Nonlinear processes in engineering

Nonlinear Processes in kngineering Dynamic Programming Invariant Imbedding Quasilinearization Finite Elements System Identification Optimization

Ikpnrlmcn ts of (’ivil Ilnginccring and Arrliitcrturc University of (hlifornia, Ikrkclcy

ACADEMIC PRESS New York and London A Subsidiary of Harcourt Bracc Jovanovich, I’tiblishcrs

1974

COPYRIGHT 6 1974,BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY A N Y MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New Ymk, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NWl

Library of Congress Cataloging in Publication Data Distifano, Nistor. Nonlinear processes in engineering. (Mathematics in science and engineering, v. 110) Includes bibliographies. 2. Nonlinear theories. 1. Engineering mathematics. I. Title. 11. Series. TA33 1 .D57 620.1'001'5 15 73-7437 ISBN 0-12-218050-X AMS (MOS)1970 Subject Classifications: 731(99,77F99,73K25,65D15

PRINTED IN THE UNITED STATES OF

AMERICA

Contents

Preface

xv

Acknowledgments

xvii

Chapter 1. Invariant Imbedding 1. Introduction 2. An Elastic String 3. Classical Imbedding 4. Representation of the Solution 5. An Analytical Example 6. Variable Coefficients 7. Numerical Aspects, Stability 8. Invariant Imbedding 9. Discussion 10. Semigroup Properties 11. Reciprocity Relation 12. Computation of u l . A Two-Sweep Method vii

1 2 3 4 4

6 8 9 12 12 16 16

...

CONTENTS

Vlll

13. A One-Sweep Method 14. A Continuous Analog 15. An Infinite String on Elastic Foundation

16. Buckling. Classical Approach 17. Buckling. Invariant Imbedding

NOTES, COMMENTS, A N D BIBLIOGRAPHY

17 18 20 21 24 26

Chapter 2. Differential Equations I . Introduction 2. Vector-Matrix Notation 3. Systems of Ordinary Differential Equations 4. Linear Homogeneous Systems. Superposition 5 . Linear lnhomogeneous System 6. Variation of Parameters 7. Constant Coefficients 8. Higher-Order Equations 9. Discussion 10. Physical Systems. State Variables 11. Initial-Value Systems 12. Numerical Solutions 13. Stability of the Numerical Process 14. Two-Point-Value Problems 15. Linear Two-Point-Value Problems NOTES, COMMENTS, AND BIBLIOGRAPHY

29 29 32 34 35 36 37 37 38 39 40 41 42 43 44 45

Chapter 3. Beamlike Structures 1 . Introduction 2 . The Elementary Beam

3. Higher-Dimensional Mode& Cylindrical Shells 4. Semigroup Properties of Generalized Beams 5 . Invariant Imbedding. Perturbation Analysis 6 . Matrix Riccati Transformations 7. Validation of the Results 8. Discussion 9. A Two-Sweep Computational Method 10. Invariant Imbedding. A One-Sweep Method 1 1. Self-Adjoint Systems 12. Numerical Stability 13. General Two-Point Boundary Conditions 14. Difference Equations 15. Transfer Matrices

48 49 50 54 56 58 61 62 63 63 65 67 70 73 74

ix

CONTENTS

16. The Scattering Matrix 17. The Flexibility Matrix NOTES, COMMENTS, A N D BIBLIOGRAPHY

Chapter 4.

Partial Differential Equations I

1 . Introduction 2. Poisson’s Equation 3. Finite Elements 4. Difference Equations 5 . Solution of the Equations 6. Discussion 7. Semidiscretization. The Elastic Plate 8. A Stiffness Matrix 9. Prescribed Displacements at Opposite Edges 10. A Flexibility Matrix 11. Numerical Example 12. Buckling Problems. Critical Lengths NOTES, COMMENTS, A N D BIBLIOCRAI’HY

Chupter 5.

84 85 86 88 90 92 94 97 99

100 101

102 103

Partial Differential Equations I 1

1 . Introduction 2. Formulation of the Problem 3. Initial-Value Formulation 4. Reciprocity Relations 5 . Integral Riccati Transformations 6. Computational Aspects 7. Influence of Poisson’s Ratio 8. Invariant Imbedding Solution NOTES, COMMENTS, A N D BIBLIOGRAPHY

Chapter 6.

75 78 79

106 107 109 113 114 1 I6 117 119 121

Dynamic Programming

1 . Introduction 2 . A Linear String 3. The Principle of Optimality 4. An Analytical Solution 5 . Constraints 6. A Nonlinear String 7. A Routing Problem 8. A Continuous Analog 9. Euler Equations

123 124 125 126 128 128 130 133 135

CONTENTS

X

10. 11. 12. 13. 14. 15.

16. 17. 18. 19. 20. 21. 22. 23. 24.

25.

Convexity Numerical Aspects Discussion Partial Differential Equations The Biharmonic Equation Theorem of Castigliano Euler Equations Natural Boundary Conditions Solution of (14.17). Betti’s Theorem A Statically Indeterminate Structure The Theorem of Complementary Energy Linearity An Identification Problem The Error Measure A Numerical Example Discussion NOTES, COMMENTS, AND BIBLIOGRAPHY

137 139 141 142 144 146 147 148 148 149 152 152 155 156 157 158 159

Chapter 7. Quasilinearization 1. Introduction 2. Approximation in Policy Space 3. Quasilinearization 4. A Nonlinear Fredholm Operator 5. Successive Approximations-Monotonicity 6. Quadratic Convergence 7. Green’s Function 8. Beam on Nonlinear Foundation 9. Numerical Example 10. Lower and Upper Bounds 11. Nonlinear Systems 12. The Linear Associated Problem 13. Computational Aspects. Storage 14. An Alternative Quasilinearization Procedure 15. A Matrix Riccati Equation 16. A Parametric Design Problem NOTES, COMMENTS, AND BIBLIOGRAPY

161 162 164 166 169 170 171 173 177 179 181 183 184 185 185 187 189

Chapter 8. Elements of Optimal Structural Design 1. 2. 3. 4.

Introduction Objective Functions and Constraints Elastic Trusses Sufficiency. Prager’s Device

192 193 193 195

xi

CONTENTS

5. Discussion 6. Dual Loading 7. Kuhn-Tucker Conditions

8. Prescribed Displacements 9. Dual Loading in Statically Determinate Elastic Trusses Subject to Displacement Constraints 10. Beam Optimization 11. Sandwich Beams 12. Discussion 13. Example 14. Piecewise Continuous Stiffness 15. Prescribed Deflections 16. An Example 17. Rotating Disks NOTES, COMMENTS, AND BIBLIOGRAPHY

196 196 197 198 199 201 203 204 205 207 208 210

212

214

Chapter 9. Control Theory Introduction Dynamical Equations Control Variables. Objective Functionals Constraints A Control Problem 6. A Fundamental Equation 7. The Optimality Condition 8. Optimal Loading of a Creeping Bar Axially Stretched 9. Linear Dynamics and Quadratic Criterion 10. Smoothing, Filtering, and Prediction 11. Euler-Lagrange Equations 12. The Minimum Principle 13. Optimal Loading Policy for a Voigt Model 14. Optimal Design as a Control Process 15. Discussion 16. The Necessary Conditions 1. 2. 3. 4. 5.

NOTES, COMMENTS, AND BIBLIOGRAPHY

217 219 22 1 22 1 222 222 224 224 227 229 232 234 236 237 239 239 240

Chapter 10. Optimal Beam Design 1. Introduction

244

I. Dynamic Programming 2. Max-Min Optimization Problem 3. Dynamic Programming

245 246

C 0N TEN TS

xii 4. Constant Curvature 5. Euler-Lagrange Equations

6. Design Constraints. Successive Approximations 7. Numerical Example

249 250 251 253

11. Invariant Imbedding 8. Alternative Formulation 9. An Alternative Derivation of the Optimality Condition 10. Optimum Cantilever under Prescribed Displacement 11. Invariant Imbedding 12. A Two-Sweep Iterative Procedure 13. Inertia Forces 14. Stability Considerations 15. Numerical Examples NOTES, COMMENTS, AND BIBLIOGRAPHY

256 259 260 261 263 265 265 266 273

Chapter I I . Optimal Design of Rotating Disks 1. Introduction

276

I. Minimum Potential Energy 2. The Problem 3. Dynamic Programming 4. Numerical Aspects 5. Design Constraints. Successive Approximations 6. Equilibrium and Compatibility-Euler Equations

277 279 282 282 285

11. Prescribed Displacements 7. Optimization 8. Successive Approximations 9. Numerical Examples 10. Conclusions NO1 ES, COMMENTS, AND BIBLIOGRAPHY

286 290 292 296 296

Chapter I2A. Modeling and Identification of Hereditary Processes 1. 2. 3. 4. 5.

Introduction Modeling Aspects in Viscoelasticity Hereditary Processes Dissipation of Hereditary Effects Time-Invariant Processes

298 299 307 310 311

CONTENTS

6. Aging Processes 7. Bounds and Asymptotic Behavior 8. Asymptotic Stability of Linear Viscoelastic Structures 9. Aging Spring 10. Nonconstant Loading, Bounds 11. Almost Periodic Loading. Time Averages 12. Stochastic Aspects 13. Nonlinear Models. Generalities 14. System Identification 15. Formulation of the Identification Problem 16. Identifiability. Hidden Variables 17. An Alternative Formulation 18. Differential Approximation of Functions 19. Controlled Inputs. Discussion 20. Nonlinear Filtering and Sequential Identification NOTES, COMMENTS, AND BIBLIOGRAPHY

xiii 312 313 316 319 32 1 323 324 325 325 327 328 329 330 332 332 335

Chapter 12B. The Identification Problem in Viscoelasticity 1. Introduction 2. The Identification Problem 3. Quasilinearization 4. Computational Aspects 5. Computation of the Derivatives 6. Computation of the Convolutions 7. Prediction. Inversion of a Nonlinear Volterra Equation 8. Identification of Solid Polyurethane from General o-EData 9. Another Example: Low-Density Polyethylene 10. Discussion NOTES, COMMENTS, AND BIBLIOGRAPHY

Index

339 340 341 342 343 344 345 341 3 52 356 3 58 361

Preface

The advent of the digital computer deeply affected attitudes toward the formulation and solution of mathematical problems arising in engineering applications. Old and new mathematical ideas quickly combined to give rise to a host of powerful and sophisticated postcomputer methods that recognized the intrinsic features of the new computational device. With them, a new methodology for the modeling of physical phenomena started to develop. This bookis intended to be an introduction to a number of methods and techniques for the study of nonlinear processes, developed in this methodological framework. More specifically, it may be considered an introduction to the study of a class of dynamical processes in engineering, using modern notions of imbedding, control, and optimization. Although the presentation concentrates on engineering mechanics applications, many of the ideas and the methods may be easily applied to a broad class of engineering problems. This book is an outgrowth of courses on analytical techniques for design that the author has been teaching at Berkeley during the past five years. The title emphasizes the main theme of the book, i.e., the study of nonlinear processes in engineering applications. Nonlinearities enter into the description of engineering processes in a number of different ways. In some cases we observe the process through variables clearly related by nonlinear xv

xvi

PREFACE

relations. In this fashion we are naturally led to the formulation of nonlinear mathematical models. In other cases we may transform a linear mathematical model of a certain process into a nonlinear one by a suitable change of variables. There are many reasons we may wish to proceed in this direction. Computational advantages might be the principal motivation of these transformations. This is the main thrust in Chapters 1-5 and the reasons for the appearance of matrix and integral (nonlinear) Riccati equations as the pertinent models for the determination of quantities such as flexibilities, stiffnesses, etc., entering in studies of linear structural mechanics. In the remaining chapters of the book a variety of nonlinear processes are formulated and studied. It is not our intenr to examine critically other reasons for nonlinear behaviour. The reader will find in this task, if undertaken on his own, a most exciting and rewarding activity. However, we feel tempted to whisper, en passant, that it is in the areas of the major inverse problems in engineering, e.g., design, control, optimization, system identification, where he will find a most prolific and challenging source of nonlinear problems. We expect this book to be of value to graduate students, researchers, and professors. Many examples, problems, and exercises, presented as an integral part of the text, should facilitate the task of the instructor. A number of research exercises and abundant citing of the literature will hopefully assist graduate students starting research in the topics covered here.

Acknowledgments

I feel particularly indebted to Richard Bellman who inspired this book and to the University of California at Berkeley which, in so many ways, facilitated the completion of this work. Many sincere thanks to Edward Angel, Anil Jain, Estela Llinas, Amitav Rath, Avelino Samartin, Jaime Schujman, and Ricardo Todeschini who read parts of the manuscript and made invaluable suggestions. I am also grateful to Debbie Aoki and Pat Tifft who typed the manuscript in such a good spirit. Finally, I wish to acknowledge the following organizations for permission to reproduce the indicated tables and figures :Akademie-Verlag, Figures 12B.4, 12B.5 and 12B.6 and Tables 12B.3, 12B.4 and 12B.5; American Society of Civil Engineers, Table 4.1; Pergamon Press, Figures 10.2, 10.3, 10.4, 10.5, 12B.1, 12B.2, 12B.3 and Tables 10.2, 10.3, 12B.1 and 12B.2; Plenum Publishing Corporation, Figures 11.1, 11.2, 11.3, 11.4 and Table 11.1; The Franklin Institute, Figure 10.1 and Table 10.1; The Society of Rheology, Inc., for permission t o use data employed to prepare the numerical examples of Chapter 12B.

xvii

Chapter I

Invariant Imbedding

1. Introduction

The determination of configurations of equilibrium in structural mechanics usually leads to the consideration of boundary-value problems. Unfortunately, boundary-value problems cannot generally be solved directly by numerical methods, and therefore the analyst is forced to recast the problem in a manner suitable for handling by a digital computer. This can be accomplished by transforming the original boundary-value problem into an initial-value formulation for which many direct methods are available. This in turn is intimately related to the modeling of the mechanical process itself, for the reduction of boundary-value problems into stable initial-value ones by purely mathematical transformations is generally neither a trivial nor an easy task. The construction of alternative models of the mechanical boundaryvalue problem resulting in stable initial-value formulations is therefore highly desirable if a digital computer will be involved in the solution of the problem. A systematic study of these problems must deal with a number of sophisticated aspects of topology and group theory. In connection with several branches of mathematical physics, the theory of invariant imbedding has 1

2

I

INVARIANT IMBEDDING

recently emerged in an attempt to furnish answers to those fundamental questions. In this chapter, we start a systematic study of invariant imbedding in structural mechanics using a simple structural system to illustrate the application and constantly attempt to emphasize the conceptual rather than the formal aspects of the theory. 2. An Elastic String

To illustrate the basic ideas, we consider throughout this chapter, a simple structure, namely, an elastic string of length L stretched with a force S > 0, fixed at the ends, and resting on a number of equally spaced elastic springs. After deformation takes place, each node will undergo a vertical deflection ui; see Fig. 1-1.

i

i+l

Fig. 1-1

The classical way to formulate a mathematical model of this structure is by consideration of equilibrium and continuity around an internal, generic point i. Let ui be the slope of the string at the left of node i, i.e., (2.1) a quantity assumed to be small with respect to its square. We also assume that the stretching force S does not change with the deformation. Then, equilibrium of vertical forces around node i requires that vi

= (Ui - Ui-JZ,

+

s(ui+l - V i ) = -Pi k iu i , (2.2) where Piis the vertical external load and ki is the spring constant at node i. Since the string is assumed to be fixed at the ends, two additional conditions are provided by specifying the values of uo and uN . We recast Eqs. (2.1) and (2.2) in the form ‘ i = l , 2,..., N , ui = Ui- 1 h i , (2.3) i = 1,2, . , N - 1, u i + , = u i (ki/S)ui- Pi/S,

+

+

..

3

3. CLASSICAL IMBEDDING

and subject the end deflections to the specific conditions ug

=o,

UN

= c.

(2.4)

Equations (2.3) and (2.4) comprise a two-point boundary-value problem involving a system of difference equations. The existence and uniqueness of the solution of these equations can be established easily for all N when S and k, are positive. This in fact could be expected on physical grounds but, fortunately in this simple case, it can be made rigorous with little effort. See the exercises below. EXERCISES 1. Show that u l , i = 1, 2 , . . . , N - 1, given by Eqs. (2.3) and (2.4), satisfies the secondorder differenceequation U I -I

- (2

+ lkr/S)ul+ ui +

=

-IPJS,

i = 1,2, . . . ,N - 1,

(a)

subject to the end conditions uo = 0 and un = c. 2. Prove existence and uniqueness of the solution of system (2.3) and (2.4) for all N > 0 assuming that k I and S are positive. Hint: Use the second-order system of difference equations for ul given in the exercise above and show that the associated matrix possesses an inverse for all N. 3. Derive equations similar to (2.3) assuming that the springs are unequally spaced distances Ir , i = 1,2, . . . , N , and prove the proposition of Exercise 2. 4. Let

and lim ( P , / l )= p(x),

lim ( k I / l )= k(x).

I*O

1-0

Show that the limiting model that can be derived from Eqs. (2.3) and (2.4) is given by the differential system

and therefore u satisfies the second-order differential equation Su" - ku = - p , subject to the end conditions u(0) = 0, u(L)= c.

3. Classical Imbedding

We observe that the space of the solutions of Eqs. (2.3) and (2.4) is the set of all polygonals joining the endpoints uo and u N = c, i.e., all the admissible polygonals. See Fig. 1-2. We could, therefore, state our problem in the following way: Find an admissible polygonal that satisfies the equations

4

1

INVARIANT IMBEDDING

un=0

Fig. 1-2

of equilibrium (2.3). This is a classical idea that implicitly involves the notion of imbedding, i.e., the idea of generating a family of problems containing the original one as a particular member of the family. Boundary-value problems associated with difference-or differential-operators furnishing local conditions of compatibility and equilibrium offer typical examples of classical imbedding. As opposed to this classical concept, invariant imbedding offers an alternative model associated with different functional representations. Before entering into the discussion of these new models, we wish to refresh the memory of the reader in connection with the classical steps in the solution of our two-point boundary-value problems. 4. Representation of the Solution In view of the linearity of the equations, the general solution of (2.3) and (2.4) can be represented by ui = c,u{’) ~i

= C,V:’)

+ ~2 ui2), + vI”, ~2

i = 0,1, 2,

. ..,N ,

i = 1, 2, . . .,N ,

(4-1)

where u ! ’ ) ~u:’) and ui2), v!” are two arbitrary pairs of linearly independent solutions of Eqs. (2.3). Here we have considered the homogeneous case, i.e., P i = 0. Making use, of the boundary conditions (2.4), the constants c1 and c2 appearing in Eqs. (4.1) can be obtained as the solution of an algebraic system. This provides a very general method for the solution of twopoint boundary-value problems. Since this method primarily hinges on the availability of a set of linear independent solutions, we first concentrate on this aspect, and consider next the simplest case, i.e., constant coefficients. The subtleties of the numerical solution using this method are discussed in Section 7 in connection with problems of stability.

5. An Analytical Example Consider the homogeneous system with constant coefficients ui = ui-1

+ lv,,

v i + , = vi + (k/S)ui,

i = l , 2 ,..., N ,

.

i = 1,2, . . ,N - 1,

5

5. AN ANALYTICAL EXAMPLE

subject to the end conditions

=o,

c. (5.2) There is no difficulty in proving, by direct substitution, that Eqs. (5.1) and (5.2) admit solutions of the form uo

UN =

ui = Alai, vi = I , a i , (5.3) where 11,2, are arbitrary constants and c1 is a number, real or complex, that must satisfy the characteristic equation,

d e t l lklS -'

1 la -a

1

=0,

or u2 - (2

+ Ik/S)a + 1 = 0.

(5.4) Since Ik/S > 0, the roots of Eq. (5.4) are real and positive and such that one root is the inverse of the other. We call them r and l / r , respectively. Two corresponding characteristic vectors are given by

Hence, the general solution given by the representation (4.1) can be written

+

ui = clri c2 r - i , ui = cl(l - r-')i-'ri

+ c2(l - r)I-'r-i,

(5.6)

where the constants c1 and c2 must be determined using the boundary conditions (2.4), i.e., c = clrN + c2 r - N 0 = ~1 c2, (5.7) Hence

+

-C -N --N'

C

r

-N

-r"

(I - r - l ) r i - (1 - r ) r b i 0.= c, ( r N- r - N ) l

(5.9) i = 1, 2,. .

N - 1.

6

1 INVARIANT IMBEDDING

Clearly, u i ,u i given by Eqs. (5.9) are the unique solution of Eqs. (5.1) and (5.2). EXERCISES 1. Solve Eqs. (5.1) subject to end conditions u,, = 0, u N = c.

