s easy t o check t h a t

h.

W,V) = D(W,V) t (W,V)

(Ti

Comparing ( 6 12), (6.11),

for all

W,V€Sh

(4.7) and (4.8) i m p l i e s t h a t t h e CG method

applied t o

T h A t ThAh U = ThA;

(6.13)

Th f

converges w i t h a r e d u c t i o n p e r i t e r a t i o n which can be bounded independently o f t h e number of unknowns. r e s p e c t i v e l y denote t h e " s t i f f n e s s " m a t r i c e s N corresponding t o ( 6 . 3 ) and (6.5) i n a g i v e n b a s i s 8 = E8ili,l Let

for

Sh.

basis

8

M

and

M1

I f the coefficients o f a function a r e r e p r e s e n t e d by t h e v e c t o r d

in

Sh

i n terms o f t h e

then

-1 t M1 M MY;' Mc Th A;

gives the c o e f f i c i e n t s o f t h e sequen'ce o f v e c t o r s

W

c

ci

ThAh W

i n terms o f

8.

Consequently,

generated by A l g o r i t h m I 1 gives t h e

c o e f f i c i e n t s o f t h e sequence o f f u n c t i o n s generated by t h e CG method appl ied t o (6.13).

Thus t h e it e r a t i ve convergence estimates f o r

t h e CG method a p p l i e d t o (6.13) i m p l y i t e r a t i v e convergence r a t e s f o r A1 g o r i thm

II .

The above procedure i s an example o f an i t e r a t i v e convergence analysis i n

H1(,).

o p e r a t o r on

Sh

(6.14)

We a l s o n o t e t h a t i f

T,,l

i s another d i s c r e t e

which i s s p e c t r a l l y e q u i v a l e n t t o

Co(Th W,W) < (TA W,W)

5

C1(Th W,W)

Th

for all

i n t h e sense t h a t W E Sh

181

Nonselfadjoint or Indejinite Elliptic Boundary Value Problems

then

can be r e p l a c e d by

Th

1 Th

i n (6.11).

2 We n e x t c o n s i d e r an i t e r a t i v e a n a l y s i s i n L ( Q ) based on 1 2 2 H ( Q ) denote t h e s o l u t i o n o p e r a t o r Theorem 2. L e t T : L ( Q ) -f

f o r problem ( 6 . 1 )

with

B = 0, i . e . ,

T1 f

The s o l u t i o n o p e r a t o r

Eu.

T1

s a t i s f i e s an e s t i m a t e o f t h e form

We have r e s t r i c t e d t o t h e case o f

13

t h a t case.

and

Assume t h a t b o t h

T1

f i n i t e element subspaces and l e t discrete s o l u t i o n operators.

TA

0

s i n c e (6.15) i s w e l l known i n T

and

can be approximated i n t h e same Th

denote t h e c o r r e s p o n d i n g

The f o l l o w i n g convergence e s t i m a t e s a r e we1 1

known f o r a wide c l a s s o f f i n i t e element a p p l i c a t i o n s [271:

We f i n a l l y assume t h a t t h e i n v e r s e p r o p e r t i e s

are a l s o s a t i s f i e d .

Estimates o f t h e t y p e (6.17) can u s u a l l y be A p p l y i n g Theorem 2

d e r i v e d f r o m i n v e r s e assumptions f o r t h e subspaces. gives t h a t

for all E s t i m a t e (6.18) guarantees t h a t t h e CG method a p p l i e d i n

W

LL(Q)

E

Sh

.

f o r the

solution o f

A[ ThThAh X = A;

(6.19) where

An

(T:)-'

ThTh f

w i l l converge t o t h e s o l u t i o n

X

a t a rate

which

The r e s u l t i n g a l g o r i t h m

i s independent o f t h e number o f unknowns i n

Sh.

does n o t however correspond t o A l g o r i t h m I .

To guarantee r a p i d i t e r a t i v e

182

J.H. Bramble & J. E. Pasciak

convergence r a t e s f o r A l g o r i t h m I we must make a d d i t i o n a l assumptions. Again we use t h e b a s i s coefficients o f

(6.20)

W

for

63

Sh

i n t h e basis

cw )

co(cw

63.

If W

L

E

we denote by

Sh

the

Cw

We r e q u i r e t h a t

5

< (W,W)

-

,

(a

C1(Cw* Cw)

for a l l

W

E

Sh

.

Estimate (6.20) s t a t e s t h a t t h e Gram o r mass m a t r i x i s " e q u i v a l e n t " t o the coordinate i n n e r product.

for all

N

dimensional v e c t o r s

Combining (6.19) and (6.20) i m p l i e s

c.

E s t i m a t e (6.21) i s f i n a l l y an

e s t i m a t e which can be a p p l i e d t o guarantee uniform i t e r a t i v e convergence r a t e s f o r A1 g o r i thm I.

7.

AN ESTIMATE FOR THE DISCRETIZATION ERROR. I n order t o estimate the d i s c r e t i z a t i o n e r r o r

defined by (6.2) and (6.3) projection

Ph

onto

I t i s w e l l known t h a t

for

v

E

Hr(Q)

Sh(cf. [2,7]).

Sh.

Ph

and some

.

with

1 2 Rh = Rh + Rh

with

u

and

r e s p e c t i v e l y , we i n t r o d u c e t h e H (Q)v c H 1 ( n ) by

It i s defined f o r

satisfies

r > 1 which depends on t h e c h o i c e o f u-U

we need o n l y c o n s i d e r

Hence we apply (5.4) t o o b t a i n

.

U

1

I n view o f (7.2), t o e s t i m a t e

Ph u-U

u-U

From t h e d e f i n i t i o n s o f

R

1

, Rh,1

see t h a t (I+Rh)(Ph u-U) = Ph(R 1+R 2 ) ( P h - I ) u

.

R

2

and

2 Rh

we

Nonselfadjoint or Indefinite Elliptic Boundary Value Problems

183

Hence

from which i t follows immediately t h a t (7.3) Thus u s i n g ( 7 . 2 ) we obtain the estimate f o r the d i s c r e t i z a t i o n e r r o r ,

REFERENCES. 0. Axelsson; A c l a s s of i t e r a t i v e methods f o r f i n i t e element equations , Comp. Methods Appl. Mech. Engng., V. 9 , p p . 123-137.

I . Babuika and A . K . Aziz; Part I . Survey l e c t u r e s on the mathematical foundations of the f i n i t e element method , The Mathematical Foundations of the Finite Element Method w i t h Applications t o P a r t i a l D i f f e r e n t i a l Equations, A . K . Aziz, ed. Academic Press, New York, 1972. J.H. Bramble and J.E. Pasciak; An e f f i c i e n t numerical procedure f o r the computation of steady s t a t e harmonic c u r r e n t s i n f l a t p l a t e s , COMPUMAG conf., Genoa, 1983. J.H. Bramble, J.E. Pasciak, and A . H . Schatz; Preconditioners f o r i n t e r f a c e problems on mesh domains, p r e p r i n t .

B . L . Buzbee, F.W. Dorr, J.A. George, and G.H. Golub; The d i r e c t s o l u t i o n of t h e d i s c r e t e Poisson equation on i r r e g u l a r regions , SIAM J . Numer. Anal., V. 8 , 1971, pp. 722-736. R. Chandra; Conjugate gradient methods f o r p a r t i a l d i f f e r e n t i a l equations, Yale University, Dept. of Comp. S c i . Report No. 129, 1978. P.G. C i a r l e t ; The f i n i t e element method f o r e l l i p t i c problems, North-Holland, Amsterdam, 1978.

P . Concus, G. Golub, and D. O'Leary , A generalized conjugate gradient method f o r the numerical s o l u t i o n of e l l i p t i c p a r t i a l d i f f e r e n t i a l equations , i n Sparse Matrix Computation, J . Bunch and D. Rose, e d s . , Academic Press, New York, 1976, pp. 309-322.

J.H. Bramble & J.E. Pusciuk

184

[9]

S.C. E i s e n s t a t , M.C. Gursky, M.H. S c h u l t z , A.H. Sherman; Yale sparse m a t r i x package, I . t h e symmetric codes, Yale Univ. Dept. o f Comp. Sci . Report No. 112.

[lo]

H. Elman; I t e r a t i v e methods f o r l a r g e , sparse, nonsymmetric systems o f l i n e a r equations, Yale Univ. Dept. o f Comp. S c i . Report No. 229, 1978.

[ll]A. George and J.W.H.

L i u ; User Guide f o r SPARSPAK, Waterloo Oept. o f Comp. S c i . Report No. CS-78-30.

[12] J.A. M e i j e r i n k and H.A. Van d e r Vorst; An i t e r a t i v e s o l u t i o n method f o r l i n e a r systems o f which t h e c o e f f i c i e n t m a t r i x i s a symmetric M-matrix , Math. Comp. 1973, V. 31, pp. 148-162. [13] W.M. P a t t e r s o n ; I t e r a t i v e methods f o r t h e s o l u t i o n o f a l i n e a r o p e r a t o r e q u a t i o n i n H i l b e r t space - A survey, l e c t u r e notes i n mathematics, S p r i n g e r - V e r l a g , No. 394, 1974. [14] W . Proskurowski and 0. Widlund; On t h e numerical s o l u t i o n o f H e l m h o l t z ' s e q u a t i o n by t h e capacitance m a t r i x method , Math. Comp., V. 20, 1976, pp. 433-468. [15] A.H. Schatz; E f f i c i e n t f i n i t e element methods f o r t h e s o l u t i o n o f second o r d e r e l 1i p t i c boundary v a l u e problems on piecewise smooth domains , Proceedings o f t h e conference Construct! ve methods f o r s i n g u l a r problems , November 1983, Oberwolfach, West Germany, P . G r i s v a r d , W. Wendland and J. Whi teman, e d i t o r s , S p r i nger-Verl ag l e c t u r e notes i n mathematics, t o appear. [16] J . Simkin and C.W. Trowbridge; On t h e use o f t h e t o t a l s c a l a r p o t e n t i a l i n t h e numerical s o l u t i o n o f f i e l d problems i n e l e c t r o m a g n e t i c s , I n t e r . J . Numer. Math. Eng., 1979, V . 14, pp. 423-440. [17] O.C. Z i e n k i e w i c z ; The f i n i t e element method McGraw-Hill , 1977.

, 3rd e d i t i o n ,

Unification of Finite Element Methods H. Kardestuncer (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

185

CHAPTER 8 ON THE UNIFICATION OF FINITE ELEMENTS & BOUNDARY ELEMENTS

Cd.Brebbia

T h i s p a p e r r e v i e w s some of t h e a p p l i c a t i o n s of b o u n d a r y e l e m e n t methods f o r t h e s o l u t i o n of e n g i n e e r i n g p r o b l e m s . The p a p e r c o n s i d e r s how t h e new t e c h n i q u e r e l a t e s t o c l a s s i c a l f i n i t e e l e m e n t s , by r e v i e w i n g t h e f u n d a m e n t a l s of m e c h a n i c s , i n p a r t i c u l a r v i r t u a l T h i s a p p r o a c h g i v e s a common b a s i s work a n d a s s o c i a t e d p r i n c i p l e s . f o r a l l approximate techniques and h e l p s t o understand t h e r e l a t i o n s h i p between f i n i t e a n d boundary e l e m e n t method. The p a p e r s t r e s s e s t h e r a n g e of a p p l i c a t i o n s f o r which t h e boundary e l e m e n t method c a n g i v e a c c u r a t e r e s u l t s and be computat i o n a l l y e f f i c i e n t .

1.

INTRODUCTION

I n t h e l a s t few y e a r s t h e a p p l i c a t i o n s of b o u n d a r y i n t e g r a l e q u a t i o n s i n e n g i n e e r i n g have undergone i m p o r t a n t c h a n g e s . The b r a v e a t t e m p t s d u r i n g t h e s i x t i e s and e a r l y s e v e n t i e s p i o n e e r s s u c h a s Jawson [ I ] , Symm [ 2 ] , Massonet [ 3 ] , Hess [ 4 ] , C r u s e [ 5 ] and few o t h e r s , h a v e now b o r n e f r u i t i n t h e newly d e v e l o p e d boundary e l e m e n t method. I n t h i s way boundary i n t e g r a l e q u a t i o n s h a v e become a n e n g i n e e r i n g t o o l r a t h e r t h a n a mathem a t i c a l method w i t h i m p o r t a n t b u t r a t h e r r e s t r i c t i v e a p p l i c a t i o n s . S i n c e t h e e a r l y 1 9 6 0 ’ s a small g r o u p a t Southampton U n i v e r s i t y i n England s t a r t e d working on t h e a p p l i c a t i o n s of i n t e g r a l e q u a t i o n s t o s o l v e s t r e s s a n a l y s i s p r o b l e m s . Some of t h i s work h a s b e e n r e p o r t e d a t t h e f i r s t i n t e r n a t i o n a l C o n f e r e n c e on V a r i a t i o n a l Methods i n E n g i n e e r i n g , h e l d t h e r e i n 1972 [ 6 ] . More i s e x p e c t e d t o b e p r e s e n t e d d u r i n g t h e s e c o n d C o n f e r e n c e reconvened f o r 1985. These C o n f e r e n c e s a r e h e l d t o d i s c u s s t h e d i f f e r e n t t e c h n i q u e s of e n g i n e e r i n g a n a l y s i s and how t h e y a r e i n t e r r e l a t e d . The i m p o r t a n c e of t h e B I E p r e s e n t a t i o n s d u r i n g t h e 1st C o n f e r e n c e i s t h a t t h i s was t h e f i r s t t i m e t h a t boundary i n t e g r a l e q u a t i o n s w e r e i n t e r p r e t e d a s a v a r i a t i o n a l t e c h n i q u e . The work a t Southampton was c o n t i n u e d t h r o u g h o u t t h e s e v e n t i e s t h r o u g h a s e r i e s of t h e s e s m a i n l y c o n c e r n e d w i t h b o u n d a r y i n t e g r a l A t t h e Same t i n e new d e v e l o p m e n t s s o l u t i o n s of e l a s t o s t a t i c p r o b l e m s . i n f i n i t e e l e m e n t s s t a r t e d t o f i n d t h e i r way i n t o b o u n d a r y i n t e g r a l equat i o n s and t h e p r o b l e m of how t o r e l a t e t h e t e c h n i q u e t o o t h e r a p p r o x i m a t e s o l u t i o n s was s o l v e d u s i n g w e i g h t e d r e s i d u a l s [ 7 ] . T h i s work a t Southampton U n i v e r s i t y c u l m i n a t e d a r o u n d 1978 when t h e f i r s t book was p u b l i s h e d w i t h t h e t i t l e “Boundary E l e m e n t s ” [ 8 ] . The work was expanded t o encompass t i m e d e p e n d e n t and n o n - l i n e a r p r o b l e m s i n two s u b s e q u e n t books [ 9 ] ,[ l o ] , o n e of them v e r y r e c e n t l y p u b l i s h e d [ l o ] . The i m p o r t a n c e of t h i s work i s t h a t i t s t r e s s e s t h e common p r i n c i p l e s and f u n d a m e n t a l s r e l a t i o n s h i p s g o v e r n i n g

186

C A . Brebbia

t h e d i f f e r e n t t e c h n i q u e s , r a t h e r than t r y i n g t o set t h e boundary element method a s a completely s e p a r a t e c o m p u t a t i o n a l t e c h n i q u e . F i v e i m p o r t a n t i n t e r n a t i o n a l c o n f e r e n c e s have a l r e a d y been h e l d on t h e t o p i c of boundary elements i n 1978 (Southampton) [ l l ] , 1980 (Southampton) [ 121 , 1981 ( C a l i f o r n i a ) [ 131 , 1982 (Southampton) [ 141 , 1983 (Hiroshima) [ 1 5 ] and t h e n e x t one i s t o b e h e l d i n J u l y 1984 on board t h e Queen E l i z a b e t h I1 c r u i s e r . The frequency of t h e meetings and t h e i n c r e a s i n g number of p a p e r s p r e s e n t e d a t each of them i s e v i d e n c e of t h e h e a l t h y growth of t h e new method. I n a d d i t i o n , a s e r i e s of s t a t e of t h e a r t books a r e r e g u l a r l y p u b l i s h e d t o h i g h l i g h t t h e main developments of t h e t e c h n i q u e [ 161 [ 1 7 1 [ 181. The s u c c e s s and r a p i d a c c e p t a n c e of t h e new t e c h n i q u e i s due t o some important a d v a n t a g e s o v e r c l a s s i c a l f i n i t e e l e m e n t s , which a r e b e t t e r understood by reviewing t h e main c h a r a c t e r i s t i c s of t h e method. The boundary element method a s understood nowadays i s a r e d u c t i o n t e c h n i q u e based on boundary i n t e g r a l e q u a t i o n f o r m u l a t i o n s and i n t e r p o l a t i o n f u n c t i o n of t h e t y p e used i n f i n i t e e l e m e n t s , The main c h a r a c t e r i s t i c of t h e method i s t h a t i t reduces t h e d i m e n s i o n a l i t y of t h e problem by one and hence produces a much smaller system of e q u a t i o n s and more i m p o r t a n t f o r the practicing engineer, considerable reductions i n the data required t o run a problem. The l a t t e r advantage i s making boundary elements a f a v o u r i t e f o r many mechanical e n g i n e e r i n g problems when t h e n u m e r i c a l model h a s t o be i n t e r f a c e d w i t h mesh g e n e r a t o r s and o t h e r CAD f a c i l i t i e s . I n a d d i t i o n t h e numerical accuracy of t h e method i s g e n e r a l l y g r e a t e r t h a n t h a t of f i n i t e e l e m e n t s , which have l e d many e n g i n e e r s t o u s e BEM f o r problems such a s f r a c t u r e mechanics and o t h e r s where s t r e s s c o n c e n t r a t i o n can o c c u r . T h i s accuracy i s due t o u s i n g a mixed f o r m u l a t i o n . t y p e of approach f o r which a l l boundary v a l u e s a r e o b t a i n e d w i t h s i m i l a r d e g r e e of a c c u r a c y . I n t h i s r e s p e c t BEM i s c l o s e l y r e l a t e d t o t h e mixed f o r m u l a t i o n s p i o n e e r e d by R e i s s n e r [ 1 9 ] and e x c e l l e n t l y e x p l a i n e d and g e n e r a l i z e d by Washizu [ZO] and Pian and Tong [ 2 1 ] . The method i s a l s o w e l l s u i t e d t o problem s o l v i n g w i t h i n f i n i t e domains such a s t h o s e f r e q u e n t l y o c c u r r i n g i n s o i l mechanics and hydrodynamics, and f o r which t h e c l a s s i c a l domain methods a r e u n s u i t a b l e . A boundary s o l u t i o n i s f o r m u l a t e d i n terms of i n f l u e n c e f u n c t i o n s o b t a i n e d by a p p l y i n g a fundamental s o l u t i o n . I f t h e s o l u t i o n i s s u i t a b l e f o r a n i n f i n i t e domain no o u t e r b o u n d a r i e s need t o be defined. I t i s now g e n e r a l l y a c c e p t e d t h a t t h e b e s t way of f o r m u l a t i n g boundary elements f o r g e n e r a l e n g i n e e r i n g problems i s by u s i n g weighted r e s i d u a l t e c h n i q u e s , as shown i n r e f e r e n c e s [ 7 ] , [ a ] and [ 101. T h i s f o r m u l a t i o n c l o s e l y r e l a t e s t h e BEM t o t h e v a r i a t i o n a l methods and t o t h e o r i g i n a l It a l s o allows i n t e r p r e t a t i o n of v i r t u a l work proposed by B e r n o u l l i . f o r c o m p l i c a t e d n o n - l i n e a r and t i m e dependent problems t o be p r o p e r l y f o r m u l a t e d , w i t h o u t need t o f i n d a n i n t e g r a l expansion beforehand.

The term boundary element now a l s o i m p l i e s t h a t t h e s u r f a c e of t h e domain i s d i v i d e d i n t o a s e r i e s of elements o v e r which t h e f u n c t i o n s under c o n s i d e r a t i o n v a r y i n a c c o r d a n c e w i t h some i n t e r p o l a t i o n f u n c t i o n s , i n much t h e same way a s i n f i n i t e elements. By c o n t r a s t w i t h p a s t i n t e g r a l e q u a t i o n s f o r m u l a t i o n s - which were r e s t r i c t e d t o c o n c e n t r a t e d s o u r c e s t h e s e v a r i a t i o n s p e r m i t t h e p r o p e r d e s c r i p t i o n of curved s u r f a c e s i n a d d i t i o n t o working w i t h more a c c u r a t e h i g h e r o r d e r i n t e r p o l a t i o n functions.

The Unification o f Finite Elements & Boundary Elements

187

Summarizing, a f t e r y e a r s of r e s e a r c h and development t h e b o u n d a r y e l e m e n t method h a s emerged a s a p o w e r f u l m a t h e m a t i c a l t o o l f o r t h e s o l u t i o n of a l a r g e v a r i e t y of e n g i n e e r i n g p r o b l e m s . The a c c e p t a n c e of t h e t e c h n i q u e amongst p r a c t i c i n g e n g i n e e r s i s m a i n l y due t o t h e f o l l o w i n g a d v a n t a g e s : i) Simple d a t a p r e p a r a t i o n , which c o n s i d e r a b l y r e d u c e s t h e amount of manpower r e q u i r e d t o r u n a problem i i ) More a c c u r a t e r e s u l t s , which makes t h e t e c h n i q u e e s p e c i a l l y a t t r a c t i v e f o r s t r e s s c o n c e n t r a t i o n problems, f r a c t u r e mechanics a p p l i c a t i o n and o t h e r s . T h i s i n c r e a s e d a c c u r a c y a l s o a l l o w s t h e d e s i g n e r t o work w i t h c o a r s e r meshes t h a n i n f i n i t e e l e m e n t s w i t h f u r t h e r r e d u c t i o n i n manpower. i i i ) D e f i n i t i o n of s y s t e m and i n t e r p r e t a t i o n of r e s u l t s become e a s i e r which p e r m i t s a b e t t e r i n t e r f a c i n g t o s u r f a c e m o d e l l i n g and o t h e r CAD systems. i v ) Problems w i t h i n f i n i t e domains c a n b e s o l v e d a c c u r a t e l y , which makes t h e method w e l l s u i t e d f o r a p p l i c a t i o n s s u c h a s s o i l m e c h a n i c s and hydrodynamics.

2.

FUNDAMENTAL PRINCIPLES

I n what f o l l o w s we w i l l c o n s i d e r p r o b l e m s i n l i n e a r e l a s t i c i t y f o r which t h e problem c a n b e e x p r e s s e d i n f u n c t i o n of a s e t o f e q u i l i b r i u m e q u a t i o n s and a n o t h e r s e t of c o m p a t i b i l i t y r e l a t i o n s , r e l a t e d t o g e t h e r by c o n s t i t u t ive laws. These e q u a t i o n s w i l l b e w r i t t e n u s i n g t h e i n d i c i a 1 n o t a t i o n . Dynamic l o a d i n g w i l l n o t b e c o n s i d e r e d e x p l i c i t l y b u t i t c a n b e e a s i l y i n c l u d e d u s i n g D ' A l e m b e r t ' s h y p o t h e s i s , i . e . by c o n s i d e r i n g t h a t a t a g i v e n time t h e dynamic and s t a t i c f o r c e s a r e i n e q u i l i b r i u m . T h i s s i m p l e b u t b r i l l i a n t i d e a f a c i l i t a t e s t h e dynamic a n a l y s i s .

The a p p r o x i m a t e methods of s o l u t i o n u s e d i n e n g i n e e r i n g a n a l y s i s h a v e a l l a common b a s i s n o t o n l y g i v e n by t h e f u n d a m e n t a l e q u a t i o n s of p h y s i c s b u t a l s o by t h e f a c t t h a t t h e a c t u a l a p p r o x i m a t i o n s c a n b e i n t e r p r e t e d u s i n g t h e p r i n c i p l e of v i r t u a l work. The a p p l i c a t i o n of t h i s p r i n c i p l e i n d i f f e r e n t ways g i v e s r i s e t o t h e d i v e r s e t e c h n i q u e s o f e n g i n e e r i n g a n a l y s i s . I t is important t o p o i n t o u t t h a t t h e p r i n c i p l e i t s e l f i s a fundamental i d e a b a s e d on p h i l o s o p h i c a l and p h y s i c a l i n t u i t i o n r a t h e r t h a n h i g h e r mathematics. I n t h i s r e s p e c t i t i s i n t e r e s t i n g t o remark t h a t t h e p r i n c i p l e h a s b e e n d i s c u s s e d s i n c e t h e b e g i n n i n g of w e s t e r n c i v i l i z a t i o n and i s r e l a t e d t o t h e ' p o t e n t i a l i t i e s ' o f p h y s i c a l s y s t e m s a s d i s c u s s e d by A r i s t o t l e [ 2 2 ] . From c l a s s i c a l a n t i q u i t y onward t h e p r i n c i p l e h a s b e e n f r e q u e n t l y a p p l i e d and s e v e r a l w e l l known f i e l d s of m a t h e m a t i c s r e l a t e d t o i t , s u c h a s t h e C a l c u l u s of V a r i a t i o n s , F u n c t i o n a l A n a l y s i s , D i s t r i b u t i o n T h e o r y , e t c . These m a t h e m a t i c s , a l t h o u g h i m p r e s s i v e , s h o u l d n o t d i s t r a c t u s f r o m t h e e l e g a n c e , s i m p l i c i t y and g e n e r a l i t y o f t h e o r i g i n a l v i r t u a l work s t a t e m e n t . I n t h i s s e c t i o n we w i l l t r y t o p o i n t o u t how t h e p r i n c i p l e of v i r t u a l work can b e u s e d t o g e n e r a t e models i n s o l i d m e c h a n i c s . T h i s i s f i r s t done by assuming t h a t t h e same p h y s i c a l e q u a t i o n s a p p l y t o two d i f f e r e n t s t a t e s , o n e i s t h e ' a c t u a l ' and t h e o t h e r i s t h e ' v i r t u a l ' s t a t e . The a c t u a l s t a t e i s u s u a l l y d e f i n e d i n t e r m s of a n a p p r o x i m a t i o n i n t h e p r a c t i c e . The p r o d u c t s of t h e s e two s t a t e s g i v e r i s e t o v i r t u a l work s t a t e m e n t s . T h i s s e c t i o n w i l l a t t e m p t t o c l a s s i f y t h e s e s t a t e m e n t s d e p e n d i n g on which t y p e of r e l a t i o n s h i p s a r e i d e n t i c a l l y s a t i s f i e d a n d which a r e t o b e imposed on t h e a p p r o x i m a t e f u n c t i o n s . T h e s e f o r m u l a t i o n s a r e a s w e l l known a s v i r t u a l d i s p l a c e m e n t s and v i r t u a l f o r c e s , b u t c a n a l s o b e some t y p e o f mixed o r h y b r i d a p p r o a c h . We w i l l p a r t i c u l a r l y c o n s i d e r t h e p o s s i b i l i t y of p r o d u c i n g g e n e r a l i z e d f o r m u l a t i o n s and t a k i n g them t o t h e b o u n d a r y , a s i t

C.A. Brebbia

188

i s due i n boundary elements. The s i m p l i c i t y of t h e v i r t u a l work approach allows f o r t h e f o r m u l a t i o n of very g e n e r a l approximate models, v a l i d even f o r non l i n e a r and time dependent problems. The formulation of d i f f e r e n t techniques - i n c l u d i n g boundary elements - becomes then independent of t h e e x i s t e n c e o r , o t h e r w i s e of a f u n c t i o n a l o r i n t e g r a l s t a t e m e n t . These f o r m u l a t i o n s w i l l not b e d i s c u s s e d h e r e , but t h e i n t e r e s t e d r e a d e r i s r e f e r r e d t o [ 2 3 ] . VIRTUAL WORK

The V i r t u a l Work p r i n c i p l e can be i n t e r p r e t e d a s t h e work done by one s t a t e ( ' a c t u a l ' ) over a n o t h e r ( ' v i r t u a l ' ) . This work can be expressed i n d i f f e r e n t ways, depending on t h e v a r i a b l e s under c o n s i d e r a t i o n . For i n s t a n c e i f one is d e a l i n g with displacements and body and t r a c t i o n f o r c e s one can w r i t e t h e following v i r t u a l work s t a t e m e n t

Notice t h a t t h e work h a s been d e f i n e d i n terms of t h e usual i n n e r p r o d u c t , 5.e. t h e m u l t i p l i c a t . i o n of t h e v a r i a b l e s i n t e g r a t e d over t h e domain and e x t e r n a l s u r f a c e . bk a r e t h e body f o r c e s , t k t h e s u r f a c e t r a c t i o n s and u k t h e displacement components. The v i r t u a l f i e l d i s i n d i c a t e d by an asterisk

.

The same p r i n c i p l e can a l s o be expressed i n terms of t h e i n t e r n a l work, which g i v e s ,

I.

jk

€Jk

dR =

I

ufk

E~~

dR

a j k and c j k a r e t h e s t r e s s and s t r a i n components r e s p e c t i v e l y . S t i l l more i n t e r e s t i n g l y , v i r t u a l work could be given a s a r e l a t i o n s h i p between c o m p a t i b i l i t y e q u a t i o n s and s t r e s s f u n c t i o n s . I f t h e c o m p a t i b i l i t y r e l a t i o n s h i p s a r e expressed by t h e Rk components of a c o m p a t i b i l i t y v e c t o r and t h e a s s o c i a t e d s t r e s s f u n c t i o n xk one can w r i t e ,

These t h r e e s t a t e m e n t s a r e e q u a l l y v a l i d and they can even be added t o f i n d an extended v e r s i o n of v i r t u a l work a s w e w i l l s e e soon. This p r e s e n t a t i o n of v i r t u a l work h a s some advantages over t h e more c l a s s i c a l v a r i a t i o n a l t y p e of approach a s we w i l l s e e s h o r t l y . The c l a s s i c a l approach o r i g i n a t e d w i t h B e r n o u l l i c o n s t r a i n t e q u a t i o n s , u s u a l l y r e q u i r e s t h e d e f i n i t i o n of some Lagrangian m u l t i p l i e r s t o g e n e r a l i z e t h e p r i n c i p l e s . Our approach i n s t e a d i s much s i m p l e r . VIRTUAL DISPLACEMENTS I t i s now easy t o deduce d i f f e r e n t v e r s i o n s of t h e v i r t u a l work p r i n c i p l e by applying t h e above e q u a t i o n s . Let u s s t a r t with i d e n t i t y ( 1 ) i n t e g r a t i n g by p a r t s t h e s u r f a c e i n t e g r a l on t h e r i g h t hand s i d e . I n o r d e r t o do

The Unification of Finite Elements & Boundary Elements t h i s we can use t h e w e l l known Gauss theorem and f o r l i n e a r s t r a i n displacement r e l a t i o n s - which we a c c e p t a r e i d e n t i c a l l y s a t i s f i e d ob t a i n ,

189

-

where t k . = nj ujk ; n . a r e t h e d i r e c t i o n c o s i n e s of t h e normal w i t h r e s p e c t 3 t o x j , a x i s . I f f u r t h e r m o r e we a c c e p t i ) r e c i p r o c i t y a s given by e q u a t i o n (21, i i ) t h a t t h e v i r t u a l displgcements u t are-such t h a t t h e d i s p l a c e m e n t s boundary c o n d i t i o n on r l (uk = uk on r l where Uk a r e known v a l u e s ) a r e -0 on r l and i i j . ) t h a t t h e v i r t u a l f i e l d identically satisfied, i.e. s a t i s f i e s e q u i l i b r i u m , one f i n d s ,

$

which i s t h e u s u a l e x p r e s s i o n f o r v i r t u a l work. Notice t h a t t h e o t h e r p a r t of t h e boundary r 2 i s - t h a t on which t h e t r a c t i o n boundary c o n d i t i o n s a r e p r e s c r i b e d , i . e . tk = t on r

2'

k

Another form of v i r t u a l disolacements can b e o b t a i n e d by i n t e g r a t i n g by p a r t s t h e l e f t hand s i d e i n t e g r a l i n ( 5 ) . This g i v e s

1

dQ =

(tk

-

r2 The above s t a t e m e n t i s e q u i v a l e n t t o ( 5 ) provided t h a t we a c c e p t t h a t t h e s t r a i n - d i s p l a c e m e n t e q u a t i o n s and c o n s t i t u t i v e r e l a t i o n s h i p s a r e i d e n t i c ally satisfied. The above r e s t r i c t i o n s t o v i r t u a l work g i v e r i s e t o t h e p o s s i b i l i t y of d e f i n i n g a f u n c t i o n a l c a l l e d t o t a l p o t e n t i a l energy, composed of two p a r t s , i . e . t h e i n t e r n a l s t r a i n energy f u n c t i o n ,

and t h e p o t e n t i a l of t h e l o a d s (assuming they a r e c o n s e r v a t i v e

"-j

-tk uk

dr

-

1

bk uk dR

I-2 The t o t a l p o t e n t i a l energy i s then

Equilibrium s t a t e m e n t s ( 5 ) o r ( 6 ) f o r i n s t a n c e a r e now d e f i n e d by t h e ' v a r i a t i o n ' of n , i . e .

*

*

*

n = u +n = o

(10)

190

C A . Brebbia

Notice that Potential energy is function of the displacements and strains. As it is well known this principle is the basis of the stiffness finite element formulations. Principle of Virtual Forces The converse of the Principle of Virtual Displacements is the Principle of Virtual Forces which can be described in several different ways. In this paper we will start by using the virtual work relationship (equation ( 2 ) )

Accepting that

* E

ij

9:

=

l(uiyj+u

,t

j

. ) we can transform the right hand side term

of ( 1 1 ) into,

Furthermore accepting that the

G

state satisfies the equilibrium equations

one can write ( 1 2 ) as,

9<

In order to eliminate uk from ( 1 1 ) we can use another reciprocity relationship - equation ( 1 ) - this gives,

I

ij

(I

E”i

=j

j dR

uk b:dR

+

uk tt dr

(15)

Hence equation ( 1 1 ) can be written as,

1

GYj

dn

Eij

=

b t uk dR

+

I*

tk uk dr

(16)

The unknown boundary displacements on r2 can be eliminated by stipulating that the t i components vanish there and ( 1 6 ) becomes u ij

E

t: -uk d r

ij d R = I b : u k d Q +

r

1

(17)

The Unification of Finite Elements & Boundary Elements

191

where the bar on uk components indicates that these values are known. We now want to demonstrate that equation ( 1 7 ) will produce as stationarity requirement, the compatibility equations. I n order to do so consider the bc body forces which, if the virtual stresses satisfy equilibrium, (i.e. a‘’j k , j + b i = 0) give rise to,

dT 9,

Noticing that tk # 0 only on ( 1 7 ) and obtain,

r,

one can substitute (18) into equation

dT

The

=

0

(19)

stationarity requirements are compatibility, i.e. Eij =

J(Ui,j -+

U .

J

.)

(20)

9 1

plus the associated displacement boundary conditions

Uk

=

Uk

on I‘

(21)

1

For the principle of virtual displacements instead the requirements were equilibrium (equation ( 6 ) ) . This means that if one has the equilibrium o r the strain displacements set of equations one can derive the other set from one of the two principles (i.e. virtual displacements or forces). This allows u s to produce a consistent set of equations which in some cases may be difficult to find otherwise. Some researchers have applied these ideas to deduce a consistent set of equations for shell theory for instance. One can-now define a functional called complementary energy, composed of two parts, i.e. the internal complementary energy, given by

plus the potential of the surface forces assuming that the uk displacements are not functions of tk, i.e.

C A . Brebbia

192

The t o t a l complementary energy i s then Itc =

w +

Rc

The c o n d i t i o n f o r s t a t i o n a r i t y i s d e f i n e d by

*

s t a n d s a s always f o r v a r i a t i o n . T h i s p r i n c i p l e i s t h e b a s i s where t h e of t h e f l e x i b i l i t y f o r m u l a t i o n s . N o t i c e t h a t

W*b)

=

I* u

ij

u* ij

=

c

E

jkllm

ij

u

Em

dR

dR

GENERALIZED PRINCIPLE

A g r e a t d e a l of c o n f u s i o n s t i l l e x i s t s r e g a r d i n g t h e so c a l l e d g e n e r a l i z e d p r i n c i p l e s , which a r e based on a g e n e r a l i z a t i o n of t h e p r e v i o u s two c a s e s . F r e q u e n t l y r e s e a r c h e r s deduced them from a n e x t e n s i o n of t h e P r i n c i p l e We w i l l of Minimum P o t e n t i a l Energy. ( s e e Washizu [ 2 0 ] and R e i s s n e r [ 1 9 ] ) . now s e e t h a t t h i s i s n o t r e a l l y n e c e s s a r y and t h e y can e a s i l y b e o b t a i n e d from c o n s i d e r a t i o n s of v i r t u a l work. L e t u s f i r s t w r i t e t h e p r i n c i p l e of v i r t u a l work f o r t h e c a s e of v i r t u a l d i s p l a c e m e n t s b u t w i t h o u t t h e r e s t r i c t i o n s t h a t t h e y have t o be i d e n t i c a l l y z e r o on t h e r p a r t of t h e boundary. I n t h i s c a s e w e have 1

Then we w r i t e t h e e x p r e s s i o n f o r v i r t u a l f o r c e s a l s o w i t h o u t t h e r e s t r i c t i o n t h a t t* = 0 on I'2 and assuming t h a t t h e s t r a i n s a r e f u n c t i o n s of s t r e s s , i . e . E c u This g i v e s jk jkllm Rm'

1

I

j k cjkllm uLm dR

u*

=

j

b i uk dR +

I

t; uk d r +

j

t;

ik d r

One c a n now r e p l a c e t h e f i r s t i n t e g r a l on t h e r i g h t hand s i d e by

193

The Unification of Finite Elements & Boundary Elements

I

I n what f o l l o w s w e w i l l a c c e p t t h a t

= i ( ~ + ~u . , . )~. Hence we can ij J s u b t r a c t (28) f r o m ( 2 7 ) t a k i n g i n t o c o n s i d e r a t i o n ( 2 9 ) an d o b t a i n

i

* {Ujk

Elk

+ u*j k ‘jk

=

bk u;

dR +

-

‘jk

‘jkkm

i-

*

u

E

Em

t k uk d r +

} d o =

I

tk

k dr

U*

r2

T h i s i s a w e l l known ‘ g e n e r a l i z e d ’ e x p r e s s i o n . I n o r d e r t o i n v e s t i g a t e t h e s t a t i o n a r i t y c o n d i t i o n s a s s o c i a t e d with i t we can c a r r y o u t an i n t e g r a t i o n by p a r t s and o b t a i n t h e f o l l o w i n g e q u a t i o n ,

j{lojk,j

+ bkh:

+ (cjkllm u Ilm

-

sjk)uik!

(31)

dR

i ) the equilibrium equations i n R; The s t a t i o n a r y c o n d i t i o n s a r e i i ) t h e s t r e s s b o u n d a r y c o n d i t i o n s o n r2 ; i i i ) t h e s t r e s s - s t r a i n r e l a t i o n s h i p s an d i v ) t h e d i s p l a c e m e n t b o un d ar y c o n d i t i o n s on r 1’ Remembering t h e d e f i n i t i o n of W ( e q u a t i o n ( 2 2 ) ) t h e seco n d i n t e g r a l i n (31 ) c a n b e w r i t t e n a s ,

(where .M i s t h e complementary s t r a i n e n e r g y d e n s i t y ) . components i n ( 3 1 ) a r e i n d e p e n d e n t . k

N o t i c e t h a t uk an d

t

With these c o n c e p t s i n mind we can now p r o p o s e a n e n e r g y f u n c t i o n a l f o r t h e g e n e r a l i z e d p r i n c i p l e , i.e.

194

C.A. Brebbia

C U ~ -~ W(u E )Ida ~ ~ (33)

The condition oE stationarity which produces expression ( 3 1 ) is

*

rIG

=

0

(34)

This model requires expansions for both stresses and displacements and gives rise to the mixed formulation of boundary elements. HYBRID MODEL Another interesting development in recent years has been the study of the so-called "hybrid" models. In this case we can start with expression ( 3 1 ) but select stress functions a which identically satisfy the equilibrium equation, i.e. U jk,j

+ bk

=

(35)

0

Hence equation ( 3 1 ) reduces to

Integrating by parts the term in finds,

E

*

jk

and assuming that a = 0 one jk,k -

Note that expressions for the Uk displacements are required only on the boundaries. This means that the only expressions needed on the volume which has to satisfy the equilibrium equations. are the stress u jk If the displacement boundary conditions are made to satisfy the boundary conditions on r , , equation ( 3 7 ) can be written

W* dR 1

=

-

(38) L

which is the form usually presented in the literature 1201.

The integral

The Unification o f Finite Elements & Boundary Elements

195

on t h e l e f t hand s i d e of t h i s e q u a t i o n p r o d u c e s a f l e x i b i l i t y m a t r i x . A f t e r c e r t a i n manipulations t h i s can be transformed i n t o a s t i f f n e s s matrix and u s e d i n t h e same manner a s t h e m a t r i c e s deduced u s i n g t h e p r i n c i p l e of v i r t u a l displacements. BOUNDARY SOLUTIONS

To o b t a i n t h e t y p e of b o u n d a r y s o l u t i o n s u s e d i n b o u n d a r y e l e m e n t s one c a n a l s o s t a r t w i t h e q u a t i o n (31) b u t t h i s t i m e s a t i s f y i n g i d e n t i c a l l y t h e s t r e s s - s t r a i n r e l a t i o n s . Here w e h a v e ,

' 2

ll

The a i m i s now t o t r y t o r e d u c e t h e s o l u t i o n t o t h e b o u n d a r y . One p o s s i b i l i t y i s t o p r o p o s e 0 . k f u n c t i o n s which s a t i s f y t h e e q u i l i b r i u m e q u a t i o n s as we h a v e done i n t $ e c a s e of " h y b r i d " f o r m u l a t i o n s . A n o t h e r i s t o f i n d v i r t u a l work f i e l d uk,uk which a r e i n e q u i l i b r i u m . To t h i s end we c a n i n t e g r a t e t w i c e by p a r t s t h e f i r s t i n t e g r a l i n ( 3 9 ) which g i v e s

* *

I* One c a n now l o o k f o r f u n c t i o n s s u c h as t h e f u n d a m e n t a l s o l u t i o n s which s a t i s f y t h e e q u i l i b r i u m equation, such t h a t ,

i where A i s t h e D i r a c d e l t a f u n c t i o n and r e p r e s e n t s a u n i t l o a d a t t h e p o i n t '!' a c t i n g i n t h e R d i r e c t i o n . T h i s s o l u t i o n i s sometimes c a l l e d K e l v i n ' s s o l u t i o n a n d w i l l p r o d u c e f o r e a c h d i r e c t i o n 'R' t h e f o l l o w i n g equation ui +

1

uk

ti

dr

=

i u R r e p r e s e n t s t h e d i s p l a c e m e n t a t ' i ' i n t h e 'R' d i r e c t i o n .

Notice t h a t

f o r s i m p l i c i t y we h a v e added t o g e t h e r t h e two t y p e s of b o u n d a r y ,

*

\

*

r

=

r 1+r2

a n d tk a r e components of t h e f u n d a m e n t a l s o l u t i o n , i . e . d i s p l a c e m e n t a n d

t r a c t i o n s d u e t o a u n i t c o n c e n t r a t e d l o a d a t t h e p o i n t 'i' a c t i n g i n t h e '9.' d i r e c t i o n . I f we c o n s i d e r u n i t f o r c e s a c t i n g i n t h e t h r e e d i r e c t i o n s , equation ( 4 2 ) can b e w r i t t e n a s ,

C.A. Brebbia

196

*

a n d u* r e p r e s e n t t h e t r a c t i o n s and d i s p l a c e m e n t s i n t h e k Lk Lk d i r e c t i o n d u e t o u n i t f o r c e s a c t i n g i n t h e il d i r e c t i o n . E q u a t i o n ( 4 3 ) i s v a l i d f o r t h e p a r t i c u l a r p o i n t ' i ' where t h o s e f o r c e s a r e a p p l i e d .

where t

E x p r e s s i o n ( 4 3 ) g i v e s r i s e t o t h e so c a l l e d d i r e c t boundary e l e m e n t method which i s d e s c r i b e d i n d e t a i l i n r e f e r e n c e [ 101. F o r t h e p u r p o s e s of t h i s p a p e r i t i s i m p o r t a n t t o p o i n t o u t t h a t t h e Uk a n d t k unknowns are a l l d e f i n e d on t h e boundary. F u r t h e r m o r e i f t h e domain t e r m i n body f o r c e s which d o e s n o t c o n t a i n a n y unknown - i s t a k e n t o t h e b o u n d a r y , o n e o n l y n e e d s t o compute boundary i n t e g r a l s which e f f e c t i v e l y r e d u c e s t h e dimensiona l i t y of t h e problem b y o n e . S e v e r a l ways i n which t h e body f o r c e t e r m c a n b e t a k e n t o t h e boundary a r e d e s c r i b e d i n r e f e r e n c e [ l o ] a n d [ 2 4 ] . The boundary e l e m e n t as d e s c r i b e d above i s b a s i c a l l y a p o i n t c o l l o c a t i o n technique a s t h e fundamental s o l u t i o n s a r e a p p l i e d a t d i f f e r e n t p o i n t s ' i ' on t h e boundary. I t i s a l s o p o s s i b l e t o d i s t r i b u t e t h e s e f u n d a m e n t a l s o l u t i o n s o v e r p o r t i o n s of t h e b o u n d a r y o r e l e m e n t s b u t i n t h i s c a s e a d o u b l e i n t e g r a t i o n w i l l b e r e q u i r e d which c o m p l i c a t e s t h e s o l u t i o n and r e d u c e s t h e e f f i c i e n c y of t h e new t e c h n i q u e . T h i s t e c h n i q u e i s d e s c r i b e d b y Wendland i n r e f e r e n c e [ 2 5 ] . SUMMARY T a b l e I summarizes t h e main c h a r a c t e r i s t i c s of e a c h of t h e f i v e e n g i n e e r i n g a n a l y s i s methods d i s c u s s e d e a r l i e r , t o g e t h e r w i t h t h e most u s u a l t y p e of s t a t e m e n t which g i v e s o r i g i n t o t h e t e c h n i q u e s t o b e d e s c r i b e d i n d e t a i l i n p a r t 3 of t h i s p a p e r . The g e n e r a l i z e d f u n c t i o n a l IIG c a n b e c o n s i d e r e d as t h e s t a r t i n g p o i n t f o r a l l t h e f o r m u l a t i o n s . N o t i c e t h a t t h e r e a r e many o t h e r s t a t i o n a r y c o n d i t i o n s t h a t we c o u l d i n c l u d e i n IIG b u t t h e y h a v e n o t y e t produced p r a c t i c a l methods of e n g i n e e r i n g a n a l y s i s . By c o n t r a s t t h e f i v e formulat i o n s shown i n t h e t a b l e a r e w e l l known i n e n g i n e e r i n g , a l t h o u g h most of t h e e n g i n e e r i n g c o d e s a r e b a s e d on t h e d i s p l a c e m e n t f o r m u l a t i o n . More r e c e n t l y a s u b s t a n t i a l number of c o d e s h a v e s t a r t e d t o a p p e a r b a s e d on boundary methods u s i n g t h e f u n d a m e n t a l s o l u t i o n d u e t o K e l v i n o r similar. Although T a b l e I h e l p s t o u n d e r s t a n d the common b a s i s of t h e s e methods of e n g i n e e r i n g a n a l y s i s i t i s n e c e s s a r y , i f o n e w i s h e s t o combine them, t o s e e t h e form t h a t t h e e l e m e n t m a t r i c e s t a k e f o r e a c h of t h e f o r m u l a t i o n s . T h i s analysis i s c a r r i e d out i n the next section. 3.