2. Solve the following nonhomogeneous equations u,=Ul-l$ul,

uIfl

= u,

+ ul + i,

u,=o uN = 0.

3. Show that the limiting form of the solution of problem 1 is u(x) =

sinh[(kS/)’/’x] (k/S)’” co~h[(k/S)~’~L] ”

v(x) =

cosh[(k/S)’lZx] co~h[(k/S)~’” ~L]

where L is the length of the string. 4. Show that the solution of the continuous analog of problem 2 is given by u = -x

+ L(sinh xlsinh L).

(4

6. Variable Coefficients The elegant analytical solution obtained in Section 5 owes its existence to our ability to obtain two independent solutions in closed form plus the possibility of explicitly solving the associated algebraic problem to determine c1 and c 2 . None of this will occur in the general case. Either we shall encounter larger and more involved algebraic problems or no explicit set of independent solutions will in general be available. The latter is particularly true when the general case of variable coefficients is considered. It is therefore convenient to abandon any hope of closed-form solution in terms of analytical functions in favor of a different, numerical approach. This can be easily done. Consider, for example, the homogeneous case with variable coefficients : ui = ui-1

ui+l

= vi

+ hi,

+ (ki/S)ui,

i = l , 2 ,..., N, i = 1,2, . . . , N - 1 ,

(6.1)

subject to the nonhomogeneous boundary conditions uo = 0,

U N = c.

(6.2)

It is clear that if instead of boundary conditions (6.2), we prescribe initial conditions, i e . , we give the values of uo and u l , then the sequence ui,ui can be recursively computed using Eqs. (6.1). Hence the determination of

7

EXERCISES

two independent solutions amounts to the solution of two linearly independent initial-value problems. This can be done, for example, by prescribing as initial conditions

MI'),

i.e., two linearly independent vectors. The solutions 0s') and uj2) obtained in this way are called principal solutions. The general solution of (6.1) can then be written as the linear combination, i.e., superposition, of the principal solutions,

+ c2 ui2), vi = c,vl') + c2 vj'),

ui = c,uI')

. .

i = 0,1, . .,N , i = 1,2, . . ,N.

(6.4)

Making use of the boundary conditions (6.2), we can easily determine cI and c2. In fact, combining Eqs. (6.2)-(6.4), we have c1 = 0,

+ c2up,

(6.5)

i = 0, 1, ..., N, i = l , 2 ,...,N.

(6.6)

c =c,up

Hence ui = (ui (2) /uN ( 2 ))c,

ui=(v,

(2,

(2)

/uN)c,

EXERCISES 1. Consider the nonhomogeneous difference equation ut+ 1

- (2 + Iktls)ut + UI+

1

=

+/WI.

(a)

Show that if u:') and u:') satisfy the associated homogeneous equation and the initial

conditions

then the general solution of Eq. (a) subject to boundary conditions u o = u N = O is given by

where A1 and A, are two arbitrary constants and ujp) is a particular solution of (a), subject to the initial conditions u$) = hi and u y ) = & . 2. Solve problem 1 by considering boundary conditions uo=O, u N = 1 and principal solutions u l l ) and u!') subject to the initial conditions @ = 1, u(l)

1

=o,

= 0, u(2)-.

1-11.

8

1 INVARIANT IMBEDDING

7. Numerical Aspects, Stability

The method of superposition of principal solutions presented in the last section exploits the linear structure of the problem and our ability to determine numerically linear independent solutions as the solution of initial-value problems. This is very convenient computationally, but unfortunately the method is severely handicapped by its intrinsic instability. What this entails is that any small error, inevitably introduced because of the limited accuracy of our computer, will grow in an exponential fashion and reach arbitrarily larger values provided the number of steps in the computation is large enough. To illustrate this point we consider the familiar second-order difference equation

ui+1 - (2

+ lk/S)ui + ui-1 = 0,

(7.1)

where IklS is a positive quantity, subject to the boundary conditions 240

=o,

(7.2)

U N = c.

Let us now assume that, by using two linearly independent solutions such as ri and r - i , where r a n d r - l are the two positive solutions of the associated characteristic equation

r z - (2 + Ik/S)r

+ 1 = 0,

(7.3)

we have determined the “missing” initial condition of Eq. (7.1), namely the quantity u1 given by

r - r-l NAr-N r

(7.4)

C.

Once u1 has been determined, our task is to compute u irecursively in Eq. (7.1) by using the given initial value uo = 0 and the previously computed value of u1 given by (7.4). We wish to investigate now the effect of a small error introduced in the computation of ul. To this end we subject (7.1) to the initial conditions r - r-’ uo = 0, 241 = c E, (7.5) r -r-N

+

where E is a small quantity. We leave as an exercise for the reader to show that u isatisfying (7.1) and (7.5) is given by ri - r - i ri - r - i u’. = r * - r - N C+- r - r - 1 E. (7.6) It is clear now that the error term, no matter how small E is, will reach any prescribed value for i large enough. The dominant solution is ri, if r > 1, and

9

8. INVARIANT IMBEDDING

r-i, if r < 1. We shall be concerned with a number of stability problems of this type in subsequent applications. Here they are only mentioned as a valid reason to seek alternative methods of solution. EXERCISES 1. Derive Eqs. (7.4) and (7.6).

+

+ +

2. Assume 2 lk/S = 10.1 in Eq. (7.1). Show that if i > K L 1 and E term [(r' - r-')/(r - r - ' ) ] s in Eq. (7.6) will be larger than loL.

=

the error

8. Invariant Imbedding

In Section 3 we have shown that the notion of solution cannot be separated from the imbedding used in the formulation of the problem. In other words, a particular imbedding automatically defines the space of the solutions. It is therefore understandable that different questions relevant to a given mechanical system should be studied in different solution spaces. This is a very convenient conceptual framework which is the ruison d'ttre of invariant imbedding. A number of meaningful and interesting questions associated with modeIing and perturbations of the mathematical structure of mechanical processes can be systematically studied by exploiting the notion of imbedding. In order to illustrate the basic ideas, we return once more to the example used in previous sections. We consider here a homogeneous string, i.e., with no external loading acting on it, fixed at the left end, uo = 0, and subject it to a vertical deflection, uN = 1, at the right end. Our problem is to determine the vertical component of the string at node N or, in other words, we are interested in the slope v,. Clearly one way to treat this problem is by solving the resulting boundary-value problem along with the classical lines outlined in previous sections. We realize, however, that this procedure would imply the computation of the whole sequence ul, u 2 , ... , u ~ - ~ , ul, u 2 , . .., u,, in order to obtain the desired value of u N . A more direct answer should be obtained if we imbed our problem in an appropriate space. Quite arbitrarily at this point, we consider the family of strings with variable length J I N and subject it to a vertical deflection u, = c at the right end. Assuming the problem to be linear, we consider the slope u, given by the linear homogeneous expression

where Z, is a quantity depending on the length J but not on the deflection u J . We realize that the particular member of this family corresponding to J = N and uN = 1, i.e., Z,, is the desired value of the slope v N .It is worth

10

1

INVARIANT IMBEDDING

noting that formulated in this fashion, 2, turns out to be a “stiffness” in the parlance of structural engineering since it transforms (by multiplication) deflections into forces. In order to find an equation for Z , , we consider a perturbation with respect to the spatial variable J, a technique at the very basis of invariant imbedding. To this end, we consider an extension of one link as indicated = 1. in Fig. 1-3. With no loss in generality, we can assume that J-I

E

J

.t

J+I

Fig. 1-3

We define V- = vertical forces at the left of node J , V , = vertical forces at the right of node J .

Taking into account Eq. (8.1), we have

v+ = SVJ+1= SZJ+1,

(8.2)

V- = S V= ~ SZj U J = SZj( 1 - lZj+i).

(8.3)

and

An equilibrium equation is finally obtained by making the balance of vertical forces V+ = V-

+ kJ(l - fzj+l)

(8.4)

or, substituting V - and V , given by Eqs. (8.2) and (8.3) into Eq. (8.4), ZJ + 1

= (2,

+ k J / W - G+

117

(8.5)

from which we derive

a recurrence equation for the quantity 2, subject to the initial condition 2, = 111.

(8.7)

What is remarkable about Eq. (8.6) is that it is afirst-order nonlinear difference equation subject to the initial condition (8.7) in contrast with the

11

EXERCISES

system of linear difference equations (6.1) subject to two-point boundary condition (6.2). We wish to prove now that the quantities 2, computed recursively with Eqs. (8.6) and (8.7) are stable, i.e., that a small error introduced in step J will not grow larger in subsequent steps. To this end, let be the approximate value of Z , given by

z,

z,=z,

(8.8)

+EJ,

,

,

where E is the error at step J . The quantity Z satisfies the recurrent relation

k,/S + Z, z,+, = 1 + I(k,/S + Z,)’ -

=Z Therefore, since E, using (8.Q we obtain

,

+1

- Z , +1, by subtracting (8.6) from (8.9) and

+ + + +

kj/S Zj 1 /(kj/S Zj

+

EJ+1=

-

EJ

Ej)

k f / S + zJ 1 I(kj/S Zj)’

+

+

(8.10)

i.e., an equation relating the error in two consecutive steps. Assuming that E~ is small, Eq. (8.10) may be conveniently linearized in E, furnishing, after some transformation,

+ ZJ2E, = 1 + l ( k J / s+ Z J )- [I 4kJlS + f ( k J / Si-2,)12’ EJ

Eft1

an approximate equation for the error which in turn may be reduced to the more convenient one

(8.1 1) from which we may readily show that IEJ+ll

5

IEJl,

(8.12)

since the denominator in (8.1 1) is a quantity always greater than one. Clearly, (8.12) establishes the required property. EXERCISES 1. Instead of boundary conditions uo = 0 and uN = 1, consider uo = 0 and uN = 1. Using a procedure similar to that employed in this section, find a recursive equation for Rj in the equation uj=Rj~j. 2. Show that Rj

= 2;

’.

(a)

12

1 INVARIANT IMBEDDING

9. Discussion In the previous section we presented a different approach to the study of the equilibrium of an elastic string. Here the imbedding was quite different to that used in classical mechanics. It is similar to that introduced by R. Bellman in the study of transport processes and radiative transfer, and it is called invariant imbedding. In the following sections we enlarge and refine the notion of this kind of imbedding by considering a number of illustrative applications. Historical and general references are in the Bibliography at the end of the chapter. 10. Semigroup Properties

The solution obtained in Section 8 is a particular version of a more general result. In fact, we can use invariant imbedding ideas to derive semigroup relations that in turn will furnish a number of useful results. Consider, for example, a string of length N where at the ends we prescribe the values of the deflection uN and the initial slope q.Our objective will be, first, the determination of the missing boundary conditions, i.e., the values of vN and uo . If we admit existence and uniqueness of these values, they must be determined as functions of the data values u1 and u N .In formulas this fact can be expressed (10.1) UN =Z ( ~ N u1, , N), U o = T(uN 01 N ) . 9

9

Considering now the portion of the string between i = 0 and i = J , we can write (see Fig. 1-4) uJ

= z(uJ

5

vl,

J),

UO

= T(uJ9

J),

ul,

(10.2)

Fig. 1-4

and, similarly, considering the remaining portion of the string between i =J a n d i = N , uN=Z(UN,

u J + ~N, - J ) ,

U J = T ( U Nu, J + ~N, - J ) .

(10.3)

The intuitive results given by (10.2) and (10.3), derived from invariant imbedding considerations, express some semigroup relations. To see this

13

10. SEMIGROUP PROPERTIES

it is enough to consider first uN given by the first equation of (10.3) and uo given by the second equation of (10.2), i.e.,

and the values of

U,

and

v J + 1 given

(a)

uO=T(uJ?ul,J),

uN=Z(uN,uJ+l,N-J),

by

uJ = R(T(uN VJ + 1 = s(z(uN

9

ul,

N), u l , J ) ,

9

Ul?

N ) , uN J ) ?

(b)

9

obtained from solving for uJ and u J + l in the second equation appearing in (10.2) and the first equation in (10.3), respectively, where u,, and uN were eliminated using (10.1). Clearly, Eqs. (a) and (b) are the desired semigroup relations associated with the original boundary-value problem. In other words, they provide the missing boundary conditions uN and uo in terms of the state variables U, and vr+ of the system at an intermediate point J . We discuss more general semigroup properties in Chapter 3, in connection with nonlinear two-pointvalue systems. Our aim at present is to use (10.1)-(10.3) in the context of linear problems. To this end, if linearity is postulated, Eqs. (10.1)-(10.3) will be linear expressions of the deflections and slopes, the coefficients of proportionality being functions of the length of the string. For example, Eqs. (10.1)-(10.3) will read uN = Z l l ( N ) u N uO

=Z21(N)UN

+ Z12(N)u1 + wl(N),

(10.4)

+ Z22(N)u1 + w2(N),

(10.5) uN = Z11(N - J)uN

uJ = Z 2 1 ( N - J ) u N

+ Z12(N - JkJ+l+ w l ( N - 4, +Z22(N-J)uJ+1 + w2(N - J ) ,

(10.6)

, 2 2 ,w l , and w2 are functions of the string respectively, where Z , , , Z , , , Z Z l Z length but do not depend on the deflections and slopes. We shall use limiting forms of Eqs. (10.5) and (10.6) to derive recurrence relations for the Z and the w functions. Consider, for example, J = N - 1 in Eqs. (10.5). Subtraction of (10.5) from (10.4) yields uN-vN-l

=Zll(N)uN-Zll(N-

l b N - 1 +(Z12(N)-Z12(N-

+ w , ( N ) - w,(N - 11,

= Z21(N)uN

- Z21(N

-

l h N - 1

+ w , ( N ) - w2(N - 1).

+ (Z22(N)

l))ul

- Z22(N -

(10.7)

14

1 INVARIANT IMBEDDING

Equilibrium at node N - 1 requires that ON - v N - l

(10.8)

=(k~V-l/~)uN-l -PN-l/s,

where vN must satisfy the continuity relation UN

= UN-1

+ IvN

(10.9)

Substituting the values of vN - vN-l and uN given by Eqs. (10.8) and (10.9) respectively, into Eqs. (10.7) and collecting terms in UN-1 and ul, the following equations for Z i j, i, j = 1,2, and w 1 and w2 are found : (10.10)

(10.12) Z,,(N) = Zzz(N - 1) - G ( N ) Z l , ( N - 11, w,(N) = (wl(N - 1) 2PN-1/s)(l

w,(N) = wz(N - 1)

(10.13)

- IZ11(N - I)),

(10.14)

+ Z,,(N)w,(N - 1).

(10.15)

These equations must hold for N = 1, 2, . . . , i.e., for strings of various lengths. In particular, the string of length N = 1 yields the obvious initial conditions Zll(1) = 0, Z120) = 1, ZZl(1) = 1, Z Z Z ( 1 ) = 0, w,(l) = W,(l) = 0.

(10.16)

The recurrence relations (10.10)-(10.15) can be made simultaneous by substituting Z,,(N), Z 2 , ( N )appearing in Eqs. (lO.ll), (10.13), and (10.15) with the values given by Eqs. (10.10) and (10.12), respectively, yielding Zll(l) = 0, (10.17) Zlz(l) = 1, (10.18) &(l)

=

1, (10.19)

15

EXERCISES

Wl(1) = 0, (10.21)

a set of simultaneous Riccati difference equations subject to initial values. Similar recurrence relations can be obtained as limiting forms of Eqs. (10.6). See the exercises at the end of the section. It is worth noting that Eq. (10.17) is uncoupled from the rest of the system (10.18)-(10.22). This was already found in Section 8 where a direct procedure to derive an equation for Z, in the equation V, = Z,U, was developed. We notice that the recurrence relations (8.6) and (10.1") are the same, but the quantities defined by them have different initial conditions. The reason for this is that in the problem of Section 8 we prescribed the deflection u o , whereas in Section 10 we prescribed the initial slope ul. EXERCISES 1. Instead of Eqs. (10.4), consider

and find recurrence relations for the R's and x's. 2. Show that R l z ( N )= R Z 1 ( N ) . 3. Show that R i i ( N ) = Zi;(N),

(b)

where Z 1, ( N ) is given by Eq. (10.17). 4. Show that if kNis a positive constant k , and S > 0, then ZL1(N) in Eq. (10.17) is bounded and has a limit as N + 00. 5. Show that

Z = lim Zll(N) N-W

is the positive root of the algebraic equation

IZz

+ I(k/S)Z - k / S = 0.

(d)

6. The statically indeterminate moments Ml at the supports i = 1,2, . . . , N - 1 of a continuous beam of N spans of equal length, loaded with a unit moment at node i = 0, satisfy the second-order difference equations

+

Mi-1 4Mt f Mi+1 = 0,

(4

subject to boundary conditions MO=l,

MN=O.

(f)

16

1 INVARIANT IMBEDDING

Clearly, if MI were known,Eq. (e) could be solved recursively, M2 = -1 Show that Mi admits a solution of the form

M f + *= T i M i ,

i = O , 1,2,

..., N-

- 4Mi, , .. .

1,

(g)

and find the recurrence relation for the Tf’s. 7. What is the value of M I as N co ? 8. Using Eqs. (10.4) and (10.6), derive the pertinent equations for the quantities Zi1(N) and wf(N),i = 1,2.

11. Reciprocity Relation Inspection of Eqs. (10.18) and (10.19) shows that z12

= 221,

(11.1)

a symmetry relation. If we regard u1 and uN as quantities proportional to the vertical components of the forces acting along the displacements u,, and uN , respectively, then conservation of energy will demand the symmetry of the cross quantities in Eqs. (10.4). In other words, Eq. (11.1) is the equation of reciprocity of Maxwell derived from invariant imbedding considerations. 12. Computation of ui,i = 0, 1, 2,

. , . , N - 1. A Two-Sweep Method

Once the Zij(N), i, j = 1,2, and w,(N), w2(N) have been computed, Eqs. (10.4) yield the “missing” boundaryconditions vN and uo . If in addition, to this information, the value of all the deflections ui,i = 0, 1, . .. ,N - 1 is required, we can proceed as follows. (To fix ideas assume u N = 1.) Substituting u, given by Eq. (10.5) in the continuity equation uJ = u ~ - ~ hJ,we obtain

+

UJ-1

= (1

- lzI,(J))uJ - lz1Z(J)vl - Iw,(J),

uN =

1,

(12.1)

a backward recurrence equation for the deflections at the interior points i = 0, 1,2, . ..,N - 1. We note that (1 2.1) is a backward recurrence relation in contrast to Eqs. (10.16)-(10.22), which provide Zij(N), i, j = 1,2, and w,(N), w,(N) in the forward direction. The combination of Eqs. (10.13,

17

13. A ONE-SWEEP METHOD

(10.21), and (12.1) to compute the deflections is therefore a two-sweep procedure. In practice, it will generally require the storage in core of all the quantities computed in the first sweep, i.e., the 2 ’ s and the w’s.

13. A One-Sweep Method The two-sweep method outlined in the last section is particularly remarkable in that it furnishes all the deflections in a sequential fashion. However, when not all of this information is needed, for example, if only the deflection at the center is desired, then a more direct procedure can be developed. In fact, invariant imbedding leads naturally to the formulation of a onesweep method that proves to be efficient when only partial information is needed. We shall illustrate the method with an example. Let, for instance, u, for a particular value of J < N be required. Then from the linearity of the problem we can establish that UJ

+

= ri(J, NUN rz(J, N)o,

+ x(J, N ) ,

(13.1)

i.e., the deflection at a point J is a linear combination of the data values uN and ui, the coefficients of proportionality being functions not only of the length N of the string but also of the point J under consideration. The perturbation technique used in Section 10, i.e., consideration of the reduced string of length N - 1, yields the equation UJ

= ri(Jy N

- 1 ) ~ N - l + rz(J, N - I)o,

+ x(J, N - 1).

(13.2)

Subtracting (13.2) from (13.1), substituting uN with the value given by Eq. (10.9), eliminating uNdl and U, in the resulting equations [using (10.4) and (10.8)], and collecting terms in u,-, and ulY we finally obtain the following recurrence relations : r1(J,

N)=

ri(J9

1

N

- 1)

+ l(Z,,(N - 1) + kN-1/S)’

subject to the obvious initial conditions, rl(JyJ ) = 1,

r2(J, J> = 0,

x(J, J ) = 0.

(13.4)

Naturally, functions rl(J, N ) , rz(J, A’),and x(J, N ) are not defined for N < J.

18

1 INVARIANT IMBEDDING

Thus, the one-sweep method consists of the sequential computation of Zij(N),i, j = 1,2,and w,(N), w2(N), using the forward equations (10.17)(10.22)from N = 1 to N = J. At that point, we introduce the system (13.3) subject to the initial conditions (13.4)and proceed with the computation until N reaches the desired value. Clearly r,(J, N), rz(J, N ) , and x(J, N) computed in this way and substituted into Eq. (13.1)yield the value of uJ. EXERCISES

1. Let VJ

= ll(J, N)uN

+

fZ(J,

N)ol

+ y(J, N),

J _< N .

(a)

Using the same method employed in this section, derive recurrence relations for t l ( J , N ) , tz(J, N),and y(J, N).What are the initial conditions of those relations?

2. Develop a one-sweep method for the computation of M,,j < N,in Exercise 6 of See tion 10.

14. A Continuous Analog

It is not difficult to construct a continuous analog of the results derived in Section 10. A continuous version of the equation of equilibrium (2.3) is duldx = V ,

S dvldx

= k(x)u

- P(x),

(14.1)

where u, u, k, and p are the limiting quantities of u i , u i , ki , and p i , when the distance I between springs tends to zero. Assuming the string to be of length L and subjected to the same end conditions of Section 10, i.e., u(L) and u(0) prescribed, we seek solutions of the form

+ z, z(L)v(O) + w , m ,

=Zll(~)U(L)

(14.2)

u(0) = Z21(L)4L) + Z 2 , ( ~ ) 4 0 + ) wz(0,

a continuous version of (10.4). Clearly, perturbation with respect to the length of the interval L can be achieved in the continuous case by differentiation. In fact, taking derivatives with respect to L in (14.2), we obtain

+

+ +

+

v’(L) = 2;,(L)u(L) 2,,(L)u’(L) z;,(L)V(O) w,yL), 0 = z;,(L)u(L) + Z,,(L)u’(L) z;,(L)v(o) + wz’(L),

(14.3)

where a prime indicates derivative with respect to the length of the string. Clearly (14.3)is the analog of Eq. (10.7). Substituting the derivatives by the values given by the equilibrium equation (14.l), and collecting terms common in the boundary conditions v(0) and u(L), we finally obtain Z;, =k/S-Z:,, Wlf

Z ; , = -Z11Z12,

= p / s - Z,,Wl,

Zj’,

W2) = - z 1 2

=

w,,

-Z;2,

(1 4.4)

19

EXERCISES

a system of Riccati differential equations subject to the initial conditions

z , m = 1,

Z,,(O) = 0,

W,(O) = W Z ( 0 )

zzm = 0,

= 0,

(14.5)

and where Z,,(L) = Z,,(L) as was expected. EXERCISES 1. Show that u in Eq. (14.1) satisfies the second-order differential equation

SU"- ku = - p . 2. Show that if

u

(a)

satisfies Eq. (a) in problem 1, then Z. w and R , x in the equations

+ u(L) = R(L)u'(L) + x(L),

u'(L) = Z(L)U(L) w(L),

(b) (4

satisfy the Riccati differential equations Z ( L ) = k(L)/S - ZZ(L),

(dl

w'(L) = - p(L)/S - Z(L)w(L),

(4

and R'(L) == 1

- (k(L)/S')R'(L),

respectively. 3. Show that Z(L) = R-'(L) and w(L) = Z(L)x(L). 4. If Eqs. (f) and (g) in Exercise 2 are subject to the initial conditions R(0) = 0 and x(0) = 0, show that Z and w in Eqs. (d) and (e) satisfy the following asymptotic properties:

lirn Z(L) = 1/L,

lim w(L) = L.

L-t 0

L-0 0

5. Let u(x, L), x S L, be given by u(x, L) = Sl(X, L)u(L)

+

SAX,

LMO)

+ Ax, L),

x IL.

Show that sI(x, L), sz(x, L), and y(x, L ) satisfy the differential equations

dsdx, L)IdL = -S1(X, L)Zll(L), L)Z,,(L), d Y k N d L = -s1(x, L)WdL)),

d d x , LYdL = --SI(X,

where Z 1l(L), Z l z ( t ) ,and w l ( L ) in Eqs. (i) are given by Eqs. (14.4)and (14.5), subject to the initial conditions Sl(X,

x ) = 1,

SZ(X,

x) = 0,

y(x. x) = 0.

(j)

6. Show that Eqs. (i) and (j) of the preceding exercise are limiting forms of Eqs. (13.3) and (13.4) as the distance I between springs tends to zero.

20

1

INVARIANT IMBEDDING

7. Consider the Riccati differential equation

Z' = K - Zz,

Z(0) = 0,

where K is a positive quantity. Show that Z is stable with respect to small perturbations in the initial conditions. Hint: First prove that Z is positive and uniformly bounded. Then consider the behavior of the approximate differential equation

6 z = -2282,

(1)

for the perturbation term 62.

15. An Infinite String on Elastic Foundation

We consider now the problem of determining the deflection under a concentrated load 2P of an infinite string resting on a uniform foundation.

Fig. 1-5

To this end we first consider the finite string of length 2L (Fig. 1-5). We set

u(L)= W M L ) ,

(15.1)

where v(L)is related to the vertical load by means of (15.2)

V(L) =P/S,

S being the axial force in the string. Now, if the foundation coefficient k is constant and L is large enough, a small increment in the length of the string will not result in a variation of the compliance R in Eq. (15.1). Therefore we can write

u(L

+ A) z R(L)v(L+ A).

(15.3)

Combining (1 5.1) and (1 5.3) we obtain

u(L + A) - u(L)= R(L)[v(L+ A)

- u(L)],

L large.

(15.4)

Retaining only linear terms of a Taylor's expansion of u(L + A) and v(L + A) in (15.4), and using the equilibrium equations U' = 0,

V' = (k/S)u,

(15.5)

we find 1 - (k/S)R2= 0,

(15.6)

16. BUCKLING. CLASSICAL APPROACH

21

an algebraic equation for R, the limiting value of R(L). Of course, we could have derived (1 5.6) directly from the asymptotic behavior of the Riccati equation (15.7) R’(L) = 1 - (k/S)R2(L). Combining (15.1 .), (1 5.2), and (15.6) we obtain u = (P/S)(S/k)”2,

(15.8)

the desired asymptotic deflection. The simplicity of this result is emphasized by the algebraic character of Eq. (15.6), from which this originated. The connection between the algebraic problem (1 5.6) and the differential equation (15.7) is the basis for a number of gradient methods for solving nonlinear equations. As the reader possibly suspects at this point, (15.7) is not the only differential equation whose asymptotic behavior tends to the solution of (15.6). In general we can use

R‘ = F[1 - ( k / S ) R Z ] , where F is a function introduced to accelerate the convergence. The same procedure can in general be used to solve more general functional equations. EXERCISES 1. In the problem of the infinite string considered above, how would you proceed to compute the deffection u(z) at a distance z of the concentrated load P? If necessary, use the value u(0) given by (15.8). Prove the stability of the proposed method 2. Formulate the solution of (15.6) by Newton’s method.

16. Buckling. Classical Approach Before closing this chapter we wish to present an example in which the lack of uniqueness plays the major role. As it is well known, examples of this kind in the context of structural mechanics are elegantly furnished by the elementary theory of buckling. This in turn will provide us with an example to compare the classical and the invariant imbedding approach in the realm of characteristic value problems. Consider the chain of rigid bars of length 1 supported on springs with elastic coefficients ki> 0 and subject to a compressive axial load S < 0. The ends are assumed to be fixed in the lateral direction as indicated in Fig. 1-6. We consider first the classical version of the problem. To this end we

s-~T~”-’ Fig, 1-6

22

1

INVARIANT IMBEDDING

write the equilibrium equations in the form U, - 1

- (2 + Ik,/S)u, + U, + 1 = 0.

(16.1)

Clearly, (16.1) follows directly from the equilibrium equations of the elastic string, by making the lateral loads P, = 0 and by eliminating the slopes u, between the two equations of (2.3). The present problem of buckling can now be mathematically formulated as the problem of finding a nontrivial solution u, # 0, n = 1, 2, . . . ,N - 1, to Eq. (16.1) subject to the fixed end conditions ug

(16.2)

= u N = 0.

In order to examine this matter in more detail it is convenient to write Eqs. (16.1) and (16.2) in the matrix-vector form

TU= -(I/S)Ku,

(16.3)

where u is the ( N - 1)-dimensional vector

(16.4)

u =

K is the diagonal matrix

(16.5)

and T is the ( N - 1)-dimensional Jacobi matrix

(1 6.6)

The homogeneous system (16.3) possesses the unique solution u, = 0, n = 1, 2, . .., N - 1, if and only if the associated matrix T (I/S)K is nonsingular. This is the case when S 2 0, i.e., when the chain is subject to a

+

23

EXERCISES

tensile axial force or to no force at all. We leave to the reader the exercise of proving that for S 2 0, T + (1/S)K is positive definite, thus having an inverse. However, when S < 0, we cannot ensure that this property will hold. In fact, there will be values of S < 0 for which the system (16.3) will have infinite solutions, in addition to the trivial one u, = 0, n = 1,2, .., N - 1. This will occur for those values of S for which the characteristic equation

.

det[T

+ (1/S)K]= 0

(16.7)

is satisfied. The minimum absolute value of S satisfying (16.7) is known as the Euler critical load. Higher characteristic values are associated with higher order buckling configurations. When k , = k , = . * . = kN-l= k , Eq. (16.7) reduces to det[T

+ (l/S)klZ]= 0,

(16.8)

a characteristic equation that can be solved in closed form. See the exercises below. In general, the solution of (16.7) will be obtained employing numerical procedures. EXERCISES 1. Show that if kI = k 1 = .* . written in the form

=kN- 1 =k

> 0, the general solution of (16.1) may be

u. = cI sinh na

+ cl cosh na,

(a)

where a is a number, real or complex, that satisfies the characteristic equation cosh a = 1

+ lk/2S,

(b)

and where cl, c2 are arbitrary constants.

2. If uo = 0, then c2 = 0. Thus Eq. (a) reduces to u. = c1 sinh na.

(4

Show that the minimum absolute value of S [from Eq. (b)], such that uN = c1 sinh Na = 0, is given by

3. Using the determinantal condition (16.8) show that the characteristic values are given by

s.=

-

3lk

1

+ cos ( n a / N ) ’

n = 1,2, . . .,N

-

24

1

Hint: Set 1

INVARIANT IMBEDDING

+

k1/2S = cos y in (16.8), expand the determinant about the first row, and study the resulting difference equation. 4. What is the buckling load when N + co?

17. Buckling. Invariant Imbedding

We look now at the same problem from an invariant imbedding point of view. As usual we imbed the problem in terms of length. In particular, we consider the chain of n 1 links fixed at n = 0 as indicated in Fig. 1-7.

+

Fig. 1-7

The displacements must satisfy the equilibrium equations and end condition ~,-1

+

- (2 + Ikn/S)un

~ , += l 0,

UO

= 0.

(17.1)

We seek solutions of the type un+1 = R n U n .

(17.2)

Combining (17.1) and (17.2) for n = 1, we readily obtain

R, = 2 +- IkJS,

(17.3)

R, = 2 + lk,lS - R,=',,

(1 7.4)

and for n # 1,

a recurrence relation for the quantities R, subject to the initial condition (17.3). Now consider the chain of two links. According to (17.2), when R, = 0, for any value of u1 we have u2 = 0. In other words, R, = 0 furnishes the buckling condition for a chain of two links. Similarly, the buckling condition for a chain of n + 1 links fixed at the two ends is provided by R, = 0. We can make use of this condition in a number of ways. We note, however, that Eq. (17.4) is particularly well adapted to determine the so-called critical lengths. In fact, starting with a value 1 So1 < #,I, i.e., below the critical load for n = 1, we compute the sequence R 2 ,R , ,.. . ,using (17.4) and record the values n, < n2 < . . * for which R, changes the sign, i.e., when sign R,,# sign R,+ n i .

(17.5)

25

EXERCISES

+

We agree to call the numbers 1 n, the critical lengths of the chain. Let N be the smallest critical length associated with an axial load So, i.e., O,

RN(SO)

> 0.

RN+I(SO)

If RN(SO)= 0, So is the critical load associated with the chain of length N + 1. If R N ( S O< ) 0, [ SoI is clearly a lower bound. To compute the critical load I S* I > I So1, we can proceed in several ways. A direct procedure consists in computing R,, n = 1, 2, . . . , N , by means of (17.3) and (17.4) using increasingly large values of I SI such as S , = So i AS, where A S is an appropriate increment of the load, until R , = 0. Other procedures are discussed in the following chapters.

+

EXERCISES 1. Show that the critical loads of a chain of n

+ 1 links are given by the roots of the equation

0 = K,,(S) -

where K,(S) = 2

+ iki/S.

+

2. Consider the chain of N 1 links shown in the figure, subject to a compressive force S
uo = 0.

=Znu.,

Prove that Z , satisfies Z. = 2

+ ik./S - l/Zn-

1,

Z o = a,

and show that the condition for criticality is given in this case by

Z,v = (1

+

'.

NCN+ 11s)-

Hint: Consider the condition for a free end given by

s-v -s

3. Assuming k l = k z = ... - k , compute the critical loads for N using (c) and (d).

= 1,

N

= 2,

and N

= 3,

26

1 INVARIANT IMBEDDING

4. Using the D’Alembert principle, it is easy to write the equations of motion of a chain subject to a compressive force S below the critical value. Assuming the masses m, to

be concentrated at the nodes, we have

uf- - (2

+ lkl/S)uf+ uI+

= m ld’u,ldt2.

(f)

Setting uI = Zfsin w l t, we are led to the frequencies equation Zi- - (2

+ lkl/S - mi wiz)Z1 + Z I ,

= 0.

(9)

Assuming Zo = ZN= 0, discuss the solution of (g) by classical and invariant imbedding procedures. * = kN = k,and mi = rn2 = mN = m in Eq. (g) of Exercise 4, find the dependence between the natural frequencies and the axial load S.

5. Assuming k l = kZ =

NOTES, COMMENTS, AND BIBLIOGRAPHY

The purpose of this chapter has been to introduce the reader to invariant imbedding ideas using one of the simplest two-point boundary-value problems occurring in structural mechanics. Difference models, as opposed to differential ones, were selected to illustrate the basic ideas. Simplicity of exposition is partially responsible for this selection of methodology. The main reason, however, should be associated with the increasing interest that discrete formulations are acquiring on their own. The classical treatment at the beginning of the chapter, Sections 2-7, serves to emphasize to the reader the dependence of the solution on the boundary conditions. This chapter follows closely the first part of N. DistCfano, Dynamic Programming and Invariant Imbedding in Structural Mechanics, in “Invariant Imbedding” (R. Bellman and E. Denman, eds.), Chapter 9. Lecture Notes in Operations Res. and Math. Syst., No. 52, Springer-Verlag, Berlin and New York, 1971.

The following comments, notes, and bibliography serve to expand the basic material covered in the corresponding sections. 2-6. A treatment of the string and other linear second-order boundaryvalue problems in structural mechanics along classical lines can be found in T. von Karman and M. A. Biot, “ Mathematical in Methods Engineering.” McGraw-Hill, New York, 1940.

A systematic treatment of difference equations at an elementary level is available in K. S. Miller, “Linear Difference Equations.” Benjamin, New York, 1968;

and for the more advanced subject of difference-differential equations R. Bellman and K. Cooke, York, 1963.

‘I

Differential-Difference Equations.” Academic Press, New

NOTES, COMMENTS, AND BIBLIOGRAPHY

27

7. For the subtleties and difficulties in the application of the method of superposition, see L. Fox, “The Numerical Solution of Two-Point Problems in Ordinary Differential Equations.” Oxford Univ. Press (Clarendon), London and New York, 1957.

8. The term invariant imbedding was coined by Richard Bellman in the 1950s to designate a set of techniques that he and his coworkers developed to

attack a number of boundary-value problems in mathematical physics by reduction to initial-value formulations. See R. Bellman and R. Kalaba, On the Fundamental Equations of Invariant Imbedding, I. Proc. Nar. Acad. Sci. U.S.41, 336, 1961.

See also the book R. Bellman and M. Wing, “Introduction to Invariant Imbedding” (to appear).

Historically, the roots of the method can be traced back to Stokes and the problem of determining the reflection of N + 1 plates when the reflection of the first N plates is known. See M. Wing, The Method of Invariant Imbedding with Applications to Transport Theory and Other Areas of Mathematical Physics. Colloq. Lectures in Pure and Appl. Sci., Number 10, March 1965, Socony Mobil Oil Company; R. Redheffer, Difference Equations and Functional Equations inTransmission-Line Theory, “Modern Mathematics for the Engineer,” 2nd Ser. McGraw-Hill, New York. 1962.

More recently, invariant imbedding can be associated with, and was certainly strongly influenced by, Ambarzumian in the context of a number of problems in astrophysics. See, for example, S. Chandrasekhar, New York, 1950.

“

Radiative Transfer.” Oxford Univ. Press (Clarendon), London and

10. Semigroup properties of differential equations in connection with invariant imbedding were first given in

R. Bellman and T. Brown, A Note on Invariant Imbedding and Generalized Semigroups. J. Math. Anal. Appl. 9, 1964.

The perturbation character of invariant imbedding has been emphasized in M. Wing, “An Introduction to Transport Theory.” Wiley, New York, 1962;

and in the first reference of Section 8. 13. Reduction of two-point boundary-value problems to one-sweep initial-value formulations is one of the keystones of invariant imbedding. The name itself can be easily justified in the context of the example presented

28

1

INVARIANT IMBEDDING

in this section. In fact, here the variable J of the classical imbedding remains fixed while the effective independent variable of our problem is N , the length of the string. The treatment of this section follows the introductory paper of the author quoted at the beginning of these notes. See also R. Kalaba, A One-Sweep Method for Linear Difference Equations with Two-Point Boundary Conditions. USCEE Tech. Rep. 69-23, Univ. of Southern California, Los Angeles, 1969.

16. A good introduction to characteristic value problems in mechanics using classical methods can be found in S . Timoshenko and J. Gere, ‘‘ Elastic Stability.” McGraw-Hill, New York, 1961 ; Ziegler, H., “ Principles of Structural Stability.” Ginn (Blaisdell), Boston, Massachusetts, 1968.

17. The first invariant imbedding treatment of buckling problems appears

to be E. M. Shoemaker, Invariant Imbedding Applied to Eigenvalue Problems in Mechanics, J. Appl. Mech. 32 (1965), 41.

Differential Equations

1. Introduction

In this chapter we introduce some topics in differential equations that are used systematically throughout this book. We assume that the reader is acquainted with basic notions in the theory of ordinary differential equations. Here we present only a brief survey of the main properties of systems that wjff be repeatedly used in the applications. In order to provide a methodological link with Chapter 3, we introduce the notions of initial-value and two-point-value problems in the context of physical systems and discuss the difficulties associated with the numerical solution of such problems. 2. Vector-Matrix Notation

An array of numbers in column form such as

29

30

2 DIFFERENTIAL EQUATIONS

will be called a vector of dimension N, or simply an N vector. We shall always refer to column vectors unless otherwise noted. The quantities y i are called the components of the vector. The square array of numbers given by

is called an N-dimensional matrix or N x N matrix. The quantities aij are called the elements of A . We define the sum of two vectors y and z

where y i and zi are the components of the vectors y and z, respectively. Similarly we define the sum of two matrices A

+B =

(ajj

+

bij),

(2.4)

where a j j and bij are the elements of A and B, respectively. The multiplication of a vector y, or a matrix A , by a scalar c1 is defined

c l A = A c , = (c,aij).

(2.6)

The multiplication of a vector y by a matrix A is the vector b = (bi), i = 1, 2, .. ., N , such that N

bi = C a i j y j . j= 1

In compact notation we write

A y = b. The multiplication of two matrices A and B is a matrix C = (c,)

(2.8)

such that

In compact form AB = C.

(2.10)

31

2. VECTOR-MATRIX NOTATION

Clearly the product of two matrices is not commutative, i.e., AB # BA, in general. Given the system Ay = b,

(2.1 1)

where A is an N x N matrix such that det A # 0 and b is a given N vector, we can solve for y and obtain y = Cb,

(2.12)

where C is an N x N matrix whose elements can be obtained in principle using Cramer’s rule. We call C the inverse of A and write symbolically C = A-’.

(2.13)

Substituting (2.13) in (2.12) and the result in (2.1 1) we obtain (AA-’ - Z)b = 0, where I is the identity matrix given by (2.14)

Therefore

AA-’ = I .

(2.15)

Given the matrix A = (aij),the matrix AT = (aji),obtained by transposing rows and columns, is called the transpose of A. If A = AT, we say that A is symmetric. Given two N vectorsy and z we define (2.16) to be the inner product of the two vectors. We consider naw the inneq ptaduct

of y and Az, i.e., (2.17)

Therefore, interchanging subscripts we find

If A is an N x N real and symmetric matrix such that the quadratic form N

N

(2.19)

32

2 DIFFERENTIAL EQUATIONS

is positive for all real vectors y # 0, we say that A is positive definite and write A > 0.