THE DISCRETE ELEMENT METHODS

I n t h i s s e c t i o n we w i l l t r y t o deduce t h e m a t r i c e s c o r r e s p o n d i n g t o t h e d i f f e r e n t methods s e e n i n S e c t i o n 2 , s t a r t i n g w i t h t h e s i m p l e r - t h e f i n i t e e l e m e n t d i s p l a c e m e n t method f o r completeness.

-

i)

D i s p l a c e m e n t Model

I n t h i s c a s e one s t a r t s w i t h t h e following e x p r e s s i o n ,

TABLE I GENERAL I ZED FUNCTIONAL

METHOD

S TAT1ONARY

INDEPENDENT

USUAL STATEMENT

IDENTICALLY SAT I SFI ES

+ b = O

DISPLACEMENT ~

,

-

= o tn r

k

2 I

(with fundaental solution)

k

on

r2

= uk on

rl

tk = t

\

-k

uI1 i +

\t*QkdT

tku*QkdT+

=

1 bkulkdQ

r

XED

t

k

=tk

on

r 2 ; \=uk

On

rll I

JW*dR

-

\=uk

=

*

-I f , u i d r

r

r2

on

rl

(or p a r t )

*

+ I(tkuk+uktk)dT

U

LMIBILITY

1 uikEijdQ

*-

tkuk d r +

=

rl

b:ukdR

198

C.A. Brebbia

jk

E;~

-

j

dR =

*

tk u k d r +

r

I

bk u l dR

(44)

I2 which can be rendered i n m a t r i x form a s f o l l o w s ,

We can now propose t o u s e displacement f u n c t i o n s such t h a t u = $ T U

-

-

_e

;

u* = $T u* .e I

where t h e 9 a r e t h e i n t e r p o l a t i o n f u n c t i o n s f o r t h e d i s p l a c e m e n t s o v e r one element and u t h e nodal unknowns. D i f f e r e n t i a t i n g we o b t a i n t h e s t r a i n s , i.e.

-

E

= B u -e

(47)

-

and a c c e p t i n g t h e s t r a i n - s t r e s s r e l a t i o n s h i p s w e c a n w r i t e , O = D E

-.

-..

D i s t h e m a t r i x of e l a s t i c c o n s t a n t . Under t h e s e c o n d i t i o n s e q u a t i o n ( 4 5 ) becomes,

where , K- e = I B -T DI -B .d R

A s t h e v i r t u a l d i s p l a c e m e n t s a r e a r b i t r a r y , e q u a t i o n ( 4 9 ) can b e w r i t t e n simply a s (51)

K- e u_ e = P,e

The f u n c t i o n s y i n ( 4 6 ) a r e assumed t o b e a d m i s s i b l e and $ p a r t of a complete s e t of f u n c t i o n s . N g t i c e t h a t they have t o s a t i s f y i d e n t i c a l l y t h e boundary c o n d i t i o n s uk = uk on r , i n c l u d i n g t h e i n t e r e l e m e n t s u r f a c e s . ii)

Mixed Models

*

Here we s t a r t w i t h t h e complete IIG e x p r e s s i o n , i . e .

The Unification o f Finite Elements & Boundary Elements

]{ajk Elk + a*jk

E~~

-

W*}dR

199

=

(52)

which can be written in matrix form as,

+

(u-;)~ t* dr +

- -

I

j-

bT

-

U*

dn

We now adopt expressions for both displacements and stress over an element, i.e.

-

-

u = $ T U- e '.

o- = J , T _e o

(54)

I

(Notice that we will assume displacement and stress continuity here for simplicity. Otherwise extra "jump" terms should be included in II"). G The expressions for

where

C

- -t

E,

and

w

become

is the elastic compliance matrix (C

-

direction cosines on the boundary.

where, AT

-e

=

1

BT -

J, dR I

-

I-

I

-

N is a matrix of

Substituting ( 5 5 ) into ( 5 3 ) we obtain,

@ T N J,T dr I

= D-l).

C A. Brebbia

200

One can t h e n assemble a l l t h e elements t o g e t h e r and o b t a i n t h e f o l l o w i n g m a t r i c e s f o r t h e whole s t r u c t u r e ,

U*’T(AT

CI

-

P) + a *,T ( A U - F C I - Q ) = O

(58)

where a n d g a r e t h e nodal s t r e s s e s and d i s p l a c e m e n t s f o r t h e whole s t r u c t l i r e . The f i n a l system of e q u a t i o n s can b e w r i t t e n a s ,

N o t i c e t h a t t h e system i s symmetric b u t n o t p o s i t i v e d e f i n i t e . i i i ) Hybrid Models I n t h e c a s e of h y b r i d models w e can s t a r t w i t h t h e f o l l o w i n g s t a t e m e n t ,

r

r2

which i n m a t r i x form can b e w r i t t e n a s

The LJ v e c t o r r e f e r s t o t h e bouqdary d i s p l a c e m e n t s o n l y and t h e g needs t o s a t i s f y t h e e q u i l i b r i u m e q u a t i o n s . Furthermore on t h e c x t e r n a l r l b o d n d a r i e s t h e d i s p l a c e m e n t s u s u a l l y i d e n t i c a l l y s a t i s f y t h e uk = U, boundary c o n d i t i o n s . With t h e s e c o n d i t i o n s i n mind we c a n d e f i n e , a = &

-

-

_e

-

_e

u=$ITLl I

The boundary t r a c t ons a r e w r i t t e n i n f u n c t i o n of t h e nodal stresses t = N $

-

T

0

-e

(63)

The Unification of Finite Elements &Boundary Elements

20 1

Now w e c a n w r i t e e q u a t i o n (61) i n t h e f o l l o w i n g d i s c r e t i z e d manner; i . e .

A = -e

1--

$ N

T

-$

T

dr

(boundary i n t e g r a l )

Equations (61) can now b e w r i t t e n a t t h e element l e v e l a s ,

This system of e q u a t i o n s can now b e s o l v e d , a l s o a t t h e element l e v e l by p a r t i t i o n i n g , i . e .

where Ke i s t h e s t i f f n e s s m a t r i x f o r t h e element r e l a t i n g t h e n o d a l d i s placemSnts t o t h e e q u i v a l e n t nodal f o r c e s . Consequently t h e r e s u l t i n g model can b e used i n a s t i f f n e s s code w i t h o u t any s p e c i a l problems. iv)

F l e x i b i l i t y Model

I f t h e s t r e s s e s s a t i s f y e q u i l i b r i u m w i t h i n t h e body and we r e q u i r e t h e s u r f a c e t r a c t i o n s t o i d e n t i c a l l y s a t i s f y t h e boundary c o n d i t i o n s tk = tk on r2, we c a n u s e t h e f o l l o w i n g e x p r e s s i o n as a s t a r t i n g p o i n t f o r a f l e x i b i l i t y model,

This i s a form of t h e p r i n c i p l e of v i r t u a l f o r c e s . The development f o l l o w s t h e p r e v i o u s l i n e s and w e w i l l o b t a i n a f l e x i b i l i t y m a t r i x F f o r o u r problem. U n f o r t u n a t e l y t h e e q u i l i b r i u m c o n d i t i o n s r e q u i r e d - f o r t h i s p r i n c i p l e a r e g e n e r a l l y d i f f i c u l t t o s a t i s f y . I n many c a s e s i t i s e a s i e r t o work i n terms of s t r e s s f u n c t i o n s which i d e n t i c a l l y s a t i s f y t h e e q u i l i b r i u m e q u a t i o n and t h e n t h e s t a r t i n g p o i n t i s e q u a t i o n (3), i . e .

C A . Brebbia

202

and x define the governing compatibility equations and stress where the function respectively. Flexibility models with a few exceptions, are not popular in solid mechanics, consequently they will not be discussed here. The interested reader is referred to [23]. v)

Boundary Model I I _ _

The starting point for the boundary element model is given by the following principle,

I

I

The equation can be written in matrix form as follows, (71)

One can assume that the boundary of the domain is divided into elements and that the u and t functions can be approximated on each element, i.e.

The 4 interpolation functions are similar to those of finite elements but with-one degree of dimensionality. They are usually taken to be of the same order of u and t for simplicity. Application of equation (70) at different points on the boundary produces a system of equations, i.e. H U = G T + B

-..

.-

-

(73)

are the values of displacements and T the values of tractions taken at the boundary nodes. After having assembled equation (73) one can apply the boundary conditions on r l and r2 and solve the system for unknown values of body tractions and displacements. The resulting system of equations is not generally symmetric

.

4.

COMBINATION OF MODELS

In many practical applications it is important to be able to combine some of the above techniques. We have already described how hybrid models can be easily incorporated into displacement finite element models. These

203

The Unification of Finite Elements & Boundary Elements have many p r a c t i c a l a d v a n t a g e s as t h e h y b r i d f i n i t e e l e m e n t s a r e i n many c a s e s more a c c u r a t e than t h e d i s p l a c e m e n t models.

I n s p i t e of t h e many p o s s i b i l i t i e s o f f e r e d by t h e combination of d i f f e r e n t Sometimes methods, few r e s e a r c h e r s u s e more t h a n one t e c h n i q u e a t a t i m e . however, e n g i n e e r s a r e f o r c e d t o look i n t o some model combination b e c a u s e of t h e s p e c i a l c h a r a c t e r i s t i c s of t h e problem. Cases such a s o f f s h o r e s t r u c t u r e s , b u i l d i n g s on s o i l f o u n d a t i o n s , e t c . may r e q u i r e a f i n i t e element a n a l y s i s coupled w i t h a s p e c i a l a n a l y s i s f o r t h e w a t e r o r s o i l . Many of t h e s e a n a l y s e s a r e nowadays c a r r i e d o u t by c o u p l i n g f i n i t e and boundary element s o l u t i o n s . These c o m b i n a t i o n s and t h e u s e of approximate solutions w i l l be discussed in t h i s section.

I n o r d e r t o e f f e c t t h e combination one should f i r s t n o t i c e t h a t t h e v a l u e s In of T i n e q u a t i o n ( 7 2 ) a r e t h e a c t u a l s u r f a c e t r a c t i o n s a t t h e nodes. f i n l t e elements i n s t e a d t h e s e v a l u e s a r e weighted a s shown i n t h e r i g h t hand s i d e of e q u a t i o n (51) and t h e " i n t e g r a t e d t r a c t i o n s " a r e c o n c e n t r a t e d a t t h e nodes. These v a l u e s a r e r e p e a t e d by t h e v e c t o r p. I t i s now and T through a d i s t t i b u t i o n m a t r i x M p o s s i b l e t o r e l a t e t h e v a l u e s of whose terms r e p r e s e n t t h e w e i g h t i n g of t h e boundary v a l u e s of t h e t r a c t i o n s by t h e i n t e r p o l a t i o n f u n c t i o n s , i . e . P = M T

-

I

(73)

-

I n o r d e r t o combine t h e boundary element r e g i o n w i t h t h e f i n i t e element p a r t , one c a n deduce a m a t r i x which can b e e a s i l y implemented i n f i n i t e element codes. We s t a r t by t r a n s f o r m i n g e q u a t i o n (72) and i n v e r t i n g G , i.e.

G-'(H -

u - B) -

T

=

(74)

I

t o obtain Next one m u l t i p l i e s b o t h s i d e s by t h e d i s t r i b u t i o n m a t r i x t h e weighted t r a c t i o n v e c t o r s , p of f i n i t e e l e m e n t s , as f o l l o w s ,

These terms can b e r e d e f i n e d u s i n g K'

=

M G-' I

P'

-

H ~

= M T + M G - ' B

- -

_

I

-

Hence e q u a t i o n (75) p r e s e n t s now t h e same form a s f i n i t e e l e m e n t s , i . e . K' U

- -

=

P'

-

(77)

The main d i s c r e p a n c y t h a t a r i s e s now between t h i s f o r m u l a t i o n and f i n i t e element d i s p l a c e m e n t models i s t h a t t h e m a t r i x K ' i s g e n e r a l l y asymmetric. The asymmetry i s due t o t h e a p p r o x i m a t i o n i n v o l v e d i n t h e d i s c r e t i z a t i o n p r o c e s s and t h e c h o i c e of t h e assumed s o l u t i o n . The m a t r i x can b e made symmetric by minimizing t h e s q u a r e of t h e e r r o r s i n t h e non-symmetric o f f d i a g o n a l terms a s t h e asymmetry i s small i n most p r a c t i c a l a p p l i c a t i o n s .

204

C A. Brebbia

T h i s g i v e s a new m a t r i x whose c o e f f i c i e n t s a r e d e f i n e d by

T h i s m a t r i x c a n now b e assembled w i t h t h e f i n i t e e l e m e n t d i s p l a c e m e n t m a t r i c e s a s u s u a l . The d i s a d v a n t a g e of t h i s t e c h n i q u e i s t h a t t h e e q u a t i o n s w i t h i n t h e b o u n d a r y e l e m e n t r e g i o n a r e a l l c o u p l e d , which g i v e s a f u l l s y s t e m of e q u a t i o n s . Because of t h i s i t i s sometimes p r e f e r a b l e t o u s e a p p r o x i m a t e boundary e l e m e n t s . T y p i c a l c a s e s w h e r e t h e r e s e a r c h e r s may p r e f e r t h e a p p r o x i m a t e o n e s a r e when t h e i n f i n i t e medium i s d e s c r i b e d u s i n g boundary e l e m e n t s . Approximate boundary e l e m e n t s a r e b a s e d on t h e a s s u m p t i o n t h a t s u f f i c i e n t l y f a r from t h e r e g i o n u n d e r c o n s i d e r a t i o n - which c a n b e assumed t o b e d i s c r e t i z e d u s i n g f i n i t e e l e m e n t s - t h e b e h a v i o u r of t h e f u n d a m e n t a l s o l u t i o n can b e approximated. This approximation r e s u l t s i n a simple e x p r e ssi o n f o r t h e boundary s o l u t i o n a t t h e i n t e r f a c e w i t h o u t i n v o l v i n g t h e n e i g h b o u r i n g p o i n t s . The f u l l e x p l a n a t i o n of t h e way t h e s e s o l u t i o n s c a n b e deduced i s g i v e n i n r e f e r e n c e s [ 2 6 ] [ 2 7 ] . The methodology t o d e t e r m i n e t h e s e cond i t i o n s c a n b e a p p l i e d t o a l a r g e number o f . p r o b l e m s and i t i s i n t e r e s t i n g t o mention t h a t when a p p l i e d t o c e r t a i n c l a s s i c a l p r o b l e m s , i t p r o d u c e s some w e l l known e q u a t i o n s , s u c h a s t h e Somerfeld r a d i a t i o n c o n d i t i o n s

5.

CONCLUSIONS

The p u r p o s e of t h i s c h a p t e r h a s been t o p r e s e n t t h e common b a s i s of t h e more i m p o r t a n t methods of s o l u t i o n s u s e d i n e n g i n e e r i n g a n a l y s i s . These methods which c a n b e c a l l e d DISCRETE ELEMENT TECHNIQUES encompass t h e c l a s s i c a l s t i f f n e s s f i n i t e element formulat i o n , mixed, h y b r i d and f l e x i b i l i t v t e c h n i q u e s a n d t h e newly d e v e l o p e d boundary e l e m e n t method. I n t h i s c h a p t e r t h e common b a s i s of a l l t h e t e c h n i q u e s i s s t r e s s e d by p o i n t i n g o u t how t h e p r i n c i p l e of v i r t u a l work c a n b e u s e d t o g e n e r a t e s o l i d mechanics models. The s i m p l i c i t y of t h e v i r t u a l work a p p r o a c h a l l o w s f o r t h e f o r m u l a t i o n of v e r y g e n e r a l a p p r o x i m a t e m e t h o d s , v a l i d even f o r n o n l i n e a r and time d e p e n d e n t p r o b l e m s . The f o r m u l a t i o n of t h e d i f f e r e n t t e c h n i q u e s - i n c l u d i n g boundary e l e m e n t s - becomes t h e n i n d e p e n d e n t of t h e e x i s t e n c e o r o t h e r w i s e of a known i n t e g r a l o r v a r i a t i o n a l s t a t e m e n t . A g e n e r a l i z e d p r i n c i p l e i s p r e s e n t e d f r o m which a l l t h e d i f f e r e n t models a r e deduced. The p r i n c i p l e h a s been p r e s e n t e d u s i n g c o n s i d e r a t i o n s of v i r t u a l work o n l y , r a t h e r t h a n b y e x t e n d i n g t h e P r i n c i p l e of Minimum P o t e n t i a l Energy. T h i s new d e d u c t i o n may have i m p o r t a n t a p p l i c a t i o n s i n o t h e r c a s e s a s v i r t u a l work i s more g e n e r a l t h a n t h e e n e r g y p r i n c i p l e s . By i d e n t i c a l l y s a t i s f y i n g some of t h e s t a t i o n a r y c o n d i t i o n s i m p l i e d b y t h e g e n e r a l i z e d f u n c t i o n a l one can o b t a i n t h e d i f f e r e n t d i s c r e t e element t e c h n i q u e s.

The c h a p t e r ends by p o i n t i n g o u t a way of combining f i n i t e a n d boundary e l e m e n t s s t r e s s i n g t h e p o s s i b i l i t y of u s i n g a p p r o x i m a t e boundary e l e m e n t s a s described i n references [10][26].

The Unification o f Finite Elements &Boundary Elements

205

REFERENCES JASWON, M.A. "Integral Equation Methods in Potential Theory, I". Proc. R. SOC. A., 1963, 275. SYMM, G.T. "Integral Equation Methods in Potential Theory, 11" Proc. R. SOC., A, 1963, 275. MASSONET, C.E. "Numerical Use of Integral Procedures" in Stress Analysis, Zienkiewicz, O.C. and Holister, G.S. (Eds) Wiley, 1966. HESS, J.L. and SMITH, A.M.O. "Calculation of Potential F l o w about Arbitrary Bodies" Progress in Aeronautical Science, 8 , Kuchemann, D (Ed.) Pergamon Press, 1967. CRUSE, T. "Application of the Boundary-Integral Equation Solution Method in Solid Mechanics" in "Variational Methods in Engineering, Vol. 11" (C.A. Brebbia and H. Tottenham, Eds) Southampton University Press, 1973 and 1975. BREBBIA, C.A. and TOTTENHAM, H. (Eds) "Variational Methods in Engineering" ( 2 volumes) Southampton University Press, England, 1973. Reprinted 1975. BREBBIA, C.A. "Weighted Residual Classification of Approximate Methods" Applied Mathematical Modelling, 2 , September 1978. BREBBIA, C.A. "The Boundary Element Method for Engineers" Pentech Press, London, Halstead Press, NY, 1978. Reprinted 1980,1982. BREBBIA, C.A. and WALKER, S. "Boundary Element Techniques in Engineering" Butterworths, London, 1979. BREBBIA, C.A., TELLES, J. and WROBEL, L. "The Boundary Element Technique - Theory and Applications in Engineering" Springer-Verlag, Berlin and NY, 1984. [ 1 1 1 BREBBIA, C.A. (Ed.) "Recent Advances in Boundary Element Methods'' Proc. 1 s t Int. Conf. on BEM, Southampton University, 1978. Pentech

Press, London, 1978. [ 121 BREBBIA, C.A. (Ed.)

"New Developments in Boundary Element Methods" Proc. 2nd Int. Conf. on BEM, Southampton University, 1980. CML Publications, Southampton, 1980.

[ 131 BREBBIA, C.A. (Ed.)

"Boundary Element Methods" Proc. 3rd Int. Conf. on BEM, Salifornia, 1981. Springer-Verlag, Berlin & NY 1981.

[ 1 4 ] BREBBIA, C.A. (Ed.)

"Boundary Element Methods in Engineering" Proc. 4th Int. Conf. on BEM, Southampton, 1982. Springer-Verlag, Berlin & NY, 1982.

[ 151 BREBBIA, C.A., FUTAGAMI, T. AND TANAKA, M. (Eds)

"Boundary Elements" Proc. of 5th Int. Conf. in BEM, Hiroshima, 1983. Springer-Verlag, Berlin-Ny, 1983.

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[ 1 6 ] BREBBIA, C . A . (Ed.) " P r o g r e s s i n Boundary Element Methods, V 0 1 . l " P e n t e c h P r e s s , London, H a l s t e a d P r e s s , N Y , 1982. [ 171 BREBBIA, C . A . (Ed.) " P r o g r e s s i n Boundary Element Methods, V 0 1 . 2 " P e n t e c h P r e s s , London, S p r i n g e r - V e r l a g , N Y , 1983.

(Ed.) " P r o g r e s s i n Boundary Element Methods, V 0 1 . 3 " S p r i n g e r - V e r l a g , B e r l i n and N Y , 1984.

[ 181 BREBBIA, C . A .

[ 1 9 ] REISSNER, E. "A Note on V a r i a t i o n a l P r i n c i p l e s i n E l a s t i c i t y " I n t . J . S o l i d s a n d S t r u c t u r e s , 1 , 1965, pp.93-95 a n d 357. " V a r i a t i o n a l Methods i n E l a s t i c i t y a n d P l a s t i c i t y " [ 2 0 ] WASHIZU, K. 2nd E d i t i o n . Pergamon P r e s s , N Y , 1975. [ 2 1 ] PIAN, T.H.H. and TONG, P. " B a s i s of F i n i t e Elements f o r S o l i d Continua" I n t . J . Num. Method. Engg., 1 , 1969, pp.3-28. [ 2 2 ] ARISTOTLE

"Physics"

Oxford U n i v e r s i t y P r e s s , 1942.

" B a s i c P r i n c i p l e s " Opening A d d r e s s . 5 t h I n t . Conf. [ 2 3 ] BREBBIA, C . A . on BEM, H i r o s h i m a , Nov. 1983. P u b l i s h e d i n "Boundary Elements" N Y , 1983. (C.A. B r e b b i a , e d . ) S p r i n g e r V e r l a g , B e r l i n

-

[ 241 DANSON, D . " L i n e a r I s o t r o p i c E l a s t i c i t y w i t h Body F o r c e s " C h a p t e r i n " P r o g r e s s i n Boundary Elements'' Vol. 2 (C.A. B r e b b i a , Ed.) P e n t e c h P r e s s , London and S p r i n g e r - V e r l a g , N Y , 1983. [ 251 WENDLAND, W.

"Asymptotic Accuracy and Convergence" C h a p t e r i n " P r o g r e s s i n Boundary Element Methods" Vol. 1 (C.A. B r e b b i a , e d . ) P e n t e c h P r e s s , London, H a l s t e a d P r e s s , N Y , pp.289-313, 1981.

[ 261 BREBBIA, C . A . "Coupled Systems" S e c t i o n i n "Handbook of F i n i t e Elements" (H. K a r d e s t u n c e r a n d J . N o r r i e , Eds.) MacGraw H i l l , N Y , 1984. [ 271 BREBBIA, C . A . "New Developments" I n v i t e d L e c t u r e p u b l i s h e d i n P r o c e e d i n g s of t h e I n t e r n a t i o n a l C o n f e r e n c e on F i n i t e Element Methods, August 1982, S h a n g h a i , C h i n a . P u b l i s h e d by Gordon & B r e a c h , S c i e n c e P u b l i s h e r s , N Y , 1982.

Unification of Finite Element Methods H. Kardestuncer (Editor) 0 Elsevier Science Publishers B.V.(North-Holland), 1984

207

CHAPTER 9 UNIFICATION OF FEM WITH LASER EXPERIMENTATION

H. Kardestuncer & R.J. Pryputniewicz

Unification of finite element methods with laser experimentation is presented. It is pointed out that most engineering problems contain regions in which finite element modeling encounters difficulties due to nonlinearities, irregular boundaries, ambiguous energy functionals, etc.

Measurements

obtained

by

laser experimentation, particularly in these regions, can be digitized and automatically incorporated into the finite element modeling to improve results. Unification is possible in solid mechanics, fluid mechanics, gas dynamics, heat transfer, and in an everincreasing number fields.

of

other

INTRODUCTION Solution methodologies for engineering problems can, in general, be categorized as experimental, analytical, and numerical. In the recent past, the emphasis appears to have shifted from the first to the last. Certainly, each methodology has considerable advantage over the others for a given class of problems and each makes use of the others for verification of the results. In many cases, even the data furnished by one methodology has been utilized b y the others.

In spite of recent advances in number crunching equipment which have drawn considerable attention to numerical

208

H. Kardestuncer & R. J. Pryputniewicz

methodologies, in particular to the finite element methods, the importance of experimental, analytical, or semi-analytical methods has not diminished. Today's demands for optimum and reliable design are, to a great extent, satisfied b y application of finite element methods. In these applications, the finite element methods are used to solve problems for which exact solutions are nonexisting, or are very difficult to obtain. Also, the finite element methods are the only way to analyze complex three-dimensional structures, response of which to applied load system cannot be predicted in any other way. However, results obtained b y the finite element methods are subject to the boundary conditions used, rely greatly on the accurate knowledge of material properties, depend on accurate representation of the object's geometry, and are sensitive to the shape and size of elements employed in modeling. All of the information necessary to "run" the finite element models is obtained either from published data (for example, material properties), from design specifications (object's geometry), or from experimental studies (boundary conditions, shape and size of elements).

As is often the case with new and powerful methods, the finite element method has been over-used, perhaps even misused. Only recently have we begun to realize that virtually all versions of FEM contain some shortcomings. As a result, the need for unifying (merging, coupling) FEM in the physical and time domain with other methods has begun to manifest itself (see, for instance, Kardestuncer (1975, 1978, 1979, 1980, 1982), and Zienkiewicz et al. (1977, 1980). Here we are interested in exploring the unification of laser experimentation with FEU in space and time simultaneously. Other experimental techniques which can be used in conjunction with the finite element methods employ strain gauges, photoelastic procedures, etc. These experimental procedures,

Unification o f FEM with Laser Experimentution

209

although conventionally used, do not provide all the information necessary to reliably model an object's response to the applied load system by the finite element methods. For example, strain gauges give only pointwise information for the surface of the object directly under them; to obtain a complete strain mapping a large number of strain gauges must be bonded to the surface of interest. The procedure, moreover, is invasive and interferes with the object's performance. In photoelastic modeling, on the other hand, an oversimplification is made because the object is formed in plastic which has properties totally different from those of the actual material the object is made of. Although such a model, when observed under polarized light, is very useful in identification of stress fields, it does not represent the true response of the object to the applied force system. Ideally, what is needed is an experimental method that would provide necessary displacements and/or deformations at any point on the investigated object. Also, the results should be provided in three dimensions with high accuracy and precision in such a way as not to interfere with the object's performance. Recent advances in the field of optoelectronics have led to development of methods satisfying the above requirements. These methods utilize lasers as a source of light and as such can be called the laser methods. Although there are several laser methods available today, of particular interest to finite element analysis are: ( i ) hologram interferometry, ( i i ) heterodyne hologram interferometry, ( i i i ) laser speckle metrology, (iv) fiber optic metrology, and ( v ) directed light beam metrology. Each of these methods has certain characteristics which make it particularly useful in specific experiments. In general, however, all of these methods allow highly accurate, precise, noninvasive, rapid determination of the object's response to the applied load system.

2 10

H. Kurdestuncer & R. J. Pryputniewicz

In this paper, laser methods are described, including their representative applications, with particular emphasis on their unification with the finite element methods to improve the results.

UNIFICATION IN ERROR ANALYSIS The most important issue in any approximate procedure in engineering is the accuracy of the results. they? What are the upper bounds of errors?

How good are Such questions

have always been asked though answers have not always been found. Nevertheless, problems were solved and systems were put into service.

The easiest response to these questions has

always been the use of a factor of safety (FS) big enough to accommodate all uncertainties. How big should it be has, of course, been another question. If it was big enough, the engineer was successful; if not, he was doomed. An alternative approach to these questions has been to experiment (full scale, half scale, whatever) before putting the system into service. Recognizing that things designed and built yesterday are not as complex as those designed and built today, experimentation and the choice of FS were relatively easier tasks than they are today.

In recent years, however,

the availability of numerical tools (both in respect to methodology and equipment) has enabled engineers to design very complicated systems by successfully solving very difficult problems. Nevertheless, one question raised above still remains: how good are the results? When we examine the finite element methods, for instance, we find that error sources are quite numerous. Basically, these sources can be categorized as mathematical modeling, discretization, and manipulation (Melosh and Utku (in print)). In addition, each of these has many subdivisions of error sources. To address all and come up with a generally

21 1

Unification of FEM with Laser Experimentation

acceptable methodology for error bounds might very well be the most difficult task in numerical methods today. Some of the error sources are rather general--tool-dependent ( i . e., they include errors due to equipment, methodology, solution procedures, etc.); others are very

specific--problem-dependent

( i . e., they include physical and

geometric characteristics of the domain). difficult to deal with.

The latter are more

Many have addressed the question of error bounds for problems of the first kind in finite element methods; in particular, are works by Babuska and Rheinboldt (1977, 1978, 1980), Szabo and Mehta (1978), Peano et al. (1979), Utku and Melosh (1984), and a very fine work on a Dosteriori error analysis by Kelly et al. (1983). Error bounds and controls for problems of the second kind, in particular for those which are time and

a

dependent, have yet to be established. When it comes to the finite element methods, certainly

h and E

(mesh size and order of polynomials, respectively) are the more important (or, the easier) parameters to play with in estimating or even minimizing error bounds which are due to discretization only. The work b y Kelly et al. (1983), cited earlier, estimates and minimizes error bounds based on information obtained during the solution process itself. Using two independent error measures consisting of an error indication and an error estimation, they establish certain criteria for where to refine a given mesh and when t o stop adaptive processes. The programs developed using either or both (the latter, often referred to as the pony express policy, is claimed to be the better) are called self-adaotive processes because they require no interaction on the part of the user. Supposedly, it is also more practical and less expensive than theoretical a Driori error estimates and classical approaches requiring multiple analyses. Self-adaptive processes, however, are tested for linear and

212

H. Kardestuncer & R.J. Prvputrriewicz

self-adjoint boundary value problems only. One of the main features of a Dosteriori error analysis presented by Kelly et al. (1983) is that it involves local rather than global computations. It also necessitates establishing an energy functional beforehand for a given problem. In spite of many useful properties of energy functionals, there are a good many problems for which one can not come up with a functional which is valid for all stages of the problem. Furthermore, in self-adaptive processes, the coefficients characterizing the physical domain must be well-defined and their variations in respect to time and path must be sufficiently smooth. Otherwise, codes developed based on h- and p-processes will be insufficient. Nevertheless, they cover at least one aspect of error analysis and minimization. The fact remains, however, that development of fully automatic self-adaptive processes is one of the most crucial needs of finite element computations today. To achieve this, one must not develop algorithms based on the computed information alone. Instead, information based on actual measurements made during the processing must also be incorporated into the algorithm. These measurements (observations) should be employed not for veryfying the computed values of the unknown function ( a s is often done) but for estimating and even controlling errors. When discretizing the domain, engineers generally pay attention to certain regions of the domain which are critical or very sensitive to changes in h- and p-refinement parameters. Localized error norms in these regions may fluctuate drastically or even diverge as in the case of ill-conditioned systems of equations. If uc and u, represent the computed and measured values of the unknown function, respectively, then the error can be defined as either ec = u - uc or em = u - u,, where u ia the correct answer. Moreover, if the measurements are of very high precision, then we suggest that the latter be employed for error estimates and

Unification of FEM with Laser Experimentation

213

for adaptive processes in the critical regions of the domain. In structural mechanics problems, the energy of the error corresponding to a particular solution is given as 1

T2= j e r d R , n

(1)

where r represents the residual forces (Kelly et al. (1983)). Following the same procedure as in Kelly et al. (1983) one can obtain the possible refinement on u by using a hierarchic mode (the finite element basis function for the polynomial of

Np+l

degree ptl). Since the energy absorbed by this additional mode is assumed to be directly proportional to the corresponding force and inversely proportional to the stiffness, the above equation €or the ith element becomes

Hence, this procedure suggests that among all the available

Np+l

,

the one that gives the greatest error decrease should be

chosen as the new refinement. One should, however, make sure that Np+l is not orthogonal to ri otherwise ni = 0 indicating that the estimate may be deceptive.

UNIFICATION IN EVALUATING ELEMENT MATRICES Experimental techniques can also be used for direct evaluation of the element stiffness and/or flexibility coefficients. particular, when an element's

In

shape is irregular ( i . e.,

possesses curved lines and surfaces, which is often the case at the free boundaries) or when the material is anisotropic, composite, nonlinear, or stratified, the computed stiffness matrix, even using higher order isoparametric elements (implying p-refinements), may not yield the accuracy desired.

H, Kardestuncer & R.J. Pryputniewicz

214

The h-refinements for those elements would, on the other hand, increase the number of equations to be solved, thereby decreasing the accuracy of the results. In the case of solid elements, for instance, (whether one-, two-, or three-dimensional), the stiffness matrices can be obtained experimentally. For this we shall refer to Castigliano's second theorem in tensor notation (Kardestuncer 1977 ) Thus,

.

where

represents strain energy stored in the element. Eq. 4 into Eq. 3 and keeping in mind that aujr

- I

for i,q = j,r, respectively;

Substituting

zero otherwise,

duiq

the result is

which can be written as

Kardestuncer (1969) has investigated the tensorial properties of Eq. 6 and the similarities with the following well-known tensorial equation in solid mechanics

Note that Eq. 7 is a physical equation (Hooke's Law) without geometry (i. e., direction but no distance) whereas Eq. 6 contains geometrv as well as phvsics.

Unification of FEM with Laser Experimentation

215

The bivalent version of the quadrivalent tensor on the right hand side of Eq. 6 is the stiffness matrix of the element. This equation indicates that the stiffness (or flexibility) matrix coefficients can be determined by observing (measuring) the variation of piq in respect to uj,. or vice versa. Note that in this equation i and j represent the nodes (the integration points in the standard FEM) of the element and q and r are the directions of coordinate axes (local and global). Today, there are many high precision instruments that can evaluate the stiffness (or flexibility) matrix coefficients of an element of any shape and material.

Here,

we emphasize the use of laser technology for such evaluation. Since the measurements are continuous (independent of time and path), the stiffness matrix coefficients for those elements (highly nonlinear both in respect to time and path) can be determined at any increment of time and/or load. These coefficients can then be incorporated into the global K prior to the solution procedure. Let us assume that the overall final stiffness matrix is partitioned as follow5

SYMMETRK

where the subscripts c and m identify the computed and measured entities, respectively. Note that in certain portions of K , the measured and computed elements are coupled and identified with subscripts c,m. Equation 8 can be further

216

H. Kardestuncer & R. J. Pryputniewicz

reduced to

Since the left hand side of this equation contains all the known entities (whether given, computed, or measured), its solution is possible and will yield the unknown values of the function at the nodes where no measurements have been taken. Once we have determined uIlc and residual force vector as

Ull,c,

we can compute the

The components of this vector corresponding to element i can then be utilized in Eq. 2 to determine the next refinement for the adaptive processes presented by Kelly et al. (1983).

We

shall now present various laser methods to obtain the measurements mentioned above.

HOLOGRAM INTERFEROMETRY The most useful of all methods of ho-Dgram interferome-ry, for finite element applications, is the two-beam, off-axis method (Pryputniewicz (1979, 1982a)).

In this method, the laser beam

is divided into two beams, as shown in Fig. 1.

One of the

beams is made to interact with the object, or scene being recorded (the so-called object beam) while the other beam does not interact with the object at all. In fact, the second beam

is a reference beam against which the object beam is recorded. A setup for recording holograms is made so that the object beam and the reference beam overlap in a given region of

Unification o f FEM with Laser Experimentation

space.

217

A s a result of this, the two beams interfere with each

other and the resulting interference pattern is recorded in a suitable medium (Smith (1977)). The exposed medium, upon processing, becomes a hologram.

The hologram is reconstructed

with the same setup that was used in recording, except that now it is illuminated with the reference beam alone. Of the images produced during reconstruction, the most applicable to finite element analysis is the virtual image. To observe the virtual image, the reconstruction should be viewed through the hologram as if it were a window. The image is seen in the space which was occupied by the object while the hologram was recorded, even though the original object had since been removed. The image observed has all the visual characteristics of the original object.

In fact, there is no visual test that can differentiate between the two. FRONT SURFACE MIRROR, (3) PLACES

SPATIAL FILTER, (2)PLAC ES I

’%OTOGRAPHiC PLATE Figure 1. Setup €or recording and reconstruction of holograms. Directions of propagation of the object beam and the reference beam are defined by position vectors b specified in respect to the x-y-z coordinate system.

H Kardestuncer & R. J. Pryputniewicz

218

There are three basic variations of hologram interferometry:

( i ) real-time, ( i i ) time-average, and (iii) double-exposure. Real-time holooram interferometry involves recording a single exposure hologram as shown in Fig. 1, processing it, and reconstructing it by illumination with the original reference beam.

The reconstructed image is superimposed onto the

original object which is also illuminated with the same beam as used in recording the hologram. If the object is now even slightly displaced and/or deformed, interferometric comparison between the holographically reconstructed image and the new state of the object occurs instantaneously (Fig. 2). The particular advantage of the real-time method is that different types of motion, dynamic as well as static, can be studied with a single holographic exposure.

Figure 2. Images obtained using real-time hologram interferometry: studies of microcracks in porous, ceramic components.

Unification o f FEM with Laser Experimentation

219

In time-averase interferometrv a single holographic recording of an object undergoing a cyclic vibration is made. With the exposure time long compared to one period of the vibration cycle, the hologram effectively records an ensemble of images corresponding to the time-average of all positions of the object during its vibration. the reconstruction of such a

In

hologram, interference occurs between the entire ensemble of the recorded images, with the images recorded near zero velocity contributing most strongly. As such, images reconstructed from the time-average

Figure 3. Time-average hologram of a vibrating cantilever beam: the first flexure mode.

hologram have intensity distribution given by the zero-order Bessel function (Fig. 3). In the case of stroboscopic illumination of a vibrating object, however, cosinusoidal intensity distributions are obtained.

The double-exoosure holouram interferometrv, which can be considered to be a special case of the time-average method (where only two exposures of the object are made in the same medium), is the most widely used of all holographic methods. In this method, the object is displaced and/or deformed between the two exposures. Therefore, the object beam during the second exposure is different from that used in making the first exposure. During reconstruction of the double-exposure hologram, both object beams are faithfully reconstructed, forming images of the object's initial and final positions. Since these images are formed in coherent laser light, they interfere with each other forming a pattern of bright and dark

220

H. Kardestuncer & R.J. Pryputniewicz

fringes resulting in cosinusoidal intensity variation of the image (Fig. 4). These fringes are a direct measure of changes in the object's position and/or shape which occured between the two exposures.

Figure 4. Double-exposure hologram of a hydraulic cylinder: pressure between the exposures was increased from 5,100 psi to 5,800 psi,

QUANTITATIVE INTERPRETATION OF HOLOGRAMS There are a number of methods dealing with interpretation of the fringes observed within the holographically reconstructed image (Stetson (1979),Schuman and Dubas (1979), Vest (1978), Pryputniewicz (1980a), Pryputniewicz and Stetson (1976, 1980)). The most general of these methods employs multiple observations of the holographic image. It results in displacement vector u expressed as a product of the inverse of the matrix formed by the sum of the projection matrices B with the matrix representing the sum of the observed vectors uOb (Stetson (1979), Pryputniewicz and Stetson (1980), Pryputniewicz (1980a))

In Eq. 11, i denotes the observation number with n being the

Unification of FEM with Laser Experintentation

22 1

total number of observations while uob is measured in the plane normal to the direction of observation and is defined by the corresponding B. The projection matrix B ' , for the ith direction of observation, can be either one of the following two types. If the projection is made in a direction parallel to the direction of observation, then the projection is normal; if it is not, the projection is oblique. The normal projection is defined as a difference between the identity matrix and the matrix resulting from the dyadic product of the ith unit observation vector with itself. In the case of the oblique projection, the matrix is formed by the dyadic product of the object's surface unit normal vector with the unit vector defining the particular direction of observation.

Figure 5 . Typical finite element breakup of an airfoil.

Of particular interest in FEM modeling (Fig. 5 ) is the application of double-exposure hologram interferometry in determination of strains

and rotations (Pryputniewicz and Stetson (1976)). In this case, the strain-rotation matrix f is determined directly from the parameters S (defining illumination and observation

H. Kardestuncer & R. J. Pryputniewicz

222

geometry) and S f (defining shape and distribution of fringes seen during reconstruction of a hologram) f =

.

[s's]-l[sTs1]

Decomposition of the matrix f into the symmetric part e and the antisymmetric part 0 gives strains and rotations, respectively.

c

-0.3

I

-0.5 \

-

INITIAL POSITION

---FINAL

..-..-

0.2

POSITIONHOLOGRAPHIC ANALYSIS 0'3

-

FINAL POSITION FINITE ELEMENT ANALYSIS

0.5

',

t

Figure 6. Displacements of a radially loaded airfoil: the finite element computations were subject to the boundary conditions obtained from the double-exposure holograms.

The matrix Sf appearing in Eq. 12 consists of fringe vectors, one for each direction of observation, which can be computed from the fringe patterns produced during reconstruction of the holograms, that is,

w = S,.D

,

(13)

Unification of FEM with Laser Experimentation

223

where w is the fringe-locus function, constant values of which define fringe loci on the object's surface, and 0 specifies coordinates at which the specific values of w were determined. Knowledge of the fringe vector is essential in quantitative interpretation of holograms (Fig. 6).

The fringe vector, as

expressed in Eq. 13 and used in Eq. 12, is also helpful is determining the system's optimum geometry for recording of holograms. It should be noted that analysis of holographically produced fringes does not depend on material properties at all.

In

fact, the holographic procedures are particularly suited for quantitative determination of a material's behavior.

constitutive

Figure 7. Double-exposure holograms of: ( a ) heated inclined plate, ( b ) heated horizontal rod.

Hologram interferometry is also very useful in heat transfer studies.

For example, Fig. 7 shows typical images recorded

during studies of heat transfer characteristics of flat and

H. Kardestuncer & R. J. Pryputniewicz

224

curved surfaces. From reconstructed images, temperature distributions can be determined to within a fraction of one degree, anywhere within the image, without any interference whatsoever with the studied “space’1.

160

0

-

HOLOGRAMS

THEORY

E =O E

$ 40 t v)

8 I

-1.0

l

l

20

,

-0.5

I

I

0.5 MODE SHAPE, ym

I

,

I .o

,

I

1.5

Figure 8. Quantitative study of the vibrating beams: ( a ) the time average hologram of the cantilever beam, ( b ) displacements corresponding to the third flexure mode shown in (a).

In the case of time-average hologram interferometry, displacements are found from (Pryputniewicz (in print))

where

is the laser wavelength,

btl

is the argument of the

Unification of FEM with Laser Experimentation h

225

A

zero order Bessel function, while S1z and S Z z are components of the unit vectors defining directions of illumination and observation, respectively. Typical results for a vibrating cantilever beam are given in Fig. 8 showing good agreement with the theoretical predictions. In the case shown, the beam

is vibrating in the third flexure mode as vividly depicted by the hologram (Fig. 8-a) where nodes are demarcated by the brightest fringes and antinodes by the darkest fringes; for this mode, the theoretically predicted frequency was 1772 cps, while that determined experimentally was 1733 cps. Also, Fig. 8-b shows very good agreement between the mode shape determined from the hologram and the mode shape predicted by a theoretical model, which was developed to simulate beam vibrations. However, it should be noted that the theoretical computation was subject to the boundary conditions which were provided from the results obtained

A 5 shown in Fig. 8-b, the maximum displacement of the beam, vibrating in the third flexure mode, is 1.05 microns. directly from the holograms.

In the manner similar to that described above, mode shapes at other frequencies can be studied using methods of hologram interferometry.

HETERODYNE HOLOGRAM INTERFEROMETRY Heterodyne hologram interferometry is similar to the double-exposure hologram interferometry in that, there also, two images of an object, at different states of stress, are recorded in the same medium. However, each of these images is recorded with a different reference beam, in such a way that the reference beams can later be reconstructed independently (Dandliker et al. (1976), Pryputniewicz (1982b)). This allows introduction, during the reconstruction process, of a small frequency shift between the two reconstructed and interfering light fields, resulting in an intensity modulation at a beat

H. Kardestuncer & R.J. Pryputniewicz

226

frequency of these light fields, for any point within the interference pattern.

I50

-

I40 130

0

120

THEOW EXPERIMENTS

I10 100 E E 90

-3 80 E! 70

$

60

50 40

30

20 10 0

0

0.5

1.0

1.5

STRAIN,

2.0

2.5

3.0

W h

Figure 9. Strains determined from a heterodyne hologram of a loaded cantilever beam.

The optical phase difference, corresponding to the displacement and/or deformation recorded within the hologram being reconstructed, is converted into the phase of the beat frequency of the t w o interfering light fields.

This phase, in

turn, is interpolated optoelectronically, resulting in determination of fringe orders to within 1/1000 of one fringe.

This high accuracy in determining fringe orders leads to determination of displacements to within 0.3 nm, and strains to within 0.000,02 X (Pryputniewicz (1982a, 1982b)).

Unification of FEM with Laser Experimentation

227

Representative results obtained using heterodyne hologram interferometry are shown in Fig, 9. In this case, a prismatic cantilever beam was loaded in the direction normal to its neutral plane, between the exposures of the heterodyne hologram. Resulting interferograms were scanned b y placing a fiber-optic detector probe in the image plane formed b y a lens placed between the hologram and the detector (Pryputniewicz (198Zb). The resulting phase measurements were then processed using the equations relating them to parameters characterizing the system used to record, reconstruct, and scan the heterodyne hologram. Figure 9 shows that the results obtained from the heterodyne holograms correlate very well with the theory. It should be noted that the measured strains ranged from 0.3 microns/m to 2.5 microns/m and were well below the values that can be reliably detected by conventionally used strain measuring devices. Also, the results presented in Fig. 9 were obtained without contacting the object at all, and without interfering with it in any other way. All measurements were made remotely by scanning the object's image, thus producing the results in a truly noninvasive manner.

SPECKLE METROLOGY Any object illuminated with laser light will seem to have a granular appearance. That is, its surface will appear to be covered with fine randomly distributed light and dark irregular spots. If the observer moves, these spots appear to twinkle and move relative to the object. This phenomenon is caused by each point on the object scattering some light toward the observer. In fact, the laser light scattered b y one point on the object's surface interferes with the light scattered by other points. In any region of space where these light fields overlap, a random pattern of interference spots is observed, These interference spots are known as

H. Kardesturicer & R.J. Pryputniewicz

228

'ispeckles". The size of the speckles depends on optical properties of the imaging system and directly influences the accuracy of measurements: the finer the speckles the higher the accuracy. Y

X c_)

OBJECT

Specklegrams are recorded by illuminating an object with a single laser beam; no reference beam is used (Fig. 10). The light scattered by the object (or transmitting medium in the case of fluid flow or gas dynamics applications) is imaged from one or more

directions onto a high resolution recording medium. For interferometric purposesr two exposures are made in SPECKLEGRAM N o 2

i \ *

SPECKLEGRAM N. I

Figure 10. Setup for simultaneous recording of two specklegrams from two different directions.

the same medium to record the object's initial and final configurations, unless tandem specklegrams are used where each configuration is recorded in separate media which are 1at er '' sa ndw i ch ed together. It

Developed specklegrams are analyzed b y sending a narrow laser beam directly through the specklegram (Fig. 11). Upon passing through the specklegram, the illuminating beam diffracts and forms a halo which is modulated by Young's fringes (Fig. 12). The frequency of Young's fringes is directly proportional to the magnitude of the displacement recorded by the specklegram,

Unification of FEM with Laser Experimentation

229

while their direction is normal to the direction of this displacement.

t"

Figure 11. Setup for reconstruction of specklegrams.