(2.20)

If (y,Ay) 2 0 for all real vectors y # 0, we say that A is nonnegative definite and write (2.21)

A2O.

We shall use lowercase symbols to designate vectors and uppercase ones to designate matrices, unless otherwise stated. EXERCISES 1. Show that (A+B)y = Ay 2. Let A = (alJ), i, j = 1, 2, ail # 0, A - ' = ( 0 ; ' ) .

3. 4. 5. 6.

Show that (AB)'

+ By,

A(y+

Z) = A y

+ Az.

.. . ,N, be a diagonal matrix, i.e., ail = 0 if i # j . Show that if

= ETAT.

Show that if A and Bare nonsingular, (A@-' = B-'A-'. Show that A-lA = I . Show that A'A is a symmetric matrix.

7. Show that matrix T given by Eq. (16.6) of Chapter 1 is positive definite.

8. As norm of a matrix A = (al,), i, j = 1,2, ...,N,we can use the quantity

3. Systems of Ordinary Differential Equations We consider now vectors and matrices whose elements are fmctions of the real variable x , e.g., 4 x 1 = (ui(x)), i, j = 1, 2,

A(x)

(aij(x)),

(3.1)

...,N. We introduce the derivatives given by du/dx = (dui/dx),

dA/dx = (daij/dx),

(3.2)

S A dx = ( ! a i j d x ) .

(3.3)

and the integrals

I u dx = ( s u i dx),

3. SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

33

Consider now the nonlinear system of ordinary differential equations d ’ l / d x = g l ( u l , u 2 , . . * )U N , x ) d‘2/dx = gZ(U1r u2 > . . * uN x ) f

d‘N/dx

= gN(U1f U 2 9

...

9

9

uN

(3.4)

x).

Using vector notation we can write (3.4) in the compact form dujdx

(3.5)

= g(u, x ) ,

where u is the vector whose components are u i ,i = 1, 2, . . . , N , and g is an N vector given by Sl(U9 x ) g(u,x)=

x ) ) =(g,(u,x)).

f2(;

(3.6)

SNh4

I),

It is of interest to note that we could, if desired, remove the dependence of g with x in (3.5) by considering the N + 1 vectors

U =

UN

(3.7)

X

and

h(u) = 9 N W

With this notation (3.5) reduces to

(3.9)

dv/dx = h(0).

A system such as (3.9) exhibiting no dependence with the independent variable is called auronomous. An important particular case of Eqs. (3.4) is the linear system dU,/dx=ailui duZ/dx =

+ + + +P~(x), + a22 U 2 + + & N U N + p2(X), ~

1

2

~

2

*. *

u~NuN

(3.10)

34

2

DIFFERENTIAL EQUATIONS

where the coefficients aij are generally functions of x . Using matrix notation we can write (3.10) in the compact form du/dx = A(x)u

+p ,

(3.11)

where A(x) = (a,(x)) a n d p = (pi), i, j = 1,2, .. .,N. In what follows we shall recall some of the fundamental properties of linear systems. 4. Linear Homogeneous Systems. Superposition Let us consider the linear homogeneous system dYWX

=m

lY,

(4.1)

where y = (yi) and A = (aij(x)),i, j = 1, 2, . . , , N. If y ( l )denotes a particular solution of (4.1), the function c,y"', where cl is a scalar, is also a solution, as may be proved by substitution. Let J J ( ~ ) , i = 1, 2, . . . , N, be N linearly independent solutions of (4.1) and let ci, i = 1, 2, . . , , N, be N arbitrary constants. Then the linear form y

= qy")

+ c2y(2) + - - * + c,

y'"

(4-2)

is the most general solution of (4.1). We shall now show how these solutions are elegantly furnished by the solution of a certain matrix differential system. For this purpose, we construct a matrix Y(x) whose columns are the desired linearly independent solutions of (4.1), i.e., Y(x) = (y'"(x) y'2'(x)

* * *

y("(x)).

(4.3)

It may be easily verified that the matrix Y(x) satisfies the matrix differential equation d Y/dx = A Y.

(4.4)

Now, the simplest way to ensure that the vector functions ~ ( ~ ) (i x=) 1,2, , . . ., N, are linearly independent is to prescribe Y ( x , ) = I, where x i is any point in the interval of definition of (4.1). In particular, we may prescribe Y(0)= I.

(4.5)

Therefore, the solution of a linear homogeneous system of differential equations reduces to the integration of a matrix differential system (4.4) subject to initial condition (4.5). The matrix Y is called the matrix of the principal solutions, or the fundamental matrix. Now we can write y given by (4.2) in terms of the fundamental matrix Y , i.e., y = Yc,

(4-6)

35

EXERCISES

where c = (ci)is an arbitrary constant vector. Since Y(0)= I it follows that

(4.7)

c = y(0). EXERCISES 1. Let Y satisfy the matrix differential equation

d Y/dx = A Y,

Y(0)= B,

where B is a nonsingular N x N matrix. Show that Y is a fundamental matrix of the linear homogeneous system (4.1). 2. Prove that if

dZ/dx = AZ,

Z(0) = I ,

then Y = ZB, where Y satisfies Eq. (a) in Exercise 1.

5. Linear Inhomogeneous System

We consider the inhomogeneous system

dyldx = A y

+ p,

(5.1)

and introduce the associated matrix differential equation dYjdx = A Y, Y(0) = I. If y(P) denotes a particular solution of (5. I), then y given by

y = Yc

(5.2)

+ y(p),

(5.3) where c is an arbitrary constant N vector, is the most general solution of (5.1). This, a result going back to D’Alembert, is a consequence of the linear structure of the system (5.1). We can construct a particular solution y(p)by integrating the initial-value system

+

dy‘p’/dx = Ay”’ p, y(P’(0) = 0, (5.4) numerically or otherwise. In the next section we present the elegant method of variation of parameters due to Lagrange, which permits the construction of a particular solution in terms of the fundamental matrix. Clearly, when y ( p )in (5.3) satisfies (5.4), c in (5.3) is given by

c = y(0).

(5.5)

EXERCISES 1. Show that i f p is a constant vector and A is a constant, nonsingular matrix, a particular solution of Eq. (5.1) is given by y(”)= - A - ’ P.

36

2

DIFFERENTIAL EQUATIONS

2. Show that a particular solution of the system u'=xu+x

u'=u,

isgivenbyu=-1

andu=O.

6. Variation of Parameters To construct a particular solution of (5.1), we can alternatively resort to the elegant method of variation of parameters due to Lagrange. In this method we seek solutions of the inhomogeneous system (5.1) of the form y = Yu,

(6.1)

where Y is the matrix of the fundamental solution, i.e., it satisfies (4.4) and (43,and u is an unknown vector function to be determined. Differentiating (6. I), we obtain dy/dx = (d Y/dx)u

+ Y duldx.

(6.2)

Eliminating dyldx and dY/dx from among (5.1), (5.2), and (6.2) and using (6.1) conveniently we obtain p

=

Y du/dx

(6.3)

from which

where c is an arbitrary constant vector. Substitution of (6.4) into (6.1) yields

the desired general solution of (5.1) in terms of an arbitrary constant vector c. We recognize the integral in (6.5) as a particular solution of (6.1). That Y ( x )is nonsingular follows from a classical result due to Jacobi, namely,

(

det Y = exp jOxtr(A)d o , where tr(A) is the trace of matrix A given by N

tr(A) =

11a i i.

i=

(6.6)

37

8. HIGHER-ORDER EQUATIONS

EXERCISES 1. Show that dY-'/dx = - Y-'(dY/dx)Y-'. 2. Prove Jacobi's formula given by Eq. (6.6). 3. If Y satisfies the matrix differential equation

d Y/dx = A Y,

Y(0) = I ,

the equation dZ/dx = -ZA,

Z(0) = I,

is called the adjoint. Prove that Z = Y-'.

7. Constant Coefficients A particular case occurs when A in (5.1) is a constant matrix. Then Y ( x ) satisfying (5.2) is given by the matrix exponential 1 +A2x2 + . 2!

Y(x) = eAx = Z + A x

* *.

(7.1)

Using this representation, (6.5) reduces to

(7.2) EXERCISES 1. Show that e A ( x + y= ) eAxeAY.

2. Derive Eq. (7.2). 3. Show that (elx)-'

= e--IX.

4. Show that (d/dx)eAx= Ae"". 5. Let A be a constant N x N matrix. Show that if h is a root of the characteristic equation

det(A - h1) = 0, then y = eAXcis a solution of the equation y(0) = C.

dy/dx = Ay,

8. Higher-Order Equations Our apparent lack of interest in equations of higher order, so often appearing in the applications, can be justified in view of the possibility of reducing any differential equation of arbitrary order to first-order differential systems. Consider, for example, the Nth-order differential equation given by

. . . , dNU/dXN)= 0,

g(u, duldx, d2UldX2,

(8.1)

explicitly solved in terms of the higher derivative, i.e.,

dNU/dXN= h(u, duldx,. .. , dN-IU/dXN-').

(8.2)

38

2 DIFFERENTIAL EQUATIONS

Introducing the new variables

Y N = dN-'u/dxN-',

we can transform (8.2) into the differential system of dimension N :

The linear version of @.I), given by

a, u + al duldx + *

+ aN-

dN-'u/dxN-' + dNu/dxN= 0,

(8.5)

may be readily shown to be equivalent to the system

where y is the vector

and A is the matrix

9. Discussion Clearly, there is no unique way of transforming a higher-order equation into a system of first-order equations. In other words, (8.4) is a convenient but not a unique set of auxiliary variables performing the transformation.

39

10. PHYSICAL SYSTEMS. STATE VARIABLES

As arule of thumb, the best results may be obtained by introducing as auxiliary variables, quantities exhibiting a clear physical meaning. There is much to gain, computationally and analytically, in so doing. We note in passing that a natural way to proceed in this respect is to trace back the derivation of the model equation. In problems of structural mechanics, for example, we could start with the system formed by the constitutive equations in addition to the equations of equilibrium and compatibility. See the exercises below. EXERCISES 1. Using the slope, shear force, and bending moment in addition to the deflection, transform the fourth-order differential equation of a beam in elastic foundation (d2/dx2)(EZd2u/dx')

+ ku =p

into a first-order linear system. 2. Write the linear equations of bending of an elastic circular plate subject to axisymmetric loading in terms of a system of ordinary first-order linear differential equations. 3. Write the equations of bending for shells of revolution in terms of a system of firstorder nonlinear differential equations. See the two papers by Kalnins in the Bibliography at the end of the chapter.

10. Physical Systems. State Variables

The preceding discussion highlights the interest in the study of some structural properties of alternative mathematical models used to describe a physical system. We have already emphasized this aspect in Chapter 1 in connection with the modeling of a simple structural system. The same concept is systematically emphasized throughout this book in connection with a number of models in applied mechanics. Here we only wish the reader to recall the basic steps usually taken in formulating a mathematical model, namely: 1. We pick descriptive quantities known as state variables. 2. We define relationships between the state variables. In other words, we define the basic laws followed by the components of the system. 3. We apply mathematical reasoning to determine a solution of the model using a prescribed computational device such as a digital, analog, or a hybrid computer. Clearly, the choice of appropriate state variables is a crucial step in the formulation of the model. The remaining steps will depend heavily on the first one. Once the state variables are chosen, the nature of the relationships between the variables will define the type of model. We can think in terms of differential (local) or integral (global) relations. Realistic description of

40

2

DIFFERENTIAL EQUATIONS

processes entails frequently the incorporation of difference, differencedifferential, or integrodifferential relations. We are presently interested only in differential models. In the remainder of this chapter we discuss two important types of differential models, namely, initial-value and boundary-value models. Both are shown to be completely different from the third modeling stage point of view, i.e., with respect to the construction of computational algorithms. This discussion provides enough motivation for the developments presented in Chapter 3 and those following. 11. Initial-Value Systems

If a system of differential equations is to represent the behavior of a physical system, we must require at least two crucial properties : existence and uniqueness of the solution, and stability with respect to the structure of the equations and the data furnished to solve the equations. In the preceding sections we have studied some basic properties of the solutions of differential systems. In order to ensure uniqueness, however, we must prescribe the values of the variables at one or various points. When all the components of the vector representing the state variables of the system are prescribed at a single point, we say that the system is initial valued. We consider, for example, the nonlinear system discussed in Section 3, namely, dujdx = g(u, x ) ,

where u is the vector

U =

("),

(11.1)

(11.2)

UN

and g is given by (11.3)

where gi(u, x), i = 1,2, .. . ,N,are scalar functions of the vector u, continuous in u and x , and enjoying a Lipschitz condition of the type

Ilgi(u, X )

-gi(u,

x)II I M l l u - 011,

(11.4)

where llull is the norm given by (11.5)

41

12. NUMERICAL SOLUTIONS

Now, if all the components of vector u are prescribed at a single point, say x = 0, i.e., if

(11.6)

u(0) = c,

where c is a known vector, the preceding conditions are sufficient to ensure the existence and uniqueness of the differential system (1 1.1). We observe that when the elements of the Jacobian matrix (agJdu,), i , j = 1, 2, , . . , N , are continuous in u, the Lipschitz condition (I 1.4) is automatically satisfied. This, a more restrictive condition, is frequently encountered in the applications. Problems of stability are far more involved and require a special treatment. The reader can find a number of appropriate references at the end of the chapter. 12. Numerical Solutions The concept of solution of ordinary differential equations subject to initial conditions is by no means a statical concept. It has in fact been changing progressively with the advancement of the mathematical buildup. The last important step in this progress occurred with the advent of the digital computer. The ideas and methods currently employed in the numerical solution of differential equations subject to initial conditions can be traced back as far as Euler and Cauchy, if not to Newton himself. But the number of equations that can be handled by these numerical procedures has increased astronomically during the last 25 years. We can in fact accurately integrate systems of 5000-10,000 nonlinear equations using standard routines in a matter of minutes. This change in quantity has induced a change in quality, a manifestation of a Hegelian proposition that is occasionally invoked to point out the qualitative nature of the changes introduced by grand-scale, accurate arithmetic. We illustrate the main ideas associated with the numerical solution of initial-valued systems using a simple example, namely the scalar equation du/dx = g(u),

~ ( 0= ) C,

(12.1)

where g(u) is a continuous function of the scalar function u and c is a prescribed initial value. The simplest numerical scheme for the solution of (12.1) is due to Euler. We seek the numerical values of u ( x ) at the points x = iA, i = 1, 2, ..., where A is usually called the step size. Let ui = u(iA),

(12.2)

and consider an expansion of ui+ linear in u iand the first derivative, i.e., (1 2.3)

42

2 DIFFERENTIAL EQUATIONS

Using (12.1) we can write (12.3) in the form u i + l E ui + Ag(ui).

(12.4)

Since uo = c, the sequence u l r u 2 , . . . , is uniquely determined by the recurrence relation (12.4). The algorithm given by (12.4) and the initial condition are frequently called a numerical process. We can think now of a number of possible generalizations of this numerical process. The names of Runge-Kutta, Adams-Moulton, and so on, are associated with these generalizations. A geometrical interpretation of the process given by (12.4) helps to understand the nature of possible generalizations. In Fig. 2-1, the solid line represents the solution of (12.1) while the

Fig. 2-1

vertices of the polygonal are the values of ui obtained using the recursive relation (12.4). Clearly, the slopes of the polygonal are the values of the derivatives u’(iA). We encourage the reader to develop these notions further using appropriate literature such as that indicated at the end of the chapter. 13. Stability of the Numerical Process Despite the particular method employed to discretize the original equation, in general what we do is to replace the original equation (12.1) by an approximate difference equation of the form

where A i is the operator of elementary operations.

43

14. TWO-POINT-VALUE PROBLEMS

Using a machine with finite word length, we cannot guarantee an exact evaluation of ui at each step of the calculations. Rather, we shall obtain a sequence of values u i , i = 1, 2, . . . , that satisfies the equation Vi+l

= Ai(Ui, u i - 1 ,

. . . , Vk)

+ Si,

(13.2)

where di is presumably an element of small norm. Since zli is the output of our computational device, we must compare it with the solution of (12.1), the original equation. We generally proceed in two parts. First we can study how well the difference equation (13.1) approximates the solution of (12.1). In particular we are interested in the dependence of ui with A, the step size of the process. For a number of numerical processes we can derive a number of results establishing the degree of approximation obtained by using (13.1) rather than the original equation (12.1). We note in passing that although this is a problem in approximation theory, error analysis has only recently been tackled using the powerful ideas and techniques of the theory of approximation. The second problem, a consequence of the limited accuracy of digital computers (finite word length), is to study the behavior of di in (13.2) as i-, co. In general the problem consists of establishing conditions under which I d i I < 6, for i 2 N , where 6 and N are two prescribed positive numbers. This is clearly a problem in stability theory. We have so far implicitly assumed that A, the step size, is a constant. In some cases this is a convenient simplification but in general we would expect better results using a variable step size Ai that might account for rapid changes of the solution, and so forth. An interesting proposition here appears to be the determination of the sequence Ai that minimizes the computational error, subject to an appropriate constraint that accounts for the computational effort (time, storage, etc.). This is a problem in control theory, offering interesting research possibilities. Although the stability properties of numerical processes comprise an area still in vigorous development, we can safely assume that at present, a blend of intuition, analytical ability, and an accurate computer are sufficient ingredients to tackle the solution of a broad class of large systems of ordinary differential equations subject to initial conditions. 14. Two-Point-Value Problems

The study of problems of equilibrium in structural mechanics is a prolific source of two-point-, or more generally, multipoint-value problems. These problems differ from the initial-value ones discussed in Sections 11-13 in that the components of the state variable vector, instead of being prescribed at a single point, are partially known at two or more points in a given interval.

44

2

DIFFERENTIAL EQUATIONS

For example the problem of the string studied in the first chapter is a classical two-point-value problem; a continuous elastic beam furnishes a good example of a multipoint-value problem. The reader can find a number of additional examples with little effort. The treatment of problems of this sort is considerably more difficult than the treatment of initial-value problems. From a numerical point of view the difficulty is of a structural nature: Since we do not know completely the state vector at any point of the interval, we cannot initiate the numerical integration. Thus, to construct a numerical solution we must resort to more involved procedures. To fix ideas we consider the following two-point-value problem

du/dx= g(u, u), dv/dx = h(u, v),

~ ( 0=) C, (14.1) u(L)= d, defined in the interval [0, L ] .Assuming L < 1, we can ensure, by fixed-point techniques or otherwise, the existence and uniqueness of the solution for a large class of functions g and h. However, the determination of the largest interval for which this property holds is a matter of some difficulty that requires special treatment. This is a difficulty of an analytical nature, invariably attached to two-point-value problems. The construction of a numerical solution of (14.1) cannot follow a routine path since we do not have a complete set of intial conditions to start the numerical integration. Therefore, we are forced to implement ad hoc methods (and theory) to overcome this difficulty.In general we are led to use the method of successive approximations in conjunction with a convenient linearization of the equations. Thus, in a final analysis, the construction of a numerical solution will rely on our ability to solve systematically linear systems of differential equations subject to two-point (and multipoint)-value conditions. This is not, unfortunately, a routine problem. In the next section we show how to use the method of superposition to solve problems of this kind and indicate some of the difficulties that arise in connection with some numerical aspects. This in turn is the main motivation for the developments of Chapter 3.

15. Linear Two-Point-Value Problems We consider the two-point-value system

dy/dx= A y + Bz, dz/dx= Cy + Dz,

y(0) = 0

z(L) = ZO

(15.1)

where y , z are N-dimensional vectors, zo is a given N vector, and A , B, C, D are N x N matrices. For small L we can guarantee the existence and uniqueness of the solution of (1 5. I). Assume that this is the case. We introduce now

45

NOTES, COMMENTS, A N D BIBLIOGRAPHY

the associated principal matrices Y c l )Z, ( l )and Y(2),Z(’) satisfying the matrix differential equations dY“’/dx = A Y ( l )+ BZ“), dZ‘”/dX = CY“’ DZ“’,

Y(”(0) = I , Z“’(0) = 0,

(15.2)

dY‘2’/dx = A Y(2) + B z ( 2 ) , dZ‘2’ldX = c Y ‘ 2 ’ Dz‘2’,

Y(2)(0)= 0, Z‘2’(0) = I ,

(15.3)

+

and

+

respectively. Now, the solution of (15.1) can be written by superposition in the form y = y(1)c+ Y(2)d,

z = Z“’C

+ Z‘Z’d,

(1 5.4)

where c and d a r e two vectors that satisfy the algebraic system Y‘”(0)c + Y”’(O)d, zo = Z“’(L)c + Z‘2’(L)d. 0

=

(1 5.5)

Since Y(2)(0)= 0, (15.5) yields c = 0. Thus

d = [Z(2’(L>] - lzo .