Recent studies (Stetson (1978), Pryputniewicz and Stetson (1980), Pryputniewicz (1980b)) show that the equations governing determination of displacements from specklegrams are exactly the same as those used €or quantitative interpretations of holograms. That is, Eq. 11 applies directly in quantitative speckle Figure 12.. ~ _ _ ~ ~ Young's i ~ afringe l pattern observed during reconstruction of a double-exposure specklegram.

metrology. This equation indicates that two specklegrams recorded from different directions are sufficient to compute three-dimensional displacements of loaded objects.

230

H. Kardestuncer & R. J. Pryputniewicz

The parameters necessary to interpret specklegrams are obtained directly from the geometry of the recording and reconstructing systems (Figs 10 and 11, respectively) and from the observed Young's fringes (Fig. 12).

Figure 13. Setup for simultaneous recording of two specklegrams in fluid flow analysis.

The speckle metrology finds particular applications in studies of three-dimensional displacements of solid objects, in studies of fluid flow (Fig. 13), and in gas dynamics. In these applications, the speckle methods allow recording of the displacement and/or deformation pattern over the entire surface of the object, permit recording of the entire velocity profiles or the thermal profiles and are particularly suited to studies of dynamic as well as transient behaviors.

Unification o f FEM with Laser Experimentation

23 1

COMPUTER AIDED INTERPRETATION OF LASER IMAGES In FEM modeling, coordinates of nodal points are known. To specify boundary conditions at these nodes, their position has to be established and reproduced while using the experimental methods.

One of such

methods involves scanning the holographically reconstructed image (or a diffraction halo obtained during reconstruction of a specklegram) with a computer compatible video digitizer, as shown in Fig. 14.

The

digitizer, in addition to converting the scene being observed into a composite analog video signal which is viewed on a monitor, produces a Figure 14. Schematic of a computer controlled system for automated interpretation of holograms.

digital signal that is transmitted directly to a computer.

The

computer, in turn, rapidly reads the

electronic signal corresponding to the video image being digitized.

It processes the digitized data, producing plots

of intensity distribution within the image plane. Data characterizing these intensity distributions, together with other pertinent parameters, are used in quantitative interpretation of laser images. These results can be obtained

H. Kardestuncer & R.J. Pryputniewicz

232

for any point within the reconstructed image by simply instructing the computer to perform calculations for a point, or a number of points, at specified coordinates. A system such as that shown in Fig. 14 will provide a unique capability for unification of finite element methods with laser experimentation. As such, it will lead to the development of a fully automated system for quantitative analysis of structural deformations, which will provide highly accurate and precise results at any point on the surface of the studied objects.

REFERENCES 1.

Babuska, I., and Rheinboldt, W. C., Computational aspects of the finite element method, in: Mathematical Software, Vol. I11 (Academic Press, New York, 1977).

2.

Babuska, I., and Rheinboldt, W. C., A posteriori error estimates for the finite element method, Int. J. Num. Meth. Engr., 12 (1978) 1597-1615.

3.

Babuska, I., and Rheinboldt, W. C., Reliable error estimation and mesh adaptation for the finite element method, in: Oden, J. T. (ed.), Computational Methods in Nonlinear Mechanics (1980) 67-108.

4.

Dandliker, R., Marom, E., and Mottier, F. M., Two-reference beam holographic interferometry, J. Opt. SOC. Am., 66 (1976) 23-30.

5.

Kardestuncer, H., Tensors in discrete mechanics, Tensor Quarterly - TSGB, 20 (1969) 1-9.

6.

Kardestuncer, H., Descrete Mechanics: Springer-Verlag, Vienna, 1975).

7.

Kardestuncer, H., Proceedings of the UFEM Symposium Series (University of Connecticut, Storrs, CT, 1978,

A Unified Approach

1979, 1980, 1982). 8.

Kardestuncer, H., Tensors versus matrices in discrete mechanics, in: Branin, F. H., Jr., and Huseyin, K. (eds.), Problem Analysis in Science and Engineering (Academic Press, New York, 1977).

9.

Kelly, D. W., de Gago, J. P., Zienkiewicz, 0. C., and

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233

Babuska, I., A posteriori error analysis and adaptive processes in the finite element method: Part I -- Error analysis, Part I1 -- Adaptive mesh refinement, Int. J. Num. Meth. Engr., 19 (1983) 1593-1619. 10.

Melosh, R. J., and Utku, S., Efficient finite element analysis, to appear in: Kardestuncer, H. (ed.), Finite Element Handbook (McGraw-Hill, New York).

11.

Peano, A. G., Pasini, A , , Riccioni, R. , and Sardella, L., Adaptive approximation in finite element structural analysis, Comp. & Struct., 10 (1979) 332-342.

12 *

Pryputniewicz, R. J., Laser Holography (Worcester Polytechnic Institute, Worcester, MA, 1979).

13.

Pryputniewicz, R. J., State-of-the-art in hologrammetry and related fields, Internat. Arch. Photogram., 23 (1980a) 620-629.

14.

Pryputniewicz, R. J., Projection matrices in specklegraphic analysis, SPIE, 243 (1980b) 158-164.

15.

Pryputniewicz, R. J., Unification of FEM modeling with laser experimentation, in: Kardestuncer, H. (ed.), Finite Elements - Finite Differences and Calculus of Variations, (University of Connecticut, Storrs, CT, 1982a).

16.

Pryputniewicz, R. J., High precision hologrammetry, Internat. Arch. Photogram., 24 (1982b) 377-386.

17.

Pryputniewicz, R. J., Quantitative interpretation of time-average holograms in vibration analysis, in print,

18.

Pryputniewicz, R. J., and Stetson, K. A , , Holographic strain analysis: extension of fringe-vector method to include perspective, Appl. Opt., 15 (1976) 725-728.

19.

Pryputniewicz, R. J., and Stetson, K. A., Fundamentals and Applications of Laser Speckle and Hologram Interferometry (Worcester Polytechnic Institute, Worcester, MA, 1980).

20.

Schuman, W., and Dubas, M., Holographic Interferometry (Springer-Verlag, Berlin, 1979).

21.

Smith, H. M., Holographic Recording Materials (Springer-Verlag, Berlin, 1977).

22.

Stetson, K. A., Miscellaneous topics in speckle metrology, in: Erf, R. K. (ed.), Speckle Metrology (Academic Press, New York, 1978).

23.

Stetson, K. A., The use of projection matrices in hologram interferometry, J. Opt. SOC. Am., 69 (1979)

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234 1705-1710.

24.

Szabo, B. A . , and Mehta, A . U., P-convergence finite element approximations in fracture mechanics, Int. J. Num. Meth. Engr., 12 (1978) 551-560.

25.

Utku, S., and Melosh, R. J., Solution errors in finite element analysis, Comp. & Struct., 18 (1984) 379-393.

26.

Vest, C. M., Holographic Interferometry (Wiley, New York, 1978).

27.

Zienkiewicz, 0. C., Kelly, D. W., and Bettess, P., The coupling of the finite element method and boundary solution procedures, Int. J. Num. Meth. Engr., 11 (1977) 355-373.

28.

Zienkiewicz, 0. C., Kelly, D. W., and Bettess, P., Marriage a la mode -- the best of both worlds (Finite elements and boundary integrals) in: Glowinski, R., Rodin, E. Y., and Zienkiewicz, 0. C. (eds.), Energy Methods in Finite Element Methods, Ch. 5 (John Wiley, New York, 1980).

Unification of Finite Element Methods H. Kardestuncer (Editor) 0 Elsevier Science Publishers B.V.(North-Holland), 1984

235

CHAPTER 10 LINEAR CROSSED TRIANGLES FOR INCOMPRESSIBLE MEDIA

D.S.Malkus & E. i? Olsen

This paper examines the error analysis for a rather remarluble type of finite element, which seems to be ideally suited for solving steady flow problems involving fluids with integral constitutive equations. The element is a quadrilateral macroelement of four linear trianglee, arranged so that their interior edges form the diagonals of the quadrilateral. The properties which are most useful in such calculations are the conctancy of velocity-gradients on the aubtriangles and exact incompressibility of the weakly constrained Lagrange multiplier solution. On the other hand, these elements have abundant 'spurious pressure modes" and thus fail to satiafy the requirements of the Brerzi - Babuska convergence theory, thought to be necessary t o establish convergence of the finite element solutions in simple, Stokesian flows. There is an apparent paradox in this, because without the spurious modes, a simple count of unconstrained degrees of freedom would predict that the element is useless for incompressible media. This paper discusses a new approach to error analysis for finite elements for incompressible media. Though error estimates can only be obtained for a rather restrictive class of problems at present, our results and those in a similar vein by other investigators seem t o resolve the apparent paradox of the crossed triangle macroelement: the reason for its success seemed to be the very same as the reason for its expected failure. While these results do not apply rigorously to nowNewtonian flow, they give us reason to expect that the good resultcl so far obtained in such problems are more than fortuitous coincidence.

1. INTRODUCTION The computation of steady-flow solutions for problems involving fluids with integral constitutive equatiom has attracted mnch interest among numerical modellerr recently [l-71.Much of this interest derires from the relevance of such calculations to the modelling of industrial polymer processing, and the potential usefulness of numerical modelling in sorting out which among the many proposed constitutive theories for riscwlastic materials gives the most faithful representation of observed polymer behavior. But there has also been a meat deal of interest in such problems because of the peculiar challenges and difficulties encountered in attempts to compute apparently very simple, twc-dimensional flows. Part of the challenge is that, for fluids w i t h intrgral constitutive equations, the stress involves a path integral along the historical path followed by the particle a t which the stress is evaluzted. The purpose of this paper is t o examine the error analysis of a particulsr finite element proposed for the computation of such solutions. In ref. 2 it is argued that this finite element is ideal for such computations-allowing the determination of exact relative strains (apart frcim rounding errors) in 3 finite element trial velocity field. The element is the crossed-triangle inacroelement, discovered by Nag?ignal, et al. (81 and analyced by hlercier [a]. In ref. 8, the element was found to be effective lor modelling elasto-plastic materials, and in ref. 10, was sucessfully employed in probleins involving large inelastic deformation. We refer to this element as the NRC element (for 'Nagtigaal redundant constraint').

D.S. Malkus & E. T. Olsen

236

This element fails t o satisfy the "discrete LBR condition" (the primary requirement of the Brezzi - Babuska theory [2,11-14,21,22,24,25]) in a most dramatic way. Therefore the discussion in this payer centers on the deceptively simple question as to whether the element can be expected t o woik even in Stokes-flow. This paper builds upon arguments presented in ref. 2, where the usefulness of the crossed triangle elemeot for non-Newtonian flows in argued in iome detail. We discnss in more detail the theorems proved in ref. 13, which lead to the establishment of error estimates for the NRC element in Stokes flow on simple meshes. We will show that the failure of the elemeot t o satisfy the LBB condition and the reasons for its success are two sides of the lame coin. and are explainable in terms of an error analysis which does not require the LBB condition. In non-Newtonian flows, normal forces are crucially important, which implies that accurate pressures must be obtainable from elements chosen for these,flows. Because the NRC element fails the LBB condition, it needs a poseprocessing of the computed pressures t o remove unstable modes (141. In ref. 2 a practical pressure-smoothing scheme is discussed, which seems to work well. An error analysis for the practical scheme is unknown to us at this time. Here we show t h a t there is a t least one smoothing scheme for which error estimates can actually be proved, even though it is not as computationally convenient as the scheme employed in practice. 2. STEADY F L O WS

OF MEMORY FLUTDS

2.1 Equations of Motion We solve the equations of steady flow, V.g+f=p(g.V)g,

for a velocity field, y. Incompressibility implies p = constant and !7=b-p€

v.g=o, for a suitably chosen hydrostatic pressure function, p. 8.2 The Covstitutive Equations

The constitutive equations we employ are of the following form proposed by Curtiss and Bird 1151:

-

d' ~ ~ [ T ; i ' f , A ( r ) m ( r ) d r +6T;i' 0' = gJ' - t ( U y l + d2)!.

f,&(r)mz(r)dr]

(3)

Td is the disengagement time. Its magnitude determines the effective memory of the fluid. t is t i e link-tension coeBcient of the Curtiss-Bird model [15]. The kinsmatic tensors and & are functions of the Cauchy and Finger-strain tensors of the deformation which carries the fluid from its reference state at time r -- 0 to ita conGguration -,t historical time r. is also a function of the strain-rate, <(O), a t the present time r = 0. The memory functiona of eq. (3) are given by (keeping in mind that r LO):

231

Linear Crossed Trianglesfor Incompressible Media

With this normalization, the parameter 1 . may be nsed t o adjust the zero-shear viscosity [15]. It should be noted that both &, and &?, are functions of the position variable and are, in principle, defined everywhere in the domain of the problem. The time dependence implied by eq. (3) arises by evaluating these tensors along the historical path followed by a particle which is at point I . at time r = 0. Thus,both of the integrals in eq. (3) are path integrals along a path parameterized by r . Construction of the path and evaluation of the path integral are numerical procedures crncial t o the success of our methods and are discussed in ref. 2. In ref. 2 it is shown t h a t the properties of the NRC element are vital t o the efficient evaluation of the path intrgral in eq.(3). Many other constitutire equations have a form similar to that described here [I] and can be treated by techniques described in ref. 2. Ref. 2 describes the calculation of the integrand of eq. (3) at various historical times, given a h i t e element trial velocity field. Quadrature formulas with specified degrees of precision with respect to the weighting functions m l ( r ) and m*(r) can be generated by classical orthogonal polynomial techniques 131. The time integral in eq. (3) is replaced by a finite weighted sum:

1:

f(a)m(a)da

C uif(ri),

(5)

I

where / ( a ) can be either integrand of eq. (3) and m(8) t he corresponding memory function. Then are th e weights and points computed once and for all for the appropriate memory function.

wi

and

ti

2.3 The Ga l e r k i n Equation As in ref. 2, we employ a standard Galerkin form of t he equations of motion, in which the incompres sibility is enforced by a penalty [2,3,13,14]

where I is the penalty parameter and fl is the spatial domain of the problem. Pressure is computed by (7)

p=-2zV.c Eq. (6) is restricted t o g,! drawn from a finite element trial space,

Sh. Given any estimate

#

E Sh for

the solution, we need t o be able to compute the residual of eq. (0) on which to base an iterative correction scheme. In order t o do this, a spatial numerical integration scheme must be employed, which results in a discrete finite sum in place of the space integral in eq. (9). 3.

THE NRC MACROELEMENT

As illustrated in Figure I, the NRC element is a quadrilateral formed by four linear triangles [IS]whose intcrior sides define the diagonals of the quadrilateral. A standard area-coordinate [16] traidorm;rt ion is used lor generation of all element quantities from a refcrence triangle. The geometry of each r x c r o is is uniquely uniquely specified by specifying the corner coordinates, 3;"(the location of the central node,

r,

determined by

Sic). The rectangle pictured in reference configuration in Figure

arbitrary quadrilateral.

1 can be mapped to an

238

D.S. Malkus & E. T. Olsen

t'

Figure 1

We employ a one-point centroidal integration formula on all terms of eq. (6). Note that for Stokes flow (i.e. let p = 0 and take the limit of eq. ( 8 ) a# Td n 0), t h h formula ia exact. Let T h denote the trial space of piecewise constant pressnres, based on the individual triangles. Consider the following related Galerkin equations: 1L Lagrnnge Multipller

In@ .v g h - p h v .c" - #V

The equivalence theorem of ref.

*

yh

+ p [ ( y h . V ) l p ].g"

VghESh, @€TI;

-e*f]dn=O,

17 which generalizes t h a t ref. 18 t o non-self adjoint problems implies that

when the one-point formda is applied uniformly ta A, R and eq. (8), eq. (6) and B produce the same solutions.

We assume that conditions are satisfied that guarantee the convergence of these solutions t o the solution of A an z H 00. Note that the integration is exact for Stokes-flow and for the presaure terns for any Td end p. One may easily deduce that for A

and V . S h

C T h .Thus V .yh = 0 pointwise for any solution to A. For B or cq. (8)

With the NRC element, we can set @ = V . yh

I,

and thus deduce from eq. (9) that

Linear Crossed Triangles for Incompressible Media

239

pointwise in element interiors. It is the exact incompressibility of the Lagraoge multiplier solution and the exact incompressibility of the penalty solution in the infinite limit of the penalty that is the kcy to the element's success in modelling flows of fluids with constitutive equations like eq. (3) 121. It means that such a solution has a stream-function, and since its curl gives the velocity field, which is demonstrably continuous, this stream-function is evidently continuously differentiable. Furthermore, since the derivatives of the stream-function are linear, the stream-function is a piccewisc quadratic (see ref. 2 for an expression for it). Thus the stream-functions associated with weakly incompressible velocity fields of the NRC element are quadratic splines [19]. We should point out that only very special elements like the NRC have discrete Lagrange multiplier solutions which satisfy eq. (8) and discrete penalty solutions which satisfy eq. (10). It is more usual for the right-hand sides of these equations (8) and (10) to involve additional terms which arc governed by mesh spacing and do not vanish in the infinite limit of the penalty. The bilinear / constmt-p elements of ref. 4 have this feature, and the larger compressibility of the discrete solutions obtained using this element probably contributed t o its less than adequate performance [1,4]in non-Newtonian flow calculations. But the NRC fails to satisfy the discrete LBB condition [ll-141. There is then a serious question as to whether thc element is stable, even for Stokes-flow. This is the question which is addressed in the remainder of this paper. 4.

THE ILL-DISPOSED PRESSURES OF THE NRC ELEMENT In ref. 20 the term "spurious mode' is applied t o the pressures ch E T h which satisfy

and differ from a constant 'hydrostatic mode" which occurs in some problems. These ch have also been mferred t o as 'chcssboardr" [Zl]or "checkerboards" owing to their geometric pattern. We prefer t o refer to these modes as 'ill-disposed" following ref. 22 - without the negative connotation which has become attached to such eh. In rcf. 22 it is argued that ill-disposed modes play. no role in determining natural or physical modes; likewise they may lead to inconsistent algebraic systems. But alternatively each ill-disposed mode has a dual velocity mode which is weakly incompressible. A prime example of this simple algebraic consequence of ill-disposed pressures is provided by the NRC element. A aimple count of unconstrained degrees of freedom [la]for the NRC element according t o

-

N d f = dim S" dim

e,

(12)

on requences of regular meshes leads t o the conclusion that the NRC in grossly overconstrained - as are other arrangements of linear triangles [IS].Let Uh C Th be the subspace satisfying eq. (11) of ill-disposed modes and possibly the hydrostatic mode. What the algebraic argument of ref. 22 is saying is that a careful coostraint count gives

N h f = dim Sh

- dim T" + dim Uh.

(13)

blercior ['J] recognized that dimUh was significant for the NRC. Given y h E Sh, considcr a typical NKC macro in any mesh, and let e l , cz, ea, er denote the four subtriangles, ordered as in Figure 1. Mercier proved that

v . y h I.1 -v

yh

,.I +v .gh Ic, -v .gh =.I

0.

(14)

tlcrcier then argues tbnt for Th as we have chosen, only three of the four incompressibility constraints are independent, since setting V , y h = 0 on three triangles forces it t o be zero on the fourth. This argument which shows that the NRC is not overconstrained can be turned around to show that thc NRC also has many ill-disposed modes. Consider again our typical macro, labelled ' W ,and define

ebI [;l/aeJ, whcre a'' h the area of triangle c i . Now

on triangle c;[+, i= 1,3 and outside of macro M.

-, i= 2,4];

(15)

D.S. Malkus & E. T. Olsen

240

since &I

and V . p" are constant on triangles. But substituting eqs. (14) and (15) into eq. (16) shows that

satisfies eq. (ll),thus cb E II". Note that there is one such ill-disposed mode for each macro. It is not hard to show that on regular rectangular meshes with boundary conditions which cause the bilinear/ constant- p element [ZO]to have a checkerboard mode, the NRC has a similar checkerboard mode satisfying eq. (ll),constant on each macro and alternating in sign between macros 1131. We refer to this as the "global ill-disposed mode" and those of eq. (15) as "local ill-disposed modes" for obvious reaqons. What we have seen is an instance of a simple algebraic truth [22] - redundant constraints and ill-disposed modes are one in the same. The NRC has a favorable constraint c o u t cnly because it has 'spurious pressure modes". The global mode of the NRC can cause algebraic consistency problems related to inhomogeneous boundary conditions [ZO]. This must be avoided the way it is with bilinear/ constant-p rectangles. We shall now prove that the local ill-disposed modes cannot lead to inconsistency. Inhomogeneous boundary conditions are often imposed as described in ref. 23. S" is taken to be the trial space satisfying homogeneous boundary conditions wherever essential b.c. are specified. tit is chosen to be wro at all mesh nodes except those which coincide with boundaries on which inhomogeneous essential b.c. are specified. At those nodes ti? interpolates t o the inhomogeneous data. Thus y" = W" + $1is sought

,c:

with yh E S" determined by the Galerkin equation. Constraint equation (8) becomes

Note that eq. (8) has the following algebraic iolwbility condition [20,22]. There is a solution

E S"

to eq. (8) only if (18)

We hare Theorem I: Let c% be a local ill-disposed mode of the NRC element, i.e.

cb in as in eq.

(1 1). Then for 9.h as defined above, the iolvahility equation (18) is satisfled.

P r o d :We note that the only reason eq. (18) does not follow from eq. (11) is that yk 6 Sh. Consider the Izrgcr space sh 2 S" of functions free on all boundaries: 5: E 3".Now observe that the argument of eq. (la) applies equally well to all p" E S", since it involves only one macroelement and does not depend on the

Imindary conditions. QED 1.1; The above argument does not apply to the global ill-disposed mode. As with :he bilinear/ constant-pressure rectangle, the global mode occurs on some meshes because there is a C" which is orthogonal to V . SA hut not V -B*. Remark 1.2; The local ill-disposed modes of the NRC are pieced together from ill-disposed modes of the element weak-gradient matrix [ZZ] and are not related to boundary conditions. Theorem 1 says that inhomogeneous boundary conditions cause no more difficulty for the NRC element than with the bilinear/ constant-p element. Boundary conditions chosen so that yk is such that eq. (18) is satisfied for the global ill-disposed mode will avoid the pathologies described in ref. 20. This, however, is a 10116way from guaranteeing convergence of the NRC approximation, even in Stokes-Bow. We now turn OUT attention to that question. I:crnnrk

24 1

Linear Crossed Triangles f o r Incompressible Media

5. APPROXIMATION ERRORS U S N G TBE NRC IN STOICES-FLOW The ill-disposed pressures lead to a non-uniqueness of the pressure solution from the Lagrange Multiplier mothod A for Stokes-flow. This assumes that any dEebraic inconsistency with thc global mode haa been avoided and results from the fact that if pk is a solution, so is pk + eh for ch E Uh.The velocity solution is utiiquo (201. The penalty method gives a unique pressure solution which tends to a representative pressure ioliition of the Lagrange multiplier method as I H 00 [24]. The question 01 convergence as the mesh is refined :an be resolved by resolving the question for the Lagrange multiplier solutions (gh ,p:), where

which is unique. To determine whether

(sh ,p$) converges to the exact solution (5. ,p') rn the mesh is refined, we 5rst

2oosider a generalized LBB condition [2,11-14,21,22,24,25]: C'or qh orthogonal to U h in h(0):

whcre 11. /I1 denotes the energy norm [22] (we assume essential boundary conditions imply that the energynorm is equivalent to the W1s2(n)x W1vZ(R)-norm).11. ]I, denotes the &(R)-norm. The desired rcsult for (gh , p i ) could be established if kh were bounded away from cero independent of h; unfortunately for many

NRC meshes, it is not. This has been rigorously established on some simple meshes [13]. 5.1 Velocity E s t i m a t e s without t h e Discrete

LBB C o n d i t t o n

It will be convenient in what follows to use t h e energy inner product [13,14,22]:

wlicre iij = $(ui,j+uj,i) and /ij = ~(ui,j+wj,i)for i , j = 1,2. We consider the subspace kVh functions satisfying the incompressibility constraint equation (8) and deline the operator zh

Sh of trial

:HiH W h

(22)

which is the projection from the space H I onto @ with respect to the energy norm. H I is a subspace of W',2(n)X W1s2(n) satisfying homogcneous essential boundary conditions on some or all of the boundary of n. For admissible pressures in the Lagrange multiplier method we take He= &(n). The exact solution (8' , p * ) satisfies the continuous analogue to A above:

where (.,.).

denotes the &(R) inner product. We assume that the domain, boundary conditions and

h ( n )X h(n)are such that 9'

E Wksn(n) X Wko2(R)for k 2 2 with go unique, and p'

F

E

E Wm-z(n)for rn>l

[12]. p' may not he unique when there is a hydrostatic mode [ZO], but hereafier we assume that in such c a e s p' is the unique representative pressure solution for which (pa,.)1 = 0.We now prove

Theorem 21 If (tj',p') is the solution toeq. (23)and ( g h , p L ) is the finite element solution

to the Lagrange multiplier method A in the Stokes- flow caae,

D.S. Malkus & E. T. Olsen

2 42 Proof8 Let

fl = 0 in problem A and q = 0 in eq. (23). Also in eq. (23) let p = vh E S h , a(g', a(#,

Subtracting gives

tp)- @*, v .g"). y h ) - @", v . $1.

-

a(yA y', g h ) = (p'

. F,l)* = ( y h . F,110

= (gh

- ph, v * yh)e

Now observe that yh E w" so that Zhgh = gh thus letting vh = gh-Zhg' and that

and using the fact that Z i = z h

is self-adjoint in a(., .) implies O(gh

- <,Uh - Zh<)

= a(gh

- z&g', - 49') = @'

- ph$ v

'

[

gh - ZhG

I

)o

(25)

We note that the righthand side of eq. (25) may not be zero since p' - p h Th,but it is amall since we p = best &(n) approximation to p' and add zero t o the right-hand side of eq. (25) in the form of

can let

[

I

@', v * gh - ,%'by

[ I

). Also (ph, v * gh - Zhy* )* = 0, which gives

-zh<,yh-

&yo) = @'

-'I

-p", v ' uh - Zhu

[-

-fl11011sh- ZhU' 11

)e < C l l P '

The inequality follows from the boundedness of the weak divergence, whose operator norm gives c 1121 Division by 11 9' &Yo 11 gives the desired result. QED

-

1

R e m a r k 2.1: This result should be compared t o the classical result for unconstraincd problelns, which ahows that the Bnite element solution is the best energy approximation to the exact solution. liere tLe FEhl solution is as close to the best weakly imcompressible approximation to the exact solution as the pressures are accurate. Remark 2.2: Theorem 2 shows that the primary role of the Lagrange multiplier is to constrain the FEM soliltion to be close to &go.

Remark 2.38 Only accuracy of the Lagrange multiplier space T h is required, not the stability of the pressure approximation. 'Theorem 2 is quite easily turned into an estimate for

11 yh - y' 11

Theorem 3:

which does not require the LBR

1

condition:

11 uh - g' /Il

11 g'

- &go 11 + e infqhCTh 11 p' - fill,, 1

Prooh Add and subtract g' and use the triangle inequality in Theorem 2.

QED Remark 3.1: The idea of constraint counting [IS]was designed to give a heuristic ideaof whether 11 g'

- zhg'

could be expected to be small based on an estimate of Nud.f = dimWh. Reaults similar to Theorem 3 have been established by others. We point out particularly the work of B. Mercier in ref. 24. He proves a result which differs from ours in the present application only in that he uses the standard norm on Wl**(n)X W'8z(fl) rather than the energy norm. This gives different absolute constants in the estimate. Mercier also points out an interesting sharpening of our estimate which applies to the NRC element:

Corollary 3.1 (Mercier): If the functions in W" are weakly incompressible with respect to multipliers in H., then

11 (I" - I'11,

11 Yo - zh9* 11,

Prooh The right-hand side of eq. (23) is rero in this case, implying the righthand side in Theorem 2 is bero. QED

11,

243

Linear Crossed Triangles f o r Incompressible Mediu

Remark 3.2: The result applies t o the NRC since weakly incompressibility implies pointwise incompressibility implies incompressibility w.r.t. E. multipliers. The work of Mercier predates ours but seems to have received less attention than it deserves because emphasis has since been placed on assuring the optimality of 11 9' - ZAY. 11 by choice of elements which 1

iatisfy the LBB conditon. Satisfaction of the LBB condition guarantees that 11 y'

- ZAP' I[

is on the order

1

of optimal approximation of strain-rates by S". This follows directly from the work of Brerzi [ll].There are ieveral ways which 11 ' u - 2~9.I( can be shown to be small when the LBB condition is not satisfied. The 1

intereated reader is referred to ref. 14. In ref. 24 a construction is rketched which is intended t o show that there is a @ E W hwith

1 I P" - Y* Ill iChll u* Ila on rectangular domain discreticed by aquare elements. Since the transformation of one rectangular domain to another b inRnitely differentiable, a simple change of variable would lead t o the same conclusion for a rectangular domain with rectangular elements. The construction in ref. 24 is very brief and contains a confusing misprint, and most important does not ehow how the construction can be carried out near nwslip hoiindaries 121. Mercier's construction appears to be correct if no b.c. ere imposed. It can be modiEed to ruake ph incomprcssible quite easily, but this seems to require the sacrifice of approximation accuracy. Thc most desireable result would be to be able to extend the construction oi Mercier to other boundary conditions. This would be desireable because the construction requires only that I[ g* 112 < m, but such an extcnsion of Mercier's work does not seem obvious to us at this time. Instead, we turn to techrliques inspired by Johnson and Pitkiranta [25] for the bilinear/conttant-p element. We will refer to this element by the scronym, 'BCP," in what followa. The arguments of ref. 25 for the BCP element rely on a superconvergence rcsult; our arguments for the NRC will directly use the results for the BCP element, and both estimates will require 11 g' 113 < 00. This is not optimal in the smoothness required of the exact solution in a linear element method, but is the Scst we have been able to argxe rigorously, at present. Numerical experiments seem to suggest that the extra smoothness required is only an artifact of proof techniqw [2,13,14]. 5.2

The NRC-BCP

Correspondence We continue to consider the Lagrange multiplier method A. In this subsection we present several results from ref. 13 which establish that there exists a yAE S" such that

This, of course, establishes that

We r.efer to eq. (26b) as satisfaction of the constrained approximation condition (WAC") [14]. We will deal with 'triangulations" of domains into rectangles which have the usual restrictions [24], the most important 31 which is that the rectangles meet vertex to vertex. We shall say that a domain n, has the rectanguh regularity property if:

1. n is the interior of its closure. 2. There exists a bounded rectangle, R, and a uniform triangulation, T A ,of R by rectangles of dimension h x k such that n 5: R and nA n is either empty or equal nofor all

n

R* E 7".It then follows that

[n'n nl is a uniform triangulation of n.

.

D.S.Malkus & E. T. Olsen

244

The rectangular regularity property assures that the domain, fl has no c r a c h and that it can be triangulated by triangulations which can be extended to triangulations of the rectangle, R, by extending the lines which forio interelement boundaries. Some of the results below require the rectangular regularity property, and wllilo it is somewhat restrictive, it does apply to triangulations of T-shaped and Lshaped domaina and other unions of rectangles. We Orst state a result for the BCP element which extends a similar result of Johnson and Pitkiranta 1251 to domains with the rectangular regularity property. It is the fundamental building block o f our arguments for the NRC elcrnent. As with all of the remaining theorems we state here, this theorem is given without proof. The interested reader is referrcd to ref. 13 for the proofs and further discussion of the results. The nurnhcrs in parcntheses which immediately follow the theorem numbering of this paper give the theorem(s) :ind page numbers of the corresponding theorem(s) in ref. 13. The ore m 4(4.12, p. 74)(The CAC for the BCP element): Let fl be a connectcd domain with Ibe rectangular regularity property, and let I T h ] be a collection of uniform triangulations of ( I . Let S h Wi'*(fl) X W:l2(n)= HI be the velocity trial space constructed on 7" using the BCP element, and Ict W h S h be the subspace consisting of weakly incompressible velocity fields. Thcn given an incompressible g' E H I [W'**(n)X lV2ffl)], there exists

n

y h E Wh satisfying eq. (ZSa) with C independent of h and y'.

This theorem is of use in obtaining estimates for the NRC element because of the following two results which make a correspondence between the NBC and BCP elements:

Theorem S(5.1, p. 80): Let S h be the FEM velocity trial space comtructed usiog the NRC element for some domain, n, and some uniform triangulation, T h = ine],of fl by rectangles. A y" E S h is incompressible iff the nodal values, (U!~U:), satisfy

for each fl. in Th, where fl' has dimension8 h Figure 1.

xk

and the nodes of W'are labelled M in

Thcorem 8(5.3, p. 83): Let fl be any domain which can be triangulated by rectangles, and let T h = [riel be a triangulation of fl by rectangles (7" need not be uniform, but we still

: X T: be the FEM require that elements in the triangulation join vertex t o vertex). Let S hid space constructed on T h using the BCP element, and let S$ X 7'5 be the FEM trial space constructed on T h using the NRC element. Denote by f i the h(n) projection onto T!, Then: (1) II yh is any weakly incompressible element of S : , i. e. R h v . yh = 0, there i8 an element, y h E Sp, which is incompressible and agrees with 9" on Bfl' for all fl* E Th.

ch is any incompressible element of Sg,the element, sh of 3 : which agrees with on Bfl' for all fl* E T h is weakly incompressible.

(2) If

zh

245

Liriear Crossed Triangles f o r Incompressible Media

R e m a r k 6.1: The implication of Theorems 5 and 6 which is crucial to obtainining estimates for the NRC is that, given a BCP weakly incompressible velocity field (which is not necessarily exactly incompressible), there is a corresponding exacltyincompressible NRC field. The NRC field is obtained by interpolating t o the BCP at the corncr nodes of the NRC and assigning the central nodal values according to eq. (27). A technical result not cited here (Theorem 5.4, p. 85 [13]) implies t h a t if the BCP field is a good %woximation to y', then the assignment of the central nodal values according to eq. (27) carries that Property over to the corresponding NRC field. This leads t o

T h e o r e m 7(5.5, p. 87 and 5.6, p. 89) (The CAC for the NRC element): The NRC satisfies thc CAC t o the same order as the RCP element, with the same requirernenta of smoothness on T!', on appropriate domains and meshes. In particular, let n be a connected domain with the rcctangular regularity property, and let [ T h ]be a collection of uniform triangulations of R. Let S" C Wi82(ll)X W ~ s z ( f = l ) HI be the velocity trial space constructed on T h using the NRC element, and let Wh C Sh be the subspace consisting of incompressible velocity fields. Then given a divergenceless g' E [HI nWT2(n) X W'*2(n)], there exists a F~E W h satisfying eq. (263), with a C independent of h. Iu the light of Theorem 3,Theorem 7 establishes that the velocity estimate in Stokes flow on a domain satisfying the recatangular regularity property is O(h),when uniform triangulations of NRC rectangles are employed, and the exact solution has at least three & derivatives. This has been done without obtaining any estimate for the pressures. We turn our attention t o the question of the pressures in the remainder of thiE paper. 5.3 P r e s s u r e E s t i m a t e s w i t h o u t the Discrete LBB Condition

Let us assume that velocity estimates along the lines of the previous subsection have been obtained. Subtracting discrete and continuous weak equations as in the proof of Theorem 2 gives for d1 c" E S" a($"

- g',ch) = (p' - p",, .g").

wlwe, as before, ph is the best

= Ip'

h(n) approximation top'

- f + p" - p:,

v.

from T h . Thus

Under usual assumptions of approximation accuracy, it sn5ces t o show that IIp(: -fl j c f s not immediately follow from eq. (28) even though the right-hand side is small when riglrt-hand side is indeed small because

11 ?h - 9' 11

and

II$

-p'

[lo are small.] The

[lo is small. This 11 ch 11 = 1 [The

problem !s twc-fold:

1

First, ph, - ph may have a component in U". The smallness of the right-hand side might then only show that this component is substantial, not that 11 p i - $' Ilo is small. In ref. 14 this problem is addressed. There it i.i s!iown that with some reasonable assumptions there is an approximation E [ U h J j l T h such that 11 p' - fih I(, is as small as the maximum of 11 g t y' Ill and 11 p" p' 1l0 , where 9 : is the nodal interpolate

-

to 5.. Thus proving that IIp" (ph

- Bh, v . yh).

n

ch

-

- f 1l0 is small will s u 5 c e t o produce a pressure estimate. Furthermore, since

= 0 for all ph E S", f can be substituted for $ in eq. (28).

Second, since the constant kh of eq. (20) can evidently only be bounded by kh 2 C1h on many meshes, the best that follows from eq. (28) by taking the supremum of both sides with 11 p" 11 = 1 is 1

lip" - p* ,1

is usually ~ C h lp'l

Ill.

We believe t h a t

that eq. (29) gives then is that 11 ph, - f

11 9" - g' Ill

1l0 is bounded.

Chll ,u* I(p ar discussed earlier. The best

D. S Malkus & E. T Olsen

246

Fortunately, we have had good success with smoothing the pressure Eeld by projection of the raw Pressures eomputcd on each subtriangle onto an auxiliary trial space [2,13,14,20,23,25]. In ref. 2 a simple

.md easy to implement projection method is described which uses k projection of the raw pressures onto tho conforming bilinear space based on NRC macro corner nodes. Unfortunately, we do not know any error .:af,irnate for that scheme to assure us that the possible !ack of convergence pointed out by cq. (28) does not occur in practice. We have computational experience which seems to indicate that good results can hc obtained using the projection mcthod of ref. 2 12,141. We present the Enal tbcorom to illustrate that the idea of pressure projection can be well founded in theory, even though we prefer to use the scheme af ref. 2 for reasons of practicality. The Enal theorem is a composite of results from Chapter 7 nf ref. 13. T h e o r e m 8: Let fl be a rectangle and [ T h ] be a seqcence of uniform tri3nylations of R with Zrn rectangles per side for rn > 2. Denote these rectangles by UW].Let Sh X T h be the FEM trial space constructed ou T h , using the crossed triangle. Let rh denote the & projection from T h to tbe space of piecewise constant pressures based on 2 X 2 rectangular patches of W's. If p h is the raw pressure solution t o the Lagrange multiplier problem A from Th, and f is an arbitrary positive number, then the 'filtered" pressure, rhph, satisEes

8. CONCLUSIONS

We believe that the NRC crossed triangle element is an ideal element 'for steady Bows of memory Iluids. We believe that the theoretical results for linear problems given here provide a logical scenario, cxplaining why such elements can give convergent approximations when employed with care. The advantage provided by exact incompressibility of Lagrange multiplier FEM solutions is of fundamental importance to implementation of numerical procedures for memory fluids. Theae procedures also take much advantage of the fn,t that the velocity gredients are constant on each subtriangle. The several advantages just mentioned make the NRC element completely unique among flnite elements for plane and axisymmetric incompressible Bows. \% believe that this uniqueness is ample justi5cation for putting up with the added computational cost of the int~rnalmacro node. Looking beyond the application t o non-Newtonian memory fluid problems, we believe that the NRC element will prove useful in many situations where e5cient analytic construction of smooth 3trtlamlines is important. In fact, because of the BCP NRC correspondence, an exactly incompressible NRC velocity fleld can easily be computed from every wealky incompressible BCP Eeld, by assigning central ncdal values as we have described and using the corner nodal values of the BCP field. 'This seems t o show promise as a method by which t o obtain smooth streamlines from results computed using the BCP element.

-

Acknowledgement: The research described in this paper was performed while both authors were at Illinois Institute of Technology, and was partially supported by N. S. F. Grant MCS-81-02088. The manuscript was preparcd at the Mathematics Research Center, University of Wisconsin - Madison, where the 5 s t author holds a visiting poaition, and is sponsored by the United States Army under Contract No. DAAG29-80.C 0041. 7.

REFERENCES

[I] M.Crochet and K. Walters,Nums;ical methods in non-Newtonian Buid mechanics, Ann. Rev. Fluid Mech. 15 (1983) 241. [2] B. Bernstein, D. S. Malkua, and E. T. Olsen, A Enite element for incompressible plans Bows of Euids with memory, Int. J. Numer. Meths. Fluids, t o appear.

Linear Crossed Triangles f o r Incompressible Media [3]D. S. hlalkus and B. Bcrnstein, Flow of a Curtiss-Bird,fluid over a transverse slot using the finite element driht-function method, J. Non-Newtonian Fluid Mechs., t o appear.

[I] B. Bernstein, M. K. Kadivar, and D.

S. Malkus, Steady flows of memory fluids with finite elements: Two test problems, Comp. hlatha. Appl. Mech. Eng. 27 (1981)279-302.

[5] M Viriyayuthakorn and B. Caswcll Finite element simulation of viscoelastic flows, J. Nc Newtonian Fluid Mechs. 8 (1981)245-267.

and R. Davies, Long-range memory effects in flows involving abrupt changes in geometry. Part 4: Numerical simulation using integral rheological models, J. Non-Newtonian Fluid Mechs. 8 (1981)95.

[G] H. Court, K. \+‘alters

p] 0. Hassager,

A Lagrangian finite element method for the simulation of flow of nonNewtonian liquids, J. Non-Newtonian Fluid Mechs. 12 (1983)153.

[S]J. Nagtigaal, D. M. Parks, and J. R. Rice, On numerically accurate finite element solutiom iu the fully plastic range, Comp. Meths. Appl. Mech. Eng. 4 (1974) 153-178.

B. Mericier, A conforming finite element method for two dimensional, incompressible elasticity, Int. J. Numer. Meths. Eng. 14 (1979)942-945.

191

[lo] J. H. Argyris, J. St. Doltsinis, W. C. Knudson, J. Szimmat, H. Wiistenberg, and K. Willam, Eulerian and Lagrangian techniques for elastic and inelastic large degormation processes, I.S.D. Report No. 256 (1978).

[II] F. Breczi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers, R A . I . R . 0 Analyse Numerique 8 (1974)129-151.

[ I ? I. Babuska and A. K. Aziz, Mathematical Foundations of the Finite Element Method

/Academic Press, New York, 1972).

[13]E. T.Olsen, Stable finite elements for non-Newtonian Bows: First order elements which fail the LRB condition, Ph. D. Thesis, Department of Mathematics, Illinois Institute of Technology, Chicago (1983).

[14] D. S. Milkus and E. T. Olsen, Obtaining error estimates for optimally constrained incompressible finite elements, Comp. Meths. Appl. Mech. Eng. 42 (1984). [15 C. Curtiss and R. B. Bird, Kinetic theory for polymer melts. Parts I and dhys. 74 (1981)2016-2033.

II, J. Chem.

[lo] P.Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978).

[17 D. S. Malkus, Penalty methods in finite element analysis of fluids and structures, Nuc. dng. Design. 57 (1980)441-448. [I81D. S. Malkus and T. J. R. Hughes, Mixed finite element methods - reduced and selective illtcgration techniques: A unification of concepts, Comp. Meths. Appl. Mech. Eng. 15 (1978)

83-81. [19]A. Rslston and P. Rabinowita, York, 1978).

A First.Course in Numerical Analys~s(McGraw-Hill, New

247

248

D.S. Malkus & E. T. Olsen

R. L. Sani, P. M.Cresho, R. L. Lee, D. F. Griffiths, and M.Engelman, The cause and cure (?) of spurious pressures generated by certain FEM solutions to the incompressible Navier-Stokes equations. Parts I and 11, Int. J. Numer. Meths. Fluids 1 (1981) 17-43 (part I), 171-204 (part n).

[ZO]

An analysis of convergence of mixed 5 i t e element methods, k a i y s e Numerique 8 (1977) 341-254.

[21 M Fortin,

R.A.I.R.O.

(221 D. S. Malkus, Eigenproblems associated with the discrete LBB condition for incompree sible finite elements, Int. J . Eng. Sci. 18 (1981) 1299-1310. [23] T. J. R. Hughes, W.K. Liu, and A. Brooks, Finite element analysis of incompressible viscous Bows by the penalty function formulation, J. Comp. Phys. 30 (1776) 1-60.

[241 B. Mercier, Topics on finite element solution of elliptic problems (Tata Institute of fundamental Research in Bombay Lecture Series, Springer-Verlag, Berlin, 1979). [25] C. Johnson and J. Pitkiiranta, Analysis of nome mixed finite element methods related to reduced integration, Math. Comp. 38 (1982) 375-400.

Unification of Finite Element Methods H. Kardestuncer (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

CHAPTER 1 1 THE NUMERICAL ANALYSIS OF NECKING INSTABILITIES A. Needleman

Some i s s u e s a r i s i n g i n t h e f i n i t e element a n a l y s i s of n e c k i n g i n s t a b i l i t i e s a r e a d d r e s s e d w i t h i n t h e c o n t e x t o f s p e c i f i c problems. In a d d i t i o n t o a n a l y s e s based on c l a s s i c a l Mises type c o n s t i t u t i v e l a w s , we d i s c u s s n u m e r i c a l s o l u t i o n s u s i n g c o n s t i t u t i v e r e l a t i o n s t h a t more a c c u r a t e l y model p l a s t i c s l i p p r o c e s s e s and p r o g r e s s i v e r u p t u r e on t h e m i c r o s c a l e . We i l l u s t r a t e t h e a b i l i t y of f i n i t e element s o l u t i o n s based on t h e s e n o n c l a s s i c a l c o n s t i t u t i v e r e l a t i o n s t o reproduce e s s e n t i a l f e a t u r e s o f t h e development o f f a i l u r e i n t h e necked down r e g i o n .

1. INTRODUCTION When a d u c t i l e m e t a l i s deformed p a s t t h e maximum load p o i n t i n t e n s i o n , t h e deformations l o c a l i z e i n t o a neck-like region. F a i l u r e u l t i m a t e l y o c c u r s w i t h i n t h i s "neck," a l t h o u g h t h e mode of f a i l u r e i s b o t h m a t e r i a l and geometry dependent. "Necking" h a s a l s o come t o be used a s a g e n e r i c term d e n o t i n g a n y t e n s i l e i n s t a b i l i t y t h a t l e a d s t o l o c a l i z e d t h i n n i n g . Necking i n s t a b i l i t i e s p l a y an i m p o r t a n t r o l e i n s e t t i n g d e f o r m a t i o n and stress patterns i n material testing situations, i n limiting ductility i n s h e e t forming o p e r a t i o n s and i n s e t t i n g macroscopic c o n d i t i o n s f o r d u c t i l e r u p t u r e . Hence, t h e r e i s s t r o n g m o t i v a t i o n f o r d e v e l o p i n g a q u a n t a t a t i v e d e s c r i p t i o n of n e c k i n g phenomena. Much p r o g r e s s toward t h i s end h a s been made i n t h e p a s t decade o r so and numerical s o l u t i o n s have played a major r o l e i n t h i s development.