(15.6)

The solution (15.4) may now be written y(x)= Y~2’(X)[z~2~(L)]-’zo, z ( x ) = Z(2’(X)[Z(2’(L)] - 1Zo .

(15.7)

We note that the possibility of using the representation (15.7) for effective numerical solutions depends on the possibility to compute the inverse of Z@)(L)accurately. The inverse certainly exists if (15.1) is to have a unique solution. The problem, however, lies in the difficulty in computing the ) This, a result of the limited accuracy of numerical value of Z ( 2 ) ( Laccurately. the digital computer, makes the matrix Z(”(L)more and more ill-conditioned as L grows. The consequence of all this is that the error made in the computation of d given by (15.6) will grow exponentially with L . This poses a severe limitation on this method. Chapter 3 is devoted to the development of an algorithmic solution of linear two-point-value problems that is exempt from the present undesirable characteristics. NOTES, COMMENTS, A N D BIBLIOGRAPHY

1. An excellent elementary text on ordinary differential equations is R. Bellman and K. Cooke, “ Modern Elementary Differential Equations,” 2nd ed. AddisonWesley, Reading, Massachusetts, 1971.

46

2 DIFFERENTIAL EQUATIONS

Classical references at a more advanced level are E. A. Coddington and N. Levinson, “ Theory of Ordinary Differential Equations.” McGrawHill, New York, 1955; E. Ince, “Ordinary Differential Equations.” Dover, New York, 1944.

2-7. Excellent books on matrices and differential equations are R. Bellman, “ Introduction to Matrix Analysis,” 2nd ed. McGraw-Hill, New York, 1970; F. R. Gantmacher, “The Theory of Matrices.” Chelsea, Bronx, New York, 1959.

A profound, more classically oriented treatment of systems of linear differential equations in connection with the applications can be found in C. Lanczos, Linear Differential Operators.” Van Nostrand Reinhold, Princeton, New Jersey, 1961. I‘

8. A classical treatment of ordinary differential equations of higher order can be found in the book by Ince quoted above. 9. A number of structural systems can be modeled in terms of first-order differential equations. For beams and frames see E. C. Pestel and F. A. Leckie, “Matrix Methods in Elastomechanics.” McGraw-Hill, New York, 1963.

For shells of revolution, linear and nonlinear, see the articles A. Kalnins, Analysis of Shells of Revolution Subjected to Symmetrical and Nonsymmetrical Loads, J. Appl. Mech. September 1964, pp. 467476; A. Kalnins and J. F. Lestingi, On Nonlinear Analysis of Elastic Shells of Revolution, J. Appl. Mech. March 1967, pp. 59-64.

10. An excellent exposition of various topics in systems theory may be found in R. Bellman, “Vistas in Modern Mathematics.” Kentucky Univ. Press,1968.

11. The stability of dynamical systems has been thoroughly investigated. Introductory to the field are J. Lasalle and S. Lefschetz, “ Stability by Liapunov’s Direct Method with Applications.” Academic Press,New York, 1961; S. Lefschetz, “ Differential Equations, Geometric Theory.” Wiley (Interscience), New York, 1957.

Classical references containing a number of results are R. Bellman, “Stability Theory of Differential Equations.” McGraw-Hill, New York, 1953. Also available in Dover, New York, 1969; L. C w i , ‘ 2 s ~ ~ @ uBehavior tk aod S¶abilityProblems jn Ordinary Differential q u a tions.” Springer-Verlag, Berlin and New York, 1963.

NOTES, COMMENTS, AND BIBLIOGRAPHY

47

12. Difference methods for the solution of initial-value systems are treated in detail in I. BabuSka, M. Prager, and E. Vitisek, ‘‘ Numerical Processes in Differential Equations.” Wiley (Interscience), New York, 1966; P. Henrici, “ Discrete Variable Methods in Ordinary Differential Equations.” Wiley, New York, 1962; R. D. Richtmyer, “ Difference Methods for Initial-Value Problems.” Wiley (Interscience), New York, 1957.

13. The stability of numerical processes is thoroughly treated in the book by BabuSka, Prhger, and Vithsek quoted in Section 12, and in P. Henrici, “ Error Propagation for Difference Methods.” Wiley, New York, 1963.

When the characteristic values of a given system are separated, i.e., their magnitudes vary greatly, the system is called stifl. The numerical solution of such systems, using single- or multiple-step methods, presents some difficulties associated with stability and accuracy. The problem is usually formulated so as to find the largest possible step size for which both the stability and the accuracy criteria are satisfied for a reasonable computing time. Examples of stiff systems abound in applications of dynamics, system identification, and so on. An introductory reading is provided in L. Lapidus and J. H. Seinfeld, “ Numerical Solution of Ordinary Differential Equations.” Academic Press, New York, 1971.

14. A number of results in connection with numerical applications can be found in P. B. Bailey, L. F. Shampine, and P. E. Waltman, “Nonlinear Two Point Boundary Value Problems.” Academic Press, New York, 1968.

The use of quasilinearization to solve two-point boundary-value problems is well illustrated in R. Bellman and R. Kalaba, “ Quasilinearization and Nonlinear Boundary Value Problems.” American Elsevier, New York, 1965.

15. For the numerical difficulties associated with the method of superposition of principal solutions, see the book by L. Fox quoted in the Bibliography of Section 1-7.

Chapter 3

Beamlike Structures

1. Introduction

In this chapter we discuss the solution of vector-matrix differential equations such as

du(x)/dx = B(x)u(x) + A(x)u(x) + p(x), - du(x)/dx = D(x)u(x) + C(x)u(x) + q(x),

(1.1)

where x is a real variable in the interval [0, L],and vector-matrix difference systems such as u(i + 1) = B(i)u(i)+ A(i)v(i) + p ( i ) , + 1) = D(i)u(i) C(i)u(i)+ q(i),

+

-u(i

(1.2)

where i is an integer variable, subject to a number of two-point-value conditions. Mathematical systems such as (1.1) .and (1.2), associated with boundary conditions, usually arise in applications of structural mechanics, in particular in connection with a linearized treatment of beamlike structures. The simplest example in this class is the well-known Euler-Bernoulli beam and its elementary generalizations. Examples of higher-dimensional problems arise 48

49

2. THE ELEMENTARY BEAM

very naturally considering more complex structures with one predominant dimension such as long cylindrical shells, folded plates, and thin-walled sections. Framed structures such as towers usually lead to consideration of discrete systems such as (1.2), although a distinction between difference and differential models does not need to be associated with any structural characteristic of the mechanical system. In fact, on analytical or computational grounds, it might be more convenient to use difference models for structural systems of the “continuous” type, and vice versa. The solution of two-point boundary-value problems governed by equations of the type (1. l) or (1.2) can be obtained in principle by using the method of superposition of principal solutions outlined in the previous chapters. There are, however, a number of reasons for developing alternative solution procedures. The basic one of course is the well-known inherent numerical instability of the method of superposition. In this chapter we construct solutions to linear two-point-value problems by using ideas of invariant imbedding. In this sense we are enlarging the scope and content of the introductory chapter. 2. The Elementary Beam

The simplest model for the bending of an elastic beam is furnished by the linear, fourth-order differential equation

known as the Euler-Bernoulli beam, where w is the displacement, EZ is the stiffness, and f is the load. Slightly more general and Guch more interesting is the system of four first-order differential equations

where rp = (d/dx)w is the slope and t and m are the shear force and bending moment, respectively. The first two equations of (2.2), i.e., those containing the derivatives of w and cp, are the constitutive equations of the bar, involving the flexibility coefficients aij(x); the remaining two are the equilibrium equations. It is easy to verify that (2.2) contains, as special cases, some of the classical formulations. For example, taking all = at2 = azl = 0,

aZ2= l/EZ,

(2.3)

50

3

BEAMLIKE STRUCTURES

and eliminating the three variables cp, t , and m from among the four equations of (2.2), we recover the classical Euler-Bernouilli equation (2.1). Similarly, the Timoshenko beam is found by taking

where A is the cross-sectional area, G is the shear modulus, and p is a formdependent coefficient whose value is 2 for a thin-walled circular section. In the general case [Eq. (2.2)], the coefficients a i j must satisfy the symmetry condition

a consequence of Betti’s theorem. On the other hand, the positive-definite character of the strain energy implies a,,

> 0,

a22 > 0,

a11a22

-4

2

> 0,

(2.6)

i.e., the positive definiteness of the matrix ( a i j ) .Clearly, if a,, = 0, the condition for positive definiteness of the strain energy reduces to a,, = aZ1= 0 and a,, > 0. Similar considerations hold when a 2 , = 0. 3. Higher-Dimensional Models. Cylindrical Shells The vector of the four components w , cp, t, and rn employed in the previous section to formulate the equations of the elementary beam proves to be inadequate when more involved structures are considered. A realistic treatment of more elaborate beam structures requires higher-dimensional descriptions. In order to illustrate the ideas, we present one of the most representative examples in this class involving cylindrical shells. Only the model equations are presented, not the derivation. Specific technical versions of cylindrical shells, such as various kinds of prismatic folded plates leading in a very natural way to equations of the type (1.1) and (l.2), are not considered here. Some references can be found at the end of the chapter. In addition to these examples, the reader should bear in mind that one of the most prolific sources of equations such as ( I . 1) or (1.2) is the method of finite elements in various discrete and semidiscrete versions, when applied to the solution of equilibrium problems in continuum mechanics. This will be illustrated i n Chapter 4, where adequate bibliography will be given. The equations of the theory of cylindrical shells can be written in terms of systems of first-order partial differential equations with respect to either x, the variable along the longitudinal direction, or y , the variable along the directrix. When we use x for the first-order derivatives, the equations reduce to (for notation and geometry see Fig. 3-1) Eq. (3.1) on p. 51, where :8

3. HIGHER-DIMENSIONAL MODELS. CYLINDRICAL SHELLS

0

3

0

Q

S l Q

I

Q"

51

ax

Q

0

n

N

Q" N

Q I

ax > l Q Q A

I

r"

F"

a

G

I

0

.

3

3

I

",.. >

I

.

52

3

BEAMLIKE STRUCTURES

Fig. 3-1

with respect to

denotes partial derivative of order

+ ( 1lP)mxy sx = 4 x + aym x y

sxy

= nxy

9

3

syx

= nyx

CI

and where

3

x, = -a,w.

(3.2)

If instead of x we use y for the first-order derivatives, then the equations read 0 0 aY

Ehu

-a x

Eh w

-

1

0 20 I + v

--

ax2

0

-1

P

where sy = qy

+ axmy,,

xy =

-ay w + ( i / p ) ~ .

(3.4)

3. HIGHER-DIMENSIONAL MODELS. CYLINDRICAL SHELLS

53

With the introduction of a specific behavioral hypothesis such as neglecting the longitudinal bending of the shell, we can achieve a simplification of the equations. We are, however, not interested in pursuing that direction. Our scope is to show that (3.3) and (3.1) may be satisfactorily transformed into a set of ordinary differential equations. In order to operate such a reduction in transcendentality we can resort to a number of procedures. A standard, very convenient method consists in replacing the partial differential operators appearing in the matrices by an approximate, finite-dimensional operator. In other words, by discretizing with respect to one variable. This, a semidiscretization procedure usually known as the method of lines, leads to a system of ordinary differential equations of order 8 M , where M is the order of the semidiscretization and the factor 8 represents the number of equations in the systems (3.1) or (3.3). We discuss this method in the next chapter in connection with the solution of partial differential equations. Another classical device to reduce the transcendentality of the partial differential equations is by means of Fourier decomposition in one direction. Suppose, for example, that the shell is supported on two diaphragms located at x = 0 and x = L and such that u = w = m, = xy = 0 on them. Assuming p x , p y , p z given by the Fourier expansions

where c(k

= kn/L, we

can write

ny = nk(y)sin akx, my = mk(y)sin c(k x,

Ehu = vk(y)sin akx, Ehx,,= xk(y)sin c(k x,

u,(y)cos c(k x, Ehw = Wk(y)Sin c(kX, Ehu

tk(y)cos @kX, sy = sk(y)sin ak x,

=

sxy

=

(3.6)

where the coefficients of the expansion are functions of the transverse coordinate y and satisfy the set of ordinary differential equations

: \I -1

0

O I \ (3.7a)

54

3 BEAMLIKE STRUCTURES

In Eqs. (3.1)-(3.7), E and v denote Young’s modulus and Poisson’s ratio, respectively, h is the thickness, and D = h2/12. 4. Semigroup Properties of Generalized Beams

Our main objective in this chapter is to discuss the solution of twopoint boundary-value problems associated with Eqs. (1.1) and (1 -2) using invariant imbedding ideas. There are several ways to do this. We have chosen to derive the pertinent results from consideration of principles of invariance on a general nonlinear beam. In so doing we shall formulate some semigroup properties of the solutions of generalized beams that will prove to be of interest analytically and computationally. To this end consider a generalized nonlinear beam described by the system duldx = g(u, U, x),

duldx = h(u, U, x),

(4.1)

where g and h are two generally nonlinear vector functions of the N-dimensional vectors u and u, the state variables of the beam. To fix ideas, and with no loss in generality, we shall identify u with displacements and u with forces, subject to the end conditions ~ ( 0=) a,

u(L) = 6,

(44

where L is the length of the beam and a and b are two arbitrary, N-dimensional vectors. We shall assume that Eqs. (4.1),subject to boundary conditions (4.2), have a unique solution, Now in the tradition of statics we make a cut in the beam as indicated in Fig. 3-2. Invoking a well-known principle in statics we know that forces and deformations in the remaining portions (0, X) and (X, L) of the beam will remain undisturbed if and only if we introduce the internal vector u(X), v ( X ) associated with the equilibrium of the entire beam, as an external effect, at both faces of the section x = X. This is indeed a consequence of the principle of causality in physics. Observe that the fi and only if statement is equivalent to the assumption of the existence and uniqueness of the solution

55

4. SEMIGROUP PROPERTIES OF GENERALIZED BEAMS

X

0

4x1

u(o)=a

L

v(L)=b

Fig. 3-2

of the underlining differential equations (4.1) and (4.2). We shall now construct some functional equations in the theory of beams. To this end let d(u(L), u(O), L) be a yet unknown N-dimensional vector function defined by d(u(L),u(O), L ) = the deformation vector at x = L of a beam of length L subject to forces u(L) at x = L and to de€ormationsu(0) at x = 0.

(4.3)

Clearly by definition (4.3) we have

449,401, L ) = u(L).

(4.4)

Using (4.3) in the portion (0, X ) of the beam we have

4x1= d(v(X), u(O), XI,

(4.5)

while consideration of the remaining portion ( X , L ) results in

u(L) = d(v(L),u(X),L - X ) .

(4.6)

Similarly, defining the N-dimensional vector function f ( v ( L ) ,u(O), L ) = the force vector occurring at x = 0 of a beam of length L subject to force v(L) at x = L and deformations u(0) at x = 0,

(4.7)

and considering the two portions of the beam we find the functional relations and We now observe, by inspecting Eqs. (4.5) and (4.6), that in order to determine u ( X ) and v ( X ) , it is enough to know function d. Let u ( X ) and v ( X ) , obtained from the solution of the simultaneous equations (4.5) and (4.6), be given by

u ( X ) = l(V(L), u(O), x,L )

(4.10)

NX) = k ( W , 4 0 ) , X , L),

(4.1 1)

and

56

3 BEAMLIKE STRUCTURES

respectively. Now, Eqs. (4.6) and (4.8), where u ( X ) and u ( X ) are given by (4.10) and (4.11), respectively, are the desired semigroup relations of the generalized beam described by the two-point boundary-value problem (4.1) and (4.2). In other words, those equations furnish functional equations that relate the missing boundary conditions u(L)and u(0) with the known end conditions u(0) and u(L) through the value of u and u at an interior point .'A These relations are a natural device for the derivation of a number of procedures devoted to compute functions g and$ In the next section we derive differential equations for g and f using a perturbation procedure. We finally note that if the beam is homogeneous, i.e., if we can remove the x dependence in the right-hand side of Eqs. (4.1), then functions d and f defined by (4.3) and (4.7), respectively, enjoy the obvious symmetry relations (4.12)

5. Invariant Imbedding. Perturbation Analysis In this section we use the functional relations previously derived to construct partial differential equations for the unknown functions d and f defined by (4.3) and (4.7), respectively. The limiting form of (4.5) as X - r L will furnish an equation for d while the limiting form of (4.8) yields an equation for f.Alternatively, we could consider limiting forms of (4.6) and (4.9), as X - t 0. In either case, the study of limiting forms is indeed a perturbation problem. The simplest way to proceed is to differentiate (4.5) and (4.8) with respect to X , the length of the imbedded beam, keeping in mind that the equilibrium of the beam should remain unaltered. This implies that u(0) and u(0) should be regarded as constants. In this fashion we obtain

du(X)/dX= dx 0 =fx

+ U du(X)/dX, + Vdu(X)/dX,

(5.11

(5.2)

where the subscript X stands for partial derivative with respect to X and where U and I/ are the Jacobian matrices U = (adifavj),

i, j = 1 , 2 , . .., N,

V = (8fi/auj),

i, j = 1,2,,..

(5.3)

and

.,N ,

(5.4) where di and fi are the components,-of vectors d and f, respectively, and where the u;s are the corresponding components of vector v. Now, a condition for the beam to remain unaltered after the perturbation at the end

57

EXERCISES

x = Xis that the total derivatives du/dXand du/dX appearing in the perturbed equations (5.1) and (5.2) satisfy the conditions of equilibrium and compatibility given by (4.1). Thus substituting (4.1) into (5.1) and (5.2) we obtain

( W W ( u , a, X ) = - W o , a, X)h(d(u,a, X ) , v, X ) + g(d(u, a, XI, U,

XI, (5.5)

and

( W W ( u , a, XI = - U u , a, W h M u , a, W , u, XI,

(5.6)

respectively, a system of quasilinear partial differential equations for the vector functions d and f in terms of u and X. We can easily derive initial conditions for d andf. In effect, making X = 0 in (4.5) and (4.8) we obtain and the pertinent initial conditions of Eqs. (5.5) and (5.6), respectively. We observe that dcan be determined using (5.5) above but the determination offdemands the solution of the coupled system (5.5) and (5.6). What is remarkable about the present analysis is that the invariant imbedding approach leads to the solution of quasilinear partial differential equations (5.5) and (5.6) subject to initial conditions (5.7) and (5.8), in contrast with the system of ordinary diflerential equations (4.1) subject to two-point boundary conditions (4.2) afforded by the formulation based in a classical imbedding. EXERCISES 1. Let u(x) satisfy the nonlinear boundary-value problem d2uldx2= au

+ bu3,

u(0)= 0,

u’(L) = v.

(a)

Show that din the equation

u(L)= d(u’(L),L) satisfies the quasilinear partial differential equation ad ad ( a d + b d 3 ) ~ =+u ,~

subject to the initial condition d(v, 0) = 0.

2. Consider Hamilton’s equations

(b)

58

3

BEAMLIKE STRUCTURES

where 4 = ( 9 i ) and p = ( p , ) are the generalized displacement and momentum vector, respectively, subject to boundary conditions 4(O)= a ,

p ( T ) = b.

(f)

Using the principles of invariance outlined in Section 5 derive partial differential equations and associated initial conditions for Y and T in the equations 9 ( T ) = W T ) , do), TI,

P(0) = T ( P ( T ) , 9(0), T ) . See the paper by Bellman et a/. in the Bibliography. 3. Consider the nonlinear two-point-value problem du/dx = g(u, u), dV/dx = h(u, v ) ,

u(0) = a ,

v(L) = b,

where u and u are scalar functions. Using invariant imbedding it is easy to show that r in the equation

u ( D = r(u(L), L ) satisfies the quasilinear partial differential equation (ar/au)h(r, u )

+ a r / Z = g ( r , L),

r(u, 0) = a.

(9

Show that (h) are the differential equations of the characteristic curves associated with Eq. (i).

6. Matrix Riccati Transformations

We now apply some of the preceding ideas to the solution of the linear two-point boundary-value system

+ Au + p , = DU + CV + 4,

duldx = Bu

-duldx

u(0) = a,

v(L) = b,

(6.1)

where u, u, p , q, a. and b are N-dimensional vectors and A , B, C, D are N x N matrices. In view of the linearity of the system we assume the missing boundary conditions, i.e., u(L) and v(O), to be linear combinations of the available data u(0) and v(L). Under this assumption we can write

+ a),

u(L) = R(L)U(L)4- T(L)U(O)

~ ( 0=) S(L)v(L)+ Q(L)NO) + W ) ,

(6.2)

where matrices R, T, S, Q and vectors a, p depend on L, the length of the structure, but not on the boundary conditions. In order to derive equations for R, T,S, Q , a, and p, we modify the length of the beam a small amount dL while keeping the equilibrium of the structure unaltered, and see how

59

6. MATRIX RICCATI TRANSFORMATIONS

those quantities change. We can study infinitesimal changes by taking the derivatives of u(L) and u(0) given by (6.2) with respect to L.In this manner we obtain

+

+

+

u’(L)= R’(L)v(L) R(L)d(L) T’(L)u(O) “(I.), 0 = s’(L)o(L) S(L)v’(L) Q(L)u(O) pyL),

+

+

+

(6.3)

where the primes indicate derivative with respect to L. Now, in order for u(x) and u(x), 0 5 x 5 L, to remain unaltered while enlarging infinitesimally

the length of the beam, the derivatives u’(L) and u’(L) must satisfy the equilibrium and compatibility equations (6.1). Thus, substituting u’(L) and u‘(L) given by (6.1) into (6.3) and eliminating u(L)and v ( 0 ) from the resulting equations by using (6.2), we obtain (R‘ - A

- BR - RC - BDR)u(L)+ (T’ - BT - RDT)u(O)

+ a’ - ( B + RD)a - p

- Rq = 0,

(6.4)

and (S‘ - SDR - SC)U(L)+ (Q’ - SDT)u(O)

+ 8‘ - S(q + Du) = 0,

(6.5)

where we purposely collected terms common in the boundary conditions u(0) and o(L).Since these values are arbitrary, (6.4) and (6.5) yield

+ BR + R C + RDR T’ = ( B + RD)T S‘ = S(DR + C) R’= A

(6.6)

Q’ = SDT a’ = ( B

+ RD)a + p + Rq

8‘ = S(DU + q),

i.e., a system of Riccati matrix differential equations for the quantities R, T, S, Q, u, and 8 subject to the initial conditions R(0) = Q(0) = 0,

T(0) = S(0) = I,

~(0) = B(0)

= 0,

(6.7)

where I is the N x N identity matrix. The initial conditions (6.7) follow from consideration of Eqs. (6.2) where L, the length of the beam, is set equal to zero. Clearly, Eqs. (6.6) hold for the general case where the components of the matrices A, B, C, and D are functions of x, i.e., when (6.1) is a system with variable coefficients. The reader should carefully follow the steps in the derivation of (6.6) to convince himself that no assumption on the constancy of the coefficients has been made.

60

3

BEAMLIKE STRUCTURES

EXERCISES 1. Consider the differential equation

d2UldX2= U,

where u is an N-dimensional vector subject to two-point value conditions u(0) = 0 and u'(L) = c. Show that the N x N matrix R(L) in the equation

u(L)= R(L)u'(L) satisfies the matrix Riccati differential equation

(b)

dRldL = I - R Z , R(0) = 0. (4 2. Consider the differential equations of a cantilever on elastic foundation subject to a concentrated load P and moment M at the end x = L, given by dwldx=p,+allt+alzm,

w(O)=O,

d~ldx=a12t+azzm, dtldx -kw, dmldx = t,

F(0) = 0, t(L) = P,

(d)

m(L) = M,

where k is the coefficient of the foundation and all are the flexibility coefficients defined in Section 2. Show that r l l , r 1 2 ,and r Z 2in the equations

+ +

w(L) = rll(L)t(L) rl2(L)m(L), y(L)= rlz(L)t(L) rZAL)m(L)

satisfy the differential equations drtl/dL=all + 2 r 1 ~ - k r : ~ , d r l ~ / d L = a 1+r22 2 -krllr,2,

rl1(0)=O, rl2(0)=O,

drZz/dL= aZ2- k&,

rz2(0)= 0.

3. Show that the flexibility matrix

'-"')

where ri, is given by Eqs. (f), satisfies the Riccati equation dx-

a12 a 2 2

-k

(" ')

("

0 0 R+R 1 0 ')-R(;

:)R,

R(O)=O.

(g)

4. Set k = 0 and consider a,, , i, j = 1,2, constants in Eq. (d). Under these assumptions, integrate Eqs. (f) in closed form.

5. Consider the two-point-value problem

+ +

dU/dX = BU Au, -dv/dX = DU CV,

~ ( 0-) Av(0) = 0,

u(L) = b, where A is a prescribed matrix. Show that R(L) in the equation

(h)

(0

u(L) = R(L)u(L)

satisfies the Riccati equation

R = A + BR + R C + RDR,

R(0) = A.

(j)

61

7. VALIDATION OF THE RESULTS

6. Instead of the boundary condition u(L) = b in (h), consider u(L) = 6. Set

(k)

u(L) = Z(L)u(L)

and show that Z(L) satisfies the equation -Z’=ZAZ+ZB+CZ+D,

Z(O)=A-’.

(1)

7. Validation of the Results The results of the previous section must be validated since they were obtained under the assumption that the linear relations (6.2) hold. We can easily prove the validity of that assumption. To this end we introduce the matrices of the principal solutions U ( l ) , V ( ’ )and U c 2 ) ,V ( 2 )such that U‘”(0) = z, (d/dx)U“’ = BU“’ + AV“’, (7.11 V‘l)(O)= 0, -(d/dx)V‘” = DU‘” + CV‘” , and (d/dx)U”’ = BU‘” + A V ” ) , U”’(0) = 0, (7.2) V‘Z’(0) = I . -(d/dX)V‘Z’ = DU‘2’ + cV‘z’, If u p , up denotes the particular solution that satisfies the equation u(P’(0) = 0, (d/dx)u‘P’ = BU(P) + Au‘” + p , u(P)(O) = 0, -(d/dX)U‘P’ = Du‘P) CU(P)+ q,

+

(7.3)

then u(L) and u(L) may be expressed by superposition as the linear combinations u(L) = U(”(L)u(O)+ U(Z’(L)U(O)+ u‘P)(L), (7.4) u(L) = V(”(L)u(O)+ V(2’(L)V(O)+ U ( P ’ ( L ) . Since u(0) and v(L) are given boundary conditions, we solve (7.4) for the missing boundary values u(L) and v(O), i.e., u(L) = U‘2’(V‘2’)-1u(L)+ [U”’ - 1/‘2’(v‘2’)-~V(1’]u(o) - U‘2’(V(2’)-1U(P’ + u(P’ 3

(7.5)

u(0) = (V(2))-’u(L)- (V(Z))-1V(1’u(O)- (V(2))-1U‘p’,

are evaluated at x = L. Comparison where U ( l ) ,V ( l ) ,U ” ) , V(’), u(J’),and dP) of (7.5) and (6.2) proves the desired property. Here we could go a little further and prove that the coefficients in (7.5) satisfy the matrix Riccati equations (6.6) subject to the initial conditions (6.7). For example, let R(L) be the matrix R(L) = u ( 2 ) [ V ( 2 ) ( L ) ] - 1 . Differentiating with respect to L, taking into account that

(d/dx)(V‘Z’)-1 = - ( V‘Z’)- ‘(d/dx)V‘Z’( V ( 2 ) ) -1 ,

(7.6) (7.7)

62

BEAMLIKE STRUCTURES

3

and that U"), V(') satisfies (7.2), we obtain dR,/dL= A + BU(2)(J,d2))-1 + (j(z)(V(2))-1C+

,7J(2)(V(2))-IDu(2)( v(2))-1,

(7.8) a result which shows that R satisfies the equation dRldL = A

+ BR + RC + RDR,

R(0) = 0.

We can show in a similar way that the remaining coefficients in (7.5) satisfy (6.6)-( 6,7), We finally note that the existence of the inverse (V('))-' in (7.5) and subsequent equations is guaranteed by the existence and uniqueness of the solution of the original two-point-value problem.

8. Discussion In Section 4 we pointed out that in the general nonlinear case, vectors < L, can be determined by using only function d. This property naturally carries over to the linear case studied in Sections 6 and 7. In fact, R , T, and LY in the first equation of (6.2), the linear counterpart of (4.4), can be determined by using the first, second, and fifth equations of (6.6), subject to the corresponding initial conditions (6.7), independently from the evaluation of S, Q , and p. As should be expected, the linear case offers further simplifications. For example, we observe that the first equation of (6.6) decouples from the rest of the system, indicating that matrix R can be evaluated independently from T, S , Q, LY, and p. This property is of analytical and computational interest. Additional properties can be found if we further specify the nature of matrices A , B, C, and D . For example, we may assume homogeneity along the structure, i.e., that matrices A , B, C, and D are x independent. In this case matrices R, T, S, Q and vectors LY, j? will be independent of which end of the structure is considered. Thus we can interchange zero and L in u and v appearing in (6.2), namely,

u ( x ) and v(x), 0 < x

+ T(L)u(L)+ u(L),

u(0) = R(L)v(O)

4 L ) = S(L)v(O)+ Q(L)u(L)+ LW).

(8.1)

Substituting u(L)and u(0) from (6.2) into (8.I), we find the following symmetry conditions RQ + T 2 = I , RS -k TR = 0,

+ QT=O, + + I). = 0,

SQ Rp ( T

Q.

QR+S~=I, ( S I)B = 0,

+ +

(8.2)

which hold if the structure is homogeneous. Clearly, (8.1) is the linear version of (4.12).

10. INVARIANT IMBEDDING. A ONE-SWEEP METHOD

63

9. A Two-Sweep Computational Method

Equations (6.6), a system of matrix differential equations of the Riccati type subject to initial conditions (6.7), can be integrated numerically, furand p in terms of the imnishing in this manner the values of R,S , T, Q, CL, bedding variable L, the length of the structure. Hence, substitution of those values in Eqs. (6.2) provides the missing boundary conditions of a family of structures with varying length L. The information furnished by this procedure might eventually satisfy our needs but in general our purpose will be to determine u and u at a number of points 0 I x1 < x2 < . . < x, I L within a fixed interval [0, L ] and for fixed values of the boundary conditions u(0) and u(L). When this is the case we can proceed as follows: We substitute u(x) given by the first equation of (6.2), i.e., U(X)

+

+

= R(X)U(X) T(x)u(O)C L ( X ) ,

(9.11

into the second equation of (6.1), to obtain (d/dx)= ~ -(DR C)V- D(Ta a) - 4,

+

+

u(L)= b, (9.2) where a stands for the prescribed value of u(0) and L is the required length of the structure. Equation (9.2) may be integrated backward, furnishing the values of u(x) at every point of the integration mesh. At the same time that we compute u by the backward integration of (9.2), we can compute u(x) using (9. I), where u(0) = a. Clearly, the integration of (9.2) requires the availability (storage) of R, T, and CL, quantities that have been computed in a previous run in the forward direction using the corresponding equations, (6.6) and (6.7). For this reason, we call the method a two-sweep procedure.

10. Invariant Imbedding. A One-Sweep Method We have shown above how to reduce the original boundary-value problem to an initial formulation by proceeding in two stages. First we construct a forward initial-value problem that yields the missing boundary conditions of a family of structures of variable length L. Second, by using that information, we integrate a backward initial-value problem that provides the complete solution of the problem, i.e., the values of u(x) and u(x) in [0, L]. In this section we wish to construct an initial-value problem that furnishes the values of u(x) and u(x) directly at any given point 0 < x 5 L without resorting to a second sweep or matrix inversions. To this end we shall use ideas of invariant imbedding. Using the same conceptual framework that led us to the formulation of Eqs. (6.2), we seek solutions of the form

u(x> = E(x, L)@) + m,0 4 0 ) + Y(X, L), V(X) = G(x, L)u(L) H ( x , L)u(O) d(x, L),

+

+

(10.1)

64

3

BEAMLIKE STRUCTURES

where the matrices E, F, G, H a n d the vectors y, 6 are defined only for x I L. Equations (10.1) can be justified easily on account of the linearity of the problem. Comparison of (10.1)with (6.2) shows that

E(L, L) = R(L), G(L, L) = I, y(L, L ) = a(L),

F(L, L) = T(L), H(L, L ) = 0,

(10.2)

6(L, L ) = 0.

We can now proceed to the derivation of pertinent differential equations for the quantities E, F, G , H, y, and 6 appearing in (10.1). To this end we use a perturbation technique, similar to the one employed in Section 6. We differentiate (10.1) with respect to the length of the structure L. Note that during this process u(x) and v(x) remain constant. We obtain

+ +

+ +

+

0 = E'(x, L)v(L) E(x, L)v'(L) F'(x, L)u(O) y'(x, L), 0 = G'(x, L)v(L) G(x, L)v'(L) H'(x, L)u(O)+ 6'(x, L),

(10.3)

where the primes indicate derivative with respect to L. For u(x) and v(x) to remain undisturbed while changing the length of the structure, u' and v' appearing in (10.3) must satisfy the equations of equilibrium and compatibility given by (6. I). Thus, using (6.1) in (10.3) and collecting terms common in the boundary conditions we obtain

+

( d / W E ( x ,L) = E(x, L)[D(L)NL) W)I, VldL)F(X,L) = E(x, L ) W ) W ) , ( d / W G ( x ,L) = G(x, L ) [ W W ( L ) W ) I , (d/dL)H(x, = G(x, L ) W ) W ) , (WL)Y(X,L) = E(x, L)twJa(L) 4W)1, ( W M X , L) = G(x, ~)tD(L)a(L) 4 w 1 ,

+

(10.4)

+ +

a system of equations for E, F, G, H, y, and 6 subject to the obvious initial conditions E(x, x ) = R(x), G(x, X ) = I, y(x, x ) = a(x),

F(X, x ) = T(x), H(x, x ) = 0, 6(x, x ) = 0.

(10.5)

We observe now that the computation of u(x) and v(x) using (10.1) requires the integration of (10.4), which in turn requires the values of R(L), T(L),S(L), Q(L),a(L),and p(L)for L 2 x . The actual computation is carried out in the following way: We first integrate (6.6)-(6.7) in the forward direction until L reaches the value x , the point at which we wish to evaluate u and v. At this point we adjoin the systems (10.4) and (10.5) and continue the

65

11. SELF-ADJOINT SYSTEMS

integration in the forward direction until the value of L has been reached. The vectors u ( x ) and u(x) can be computed using (10.1) evaluated at the prescribed values of u(0) and v(L), i.e.,

+ F(x, L)u+ Y(X, L), V ( X ) = G(x, L)b + H(x,L)u+ 6 ( ~L). , U(X) = E(x,L)b

(10.6)

This is indeed a one-sweep forward procedure derived from invariant imbedding considerations. We observe that in the present method, x , the variable of the classical imbedding, is maintained fixed while the length of the structure, a constant in classical formulations, is the present variable of integration. 11. Self-Adjoint Systems

In previous sections we have shown that the missing end conditions of generalized beams described by differential equations and boundary conditions of the form

may be given in the form (11.2)

where R, T , S, Q , a, and P are given by the solution of the differential equations (6.6) and (6.7). In the foregoing analysis, no specification on the nature of matrices A , B, C , and D was made. It is known, however, that in dealing with problems in structural mechanics we can generally ensure the following properties on those matrices : A

= AT > 0,

D

=

DT 2 0 ,

B

=

CT,

(11.3)

by means of a proper choice of vectors u and u. This is particularly true when u and u are the generalized displacement and force vector, respectively, and such that there is a correspondence between the ith components u iand ui,

in an energy sense. System ( l l . l ) , subject to conditions (11.3), is called a self-udjoint system and is such that it can be derived from the variation of a certain functional. Interest in systems of this kind is apparent, because we expect that the additional information furnished by (1 1.3) will be reflected in a deeper understanding of the solution of the problem. And this is indeed the

66

3

BEAMLIKE STRUCTURES

case. Inspection of Eqs. (6.6) and (6.7), having (11.3) in mind, shows that matrices R and Q are symmetric, i.e., R

= RT,

Q = QT

(11.4)

and, additionally,

ST= T.

(11.5)

In other words, the composed 2N x 2N matrix Sc R

“=(S

T

Q)

(11.6)

is a symmetric matrix. This could be anticipated on mechanical grounds since the symmetry of S, is a consequence of Betti’s law. In addition, we could go a little further and prove that R 0 and Q 2 0. We shall see this in Section I2 in connection with problems of stability. The composed matrix S, given by (11.6) appears to have been little investigated in the literature of structural analysis and therefore bears no name. We shall refer to it as the scattering structural matrix or just the scattering matrix, by analogy with the scattering matrix appearing in wave propagation and transmission line theory. It is clear that the results outlined here for the scattering matrix still hold in the case where v and u are not forces and displacements, respectively, but generalized state vectors constructed such that the scalar product (u, u) represents a quantity proportional to the work of deformation. By exploiting this notion we can derive a number of mixed formulations of interest in various applications. We leave this as a research exercise to the reader. When u and u denote forces and displacements, respectively, then it is customary in structural analysis to introduce a stiffness matrix defined by

=-

(11.7)

where the minus sign in front of u(0) indicates that the forces denoted by the components u(0) and v(L) of the state vector are acting on the structure. The connection between the scattering matrix S, and the stiffness matrix (K,) may be shown to be K l l = R-’, K 2 , = -SR-‘,

K I 2= -R-’T, K22 = S R - ’ T - Q.

(11.8)

By making use of the fact that R is positive definite and that ST= T, the reader may easily show that ( K i j )is also positive definite, a property to be expected on mechanical grounds.

67

12. NUMERICAL STABILITY

EXERCISES 1. Write Eqs. (2.2) in the form of Eqs. (11.1). Assuming a l I , a I 2 ,and azz to be constants,

integrate the differential equations in closed form and derive expressions for the scattering matrix S, given by (11.6). 2. Consider rill = a12= uZ1= 0 in problem 1. Show that the submatrices R, T, S, and Q of the scattering matrix S, are given by =

L3/3EI Lz/2EI), (L2/2EI LIEI

=

(1L

01 ) '

Q=C 3. 3. Assume that u and u are forces and displacements, respectively. Exhibit a physical meaning for R, T,S, and Q in (1 1.6) in terms of stiffness, flexibilities, and transmission concepts.

12. Numerical Stability

$$$tion arc of theoretical

The properties just discussed in the last ifld practical (numerical) interest. We see briefly here some of the consequences of those properties in a computational context. In particular we sketch a proof of the stability of the Riccati differential equations (6.6) and (6.7) with respect to small numerical errors introduced in the computational process. We only discuss the stability of R as given by the first equation of (6.6), namely,

R' = A

-+ BR i- RBT i- RDR,

R(0) = 0,

(12.1)

where C has been replaced by BT.We consider a small perturbation on R given by (12.2)

R=R+S,

where R satisfies the perturbed equation

R' = A i- BR i- KBT i- RDR,

(12.3)

and S is a matrix, small in norm, such that S(0) = A, the initial error. Eliminating K and R' from Eqs. (12.1)-(12.3), and neglecting the higher-order term SDS, we obtain

S' = ( B + RD)S + S(BT+ DR), an equation for the perturbation matrix S.

S(0) = A,

(12.4)

68

3

BEAMLIKE STRUCTURES

The behavior of S in Eq. (12.4), a linearized version of Eq. (12.1), will establish the desired stability properties. We now leave as an exercise to the reader the task to prove that the solution of (12.4) is given by (12.5)

S ( t ) = X ( t ) AXT(t),

where X is the fundamental matrix of B + R D , i.e., X satisfies the matrix differential equation X' = ( B + RD)X, X ( 0 ) = I, ( I 2.6) and where it is tacitly assumed that D is a symmetric matrix. It follows now that the desired stability depends on the characteristic values of B RD. When B + R D is a stability matrix, i.e., when the real part of all its characteristic values is negative, then there exist two positive constants k and y such that llXll I k e - Y x , (12.7)

+

where the norm is taken to be IIAII = II(aij>II =

1

(12.8)

IaijI.

i, j

Using (12.7) in (12.5) we can easily bound the perturbation matrix S, namely, IlSll = IIXAXTII I llXllzllAll i k211Alle-2Yx.

(12.9)

Equation (12.9) now establishes the fact that, under the pertinent assumptions, the errors will propagate bounded by a negative exponential function, i.e., will tend to zero as x 4 00. This is usually called exponential stability, a very desirable property in numercial appications. When exponential stability cannot be ensured, we must look for an alternative, less demanding stability criterion. In most practical cases all we shall require is that, for x sufficiently large, llSll does not exceed a prescribed value. This will cause the accumulated error to grow not faster than linearly. We are not concerned with processes of this type here. It remains now to establish conditions under which the exponential stability holds. We note first that when A > 0 and D i 0, R ( t ) and R'(t) given by (12.1) will be positive definite and bounded for all t > 0. This follows from the representation of R in terms of X, the fundamental matrix of B + R D , given by

. which in turn follows from (12.1) written in a slightly different manner,

namely R'

= (B

+ RD)R + R(BT + DR) + A - RDR,

R(0) = 0.