Here, no a t t e m p t i s made t o g i v e a n e x t e n s i v e review o f t h e s u b j e c t . I n s t e a d , a t t e n t i o n i s d i r e c t e d toward a p a r t i c u l a r t o p i c i n o r d e r t o i l l u s t r a t e t h e p r o g r e s s t h a t h a s been made and some of t h e u n r e s o l v e d i s s u e s t h a t remain. The t o p i c t h a t w i l l s e r v e a s t h e f o c u s f o r t h i s d i s c u s s i o n i s t h e d i f f e r e n c e i n b e h a v i o r e x h i b i t e d by s t r u c t u r a l m e t a l s i n a x i s y o m e t r i c and p l a n e s t r a i n t e n s i l e t e s t i n g . T y p i c a l phenomenologies t h a t d e v e l o p i n t h e s e two t e s t s a r e i l l u s t r a t e d i n F i g . 1, from Speich and S p i t z i g [l]. In b o t h t e s t s , t h e d e f o r m a t i o n s remain e s s e n t i a l l y homogeneous up t o t h e maximum load p o i n t a f t e r which a d i f f u s e neck d e v e l o p s . In t h e round b a r t e n s i o n t e s t , F i g . l a , d i f f u s e n e c k i n g l e a d s t o a cup and cone t y p e f r a c t u r e . On t h e o t h e r hand, i n t h e p l a n e s t r a i n t e n s i l e t e s t o f t h e same m a t e r i a l , F i g . I b , t h e d e f o r m a t i o n

249

250

A. Needleman

mode s h i f t s t o one i n v o l v i n g l o c a l i z e d s h e a r i n g w h i l e t h e d i f f u s e neck i s r a t h e r s h a l l o w . It i s worth emphasizing t h a t t h e specimens shown i n F i g s . l a and Ib a r e made o f t h e same m a t e r i a l ; t h e v a r i a t i o n i n geometry accounts f o r t h e d i f f e r e n c e i n b e h a v i o r . Analyses of necking d a t e back, a t l e a s t , t o t h e one dimensional model f o r This a n a l y s i s p r e d i c t s t h a t t h e o n s e t of necking due t o Considere [ 2 ] . necking i n i t i a t e s a t t h e maximum load p o i n t , i n b o t h axisymmetric and p l a n e s t r a i n t e n s i o n . An a t t r a c t i v e f e a t u r e o f t h i s one d i m e n s i o n a l theory i s t h a t the only m a t e r i a l property required a s input i s th e u n i a x i a l s t r e s s - s t r a i n c u r v e . However, t h e development of t h e neck and t h e nonhomogeneous s t r e s s and d e f o r m a t i o n s t a t e s t h a t occur w i t h i n i t cannot be analyzed w i t h i n t h i s framework. These s t r e s s and s t r a i n d i s t r i b u t i o n s a r e needed i n o r d e r t o r e l a t e measured q u a n t i t i e s t o m a t e r i a l p r o p e r t i e s . I n t h e 1 9 4 0 ' s , Bridgman [31 developed h i s now c l a s s i c a l approximate a n a l y s i s which g i v e s e x p r e s s i o n s f o r f i e l d q u a n t i t i e s of i n t e r e s t a c r o s s t h e p l a n e of t h e neck. Bridgman's [3] a n a l y s i s i s f o r an i d e a l l y p l a s t i c s o l i d and r e q u i r e s as i n p u t t h e neck c u r v a t u r e a s a f u n c t i o n of imposed s t r a i n . A l s o , t h e question o f how t h e neck i n i t i a t e s i s n o t a d d r e s s e d . S i n c e necking i n t e n s i l e specimens i s such a cormnon and s t r i k i n g m a n i f e s t a t i o n of f i n i t e s t r a i n p l a s t i c b e h a v i o r , t h e r e h a s been c o n s i d e r a b l e i n t e r e s t i n p r e d i c t i n g t h e observed phenomenology from b a s i c p r i n c i p l e s of p l a s t i c i t y t h e o r y . This i s n o t e n t i r e l y s t r a i g h t f o r w a r d , s i n c e i t was demonstrated [4] t h a t a necking t y p e b i f u r c a t i o n i s n o t p o s s i b l e i n axisymmetric t e n s i o n f o r a c l a s s i c a l r i g i d - p l a s t i c s o l i d . However, a necking b i f u r c a t i o n does o c c u r i n p l a n e s t r a i n t e n s i o n , Cowper apd Onat [ 5 ] . Even though e l a s t i c s t r a i n s a r e v e r y s m a l l compared w i t h p l a s t i c s t r a i n s , i t t u r n s o u t t h a t t h e r i g i d p l a s t i c framework i s t o o r e s t r i c t i v e f o r a n a l yz i n g necking i n s t a b i li t i e s

.

A g e n e r a l framework f o r a n a l y z i n g f i n i t e s t r a i n e l a s t i c - p l a s t i c problems, i n c l u d i n g v a r i a t i o n a l p r i n c i p l e s and uniqueness and b i f u r c a t i o n c r i t e r i a is l a r g e l y due t o H i l l [6,71. T h i s framework p e r m i t s a u n i f i e d t r e a t m e n t of p l a s t i c i n s t a b i l i t i e s , i n c l u d i n g t e n s i l e n e c k i n g a s w e l l a s t h e more thoroughly e x p l o r e d problems of s t r u c t u r a l b u c k l i n g . For b o t h t h e axisymmetric and p l a n e s t r a i n specimen g e o m e t r i e s , t h e o n s e t of a necking b i f u r c a t i o n i s found t o b e delayed beyond t h e maximum l o a d p o i n t , w i t h t h e d e l a y b e i n g g r e a t e r f o r s t u b b i e r specimens, Cheng e t a l . [81, Needleman [ 9 ] , Hutchinson and M i l e s [ l o ] and H i l l and Hutchinson [ l l ] . For axisymmetric specimens t h e magnitude of t h e d e l a y i n c r e a s e s w i t h t h e e l a s t i c s h e a r modulus and becomes i n f i n i t e i n t h e r i g i d - p l a s t i c l i m i t , [lo]. The v a r i a t i o n a l s t r u c t u r e of t h e g o v e r n i n g e q u a t i o n s p e r m i t s c o n v e n i e n t a p p l i c a t i o n of f i n i t e element methods. I n f a c t , one of t h e f i r s t a p p l i c a t i o n s of t h e f i n i t e element method t o a f i n i t e s t r a i n p l a s t i c i t y problem was an a n a l y s i s o f neck development i n a n a x i s p e t r i c t e n s i l e b a r [91. This s o l u t i o n and o t h e r numerical s o l u t i o n s [12-141 f o r t h e stress and s t r a i n d i s t r i b u t i o n s i n t h e neck make i t p o s s i b l e t o a d d r e s s i s s u e s such a s t h e e f f e c t of s t r a i n h a r d e n i n g on neck developnent. Such numerical s o l u t i o n s a l s o p l a y a u s e f u l r o l e i n a s s e s s i n g c o n d i t i o n s governing f r a c t u r e i n i t i a t i o n [15-181. However, a n a l y s e s which employ

The Numerical Analysis of Necking Instabilities

25 1

c l a s s i c a l e l a s t i c - p l a s t i c flow theory t o c h a r a c t e r i z e t h e m a t e r i a l b e h a v i o r a r e i n c a p a b l e of m o d e l l i n g t h e t e n s i l e f r a c t u r e shown i n F i g . l a . N e v e r t h e l e s s , t h e s u c c e s s e s of t h e s e n u m e r i c a l a n a l y s e s played a u s e f u l r o l e i n c l a r i f y i n g t h e f i n i t e s t r a i n ( a s opposed t o m a t e r i a l m o d e l l i n g ) issues a r i s i n g i n formulating c o n s t i t u t i v e r e l a t i o n s f o r e l a s t i c - p l a s t i c Mises-type s o l i d s . I n f a c t , t h e f o r m u l a t i o n and use of n u m e r i c a l methods f o r f i n i t e s t r a i n p l a s t i c i t y have advanced t o t h e p o i n t where t h e g e n e r a t i o n of s o l u t i o n s f o r neck d e v e l o p n e n t , based on c l a s s i c a l flow l a w s , is v i r t u a l l y r o u t i n e . Numerical s o l u t i o n s f o r neck development, u s i n g i s o t r o p i c a l l y h a r d e n i n g Mises-type p l a s t i c i t y t h e o r y , have a l s o been c a r r i e d o u t f o r t h e p l a n e s t r a i n t e n s i o n t e s t [19,20]. These p l a n e s t r a i n c a l c u l a t i o n s have found c o n t i n u e d growth o f t h e d i f f u s e neck w i t h o u t any tendency f o r t h e flow p a t t e r n t o s h i f t t o one i n v o l v i n g t h e l o c a l i z e d s h e a r i n g shown i n F i g . l b . T h i s is a consequence o f t h e c o n s t i t u t i v e assumption, s i n c e " m a t e r i a l i n s t a b i l i t y " a n a l y s e s r e v e a l t h a t t h e c l a s s i c a l smooth y i e l d s u r f a c e e l a s t i c - p l a s t i c s o l i d is q u i t e r e s i s t a n t t o l o c a l i z e d s h e a r i n g , Rice [21]. D e v i a t i o n s from t h e c l a s s i c a l smooth y i e l d s u r f a c e i d e a l i z a t i o n d o p e r m i t s h e a r bands t o emerge a t a c h i e v a b l e s t r a i n l e v e l s . Y i e l d s u r f a c e v e r t e x e f f e c t s , which a r i s e from t h e d i s c r e t e n a t u r e of c r y s t a l l i n e s l i p , and t h e d i l a t i o n a l p l a s t i c flow induced by t h e n u c l e a t i o n and growth of microv o i d s a r e p a r t i c u l a r l y s i g n i f i c a n t i n t h i s r e g a r d , [21,22]. When f i n i t e element a n a l y s e s of neck development a r e c a r r i e d o u t u s i n g c o n s t i t u t i v e r e l a t i o n s i n c o r p o r a t i n g t h e s e e f f e c t s , t h e numerical s o l u t i o n s reproduce t h e observed phenomena i n c o n s i d e r a b l e d e t a i l 123-251.

We b e g i n by o u t l i n i n g a Lagrangian convected c o o r d i n a t e f o r m u l a t i o n of t h e g o v e r n i n g e q u a t i o n s t h a t h a s been e x t e n s i v e l y used i n a n a l y z i n g f i n i t e d e f o r m a t i o n p l a s t i c i t y problems. This f o r m u l a t i o n i s r e a d i l y adapted f o r a v a r i e t y of p l a s t i c c o n s t i t u t i v e r e l a t i o n s . Next, p l a s t i c flow r u l e s t h a t i n c o r p o r a t e models of y i e l d s u r f a c e v e r t e x e f f e c t s and p r o g r e s s i v e c a v i t a t i o n a r e d i s c u s s e d i n a d d i t i o n t o a f i n i t e d e f o r m a t i o n v e r s i o n of t h e c l a s s i c a l i s o t r o p i c h a r d e n i n g Mises s o l i d . A b r i e f resume i s t h e n g i v e n of some a n a l y t i c a l r e s u l t s on n e c k i n g and s h e a r l o c a l i z a t i o n . The a n a l y s e s g i v e i n f o r m a t i o n needed f o r an e f f e c t i v e mesh d e s i g n a s w e l l a s Finally, provide a p e r s p e c t i v e f o r i n t e r p r e t i n g t h e numerical r e s u l t s . n u m e r i c a l s t u d i e s of n e c k i n g are reviewed w i t h a f o c u s on r e c e n t c a l c u l a t i o n s of t h e development of f a i l u r e i n t h e necked down r e g i o n . 2.

FIELD EQUATIONS

R e l a t i v e t o a f i x e d C a r t e s i a n frame, t h e p o s i t i o n of a m a t e r i a l p o i n t i n t h e i n i t i a l c o n f i g u r a t i o n i s d e n o t e d by i,. I n t h e c u r r e n t c o n f i g u r a t i o n the material point i n i t i a l l y a t is at The d i s p l a c e m e n t v e c t o r u% and d e f o r m a t i o n g r a d i e n t F a r e d e f i n e d by %

A . Needleman

252

To e x p r e s s e q u i l i b r i u m , w e f i r s t i n t r o d u c e t h e f o r c e t r a n s m i t t e d a c r o s s a m a t e r i a l element, $3, which c a n be r e l a t e d t o e i t h e r t h e symmetric Cauchy s t r e s s t e n s o r u or t h e unsymmetric nominal stress t e n s o r

t

%

(2.2) where d S and a r e t h e a r e a and o r i e n t a t i o n i n t h e c u r r e n t c o n f i g u r a t i o n of a m a t e r i a l element t h a t had a r e a , dS, and o r i e n t a t i o n , v i n t h e a’ I t i s c o n v e n i e n t t o i n t r o d u c e t h e Kirchhoff stress i n i t i a l c o n f i g- u r a t i o n . d e f i n e d by

T =

‘L

det(x)g

where d e t ( ) d e n o t e s t h e d e t e r m i n a n t . Although g e n e r a l t h e o r e t i c a l i s s u e s a r e c o n v e n - i n t l y a l r e s s e d u s i n g a c o o r d i n a t e f r e e n o t a t i o n , a c t u a l l y c a r r y i n g o u t a computation r e q u i r e s d e a l i n g w i t h t h e component form of t h e e q u a t i o n s . A Lagrangian, convected c o o r d i n a t e f o r m u l a t i o n of t h e governing e q u a t i o n s , a s d e s c r i b e d i n Green and Zerna [ 2 6 ] and Budiansky [ 2 7 ] , h a s been e x t e n s i v e l y employed i n f i n i t e d e f o r m a t i o n p l a s t i c i t y a n a l y s e s . D i s c u s s i o n s of t h i s f o r m u l a t i o n , w i t h p a r t i c u l a r focus on t h e f i n i t e element implementation a r e g i v e n by Needleman [ 281 and Needleman and Tvergaard [29]

.

i are i n t r o d u c e d which s e r v e a s p a r t i c l e l a b e l s Convected c o o r d i n a t e s , y and a l l f i e l d q u a n t i t i e s are c o n s i d e r e d a s f u n c t i o n s of 9 and a time l i k e parameter t which i s i d e n t i f i e d w i t h a m o n o t o n i c a l l y i n c r e a s i n g q u a n t i t y t h a t p a r a m e t e r i z e s t h e l o a d i n g h i s t o r y . Covariant base v e c t o r s i n t h e , a r e given i n i t i a l c o n f i g u r a t i o n , g . , and i n t h e c u r r e n t c o n f i g u r a t i o n , %l %i by

,

(2.4) so t h a t from ( 2 . 1 ) (2.5) The m e t r i c t e n s o r i n t h e L n i t i a l C o n f i g u r a t i o n i s 6 = and i n t h e i‘ c u r r e n t c o n f i g u r a t i o n i s gij = gi.gj, w i t h i n v e r s e s i i j and @ j , r e s p e c t i v e l y . The r e c i p r o c a l ( o r c o n t r a v a r i a n t ) b a s e v e c t o r s a r e g i v e n by

Ei*&

(2.6) Components of v e c t o r s and t e n s o r s on t h e convected c o o r d i n a t e s a r e o b t a i n e d simply v i a “ d o t “ p r o d u c t s w i t h t h e a p p r o p r i a t e b a s e v e c t o r example t h e components of Q on t h e i n i t i a l base v e c t o r s a r e nlJ = 4 .n. %

253

The Numerical Analysis of Necking Instabilities

w h i l e t h e components o f $ on t h e c u r r e n t b a s e v e c t o r s a r e

T

ij

These components a r e r e l a t e d by nij = ? i k F j

- E ~ ' ~ , ~ E ~ . (2.7)

.k

With body f o r c e s n e g l e c t e d and g i v e n t h a t t h e body i s i n e q u i l i b r i u m a t t , e q u i l i b r i u m a t t + d t r e q u i r e s t h e i n c r e m e n t a l p r i n c i p l e o f v i r t u a l work t o be s a t i s f i e d .

V

where f =

S

and

('1

= d( )/dt a t fixed y i .

For t h e r a t e independent e l a s t i c - p l a s t i c s o l i d s c o n s i d e r e d h e r e , t h e c o n s t i t u t i v e r e l a t i o n s a r e s p e c i f i e d a s a piecewise l i n e a r r e l a t i o n between t h e Jaumann d e r i v a t i v e of K i r c h h o f f str$ssll;, and t h e r a t e of 1, SO t h a t w e w r i t e d e f o r m a t i o n t e n s o r , $ (= t h e symmetric p a r t of [ a [

h e r e : denotes t h e dyadic product, i . e . ClJkRd Rk'

t h e component form of

$:$ i s

,

S i n c e ;I = E.,$*where n i . a r e t h e components of L a g r a n g i a n s t r a i n r a t e on t h e &hdef?&me$jbase v e c t o J s , t h e component form of ( 2 . 9 ) on t h e c u r r e n t base vect ors i s

(2.10)

For u s e i n t h e incyemental p r i n c i p l e of v i r t u a l work (2.8) we need t h e r e l a t i o n between nlJ (which a r e t h e c o n t r a v a r i a n t components of t h e nominal stress r a t e on t h e i E i t i a l undeformed b a s e v e c t o r s ) and t h e c o n t r a v a r i a n t components of ;on t h e c u r r e n t deformed b a s e v e c t o r s . This r e l a t i o n i s c o n v e n i e n t l y s p e c i f i e d i n two s t e p s . F i r s t , (2.11)

t h e n from t h e i n c r e m e n t a l form of ( 2 . 7 )

A. Needleman

25 4

(2.12) Now we can w r i t e (2.13) where i j k L = Lmink L

ik j k

j

YnKrn

K

(2.14)

g

+

with L

i j k L = ,ijkL

-

1 -ik

?[g

j L -jk

'I

+g

T

ill -iL +g

T

j k -jL +g

T

ik

(2.15)

The o n l y r e l a t i o n t h a t changes w i t h m a t e r i a l model is ( 2 . 9 ) . The r e l a t i o n s (2.14) and (2.15) a r e model independent and c a n be i n c o r p o r a t e d i n t o a f i n i t e element code i n a modular f a s h i o n . We a l s o n o t e t h a t t h e components of nominal stress and d e f o r m a t i o n g r a d i e n t a p p e a r i n g i n (2.8) a r e d e f i n e d on t h e undeformed c o o r d i n a t e n e t which i s chosen f o r convenience, e . g . a s a C a r t e s i a n system o r a s a c y l i n d r i c a l c o o r d i n a t e system f o r axisymmetric problems.

then I f t h e i n c r e m e n t a l moduli (2.9) p o s e s s t h e symmetries C i j k L = $ L i j , f o r a s t a n d a r d boundary v a l u e problem, p r e s c r i b e d t r a c t i o n r a t e s T on S and d i s p l a c e m e n t rates 6 on Su, t h e f o l l o w i n g v a r i a t i o n a l p r i n c i p l e h o l s s [6,7] : Among a l l d i s p l k e m e n t r a t e f i e l d s t h a t s a t i s f y t h e d i s p l a c e m e n t r a t e boundary c o n d i t i o n s , f o r any a c t u a l f i e l d 61 = 0

(2.16)

3.

CONSTITUTIVE RELATIONS

Here, we d i s c u s s a f i n i t e s t r a i n f o r m u l a t i o n of t h e c o n s t i t u t i v e r e l a t i o n s f o r an i s o t r o p i c a l l y h a r d e n i n g Mises s o l i d and g e n e r a l i z a t i o n s of t h i s r e l a t i o n t h a t permit i n v e s t i g a t i o n of ( i ) y i e l d s u r f a c e v e r t e x e f f e c t s as p r e d i c t e d by p h y s i c a l models of t h e c r y s t a l l i n e s l i p p r o c e s s and ( i i ) t h e d i l a t i o n a l and p r e s s u r e s e n s i t i v e p l a s t i c flow due t o t h e n u c l e a t i o n and We c o n f i n e a t t e n t i o n t o i s o t h e r m a l , r a t e growth of micro-voids. i n s e n s i t i v e and i s o t r o p i c m a t e r i a l r e s p o n s e . We a l s o assume t h a t e l a s t i c s t r a i n s remain small. For s t r u c t u r a l m e t a l s t h i s is q u i t e an a p p r o p r i t e s i m p l i f i c a t i o n , e x c e p t p o s s i b l y when t h e h y d r o s t a t i c stress is v e r y l a r g e .

255

The Numerical Analysis of Necking Instabilities

We aim h e r e o n l y a t p o i n t i n g o u t f e a t u r e s of t h e s e m a t e r i a l models of main s i g n i f i c a n c e f o r t h e a n a l y s i s of n e c k i n g i n s t a b i l i t i e s . More complete d e s c r i p t i o n s c a n be found i n t h e r e f e r e n c e s c i t e d . Also, Tvergaard [30] h a s reviewed i s s u e s a r i s i n g i n t h e implementation o f t h e s e c o n s t i t u t i v e r e l a t i o n s i n f i n i t e element a n a l y s e s . 3 . 1 C l a s s i c a l Smooth Y i e l d S u r f a c e P l a s t i c i t y Theory The e l a s t i c - p l a s t i c c o n s t i t u t i v e r e l a t i o n most commonly used i n f i n i t e element a n a l y s e s i s t h e i s o t r o p i c a l l y h a r d e n i n g Mises s o l i d , a l s o c a l l e d J 2 flow t h e o r y o r Prandtl-Reuss t h e o r y . For i n f i n i t e s i m a l s t r a i n s t h e flow r u l e t a k e s t h e form (3.1) where 4. is t h e s t r e s s d e v i a t o r , 0,is t h e Mises e f f e c t i v e s t r e s s , E i s Young's modulus and E t is t h e t a n g e n t modulus, t h e s l o p e o f t h e u n i a x i a l e f f e c t i v e s t r e s s - e f f e c t i v e s t r a i n curve. Of c o u r s e , ( 3 . 1 ) is u n d e r s t o o d t o a p p l y o n l y when t h e c u r r e n t s t a t e is a t y i e l d and t h e imposed d e f o r m a t i o n s a r e such a s t o e n f o r c e c o n t i n u e d p l a s t i c d e f o r m a t i o n . The t o t a l s t r a i n r a t e i s t h e n w r i t t e n a s t h e sum o f t h e p l a s t i c s t r a i n r a t e r a t e from (3.1) and t h e e l a s t i c s t r a i n r a t e from t h e i n c r e m e n t a l v e r s i o n o f Hooke's law. Several issues arose i n formulating t h i s c o n s t i t u t i v e r e l a t i o n f o r f i n i t e deformations, including: ( i ) t h e c h o i c e o f a p p r o p r i a t e stress and s t r a i n measures t o u s e i n ( 3 . 1 ) . Note t h a t t h e q u e s t i o n of a n i n v a r i a n t stress r a t e d o e s n o t a r i s e i n r e l a t i o n t o t h e p l a s t i c f l o w r u l e s i n c e t h e o n l y r a t e t h a t a p p e a r s is t h e "time" d e r i v a t i v e of a s c a l a r . For f i n i t e d e f o r m a t i o n s , t h e p l a s t i c s t r a i n rate i n (3.1) is i d e n t i f i e d with t h e p l a s t i c p a r t of t h e r a t e of d e f o r m a t i o n t e n s o r d and t h e stress d e v i a t o r and e f f e c t i v e s t r e s s c a n be based on e i t h e r t h e Cauchy s t r e s s o r t h e K i r c h h o f f stress, Budiansky [ 3 1 ] , Hutchinson [32]. The f o r m u l a t i o n s a r e p h y s i c a l l y e q u i v a l e n t f o r s m a l l e l a s t i c s t r a i n s and t h e one based on K i r c h h o f f stress is p r e f e r r e d s i n c e i t g i v e s a symmetric t e n s o r of moduli i n ( 2 . 9 ) and hence l e a d s t o t h e v a r i a t i o n a l s t r u c t u r e (2.16). This i n t u r n h a s t h e c o m p u t a t i o n a l advantage of a symmetric f i n i t e element s t i f f n e s s m a t r i x . With t h i s c h o i c e o f v a r i a b l e s t h e f l o w r u l e t a k e s t h e form

F

3 s.s 62 = e 2 %'%

where

2:s

is

k

T ~ .

A. Needleman

25 6

( i i ) t h e a p p r o p r i a t e f i n i t e s t r a i n g e n e r a l i z a t i o n of t h e i n c r e m e n t a l s t a t e m e n t o f Hooke's law. While i t is p o s s i b l e t o use a f u l l y c o n s i s t e n t f i n i t e e l a s t i c i t y f o r m u l a t i o n f o r t h e e l a s t i c p a r t of t h e s t r a i n r a t e , i t t u r n s o u t t h a t t h i s l e a d s t o an awkward f o r m u l a t i o n of t h e i n c r e m e n t a l r e l a t i o n s . A convenient approximation i n v o l v e s u s i n g t h e l i n e a r hypoe l a s t i c r e l a t i o n [31,32] e

$

l+u A E 'L

= -T

-

u A E (.C:I)I % % ' L

-

(3.3)

A s long t h e s t r e s s e s remain small compared t o Young's modulus, which i s e q u i v a l e n t t o s m a l l e l a s t i c s t r a i n s , ( 3 . 3 ) i s an a c c e p t a b l e approximation. I t i s (3.3) t h a t i n v o l v e s a c h o i c e o f s t r e s s r a t e , o t h e r c h o i c e s t h a n t h e Jaumann r a t e c o u l d be used b u t t h e s e would l e a d t o a l e s s c o n v e n i e n t f o r m u l a t i o n . Again we emphasize t h a t t h e r e i s no q u e s t i o n c o n c e r n i n g a c h o i c e of s t r e s s r a t e w i t h r e g a r d t o t h e p l a s t i c f l o w r u l e ( 3 . 2 ) . On t h e o t h e r hand f o r p l a s t i c c o n s t i t u t i v e r e l a t i o n s t h a t have a t e n s o r i n t e r n a l v a r i a b l e , a s f o r example k i n e m a t i c h a r d e n i n g ( t h e t e n s o r i n t e r n a l v a r i a b l e b e i n g t h e y i e l d s u r f a c e c e n t e r ) , t h e r e is a c h o i c e o f r a t e i n w r i t i n g t h e evolution equation for t h e internal variable. I n t h i s context, the q u e s t i o n of c h o i c e of stress r a t e i s p a r t of a l a r g e r i s s u e c o n c e r n i n g a p p r o p r i a t e e v o l u t i o n laws f o r t e n s o r i n t e r n a l v a r i a b l e s .

( i i i ) a t v a r i o u s times t h e a p p r o p r i a t e n e s s of summing t h e e l a s t i c and p l a s t i c s t r a i n r a t e s h a s been q u e s t i o n e d . A key d i s t i n c t i o n h e r e i s between t h e r a t e of accumulation of p l a s t i c s t r a i n , where by p l a s t i c s t r a i n i s meant t h e r e s i d u a l s t r a i n remaining on u n l o a d i n g , and t h e p l a s t i c p a r t o f t h e r a t e of deformation, i . e . t h e p l a s t i c p a r t of t h e s t r a i n increment from t h e c u r r e n t loaded s t a t e . W r i t i n g a flow r u l e f o r t h e former q u a n t i t y does n o t , i n g e n e r a l , lead t o a n a d d i t a t i v e r e l a t i o n , e x p r e s s i n g t h e flow r u l e i n terms of t h e l a t t e r , a a i n (3.21, d o e s . W r i t i n g t h e r a t e o f d e f o r m a t i o n , $, a s t h e sum of (3.2) and (3.3) and i n v e r t i n g g i v e s a r e l a t i o n i n t h e form (2.10) f o r which t h e t e n s o r of moduli have t h e symmetries CijkP, = CkLij, r e q u i r e d f o r t h e v a r i a t i o n a l f ormul a t i o n . Basic p h y s i c a l assumptions u n d e r l y i n g t h i s c o n s t i t u t i v e r e l a t i o n a r e t h a t t h e p l a s t i c s t r a i n r a t e t h a t h a s a d i r e c t i o n normal t o t h e y i e l d s u r f a c e i n stress s p a c e and, f u r t h e r m o r e , t h a t t h i s s u r f a c e i s smooth, w i t h a unique normal a t t h e c u r r e n t s t r e s s p o i n t . A l s o , t h e p l a s t i c s t r a i n r a t e is volume p r e s e r v i n g and p r e s s u r e i n s e n s i t i v e . These f e a t u r e s have a s t r o n g i n f l u e n c e on t h e p r e d i c t e d c o u r s e of neck development. T h e r e f o r e , we c o n s i d e r c o n s t i t u t i v e r e l a t i o n s which i n c o r p o r a t e , i n a p h y s i c a l l y a p p r o p r i a t e way, d e v i a t i o n s from t h e s e a t t r i b u t e s . 3.2

J 2 Corner Theory

A consequence o f d e r i v i n g t h e flow r u l e (3.2) from a smooth y i e l d s u r f a c e is t h a t t h e p l a s t i c s t r a i n r a t e d i r e c t i o n is not i n f l u e n c e d by t h e s t r e s s r a t e . T h i s i s a t v a r i a n c e w i t h p r e d i c t i o n s of p h y s i c a l p l a s t i c i t y models f o r p o l y c r y s t a l l i n e a g g r e g a t e s based on t h e concept of s i n g l e c r y s t a l s l i p . Such models lead i n e v i t a b l y t o t h e p r e d i c t i o n of a y i e l d s u r f a c e

25 7

The Numerical Analysis of Necking Instabilities

corner a t t h e c u r r e n t loading point [33,34], w i t h t h e p l a s t i c s t r a i n r a t e d i r e c t i o n depending, w i t h i n l i m i t s , on t h e s t r e s s r a t e . D i r e c t e x p e r i m e n t a l e v i d e n c e f o r c o r n e r s i s c o n f l i c t i n g , a l t h o u g h i n some c a s e s t h e r e i s e v i d e n c e f o r a r e g i o n of h i g h c u r v a t u r e a t t h e c u r r e n t l o a d i n g p o i n t [35]. A r e c e n t a n a l y s i s by Pan and Rice [36] shows t h a t s l i g h t m a t e r i a l r a t e s e n s i t i v i t y can account f o r t h e e x p e r i m e n t a l ambiguity. The main s i g n i f i c a n c e o f a y i e l d s u r f a c e c o r n e r f o r b i f u r c a t i o n phenomena l i e s i n t h e s o f t e r r e s p o n s e t o an a b r u p t change of l o a d i n g p a t h t h a t o c c u r s when t h e p l a s t i c s t r a i n r a t e can f o l l o w t h e stress r a t e . S t r u c t u r a l b u c k l i n g p r e d i c t i o n s based on J 2 d e f o r m a t i o n t h e o r y o f p l a s t i c i t y o f t e n g i v e much b e t t e r agreement w i t h e x p e r i m e n t a l r e s u l t s t h a n p r e d i c t i o n s based on J 2 flow t h e o r y [ 3 7 ] . Very e a r l y , Batdorf [381 noted t h a t b i f u r c a t i o n p r e d i c t i o n s of d e f o r m a t i o n t h e o r y from p r e - b i f u r c a t i o n s t a t e s a r r i v e d a t v i a p r o p o r t i o n a l l o a d i n g c o u l d be j u s t i f i e d b y a p p e a l i n g t o t h e p r e s e n c e o f a c o r n e r on t h e y i e l d s u r f a c e . I n t h e p o s t - b i f u r c a t i o n regime s t r o n g l y n o n - p r o p o r t i o n a l l o a d i n g t a k e s p l a c e and use o f J 2 d e f o r m a t i o n t h e o r y , which i s a p a t h independent n o n - l i n e a r e l a s t i c c o n s t i t u t i v e r e l a t i o n , a s a p l a s t i c i t y t h e o r y is not acceptable. The J 2 c o r n e r t h e o r y of C h r i s t o f f e r s e n and Hutchinson [39I i s a n a n a l y t i c a l l y t r a c t a b l e phenomenological t h e o r y of p l a s t i c i t y t h a t i n c o r p o r a t e s key f e a t u r e s e x h i b i t e d by p h y s i c a l models o f p o l y c r y s t a l l i n e a g g r e g a t e s . I n J 2 c o r n e r t h e o r y t h e moduli f o r n e a r l y p r o p o r t i o n a l l o a d i n g a r e t a k e n t o be t h o s e o f a f i n i t e s t r a i n d e f o r m a t i o n t h e o r y s o l i d . For i n c r e a s i n g d e v i a t i o n from p r o p o r t i o n a l l o a d i n g t h e moduli s t i f f e n m o n o t o n i c a l l y u n t i l t h e y c o i n c i d e w i t h t h e l i n e a r e l a s t i c moduli f o r stress r a t e s d i r e c t e d a l o n g o r w i t h i n t h e y i e l d s u r f a c e . The y i e l d s u r f a c e i n t h e neighborhood o f t h e l o a d i n g p o i n t is t a k e n t o be a cone i n s t r e s s d e v i a t o r s p a c e w i t h t h e cone a x i s i n t h e d i r e c t i o n

A =

(3.4) [ p p ]1 1 2 P and a r e t h e J 2 d e f o r m a t i o n t h e o r y p l a s t i c compliances, $ = Here, a n g u l a r measure The p o s i t i v e i s t h e K i r c h h o f f stress d e v i a t o r from ( 3 . 2 ) . of t h e s t r e s s r a t e r e l a t i v e t o t h e cone a x i s i s d e f i n e d by %

K:$

c

(3.5)

A t t h e c o r n e r t h e p l a s t i c p o t e n t i a l is g i v e n

and t h e p l a s t i c s t r a i n r a t e i s o b t a i n e d from t h e g r a d i e n t o f t h e p o t e n t i a l as

A. Needleman

258

(3.7) Combining ( 3 . 3 ) and ( 3 . 7 ) g i v e s t h e t o t a l i n c r e m e n t a l compliance t e n s o r . Then, i n v e r t i n g g i v e s t h e i n c r e m e n t a l moduli $ t o be used i n ( 2 . 9 ) . The i n c r e m e n t a l s t i f f n e s s e s depend on t h e a n g l e 8 d e f i n e d i n ( 3 . 5 1 , w i t h f ( 8 ) = 1 i n t h e t o t a l l o a d i n g regime ( " s o f t e s t response") and d e c r e a s i n g m o n o t o n i c a l l y t o z e r o f o r e l a s t i c u n l o a d i n g ( " s t i f f e s t response"). In the f i n i t e element c a l c u l a t i o n s t o be reviewed s u b s e q u e n t l y t h e t r a n s i t i o n f u n c t i o n t h a t has been used is t h e one found i n [ 3 9 ] t o g i v e b e h a v i o r i n good agreement w i t h p r e d i c t i o n s based on t h e p h y s i c a l model [ 3 4 1 . The i n t e r e s t e d r e a d e r is r e f e r r e d t o [ 4 0 ] , a s w e l l a s t o t h e o r i g i n a l paper of C h r i s t o f f e r s e n and Hutchinson [ 3 9 ] , f o r a complete s p e c i f i c a t i o n of J 2 c o r n e r t h e o r y . T h i s t h e o r y i s more i n v o l v e d t o implement i n f i n i t e element program than i s t h e c o n s t i t u t i v e r e l a t i o n f o r a Mises s o l i d , b u t a r e l a t i v e l y s t r a i g h t f o r w a r d procedures h a s proved e f f e c t i v e i n p r a c t i c e [30,40]. The f e a t u r e s of t h i s c o n s t i t u t i v e r e l a t i o n of g r e a t e s t s i g n i f i c a n c e f o r neck development a r e t h a t w i t h i n t h e t o t a l l o a d i n g regime t h e r e s p o n s e t o a change i n l o a d i n g p a t h i s much s o f t e r t h a n f o r t h e Mises s o l i d and t h a t a s t h e l o a d i n g p a t h l e a v e s t h e t o t a l l o a d i n g regime t h e response becomes stiffer.

3.3

Continuum Model of a D u c t i l e Porous S o l i d

Based on an approximate r i g i d - p l a s t i c a n a l y s i s of a s i n g l e s p h e r i c a l v o i d , Gurson [ 4 1 , 4 2 ] proposed a y i e l d f u n c t i o n f o r a s o l i d w i t h a randomly d i s t r i b u t e d volume f r a c t i o n , f , of t h e form

2

0

=

3 + 2fq cosh -2 1

(3.8)

where 8 is t h e c u r r e n t m a t r i x flow s t r e n g t h , oe is t h e macroscopic e f f e c t i v e s t r e s s , d e f i n e d i n terms of t h e Cauchy s t r e s s and q1 i s a n a d d i t i o n a l parameter i n t r o d u c e d by Tvergaard [ 4 3 , 4 4 ] from comparisons of p r e d i c t i o n s of t h i s model w i t h f u l l f i e l d numerical r e s u l t s . A f u r t h e r m o d i f i c a t i o n of t h e y i e l d f u n c t i o n by Tvergaard [ 4 5 ] and Tvergaard and Needleman [25] a c c o u n t s f o r t h e l o s s of load c a r r y i n g c a p a c i t y a s s o c i a t e d w i t h void c o a l e s c e n c e a t v o i d s p a c i n g s of t h e o r d e r of t h e v o i d r a d i u s . This m o d i f i c a t i o n i n t r o d u c e s a d d i t i o n a l parameters t o s p e c i f y t h e c o a l e s c e n c e r a t e and is n e c e s s a r y t o a c c o u n t f o r complete l o s s o f stress c a r r y i n g c a p a c i t y a t r e a l i s t i c v o i d volume f r a c t i o n s . However, even w i t h t h i s m o d i f i c a t i o n t h e flow r u l e m a i n t a i n s t h e s t r u c t u r e d e s c r i b e d h e r e . The m a t r i x m a t e r i a l is c h a r a c t e r i z e d a s a Mises s o l i d (when f = 0 ( 3 . 8 ) reduces t o t h e Mises y i e l d s u r f a c e ) and t h e m a t r i x e f f e c t i v e p l a s t i c s t r a i n r a t e and e f f e c t i v e s t r e s s r a t e are r e l a t e d by a u n i a x i a l stresss t r a i n r e l a t i o n . These q u a n t i t i e s a r e r e l a t e d t o macroscopic q u a n t i t i e s v i a t h e e q u i v a l e n c e of macroscopic and m i c r o s c o p i c p l a s t i c work

259

The Numerical Analysis of Necking Instabilities

(3.9)

The i n c r e a s e i n t h e void volume f r a c t i o n , f , a r i s e s p a r t l y from t h e growth of e x i s t i n g v o i d s and p a r t l y from t h e n u c l e a t i o n of new v o i d s , so t h a t growth + f n u c l e a t i o n

;=;

(3.10)

Since t h e m a t r i x m a t e r i a l is p l a s t i c a l l y incompressible fgrowth = (l-f)&$ One n u c l e a t i o n c r i t e r i o n , s u g g e s t e d by Gurson's o b t a i n e d by Gurland [ 4 7 ] , i s fnucleation =

(3.11) [41,42] a n a l y s i s of d a t a

(3.12)

P

u s i n g f, =. A a $ J/ au t o g e t h e r w i t h ( 3 . 8 ) through ( 3 . 1 2 ) and t h e c o n s i s t e n c y c o n d i t i o n $J = 0 a u r i n g p l a s t i c l o a d i n g , g i v e s a flow r u l e of t h e form

gP

1 =

A

[(X:g)

(3.13)

where e x p l i c i t e x p r e s s i o n s f o r t h e terms i n (3.13) a r e given i n t h e r e f e r e n c e s c i t e d and i n [46,48]. Here, we n o t e t h a t p l a s t i c f l o w i s d i l a t i o n a l , w i t h p o s s i b l y l a r g e p l a s t i c volume changes, and p r e s s u r e s e n s i t i v e due t o t h e p r e s e n c e of t h e micro-voids. Furthermore, a l t h o u g h t h e m a t r i x m a t e r i a l c o n t i n u e s t o h a r d e n , t h e macroscopic h a r d e n i n g H i n (3.13) can become n e g a t i v e due t o t h e weakening e f f e c t s of v o i d n u c l e a t i o n and growth. The flow r u l e (3.13) i s n e c e s s a r i l y phrased i n terms of Cauchy s t r e s s and n o t Kirchhoff s t r e s s ( t h e Cauchy s t r e s s a p p e a r s from a v e r a g i n g o v e r t h e p l a s t i c a l l y i n c o m p r e s s i b l e m a t r i x ) . In t h i s c a s e an e l a s t i c s t r a i n r a t e e x p r e s s i o n of t h e form ( 3 . 3 ) i s used b u t i s e x p r e s s e d i n terms of Cauchy stress q u a n t i t i e s . Although t h e p l a s t i c s t r a i n r a t e i n (3.13) h a s a d i r e c t i o n normal t o t h e y i e l d s u r f a c e i n Cauchy s t r e s s space i t i s n o t normal t o t h e y i e l d s u r f a c e i n Kirchhoff stress space. Hence, t h e t e n s o r of i n c r e m e n t a l moduli, i n (2.91, do not s a t i s f y t h e symmetry c o n d i t i o n s n e c e s s a r y f o r t h e v a r i a t i o n a l f o r m u l a t i o n t o h o l d and t h e r e s u l t i n g f i n i t e element s t i f f n e s s m a t r i x is unsynmetric. 4.

BIFURCATION ANALYSES

We c o n s i d e r a uniform t e n s i l e specimen s u b j e c t t o a p r e s c r i b e d a x i a l displacement a t ends t h a t remain s h e a r f r e e . Also, t h e l a t e r a l f a c e s of t h e specimen are t a k e n t o be t r a c t i o n f r e e , e x c e p t , of c o u r s e , f o r t h e normal t r a c t i o n s r e q u i r e d t o e n f o r c e t h e p l a n e s t r a i n c o n s t r a i n t i n t h e

2 60

A. Needleman

plane s t r a i n t e n s i o n specimen. For t h e s e boundary c o n d i t i o n s one p o s s i b l e s o l u t i o n i s a homogeneous one. When t h e e l a s t i c - p l a s t i c c o n s t i t u t i v e r e l a t i o n s admit t h e v a r i a t i o n a l f o r m u l a t i o n ( 2 . 1 6 ) , b i f u r c a t i o n from t h i s homogeneous s t a t e of d e f o r m a t i o n c a n be addressed w i t h i n t h e framework l a i d out by H i l l [ 6 , 7 ] . A c e n t r a l r o l e i s played i n H i l l ’ s t h e o r y by t h e concept of a l i n e a r comparison s o l i d . For t h e l o a d i n g h i s t o r i e s e n v i s i o n e d h e r e , t h e l i n e a r comparison s o l i d h a s , a t each m a t e r i a l p o i n t , moduli e q u a l t o t h e p l a s t i c l o a d i n g moduli. The o n s e t of b i f u r c a t i o n f o r t h e l i n e a r comparison s o l i d i s governed by a c o n v e n t i o n a l e i g e n v a l u e problem and t h e n t h i s eigenmode can be r e l a t e d t o t h e b i f u r c a t i o n mode of t h e a c t u a l e l a s t i c - p l a s t i c s o l i d [ 3 7 ] . When t h e v a r i a t i o n a l s t r u c t u r e i s a b s e n t , a s i s t h e c a s e f o r t h e porous p l a s t i c s o l i d , t h e r e is no analogous framework f o r a d d r e s s i n g q u e s t i o n s of b i f u r c a t i o n and u n i q u e n e s s , a l t h o u g h s t e p s toward developing such a framework have been taken [49]. Here, we f i r s t p r e s e n t a one-dimensional Considere type argument f o r t h e o n s e t of necking. Next, we surrmarize r e s u l t s on d i f f u s e necking i n p l a n e s t r a i n and axisymmetric t e n s i o n which e x h i b i t t h e d e l a y i n t h e o n s e t of necking due t o t h r e e dimensional e f f e c t s . F i n a l l y , we mention some r e s u l t s on s h e a r l o c a l i z a t i o n f o r t h e m a t e r i a l models d i s c u s s e d i n S e c t i o n 3. Shear l o c a l i z a t i o n s p l a y a c e n t r a l r o l e i n d e t e r m i n i n g t h e f a i l u r e b e h a v i o r s shown i n F i g . 1 .

4.1

One-Dimensional A n a l y s i s

The Considere t r e a t m e n t of necking c a n be viewed a s a one dimensional b i f u r c a t i o n (and i m p e r f e c t i o n ) a n a l y s i s . To do t h i s , f o c u s a t t e n t i o n on two c r o s s - s e c t i o n s , A and B, o f a t e n s i l e b a r . Cross s e c t i o n B i s t h e minimum s e c t i o n and c r o s s s e c t i o n A i s f a r from t h e neck. E q u i l i b r i u m requires

(4.1) where u is t h e average t r u e s t r e s s , A is t h e c r o s s - s e c t i o n a l a r e a and q u a n t i t i e s a s s o c i a t e d w i t h each c r o s s - s e c t i o n a r e denoted by t h e appropriate subscript. W r i t i n g ( 4 . 1 ) i n i n c r e m e n t a l form and n e g l e c t i n g e l a s t i c volume changes ( s o t h a t c r o s s - s e c t i o n a r e a changes c a n be simply r e l a t e d t o l e n g t h changes 1, g i v e s

(4.2) t

with E t h e l o g a r i t h i m i c t e n s i l e s t r a i n and K t h e s l o p e of t h e a-E c u r v e ; Kt= Et f o r axisymmetric t e n s i o n and Kt = 4Et/3 f o r p l a n e s t r a i n t e n s i o n . For a homogeneous b a r ( 4 . 2 ) reduces to t A (K -u

A

A

. .

A

) ( E ~ - E ~ )=

0

(4.3)

The Numericul Analysis of Necking Instabilities

26 1

Hence, a b i f u r c a t i o n i s p o s s i b l e when t aA = KA

(4.4)

T h i s i s , of c o u r s e , t h e c l a s s i c a l Considere r e s u l t which c a n be p h r a s e d as s t a t i n g t h a t t h e necking b i f u r c a t i o n p o i n t and t h e maximum load p o i n t coincide.