(12.11)

69

EXERCISES

If Y(x) denotes the fundamental matrix of B, i.e., Y' = BY,

Y(0) = I ,

( 12.1 2)

then R in (12.1) may also be represented by R(x) =

j:

Y ( x )Y - ' ( O [ 4 5 ) + R(OD(OR(41Y - ' ( < )Y ( x ) d5. (12.13)

Since A > 0 and D 5 0, it follows from (12.10) and (12.13) that

0 < R(x) 5 JxY(x)Y-'(&l(5)Y-'(5)Y(x)d5,

x > 0,

(12.14)

0

the desired result. To prove now that B + R D is a stability matrix amounts to proving the exponential stability of the differential system Z' = ( B

+ RD)z,

~ ( 0=) C .

( 12.1 5)

This can easily be done by constructing a Liapunov function V(z) such as V ( Z )= (z,R - ~ Z ) .

(12.16)

In fact, taking into account that R-' satisfies the Riccati differential equation -dR-'/dx

=

D -I- BTR-'

+ R - ' B + R-IAR-',

x > 0, (12.17)

and that z in (12.16) satisfies (12.15), we find dvjnx

= (QZ, z ) ,

Q

=

D - R - ~ A R - '< 0 ,

x > 0, (12.18)

establishing the desired asymptotic property. An immediate consequence of this result is the stability of the original two-point-value system (11.1). This can be shown in a number of ways. A very expressive one follows from consideration of the transmissibility matrices T (or S ) given by the second (or third) equations of (6.6)-(6.7). In fact, since B + R D is a stability matrix, T will be asymptotically stable, thus establishing the desired stability of the model with respect to boundary conditions. EXERCISES

1. Show that the solution of the matrix equation dZ/dx = A 2 is given by Z

= XCY,

+ ZB,

Z(0) = C,

where X and Y satisfy dXjdx = A X ,

X(0)= I,

dY/dx = YB,

Y(0) = I .

70

3

BEAMLIKE STRUCTURES

where the norm is given by (12.8). 3. Show that if C > 0, then BCBT > 0.

> 0 and D 5 0, show that the numerical process indicated by Eqs. (6.6)(6.7) and (10.4)-(10.5) is exponentially stable.

4. Assuming A

5 . Under similar assumptions show that u, given by the differential equation (9.2), namely,

-du/dx

=(DR

+ BT)u + D(Tu + a ) + q,

u(L)= b,

(4

is exponentially stable in the backward direction. 6. Show that if instead of A

> 0, we consider

where A , > 0, the results of this section hold.

13. General Two-Point Boundary Conditions

In previous sections, in an effort to clarify the exposition of the main ideas, we have systematically discussed end conditions of the type given in (6.1). If the problem demands consideration of other boundary conditions, we have at least two options. In the first place we could use R, T, S, Q, CI, and p computed with (6.6)-(6.7) in (6.2) and solve the resulting system of algebraic equations in the corresponding unknowns. Or, alternatively, we can derive appropriate differential equations to determine the missing end conditions directly. We shall indicate how to proceed in the second case. To this end, we rewrite system (6.1) in the form of a 2N-dimensional system (d/dx)z =

wz + w,

(13.1)

where

and subject z to the most general linear two-point boundary conditions Kz(0) + Mz(L) = m,

(1 3.3)

where m is a given 2N-dimensional vector and K, M are prescribed 2N x 2N matrices. We assume that a solution of (13.1) and (13.3) exists and is unique for all L in some interval 0 2 L 2 L,. Uniqueness of this system requires at least the existence of 2N linearly independent boundary conditions. This imposes a condition on matrices K and M in (13.3), namely, rank(KM) 2 2N,

(13.4)

71

13. GENERAL TWO-POINT BOUNDARY CONDITIONS

where ( K M ) is the 2 N x 4N rectangular matrix formed with the 2 N columns of K and the 2N columns of M . With these restrictions in mind we consider the linear transformation

z(L) = P(L)m + r(L)

(13.5)

in terms of the unknown matrix P and vector r. In order to obtain equations for P and r we differentiate (13.5) with respect to L, i.e., z'(L) = P'(L)m

+ P(L)rn' + r'(L),

(13.6)

where the primes indicate derivative with respect to L. Differentiating Eq. (13.3) with respect to L, we obtain

m'

(13.7)

= Mz'(L),

an equation that substituted into (1 3.6) yields

(P' - WP + PMWP)m + r'

+ PMw - Wr + PMWr = 0.

(13.8)

Equation (1 3.8) must hold for any vector m, the prescribed " end " conditions. Thus

P' = WP - PMWP,

r'

=

Wr - PMWr - PMw,

(1 3.9)

a system of Riccati-like equations that generalizes Eqs. (6.6). Unfortunately, we cannot ensure in this general case that a set of initial conditions at L = 0 will exist. In fact, P(L) and r(L) appearing in (13.5) need not be defined at L = 0, thus preventing a numerical integration from the origin. When this is the case we define our initial conditions in a point L = s near the origin, and start the numerical integration at that point. This can be certainly done because we can always evaluate P ( s ) and r(s) by means of superposition of principal solution, asymptotic methods, etc. A case of some particular interest arises when the same components of the state vector are prescribed at both ends of the interval. To fix ideas, consider the system of differential equations U'

= BU

+ Av,

-v' = DU

+ CV,

( 1 3.10)

subject to boundary conditions v(0) = 0,

v(L) = b,

(13.11)

where u and v are N-dimensional vectors. We set as usual U(L) = R ( W ( L ) ,

(13.12)

and find for R the well-known Riccati equation

R'= A

+ BR + R C + RDR.

( 1 3.13)

12

3

BEAMLIKE STRUCTURES

The difficulty here lies in the fact that R is not defined at L = 0. This can be seen by inspection of (13.11) and (13.12). In order to integrate (13.13) we shall find an initial condition at a point s near the origin. To this end we consider the auxiliary boundary conditions u(0) = 0, u(L) = a such that the associated Riccati equation is well behaved at L

= 0.

( 13.14) We set ( 13.1 5)

u(L) = Z(L)u(L)

and find with no difficulty -Z’ = ZAZ + ZB + CZ + D, Z(0) = 0, (1 3.16) a Riccati differential equation for Z subject to initial conditions at L = 0. Comparison of (13.12) and (13.15) shows that

R(s) = z ( s ) - ’ ,

( 13.17)

an equation that furnishes the desired initial condition for (13.13). The term Z ( s ) in (13.16) may be obtained by integration of (13.16) or, if s < 1, by means of approximate asymptotic expansions. EXERCISES Consider the boundary-value problem U’ = BU

-u’

=

+ Au,

DU + CU,

~(0) = a, u(L)= 6 ,

where u and u are N-dimensional vectors. Show that the scattering matrix

in the equation

satisfies the 2 N x 2 N Riccati equation

Consider the two-point-value problem

u” - u = 0,

u(0) = 0,

u(L) = c.

Show that r(L) in the equation u’(L) = r(L)u(L) satisfies the Riccati equation r’(L) = 1 - r2(L),

where s< 1.

r(s) 2 I/s,

73

14. DIFFERENCE EQUATIONS

14. Difference Equations

In this section we show how to generalize some of the results derived in Chapter 1 for difference equations. To this end it is enough to consider the homogeneous system u(i

-u(i

+ 1) = B(i)u(i) + A(i)u(i), + 1) = D ( i ) u ( i )+ C(i)v(i),

u(1) = 0,

u ( N ) = b,

(14.1)

where u(i) and u ( i ) are N-dimensional vectors. Assuming existence and uniqueness of the solution, we express the missing boundary conditions in terms of the given data, i.e.,

4 N )= R ( N ) W ) ,

(14.2)

where R ( N ) is an N x N matrix to be determined. Proceeding as usual we consider u(N + 1)

= R(N

+ l)u(N + 1).

(14.3)

Eliminating u(N + l), u(N + l), and u ( N ) from Eqs. (14. I)-( 14.3), we obtain

R(N

+ 1) = - [ A ( N ) + B ( N ) R ( N ) ] [ C ( N+) D ( N ) R ( N ) ] - ' ,

(14.4)

a recurrence relation for R ( N ) subject to the obvious initial condition R(1)

(14.5)

= 0.

Similarly, if we wish to determine u(i),i < N , by using a one-sweep procedure, we set u(i) = S(i, N ) u ( N ) ,

(14.6)

where S(i, N ) is an N x N matrix to be determined using the well-known technique based on principles of invariance. To this end, enlarging the interval to N + 1 while keeping u ( i ) constant, we have u(i) = S(i, N

+ l)u(N + 1).

(14.7)

+

Eliminating u(i), u ( N ) , and v(N 1) from the second equation of (14.1) and Eqs. (14.2), (14.6), and (14.7), we obtajn the recurrence relation for S(i, N ) : S(i, N

+ 1) = -S(i,

N)[C(N)

+ D(N)R(N)]-I,

(14.8)

subject to the initial condition S(i, i ) = R(i).

(1 4.9)

74

3

BEAMLIKE S T R U C T U R E S

EXERCISES 1 . If in Eq. (14.1) ~ ( 1 = ) 0 is substituted by the boundary conditions u ( l ) = 0, which are the pertinent initial conditions for R ( N ) given by (14.4)? 2. Let u ( i ) = T(i,N ) u ( N ) .

Find a recurrence relation for T(i, N ) and the associated initial condition. 3. Consider o(1)

=0

and u ( N ) = b in (14.1). Set

u ( N ) = Z(N)u(N). Find a recurrence relation for Z ( N ) and show that Z ( N ) = R ( N ) - ' , where R ( N ) is given by (14.4).

15. Transfer Matrices

At this point the reader is well aware that a class of structural systems can be suitably described in terms of matrix-difference systems of the type u(i + 1) = B ( i ) u ( i )+ A ( i ) v ( i )+ p ( i ) , - v ( i + I ) = D(i)u(i)+ C(i)v(i)+ q(i),

(15.1)

subject to a number of two-point boundary conditions. To fix ideas and with no loss in generality, we shall assume that u ( i ) represents the generalized displacements and v(i) the generalized forces. In the same vein, we consider boundary conditions ~(l= ) a,

v ( N ) = b,

( 1 5.2)

where a and b are the prescribed values of displacements and forces, respectively, at both ends of the structure. We can alternatively write ( 1 5.1) in the form

z(i+ I)

= Z(i)z(i),

(15.3)

by the simple device of making B(i) A(i) p(i) (15.4) - D ( i ) -C(i> q(i) z(i)= _________________ _____________, ____... 0 0 / 1 In view of (15.2) we know z(1) incompletely because the value of v(1) is missing. However, we can write formally ~

I

~

4 2 ) =Z(l)Z(l), z(3) = Z(2)Z(2)= Z(2)2(1)Z(I), (15.5)

z ( N ) = Z ( l ) 2 ( 2 ) . . . Z ( N - l)z(l).

75

16. THE SCATTERING MATRIX

If n ( N ) denotes the matrix product r I ( N ) = Z( 1)Z(2). . .Z(N - I),

(15.6)

we can write the last equation of (15.5) under the form z ( N ) = rI(N)z(l).

(15.7)

Partitioning the system (1 5.7) conveniently and imposing the boundary conditions (15.2), we can solve for u ( N ) and u(l), the missing boundary conditions. After this linear algebraic problem is carried out, we can use the recurrence relation ( 15.3) directly because now z( 1) is known. Matrices Z(i ) are called state transition matrices or transmission matrices in a structural engineering context. The present method is known as the transfer matrix method. The difficulty with this method lies in the ill-conditioning of the matrix product r I ( N ) as N increases. The reason for this should be traced back to the limited accuracy of the digital computer. In other words, for a fixed round-off error, and a prescribed accuracy in the elements of r I ( N ) , there exists a value of N for which this accuracy is reached. There are several methods of preventing the fatal numerical instability associated with the transfer matrix method. The reader is urged to consult the Bibliography at the end of the chapter. We devote our next section to the description of a method that is exempt from such an undesirable feature. EXERCISE

+

Consider the element i, i 1 of a generalized beamlike structure and let u(i) and u(i) be the force and displacement vector, respectively, at section i. A stiffness matrix for the element i, i 1 is defined by the equation

+

Show that the coefficients A(i), B(i), C ( i ) , and D(i) of the transfer matrix (15.1) are given in terms of the stiffness matrices K,, by means of the relations B = -K- ZI'KZ,, D = KllKF:KZz - K12,

A = - K - 2I 1 ,

C = K11KC1'

(b)

16. The Scattering Matrix

Instead of the invariant imbedding approach outlined in Section 14 for solving difference equations, we can employ alternative procedures based on invariant imbedding ideas. Here we present one of them based on the scattering matrix. In the next section we outline another one based on the flexibility matrix.

76

3

BEAMLJKE STRUCTURES

In order to explain the method it is enough to consider the homogeneous case, i.e., p ( i ) = q ( i ) = 0. We assume boundary conditions (15.2). Instead of starting from the state transition matrix (16.1) we start with the scattering matrix X ( i ) of the element i, i + 1 of the structural system, given by (1 6.2)

which is defined such that (16.3)

We assume that matrices B(i),F(i), Y(i),and I ( i ) are known for each one of the elements composing the structure. We now define the global scattering matrix of the system of i elements by means of the relations (16.4) Our task is to derive recurrence relations for R ( i ) , T ( i ) ,S ( i ) , and Q(i), i.e., for the global scattering matrix. This is a well-known problem for us. We solve it by considering the structure of i + 1 elements and associated scattering matrix (16.5) Mathematically our problem now amounts to eliminating the two intermediate vectors u(i) and u(i) between Eqs. (16.3) and (16.4) and to compare the result with (16.5). Since there are four equations to eliminate two quantities, we can operate in a number of ways. For example, eliminating u(i) between the first equation of (16.4) and the second of (16.3), we obtain

u ( i ) = A(i)Y(i)u(i+ 1) + A(i)b(i)T(i)u(l),

(16.6)

where A(i)

= (I -

I(i)R(i))-'.

(16.7)

If, on the other hand, we eliminate u(i) between the same two equations, we obtain u ( i ) = T(i)R(i)Y(i)u(i + 1) + T(i)T(i)u(l), (16.8) where (16.9) T(i) = ( I - R(i)d(i))-'.

77

16. THE SCATTERING MATRIX

We can now use v ( i ) given by (16.6), or u ( i ) given by (16.8), or both, to derive the desired recurrence relations. For example, substituting (16.6) in the first equation of (16.3), and (16.8) in the second equation of (16.4), and comparing the result with (16.5), we obtain R(i + 1) = W ( i ) F(i)T(i)R(i)Y(i) T ( i + 1) = F(i)T(i)T(i) (16.10 S(i + 1) = S ( i ) A ( i ) Y ( i ) Q(i + 1) = S(i)A(i)9(i)T(i) + 9ii) a system of recurrence relations for the matrices R ( i ) , T ( i ) ,S(i), and Q ( i ) subject to the obvious initial conditions

+

R( 1) = 0 , S( 1) = I, (16.11) T(1) = I , Q(1) = 0. By an appropriate choice of the state vectors u and u we can always make the scattering matrix of the i, i + 1 element symmetric, i.e., 9 ( i )= WT(i), 9(i)= 21T(i), Y ( i )= F T ( i ) . (16.12) Using (16.12) in ( 16.10) and (16.1 l), the reader will find no difficulty in proving that R ( i ) = RT(i), Q ( i ) = QT(i), S(i) = T T ( i ) , (16.13) i.e., the global scattering matrix associated with the structure of i elements is also symmetric. This, a manifestation of Betti’s law derived from invariant imbedding considerations, permits the deletion of one equation in (16.10) and (16.1 1). We may go a little further and verify by inspection of Eqs. (16.10)-(16.11) that W ( i )> 0 and 9(i)2 0 imply R ( i ) > 0 and Q(i) 2 0. This property is of the utmost importance since it guarantees the numerical stability of the process given by (16.10)-(16.1 I). Finally, we wish to note that the semigroup properties associated with two-point-value conditions studied for the general nonlinear case in Section 4 acquire particular relevance in the present linear case. Combining (16.3)(16.5) and ( 1 6. lo), we can symbolically write R ( i + 1) T(i+ 1) R(i) T(i) g(i) F ( i ) (s(i 1) Qci 1)) = (s(i) Qci)) * ( Y ( i ) 9(i)) I ) + F(i)T( i)R(i)Y(i) Y(i)T(i)T(i) S(i)A(i)Y(i) 9(i) S(i)A(i)2(i)T(i) ’ = ( 1 6.14) where the star symbol (*) is used in order to avoid confusion with the ordinary product of matrices. In other words, the rule of composition of scattering matrices is given by the star product while the rule of composition of transfer matrices is given by the ordinary product of matrices.

+

+

?(.

+

1

78

3 BEAMLIKE STRUCTURES

EXERCISES 1. Find the components A , B, C, D of the transfer matrix given by (15.1) in terms of the and 9 of the scattering matrix given by (16.3). components 22, 3, 9, 2. Find the components 9?,F, 9,and 9 of the scattering matrix in terms of thecomponents Ki, of the stiffness matrix given by Eq. (a) in Exercise 1 of Section 15. 3. instead of boundary conditions (15.2), assume u(1) = a,

u ( N ) = b.

Find recursive equations for 3, F, 3, and

0 in Eq. (c).

(a)

Set

and

4. Show that R(i) = R - ‘(i).

5. Consider the nonhomogeneous case p ( i ) # 0 and q(i) # 0 in (15.1), together with boundary conditions (15.2). We can generalize Eqs. (16.3) and (16.4) by writing

and

Show that y(i) and 6(i) are given by the recurrence relations

y(i

6(i

+ 1) = 4)+ F(i)r(i)[y(i)+ R(i)B(i)], + 1) S(i)h(i)[P(i) + 2(i)y(i)], =

y(1) = 0, S(1) = 0.

17. The Flexibility Matrix

If, instead of boundary conditions (1 5.2), we consider ~ ( 1 )= U ,

u ( N ) = b,

(17.1)

it might be more convenient to start the analysis with a flexibility matrix for the structural element i, i + 1:

(17.2)

79

NOTES, COMMENTS, A N D BIBLIOGRAPHY

Our purpose here is to derive recurrence relations for the components of the global flexibility matrix

(17.4) in terms of the components of 9 given by (17.2). We proceed here in exactly the same way we did in Section 16. We leave to the reader the task of proving the following recurrence relations subject to initial conditions, for the components of the global flexibility matrix:

F1l(i+ = 911(i)- 912(i)K(i)F21(i)7 F11(2)= F1l(l), F12(i 1)= F12(i)K(i)F12(i), FlZ(2)= 9 1 2 ( 1 ) ? (17.5) FZ1(i+ 1) = Fzl(i)K(i)Fz1(i), F21(2)= 921(1)7 F22(i + 1) = F22(i) - FZl(i)K(i)FIZ(i), F22(2)= ~ 2 2 ( ~ ) 9

+

where the matrix K ( i ) is given by

+

(17.6) ~ ( i=)(Fll(i)FZ2(i))-'. We leave as an exercise to the reader the proof that 9 > 0 implies F = (Fij)> 0. We also leave to the reader the task of developing a similar procedure by employing a global stiffness matrix instead of a flexibility one. EXERCISES 1. Assuming that F given by (17.2) is positive definite, show that F = (Ftj) > 0. 2. Under the assumption of Exercise 1, show that F, I ( i )

+ F2&) in (17.6) is never singular.

3. Show that the components Fijof the flexibility matrix are related to the components of the scattering matrix through the relations - TQ-lS, -Q-'S,

F11

=R

Fz,

=

F12 = -TQ-', F22 = - Q - l .

(a)

NOTES, COMMENTS, A N D BIBLIOGRAPHY

2. A number of references and historical remarks on the classical beam may be found in S. Timoshenko, History of Strength of Materials." McGraw-Hill, New York, 1953. "

The formulation of the beam presented in this section, which is the most general linear model attainable without increasing the dimensionality of the state vector of the classical beam, was employed by V. Tvergaard in V. Tvergaard, Free Vibrations of Beam-like Structures, Internat. J. Solids and Structures 7 (1971), 789-803; V. Tvergaard, Beam Analysis of Axisymmetrical Shells, The Danish Center for Appl. Math. and Mech., The Tech. Univ. of Denmark, Rep. No. 10, January 1971.

80

3

BEAMLIKE STRUCTURES

This formulation, in turn, is a generalization of J. G. Simmonds, Modification of the Timoshenko Beam Equation Necessary for ThinWalled Circular Tubes, Inrernat. J. Mech. Sci. 9 (1967), 237-244.