+

I f AA and d i f f e r i n i t i a l l y or i f g r i p conditions give rise t o inhomogeneous s t r a i n i n g , t h e n an i n c r e m e n t a l s o l u t i o n of ( 4 . 2 ) i s needed t o c a l c u l a t e t h e e v o l u t i o n of t h e neck. The s o l u t i o n c o n t i n u e s u n t i l

uB

t

(4.5)

= KB

a t which p o i n t d c /dEA + m; u n l o a d i n g b e g i n s a t c r o s s - s e c t i o n A and t h i s marks t h e o n s e t o f n e c k i n g . The r a t e a t which t h e neck s u b s e q u e n t l y d e v e l o p s i n t h e neck i s n o t d e t e r m i n e d by t h i s a n a l y s i s . A l s o , t h e s p a t i a l e x t e n t o f t h e neck i s determined by f a c t o r s n o t i n c o r p o r a t e d i n t o t h e one d i m e n s i o n a l model. 4.2

D i f f u s e Necking

Accounting f o r t h e a c t u a l t h r e e d i m e n s i o n a l necking d e f o r m a t i o n s d e l a y s t h e o n s e t of b i f u r c a t i o n t o a p o i n t s u b s e q u e n t t o t h e a t t a i n m e n t of t h e maximum l o a d . I n o r d e r t o c a r r y o u t a t h r e e d i m e n s i o n a l a n a l y s i s a c h o i c e of p l a s t i c i t y t h e o r y h a s t o be made and t h e d e l a y between t h e maximum load p o i n t and t h e b i f u r c a t i o n p o i n t c a n be c o n s t i t u t i v e r e l a t i o n a s w e l l a s geometry dependent. A n a l y t i c a l b i f u r c a t i o n a n a l y s e s have been c a r r i e d o u t f o r t h e s p e c i a l c a s e of i n c o m p r e s s i b l e e l a s t i c a s w e l l a s p l a s t i c m a t e r i a l b e h a v i o r . The small e f f e c t of e l a s t i c c o m p r e s s i b i l i t y i s c o m p l e t e l y n e g l i g i b l e . Although t h e s e b i f u r c a t i o n r e s u l t s do n o t d i r e c t l y p e r t a i n t o t h e porous p l a s t i c s o l i d , f o r s t r u c t u r a l m e t a l s , v o i d volume f r a c t i o n s p r i o r t o t h e o n s e t of necking a r e g e n e r a l l y small, and t h e o n s e t o f n e c k i n g i n t h e s e c i r c u m s t a n c e s i s well r e p r e s e n t e d by t h e a n a l y s i s based on i n c o m p r e s s i b l e m a t e r i a l behavior. Necking t y p e b i f u r c a t i o n modes c a n be w r i t t e n i n t h e form

Ai

= gicos(mTz/L

+ $.I

(4.6)

where z is t h e d i r e c t i o n of t h e t e n s i l e a x i s , m i s a wave number, $i is 0 o r n / 2 , L is t h e c u r r e n t specimen l e n g t h and t h e f u n c t i o n s g i have a s independent v a r i a b l e s c o o r d i n a t e s i n t h e c r o s s - s e c t i o n a l a r e a . The c r i t i c a l mode is t h e l o n g e s t wavelength mode c o n s i s t e n t w i t h t h e boundary c o n d i t i o n s [10,11]. When symmetry a b o u t t h e mid-plane o f t h e specimen i s r e q u i r e d t h i s g i v e s t h e f a m i l i a r hour g l a s s shaped n e c k i n g mode [10,11], m = 2 i n (4.6).

A. Needleman

262

Hutchinson and Miles [ l o ] analyzed t h e o n s e t of b i f u r c a t i o n f o r a c l a s s of i n c o m p r e s s i b l e e l a s t i c - p l a s t i c s o l i d s w i t h smooth y i e l d s u r f a c e s . For a Mises s o l i d , t h e c r i t i c a l b i f u r c a t i o n stress i s

(4.7) a s y m p t o t i c a l l y as R/L + 0. Here, R i s t h e c u r r e n t t e n s i l e b a r r a d i u s and G is t h e e l a s t i c s h e a r modulus. This is t h e + m , t h e o n s e t of necking i s d e l a y e d i n d e f i n i t e l y . r i g i d p l a s t i c l i m i t . For f i n i t e G / E t , b i f u r c a t i o n i s d e l a y e d o n l y s l i g h t l y beyond t h e maximum load p o i n t i f t h e b a r i s s u f f i c i e n t l y s l e n d e r . For r e p r e s e n t a t i v e m a t e r i a l p r o p e r t i e s and r e a s o n a b l y s l e n d e r specimens, say R/L = 4, t h e necking b i f u r c a t i o n o c c u r s a t a n imposed a x i a l s t r a i n 10-20 p e r c e n t g r e a t e r t h a n t h e imposed a x i a l s t r a i n a t maximum l o a d , a s found from f i n i t e element b i f u r c a t i o n r e s u l t s 191. For stubby specimens t h e d e l a y c a n be c o n s i d e r a b l y g r e a t e r [9,10]. The appearance o f t h e e l a s t i c s h e a r modulus i n (4.7) i s a consequence of u s i n g a smooth y i e l d s u r f a c e p l a s t i c i t y t h e o r y . As d i s c u s s e d by Hutchinson and Miles [ l o ] , a c o r n e r t h e o r y of p l a s t i c i t y l e a d s t o a v e r t e x reduced s h e a r modulus a p p e a r i n g i n (4.7); f o r J 2 c o r n e r t h e o r y G i s r e p l a c e d by t h e s e c a n t s h e a r modulus. Except i n l i m i t i n g c a s e s , s t u b b y specimens o r n e a r l y r i g i d p l a s t i c b e h a v i o r , t h e p r e d i c t e d o n s e t of necking is n o t v e r y much affected.

As G/Et

A similar e i g e n v a l u e problem can be formulated f o r a p l a n e s t r a i n r e c t a n g u l a r s l a b . The a s y m p t o t i c e x p r e s s i o n analogous t o (4.7) i s ,

[ll],

where w i s t h e c u r r e n t specimen width i n t h e p l a n e of d e f o r m a t i o n . The e l a s t i c s h e a r modulus does not a p p e a r i n (4.7) and t h e r e i s a f i n i t e b i f u r c a t i o n s t r e s s f o r r i g i d - p l a s t i c s o l i d s [5,11]. However, i n a d d i t i o n t o t h e l o n g wavelength n e c k i n g mode, s h o r t wavelength modes ( l a r g e r v a l u e s of m i n (4.6)) a r e a v a i l a b l e . The c r i t i c a l c o n d i t i o n s f o r b i f u r c a t i o n i n t o t h e s e modes a r e q u i t e c o n s t i t u t i v e model s e n s i t i v e , w i t h s h o r t wavelength modes e s s e n t i a l l y r u l e d o u t f o r t h e Mises t y p e s o l i d [111. For e l a s t i c - p l a s t i c s o l i d s , t h e one dimensional r e s u l t emerges from b o t h (4.7) and (4.8) a s t h e l i m i t f o r i n f i n i t e l y long specimens. For f i n i t e l e n g t h specimens, t h e o n s e t of t h e l o n g wavelength d i f f u s e n e c k i n g mode i s delayed beyond t h e maximum l o a d p o i n t and, f o r r e p r e s e n t a t i v e specimen geometries and m a t e r i a l p r o p e r t i e s , t h e magnitude of t h i s d e l a y i s n o t v e r y c o n s t i t u t i v e model s e n s i t i v e .

4.3

L o c a l i z e d Necking

L o c a l i z e d necking i n t h e form of a s h e a r band can be analyzed w i t h i n a framework t h a t r e g a r d s l o c a l i z a t i o n a s a "materiel i n s t a b i l i t y , " where a m a t e r i a l element is c o n s i d e r e d t o be s u b j e c t t o p r e s c r i b e d a l l around

The Numerical Analysis of Necking Instabilities

263

d i s p l a c e m e n t s t h a t are c o n s i s t e n t w i t h a homogeneous d e f o r m a t i o n . Deformations i n a l o c a l i z e d band a r e p e r m i t t e d provided t h e v e l o c i t y f i e l d remains c o n t i n u o u s and c o n t i n u i n g e q u i l i b r i u m a t t h e band i n t e r f a c e i s satisfied, H i l l [ ~ o I R , i c e 1213. The e f f e c t s of i n i t i a l i m p e r f e c t i o n s on l o c a l i z a t i o n c a n a l s o be analyzed w i t h i n t h i s m a t e r i a l i n s t a b i l i t y framework, Rice 1211. An e l a s t i c - p l a s t i c s o l i d w i t h a smooth y i e l d s u r f a c e , s u c h as a Mises s o l i d , is q u i t e r e s i s t a n t t o l o c a l i z a t i o n . I n plane s t r a i n a n e a r l y i d e a l l y p l a s t i c s t a t e is required, exactly i d e a l l y p l a s t i c i n the rigidp l a s t i c l i m i t , while i n axisynmetric t e n s i o n t h e s t r a i n hardening r a t e must be s t r o n g l y n e g a t i v e f o r a l o c a l i z a t i o n b i f u r c a t i o n t o be p o s s i b l e [21]. Y i e l d s u r f a c e v e r t e x e f f e c t s , a s modeled by J 2 c o r n e r t h e o r y [39] and t h e d i l a t i o n a l and p r e s s u r e s e n s i t i v e p l a s t i c f l o w i n c o r p o r a t e d i n Gurson’s [41,42] model (when s m a l l i n i t i a l i m p e r f e c t i o n s a r e accounted f o r , [48]), l e a d t o s h e a r l o c a l i z a t i o n a t s t r a i n l e v e l s a c h i e v e d i n t h e neck of t e n s i l e specimens. However, t h e g r e a t e r r e s i s t a n c e t o s h e a r l o c a l i z a t i o n under axisymmetric c o n d i t i o n s t h a n under p l a n e s t r a i n c o n d i t i o n s remains [21,22]. T h i s i s i l l u s t r a t e d by some s i m p l e f o r m u l a s of Needleman and Rice [22] f o r s h e a r band b i f u r c a t i o n s i n a f i n i t e s t r a i n J 2 d e f o r m a t i o n t h e o r y s o l i d . These s h e a r band b i f u r c a t i o n r e s u l t s p e r t a i n t o a J 2 c o r n e r t h e o r y s o l i d , s i n c e t h e J 2 d e f o r m a t i o n t h e o r y moduli a r e t h o s e o f t h e l i n e a r comparison s o l i d f o r a J 2 c o r n e r t h e o r y s o l i d . For a m a t e r i a l w i t h a power h a r d e n i n g e f f e c t i v e stress v e r s u s e f f e c t i v e l o g a r i t h m i c s t r a i n r e l a t i o n of t h e form

u

N

(4.9)

= KE

t h e c r i t i c a l a x i a l t e n s i l e s t r a i n s f o r t h e o n s e t o f s h e a r bands a r e g i v e n by

crit ‘crit

=mGiq =

J(1+3N) (1-N) / 3

plane s t r a i n tension N

5 112

axisyrmnetric t e n s i o n N

5 1/3

(4.10)

With N = 0.1 t h e c r i t i c a l s t r a i n i n p l a n e s t r a i n t e n s i o n i s 0.30 and t h e c r i t i c a l s t r a i n i n a x i s y m n e t r i c t e n s i o n is 0.62. The f i n i t e s t r a i n g e n e r a l i z a t i o n of J 2 d e f o r m a t i o n t h e o r y used i n 1221 i s n o t u n i q u e ; a n a l t e r n a t i v e f i n i t e s t r a i n J 2 d e f o r m a t i o n t h e o r y can be c o n s t r u c t e d 1511. This a l t e r n a t i v e d e f o r m a t i o n t h e o r y h a s t h e advantage of b e i n g a f i n i t e s t r a i n h y p e r - e l a s t i c s o l i d b u t i t s b i f u r c a t i o n f o r m u l a s d o n o t t a k e on as s i m p l e a form a s (4.10). I n any c a s e , f o r N = 0.1 t h e s h e a r band b i f u r c a t i o n p r e d i c t i o n s o b t a i n e d from t h e s e two f o r m u l a t i o n s d i f f e r little. The v e r t e x t h e o r y model a t t r i b u t e s t h e o n s e t of l o c a l i z a t i o n t o a n i n h e r e n t f e a t u r e o f t h e p l a s t i c flow p r o c e s s w h i l e t h e porous p l a s t i c s o l i d model t i e s s h e a r l o c a l i z a t i o n t o t h e f a c t o r s r e s p o n s i b l e f o r t h e i n i t i a t i o n of d u c t i l e r u p t u r e . C l e a r l y , t h e d u c t i l e f a i l u r e s i n F i g . 1 i n v o l v e b o t h i n t e n s e s h e a r i n g and v o i d n u c l e a t i o n and growth w i t h i n t h e s h e a r band. What i s not c l e a r from l o o k i n g a t F i g . l b , however, i s whether l o c a l i z e d s h e a r i n g o c c u r r e d f i r s t , l e a d i n g t o l a r g e p l a s t i c

264

A. Needleman

s t r a i n s which then induce v o i d n u c l e a t i o n and growth o r whether t h e micror u p t u r e p r o c e s s i t s e l f p r e c i p i t a t e d t h e observed l o c a l i z a t i o n . While n o t a s s t r i k i n g l y e v i d e n t from t h e f i g u r e , l o c a l i z e d s h e a r i n g a l s o p l a y s a prominent r o l e i n t h e cup-cone f r a c t u r e p r o c e s s d e p i c t e d i n F i g . l a . When s h e a r bands d e v e l o p from nonhomogeneous d e f o r m a t i o n s t a t e s , as i n t h e neck of a t e n s i o n specimen, t h e a n a l y t i c a l procedures r e f e r r e d t o above a r e not d i r e c t l y a p p l i c a b l e . The o n s e t of l o c a l i z a t i o n and i t s i n i t i a l d i r e c t i o n of p r o p a g a t i o n can be determined by a m a t e r i a l i n s t a b i l i t y a n a l y s i s [ 5 2 ] ; but t h e d e t e r m i n a t i o n of s h e a r band p r o p a g a t i o n and of t h e v a r y i n g s h e a r i n t e n s i t y w i t h i n t h e band i n v o l v e s a f u l l boundary problem s o l u t i o n . This g e n e r a l l y r e q u i r e s a l a r g e s c a l e numerical c a l c u l a t i o n and a c c u r a t e numerical a n a l y s e s of l o c a l i z e d s h e a r i n g r e q u i r e p r o p e r mesh d e s i g n . A s d e s c r i b e d by Tvergaard e t a l . [ 2 3 ] , knowledge of t h e p r e f e r r e d s h e a r band o r i e n t a t i o n c a n be used t o d e s i g n a mesh t h a t i s c a p a b l e of r e s o l v i n g narrow bands. 5.

NUMERICAL ANALYSES a F NECKING

Chen [53] p u b l i s h e d t h e f i r s t a n a l y s i s of neck development i n a n a x i s y m n e t r i c t e n s i l e specimen. He i n i t i a t e d neck development by s p e c i f y i n g a n i n i t i a l reduced c r o s s s e c t i o n a l a r e a and used a Kantorovich method i n c o n j u n c t i o n w i t h t h e p r i n c i p l e of v i r t u a l work (2.8) t o s o l v e f o r f i e l d q u a n t i t i e s a t each s t a g e of t h e l o a d i n g h i s t o r y . The m a t e r i a l was c h a r a c t e r i z e d by t h e f i n i t e s t r a i n Mises s o l i d d e s c r i b e d i n S e c t i o n 3.1. Needleman [9] used t h e same f o r m u l a t i o n of t h e f i e l d and c o n s t i t u t i v e e q u a t i o n s a s Chen [53] b u t employed t h e f i n i t e element method f o r t h e s p a t i a l d i s c r e t i z a t i o n . Also t h e e f f e c t of g r i p c o n d i t i o n s on neck development was s t u d i e d i n [9]. Subsequently, N o r r i s e t a l . [13] and S a j e [14] have used v a r i o u s f i n i t e d i f f e r e n c e f o r m u l a t i o n s t o s t u d y neck development. Before d i s c u s s i n g t h e r e s u l t s of numerical a n a l y s e s of necking, a few c o w e n t s w i l l be made on f i n i t e element methods f o r t h i s c l a s s of problems. Although d i f f u s e necking c a n be analyzed u s i n g a Mises c o n s t i t u t i v e d e s c r i p t i o n , a n a l y z i n g t h e f a i l u r e phenomena shown i n F i g . 1 requires incorporating a material description i n t o the analysis that models m i c r o s t r u c t u r a l f e a t u r e s , such a s y i e l d s u r f a c e v e r t i c e s a n d / o r v o i d n u c l e a t i o n and growth. The a l g o r i t h i m s employed need t o be f l e x i b l e enough t o a c c o m o d a t e models of t h e s e phenomena. A l s o , s i n c e t h e systems b e i n g analyzed a r e m a r g i n a l l y s t a b l e p h y s i c a l l y , t h e numerical methods should n e i t h e r i n t r o d u c e a r t i f i c i a l n u m e r i c a l i n s t a b i l i t i e s n o r mask any a c t u a l p h y s i c a l i n s t a b i l i t i e s t h a t develop. Hence, i t should n o t be s u r p r i s i n g t h a t r e l a t i v e l y simple methods have had t h e g r e a t e s t s u c c e s s i n t h i s c l a s s of problems; a s p a t i a l d i s c r e t i z a t i o n u s i n g t h e d i s p l a c e m e n t f i n i t e element method w i t h a g r i d of l i n e a r displacement t r i a n g u l a r elements and simple i n c r e m e n t a l "time" i n t e g r a t i o n p r o c e d u r e s . It is important f o r t h e t r i a n g u l a r elements t o be a r r a n g e d i n q u a d r i l a t e r a l s c o n s i s t i n g of f o u r t r i a n g l e s formed by t h e q u a d r i l a t e r a l d i a g o n a l s . Such a g r i d can accomnodate t h e n e a r l y i s o c h o r i c d e f o r m a t i o n s c h a r a c t e r i s t i c o f p l a s t i c d e f o r m a t i o n (when v o i d n u c l e a t i o n and growth a r e a b s e n t ) , Nagtegaal e t a l . [54]. Also, such a mesh, when d e s i g n e d i n l i g h t of c r i t i c a l s h e a r band o r i e n t a t i o n s , i s w e l l s u i t e d f o r r e s o l v i n g l o c a l i z e d deformation i n a narrow band [23-251.

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Some of t h e r e s u l t s from [9] w i l l now be d e s c r i b e d . Two d i f f e r e n t t y p e s of boundary c o n d i t i o n were used. An a x i a l d i s p l a c e m e n t is p r e s c r i b e d a t t h e ends of t h e specimen. I n one c a s e , s h e a r f r e e end c o n d i t i o n s a r e p r e s c r i b e d s o t h a t t h e d e f o r m a t i o n s remain homogeneous u n t i l a b i f u r c a t i o n i s e n c o u n t e r e d . I n t h e o t h e r c a s e , t h e ends a r e t a k e n t o be cemented t o r i g i d g r i p s , p r o h i b i t i n g t a n g e n t i a l d i s p l a c e m e n t s a t t h e ends and g i v i n g I n b o t h c a s e s axisymmetric inhomogeneous d e f o r m a t i o n s from t h e o u t s e t . d e f o r m a t i o n s a r e assumed and f u r t h e r m o r e symmetry a b o u t t h e mid-plane of t h e specimen i s r e q u i r e d . The u n i a x i a l r e s p o n s e i s s p e c i f i e d a s a power law of t h e form (4.9) f o r s t r e s s e s i n e x c e s s o f t h e y i e l d s t r e n g t h . Computations were c a r r i e d o u t f o r N = 0.125 and f o r an i n i t i a l specimen l e n g t h t o r a d i u s r a t i o of 4. The l o a d v e r s u s end-displacement curve f o r t h e b a r w i t h s h e a r f r e e ends i s shown i n F i g . 2. B i f u r c a t i o n t a k e s p l a c e a t an end-displacement a b o u t 10 p e r c e n t g r e a t e r t h a n t h e end-displacement a t maximum l o a d . The o n s e t of b i f u r c a t i o n was determined by a n u m e r i c a l implementation of H i l l ' s [6,7] b i f u r c a t i o n t h e o r y [9]. The development of t h e neck i s shown i n F i g . 3 f o r b o t h s e t s of end conditions. T h i s development i s measured e i t h e r by t h e r a d i a l d i s p l a c e m e n t a t t h e neck or by t h e r a t i o of c u r r e n t t o i n i t i a l neck c r o s s s e c t i o n a l area. Such a c u r v e c l e a r l y r e v e a l s t h e c o u r s e of neck development and t h e same type of p l o t h a s s u b s e q u e n t l y been used by Burke and Nix [ZO] and A r g y r i s e t a l . [55]. When c o n s i d e r e d a s a f u n c t i o n of imposed d i s p l a c e m e n t , a s i n F i g . 3, t h e a r e a of t h e neck d e c r e a s e s more r a p i d l y f o r t h e r i g i d g r i p end c o n d i t i o n . However, t h e area r e d u c t i o n v e r s u s t o t a l load r e l a t i o n i s n e a r l y t h e same f o r b o t h s e t s of end c o n d i t i o n s [9]. I n f a c t , a t a g i v e n v a l u e of a r e a r e d u c t i o n , t h e stress and s t r a i n d i s t r i b u t i o n s a t t h e neck a r e n e a r l y independent of end c o n d i t i o n , once n e c k i n g i s w e l l underway [ 9 ] . The s t r a i n d i s t r i b u t i o n s a c r o s s t h e neck t h a t emerge from t h e n u m e r i c a l c a l c u l a t i o n s [ 9 , 1 3 , 1 4 , 5 3 ] a r e i n good agreement. The stress d i s t r i b u t i o n o b t a i n e d by Chen [53] d e v e l o p s a peak stress o f f t h e neck a x i s and i s n o t r e l i a b l e i n t h e l a t t e r s t a g e s of necking. The d i s t r i b u t i o n o b t a i n e d by Needleman [ 9 ] h a s a somewhat s h a r p e r peak on t h e a x i s t h a n found by s u b s e q u e n t i n v e s t i g a t o r s , b u t o t h e r w i s e t h e l e v e l and d i s t r i b u t i o n of s t r e s s e s i n [ 9 , 1 3 , 1 4 ] are i n good agreement. Of p a r t i c u l a r i n t e r e s t i s t h e f a c t t h a t t h e c a l c u l a t i o n s a l l i n d i c a t e t h a t t h e Bridgman [3] s o l u t i o n u n d e r e s t i m a t e s t h e h y d r o s t a t i c s t r e s s i n t h e neck. T h i s i s of p a r t i c u l a r i n t e r e s t i n r e l a t i n g c o n d i t i o n s i n t h e neck of a t e n s i l e specimen t o t h e o n s e t of d u c t i l e r u p t u r e . Once d u c t i l e h o l e growth b e g i n s , however, t h e assumptions u n d e r l y i n g a n a n a l y s i s based on a Mises t y p e s o l i d a r e no l o n g e r a p p l i c a b l e . Q u i t e r e c e n t l y , Tvergaard and Needleman [25] have c a r r i e d o u t a n a n a l y s i s of t h e a x i s y m n e t r i c t e n s i l e test based on G u r s o n ' s [41,42] model of a porous p l a s t i c s o l i d , m o d i f i e d so as t o account f o r a complete l o s s o f s t r e s s c a r r y i n g c a p a c i t y a t a r e a l i s t i c v o i d volume f r a c t i o n . The m a t r i x m a t e r i a l was t a k e n t o f o l l o w a power h a r d e n i n g r e l a t i o n of t h e form (4.9) w i t h N = 0.10 and v o i d n u c l e a t i o n was p l a s t i c s t r a i n c o n t r o l l e d a s i n (3.12).

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The r e s u l t s i n [25] r e p r o d u c e t h e e s s e n t i a l f e a t u r e s of t h e cup-cone f r a c t u r e shown i n F i g . la. A c r a c k forms i n t h e c e n t e r o f t h e neck and p r o p a g a t e s a c r o s s t h e specimen as shown i n F i g . 4, where t h e shaded r e g i o n c o r r e s p o n d s t o m a t e r i a l t h a t has undergone a complete l o s s of s t r e s s carrying capacity. It c a n be seen i n t h i s f i g u r e t h a t t h e r e i s a tendency f o r t h e c r a c k t o zig-zag. A s d i s c u s s e d by Tvergaard and Needleman [25] t h i s i s a consequence of s h e a r l o c a l i z a t i o n b e i n g i n h i b i t e d by t h e a d d i t i o n a l p l a s t i c work a s s o c i a t e d w i t h t h e hoop s t r a i n s t h a t accompany s h e a r i n g i n t h e axisymmetric geometry. A s t h e f r e e s u r f a c e i s approached t h i s a x i s y m n e t r i c c o n s t r a i n t i s r e l a x e d , p e r m i t t i n g t h e cone o f t h e cupcone f r a c t u r e t o form [ 2 5 ] . This a n a l y s i s shows i n d e t a i l how t h e i n t e r a c t i o n of t h e tendency t o l o c a l i z a t i o n i n a m a t e r i a l weakened by v o i d n u c l e a t i o n and growth t o g e t h e r w i t h a c o n s t r a i n i n g g e o m e t r i c a l e f f e c t lead t o t h e cup-cone f r a c t u r e i n F i g . l a . There i s no c o r r e s p o n d i n g g e o m e t r i c a l c o n s t r a i n t i n p l a n e s t r a i n t e n s i o n s o t h a t , once i n i t i a t e d , a s h e a r band can p r o p a g a t e a c r o s s t h e e n t i r e specimen. However, whether o r n o t a s h e a r band o c c u r s i n a c a l c u l a t i o n depends on t h e c o n s t i t u t i v e r e l a t i o n employed. C a l c u l a t i o n s based on a n i s o t r o p i c a l l y h a r d e n i n g Mises s o l i d g i v e continued growth of a d i f f u s e neck w i t h no tendency f o r l o c a l i z e d s h e a r i n g t o develop. T h i s i s , of c o u r s e , e x p e c t e d based on t h e a n a l y t i c a l flow l o c a l i z a t i o n r e s u l t s d i s c u s s e d i n S e c t i o n 4.3. Numerical a n a l y s e s of t h e p l a n e s t r a i n t e n s i l e t e s t u s i n g a n i s o t r o p i c a l l y h a r d e n i n g Mises s o l i d t o c h a r a c t e r i z e t h e m a t e r i a l cannot r e p r o d u c e t h e b e h a v i o r shown i n F i g . l b . A f i n i t e element a n a l y s i s of t h e p l a n e s t r a i n t e n s i l e t e s t was c a r r i e d o u t by Tvergaard, Needleman and Lo [23] u s i n g J 2 c o r n e r t h e o r y t o c h a r a c t e r i z e t h e m a t e r i a l behavior. The u n i a x i a l stress s t r a i n curve i s a power law ( 4 . 9 ) , w i t h N = 0.10. Various small i n i t i a l t h i c k n e s s inhomogeneities a r e p r e s c r i b e d i n t h e form o f a l i n e a r combination of t h e l o n g wavelength d i f f u s e necking mode shape (m = 2 i n ( 4 . 6 ) ) and v a r i o u s s h o r t wavelength mode shapes. Tvergaard e t a l . [23] d i s p l a y v a r i o u s s h e a r band p a t t e r n s , where t h e p a r t i c u l a r p a t t e r n t h a t o c c u r s depends on t h e i n i t i a l i m p e r f e c t i o n . However, i n each c a s e t h e o r i e n t a t i o n of t h e band, o r bands, i s i n good agreement w i t h t h a t p r e d i c t e d from a s h e a r band b i f u r c a t i o n a n a l y s i s . F i g . 5 shows deformed f i n i t e element meshes f o r one of t h e i r c a l c u l a t i o n s . The a c t u a l computations were c a r r i e d o u t f o r one quadrant and symmetry boundary c o n d i t i o n s were imposed. Shear bands develop n a t u r a l l y d u r i n g t h e c o u r s e of t h e c a l c u l a t i o n and, e v e n t u a l l y , a second s h e a r band p a t t e r n develops i n t h e i n t e r i o r of t h e specimen, unconnected t o t h e o r i g i n a l one. T h i s i s a consequence of t h e v e r t e x s t i f f e n i n g i n c o r p o r a t e d i n t o t h e J 2 c o r n e r t h e o r y model. As t h e deformation p a t t e r n s h i f t s , t h e moduli i n t h e s h e a r band s t i f f e n , w h i l e t h e moduli o u t s i d e t h e s h e a r band remain t h e less s t i f f t o t a l l o a d i n g moduli. The i n c r e a s i n g s t r a i n i n t h e neck i n t e r i o r t h e n i n d u c e s t h e i n t e r n a l s h e a r bands as i l l u s t r a t e d i n F i g . 5. The e x t e n t t o which t h e s e p a r a t i o n i s a f f e c t e d by t h e d i s c r e t i z a t i o n of t h e problem i s n o t f u l l y understood. T h i s is one of s e v e r a l i s s u e s c o n c e r n i n g mesh e f f e c t s i n t h e numerical a n a l y s i s of l o c a l i z e d s h e a r i n g t h a t m e r i t i n v e s t i g a t i o n [23,29]. L o c a l i z a t i o n i n p l a n e s t r a i n can a l s o occur a s a consequence of t h e weakening induced by v o i d n u c l e a t i o n and growth. T h i s i s demonstrated i n T v e r g a a r d ' s [24] c a l c u l a t i o n on s h e a r band f o r m a t i o n a t a f r e e s u r f a c e i n

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a s o l i d s u b j e c t t o p l a n e s t r a i n t e n s i o n . Shear banding i n a p l a n e s t r a i n t e n s i l e specimen i s d e p i c t e d i n F i g . 6 , where c o n t o u r s of c o n s t a n t v o i d volume f r a c t i o n a r e shown a t two s t a g e s of l o a d i n g . The m a t e r i a l is c h a r a c t e r i z e d by t h e porous p l a s t i c c o n s t i t u t i v e r e l a t i o n d e s c r i b e d i n S e c t i o n 3.3, w i t h o u t t h e a d d i t i o n a l f a i l u r e terms used by Tvergaard [24] and Tvergaard and Needleman [25]. The m a t r i x m a t e r i a l f o l l o w s t h e power A t t h e e a r l i e r s t a g e of d e f o r m a t i o n law r e l a t i o n (4.9) w i t h N = 0.10. shown, few v o i d s have n u c l e a t e d (none a r e p r e s e n t i n i t i a l l y ) and t h e contour l i n e f o l l o w s t h e d i s t r i b u t i o n expected from t h e necking b e h a v i o r of a Mises s o l i d . A t a somewhat l a t e r s t a g e of d e f o r m a t i o n t h e v o i d volume f r a c t i o n d i s t r i b u t i o n c l e a r l y r e f l e c t s t h e presence of s h e a r bands. Both y i e l d s u r f a c e v e r t e x e f f e c t s and t h e weakening induced by micror u p t u r e phenomena can l e a d t o t h e l o c a l i z e d s h e a r f r a c t u r e d e p i c t e d i n F i g . l b . Which of t h e s e mechanisms i s o p e r a t i v e i n a p a r t i c u l a r c i r c u m s t a n c e i s m a t e r i a l (and, p o s i b l y , s t r e s s s t a t e ) dependent. Advances, w i t h i n t h e p a s t decade o r s o , i n f i n i t e element methods f o r a n a l y z i n g f i n i t e s t r a i n p l a s t i c i t y problems, i n t h e t h e o r e t i c a l u n d e r s t a n d i n g of p l a s t i c i n s t a b i l i t y phenomena and i n m a t e r i a l s m o d e l l i n g c a p a b i l i t y t o g e t h e r have made p o s s i b l e a d e t a i l e d d e s c r i p t i o n of t h e phenomena shown i n F i g . 1. Computer s i m u l a t i o n s t u d i e s of t h e t y p e d e s c r i b e d h e r e can now be used t o i n v e s t i g a t e a s p e c t s of d u c t i l e f a i l u r e mechanics t h a t a r e d i f f i c u l t , i f n o t i m p o s s i b l e , t o i n v e s t i g a t e by o t h e r means. Although h e r e o n l y a n a l y s e s of t e n s i o n t e s t s u s i n g c o n s t i t u t i v e d e s c r i p t i o n s f o r p o l y c r y s t a l l i n e m e t a l s and q u a l i t a t i v e comparisons w i t h o b s e r v a t i o n have been mentioned, a v a r i e t y of l o c a l i z a t i o n phenomena have been analysed [29] and, f u r t h e r m o r e , when q u a n t i t a t i v e comparisons c a n be made, Larsson e t e l . [56], P e i r c e e t a l . , [57,58], t h e agreement between t h e numerical r e s u l t s and experiment i s remarkably good. The agreement a r i s e s because t h e c o n s t i t u t i v e d e s c r i p t i o n s employed i n t h e a n a l y s e s model r e l e v a n t m a t e r i a l c h a r a c t e r i s t i c s . For p o l y c r y s t a l l i n e metals t h e r e is a need f o r c o n s t i t u t i v e r e l a t i o n s t h a t i n c o r p o r a t e t h e a n i s o t r o p y t h a t a r i s e s due t o c r y s t a l l o g r a p h i c t e x t u r e . I n f a c t , i t t u r n s o u t t h a t , on a more m i c r o s c o p i c l e v e l t h a n d i s c u s s e d h e r e , s h e a r l o c a l i z a t i o n can p l a y a s i g n i f i c a n t r o l e i n t e x t u r e development [59]. ACKNOWLEDGMENT

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I n t . J. F r a c t . ,

18, 237 ( 1 9 8 2 ) .

45. T v e r g a a r d , V.,

I n t . J. S o l i d s S t r u c t . , 1 8 , 659 ( 1 9 8 2 ) .

and Needleman, A.,

46. Chu, C.C.

47. Gurland, J., A c t a M e t a l l . ,

J. Engr. Mat. Tech.,

20, 735 ( 1 9 7 2 ) .

Pan, J. and Needleman, A . ,

48. S a j e , M . ,

49. R a n i e c k i , B. and Bruhns, O.T., 50. H i l l , R.,

I n t . J. F r a c t . ,

19, 163 ( 1 9 8 2 ) .

J. Mech. Phys. S o l i d s , 29, 1 5 3 ( 1 9 8 1 ) .

J. Mech. Phys. S o l i d s , 10, 1 (1962).

51. Hutchinson, J.W. (1981).

and T v e r g a a r d , V . ,

52. Iwakuma, T. and Nemat-Nasser, (1982). 53. Chen, W.

99, 2 (1980).

S.,

I n t . J. S o l i d s S t r u c t . 1 7 , 451 I n t J. S o l i d s S t r u c t . , 18, 69

I n t J. S o l i d s S t r u c t . , 7 , 685 (1971).

54. N a g t e g a a l , J.C., P a r k s , D.M., Eng., 4 , 153 (1974).

and R i c e , J . R . ,

Comp. Meths. Appl. Mech.

55. A r g y r i s , J . H . , D o l t s i n i s , J . S t . , S t r a u b , K . , Pimenta, P.M., Symeonidis, Sp. and Wustenberg, H., Comp. Meths. Appl. Mech. Eng., 3 2 , 2 ( 1 9 8 2 ) . 56. L a r s s o n , M . , Mech. Phys.

Needleman, A., T v e r g a a r d , V . , S o l i d s , 30, 121-154 ( 1 9 8 2 ) .

and S t o r a k e r s , B . ,

J.

57. P e i r c e , D . , (1982).

Asaro, R . J .

and Needleman, A . ,

Acta M e t a l l . ,

30, 1087

58. P e i r c e , D . , (1983).

Asaro, R . J .

and Needleman, A . ,

Acta M e t a l l . ,

31, 1951

59. Asaro, R . J .

and Needleman, A. S c r i p t a Metall., t o a p p e a r .

A. Needleman

270

a

b

Fig. 1 Effect of stress state on failure of steel tensile specimens (a) axisymmetric tension; (b) plane strain tension. From [l]. -

1.4

Maximum

,Fun do men t a1

4 I .2

Fig. 2

Bifurcation

1.0 -

.8 -

Load versus enddisplacement for an axisymmetric tensile specimen with shear free ends. From [9].

.6-

.4 -

.2 -

//

Rigid Grips

t

Fig. 3 Neck development versus end-displacement for axisymmetric tensile specimens. From [ 91

.

/

.04

i.774

d

The Numerical Analysis of Necking Instabilities

Fig. 4

Crack growth in the neck of an axisynrmetric tensile specimen with the material characterized by a constitutive relation for a porous plastic solid that permits a loss of stress carrying capacity at realistic void volume fractions. From [ 2 5 ] .

27 1

A. Needleman

212

Fig. 5

Two stages of shear band development in a plane s t r a i n t e n s i l e specimen with the material characterized by J2 corner theory. From [ 2 3 ] .

The Numerical Analysis of Necking Instabilities

* Fig. 6

Contours of constant void volume fraction i n a plane s t r a i n t e n s i l e specimen w i t h the material characterized by a c o n s t i t u t i v e r e l a t i o n f o r a porous p l a s t i c s o l i d . The contour l i n e corresponds to a void volume fraction of 0.005 i n (a) and 0.055 i n ( b ) .

273

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Unification of Finite Element Methods H. Kardestuncer (Editor) Elsevier Science Publishers B.V.(North-Holland), 1984

CHAPTER 12 RECENT ADVANCES IN THE APPLICATION OF VARIATIONAL METHODS TO NONLINEAR PROBLEMS

A. K. Noor

Two recent advances in the a p p l i c a t i o n o f d i r e c t varia t i o n a l methods t o nonlinear s t e a d y - s t a t e problems a r e discussed. The f i r s t i s a hybrid a n a l y s i s technique based on the combined use of regular perturbation expansion and t h e c l a s s i c a l d i r e c t v a r i a t i o n a l techniques f o r predicting t h e nonlinear response of the system. The second i s a two-stage d i r e c t v a r i a t i o n a l technique which allows a s u b s t a n t i a l reduction i n the number of degrees of freedom t o be made. The advantages of each o f t h e two techniques a r e discussed and t h e i r e f f e c t i v e n e s s f o r the s o l u t i o n of nonlinear s t e a d y - s t a t e thermal and s t r u c t u r a l problems a r e demonstrated by means of four numerical examples. NOMENCLATURE

A A, 3,

E L y ET

h

h h L m, ml

n nl

c

cross-sectional area of f i n undetermined c o e f f i c i e n t s used i n the expressions of the approxi mation functions radius of c i r c u l a r p l a t e circumference of f i n cross s e c t i o n e l a s t i c modulus of i s o t r o p i c material e l a s t i c moduli o f the individual l a y e r s i n t h e d i r e c t i o n of f i b e r s and normal t o i t , r e s p e c t i v e l y shear modulus i n the plane of f i b e r s a phi& chosen approximation functions (defined in Eqs. 8, 9 and 13) t o t a l thickness o f p l a t e ( o r s h e l l ) convective heat t r a n s f e r c o e f f i c i e n t thermal conductivity c o e f f i c i e n t length of f i n ( a l s o s i d e length of p l a t e ) ranges of indices ( s e e Eqs. 4 and 5) t o t a l number of coordinate functions range o f indices ( s e e Eqs. 3 and 1 1 )

275

A.K. Noor concentrated l o a d i n t e n s i t y o f u n i f o r m d i s t r buted l o a d i n g radius o f curvature o f the shell radial coordinate ( f o r c i r c u l a r plates) temperature f i n base temperature t o t a l s t r a i n energy o f t h e s t r u c t u r e displacement components o f t h e p l a t e ( o r s h e l l ) middle p l a n e ( o r m i d d l e s u r f a c e ) , w i t h w normal t o t h e p l a t e ( o r s h e l l ) m i d d l e plane ( o r m i d d l e s u r f a c e ) fundamental unknown ( o r v e c t o r o f fundamental unknowns) c o o r d i n a t e f u n c t i o n s (see Eqs. 2 and 3) C a r t e s i a n c o o r d i n a t e system w i t h x3 normal t o t h e m i d d l e plane o f t h e p l a t e edge a n g l e o f s p h e r i c a l cap c o n d i t i o n number o f t h e Gram m a t r i x o f t h e b a s i s v e c t o r s parameters m a t r i x o f b a s i s v e c t o r s d e f i n e d i n Eqs. 16 and 17 constant symbol o f f i r s t v a r i a t i o n nondimens i o n a l temperature c o n t r o l ( o r p e r t u r b a t i o n ) parameters Poisson's r a t i o o f i s o t r o p i c m a t e r i a l s major P o i s s o n ' s r a t i o o f t h e i n d i v i d u a l l a y e r s o f laminated p l a t e dimensionless c o o r d i n a t e functional p e r t u r b a t i o n f u n c t i o n a l s d e f i n e d i n Eqs. 4 and 5 v e c t o r o f undetermined c o e f f i c i e n t s r o t a t i o n component o f t h e m i d d l e s u r f a c e o f t h e s h e l l amplitudes o f c o o r d i n a t e f u n c t i o n s

V a r i a t i o n a l methods such as R a y l e i g h - R i t z and Bubnov-Galerkin techniques have been, and c o n t i n u e t o be, p o p u l a r t o o l s f o r n o n l i n e a r a n a l y s i s . To date t h e r e a r e t h r e e p r a c t i c a l implementations o f v a r i a t i o n a l methods. The f i r s t i s t h e d i r e c t ( o r g l o b a l ) v a r i a t i o n a l approach i n which t h e f i e l d v a r i a b l e s ( o r fundamental unknowns) a r e sought i n t h e form o f a s e r i e s o f a pkiciahi chosen c o o r d i n a t e f u n c t i o n s ( o r modes) w i t h unknown c o e f f i c i e n t s . The c o o r d i n a t e f u n c t i o n s a r e chosen t o cover t h e e n t i r e domain o f t h e f i e l d v a r i a b l e s . The second approach i s t h e f i n i t e element method ( o r l o c a l

The Application of Variational Methods t o Nonlinear Problems

277

v a r i a t i o n a l approach) based on d i v i d i n g t h e domain i n t o f i n i t e elements and u s i n g p i e c e w i s e a p p r o x i m a t i o n f o r t h e f i e l d v a r i a b l e s w i t h i n each element. The t h i r d approach i s t h e g l o b a l ( o r macro) element method which i s a compromise between t h e f i r s t two. I n t h i s t e c h n i q u e t h e r e g i o n under c o n s i d a t i o n i s d i v i d e d i n t o a small number o f subregions and a s u i t a b l e a p p r o x i m a t i o n i s made w i t h i n each subregion. The c o n t i n u i t y o f t h e f i e l d v a r i a b l e s across t h e i n t e r f a c e s i s imposed i m p l i c i t l y i n t h e v a r i a t i o n a l f u n c t i o n a l [I]. ' The d i r e c t ( o r g l o b a l ) v a r i a t i o n a l approach was t h e standard t o o l f o r n o n l i n e a r a n a l y s i s i n t h e precomputer e r a . However, t h e widespread a v a i l a b i l i t y o f computers and t h e f a s c i n a t i o n w i t h t h e f i n i t e element method, caused by i t s v e r s a t i l i t y i n h a n d l i n g complex domains and s i m p l i c i t y o f computer implementation, has r e s u l t e d i n a r e l a t i v e s t a g n a t i o n i n t h e development o f e f f e c t i v e d i r e c t v a r i a t i o n a l methods. When c o n t r a s t e d w i t h f i n i t e element methods, d i r e c t v a r i a t i o n a l methods combine t h e f o l l o w i n g two advantages: a ) t h e y p r o v i d e p h y s i c a l i n s i g h t i n t o t h e n a t u r e o f t h e s o l u t i o n o f t h e problem; and b ) t h e y g e n e r a l l y have a h i g h e r r a t e o f ( a s y m p t o t i c ) convergence and r e s u l t i n a much s m a l l e r system o f (nonsparse) e q u a t i o n s (see [2]). However, t h e i r m a j o r drawback, from a p r a c t i c a l p o i n t o f view, i s t h e d i f f i c u l t y o f s e l e c t i n g good a p p r o x i mation f u n c t i o n s f o r c o m p l i c a t e d domains and/or complex system response. The g l o b a l ( o r macro) element method a l l e v i a t e s t h i s drawback, b u t i t does n o t r e a l i z e t h e f u l l p o t e n t i a l o f v a r i a t i o n a l methods. I n t h e p a s t few y e a r s c o n s i d e r a b l e progress has been made i n combining t h e f i n i t e element method w i t h d i r e c t v a r i a t i o n a l techniques i n t o e f f e c t i v e h y b r i d numerical procedures (see, f o r example, [3]-[61). These h y b r i d numerical procedures a r e r e f e r r e d t o as /reduction rneAho& and were successf u l l y used f o r p r e d i c t i n g t h e n o n l i n e a r thermal and s t r u c t u r a l responses o f s o l i d s . Also, a p p l i c a t i o n o f r e d u c t i o n methods t o t h e p r e d i c t i o n o f nonl i n e a r response o f s t r u c t u r e s s u b j e c t e d t o m u l t i p l e independent l o a d s , and t o t r a c i n g p o s t - l i m i t - p o i n t and p o s t - b i f u r c a t i o n p o i n t paths have been The s u c c e s s f u l experience w i t h t h e h y b r i d numerical r e p o r t e d i n [7]-[9]. procedures prompted new a t t e m p t s t o be made f o r t h e r e a l i z a t i o n o f t h e f u l l p o t e n t i a l o f v a r i a t i o n a l methods i n t h e s o l u t i o n o f n o n l i n e a r problems through: 1 ) t h e combined use o f v a r i a t i o n a l methods i n c o n j u n c t i o n w i t h o t h e r a n a l y t i c a l t e c h n i ues [lo]; and 2) i n n o v a t i v e ways o f a p p l y i n g d i r e c t The p r e s e n t paper summarizes some o f t h e r e c e n t v a r i a t i o n a l methods work i n t h i s area. The t o p i c s discussed h e r e i n i n c l u d e : a ) t h e combined use o f r e g u l a r p e r t u r b a t i o n technique and d i r e c t v a r i a t i o n a l methods; and b ) t h e two-stage a p p l i c a t i o n o f d i r e c t v a r i a t i o n a l methods.

[ill.

Numerical examples a r e presented t o demonstrate t h e e f f e c t i v e n e s s o f t h e proposed two techniques. Also, r e s e a r c h areas which have h i g h p o t e n t i a l f o r novel a p p l i c a t i o n o f v a r i a t i o n a l methods a r e i d e n t i f i e d . 2.

HYBRID PERTURBATION/DIRECT VARIATIONAL TECHNIQUE

P e r t u r b a t i o n methods share w i t h d i r e c t v a r i a t i o n a l techniques t h e advantage, over f i n i t e element methods, o f p r o v i d i n g p h y s i c a l i n s i g h t i n t o t h e n a t u r e o f t h e s o l u t i o n o f t h e problem. However, i n c o n t r a s t t o d i r e c t v a r i a t i o n a l techniques, t h e r e g u l a r p e r t u r b a t i o n method c o n s i s t s o f t h e development o f t h e s o l u t i o n i n terms o f unknown f u n c t i o n s w i t h prreannigned c o e ~ ~ i c i e n t n .

278

A.K. Noor

The unknown f u n c t i o n s a r e o b t a i n e d by s o l v i n g a r e c u r s i v e s e t o f d i f f e r e n t i a l equations, o r e q u i v a l e n t l y , by m i n i m i z i n g a r e c u r s i v e s e t o f f u n c t i o n a l s which c h a r a c t e r i z e t h e response o f t h e p h y s i c a l system. The r e c u r s i v e s e t o f d i f f e r e n t i a l equations (and f u n c t i o n a l s ) are, i n general, s i m p l e r t h a n t h e o r i g i n a l governing d i f f e r e n t i a l equations (and o r i g i n a l f u n c t i o n a l ) o f t h e problem, Despite t h e i r usefulness i n s o l v i n g n o n l i n e a r problems, regul a r p e r t u r b a t i o n techniques have two m a j o r drawbacks. The f i r s t stems from t h e f a c t t h a t as t h e number o f terms i n t h e p e r t u r b a t i o n s e r i e s increases, t h e mathematical c o m p l e x i t y o f t h e d i f f e r e n t i a l equations (and t h e f u n c t i o n a l ) d e s c r i b i n g t h e response o f t h e system b u i l d s up r a p i d l y . Theref o r e , f o r p r a c t i c a l a p p l i c a t i o n s , t h e p e r t u r b a t i o n s e r i e s has t o be r e s t r i c t e d t o a few terms. The second drawback i s t h e requirement o f r e s t r i c t i n g t h e p e r t u r b a t i o n parameter t o small values i n o r d e r t o o b t a i n s o l u t i o n s o f acceptable accuracy. The aforementioned drawbacks o f t h e r e g u l a r p e r t u r b a t i o n technique have been recognized and a number o f remedial a c t i o n s were proposed. These i n cluded t h e use o f a small number o f terms (e.g., two o r t h r e e ) i n t h e p e r t u r b a t i o n expansion and e i t h e r : 1 ) g e n e r a t i n q "mimic f u n c t i o n s " which g i v e a c c u r a t e numerical e s t i m a t e s o f t h e s o l u t i o n over t h e e n t i r e p h y s i c a l domain (see [12]); o r 2 ) ap l y i n g a n o n l i n e a r t r a n s f o r m a t i o n (e.g., Shanks t r a n s f o r m a t i o n [13] and [14!) t o e s t i m a t e t h e s o l u t i o n as t h e number o f terms goes t o i n f i n i t y . However, as shown i n [15], t h e success o f these methods cannot be guaranteed, i n general, and t h e remedial a c t i o n s may f a i l t o produce s a t i s f a c t o r y r e s u l t s . I n t h i s s e c t i o n a h y b r i d p e r t u r b a t i o n / d i r e c t v a r i a t i o n a l technique i s d e s c r i b e d which a1 l e v i a t e s t h e m a j o r drawbacks o f t h e two p a r e n t techniques w h i l e r e t a i n i n g t h e i r advantages. The h y b r i d technique i s used h e r e i n i n c o n j u n c t i o n w i t h t h e v a r i a t i o n a l f o r m u l a t i o n o f t h e problem. A p p l i c a t i o n o f t h e h y b r i d technique i n conj u n c t i o n w i t h the d i f f e r e n t i a l equation formulation, t o the s o l u t i o n o f n o n l i n e a r thermal problems, i s d e s c r i b e d i n [ l o ] . Many s i t u a t i o n s can be c i t e d wherein h y b r i d i z a t i o n p r o v i d e d an i n n o v a t i v e way o f overcoming problems and proved t o be e f f e c t i v e . I n t h e case o f computational a l g o r i t h m s , examples a r e p r o v i d e d by t h e h y b r i d e x p l i c i t / i m p l i c i t temporal i n t e g r a t i o n schemes f o r t r a n s i e n t problems, and t h e combined use o f d i r e c t and i t e r a t i v e techniques f o r s o l u t i o n o f l i n e a r a l g e b r a i c equations ( p a r t i c u l a r l y those a s s o c i a t e d w i t h h i e r a r c h i c a l f i n i t e elements o r m u l t i g r i d f i n i t e d i f f e r e n c e s ) . 2.1

Basic Idea o f H y b r i d Technique

The s t e a d y - s t a t e response o f t h e system i s d e s c r i b e d by t h e s t a t i o n a r y cond i t i o n o f a f u n c t i o n a l n which c h a r a c t e r i z e s t h e system, i . e . (1 1 where X i s t h e f i e l d v a r i a b l e ( o r fundamental unknown) and 6 i s t h e symbol o f f i r s t v a r i a t i o n . For more than one fundamental unknown X i s a v e c t o r . For n o n l i n e a r problems t h e f u n c t i o n a l n ( X ) i n c l u d e q u a d r a t i c as w e l l as h i g h e r - o r d e r terms i n X (and/or i t s s p a t i a l d e r i v a t i v e s ) . Equation 1 i s e q u i v a l e n t t o t h e governing d i f f e r e n t i a l equations and t h e boundary c o n d i t i o n s o f t h e system. 61I(X) = 0

The a p p l i c a t i o n o f t h e h y b r i d p e r t u r b a t i o n / d i r e c t v a r i a t i o n a l technique t o t h e s o l u t i o n o f Eq. 1 can be c o n v e n i e n t l y d i v i d e d i n t o t h e f o l l o w i n g two

The Application of VariationalMethods to Nonlinear Problems

279

d i s t i n c t s t e p s : 1 ) g e n e r a t i o n o f c o o r d i n a t e f u n c t i o n s ( o r modes) u s i n g t h e standard r e g u l a r p e r t u r b a t i o n method, and 2 ) computation o f t h e amp1 i t u d e s o f t h e c o o r d i n a t e f u n c t i o n s v i a d i r e c t v a r i a t i o n a l technique. The procedure i s d e s c r i b e d i n d e t a i l subsequently.

2.2

Generation o f Coordinate Functions

For t h e purpose o f g e n e r a t i n g t h e r e q u i r e d c o o r d i n a t e f u n c t i o n s ( o r modes), t h e f u n c t i o n a l n ( X ) i s embedded i n a s i n g l e - o r m u l t i p l e - p a r a m e t e r f a m i l y o f f u n c t i o n a l s o f t h e form: n ( X , x ) f o r t h e s i n g l e - p a r a m e t e r case; and n ( X , xl, h 2 ) f o r t h e two-parameter case, where x, xl, and x a r e n o r m a l i z i n g

2

parameters which a r e a l s o r e f e r r e d t o as c o n t r o l o r p e r t u r b a t i o n parameters. E x t e n s i o n t o more than two parameters i s s t r a i g h t f o r w a r d and i s n o t d i s cussed h e r e i n . The f i e l d v a r i a b l e ( o r v e c t o r o f f i e l d v a r i a b l e s ) X i s r e p r e s e n t e d by t h e r e g u l a r p e r t u r b a t i o n expansion : n- 1 X = 2 x i Xi (2) i=O f o r t h e s i n g l e - p a r a m e t e r case, and n1

=

z: j=O

j

i=O

,i x j - i 1 2

f o r t h e two-parameter case, where Xi

and

(3)

Xi ,j-i are perturbation functions

which r e p r e s e n t modes; and n i s t h e t o t a l number o f terms i n t h e expansion For t h e two-parameter case n = 1 / 2 (nl + l ) ( n , + 2 ) . I f t h e p e r t u r b a t i o n expansions, Eqs. 2 o r 3, a r e s u b s t i t u t e d i n t o t h e f u n c t i o n a l II and t h e terms h a v i n g t h e same powers o f t h e p e r t u r b a t i o n parameters a r e grouped t o g e t h e r , t h e n n can be w r i t t e n i n t h e f o l l o w i n g form: in- 1 i

n = 2 x ni i=O

(4)

f o r t h e s i n g l e - p a r a m e t e r case, and

f o r t h e two-parameter case, where ni and niYj

are the perturbation

f u n c t i o n a l s , and m i s t h e t o t a l number o f terms i n t h e expansion o f IT. For t h e two-parameter case m = 1 / 2 (ml + l)(m, + 2 ) . The s t a t i o n a r y c o n d i t i o n o f n , Eq. 1, can now be r e p l a c e d by t h e f o l l o w i n g recursive set o f stationary conditions:

mi

= 0

(i

= O t o m-1)

(61

A.K. Noor

280 f o r t h e single-parameter case, and

&ni,j-i

= 0

( j = O toml,

i = O to j )

(7)

f o r t h e two-parameter case. Note t h a t whereas t h e o r i g i n a l s t a t i o n a r y c o n d i t i o n , Eq. 1, leads t o a h e 2 0 6 d i d ~ m e Me q d v m , t h e r e c u r s i v e s e t o f s t a t i o n a r y c o n d i t i o n s , Eqs. 6 and 7, l e a d t o a t l e c m i v e 0 6 fineah di6dmenLLd equativnil .in XL and XkYk-,, r e s p e c t i v e l y , where i = 2 a and j = 2 k . Only

nonLinem

even values o f t h e i n d i c e s i,j i n Eqs. 6 and 7 need t o be considered. I n most p r a c t i c a l problems, an exact s o l u t i o n f o r t h e d i f f e r e n t i a l equations r e s u l t i n g from t h e s t a t i o n a r y c o n d i t i o n s , Eqs. 6 and 7, cannot be o b t a i n e d . Approximate s o l u t i o n s a r e sought f o r X , and X through t h e a p p l i c a t i o n o f t,k R a y l e i g h - R i t z technique, i . e . ,

f o r t h e one-parameter case, and

f o r t h e two-parameter case, where g k a ) and h(')

Q,k

a r e a p d o d chosen f u n c t i o n s

o f t h e s p a t i a l c o o r d i n a t e s ( a p p r o x i m a t i o n f u n c t i o n s ) which s a t i s f y t h e e s s e n t i a l boundary c o n d i t i o n s as w e l l as t h e symmetry c o n d i t i o n s on X , and and Aka', 3(a) a r e undetermined c o e f f i c i e n t s . I n t h e case o f more 'Q,,k' k,k t h a n one f i e l d v a r i a b l e , d i f f e r e n t s e t s o f a p p r o x i m a t i o n f u n c t i o n s g and h can be assumed f o r t h e d i f f e r e n t f i e l d v a r i a b l e s ( i . e . , f o r t h e d i f f e r e n t components o f X ) . The A and 3 c o e f f i c i e n t s a r e o b t a i n e d by s u b s t i t u t i n g Eqs. 8 o r 9 i n t o t h e r e c u r s i v e s e t o f s t a t i o n a r y c o n d i t i o n s , Eqs. 6 o r 7, and s o l v i n g t h e r e s u l t i n g s e t s o f r e c u r s i v e f i n e m dgeb&c equatiann. Note t h a t t h e l e f t - h a n d s i d e s o f t h e r e c u r s i v e s e t s o f a l g e b r a i c equations, f o r a l l values o f R ( o r f o r a l l p a i r s o f R and k ) , a r e t h e same.

2.3

Computation o f Amplitudes o f Coordinate Functions

The p e r t u r b a t i o n f u n c t i o n s X ,

(or X

L,k

) a r e now chosen as c o o r d i n a t e func-

t i o n s and t h e f i e l d v a r i a b l e X i s expressed as a l i n e a r combination o f these f u n c t i o n s as f o l l o w s : n-1 X = $i Xi i=O

z

f o r t h e single-parameter case, and nl

j

The Application of Variational Methods to Nonlinear Problems

28 1

f o r t h e two-parameter case where iiand $ .

a r e unknown parameters which 1 ,j r e p r e s e n t amplitudes of t h e c o o r d i n a t e f u n c t i o n s ( o r modes) X i and X i ;; I Y J and n equals t h e t o t a l number o f modes. I n t h e two-parameter case n = 1/2 (nl + l ) ( n , + 2 ) . The parameters

IJJ~

( o r $. . ) a r e o b t a i n e d by s u b s t i t u t i n g t h e expansion o f 1,J

X i n t h e o r i g i n a l f u n c t i o n a l 11, a p p l y i n g t h e s t a t i o n a r y c o n d i t i o n , Eq. 1, and s o l v i n g t h e r e s u l t i n g small s e t o f ~ v r t e i n e we q u a t i o n s .

2.4

Comments on S e l e c t i o n o f Coordinate Functions

The chosen s e t o f c o o r d i n a t e f u n c t i o n s has t h e f o l l o w i n g t h r e e p r o p e r t i e s : 1. They a r e l i n e a r l y independent and span t h e space o f s o l u t i o n s i n t h e neighborhood o f t h e p o i n t o f t h e i r g e n e r a t i o n . T h e r e f o r e , t h e y f u l l y c h a r a c t e r i z e t h e n o n l i n e a r s o l u t i o n i n t h a t neighborhood. 2. T h e i r generation, u s i n g t h e r e g u l a r p e r t u r b a t i o n method i n conj u n c t i o n w i t h t h e R a y l e i g h - R i t z technique, r e q u i r e s t h e s o l u t i o n o f a r e c u r s i v e s e t o f l i n e a r a l g e b r a i c e q u a t i o n s . The l e f t - h a n d s i d e s o f these equations a r e t h e same.

3. They p r o v i d e a d i r e c t measure o f t h e s e n s i t i v i t y o f t h e n o n l i n e a r response o f t h e system t o changes i n t h e c o n t r o l o r p e r t u r b a t i o n parameters. The f i r s t p r o p e r t y i s necessary f o r t h e convergence of t h e d i r e c t v a r i a t i o n a l technique. The second p r o p e r t y enhances t h e e f f e c t i v e n e s s o f t h e proposed h y b r i d t e c h n i q u e f o r s o l v i n g n o n l i n e a r problems. The i m p l i c a t i o n o f t h e t h i r d p r o p e r t y i s t h a t by a p p r o p r i a t e c h o i c e o f t h e p e r t u r b a t i o n parameters, s e n s i t i v i t y o f t h e response t o changes i n t h e c h a r a c t e r i s t i c s o f t h e system can be obtained. Note t h a t t h e s e n s i t i v i t y i n f o r m a t i o n i s o b t a i n e d a t z e r o value(s) o f the p e r t u r b a t i o n parameter(s). I f the s e n s i t i v i t y i s required a t o t h e r values o f t h e p a r a m e t e r ( s ) , approximate T a y l o r s e r i e s expansions may be used. The mathematical c o m p l e x i t y o f t h e r i g h t - h a n d s i d e s of t h e r e c u r s i v e s e t o f f u n c t i o n a l s used i n g e n e r a t i n g t h e c o o r d i n a t e f u n c t i o n s b u i l d s up r a p i d l y w i t h t h e i n c r e a s e i n t h e number o f these f u n c t i o n s . T h e r e f o r e , f o r p r a c t i c a l a p p l i c a t i o n s , o n l y a few f u n c t i o n s a r e generated. 3.

TWO-STAGE DIRECT VARIATIONAL TECHNIQUE

As an a t t e m p t t o improve t h e e f f i c i e n c y o f t h e c l a s s i c a l d i r e c t v a r i a t i o n a l techniques i n n o n l i n e a r a n a l y s i s , a two-stage t e c h n i q u e was proposed f o r c o n s i d e r a b l y r e d u c i n g t h e number o f degrees o f freedom o f t h e d i s c r e t i z e d system. The t e c h n i q u e i s p a r t i c u l a r l y u s e f u l f o r p r e d i c t i n g t h e response o f n o n l i n e a r systems w i t h s i m p l e geometries b u t complex c o n s t r u c t i o n . Examples from t h e s t r u c t u r e s and s o l i d mechanics area a r e p r o v i d e d by r i n g and s t r i n g e r s t i f f e n e d c l o s e d c y l i n d r i c a l s h e l l s and s h e l l panels w i t h d i s c r e t e s t i f f e n e r s and r e c t a n g u l a r o r c i r c u l a r p l a n f o r m . The t e c h n i q u e i s o u t l i n e d i n t h i s s e c t i o n . To sharpen t h e focus o f t h e study, d i s c u s s i o n i s l i m i t e d t o s t e a d y - s t a t e n o n l i n e a r problems o f c o n s e r v a t i v e systems, w i t h two independent p a t h parameters x1 and x2. I n s t r u c t u r a l and s o l i d mechanics problems, t h e p a t h parameters can be i d e n t i f i e d w i t h l o a d , displacement o r a r c - l e n g t h parameters i n t h e s o l u t i o n space.

282

A.K. Noor

As i n t h e p r e c e d i n g s e c t i o n , t h e s t e a d y - s t a t e response o f t h e system i s c h a r a c t e r i z e d by t h e s t a t i o n a r y c o n d i t i o n o f a f u n c t i o n a l n, i . e . 6 n ( X , hl,

h2) =

(12)

0

The two stages o f t h e s o l u t i o n process a r e discussed subsequently. 3.1

Stage

I -

Spatial Discretization

I n t h e f i r s t stage o f t h e proposed technique, t h e system i s d i s c r e t i z e d by u s i n g t h e f o l l o w i n g approximate e x p r e s s i o n f o r t h e fundamental unknown ( o r v e c t o r o f fundamental unknowns) :

X =

,(a)

(1 3)

,(a)

a

where g ( a ) a r e a p t i 3 ~ Lchosen f u n c t i o n s o f t h e s p a t i a l c o o r d i n a t e s ( c o o r d i n a t e f u n c t i o n s ) which s a t i s f y t h e e s s e n t i a l boundary c o n d i t i o n s as w e l l as t h e symmetry c o n d i t i o n s on X ; and which a r e f u n c t i o n s o f t h e p a t h parameters

a r e undetermined c o e f f i c i e n t s Note t h a t f o r t h e

x1 and A2.

case o f more than one f i e l d v a r i a b l e d i f f e r e n t s e t s o f a p p r o x i m a t i o n funct i o n s g can be assumed f o r t h e d i f f e r e n t f i e l d v a r i a b l e s ( i . e . , f o r t h e d i f f e r e n t components o f X ) . The governing equations o f t h e d i s c r e t i z e d system a r e o b t a i n e d by f i r s t r e p l a c i n g X by i t s e x p r e s s i o n i n terms o f t h e undetermined c o e f f i c i e n t s and t h e n a p p l y i n g t h e s t a t i o n a r y c o n d i t i o n o f t h e f u n c t i o n a l , Eq. ( 1 2 ) .

I f t h e c o e f f i c i e n t s a(') a r e v a r i e d independently and s i m u l t a n e o u s l y , one obtains the f o l l o w i n g s e t o f nonlinear equations:

3.2

Stage I 1

-

Reduction o f t h e Number o f Degrees o f Freedom

I n t h e second stage o f t h e proposed technique, t h e v e c t o r o f undetermined c o e f f i c i e n t s { @ I o f t h e d i s c r e t i z e d system i s approximated over a range o f values o f hl and x2, by a l i n e a r combination o f { o l corresponding t o a

x1 ,

x1 = h2 0 ) and a number o f t h e i r p a t h d e r i v a t i v e s ( d e r i v a t i v e s o f { @ I w i t h r e s p e c t t o x, and A?) e v a l u a t e d a t t h e same values o f x1 and x2; i . e . p a r t i c u l a r p a i r o f values o f

h2 ( t y p i c a l l y

{ o l = [rlC$l where

[r]

i s a t r a n s f o r m a t i o n m a t r i x which i s g i v e n by:

(16)

The Application o f Variational Methods to Nonlinear Problems

283

and { $ I i s a v e c t o r o f reduced unknowns ( w i t h h components o n l y ) which a r e f u n c t i o n s of t h e p a t h parameters X 1 and x2. Note t h a t h i s c o n s i d e r a b l y s m a l l e r t h a n t h e t o t a l number o f degrees o f freedom o f t h e system (which i s equal t o t h e number o f components o f { @ } ) . The columns o f t h e m a t r i x [ r ] a r e u s u a l l y r e f e r r e d t o as t h e b a s i s v e c t o r s . The d i r e c t v a r i a t i o n a l t e c h n i q u e i s now a p p l i e d a second t i m e t o r e p l a c e t h e governing equations o b t a i n e d i n t h e f i r s t stage, Eqs. 14, by t h e f o l l o w i n g reduced system o f h n o n l i n e a r equations i n t h e reduced unknowns {$I

{?({$I, xl,

x2)1

= 0

(18)

where c ? } = [ r ~ ~ { f ( [ r l { $ } ,xl,

x2)1

(19)

I n Eqs. 19 s u p e r s c r i p t t denotes t r a n s p o s i t i o n . The equations used i n e v a l u a t i n g t h e b a s i s v e c t o r s a r e o b t a i n e d by success i v e d i f f e r e n t i a t i o n o f t h e governing e q u a t i o n s o f t h e d i s c r e t i z e d system, The l e f t - h a n d s i d e s o f t h e r e s u l t i n g Eqs. 14, w i t h r e s p e c t t o x1 and A2. r e c u r s i v e s e t o f l i n e a r a l g e b r a i c equations a r e t h e same. A c r i t e r i o n f o r s e l e c t i n g t h e number o f b a s i s v e c t o r s was proposed i n [ll]. The c r i t e r i o n i s based on m o n i t o r i n g t h e c o n d i t i o n number B o f t h e Gram m a t r i x o f t h e b a s i s v e c t o r s , and t h e g e n e r a t i o n o f these v e c t o r s i s t e r m i n a t e d when B exceeds a p r e s c r i b e d value. Also, upper and l o w e r l i m i t s f o r t h e number o f basis vectors are prescribed. The ranges o f x1 and

x 2 f o r which Eqs. 18 p r o v i d e an a c c e p t a b l e a p p r o x i -

m a t i o n f o r t h e o r i g i n a l d i s c r e t e system, Eqs. 14, depends on t h e degree o f n o n l i n e a r i t y o f t h e problem. The ranges can be i d e n t i f i e d by s e l e c t i n g an e r r o r measure (e.g., norm o f t h e r e s i d u a l v e c t o r ) ; and m o n i t o r i n g t h e accuracy o f t h e s o l u t i o n s o f t h e reduced system, Eqs. 18. When t h e e r r o r measure exceeds a p r e s c r i b e d t o l e r a n c e , t h e v e c t o r { @ I generated by t h e reduced system o f equations i s used as a p r e d i c t o r and t h e Newton-Raphson i t e r a t i v e t e c h n i q u e i s used i n c o n j u n c t i o n w i t h t h e o r i g i n a l e q u a t i o n s , Eqs. 14, t o o b t a i n a c o r r e c t e d (improved) s o l u t i o n . Then a new (updated) s e t o f b a s i s v e c t o r s i s generated. The computational procedure i s d e s c r i b e d i n [ll]and w i l l n o t be repeated h e r e i n .

3.3

Comments on t h e Proposed Two-Stage Technique

1. The essence o f t h e two-stage d i r e c t v a r i a t i o n a l t e c h n i q u e f o r t h e steady-state nonlinear analysis i s t o separate the s p a t i a l d i s t r i b u t i o n s o f t h e fundamental unknowns f o r any p a i r o f values o f x1 and h2 from t h e i r variations with

x1 and x2.

The form o f t h e s p a t i a l d i s t r i b u t i o n o f t h e

fundamental unknowns i s g i v e n by t h e assumed c o o r d i n a t e f u n c t i o n s i n t h e f i r s t stage. The use o f t h e p a t h d e r i v a t i v e s as b a s i s v e c t o r s i n t h e second stage p e r m i t s t h e known i n f o r m a t i o n about t h e n o n l i n e a r response i n I n t h i s manner t h e neighborhood o f a p o i n t t o be b r o u g h t i n t o t h e a n a l y s i s .

A.K. Noor

284

s u b s t a n t i a l l y f e w e r degrees o f freedom w i l l be r e q u i r e d t o a c h i e v e a d e s i r e d o v e r a l l a c c u r a c y i n comparison w i t h t h a t based on t h e c l a s s i c a l d i r e c t v a r i a t i o n a l t e c h n i q u e . Note t h a t t h e t i m e r e q u i r e d t o s o l v e t h e reduced system o f e q u a t i o n s i s r e l a t i v e l y s m a l l , and t h e t o t a l a n a l y s i s t i m e t o a f i r s t approximation, equals t h e time r e q u i r e d t o evaluate t h e basis vectors and g e n e r a t e t h e reduced e q u a t i o n s . 2. The e f f e c t i v e n e s s o f t h e proposed two-stage t e c h n i q u e may be a t tributed t o the f a c t that the spatial distribution o f the f i e l d variables v a r i e s s l o w l y w i t h t h e v a r i a t i o n o f t h e p a t h parameters x1 and x2. A l a r g e number o f n u m e r i c a l e x p e r i m e n t s have shown t h a t l a r g e changes i n t h e p a t h parameters a r e f r e q u e n t l y a s s o c i a t e d w i t h s m a l l changes i n t h e s p a t i a l d i s t r i b u t i o n o f t h e response q u a n t i t i e s . Gross changes i n t h e s p a t i a l d i s t r i b u t i o n r e q u i r e u p d a t i n g o f t h e b a s i s v e c t o r s i n t h e second s t a g e .

3. I f t h e proposed two-stage t e c h n i q u e i s c o n t r a s t e d w i t h t h e s t a t i c p e r t u r b a t i o n t e c h n i q u e ( s e e [ 1 6 ] ) , t h e f o l l o w i n g c a n be n o t e d . I n b o t h i s approximated by t e c h n i q u e s t h e v e c t o r o f undetermined c o e f f i c i e n t s a l i n e a r c o m b i n a t i o n o f a s m a l l number of p a t h d e r i v a t i v e s , Eqs. 16 and 17. However, t h e c o e f f i c i e n t s o f t h e l i n e a r c o m b i n a t i o n 111 i n t h e s t a t i c p e r t u r b a t i o n t e c h n i q u e a r e f i x e d and a r e equal t o : 1, axl, A x 2 , 2 (Axl I2 (Ax,) By c o n t r a s t , t h e c o e f f i c i e n t s {$I i n t h e 3 AXlAh2, ,

2

~

....

proposed approach a r e l e f t as f r e e parameters, and a r e d e t e r m i n e d by a p p l y i n g t h e d i r e c t v a r i a t i o n a l t e c h n i q u e a second t i m e . Numerical e x p e r i m e n t s i n d i c a t e t h a t t h e use o f f r e e parameters l e a d s t o a c c u r a t e s o l u t i o n s n o t o n l y w i t h i n t h e r a d i u s o f convergence o f t h e T a y l o r s e r i e s b u t a l s o w e l l beyond i t .

4. F o r t h e p r e d i c t i o n o f t h e n o n l i n e a r response o f p r a c t i c a l systems w i t h complex domains, t h e t w o - s t a g e p r o c e d u r e can b e a p p l i e d i n c o n j u n c t i o n w i t h t h e g l o b a l ( o r macro) element method, o r even w i t h t h e f i n i t e element method. I n t h e l a t t e r case, t h e t w o - s t a g e p r o c e d u r e reduces t o t h e r e d u c t i o n method d e s c r i b e d i n [91. 4.

NUMERICAL STUDIES

To e v a l u a t e t h e e f f e c t i v e n e s s o f b o t h t h e h y b r i d p e r t u r b a t i o n / d i r e c t v a r i a t i o n a l t e c h n i q u e and t h e two-stage d i r e c t v a r i a t i o n a l t e c h n i q u e , s e v e r a l n o n l i n e a r s t e a d y - s t a t e t h e r m a l and s t r u c t u r a l problems were s o l v e d . F o r each problem, t h e s o l u t i o n s o b t a i n e d b y t h e proposed t e c h n i q u e s were comp a r e d w i t h converged f i n i t e element s o l u t i o n s and w i t h o t h e r n u m e r i c a l a p p r o x i m a t i o n s , whenever a v a i l a b l e . H e r e i n t h e r e s u l t s o f f o u r t y p i c a l s t e a d y - s t a t e t h e r m a l and s t r u c t u r a l problems a r e d i s c u s s e d . The f o u r p r o blems a r e : 1 ) s t e a d y - s t a t e thermal response o f one-dimensional c o n d u c t i n g convecting f i n w i t h v a r i a b l e heat t r a n s f e r c o e f f i c i e n t ; 2) l a r g e d e f l e c t i o n analysis o f laminated a n i s o t r o p i c p l a t e subjected t o uniform transverse l o a d i n g ; 3 ) n o n l i n e a r a x i s y m m e t r i c response o f an i s o t r o p i c c i r c u l a r p l a t e s u b j e c t e d t o combined u n i f o r m and c o n c e n t r a t e d l o a d i n g s ; and 4 ) n o n l i n e a r a x i s y m m e t r i c response o f a clamped s h a l l o w s p h e r i c a l cap s u b j e c t e d t o u n i f o r m normal p r e s s u r e . The f i r s t t h r e e problems were a n a l y z e d by u s i n g t h e r e g u l a r p e r t u r b a t i o n t e c h n i q u e and were used t o assess t h e e f f e c t i v e n e s s o f t h e h y b r i d p e r t u r b a t i o d d i r e c t v a r i a t i o n a l t e c h n i q u e . The f o u r t h p r o b l e m i s used t o e v a l u a t e t h e two-stage d i r e c t v a r i a t i o n a l t e c h n i q u e .

The Application of Variational Methods to Nonlinear Problems

285

I n a l l t h e oro ble m s c o n s i d e r e d , a l l t h e a n a l y t i c a l work, namely, g e n e r a t i o n o f perturbation functionals, aeneration o f various-order perturbation e q u a t i o n s ; e v a l u a t i o n o f c o o r d i n a t e f u n c t i o n s and o f b a s i s v e c t o r s , was done b y u s i n g t h e c o m p u t e r i z e d s y m b o l i c m a n i p u l a t i o n syst em MACSYMA [ 17] .

4.1

S t e a d y - S t a t e Thermal Response o f a C o n d u c t i n g - C o n v e c t i n g F i n w i t h V a r i a b l e Hea t - T r a n s f e r C o e f f i c i e n t

The f i r s t p r o b l e m c o n s i d e r e d i s t h a t o f a s t r a i g h t c o n d u c t i n p - c o n v e c t i n g f i n o f l e n q t h L, c r o s s - s e c t i o n a l a r e a A, and p e r i m e t e r c, exposed on b o t h s i d e s t o a f r e e c o n v e c t i v e e n v i r o n m e n t o f t e m p e r a t u r e Ta = 0 . The b o u n d a r y c o n d i t i o n s a r e a c o n s t a n t base t e m p e r a t u r e and an a d i a b a t i c t i p . The t h e r mal c o n d u c t i v i t y K i s assumed t o be i n d e p e n d e n t o f t h e t e m p e r a t u r e . The c o n v e c t i v e h e a t - t r a n s f e r c o e f f i c i e n t i s t a k e n t o be o f t h e form:

h = h b eB where e = T / T b i s a n o r m a l i z e d t e m p e r a t u r e d e f i n e d i n t erms o f t h e f i n base t e m p e r a t u r e Tb; B i s a s m a l l p a r a m e t e r ( ~ = 0 . 2 5and 0.33 f o r l a m i n a r and B where 7 i s a t u r b u l e n t c o n d i t i o n s , r e s p e c t i v e l y [ 1 8 ] ) ; a nd h b = Y Tb, c o n s t a n t . The v a r i a t i o n a l f u n c t i o n a l a n d t h e g o v e r n i n g d i f f e r e n t i a l e q u a t i o n f o r t h i s p r o b l e m a r e g i v e n i n A p p e ndix I . Two d i f f e r e n t c h o i c e s a r e made f o r t h e p e r t u r b a t i o n p a r a m e t e r A . The f i r s t c h o i c e i s t h e same as t h a t o f [18], namely X = B. The second c h o i c e i s

where q2 i s a c o n v e c t i o n - c o n d u c t i o n f i n p a r a m e t e r . F o r each o f t h e t wo c h o i c e s o f t h e p e r t u r b a t i o n p a r a m e t e r , t h e c o o r d i n a t e f u n c t i o n s were obt a i n e d by s o l v i n q t h e various-order p e r t u r b a t i o n equations. I n each case, c l o s e d - f o r m s o l u t i o n s f o r t h e r e c u r s i v e s e t o f d i f f e r e n t i a l e q u a t i o n s were o b t a i n e d . The l a t t e r c h o i c e o f t h e p e r t u r b a t i o n p a r a m e t e r , x = a’, r e s u l t e d i n s i g n i f i c a n t l y s i m p l i f i e d expressions o f t h e coordinate functions. For 2 case 1, A = B , t w o c o o r d i n a t e f u n c t i o n s w e r e used, and f o r case 2, A = q , f o u r c o o r d i n a t e f u n c t i o n s w e r e g e n e r a t e d . The e x o r e s s i o n s o f t h e c o o r d i n a t e f u n c t i o n s f o r t h e two c a s e s a r e q i v e n i n [ l o ] and a r e n o t r e p e a t e d h e r e . The a m o l i t u d e s o f t h e c o o r d i n a t e f u n c t i o n s were o b t a i n e d b y a p D l y i n g t h e Bubnov-Galerkin t e c h n i q u e t o t h e o r i g i n a l qoverning d i f f e r e n t i a l e a u a t i o n , Eq. A.2. A s i n g l e f r e e p a r a m e t e r was u s e d i n case 1 and t h r e e f r e e p a r a , m e t e r s were u s ed i n c a s e 2 ( s i n c e one o f t h e f r e e p a r a m e t e r s , q ~ i ~s used t o s a t i s f y t h e p r e s c r i b e d nonzero boundary c o n d i t i o n ) . An i n d i c a t i o n o f t h e a c c u r a c y o f t h e s o l u t i o n s o b t a i n e d by t h e h y b r i d t e c h n i q u e and t h e r e q u l a r o e r t u r b a t i o n method a r e a i v e n i n F i q u r e 1 f o r q = 1 .0 and 2.0 and ~ = 0 . 3 and 3 1.0. The l a t t e r v a l u e o f B has no p h y s i c a l s i q n i f i cance and was s e l e c t e d i n o r d e r t o a m p l i f y t h e e f f e c t o f t h e m a q n i t u d e o f t h e D e r t u r b a t i o n p a r a m e t e r s o n t h e a u a l i t y o f t h e s o l u t i o n s . The s t a n d a r d f o r c omp aris o n was t a k e n t o be t h e f i n i t e e l e m e n t s o l u t i o n o b t a i n e d b y u s i n g u n i f o r m q r i d o f 15 three-noded f i n i t e elements w i t h q u a d r a t i c Laqranqian i n t e r p o l a t i o n f u n c t i o n s f o r t h e t e m p e r a t u r e . As can be seen f r o m F i g u r e 1, the accuracy o f t h e p e r t u r b a t i o n s o l u t i o n i s very s e n s i t i v e t o both t h e

28 6

A. K. Noor

c h o i c e and t h e m a g n i t u d e o f t h e p e r t u r b a t i o n p a r a m e t e r . F o r A = 8, t h e twot e r m p e r t u r b a t i o n e x p a n s i o n i s a c c u r a t e f o r B _< 0.33 b u t becomes q u i t e i n a c c u r a t e f o r B = 1.0. On t h e o t h e r hand, f o r x = q2, t h e f o u r - t e r m D e r t u r b a t i o n expansion i s g r o s s l y i n e r r o r f o r a l l q 1. 0. The D e r t u r b a t i o n s o l u t i o n s f o r q = 2 c o u l d n o t b e shown i n F i g u r e 1. By c o n t r a s t , t h e a c c u r a c y o f t h e s o l u t i o n s o b t a i n e d b y t h e h y b r i d t e c h n i q u e were f o u n d t o be i n s e n s i t i v e t o t h e c h o i c e o f t h e p e r t u r b a t i o n p a r a m e t e r . The s o l u t i o n s 2 o b t a i n e d b y u s i n g x = 8 and X = q were e q u a l l y a c c u r a t e and were a l m o s t i n d i s t i n g u i s h a b l e from t h e f i n i t e element s o l u t i o n s . 4.2

L amin at e d A n i s o t r o p i c P l a t e S u b j e c t e d t o U n i f o r m T r a n s v e r s e L o a d i n g

The second D ro bl e m c o n s i d e r e d i s t h a t o f t h e n o n l i n e a r response o f a clamped, s q uare , s y m m e t r i c a l l y l a m i n a t e d 1 6 - p l y g r a p h i t e - e p o x y p l a t e subj e c t e d t o u n i f o r m t r a n s v e r s e l o a d i n g . The m a t e r i a l and g e o m e t r i c c h a r a c t e r i s t i c s o f t h e p l a t e a r e shown i n F i g u r e 2. The p e r t u r b a t i o n p a r a 4 4 m e t e r was s e l e c t e d t o be t h e l o a d p a r a m e t e r x = p o L /(E,.h ) . A von Karman t y p e n o n l i n e a r p l a t e t h e o r y i s used a n d t h e p r o b l e m i s f o r m u l a t e d i n t erms o f t h e t h r e e d i s p l a c e m e n t components o f t h e m i d d l e p l a n e o f t h e p l a t e ; namely, t h e i n - p l a n e d i s p l a c e m e n t s u, v, and t h e t r a n s v e r s e d i s p l a c e m e n t w. The e x p l i c i t f o r m o f t h e g o v e r n i n g d i f f e r e n t i a l e q u a t i o n s and t h e f u n c t i o n a l f o r t h i s p r o b l e m a r e g i v e n i n [191. The d i s p l a c e m e n t s a r e expanded i n p e r t u r b a t i o n s e r i e s as f o l l o w s :

x

w = w1

v = v2 The p e r t u r b a t i o n f u n c t i o n s wi

w3 x 3

f

x2

+ v4

( i = 1,3,5,

f

w5

x4

x5

....

f

(22)

....

i.

...),

and uj, v j ( j =

2,4,

. . . ) were

s o u g h t i n t h e f o r m o f m o d i f i e d F o u r i e r s e r i e s as f o l l o w s :

wi =

'

mRxl ( i ) [ c o s - - ( - 1 ) m/ 2][cos L n=2,4 'mn

z

m=2,4

u

m=2,4

j

v

= j

where A:,!),

c

=

c

m=1,3

B i ) , and ):!ck

2) : B

sin

max 1

- nnx2 cos L

mRxl

nax2 sin L

n=1,3

n=2,4

nnx2

-

( - 1 ) n/2 1 (25)

(26)

a r e undetermined c o e f f i c i e n t s o b t a i n e d b y a p p l y -

i n g t h e R a y l e i g h - R i t z t e c h n i q u e t o t h e p e r t u r b a t i o n f u n c t i o n a l s nZi and and s o l v i n g t h e r e s u l t i n g s y s t e m o f l i n e a r a l g e b r a i c e q u a t i o n s i n , ,A ( i ) ,

B,$!,)

and .':C

nZj,

Note t h a t t h e l e f t - h a n d s i d e s o f t h e s e a l g e b r a i c equations

The Application of Variational Methods to Nonlinear Problems

287

a r e t h e same, r e g a r d l e s s o f t h e values o f i and j ; and t h e e q u a t i o n s f o r A),!: a r e uncoupled from those f o r B i ) and Cmn ( j 1. Three p e r t u r b a t i o n f u n c t i o n s were generated f o r w and two c o o r d i n a t e funct i o n s were generated f o r each o f u and v. The p e r t u r b a t i o n f u n c t i o n s were t h e n used as c o o r d i n a t e f u n c t i o n s and t h e amplitudes o f these f u n c t i o n s were o b t a i n e d by a p p l y i n g t h e R a y l e i g h - R i t z technique t o t h e o r i g i n a l f u n c t i o n a l ( t o t a l p o t e n t i a l energy o f t h e p l a t e ) . An i n d i c a t i o n o f t h e accuracy o f t h e t r a n s v e r s e displacements wc and t h e s t r a i n energies o b t a i n e d by t h e h y b r i d technique and t h e r e g u l a r p e r t u r b a t i o n method, f o r d i f f e r e n t values o f l o a d i n g , i s shown i n F i g u r e 3. The standard o f comparison was taken t o be t h e f i n i t e element s o l u t i o n u s i n g a u n i f o r m 6 x 6 g r i d o f b i c u b i c i n t e r p o l a t i o n f u n c t i o n s f o r t h e displacements and r o t a t i o n s . As can be seen from F i g u r e 3, t h e p e r t u r b a t i o n s o l u t i o n s a r e g r o s s l y i n e r r o r f o r x > 200 ( c o r r e s p o n d i n g t o w /h > 0 . 5 ) . By cont r a s t , t h e s o l u t i o n s o b t a i n e d w i t h t h e h y b r i d technihue a r e i n c l o s e agreement w i t h t h e f i n i t e element s o l u t i o n f o r a l l t h e range o f l o a d i n g considered. 4.3

N o n l i n e a r A x i s y m v e t r i c Response o f an I s o t r o p i c C i r c u l a r P l a t e Subjected t o Combined U n i f o r m and Concentrated Loading

The n e x t problem considered i s t h a t o f t h e n o n l i n e a r axisymmetric response o f a c i r c u l a r p l a t e s u b j e c t e d t o combined u n i f o r m d i s t r i b u t e d l o a d i n g po and a c o n c e n t r a t e d l o a d P a p p l i e d a t t h e c e n t e r o f t h e p l a t e . The p l a t e i s f r e e l y supported w i t h t h e t r a n s v e r s e displacement w = 0 and t h e r a d i a l d i s placement u u n r e s t r a i n e d a t t h e edge. The problem was considered i n [ Z O ] as an a p p l i c a t i o n o f t h e two-parameter p e r t u r b a t i o n technique. A von-Karman t y p e p l a t e t h e o r y i s used and t h e problem i s f o r m u l a t e d i n The terms o f t h e t r a n s v e r s e displacement w and t h e r a d i a l normal f o r c e Nr. governing d i f f e r e n t i a l equations a r e g i v e n i n Appendix 11. As i n [ Z O ] , t h e p e r t u r b a t i o n parameters a r e s e l e c t e d t o be t h e two normalized l o a d i n g s 4 2 Pa POa Two-parameter p e r t u r b a t i o n s e r i e s a r e used f o r and x 2 = x1 Eh4 ‘ b o t h w and Nr as f o l l o w s :

=r

w = wl0

x1 + wol x 2

f

3

2

2

w30 hl + w21 x1 x 2 + w12 x1 x 2 + wo3

2 Nr = NzO x1 + NI1 The p e r t u r b a t i o n f u n c t i o n s wl0,

wol,

X1 X2

+ NO2 X 22 +

NzO, Nll,

x23

f

....

(28)

....

NoZ, w30,

wZ1, w12 and wO3

a r e o b t a i n e d by s o l v i n g t h e r e c u r s i v e s e t o f l i n e a r o r d i n a r y d i f f e r e n t i a l equations. The e x p l i c i t f o r m o f these d i f f e r e n t i a l equations i s g i v e n i n Appendix 11. Closed-form s o l u t i o n s were o b t a i n e d f o r a l l these e q u a t i o n s S i x p e r t u r b a t i o n f u n c t i o n s were generated f o r w and t h r e e p e r t u r b a t i o n f u n c t i o n s were generated f o r N.

288

A.K. Noor

The p e r t u r b a t i o n f u n c t i o n s were t h e n u s e d as c o o r d i n a t e f u n c t i o n s and t h e a m p l i t u d e s o f t h e s e f u n c t i o n s were o b t a i n e d b y a p p l y i n g t h e R u b n o v - G a l e r k i n t e c h n i q u e t o t h e o r i g i n a l d i f f e r e n t i a l e q u a t i o n s , E q s . B.1 and 8.2. An i n d i c a t i o n o f t h e a c c u r a c y o f t h e t r a n s v e r s e d i s p l a c e m e n t s and t h e s t r a i n e n e r g i e s o b t a i n e d by t h e h y b r i d t e c h n i q u e and t h e t wo-paramet er p e r t u r b a t i o n The method i s shown i n F i g u r e 4 f o r v a r i o u s c o m b i n a t i o n s o f X1 and X 2 . s t a n d a r d o f c omp a r i s o n i s t a k e n t o be t h e f i n i t e e l e m e n t s o l u t i o n u s i n g a u n i f o r m g r i d o f 15 e l e m e n t s w i t h f i r s t - o r d e r H e r m i t i a n p o l y n o m i a l s f o r b o t h t h e r a d i a l and t r a n s v e r s e d i s p l a c e m e n t s . As can be seen f r o m F i g u r e 4, t h e s o l u t i o n s o b t a i n e d w i t h t h e p e r t u r b a t i o n method a r e g r o s s l y i n e r r o r f o r ,i2 > 2 and X1 0. By c o n t r a s t , t h e s o l u t i o n s o b t a i n e d w i t h t h e h y b r i d t e c h n i q u e a r e i n c l o s e agreement w i t h t h e f i n i t e e l e m e n t s o l u t i o n f o r a l l t h e range o f l o a d i n g s c o n s i d e r e d .

4.4

N o n l i n e a r A x i s y m m e t r i c Response o f S h a l l o w S p h e r i c a l Caps

The l a s t p r o b l e m c o n s i d e r e d i s t h a t o f clamped s h a l l o w s p h e r i c a l caps subj e c t e d t o u n i f o r m normal l o a d i n g po. The m a t e r i a l and g e o m e t r i c c h a r a c t e r i s t i c s o f t h e s h e l l a r e g i v e n i n F i g u r e 5. Two s h a l l o w caps w i t h d i f f e r e n t t h i c k n e s s e s a r e c o n s i d e r e d ; namely, h = 0.0127 and 0.005 m. The b e h a v i o r o f t h e t wo caps i s h i g h l y n o n l i n e a r w i t h a n i n i t i a l s o f t e n i n g and subsequent s t i f f e n i n g . Moreover, t h e r e s p o n s e o f t h e t h i n n e r cap e x h i b i t s l i m i t p o i n t s The p r o b l e m i s used t o a s s e s s t h e accurac,y and e f f e c t i v e n e s s o f t h e twos t a g e R a y l e i g h - R i t z t e c h n i q u e . A n a l y t i c s o l u t i o n s f o r t h e t h i c k e r cap, F i n i t e e l e m e n t s o l u t i o n s f o r t h e same h = 0 . 0 1 2 7 m., a r e g i v e n i n [ 2 1 ] . I n the present study a shear-deforp r o b l e m a r e g i v e n i n [22] and [23]. m a t i o n S anders -B u d i a n s k y t y p e s h e l l t h e o r y i s used and t h e p r o b l e m i s f o r m u l a t e d i n terms o f t h e t h r e e g e n e r a l i z e d displacements o f t h e m i d d l e s u r f a c e o f t h e s h e l l , u, w, a n d 4 . The a p p r o x i m a t i o n f u n c t i o n s used i n t h e f i r s t stage o f t h e a n a l y s i s a r e as f o l l o w s :

(30) w =

c

Bn cos

( 2 n -1 ) T U 2c10

=

C,

sin aO

The c o n t r o l p a r a m e t e r was chosen t o b e t h e g e n e r a l i z e d a r c - l e n g t h i n t h e The b a s i s v e c t o r s were g e n e r a t e d f o r t h e u n l o a d e d s o l u t i o n s pac e ( s e e [7]). cap ( q =

PoR Eh -

0, A = O ,

1 @ 3 = 0 ) ,and w e r e t h u s o b t a i n e d by s o l v i n g a l i n e a r

s e t o f f i n i t e element equations. Nine b a s i s v e c t o r s ( p a t h d e r i v a t i v e s w i t h r e s p e c t t o t h e g e n e r a l i z e d a r c - l e n g t h ) were g e n e r a t e d . The b a s i s v e c t o r s were o r t h o n o r m a l i z e d u s i n g t h e Gram-Schmidt p r o c e d u r e . An i n d i c a t i o n o f t h e a c c u r a c y o f t h e s o l u t i o n s o b t a i n e d b y t h e t w o - s t a g e R a y l e i g h - R i t z t e c h n i q u e i s g i v e n i n F i g u r e s 6 and 8 f o r t h e t wo s p h e r i c a l caps . F o r t h e t h i c k e r c a p t h e same s e t o f b a s i s v e c t o r s were used t h r o u g h o u t t h e r a n g e o f l o a d i n g c o n s i d e r e d . The s o l u t i o n s o b t a i n e d u s i n g n i n e

The Application of Variational Methods to Nonlinear Problems

289

b a s i s v e c t o r s were h i g h l y accurate. A t q = 7.95 x l o W 3 , t h e e r r o r s i n t h e t r a n s v e r s e displacement wc and t o t a l s t r a i n energy U were 0.466% and 0.722%, respectively

.

The complex n a t u r e o f t h e t h i n cap response i s d e p i c t e d i n F i g u r e 8 .

The

b a s i s v e c t o r s were updated two times i n t h i s case, c o r r e s p o n d i n g t o q = 9 . 8 8 ~ 1 0 -and ~ q=7.13~10-~. I t i s w o r t h m e n t i o n i n g t h a t t h e use o f g e n e r a l i z e d a r c - l e n g t h as t h e c o n t r o l parameter r e s u l t e d i n c o n s i d e r a b l y i m p r o v i n g t h e performance o f t h e s o l u t i o n technique f o r t h e n o n l i n e a r reduced e q u a t i o n s . The number o f NewtonRaphson i t e r a t i o n s r e q u i r e d f o r t h e convergence o f t h e reduced equations a t each increment was equal t o ( o r l e s s t h a n ) 3. I n o r d e r t o demonstrate t h e advantage o f t h e two-stage R a y l e i g h - R i t z technique over t h e s t a t i c p e r t u r b a t i o n technique, t h e t r a n s v e r s e displacement f o r t h e t h i c k e r cap o b t a i n e d by u s i n g s i x , seven, e i g h t and n i n e nonzero terms i n t h e T a y l o r s e r i e s expansion about t h e s o l u t i o n a t q = O a r e shown i n F i g u r e 7. As can be seen from F i g u r e 7, t h e s o l u t i o n d r i f t s from t h e t r u e e q u i l i b r i u m p a t h . The d r i f t i s more pronounced when t h e number o f terms i n t h e T a y l o r s e r i e s i s smal 1.

Note t h a t f o r c o n s e r v a t i v e systems t h e e f f i c i e n c y o f t h e two-stage RayleighR i t z technique can be i n c r e a s e d i f t h e e r r o r t o l e r a n c e i s i n c r e a s e d and t h e accuracy o f t h e reduced system o f equations i s m a i n t a i n e d by b a c k t r a c k i n g t h e s o l u t i o n p a t h each t i m e a new (updated) s e t o f b a s i s v e c t o r s i s generated. 5.

POTENTIAL OF THE PROPOSED H Y B R I D AND TWO-STAGE TECHNIQUES

The two techniques d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n s have h i g h p o t e n t i a l f o r s o l u t i o n o f n o n l i n e a r s t e a d y - s t a t e problems, e s p e c i a l l y f o r systems w i t h complex c o n s t r u c t i o n b u t s i m p l e geometries. Examples f r o m t h e s t r u c t u r e s area a r e p r o v i d e d by r i n g and s t r i n g e r s t i f f e n e d c l o s e d c y l i n d r i c a l s h e l l s , and s h e l l panels w i t h d i s c r e t e s t i f f e n e r s and r e c t a n g u l a r o r c i r c u l a r planform. The numerical s t u d i e s conducted c l e a r l y demonstrated t h e accuracy and e f f e c t i v e n e s s o f t h e two techniques. I n p a r t i c u l a r , t h e f o l l o w i n g two p o i n t s a r e worth mentioning: 1. The h y b r i d p e r t u r b a t i o n / d i r e c t v a r i a t i o n a l t e c h n i q u e can be thought o f as e i t h e r o f t h e f o l l o w i n g : a ) A g e n e r a l i z e d p e r t u r b a t i o n method i n which 1 ) t h e p e r t u r b a t i o n expansions o f t h e f i e l d v a r i a b l e s c o n t a i n f r e e parameters r a t h e r t h a n f i x e d c o e f f i c i e n t s ; and 2 ) t h e p e r t u r b a t i o n parameters need n o t be s m a l l . Since t h e accuracy o f t h e s o l u t i o n s o b t a i n e d w i t h t h e h y b r i d t e c h n i q u e appears t o be i n s e n s i t i v e t o t h e c h o i c e o f p e r t u r b a t i o n parameters, t h e y may be i n t r o duced a r t i f i c i a l l y t o s i m p l i f y t h e form o f t h e r e c u r s i v e s e t o f d i f f e r e n t i a l equations ( o r t h e r e c u r s i v e s e t o f f u n c t i o n a l s ) used i n e v a l u a t i n g t h e various-order p e r t u r b a t i o n solutions. b ) An extended d i r e c t v a r i a t i o n a l t e c h n i q u e w i t h t h e c o o r d i n a t e f u n c t i o n s generated by u s i n g t h e s t a n d a r d r e g u l a r p e r t u r b a t i o n technique r a t h e r than chosen a .pJLLaJLL.

2. The f o r e g o i n g h y b r i d t e c h n i q u e i s t h e a n a l y t i c c o u n t e r p a r t o f t h e two-stage d i r e c t v a r i a t i o n a l technique presented h e r e i n . The p r i m a r y obj e c t i v e o f u s i n g t h e two-stage technique i s t o reduce c o n s i d e r a b l y t h e

290

A.K. Noor

number o f degrees o f freedom i n t h e i n i t i a l d i s c r e t i z a t i o n , and hence, reduce t h e computational e f f o r t i n v o l v e d i n t h e s o l u t i o n o f t h e problem. By c o n t r a s t , t h e o b j e c t i v e s o f t h e f o r e g o i n g h y b r i d technique a r e : a ) t o extend t h e range o f v a l i d i t y o f t h e r e g u l a r p e r t u r b a t i o n method by removing t h e r e s t r i c t i o n o f a small p e r t u r b a t i o n parameter; and b ) t o enhance t h e e f f e c t i v e n e s s o f t h e d i r e c t v a r i a t i o n a l technique by removing ( o r r e d u c i n g ) the arbitrariness i n the selection o f the coordinate functions.

6.

FUTURE DIRECTIONS FOR RESEARCH

ON DIRECT VARIATIONAL METHODS

Among t h e d i f f e r e n t r e s e a r c h areas which have h i g h p o t e n t i a l f o r a p p l i c a t i o n o f d i r e c t v a r i a t i o n a l methods a r e t h e f o l l o w i n g :

a ) Development o f e f f e c t i v e a p p r o x i m a t i o n f u n c t i o n s f o r n o n l i n e a r problems. T h i s i n c l u d e s an assessment o f t h e r e l a t i v e m e r i t s o f u s i n g n o n l i n e a r forms o f c o o r d i n a t e f u n c t i o n s i n s t e a d o f l i n e a r combinations o f these f u n c t i o n s . b) Use o f the h y b r i d a n a l y t i c a l technique and t h e two-stage technique i n c o n j u n c t i o n w i t h t h e weighted r e s i d u a l - l e a s t squares procedure. Also, a s y s t e m a t i c comparison between t h e m e r i t s o f u s i n g l e a s t squares procedure versus Bubnov-Galerkin o r R a y l e i g h - R i t z technique. c ) A p p l i c a t i o n o f t h e two-stage v a r i a t i o n a l methodology t o t h e a n a l y s i s o f one- and two-dimensional s t r u c t u r a l components wherein t h e fundamental unknowns a r e o b t a i n e d from a one- o r two-dimensional t h e o r y and a r e used as c o o r d i n a t e f u n c t i o n s w i t h undetermined parameters i n t h e t h r e e dimensional t h e o r y . The undetermined parameters a r e o b t a i n e d by a p p l y i n g t h e d i r e c t v a r i a t i o n a l technique.

7.

CONCLUDING REMARKS

Two r e c e n t advances i n t h e a p p l i c a t i o n o f d i r e c t v a r i a t i o n a l methods t o n o n l i n e a r s t e a d y - s t a t e problems a r e discussed. The f i r s t i s a h y b r i d a n a l y s i s technique based on t h e combined use o f r e g u l a r p e r t u r b a t i o n expansion and t h e c l a s s i c a l d i r e c t v a r i a t i o n a l techniques f o r p r e d i c t i n g t h e n o n l i n e a r s t e a d y - s t a t e response of t h e system. The second i s a two-stage d i r e c t v a r i a t i o n a l technique. The a p p l i c a t i o n o f each o f t h e two techniques t o t h e s o l u t i o n o f n o n l i n e a r problems can be d i v i d e d i n t o two stages. For t h e h y b r i d technique t h e f i r s t stage c o n s i s t s o f g e n e r a t i n g t h e c o o r d i n a t e f u n c t i o n s ( o r modes) u s i n g t h e standard r e g u l a r p e r t u r b a t i o n method and approximating t h e f i e l d v a r i a b l e s by a l i n e a r combination o f these modes. The c l a s s i c a l d i r e c t v a r i a t i o n a l technique i s then used t o compute t h e c o e f f i c i e n t s o f t h e l i n e a r combination (amplitudes o f t h e modes). I n t h e two-stage d i r e c t v a r i a t i o n a l technique, t h e f i r s t stage c o n s i s t s o f d i s c r e t i z i n g t h e system by u s i n g c o o r d i n a t e f u n c t i o n s which cover t h e e n t i r e domain. I n t h e second stage a s u b s t a n t i a l r e d u c t i o n i n t h e number o f degrees o f freedom i s achieved by e x p r e s s i n g t h e v e c t o r o f unknown parameters as a l i n e a r combination o f a small number o f b a s i s v e c t o r s . The d i r e c t v a r i a t i o n a l technique i s a p p l i e d a second t i m e t o approximate t h e n o n l i n e a r equations o f t h e d i s c r e t i z e d system by a r e duced system o f n o n l i n e a r equations. The b a s i s v e c t o r s used i n t h e second stage a r e chosen t o be those comnonly used i n s t a t i c p e r t u r b a t i o n technique, namely, a n o n l i n e a r s o l u t i o n and a number o f i t s p a t h d e r i v a t i v e s .

29 1

The Application o f Variational Methods to Nonlinear Problems Four numerical examples a r e presented t o demonstrate t h e e f f e c t i v e n e s s o f t h e two techniques f o r t h e s o l u t i o n o f n o n l i n e a r s t e a d y - s t a t e problems. Several c o n c l u s i o n s can be made r e g a r d i n g t h e two techniques. c l u s i o n s a r e as f o l l o w s :

These con-

1 . The h y b r i d t e c h n i q u e e x p l o i t s t h e b e s t elements o f t h e r e g u l a r p e r t u r b a t i o n method and t h e d i r e c t v a r i a t i o n a l t e c h n i q u e as f o l l o w s : a ) The r e g u l a r p e r t u r b a t i o n method i s used as a s y s t e m a t i c and general approach f o r g e n e r a t i n g c o o r d i n a t e f u n c t i o n s . b) The d i r e c t v a r i a t i o n a l t e c h n i q u e i s used as an e f f i c i e n t procedure f o r m i n i m i z i n g and d i s t r i b u t i n g t h e e r r o r , i n t h e f i e l d v a r i a b l e s , t h r o u g h o u t t h e domain. 2. The h y b r i d technique extends t h e range o f a p p l i c a b i l i t y o f t h e p e r t u r b a t i o n method and enhances t h e e f f e c t i v e n e s s o f t h e d i r e c t v a r i a t i o n a l technique. I t a l s o a l l e v i a t e s t h e f o l l o w i n g major drawbacks o f t h e c l a s s i c a l techniques: a ) The requirement o f u s i n g a small parameter i n t h e r e g u l a r p e r t u r b a t i o n expansion i s avoided. b ) The method p r o v i d e s a s y s t e m a t i c s e l e c t i o n o f t h e c o o r d i n a t e f u n c t i o n s ( o r modes) needed i n t h e d i r e c t v a r i a t i o n a l technique.

3. The accuracy o f t h e s o l u t i o n s o b t a i n e d by t h e h y b r i d t e c h n i q u e i s r e l a t i v e l y i n s e n s i t i v e t o the choice o f the p e r t u r b a t i o n parameter(s). Therefore, t h e parameter(s) may be i n t r o d u c e d a r t i f i c i a l l y t o simp1 i f y t h e form o f t h e r e c u r s i v e s e t o f d i f f e r e n t i a l equations ( o r r e c u r s i v e f u n c t i o n a l s ) used i n e v a l u a t i n g t h e v a r i o u s - o r d e r p e r t u r b a t i o n s o l u t i o n s ( v i z , t h e coordinate functions). 4. The two-stage d i r e c t v a r i a t i o n a l technique g r e a t l y a1 l e v i a t e s one o f t h e major drawbacks o f t h e c l a s s i c a l d i r e c t v a r i a t i o n a l technique, namely, t h e repeated s o l u t i o n o f a l a r g e system o f nonsparse n o n l i n e a r equations. APPENDIX I - FUNCTIONAL AND GOVERNING DIFFERENTIAL EQUATION FOR THE ONEDIMENSIONAL CONDUCTING-CONVECTING FIN The f u n c t i o n a l c h a r a c t e r i z i n g t h e s t e a d y - s t a t e thermal response o f a onedimensional conducting-convecting f i n w i t h v a r i a b l e h e a t - t r a n s f e r c o e f f i c i e n t i s g i v e n by:

where e = T / T b i s a n o r m a l i z e d temperature d e f i n e d i n terms o f t h e f i n base temperature T ; 2 f i c i e n t ; q =T

Rb

i s a small parameter; c = x l / L ; cL2

k i s c o n d u c t i v i t y coef-

B i s a c o n v e c t i o n - c o n d u c t i o n f i n parameter; hb = q Tb; and

i s a constant. The governing d i f f e r e n t i a l e q u a t i o n i s g i v e n by: n

292

A.K. Noor

The boundary c o n d i t i o n s u s e d i n t h e p r e s e n t s t u d y a r e : At E = O ,

APPENDIX I 1

-

e=l

(A. 3)

GOVERNING DIFFERENTIAL EQUATIONS AND PERTURBATION EQUATIONS FOR THE CIRCULAR ISOTROPIC PLATE

The von-Karman p l a t e t h e o r y i s u s e d t o d e s c r i b e t h e n o n l i n e a r a x i s y m m e t r i c res p ons e o f t h e i s o t r o p i c c i r c u l a r p l a t e used i n t h e p r e s e n t s t u d y . The p r o b l e m i s f o r m u l a t e d i n t e r m s o f t h e t r a n s v e r s e d i s p l a c e m e n t w and t h e The t w o g o v e r n i n g d i f f e r e n t i a l e q u a t i o n s a r e g i v e n r a d i a l normal f o r c e Nr. by :

d 1 d 2 r d r [-r d r ( r N,)]

Eh3

where D =

Eh 2

dw 2 = 0 dr

t - (-)

(B.2)

i s f l e x u r a l r i g i d i t y o f t h e p l a t e ; po i s t h e i n t e n s i t y o f

12 (1 -u 2 )

u n i f o r m d i s t r i b u t k d l o a d i n g ; P i s a c o n c e n t r a t e d l o a d a t t h e c e n t e r ; and h i s the thickness o f the plate. The f o l l o w i n g r e c u r s i v e s e t s o f p e r t u r b a t i o n e q u a t i o n s a r e o b t a i n e d b y u s i n g t h e t wo -p ar a m e t e r p e r t u r b a t i o n e x p a n s i o n s f o r w and Nr, Eqs. 28 and 29. c ( w . . ) = R i j (‘ak’ 1J

C(N. . ) 1J

i j (Wak)

azi, where I;,

c

(B.3)

NQk)

k < j

(B.4)

Y

or

~
a r e l i n e a r d i f f e r e n t i a l o p e r a t o r s g i v e n by: I;= d3 - + l d 1 -d -

dr3

dr2

r 2 dr

(8.5)

2 d2 d C = 2 r ---ttrdr dr2 The e x p l i c i t f o r m o f t h e f i r s t s i x n o n z e r o components o f Rij t h r e e nonz ero components o f

Eij

a r e g i v e n i n T a b l e 1.

and t h e f i r s t

The Application of’ Variational Methods to Nonlinear Problems Table 1

-

E x p l i c i t form o f R and

i

293

functions

i

d w l 0)2 (r

2

0

-Eh

1

1

-Eh--dWOl d W I O dr dr

-The expressions f o r R12, RO3, and and

iz0 by i n t e r c h a n g i n g

io2 a r e obtained

f r o m t h o s e o f RZ1,

R30 ’

t h e two s u b s c r i p t s o f each o f w and N.

REFERENCES

[l]Delves, L.M. and P h i l l i p s , C.,

A f a s t implementation o f t h e g l o b a l element method, Journal o f t h e I n s t i t u t e o f Mathematics and I t s A p p l i c a t i o n s 25 (1980) 177-197.

[2] Delves, L.M. and Mead, K.O., On t h e convergence r a t e s o f v a r i a t i o n a l methods. I.A s y m p t o t i c a l l y diagonal systems, Mathematics o f Computation 25 (1971) 699-716. [ 3 ] Almroth, B.O., Brogan, F.A. and S t e r n , P., Automatic c h o i c e o f g l o b a l shape f u n c t i o n s i n s t r u c t u r a l a n a l y s i s , A I A A J o u r n a l 16 (1978) 525-528. [4] Noor, A.K. and Peters, J.M., Reduced b a s i s technique f o r n o n l i n e a r a n a l y s i s o f s t r u c t u r e s , A I A A Journal 18 (1980) 455-462.

[5] Noor, A.K., Balch, C.D. and S h i b u t , M.A., Reduction methods f o r non’line a r s t e a d y - s t a t e thermal a n a l y s i s , NASA TP-2098 (March 1983). [6] Noor, A.K., Recent advances i n r e d u c t i o n methods f o r n o n l i n e a r problems, Computers and S t r u c t u r e s 13 (1981) 31-44. [7] Noor, A.K. and Peters, J.M., T r a c i n g p o s t - l i m i t - p o i n t paths w i t h reduced b a s i s technique, Computer Methods i n A p p l i e d Mechanics and Engineering 28 (1981) 217-240. [8] Noor, A.K. and P e t e r s , J.M., B i f u r c a t i o n and p o s t b u c k l i n g a n a l y s i s o f l a m i n a t e d composite p l a t e s v i a reduced b a s i s technique, Computer Methods i n A p p l i e d Mechanics and E n g i n e e r i n g 29 (1981) 271-295. [9] Noor, A.K. and Peters, J.M., Recent advances i n r e d u c t i o n methods f o r i n s t a b i l i t y a n a l y s i s o f s t r u c t u r e s , Computers and S t r u c t u r e s 16 (1983) 67-80.

294

[lo]

A.K. Noor Noor, A. K. and Balch, C.D., H y b r i d perturbation/Bubnov-Galerkin t e c h n i q u e f o r n o n l i n e a r t h e r m a l a n a l y s i s , NASA TP-2145 (June 1 9 8 3 ) .

[ll] Noor, A.K., P e t e r s , J.M. and Andersen, C.M., Two-stage R a y l e i g h - R i t z t e c h n i q u e f o r n o n l i n e a r a n a l y s i s o f s t r u c t u r e s , i n : Proceedings of t h e Second I n t e r n a t i o n a l Symposium on I n n o v a t i v e Numerical A n a l y s i s i n A p p l i e d E n g i n e e r i n g Science, June 16-20, 1980, M o n t r e a l , Canada. [12] Guttmann, A.J., D e r i v a t i o n o f ' m i m i c f u n c t i o n s ' f r o m r e g u l a r p e r t u r b a t i o n expansions i n f l u i d mechanics, J o u r n a l o f t h e I n s t i t u t e o f Mathematics and I t s A p p l i c a t i o n s 15 (1975) 307-317. [13] Van Dyke, M., A n a l y s i s and improvement o f p e r t u r b a t i o n s e r i e s , Q u a r t e r l y J o u r n a l o f Mechanics and A p p l i e d Mathematics 27 (1974) 423-450. [14] Watson, L.G. and Chudobiak, J.M., S o l u t i o n o f von Karman's p l a t e e q u a t i o n s w i t h p e r t u r b a t i o n and s e r i e s summation, i n : Research i n S t r u c t u r a l and S o l i d Mechanics, NASA CP-2245 ( 1 9 8 2 ) . [15] A z i z , A. and Hamad, G . , R e g u l a r p e r t u r b a t i o n expansions i n h e a t t r a n s f e r , I n t e r n a t i o n a l J o u r n a l o f Mechanical E n g i n e e r i n g E d u c a t i o n 5 (1977) 167-182. [16] Thompson, J.M.T. and Walker, A . C . , The n o n l i n e a r p e r t u r b a t i o n a n a l y s i s o f d i s c r e t e s t r u c t u r a l systems, I n t e r n a t i o n a l J o u r n a l o f S o l i d s and S t r u c t u r e s 4 (1968) 757-768. [17] M a t h l a b Group, MACSYMA Reference Manual ( v e r s i o n Ten, f i r s t p r i n t i n g , Massachusetts I n s t i t u t e o f Technology, J a n u a r y 1983). [18] A z i z , A. and Benzies, J.Y., A p p l i c a t i o n o f p e r t u r b a t i o n t e c h n i q u e s t o h e a t t r a n s f e r problems w i t h v a r i a b l e thermal p r o p e r t i e s , I n t e r n a t i o n a l J o u r n a l o f Heat and Mass T r a n s f e r 1 9 (1976) 271-276. [19] Chia, C . Y . , 1980).

N o n l i n e a r A n a l y s i s o f P l a t e s (McGraw-Hill Book Company,

[ Z O ] N o w i n s k i , J.L. and I s m a i l , I . A . , A p p l i c a t i o n o f a m u l t i - p a r a m e t e r p e r t u r b a t i o n method t o e l a s t o s t a t i c s , i n : Shaw, W.A. ( e d . ) , Developments i n T h e o r e t i c a l and A p p l i e d Mechanics, V o l . 2, Proceedings o f t h e Second S o u t h e a s t e r n Conference on T h e o r e t i c a l and A p p l i e d Mechanics, sponsored by Georgia I n s t i t u t e of Technology, A t l a n t a , March 5-6, 1974 (Pergamon Press, New York, 1965) 35-45. [21] K o r n i s h i n , M.S. and Isanbaeva, F.S., MOSCOW, 1968; i n R u s s i a n ) .

F l e x i b l e P l a t e s and Panels (Nauka,

[22] Yaghmai , S. , I n c r e m e n t a l a n a l y s i s o f l a r g e d e f o r m a t i o n s i n mechanics o f s o l i d s w i t h applications t o axisymmetric s h e l l s o f revolution, NASA CR-1350 (1 9 6 9 ) .

[23] Bathe, K.J., Ramm, E. and Wilson, E.L., F i n i t e element f o r m u l a t i o n s f o r l a r g e d e f o r m a t i o n dynamic a n a l y s i s , I n t e r n a t i o n a l J o u r n a l f o r Numerical Methods i n E n g i n e e r i n g 9 (1975) 353-386.

The Application o f Variational Methods to Nonlinear Problems

295

Nondi mensiona I temperature,

p = 1.0 p = 0.33

e

p=0.33

p = 1.0 0.4 0

0.2

0.4

0.6

0.8

1.0

”/ a) q = 1.0 Finite-element solution 1.0

A

2 h=q,4terms

A

h =p, 2 terms

+

1 = p, 2-coordinate

PERTURBATION METHOD

0.8

Nondimensional temperature, 0.6 8

p = 1.0 0.4

p = 0.33 0.2

I

I

I

I

0.2

0.4

0.6

0.8

I 1.0

xl/L

b) q = 2.0

Figure 1 . - Comparison of s o l u t i o n s obtained by Perturbation method a n d Hybrid technique f o r one-dimensional conducting-convecting f i n with v a r i a b l e h e a t - t r a n s f e r c o e f f i c i e n t .

296 A.K. Noor

II

0

X h

0

n

I

aJ

U aJ

ru L

A

cn

n

I--

I S

aJ aJ

c, X .r

I

Ln

N

aJ L 3

W

The Application of Variational Methods to Nonlinear Problems

. . . . . . . Finite element

+

297

solution

Perturbation method Hybrid technique

0

I 0

2.0

w,/h

2000

-

1500

-

.

-

Total strain

1

ETh4

UL/( ETh4)

Figur.e 3.

-

Accuracy o f s o l u t i o n s o b t a i n e d by P e r t u r b a t i o n method and H y b r i d technique a t v a r i o u s l o a d l e v e l s . Clamped, square s i x t e e n - p l y q r a p h i te-epoxy p l a t e s u b j e c t e d t o u n i f o r m t r a n s v e r s e l o a d i n q (see F i g . 2 ) .

A1

' 0

4

812'

Perturbation method E = 6 895

X

10" Nlm'

u=0.3

(13

D

= 0. In m.

h = 1.nX

m. h2 10.0

-

'.-->'

5.0 -

wc/ h

p

A1 8

-

r

20.0

20.0

15.0

15.0

A2 10.0

A2 10. 0

5.0

5.0

0

F i g u r e 4.

12

0. m

0.14

0.21

0.28

0

0. u

O

0.24

4

8

0.36

12'

0.48

Accuracy o f s o l u t i o n s o b t a i n e d b y two-parameter P e r t u r b a t i o n method and H y b r i d t e c h n i q u e . I s o t r o p i c c i r c u l a r p l a t e s u b j e c t e d t o combined u n i f o r m and c o n c e n t r a t e d l o a d i n g .

>

I1

c

II

c

II

The Application of Variational Methods t o Nonlinear Problems

N

E

+ z 53 53

II

X

w

h

c1

3

U v)

aJ

c

L)

v)

aJ

L Q

3

0 c

I

In

299

A.K. Noor

300

-

PO

Eh

wc/R

8.0

6.0 /' 'OR

4.0

Eh

I

2.0 I

I

2.0 F i g u r e 6.

-

0

8 basis vectors

+

9 basis vectors

4.0

6.0

1

Rayleigh-Ritz techniaue

1 I

8.0

UR/(E h4) Accuracy o f t r a n s v e r s e displacements and s t r a i n energies o b t a i n e d by t h e two-stage R a y l e i g h - R i t z technique a t v a r i o u s l o a d l e v e l s . Shallow s p h e r i c a l cap shown i n F i g . 5, h = 0.0127.

solution

6.0

+

oAo+ 'OR

1

4.0

Eh

Full system

Perturbation tech n ique

2.0 9 terms I

3.5

I I

I I

7.0

10.5

I

14.10x 10-3

wc/R Fiqure 7. - Accuracy o f t r a n s v e r s e displacements obtained by t h e s t a t i c Perturbation technique a t various load l e v e l s . Shallow spherical cap shown i n Fig. 5 , h = 0.0127 m.

A.K. Noor

302

-k

0 F i g u r e 8.

-

7.0

Two-stage Rayleigh-Ritz technique (9basis vectors)

14.0

21.0

28.0

UR/Eh4 Accuracy o f t r a n s v e r s e displacements and s t r a i n energies o b t a i n e d by t h e two-stage R a y l e i g h - R i t z technique a t v a r i o u s l o a d l e v e l s - s h a l l o w s p h e r i c a l cap shown i n F i g . 5, h = 0.005 m.

Unification of Finite Element Methods H. Kardestuncer (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

303

CHAPTER 13

COLLOCATION SOLUTION OF THE TRANSPORT EQUATION USING A LOCALLY ENHANCED ALTERNATING DIRECTION FORMULATION M.A. Celia & G.F, Pinder

INTRODUCTION The a l t e r n a t i n g - d i r e c t i o n c o l l o c a t i o n (ADC) method has been shown t o be an a t t r a c t i v e technique f o r t h e s o l u t i o n o f t h e mu1 t i - d i m e n s i o n a l t r a n s p o r t e q u a t i o n ( C e l i a , 1983; C e l i a e t a l . , 1980). The method achieves h i g h o r d e r accuracy w h i l e b e n e f i t t i n g from t h e computational e f f i c i e n c i e s i n h e r e n t i n t h e a l t e r n a t i n g d i r e c t i o n procedure. In t h i s paper, we p r e s e n t a m o d i f i c a t i o n o f t h e ADC procedure designed t o achieve enhanced accuracy i n t h e neighborhood o f a sharp f r o n t . We b e g i n w i t h a r e v i e w o f t h e ADC method. On t h i s f o u n d a t i o n we b u i l d t h e enhancement procedure. F i n a l l y , t h e enhanced ADC method i s a p p l i e d t o an example problem. To s i m p l i f y o u r p r e s e n t a t i o n , we c o n s i d e r o n l y r e c t a n g u l a r subspaces. THE ADC PROCEDURE Consider t h e g e n e r a l i z e d t r a n s p o r t e q u a t i o n (1)

DtU

+ B*Vu _

-

2

CV u = 0,

I

where (x,y) designates a two-dimensional s p a t i a l c o o r d i n a t e system, t i s time, D t ( - ) i s d i f f e r e n t i a t i o n w i t h r e s p e c t t o time, and B and C a r e assumed s p a t i a l l y c o n s t a n t . f o r s i m p l i c i t y . F o r a t r e a t m e n t o f s p a t i a l v a r i a b i l i t y , see C e l i a (1983). L e t us b e g i n t h e development o f t h e b a s i c ADC procedure by w r i t i n g a f r a c t i o n a l s t e p a l g o r i t h m based upon a f i n i t e d i f f e r e n c e a p p r o x i m a t i o n o f the time derivative, t h a t i s

n+l (2b)

U

At

un+1/2

+

L un+l Y

= 0,

M.A. Celia & G.F. Pinder

304 where

and 6,

and B

Y

Lx(*)

z

BxDx(*)

-

CDxx(*)

L Y

5

6 D Y Y

-

CD ( * )

(

0

)

(

0

)

YY

denote v e c t o r components i n t h e x and y c o o r d i n a t e

d i r e c t i o n s , r e s p e c t i v e l y , and D x ( - ) , D ( * ) , D x x ( - ) and D ( - 1 a r e space Y YY d e r i v a t i v e s . The e q u i v a l e n t one-step e q u a t i o n .is o b t a i n e d by s u b s t i t u t i o n o f ( 2 b ) i n t o (2a), t h a t i s

Examination o f equations ( 3 ) and ( 1 ) r e v e a l s an e r r o r o f o r d e r A t due t o t h e f r a c t i o n a l s t e p procedure presented as equations ( 2 ) . T h i s e r r o r term may be neglected, o r a s u i t a b l e c o r r e c t i o n can be added t o t h e r i g h t hand s i d e of ( 2 ) . Such a c o r r e c t i o n i s discussed i n t h e Appendix. One must now choose a s u i t a b l e approximation f o r u. I f we choose Hermite c u b i c polynomials as o u r basis, t h e a p p r o x i a t i o n can be w r i t t e n

0 where qi(x),

1 1 q o ( y ) , $i(x), q . ( y ) a r e piecewise Hermite c u b i c p o l y n o m i a l s . j J The c u b i c Hermite b a s i s f u n c t i o n s d e f i n e d along x can be w r i t t e n i n terms o f t h e l o c a l element c o o r d i n a t e

E = 1 - 2 (

A X = Xitl

‘it1 Ax

-

-

, 5

= [-1,1],

Xi,

and ( x . , y . ) denotes a nodal l o c a t i o n , 1

J

as

where

x = [xi,xitll,

305

Collocation Solution of the Transport Equation

0

1

'

The f u n c t i o n s $ . ( y ) and $ . ( y ) a r e of t h e same f u n c t i o n a l form. These J J f u n c t i o n s and t h e ( i , j ) n o t a t i o n a r e presented i n f i g u r e 1. The parameters 1, and J o denote t h e number o f nodes i n t h e x and y d i r e c t i o n s r e s p e c t i v e l y . The undetermined c o e f f i c i e n t s Uij,

aU. ./ax, 1J

a U . .lay, 1.J

a 2 U . ./axay 1J

are

values o f u, Dxu, D u, and D u a t t h e nodal l o c a t i o n s , r e s p e c t i v e l y . Y XY S u b s t i t u t i n g u ( x , y , t ) from ( 4 ) i n t o (2a) and r e g r o u p i n g terms, one o b t a i n s n+1/2

1 and $ . a r e independent f u n c t i o n s o f y , and because t h e j J q u a n t i t i e s i n the brackets are functions o f x only, equation (7) implies t h a t anywhere a l o n g y Because $'

306

M. A. Celia & G.F, Pinder I0

=

i=1 C u?j$p(x) t

au? .

$i(x)

The s o l u t i o n o f ( 7 ) f o r t h e 41, unknown parameters r e q u i r e s t h e e v a l u a t i o n o f ( 7 ) a t 41, c o l l o c a t i o n p o i n t s . Because we employ orthogonal c o l l o c a t i o n , t h e c o l l o c a t l o n p o i n t l o c a t i o n s correspond t o t h e zeros o f t h e Leaendre polynomials, These l o c a t i o n s commonly a r e employed i n numerical i n t e g r a t i o n also, wherein they a r e known as "Gauss p o i n t s " . Each c o l l o c a t i o n p o i n t can be a s s o c i a t e d w i t h an unknown parameter i n e q u a t i o n ( 7 ) . Thus, f o r convenience, we i d e n t i f v them, as i n d i c a t e d i n f i g u r e 1, as uIJ, u x I J , The complete s e t o f equations f o r t h e x d i r e c t i o n ADC uYIJ and uxyIJ. sweep can now be w r i t t e n u s i n g t h e n o t a t i o n L x ( . ) is

=

(1 t ( a t ) L x ) ( - ) , t h a t

Collocation Solution of the Transport Equation

307

Both ( 8 a ) and ( 8 b ) r e q u i r e boundary information t o be properly posed, The imposition of boundary conditions i n the context of s p l i t t i n g schemes must be undertaken with some c a r e ; t h e procedure used herein i s given i n the Appendix. Once this s p e c i f i c a t i o n has been made, t h e r e s u l t i n g systems of 21, - 2 equations r e s u l t i n g from (8a) o r ( 8 b ) can be solved d i r e c t l y . T h i s procedure i s repeated 25, - 2 times. One next proceeds t o the second half of the time s t e p . Equations analogous t o (8a) and ( 8 b ) a r e w r i t t e n using ( 2 b ) and t h e c o l l o c a t i o n points ( x a y y a ) , a = u , u , J = 1 , 2 , ...,JO, I = 1 , 2 , ..., I0 and ( x a , y a ) . Y a = ux,uxy, J = 1 , 2 , . ..,Joy I = 1,2, I 0' These equations a r e a l s o adjusted f o r boundary conditions and each s e t of 25, - 2 equations i s solved f o r I = 1 , 2 , ..., I 0' An e n t i r e (x-y) sweep of t h e mesh i s now complete and one proceeds t o t h e next time s t e p .

...,

LOCAL ENHANCEMENT

While the ADC procedure described heretofore c o n s t i t u t e s a very e f f i c i e n t and accurate algorithm f o r t h e s o l u t i o n o f problems i n not only two b u t a l s o t h r e e space dimensions, problems e x h i b i t i n g s o l u t i o n s with s t e e p ' f r o n t s may r e q u i r e a higher-order approximation in the neighborhood of the f r o n t . This local enhancement i s achieved by u s i n g C1 q u i n t i c polynomials as i n t e r p o l a n t s i n elements where a s t e e p concentration gradient i s encountered ( s e e Mohsen and Pinder, 1983). The local use of q u i n t i c s i s p a r t i c u l a r l y a t t r a c t i v e when the additional degrees o f freedom due t o t h e q u i n t i c formulation can be condensed out a t the element l e v e l . When t h i s local condensation can be achieved, t h e overall s o l u t i o n can be enhanced without increasing the rank o r bandwidth of t h e global c o e f f i c i e n t matrix. Unfortunately, when a standard mu1 ti-dimensional collocation approximation is employed, the introduction of midside nodes f o r local enhancement couples together adjacent elements a s i l l u s t r a t e d i n f i g u r e 2. This makes local condensation a l g o r i t h m i c a l l y d i f f i c u l t and computationally i n e f f i c i e n t . However, as we i l l u s t r a t e s h o r t l y , when t h e ADC procedure is employed, local condensation is r e a d i l y achieved. Let us now formulate the approximating equations f o r an enhanced element. A point of departure i s equation ( 5 ) . The o b j e c t i v e i n t h e enhanced procedure i s t o employ C 1 q u i n t i c polynomials i n place of cubic Hermites t o represent G(x,y,t) i n elements known t o contain s o l u t i o n segments e x h i b i t i n g a sharp f r o n t . While t h e i n t e r p o l a t i n polynomial f o r such elements reads as does ( 4 ) , t h e basis functions $ %. ( x ) , @ o ( y ) , $ j ( x ) and o].(y) a r e now defined i n t h e local 5 coordinate gystem J a s ( s e e F i g . 23

M.A. Celia & G.F. Pinder

308

1

(c5

-

54 - 53 t 52)

One now f o l l o w s a development f o r t h e q u i n t i c element t h a t i s analogous t o t h a t presented above f o r t h e c u b i c s . The outcome i s an e q u a t i o n very s i m i l a r t o (8). The p r i n c i p a l d i f f e r e n c e i s t h a t now t h e r e a r e c o r n e r , mid-side, and mid-element nodes w i t h which t o contend. Moreover, one must a l s o employ more c o l l o c a t i o n p o i n t s t o accommodate t h e a d d i t i o n a l degrees of freedom a s s o c i a t e d w i t h t h e q u i n t i c polynomial i n t h e enhanced element. Note t h a t we i l l u s t r a t e i n f i g u r e 2 how t h e enhancement o f one element f o r m a l l y leads t o t h e enhancement o f t h e e n t i r e row and column o f elements t h a t i n c l u d e s t h e enhanced element. By employing t h i s extended approach t h e x and y sweeps a r e u n i q u e l y defined, and t h e r e i s no a m b i g u i t y r e g a r d i n g t h e choice of c o l l o c a t i o n p o i n t s t o be used i n each sweep. While we c o u l d a t t h i s p o i n t f o r m u l a t e and s o l v e a g l o b a l m a t r i x equation f o r each row, i t i s more c o m p u t a t i o n a l l y e f f i c i e n t t o reduce t h e q u i n t i c elements a t t h e l o c a l l e v e l . To e x p l a i n t h i s concept, we w r i t e t h e element c o l l o c a t i o n m a t r i x e q u a t i o n f o r a t y p i c a l (one-dimensional) q u i n t i c element i n t h e x - d i r e c t i o n .

Collocation Solution of the Transport Equation

309

L

E u a t i o n (10) i n s t i t u t e s f o u r equations i n s i x unknown parameters. Because a r e non-zero o n l y o v e r t h e element f o r which t h e y a r e and @o(C) d e f i n e d , t h e two v a r i a b l e s Uin'1/2 and aU?;l/Z/ax can be e l i m i n a t e d

@i(E,)

2

a l g e b r a i c a l l y from e q u a t i o n (10). E q u a t i o n (10) t h e n reduces t o two equations i n f o u r unknowns, t h e same e q u a t i o n c o n f i g u r a t i o n t h a t a r i s e s from employing a Hermite c u b i c i n t e r p o l a t i o n . One can now assemble t h e r e s u l t i n g two e q u a t i o n s i n t o t h e g l o b a l c o e f f i c i e n t m a t r i x and m a i n t a i n t h e same g l o b a l m a t r i x s t r u c t u r e as would be generated u s i n g a l l c u b i c i n t e r p o l a t e s . The v a l u e o f t h e c o e f f i c i e n t s d e r i v e d from q u i n t i c elements are, o f course, d i f f e r e n t from those d e r i v e d from c u b i c elements. Once t h e g l o b a l equations a r e solved, one can o b t a i n t h e c e n t e r node c o e f f i c i e n t s from t h e e l e m e n t - l e v e l equations. The enhanced-element concept i s a dynamic procedure. As t h e s o l u t i o n geometry changes, d i f f e r e n t elements a r e enhanced. I n o t h e r words, s e l e c t e d c u b i c elements become q u i n t i c and c e r t a i n q u i n t i c elements r e v e r t t o c u b i c . To o b t a i n s t a r t i n g values o f u(x,y,t) f o r a new t i m e s t e p a t new c e n t e r nodes i d e n t i f i e d w i t h q u i n t i c i n t e r p o l a n t s , t h e i n t e r p o l a t i o n p r o p e r t y o f t h e Hermite c u b i c polynomial i s employed i n c o n j u n c t i o n w i t h e x i s t i n g nodal c o e f f i c i e n t s . The e n t i r e procedure i s presented as a f l o w c h a r t i n f i g u r e 3. EXAMPLE CALCULATIONS

To i l l u s t r a t e t h e e f f e c t i v e n e s s o f t h e enhanced ADC procedure, we w i l l now s o l v e a convection-dominated t r a n s p o r t e q u a t i o n and i n v e s t i g a t e t h e b e n e f i t s o f adding a small number o f q u i n t i c elements a l o n g t h e p r i n c i p a l d i r e c t i o n o f f l o w . The e q u a t i o n t o be s o l v e d i s (11)

Dtu + 10Dxu

subject t o

-

02u =

0

M.A. Celia & G.F. Pinder

310

u = l

atx=0,

u = 0 a t x = 0, u = O at

3 0 2 ~ 5 5 0 a l l other y

t = O

The s o l u t i o n t o t h i s e q u a t i o n i s g i v e n by

A p l o t o f t h e s o l u t i o n , a l o n g t h e l i n e y = 40, i s shown i n f i g u r e 5. Because t h e numerical s o l u t i o n must be s o l v e d on a bounded domain, t h e system i l l u s t r a t e d i n f i g u r e 4a has been s o l v e d n u m e r i c a l l y . To assure t h a t (12) can be used t o o b t a i n an e r r o r measure, t h e numerical s o l u t i o n i s t e r m i n a t e d b e f o r e t h e f i n i t e boundaries a r e r e f l e c t e d i n t h e solution. The numerical s o l u t i o n procedure chooses t h e n elements t o be enhanced by i n t e r r o g a t i n g t h e s o l u t i o n o b t a i n e d f o r t h e l a s t t i m e s t e p . The n elements i n which t h e l a r g e s t changes i n t h e s o l u t i o n occur a r e assigned a q u i n t i c f o r m u l a t i o n . T h i s procedure i s c l e a r l y dynamic and accommodates t h e changing s o l u t i o n t o p o l o g y by i n c o r p o r a t i n g enhanced accuracy o n l y i n r e g i o n s where i t i s needed. I n t h i s example problem, l e t us c o n s i d e r t h e g r i d c o n f i g u r a t i o n o f f i g u r e 4b. Moreover, l e t us employ i n o u r computational a l g o r i t h m t h e numerical c o r r e c t i o n f a c t o r d e s c r i b e d i n ( A . 9 ) o f t h e appendix t o t h i s paper. We w i l l use as o u r e r r o r measure M (13) E = c e,(t = 0.5) + e,(t = 1.0) + e,(t = 2.0), m= 1 where e m = ( u - u l , u i s t h e e x a c t s o l u t i o n , i . e . (12), and u i s t h e c o l l o c a t i o n approximation. The parameter M i s t h e t o t a l number o f nodes a l o n g t h e l i n e s y = 40, y = 32, and y = 28. ( n o t e t h a t t h e s o l u t i o n i s symmetric about y = 40). The numerical s o l u t i o n f o r t h e cases o f n = 0 and n = 2 a r e g i v e n i n f i g u r e 5. F i g u r e 6 shows t h e r o l e o f t h e t i m e t r u n c a t i o n e r r o r i n t h i s approach. As A t becomes s m a l l e r , t h e s p a t i a l e r r o r dominates and t h e improvement i n accuracy due t o q u i n t i c

Collocation Solution o f the Transport Equation

31 I

enhancement becomes e v i d e n t . The improvement i n accuracy i s even more e v i d e n t f o r a course mesh. F o r example, when o n l y f o u r elements a r e used a l o n g t h e x - d i r e c t i o n , j u s t one q u i n t i c reduces t h e e r r o r E from 1.6 f o r t h e case o f a l l c u b i c elements t o 0.85, a 47% decrease i n e r r o r ( f o r A t = .01). I n comparison, o n l y a 16% e r r o r improvement was achieved u s i n g t h e q u i n t i c enhancement and e i g h t elements. While t h e e r r o r measure d e s c r i b e d above i s c e r t a i n l y o f i n t e r e s t , numerical d i s p e r s i o n , e x e m p l i f i e d by overshoot and undershoot, i s a l s o w o r t h i n v e s t i g a t i n g . When q u i n t i c enhancement i s employed, t h e a l r e a d y v e r y small overshoot ( a p p r o x i m a t e l y 2.5%) can be e l i m i n a t e d e n t i r e l y and t h e undershoot ( a p p r o x i m a t e l y 3.5%) can be reduced t o about 1 .O%. Numerical d i s p e r s i o n can be f u r t h e r m o d i f i e d by changing t h e s t r a t e g y used f o r t h e s e l e c t i o n o f t h e enhanced elements. F o r example, by enhancing o n l y t h e element immediately b e h i n d t h e one e x h i b i t i n g t h e maximum change, t h e undershoot can be reduced a t t h e expense o f a small ( < l % ) overshoot. ACKNOWLEDGEMENTS T h i s work was supported i n p a r t by t h e Department o f Energy under c o n t r a c t DE-AC02-83ER60170 and a l s o bv t h e N a t i o n a l Science Foundation under c o n t r a c t NSF CEE81-11240. Thanks a r e g i v e n t o D r. L i n d a !!. A b r i o l a f o r h e r h e l p f u l d i s c u s s i o n s concerning t h e s p l i t t i n g scheme. The c o n t r i b u t i o n o f D r . Robert Cleary, who s u n p l i e d t h e computed code f o r t h e a n a l y t i c a l s o l u t i o n used h e r e i n , i s a l s o a p p r e c i a t e d . REFERENCES

[11

C e l i a , M.A., C o l l o c a t i o n on Deformed F i n i t e Elenents and A l t e r n a t i n g D i r e c t i o n C o l l o c a t i o n Methods, Ph.D. Thesis, Department o f C i v i l Engi n e e r i ng, P r i n c e t o n Uni v e r s i t y (Sept. 1983)

[2]

C e l i a , M.A., P i n d e r , G.F. and Hayes, L.J., A l t e r n a t i n g D i r e c t i o n C o l l o c a t i o n S o l u t i o n t o t h e T r a n s p o r t Equation, Proc. T h i r d I n t . Conf. F i n i t e Elements i n I l a t e r Resources, Wang e t a1 (eds), Univ. o f M i s s i s s i p p i , (1980), pp. 3.36-3.48.

[3]

Gourlay, A.R. and l 4 i t c h e l 1 , A.R., The Equivalence o f C e r t a i n A l t e r n a t i n g D i r e c t i o n and L o c a l l y One-Dimensional Methods, S.I.A.M. J . Num. Anal.,6 ( 1 ) (1363) 37-46.

[4]

Mohsen, M.F.N. and Pinder, G.F., Orthogonal C o l l o c a t i o n w i t h ' A d a p t i v e ' F i n i t e Elements, s u b m i t t e d t o I n t . J . Num. Meth. Engrg., (1983).

312

M.A. Celia & G.F. Pinder

APPEND1 X

I n t h i s appendix, t h e t r e a t s e n t o f boundary c o n d i t i o n s and o f non-zero r i g h t hand s i d e f o r c i n g f u n c t i o n s w i l l be discussed, as w i l l t h e implem e n t a t i o n o f t h e c o r r e c t i o n t e r m discussed i n s e c t i o n 2. To begin, l e t us consider an e q u a t i o n o f t h e form o f ( l ) , w r i t t e n w i t h t h e d e f i n i t i o n s o f (3),as Dtu + LXu + L u = f ( x , y , t ) Y s u b j e c t t o some general D i r i c h l e t o r Neumann boundary c o n d i t i o n s . The general m u l t i - d i m e n s i o n a l , f u l l y - i m p l i c i t c o l l o c a t i o n approximation t o ( A . l ) produces an a l g e b r a i c e q u a t i o n o f t h e form (A.l)

(A.2)

( k x k y + (nt)Nxky + ( 4 t ) M k ) un+l = k k un Y X X Y + (4t)F +

B

where Kx , Mx. correspond t o mass and s t i f f n e s s m a t r i c e s , r e s p e c t i v e l y , t h e v e c t d r F Eorresponds t o t h e f o r c i n g f u n c t i o n f, and B i s a v e c t o r which accounts f o r non-zero boundary c o n d i t i o n s . When developing s p l i t equations, we w i l l r e q u i r e t h e r i g h t hand s i d e o f t h e composite ( e q u i v a l e n t one-step) e q u a t i o n t o be t h e same as t h a t o f (A.2) Consider t h e s p l i t equations U

n+1/2

(A.3a)

(A.3b)

-

un

+

A t

U

n+l

-

un+1/2

L untli2 X

+

A t

L unt' Y

=

Fl/At

= F2/At

where F1 and F2 a r e non-zero v e c t o r s which a r e t o be determined. Upon a p p l i c a t i o n o f t h e c o l l o c a t i o n method and t h e e l i m i n a t i o n o f un+1/2 to form t h e e q u i v a l e n t one-step equation, t h e r e r e s u l t s (A.4)

[kxky + A t ( M k

X Y

= kxkyu n

+ Mykx) + ( A ~ ) * M ~ M ~ ] U " ' ~

+ kyFl

+ (kx + A t M ) F

x Through a comparison o f (A.4) and (A.2),

2'

i t i s c l e a r t h a t the condition

313

Collocation Solution o f the Transport Equation should be s a t i s f i e d . s i m p l e one i s

While an i n f i n i t e number of s o l u t i o n s e x i s t , a

(A.6a)

F, = 0

(A.6b)

F2 = ( k x + AtMx)-l ( A t F + B )

or, l e t t i n g (A.7)

F = (At)F + B

one must have (A.8)

( k x + AtMx)F2 =

F.

N o t i c e t h a t t h e c o e f f i c i e n t m a t r i x o f (A.8) i s e x a c t l y t h e c o e f f i c i e n t m a t r i x f o r t h e x - d i r e c t i o n s o l u t i o n s . T h e r e f o r e , t h i s m a t r i x needs t o be reduced o n l y once t o s o l v e b o t h (A.8) and t h e x - d i r e c t i o n equations g i v e n by ( 8 ) . Furthermore, if n e i t h e r f n o r t h e boundary c o n d i t i o n s a r e t i m e v a r y i n g , t h e v e c t o r F needs t o be computed o n l y once. T h i s d i s c u s s i o n o f non-zero r i g h t hand s i d e s f o r s p l i t t i n g schemes i s s i m i l a r t o t h a t presented by Gourlay and M i t c h e l l (1969) where LOD and AD1 d i f f e r e n c e schemes were compared. The a d d i t i o R + p f a c o r r e c t i o n term t o account f o r t h e p e r t u r b a t i o n t e r m can be e a s i l y achieved through a m o d i f i c a t i o n o f t h e ( A t ) * MxMyu definition o f (A.9a)

F. S p e c i f i c a l l y , d e f i n e t h e new v e c t o r Fc as 2

Fc = ( A t ) F + B + ( A t ) C

where t h e c o r r e c t i o n v e c t o r C i s g i v e n by (A.9b)

C = M M un X Y

N o t i c e t h a t t h e a d d i t i o n o f t h i s c o r r e c t i o n t e r m reduced t h e e r r o r i n t r o d u c e d by t h e p e r t u r b a t i o n term by one o r d e r , s i n c e t h e p e r t u r b a t i o n t e r m i s now o f t h e form ( A t ) 2 M M (un+’ X Y

-

un).

With

Fc r e p l a c i n g

F,

t h e s o l u t i o n procedure f o r t h i s c o r r e c t i o n case leads d i r e c t l y t o (A.3) v i a (A.9).

M. A. Celia & G.F. Pinder

314

X

-I J

i, j t l X

X

X

X

B

A UXYIJ X

UYIJ

X

X

X

i+l,j

i X

X

X

Y

X

X

'I*! J

UXIJ

"IJ

C

D X

X

'J

X

XYJI -1

i, j -1

F i g u r e 1:

4 1

Diagrammatic r e p r e s e n t a t i o n o f a c o l l o c a t i o n - f i n i t e element network i l l u s t r a t i n g t h e nodes ( - ) and c o l l o c a t i o n p o i n t s ( x ) . The nodes c a r r y t h e s u b s c r i p t ( i , j ) , t h e c o l l o c a t i o n p o i n t s t h e s u b s c r i p t a(I,J), and t h e elements a r e i d e n t i f i e d by t h e upper case Roman l e t t e r s (A-D). The c u b i c bases a r e a l s o shown.

Collocation Solution of the Transport Equation

315

u

X

X

x

x

x

x

X

X

X

X

x

x

x

x

X

X

X

X

x

x

x

x

X

X

X

X

x

x

x

x

X

X

ijtl

II

i+lj

.ij

X

x

x

x

x

x

X

X

X

X

x

x

x

x

X

X

X

X

x

x

x

x

X

X

X

X

x

x

x

x

X

X

F i g u r e 2:

-

( a ) Diagrammatic r e p r e s e n t a t i o n o f t h e nodes and c o l l o c a t i o n p o i n t s a s s o c i a t e d w i t h l o c a l enhancement o f t h e c e n t e r element. The nodes i n d i c a t e d b y a a r e f o r m a l l y r e q u i r e d o f o r t h e l o c a l l y enhanced ADC procedure. ( b ) The q u i n t i c bases $i(5). ( c ) The quintic functions ( 5 1 , assuming A X = 2.

$1

M. A. Celiu & G. F. Pirider

316

v

m

0.4

O -1.0 d

(cc---

v

-I

0

0

0

0

L

1.1

(c)

Figure 2: (continued).

€

317

Collocation Solution of the Transport Equation

Iwo

Figure 3:

I

Abbreviated f l o w chart for the enhanced AD[: procedure.

M. A. Celia & G.F. Pinder

318

y =50

u=l

vx =I0 vy =o =LO Dy = ID

y.30

Figure 4: (a) Domain over which equation ( 1 1 ) is numerically solved, ( b ) Collocation-finite element mesh used to solve (11).

- ANALYTIC SOLUTON 0 0

Figure 5:

ALL CUBIC ELEMENTS TWO QUINTIC ELEMENTS

Various s o l u t i o n s t o equation ( 1 1 ) evaluated a t y = 40, t = 2.0. Numerical s o l u t i o n s show nodal values only.

320

M.A. Celia & G.F. Pinder

2.c

E K

w

0.5 I

0.I

Figure 6:

1

.01 TIME STEP

I

.001

Error E of (13) v s . time step for the cases of zero, one, and two quintic elements.

Unification of Finite Element Methods H. Kardestuncer (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

CHAPTER 14 NUMERICAL & BIOLOGICAL SHAPE OPTIMIZATION 4. Philpott & G. Strung

This paper describes a shape optimization code which moves the joints of the structure toward optimal positions and handles the possibilfty of degeneracy in the underlying linear program. The code i s applied to a design problem arising in biology, attempting to analyze the framework of internal fibers in the patella and eventually to predict their pattern of remodeling after fracture. The underlying problems also appear in engineering design applications, involving optimal bounds on the effective properties of composite materials and nonconvex functionals in the calculus of variations. INTRODUCTION We report here on an optimization problem which might have arisen in engineering design, but actually came from biology. It is a classical problem under given constraints, to find a structure of minimum weight. We take the structure to be a plane truss subject to forces along its boundary. The forces will roughly approximate those to which the human patella is actually subjected, and we reduce to two dimensions (plane strain) by considering a vertical cross-section taken from front to back in the knee. Figure 1 indicates the tensile forces on the upper and lower edges of the patella, from the quadriceps and the ligaments, and the equilibrating compressive force from the support provided by the femur.

-

When these forces are concentrated at single points as in Figure 2, Michell [l] found the solution. It is a truss in which the members under compression are orthogonal t o those

32 1

322

A . Pliilpott & G. Strutig

i n tension, as i s always the case i n an optimal design. "he precise geometry depends on the points a t which the forces a c t , but t h e design i t s e l f i s straightforward: t h e compress i o n members a r e d i s t r i b u t e d r a d i a l l y and t h e tension is concentrated on a s i n g l e circumferential member connecting t h e two forces. Certainly t h a t is not an accurate description of t h e fibers i n s i d e t h e p a t e l l a . On t h e other hand it is not absurd; both s t e r e o l o g i c a l d a t a and f i n i t e element analyses [2-41 confirm t h i s general p a t t e r n . Photographs of t h e trabeculae show a framework too s t r i k i n g t o be an accident. Already i n t h e l a s t century c l i n i c a l observations led t o the formulation of Wolff I s law, t h a t t h e f i b e r s a r e i n the d i r e c t i o n s of p r i n c i p a l s t r e s s . A more exact statement of t h a t law, and of the mechanisms thatgovern t h e laying down and resorbing of bone, i s badly needed. It i s a challenge t h a t s e v e r a l orthopaedic research groups have accepted, and we thank W. C. Hayes and h i s colleagues a t Beth I s r a e l Hospital f o r p a t i e n t encouragement t o explore t h e r e l a t i o n s h i p t o optimal design. A f i r s t s t e p (our immediate goal) I s t o r e f i n e Figure 2 by admitting a more r e a l i s t i c s e t of loads. This paper discusses t h e numerical optimization and not t h e biology. Even with t h i s l i l n i t a t i o n i t is incomplete: we could not pursue a l l possible approaches. Some t h a t we d i d pursue were not successful o r r a t h e r , t h e i r c o s t increased so rapidly t h a t a r e a l i s t i c three-dimensional p i c t u r e (with t h e bone allowed t o remodel a f t e r f r a c t u r e ) was inaccessible. Other algorithms a r e more promlslng. Perhaps we can l i s t here some of the a l t e r n a t i v e s , with preliminary comments:

-

1. Fixed Geometrg. Fix t h e nodes of t h e t r u s s and allow a specified subset of edges as candidates i n t h e design optimization. This y i e l d s a standard problem i n l i n e a r programming, with a sparse c o e f f i c i e n t matrix described below. The c o s t i s e f f e c t i v e l y t h e sum C a 1s I of edge lengths multiplied &Ill by member f o r c e s . The.simplex method w i l l y i e l d an optimal (and s t a t i c a l l y determinate) t r u s s .

Numerical & Biological Shape Optimizatiotz

323

It was hoped that the simplicity of t h i s model would allow a comparatively large number of nodes and edges, g i v i n g the design enough degrees of freedom t o imitate curved f i b e r s . We now believe t h a t t h i s approach i s not e f f i c i e n t . The approximation of a nonlinear design problem by l i n e a r programming can be very slowly convergent. In our experience the l i n e a r programs themselves were a l s o highly degenerate, w i t h multiple pivot steps before the cost was reduced.

Variable Geometry. The positions of the nodes a r e optimized a s well as the weight of the t r u s s t h a t connects them. Normally these steps are separate; a l i n e a r program computes the t r u s s f o r fixed nodes, and a nonlinear calculat i o n moves the nodes. O f the codes we know about (and there must be many othera) we mention three: 2.

( I ) The CONMIN code developed by (3. Vanderplaats [ S ] : This was generously made available t o us and was the basis f o r the conference paper by Snyder e t a 1 [ 6 ] on trabecular orientation i n the p a t e l l a . The code admits a much wider range of problem i n 2 and 3 dimensions, w i t h constraints on displacements and buckling s t i f f n e s s and frequency response. (11) The Michell structure code developed by McConnel [7]: This uses a d i r e c t search method of Powell t o optimize the nodal positions (with fixed node-edge incidence matrix)

.

Experimental r e s u l t s on problems of bridge design studied by Hemp [see 81 look very s a t i s f a c t o r y and we hope t o t e s t the code on biological structures. (iii) A newly developed code:

We are conpletlng a s p e c i a l purpose code t o t e s t other algorithms f o r this c l a s s of optimization problems. This paper describes the use of the dual variables from the fixed-node program t o choose the directions of movement of the nodes. It seems c l e a r t h a t t h i s variation of geometry i s valuable and probably e s s e n t i a l .

3. Michell Displacement Field. It was the great discovery of Michell that under the presribed load, the Optimsl design

324

A . Philpott & G. Strang

has p r i n c i p a l s t r a i n s of constant magnitude. ( I n a region where no members are required i n one o r both directions, t h e corresponding s t r a i n s a r e uniformly below t h a t constant value.) Lagache [ 9 ] has found a b e a u t i f u l existence proof f o r Michell t r u s s e s based on a convergent sequence of approximations t o t h i s s t r a i n f i e l d . Although those p a r t i c u l a r they a r e approximations nay be i n e f f i c i e n t i n computations based on fixed nodes, as i n 1. above t h e r e seems t o be no reason why a program t o compute f i n i t e element dlsplacements f o r Michell's s t r a i n f i e l d should not succeed.

-

-

Michell s t r e s s f i e l d . I n the s t r e s s formulation of t h e problem, t h e c o n s t r a i n t d i v cf = 0 can be removed by deriving t h e sOress u from Airy's function t(x,y): 4.

same

noted i n [lo], t h i s y i e l d s an ordinary (but nonquadratlc) problem in t h e calculus of v a r i a t i o n s . Again a combination of f i n i t e elements and nonlinear optimization would give a d i r e c t approximation of the Michell d i r e c t i o n s in t h e optimal design. The d e n s i t i e s and o r i e n t a t i o n s then come d i r e c t l y from 6 r a t h e r than i n d i r e c t l y from a d i s c r e t e t r u s s . As

This paper describes t h e new v a r i a b l e geometry code mentioned above. It was prepared i n l a t e 1983 and t e s t e d on t h e

-

b i o l o g i c a l applications although it applies equally t o engineering design. The next s e c t i o n of t h e paper r e c a l l s t h e s t e p s of the simplex method, modified only t o account f o r the absolute values i n the c o s t function; members can be i n tension or compression. Section 3 applies the simplex method t o c o s t ( o r volume) minimization, f o r a given placement of the nodes. Section 4 develops an algorithm f o r moving t h e nodes toward an improved design. There a r e numerical d i f f i c u l t i e s associated w i t h degeneracy, handled by a proj e c t i o n technique. In t h e l a s t s e c t i o n we b r i e f l y discuss the implementation and the applications.

Niitvierical& Biological Shape Optimization

325

THE FIXED GEOMETRY DESIGN PROBLEM

number of authors,startimg w i t h Dorn, Greenberg, and Gomory [U],have considered the application of l i n e a r programming t o the minimum weight problem f o r a plane t r u s s subjected t o forces along i t s boundary. "he problem is t o s e l e c t a s e t of members connecting a given collection of pin j o i n t s which a r e fixed i n number and position. Each member i n the optimal t r u s s is i n tension or compression a t i t s corresponding l i m i t s t r e s s . A

The formulation as a l i n e a r program is as follows. We consider m pin j o i n t s I n the plane, and l e t XiDyi, i = 1, .,m be the coordinates (with respect t o some o r i g i n ) of the l t h j o i n t . We l e t [ i , k ] denote the member connecting j o i n t s 1 and k . To avoid unnecessary subscripts we associa t e with each [ i , k ] a unique index j E (1,2,. .,n), and define s as the force i n member j . Here sj is positive j i f j is in tension and negative i f j is i n compression. j in "he number of members n can be a t most - , practice we r e s t r i c t our choice of members t o a r e l a t i v e l y small set, usually those having length below some chosen bound.

--

must be such t h a t each j o i n t is in s t a t i c The forces "j equilibrium. These 2 m equality constraints may be formulated by resolving each member force in the two coordinate directions and summing the forces i n each direction a t each node. The right hand sides of these constrainta are the x and y components of the applied forces. For j o i n t I, these a r e denoted by fli and fgi respectively. For many j o i n t s , including those i n t e r n a l t o the structure, the applied forces are zero. If member j = [ i , k ] makes en angle 0 w i t h the positive j x-axis a t j o i n t i, we may write the constraints f o r node i as

A . Plrilpotr & G. Strarig

326

where Ii is the set of members connected to the ith joint. Since each member connects exactly two joints, these equilibrium constraints in matrix form become

where 8 is the vector of n member forces, fl and f2 are the x and y components of the boundary forces, and A1 and A2 are m X n matrices with zero entries in column j = [ i , k ] except in row8 i and k where

-

(A21ij = -sin

6

3 ’

(A21kj = s i n 0j

I;:[

-

In what fOllOW8 we shall refer to the matrix and the right hand side vector

as

f.

A n example of a pinjolnted trusla and its constraint matrix

is given in Figure 3.

If

aj

is the length of member j, minimizing the volume of a planar truss is equivalent to minimizing

n

over those vectors 8 which satisfy the equilibrlun condition As = f. The derivation of this result requires that the l i m i t stresses in tension (a=) be the same for

327

Nu 17 i er icul & Biological Sliap e Opt irri izat io n

each member, and t h a t the l i m i t s t r e s s e s under compression (udn) be the same f o r each member. Since W is independent of the values of uand %in , we may assume without 108s of generality that uudn = u. When each member j i s f u l l y stressed i t s cross-sectional area is proportional t o

-

9:

t!i

= u

aJ

, for

j

i n tension,

for

j

i n compression.

= -6,

1s I measures the thickness required in member 3 j j the problem of minimizing the weight of a fixed j o i n t planar t r u s s may thus be formulated as

In this way

n Problem P:

subject t o As = f

C djls,l j=l

Minimize

.

(1)

As posed, t h i s objective function is nonlinear. However it is easy t o reformulate P as a l i n e a r program by defining

variables

whence W become

C 1 t

J j A(t

+

Z

-

1u

U)

j j =

is l i n e a r and the constraints

f, t

2

0,

U

2

0

.

We s h a l l write the problem in the formulation P, but make considerable use of the l i n e a r programming formulation and the large body of theory which t h i s allows us. W e can formulate the problem dual t o D: Maximize

vTf l

+

P

as follows:

wTf2 subject t o I(vTA,+wTA2),I j =

1 , e . m

an

5 dj, (21

A. Philpott & G. Strang

328

Multiplying both s i d e s by I S il duclng f l = A1s and f 2 = A2s easy half of d u a l i t y theory.

I j

on jj and i n t r 0 leads t o weak d u a l i t y the

SUmming

-

Theorem 1: If v and w a r e f e a s i b l e f o r t h e dual and s i s f e a s i b l e f o r the primal then n T T (3) v f l + w f2 5 c i j l S j l

J-1

The complementary slackness conditions, which must hold a t the minimizing 6 and maximizing v and w, a r e as follows : if

sj

>

0, then

if

8

<

0, then

j

( V TA

1

+ wTA p ) j

aj (5)

The dual problem i s o f t e n given the following physical l n t e r pretation, with E a s Young's modulus. A s e t of dual variables v i J w i J corresponds t o a v i r t u a l deformation of the t r u s s I n which each j o i n t 1 i s displaced by v i ( i ) i n the y d i r e c t i o n . The elongation e j of each member j under such a deformation i s given by

The dual c o n s t r a i n t s t h e r e f o r e require of each elongation e j that

I n other words t h e l i n e a r s t r a i n i n each member caused by t h e v i r t u a l deformation must be a t most t h e value which corresponds t o the member being a t i t s limit s t r e s s . The dual objective function vTf l

+

wTf 2 can be i n t e r p r e t e d

as E/a times t h e work done by e x t e r n a l f o r c e s t o produce t h i s v i r t u a l deformation. Since t h e optimal s o l u t i o n s w i l l s a t i s f y (4) and ( 5 ) J t h e s t r a i n in a member with sj o produced by t h e optimal displacements v and w must be

+

Numerical & Biological Shape Optimization

329

the strain which would occur with j at its limit stress (a if sj > 0 and - u if sj < 0). Members with s = 0 J and they are below do not appear in the optimal structure the limit stress when subjected to the optimal v and w.

-

AN ALUORITHM FOR THE FIXED GEOMETRY PROBUM

Since the fixed geometry design problem P may be formulated as a linear program, the simplex algorithm will solve it. This method constructs a sequence of basic feasible solutions with decreasing weight C l? I s I . The optimum does occur at J 3 a basic feasible solution (defined below) and the sequence terminates at an optimal solution after a finite number of steps. For a planar truss with n joints we have 2m equilibrium constraints. However the rows of A are not linearly independent, and for equilibrium the boundary loads must sum to zero in both coordinate directions. Furthermore the sum of moments about any point in the truss must be zero. The number of independent equilibrium constraints is therefore 2m-3, and three constraints must be removed before the simplex algorithm can be applied.

In practice these degrees of freedom are removed by fixing two or more boundary Joints. Then we remove from A1 and the rows which correspond to the anchored joints, and we remove from the vectors v and w the displacements which are no longer possible. We shall assume in what follows that the constraint matrix A has undergone this process, eqaal to the and take m to be the number of rows of A number of displacement unknowns. This removes at the 8ame t h e any restrictions on the boundary loads; they need not be self-equilibrating since any additional forces necessary for equilibrium will be taken up by the foundation. A2

-

The basic fersible solutions for problem P are found by choosing 2m linearly independent columns of A and solving (1) uniquely for the member forces sj corresponding

A. Philpott & G. Strang

330

t o these columns. A l l other member forces a r e zero. Thus each basic f e a s i b l e s o l u t i o n corresponds t o a e t a t i c a l l y e s h a l l c a l l any such truss a basic deteranate truss. W f e a s i b l e t r u s s . S t r i c t l y speaking, t h e basic f e a s i b l e s o l u t i o n t o t h e l i n e a r program is given by t h e nonnegative variables s a t i s f y i n g s = t u :

-

and With this understanding we apply t h e term "basic f e a s i b l e solution" t o the member forces 8 and also t o t h e variables

t

and u.

If t h e choeen columns of

a r e not l i n e a r l y independent, t h e corresponding truss i s no longer r i g i d under an a r b i t r a r y set of boundary loads, and it becomes a mechanism. On the other hand, i f more than 2m columns a r e chosen (with 2 m l i n e a r l y independent) then t h e corresponding t r u s s i s s t a t i c a l l y Indeterminate, and we cannot solve uniquely f o r the member f o r c e s . A t h i r d p o s s i b i l i t y is t h a t t h e 2m columns a r e l i n e a r l y independent and y e t some of t h e member forces in t h e basic f e a s i b l e truss a r e zero. I n t h i s case t h e basic f e a s i b l e s o l u t i o n i s degenerate. A

The algorithm f o r the f i x e d geometry optimal design follows t h e s t e p s of t h e slmplex method, modified to work w i t h t h e formulation P. If we assume that degenerate basic f e a s i b l e s o l u t i o n s do not occur, t h e s t e p s a r e as follows: 1. Determine a s e t of 2 m members which form a basic B = t h e 2m by 2m f e a s i b l e t r u s s . Denote by

[i;]

"basis matrix" formed by t h e corresponding columns of 2.

A.

Solve Bs = f .

3. For t h e s o l u t i o n S, construct a complementary s l a c k dual s o l u t i o n v#w. I n o t h e r words, solve t h e l i n e a r system (vTB1

+

wTB )

2 j

=

-+ 4j

(choosing t h e s i g n of

Sj)

33 1

Numerical & Biological Shape Optimization

4.

Choose the value of

'J 5. If

2

=

aJ

-

j (say

j = q)

I(vTAl + wTA )

2J

I

which minimizes

.

2 0 then stop3 the current t r u s s is optimal. Q ra < 0 then member q enters the basic f e a s i b l e t r u s s in t'ension o r compression depending on the sign of T T = ( V A1 + w If

r

9

6.

Determine the member t o leave the basic f e a s i b l e truss, given by the Index 1 = p which minimizes l q i l over indices 1 i n I. Here P

si (B'lA

iq

I = ( I ; (B-lA)iq

0

and sign

=

sign zs)

.

7.

Replace member p i n the basic f e a s i b l e truss (and the column p in the matrix B) w i t h member q . Return t o Step 2.

It is easy t o see t h a t t h i s algorithm follows exactly the same steps as the simplex method. Step 4 f i n d s a m i n i m u m reduced cost, and ti i n Step 6 i s the r a t i o t e s t t h a t determines which basic variable becomes non-basic a f t e r a pivot operation. I n t h i s step, i f no quotients of approp r i a t e sign e x i s t then the problem has an unbounded solut ion.

Since the constraint matrix A i s very sparse, the implementation of the algorithm uses an LU f a c t o r i z a t i o n of the basis matrix B and updates the triangular f a c t o r s L and U a t each i t e r a t i o n . We employ the method of Forrest and Tomlin [12]. To take f u r t h e r advantage of sparsity, the columns of A corresponding t o nonbasic variables a r e recomputed a t Step 4 i n each i t e r a t i o n . The steps described above are followed w i t h no e s s e n t i a l changes when degeneracy is encountered. If the s t a r t i n g

332

A. Philpott & G. Strarig

basic feasible t r u s s i s degenerate we a r b i t r a r i l y assign positive or negative signs t o the zero forces. Since by Step 5 every member entering the basis does so e i t h e r under tension or compression, f o r every subsequent basic feasible t r u s s we can assign positive and negative signs t o the (possibly zero) member forces. This convention gives a unique complementary slack dual solution i n Step 3. The choice of member p t o leave the basic feasible t r u s s is however not uniquely determined, since any 1 w i t h si 0 w i l l render l$il a minimum i n Step 6. In the code we choose the variable with lowest index, and although t h i s rule does not preclude cycling, no instances of cycling have been observed. (Dantzig has observed only one i n long experience with simplex calculations. ) Of course a more expensive pivot rule could ensure t h a t cycling never occurs.

-

VARIABLE GEOMETRY

For fixed j o i n t positions the above algorithm gives an optimal solution it finds the minimum weight of a plane t r u s s which supports given boundary loads. Being e s s e n t i a l l y the simplex method, the algorithm terminates a t the optimum a f t e r a f i n i t e number of I t e r a t i o n s . Unfortunately, f o r large n and m (many j o i n t s and many possible members) the time taken t o reach the optimum is prohibitive. If members are permitted between any two j o i n t s then the problem s i z e Increases rapidly with the number of j o i n t s . To avoid t h i s problem we may r e s t r i c t the allowed members t o those whose length l i e s below some fixed bound. However, even a small problem w i t h 100 j o i n t s and 800 members can take hours of CPU time

-

.

One reason f o r t h i s poor convergence behavior i s the high degree of degeneracy inherent i n the l i n e a r programs associated w i t h pin jointed structures. Often a large number of j o i n t s have no external forces, and play no p a r t i n the designs the members incident upon these j o i n t s carry zero force. Even If these j o i n t s and the members connected t o them are deleted,

333

Numerical & Biological Shape Optimization

it is often the case i n a basic f e a s i b l e truss that other zero-force members e x i s t . As observed i n [ l l ] , these cannot be deleted since the resulting truss would no longer be r i g i d . The presence of degenerate basic f e a s i b l e solutions causes poor performance in the simplex algorithm, since many pivot operations occur w i t h no change i n the objective f unc t ion. To approximate accurately the trabecular architecture of the human patella, a large number of i n t e r n a l j o i n t s would be

-

needed i n the design process i f t h e i r positions a r e fixed. This p r o l i f e r a t i o n of j o i n t s subject t o zero loads exacerbates the degeneracy problem, and the optimization becomes Idq)ract i c a b l e . A n a l t e r n a t i v e is t o admit a much smaller number of j o i n t s and then alter t h e i r positions i n such a way a8 t o improve the design. We proceed t o describe a method f o r computing the gradient of the cost W in terms of the j o i n t positions, thus giving a s e t of directions of steepest descent; s h i f t i n g the j o i n t s i n these directions produces a truss w i t h l e s s weight. To f i n d the gradient, we investigate the e f f e c t on the objective function W of a change i n position of the j o i n t s from (Xisyi) t o ( x i + 8Xj.s y i + 8yi). A t optimal solution of problem P# the cost I s n w = jc-1 aj l a3 I .

The sum is taken over members or" the optimal basic f e a s i b l e t r u s s . L e t this truss correspond t o the basis matrix

LB,'] 1 rB,

B =

where

B1

and

B2

are

m x 2 m matrices.

We

assume-for the present that this truss is not degenerate. Changing the positions of the j o i n t s w i l l change the length8 a j and a l s o a f f e c t the e n t r i e s of B implying that the member forces will a l s o change. Suppose that the forces become s j + 8 s j 8 and the changes in angles a l t e r B t o B + bB. Then equation (1) gives

-

A. Philpott & G. Strang

334

(B

+

+

bB)(s

6s) = f ,

whence t o f i r s t order

+

B6s

(6B)s = 0 .

The f i r s t order change i n W = C

a 3 1s3

+

C 0aj

a

8s

J J

given by Sjl

+

2 Jj61Sj(.

sj'o

The last term i n (7) is absent by the assumption of nondegeneracy. It then follows from the definition of the complementary slack dual variables (the displacements v w ) that r

i

L

J

(7)

and

and thus from (6) we obtain

We now express &4? and 6B i n terms of the changes in j o i n t positions, Ox and by. If j = [ i , k ] , then the is (xk + (yk Y , ) ~ . Since length

-

$

2

-

=

COB

ej,

aa 4 --

COB 0

j '

& aa = - s i n e j , 4, Yi

2 = s i n e

it follows from the d e f i n i t i o n of 6a

-

T B16x

T + B26y.

B1

and

Be

that (9)

The matrix 8B is the f i r s t order change i n the matrix B and a r i s e s from the changes i n COB e and sin 8 f o r J 3 each member a f t e r the positions o f some j o i n t s are changed. It is easy t o ehow that if j = [ i , k ] ,

335

Numerical & Biological Shape Optimization

7 -

cos e

yi

s i n ej ,

a

cos eijsin ej

cos dyk e

j

Therefore the changes i n cosine and sine a r e given t o f i r s t order by

The change ( 6 B ) j i n column j of B I s zero except i n rows I , k, i + m, and k + mj i n these four rows we have

w i t h the same formulas and opposite signs f o r ( 6B)k+m, j . The term i n brackets is j u s t

If we l e t Sjj

=

2,

S

be the

then

6W

2m x 2m

diagonal matrix w i t h

I n ( 8 ) can be written as follows:

j

8W

=I4T[ BTl .TI[(.]- 8 x

(FJB),~

[vT wT 1[82] s[B; OB1

We simplify t h i s expression by defining

-B3

E];

and

336

A. Philpott & G. Strang

leaving 6W = gT b x + hT b y .

(13)

For given loads and a given nondegenerate optimal truss, the vector [If] defined by (11) and (12) i s a direction of steepest descent f o r W when expressed as a function of the nodal positions. Therefore it i s natural t o extend the algorithm t o include a s h i f t of positions i n t h i s descent direction. This variable-geometry algorithm proceeds as follows : Step 1. For a given s e t of j o i n t coordinates the fixed geometry problem P

-x,y,-

solve

Step 2.

For the optimal basic feasible solution t o calculate g and h from (11) and (12)

Step 3.

For the given boundary loads, find the value of the stepsize 1 which minimizes W(z Xg,Y Xh)

Step 4.

If the change I n W i s l e s s than some tolerance and then terminate; otherwise update 'ji and return t o S t e p 1.

-

Pa

-

The c r u c i a l step is the l i n e search i n Step 31 It is here t h a t the Improvement In geometry I s made. There are a number of ways t o carry out t h i s search. One p o s s i b i l i t y I s t o r e t a i n the current basic feasible truss and recalculate W f o r this truss a t each point i n the l i n e search (determining t h e member forces from equilibrium Ba = f ) . Alternatively, we may return t o the simplex algorithm and find the optimal basic feasible truss a t each point of the search. We tend t o favor the former approach) re-solving the l i n e a r

331

Numerical & Biological Shape Optimization

programming problem a t each point requires a greater comput a t i o n a l e f f o r t , a s i t u a t i o n which is not improved by the presence of degeneracy. One might hope t h a t t h i s algorithm gives a reasonably robust method of improving the geometry. Unfortunately t h i s is not the case.

In practice, l i n e searches often terminate a t values of A f o r which the resulting basic f e a s i b l e t r u s s is degenerate. This seem reasonable since the cross-sectional area of the degenerate member has a l o c a l minimum exactly a t the value of X which renders it degenerate. This contradicts our assumpt i o n ofnondegeneratebasic f e a s i b l e t r u s s e s j even i f we begin the l i n e search a t a well-behaved truss, we a r e l i k e l y t o meet degeneracy. possible remedy is t o continue as i f our assumption were valid, and hope that the sum X 1 61s I over members with J J sj = 0 (which was neglected i n ( 7 ) ) is small. However it soon becomes obvious t h a t this approach is untenable. In practice, l i n e searches from current degenerate basic f e a s i b l e trusses often terminate a t a steplength A which is zero t o working accuracy, and the algorithm is j m e d a t a suboptimal solution. Clearly a t some point, t h i s neglected fourth term i n (7) has a considerable e f f e c t . A

The solution i s t o compute a more e f f e c t i v e search direction. Since the undesirable behavior ie caused by changes i n 1s I J f o r degenerate members, It makes sense t o seek a direction which minimizes these changes. We therefore search in a direction which, a t l e a s t locally, keeps degenerate members degenerate. Then one expects t h e i r contribution t o 8W t o be small. The adjusted search direction is computed as follows. we have 8s = -B-l(8B)s

,

From (6)

A. Philpott & G. Strang

338

whence, using (lo),

Let Id be the set of indices of columns of B corresponding to degenerate members, and let Bd be the matrix formed by the rows of Bol with these indices. Then if

It;] -

D = -Bd[:il]

S [BE -Bl], T any vector

_

I

in the nullspace of

D will render (6s) = 0 for j E Id. Thus to obtain a descent direction which keeps the changes in degenerate members as smell as possible, we project onto the nullspace of D :

[E]

[f] =

[g] - DT(DDT)-l

D[E]

.

Since DDT is often singular, we use a Gram-Schmidt procedure t o construct a matrix Q with orthonormal columns spanning the column space of DT, whence

Even with this projected search direction, we are not guaranteed a decrease in W(%XE,T-Ali) for large step sizes A. Indeed, if a step is so large that some member forces change sign, the current choice of direction may no longer give descent. For this reason, we compute from (14) the values of X for which a member force changes sign, and evaluate the objection function W at these values in succession. (The equilibrium equation is solved for 8.) When an increase in W is detected, this gives an interval in X over which a golden section line search minimizes W. C ONC LUS ION

The algorithm described above was implemented in FORTRAN on a VAX 780, and applied to a model of the human patella. The loads correspond to a 60’ flexion of the leg (as in

Numerical & Biological Shape Optimization

339

climbing s t a i r s ) . We worked with 40 j o i n t s , and allowed a l l edges w i t h length not exceeding one centimeter. An i n i t i a l basic feasible t r u s s i s shown i n Figure 4a. The 14 fixed nodes along the bottom provide reactions t o the 5 applied loads. Solid l i n e s denote members i n tension and dashed l i n e s indicate compression. The thickness of the members i s represented by the number of l i n e s i n the diagram, and bars denoted by single l i n e s i n Figure 4a are degenerate. They have zero thickness, as i n the two members which meet a t the unloaded node a t the top right hand side; they a r e present only t o avoid a mechanism. The weight of the i n i t i a l t r u s s i s W = 2.064 grams. Figure 4b shows the output from the fixed geometry design algorithm, s t a r t i n g from the truss of Figure 4a. Degenerate members ,are now omitted. The optimal t r u s s has a weight of W = 1.829 gram and was obtained a f t e r 25 pivot operations i n the simplex method. For a problem of t h i s size, each pivot step takes approximately .5 seconds, a value which increases substantially f o r l a r g e r problems. Figure 4c shows the output from the variable geometry algorithm. It i s the o p t i m l basic f e a s i b l e t r u s s a f t e r a succession of changes i n nodal positions. The weight i s reduced t o W = 1.708 gram a f t e r t e n l i n e searches, a f t e r each of which the fixed geometry problem was resolved and new directions of movement were computed (with projection t o control degeneracy). Subsequent l i n e searches f a i l e d t o give a significant impuovement i n the weight of the trusa and the program terminated.

ACKNOWLEDGEMENT We thank the Army Research Office and the National Science Foundation for their support under contracts DAAG 29-83K-0025 and 81-02371. We are also grateful to the National Institute for Health for the opportunity, under contract AM30875, to work in the Orthopaedics Research Laboratory at Beth Israel Hospital.

3 40

A . Plzilpott & G. Strang

REFERENCES [l] Mlchell, A.G.M., The limit of economy of material in frame structures, Phil. Mag-8 (1904) 589-597. [2] Hayes, W.C., Snyder, B., Levine, B.M. and Ramaswamy, S., Stress-morphology relationships in trabecular bone of the patella,in: R. H. Gallagher et al., eds., Finite Elements in Biomechanics ( J o h n Wiley, New York, 1983). [3] Huberti, H.H., Hayes, W.C., Stone, J . L . and Shybut, G.T., Force ratios In the quadriceps tendon and ligamentum patellae, J. Biomechanics, to appear. [ 41

Stone, J . L . , Three-dimensional stress-morphology relationships In trabecular bone, M.Sc. Thesis, M. I . T . (1983).

, Numerical Optimization Techniques for Engineering Design With Applications (McGraw-Hill, New York, 1984).

[ 5 1 Vanderplaats, G.

-

[61 Snyder, B., Strang, Go, Hayes, W.C. and Norris, G., Application of structural geometry optimiration techniques in the microstructural remodeling of trabecular bone, ASME Annual Meeting, Boston, 1983.

[TI MCCOMel, ROE., Least-weight frameworks for load across Spar

ASCE J.

H e w W.5.r 19731 191

m.MeCh.

DIv. 100 (1974) 885-901.

O;ptlmU Structures (Clarendon Press, Oxford,

Lagache, J.-M., Treillis do volume minimal dans une region donnh!, J. Mdcanique 20 (1981) 415-448.

[lo] Strang, G. and Kohn, R.V., Hencky-Prandtl nets and constrained Michell trusses, Comp. Methods in Appl. Mech. E W . 36 (1983) 207-222. 1111 Dorn, W.S., Gomory, R.E. and Greenb,erg, H . J . , Automatic design of optimal structures, J. Mecanique 3 (1964) 25-52.

1121 Forrest, J . J . H . and Tomlin, J.A., Updated triangular factors of the basis to maintain sparsit in the product form simplex method, Math. Prog. 2 (19727 263-278.

Numerical & Biological Shape Optimization

FIGURE 1. HUMAN PATELLA: ROTATED CROSS SECTION, WITH APPLIED TENSIONS AND SUPPORT REGION.

F' FIGURE 2. MICHELL TRUSS, 3-POINT LOAD.

341

A . Philpott & G. Strang

342

3

FIGURE 3. A SIMPLE PINJOINTED TRUSS.

scale:

1 cm.

FIGURE 4A. INITIAL TRUSS.

Numerical & Biological Shape Optimization

F I G U R E 4 B . OPTIMAL TRUSS WITH F I X E D NODES.

FIGURE 4C.

OPTIMAL TRUSS AFTER 10 GEOMETRY CHANGES.

3 43

This Page Intentionally Left Blank

345

INDEX

Axisymmetric t e n s i o n , 249, 260, 263 - 265

Cracks, elasto-dynamic problem, 77-79 e l l i p t i c a l , 66-75

BCP element, 243-244

propagation, 66, 75-86

Benard i n s t a b i l i t y phenomenon,

s u r f a c e , 66-67

2, 32 B e t t i ' s law, 112

Cross t r i a n g l e s f o r i n c o m p r e s s i b l e media, 235, 246

B i f u r c a t i o n , 250, 257, 259-265 Boundary element method, a p p l i c a t i o n s o f , 70, 86-94, 185-206 i n t e r a c t i v e computer g r a p h i c s f o r , 47-63

D'A1 embert I s hypothesis, 187 Data o r g a n i z a t i o n 50, 57 Data t r a n s m i s s i o n , 5 1 D i r a c d e l t a f u n c t i o n , 195 D i r i c h l e t problem, 152, 312

Boundary v a l u e problem, 167-184

D i s c r e t e t r a n s f i n i t e mapping, 53

Boussinesq approximation, 33

Displacement f u n c t i o n , 110-112,

114,

118 Command languages, 50

Dynamic f r a c t u r e toughness, 77, 81-82

Col 1oca t ion ,

Dynamic p h o t o e l a s t i c methods, 77

a1 t e r n a t i n g d i r e c t i o n , 303-320

,

Condensation, 307

E igen-function

Conjugate g r a d i e n t method, 167-

E l e c t r o m a g n e t i c 2-D problems, 160-165

171, 173-175 Conservation o f mass, 1, 7 C o n s t i t u t i v e r e l a t i o n s , 10, 254259 Contact problems ,

79-82

Element s t i f f n e s s m a t r i c e s , 213-216 e v a l u a t i o n by experiment, 214 Energy Y compl ementa ry, 191- 193 convergence t o l e r a n c e , 131-132

l a r g e deformations, 123-147

f r a c t u r e , 84

s o l u t i o n procedure, 126-132

k i n e t i c , 84

statement o f , 124-126

norm, 102-103

C o n t i n u i t y c o n d i t i o n , 6-7

p o t e n t i a l , 190, 192

C o o r d i n a t e f u n c t i o n s , 279

s t r a i n , 84, 189

g e n e r a t i o n o f , 273

Error ,

computation o f , 280

d i s c r e t i z a t i o n , 182-183

s e l e c t i o n o f , 281

estimates, 210, 213

Coupling o f methods, 86-94

f i n i t e element method, 102-105

346

F i n i t e d i f f e r e n c e method, f o r a l t e r n a t i n g d i r e c t i o n collocation procedure, 303 i n t e r a c t i v e computer graph c s f o r , 47-63 F i n i t e element method, a p p l i c a t i o n s o f , 66, 86-94 123147, 170, 196-202 coupling o f , 92-94 flow problems, 15-31 h-p methods, 101, 102 h-version method of extension, 97-98, 101, 104-105, 108, 118-119 h-p version method of extension, 97-98, 103 i n t e r a c t i ve computer graphics f o r , 47-63 Poincare problem, 169, 177-182 Fluid flow problems, c o n s t i t u t i v e r e l a t i o n s , 10 Eulerian approach, 2, 13, 21, 29 heat flow, 1, 13-15 Lagrangian approach, 29 natural approach, 1-15 penalty approach, 27, 34 streamline upwind/ PetrovGalerkin formulation, 1, 2526, 28, 32 thermal, 2 , 12-15, 32 Fracture, cup and cone type, 249 mechanics, 66, 75-86 Friction, c o e f f i c i e n t o f , 125 Coulomb's law, 125 s l i d i n g , 127-128 s t i c k i n g , 128-130

Index

Galerkin equation, 237 Gauss-Legendre formula , 159, 306 Geometric coherence, 50 Gram-Schmi d t procedure, 338 Hankel function, 150 Heterodyne hologram interferometry, 225-226 H i l l ' s theory, 260 Hologram interferometry, 216-220 Hologram i n t e r p r e t a t i o n , 220-225 Hooke's law, 255-256 Hybrid and two stage techniques, 289-290 f u t u r e d i r e c t i o n s , 290 p o t e n t i a l s o f , 289 Hybrid methods, a n a l y t i c a l -numerical , 65-67 , 73 experimental-numerical , 65-66, 75-86 modelling of , 194-195 numerical , 65-66, 86-94 Hybrid perturbation technique, 277281, 289 appl i c a t i o n o f , 284-287 I n i t i a l graphics exchange s p e c i f i cation (IGES) , 51-57 Interactive-adaptive a n a l y s i s , 48, 57-59 I n t e r a c t i v e computer graphics, a n a l y s i s models, 49, 53 application t o computer aided design, 47-50 development i n hardware/software , 48 mesh generators, 50, 53-54 substructuring, 50, 58-59

341

Index u n i f y i n g i n f l u e n c e s , 48-52 I n t e r f a c e problems, 149-165

Newton method, 126 M i c h e l l t r u s s e s , 324

I s o c h o r i c c o n d i t i o n , 16, 26-28 Navier-Lam6 equation, 93, 100, 110 52 c o r n e r theory, 256-259, 263

Navier-Stokes equation, 23, 26

Jaumann d e r i v a t i v e , 253, 256

Necking, 249-273 Networking , 48

K e l v i n s o l u t i o n , 195

Neumann problem, 173

Kol osov-Mu s k he1 is hv li formulae,

Newmark beta method , 8 1

100

NRC macroelement, 237-239

i l l - d i s p o s e d pressure, 239 LBB c o n d i t i o n , 236, 241, 243, 245

241

Numerical method,

Lagrange mu1 t i p 1 i e r s , 65, 127, 238, 241, 242

c o n j u g a t e g r a d i e n t method ( C G ) , 167-171, 173-175 implementation o f , 158-160

Lasers, 207-232 experimentation,

i n Stokes-flow,

217, 228-231

u n i f i c a t i o n w i t h FEM, 210, 213

necking i n s t a b i l i t i e s ,

249-273

nonsymmetric problems, 168

methods, 209 computer i n t e r p r e t a t i o n o f images, 231-232 L i n e a r programming, 322-323,

Optimi z a t i o n , b i o l o g y , 321 shape, 321, 343

325, 327, 336-337 L o f t i n g , 53

p - v e r s i o n method o f e x t e n s i o n , 9798, 101, 103-108, 117-119

Matrix, c o n d u c t i v i t y , 30 c o n v e c t i v i t y , 25 c o e f f i c i e n t , 29, 307, 309 d i s t r i b u t i o n , 203 f l e x i b i l i t y , 195, 201-202

Para1 1e l a n a l y s i s , 59 Plane s t r a i n , 249-250,

260, 263,

266-267 Plasticity , a n a l y s i s , 252 t h e o r y , 250, 255-267

h y d r o s t a t i c element, 22

P o i s s o n ' s r a t i o , 69

mass, 21, 24, 182

Polynomials,

s t i f f n e s s , 180, 195, 201

Hermite c u b i c , 304, 307, 309

v i s c o s i t y , 22, 26, 28

Legendre, 306

M a x w e l l ' s equation, 160 Motion, equations o f , 126

q u i n t i c , 307 -31 1 Postprocessing, 48-49,

55-57, 97-121

P o t e n t i a l f u n c t i o n , 68-69

3 48

Preprocessing, 48-50, 52-55 Pressure f i e l d , 3-4, 16, 22, 28 Rayleigh number, 33 Riesz-Schauder equation, 156 Schwarz-Newan a l t e r n a t i n g method, 66-75 Self-adaptive mesh algorithms, 50, 58 Sequential a n a l y s i s , 59 Shear , band, 264, 266-267 modulus, 69 Simplex method, 322, 333, 339 S i ngu 1 a r i t i es , inverse-square-root, 66 logarithmic, 151, 159 t e n s i l e , 249 Sol i d s , Mises type, 251-254, 258, 263267 Somerfield r a d i a t i o n condition, 204 Speckle metrology, 227-230 Steady flows of memory f l u i d s , 236 Strain, el a s t i c- pl a s t i c pro b l ems , 250 energy, 84, 189 hardening , 250 Stress , Cauchy, 9, 252-256, 258, 259 computation of , 105-110 d e v i a t o r , 255 f l u i d flow, 8-10 i n t e n s i t y f a c t o r s , 66, 7 1 , 757 7 , 80-84 i n t e r p o l a t i o n technique, 97-98 J- i nteg r a l t e c h n i qu e , 97-98

Index

Kirchoff, 252-253, 255, 257, 259 Mises e f f e c t i v e , 255 r e s i d u a l , 71 tensors, 252 Superconvergence, 118 System models, 49 Tensor , Cauchy s t r e s s , 252 metric, 252 unsymmetrical nominal s t r e s s , 252 Translating s i ngulari ty-el ement method, 79 Transverse e l e c t r i c f i e l d , 161 Transverse magnetic f i e l d , 161 T r e f f t z ' s formulation, 68 Two- s t a t e d 1rec t v a r i a t i o n technique, 281 comments on, 283 hybrid, 289 Variational met hods , 27 5-302 coup1 ing, 89-92 f u t u r e d i r e c t i o n s o f , 290 hybrid perturbation of , 277 i n t e r f a c e problems , 149-165 two-stage d i r e c t techniques, 281-284 Vector , base, 252-253 Velocity f i e l d , 4-6, 18 Wolff's law, 322 Workstations, 48