3. Higher-dimensional models are required in a number of applications. For the construction of models for elastic plates using the three-dimensional theory of elasticity as the starting point, see R. D. Mindlin, An Introduction to the Mathematical Theory of Vibrations of Elastic Plates. Monograph prepared for U.S. Army Signal Corps Eng. Lab., Fort Monmouth, New Jersey, 1955.

Cylindrical shells of various types also require higher-dimensional descriptions. For example,

Prismatic Folded Plates M. H. Gradowczyk, Dissertation, Technische Hochschule, Graz (1961); N. DistCfano and M. H. Gradowczyk, Creep Behavior of Homogeneous Anisotropic Prismatic Shells, Proc. I.A.S.S. Symp. Non-Classical Shell Probl., Warsaw September 1963, pp. 409-427; J. E. Goldberg, W. D. Glauz, and A. V. Setlur, Computer Analysis of Folded Plate Structures, Congr. Z.A.B.S.E., 7rh Preliminary Rep., 1964, p. 55.

Thin-Walled Beam Shells of Closed Sections V. Z. Vlasov, ‘‘ Thin-Walled Elastic Beams,” translated from the Russian, published for NSF and the Dept. of Commerce, USA, by the Israel Program for Scientific Translations, 1961.

Cylindrical Shells

A first step in reducing the shell equations to a Cauchy (or initial-value) system, the ultimate goal of our analysis, is to write the pertinent equations in the form given in (3.1) or (3.3). These equations, as reported here, were derived in W. Wunderlich, Zwei Halbanalytische Verfahren Zur Berechnung Von Tonnenschalen, Schriftenreihe des Lehrstuhls fur Stahlbau, T. H. Hannover, Heft 6, 1967.

from a linear theory of cylindrical shells. Similar formulations are available. See R. S. Jenkins and H. Tottenham, The Solution of Shell Problems by the Matrix Progression Method, Proc. World Conf. Shell Structures, Sun Francisco, I962 Nat. Acad. of Sci., No. 1187 (1964); J. E. Goldberg, A. V. Setlur, and D. W. Alspaugh, Computer Analysis of Non-Circular Cylindrical Shells, Paper presented at the Internat. Symp. (I.A.S.S.) on Shell Structures in Engineering Practice, Budapest, Hungary, 31 Aug. to 3 Sept., 1965.

NOTES, COMMENTS, AND BIBLIOGRAPHY

81

Similar systems of first-order equations can be constructed for general shell geometries. Shells of revolution have attracted the attention of numerical analysts particularly. For example, for conical shells see J. E. Goldberg and J. L. Bogdanoff, Static and Dynamic Analysis of Nonuniform Conical Shells Under Symmetrical and Unsymmetrical Conditions, in Proc. Symp. Ballisfic Missile Aerospace Technol., 6th Academic Press, New York, 1961.

Extensions to general meridional geometry are available : A. Kalnins, Analysis of Shells of Revolution Subjected to Symmetrical and Nonsymmetrical Loads, J. Appl. Mech. September 1964, pp. 467-476. W. Wunderlich, Zur Berechnung Von Rotationsschalen mit Ubertragungsmatrizen, Ingenieur-Arch. 1967, pp. 262-279.

Clearly, this approach is not restricted to linear systems. See A. Kalnins and J. F. Lestingi, On Nonlinear Analysis of Elastic Shells of Revolution, J. Appl. Mech. March 1967, pp. 59-64.

4. See the paper by Bellman and Brown in the references of Section 1-10. See also R. Bellman, R. Kalaba, and M. Wing, Invariant Imbedding and Mathematical Physics, I. Particle Processes, J. Math. Phys. 1 (1960), 280-308; R. Bellman and R. Kalaba, Invariant Imbedding, Random Walk and Scattering, 11. Discrete Versions, J. Math. Mech. 9, No. 3 (1960), 411-420; R. Bellman, Invariant Imbedding: Semigroups in Time, Space and Structure, Univ. of Southern California, EE Tech. Rep. No. 71-9, March 1971.

5. See the references of Section 1-8. For a connection between invariant imbedding and the theory of characteristics, see G. H. Meyer, On a General Theory of Characteristics and the Method of Invariant Imbedding, SIAM J. Appl. Math. 16 (1968), 488-509.

6-8. The technique of perturbation was extensively used by R. Bellman ef al. to derive appropriate Riccati equations from two-point boundary-

value problems. For some of the early work see R. Bellman, R. Kalaba, and M. Wing, Dissipation Functions and Invariant Imbedding, Proc. Nut. Acad. Sci. 46, No. 8 (1960), 1145-1147; R. Bellman, R. Kalaba, and M. Wing, On the Principle of Invariant Imbedding and Neutron Transport Theory. I: One Dimensional Case, J. Math. Mech. 7 (1959), 575-584.

See also the first reference of Section 1-8 and the second reference of Section 4. For an application of invariant imbedding to analytical mechanics, see R. Bellman and R. Kalaba, Invariant Imbedding and the Integration of Hamilton's Equations, Rend. Circ. Mar. Palermo Serie TI, Tom0 XI1 (1963), 1-11.

82

3

BEAMLIKE STRUCTURES

The connection between linear, second-order, ordinary differential equations and Riccati equations is well known. The connection between high-dimensional linear two-point boundary-value systems and matrix Riccati equations was extensively studied by W. T. Reid in a series of articles. For some of the early work see W. T. Reid, Solution of a Riccati Matrix Differential Equation as Functions of Initial Values, J. Math. Mech. 8 (1959), 221-230; W. T. Reid, A Class of Two Point Boundary Problems, Illinois J. Math. 2 (1958), 434-453; W. T. Reid, Riccati Matrix Differential Equations and Nonoscillation Criteria for Associated Linear Differential Systems, Pacific J . Math. 13 (1963), 665-685.

10. The results presented here are a matrix version of those given in R. Bellman, H. Kagiwada, and R. Kalaba, Invariant Imbedding and the Numerical Integration of Boundary-Value Problems for Unstable Linear Systems of Ordinary Differential Equations, Comm. ACM 10 (1967), 100-102.

See also P. Nelson, Internal Values in Particle Transport by the Method of Invariant Imbedding, J . Math. Anal. Appl. 34, No. 3 (1971), 628-643; R. Bellman, H. Kagiwada, R. Kalaba, and S . Ueno, Invariant Imbedding and the Computation of Internal Fields for Transport Processes, J . Math. Anal. Appl. 12, No. 3 (1965), 54-548.

11. See W. T. Reid, Principal Solutions of Nonoscillatory Self-Adjoint Linear Differential Systems, Pacific J . Math. 8 (1958), 147-169.

Self-adjoint systems are intimately related to dynamic programming. See, for example, R. Bellman, “Introduction to Matrix Analysis,” 2nd ed. McGraw-Hill, New York, 1970.

12. For the stability properties of matrix Riccati equations arising in filtering theory, see R. S. Bucy and P. D. Joseph, “Filtering for Stochastic Processes with Applications to Guidance.” Wiley (Interscience), New York, 1968.

13. Here we follow N. Distkfano, An Application of Invariant Imbedding to the Solution of Mixed BoundaryValue Problems, Univ. of Southern California, Rep. USCEE No. 292, July 1968, Los Angeles, California.

14. The results derived here are a matrix extension of the scalar version presented in Chapter 1.

NOTES, COMMENTS, AND BIBLIOGRAPHY

83

15. Transfer matrices, a discrete version of the state transition matrices in the theory of differential systems, have been applied to several branches of engineering. For applications to transmission line theory and wave propagation, see F. Beckenbach (ed.), ‘‘ Modern Mathematics for the Engineer,” 2nd ser. McGraw-Hill, New York, 1967.

For applications to structural engineering, see E.C. Pestel and F. A. Leckie, Matrix Methods in Elastomechanics.” McCraw-Hill, New ‘I

York, 1963.

See also the references on shells in Section 3. 16-17. The scattering and flexibility matrices are discrete versions of the Riccati transformations discussed in Section 6. The derivation of scattering and related stiffness matrices in structural mechanics from Riccati-like algorithms was originally presented in N. Distefano and J. Schujman, Numerical Solution of Boundary-Value Problems in Structural Mechanics by Reduction to an Initial-Value Formulation, EERC Res. Rep. No. 69-4, Univ. of California, Berkeley, 1969.

For application of the discrete matrix Riccati approach to wave propagation in plasmas, elastic media, and so on, see the book E. D. Denman, “ Coupled Modes in Plasmas, Elastic Media and Parametric Amplifiers.” American Elsevier, New York, 1970.

Chapter 4

Partial Differential Equations I

1. Introduction In this and the next chapter, we wish to apply some of the ideas exposed previously to the solution of boundary-value problems governed by partial differential equations. Here we shall consider discrete, and semidiscrete, versions of the original partial differential operators. In the next chapter we shall focus on the derivation of suitable initial-valued, or Cauchy, systems without resorting to a prior discretization of the operator. We shall restrict our attention to elliptic boundary-value problems. The potential and biharmonic equations in the context of some problems of equilibrium in mechanics will be the vehicle for the illustration of the main ideas and the methods. We start with the classical equation of Poisson in two dimensions over an arbitrary irregular domain. In order to discretize this equation, we employ finite elements and a weighted residuals scheme, two notions introduced for the first time in this book. Although the exposition on these topics intends to be self-contained, the reader wishing to expand this introductory treatment should resort to the literature indicated in the Bibliography. By an appropriate partition of the domain in small disjoint regions, the finite element equations are written in terms of difference equations 84

85

2. POISSON’S EQUATION

subject to two-point boundary conditions. Application of invariant imbedding techniques leads to algorithmic solutions of the equations in terms of stable, initial-valued matrix difference equations. In the rest of the chapter, we deal with the solution of an elastic rectangular plate to illustrate the fruitful combination of semidiscretization techniques with invariant imbedding ideas. The thorough treatment of this example intends to exhibit some of the techniques and applications of this accurate method. 2. Poisson’s Equation

In order to introduce the method we consider Poisson’s equation w =p(x,y)

on

V,

where w(x, y ) is a function with piecewise continuous first derivatives in the bounded domain V E R2 with boundary S, subject to Dirichlet boundary conditions

w = wo

on S, c S ,

(2.2)

i.e., over a portion of the boundary, and mixed conditions

aw -+q an

+ au = 0

on S , = S - S , ,

(2.3)

over the remaining portion of the boundary. In Eq. (2.3), n denotes the normal to the boundary and q and a 2 0 are functions defined over S, . We seek an approximate solution wh, which in addition to satisfying the essential (in this case Dirichlet) boundary conditions

w h = wo

on S,,

(2.4)

is required to satisfy the projection equations

for an appropriate set of weighting functions ui. In (2.5) the integral over S, is a weighted boundary residual added to incorporate the natural boundary conditions (2.3). This, a natural extension of the Bubnov-Galerkin method of dealing with natural boundary conditions, is a very convenient device since it relaxes the requirements on the approximating functions at a portion of the boundary.

86

4 PARTIAL DIFFERENTIAL EQUATIONS I

To reduce the differentiability requirements on the approximating functions wh, we integrate (2.5) by parts and apply the divergence theorem, obtaining

(2.6) where the boundary term over S , vanishes if the weighting functions are chosen such as to satisfy the homogeneous equations on S , . (2.7) Equations (2.6) are well suited to be discretized using finite element procedures. Before doing this, however, we can further relax the requirements on the trial functions wh over S1 given by Eq. (2.4).To this effect, instead of the Dirichlet conditions (2.4), we consider the mixed conditions ui=O

-+ an awh

l(wh - wo) = O

on S1,

where I is a constant parameter. Here we reasonably expect that wh(l) satisfying (2.6) and (2.8) will approach wh satisfying (2.6) and (2.4), as 1 + 00. This, a stratagem borrowed from variational methods, permits complete relaxation of the conditions on the approximating functions over the boundary. Substituting awh/an given by (2.8) in (2.6), we obtain

+1

(wh

- W 0 ) u i ds = 0,

(2.9)

SI

the pertinent projection equations with relaxed boundary conditions. It is important to note that a set of equations similar to (2.9) may be constructed if, instead of Eq. (2.1), we are given AW = p ,

where A is a general elliptic differential operator, not necessarily of the selfadjoint type. 3. Finite Elements

In a finite element procedure, the domain V is divided into a number of subregions or elements { V ' } such that V ' = V and I/' = 0. Such a partition may be arbitrarily designed. Triangular and quadrilateral shapes,

u

n

87

3. FINITE ELEMENTS

not necessarily with straight edges, are in general the most popular candidates for plane elements. Tetrahedrons and cubes play the same role in spatial partitions. Once the domain has been subdivided, we introduce a class of approximating functions (3.1) wh = C Nil(x,y)ui 9

i

where the u iare constants to be determined and the Ni' functions are required to form a local (or patch) basis, i.e., in addition to certain completeness requirements N i r ( x ,y ) = 0 if ( x , y ) # V' u S'. Here, a superindex denotes the element and S' the boundary of the element. Usually the constants ui are chosen to denote the values of the approximating function wh at specified locations (nodes) of the element. In particular, if ( x i , y i ) are the coordinates of node i, the quantity u i= wh(xi, y i ) is a nodal value. In this case, functions Ni' are interpolation functions that satisfy the conditions (3.2) Ni'(xj Y j ) = 6ij where aij is the Kronecker delta. Functions N i t must satisfy appropriate conditions of completeness and integrability. Possible continuity requirements on the approximating functions Wh impose additional conditions on the choice of the interpolation functions Nil. In the present example we require Co continuity on Wh along the edges of the elements, a condition not hard to satisfy in the applications. We now proceed to discretize Eq. (2.9). The weighting functions ui will be chosen B la Galerkin, i.e., u . = N.'1 . (3.3) Substitution of (3.1) and (3.3) in (2.9) yields 3

9

C C kfju j + 1I fi' = 0 , I

j

i = l , 2 , ... R ,

(3.4)

where R is the number of nodes and the sum is taken over all the elements. In (3.4), k f j given by the expression

klj = J/

aN. (-aNi a N j + aNi 2 ay ay -

v l ax ax

Ni'aNj' ds

-

+ I JsllNi'Nj'ds, (3.5)

is only defined for those values of i and j belonging to element 1. This is a consequence of the local nature of the interpolation functions N i * . Matrix ( k f j ) ,known as the stzfSness matrix of the element, is a symmetric, positive definite matrix. The quantity f i l , appearing in (3.4), and given by fi=

/ Ni'p dx dy + Is Ni'q ds V'

2'

-A

Ni'wo ds, Sl'

(3.6)

88

4 PARTIAL DIFFERENTIAL EQUATIONS I

is the loading contribution of element 1 to node i , in the structural mechanics parlance. In Eqs. (3.5) and (3.6), the boundary integrals are defined over the portions S , and S,' of the boundary such that

'

S I 1 =S , n S 1 , S,' = S , n S , , where S, is the boundary of element 1. Equations (3.4)may be compactly written in the form

(3.7)

KU +f= 0,

(3.8) where u = (ui), i = 1, ...,R , is the vector of all nodal values, K = ( K i j )is the global stiffness matrix whose generic term is given by

andf=

(A) is the global loading vector whose generic term is given by A = C fi'. (3.10) 1

In (3.9) and (3.10) the sums extended over all the elements are intended in a Boolean sense, their rules of composition clearly emanating from Eq. (3.4). We note that K in (3.8) is a sparse matrix, a direct consequence of the local basis employed to span the approximating solutions (3. I). Standard finite element procedures take advantage of this property and usually attempt to reduce the bandwidth to a minimum. Clearly, the bandwidth is intimately related to the manner in which the nodes have been numerated. In some simple cases the rules to numerate the nodes such as to achieve minimum bandwidth are easily determined. In general, however, this proposition leads to involved combinatorial problems. Once the matrix K has been conveniently conditioned, Eq. (3.8) is solved using Gauss elimination or various iterative procedures. Here we shall not pursue that path of action. Instead, we shall attempt a direct algorithmic solution of the equations conveniently written in terms of difference equations. 4. Difference Equations

Here we reformulate the finite element equations (3.4) in vector-matrixdifference form. To this end we shall first group the elements in adjacent strips as shown in the illustrative example of Fig. 4-la. Each node will be now denoted by the pair (m,n), indicating row and column, respectively. Accordingly, the value of w,,at the node (m,n ) will be denoted by u,, . The strip between column n and n + 1 will be called the nth strip. We now introduce

u(n) =

( "::),

4. DIFFERENCE EQUATIONS

89

Fig. 4-1. (a) Standard numeration, (b) strip configuration.

i.e., the vector of the nodal values along column n, where M , denotes the number of nodes in such a column. For example, in Fig. 4-1 b, M = 2 , M , = 3, M , = 5, ... , M , = 4. Using this notation it is possible to write Eq. (3.4) in the form .............. K21(1)

..............

.....................................................................

j

+ KIl(2) I ~,l(2) i~

K22(1)

K12(2) 2 ~ +2~ l) l ( 3 ) .

I

.........................

+

.........................

= 0.

........................

(4.2)

where Kij(n),i , j = I , 2, are partitions of the stiffness matrix K ( n ) = (Kij(n))of the strip (or super element) n , that may be easily constructed from the stiffness matrices of the individual elements composing the strip. Similarly, p l ( n ) and p,(n) are loading vectors at the sides n and n + 1 of the strip n, respectively. To illustrate the construction of these quantities, we compute the stiffness matrix of the strip n = 1 in the example of Fig. 4-lb, namely

I I

I

1

k....f..,..+ :..,.............k:, ....k... .................k ...:..3.............k...1..4 ...+ ...k...:..4...............0 ... k:, j k:, k3, 0 1 kz4 + k i 4 k:5

+

K(1) =

(4.3)

90

4

PARTIAL DIFFERENTIAL EQUATIONS I

and the loading vectors p , ( l ) and p 2 ( I ) given by

where the subindexes refer to the nodes and the superindexes denote the element, according to the numeration system given in Fig. 4-la. Equation (4.2) may be compactly written as the difference equation K21(n - l>u(n - 1) =

-pl(n) - ~

+ [K,,(n

- 1)

l),

n

+ Kll(n)lu(n) + K,,(n)u(n + 1) . . . , N - 1,

(4.5)

+ K12(1)42) + Pl(1) = 0,

(4.6)

2 ( n

= 2,

subject to the following end conditions Kll(IM1)

and K21(N - I)u(N - 1)

+ Kz2(n)u(N) + p 2 ( N - 1) = 0.

(4.7)

It is interesting to point out that K(n) is a square matrix of dimension ( M , + M n + 1 ) 2whose partitions Kij(n),i, j = 1,2, are of dimensions as indicated in

K(n)=

i

Kllb)

j

K12(4

1.

(Mn x M n..--.... +l) x M n ) -..........-...... _.__._ ...(..M ... ,.............._.-. ~

K21(n) (Mn+1 x Mn) j

K22(n) (Mn+1 x

(4.8)

Mn+1)

In general, K,, and K2, will be rectangular matrices that automatically take care of changes in dimensionality of vector u(n)in the difference equation (4.5). We finally observe that by construction, K,,(n) > 0, K2,(n) > 0, and Kl2(4 = K m . 5. Solution of the Equations

a. Riccati Transformations

To deal with the two-point-value problem (4.5), (4.6), and (4.7), we may resort to the method of the Riccati transformations extensively used in Chapters 1 and 3. To this end we consider a domain of variable length formed with the N - n consecutive strips between the boundaries n and N . We now look for solutions of the form u(n

+ 1) = R(n)u(n)+ r(n),

1 5 n _< N - 1,

(5.11

5 . SOLUTION

91

OF THE EQUATIONS

where R(n) is a M , , x M , matrix and r ( n ) an M,, vector to be determined. Elimination of u(n) and u(n I ) in (4.5) by making proper use of (5.1) leads to the following recurrence relations for those quantities :

+

R(n - 1) = - ( K z z ( n - 1) + K , , ( n )

+ K ~ Z ( ~ ) R ( ~ Z ) ) --' Kl),Z ~ ((5.2) ~

and r(n - 1) = -(K,,(n - 1) *

+ K , , ( n ) + Klz(n>R(n>)-l

( J G 2 ( 4 r ( n )+ P l ( 4

+ pz(n - I)),

(5.3)

subject to the initial conditions R ( N - 1) = -K,;'(N)K,,(N

-

l),

r ( N - 1) = -K;;(N)pz(N

- l),

(5.4)

and (5.5)

respectively, obtained from consideration of (5.1) and (4.7). Clearly, the computation of R ( n ) and r(n), as given by Eqs. (5.2) and (5.3), proceeds recursively in the backward direction starting from the known conditions (5.4) and (5.5) at N - 1. Once these quantities have been determined, we may compute u(n) using (5.1) starting with the initial condition 4 1 ) = -(41(1)

+~

l

z

~

~

~

R+ PIU)), ~ ~

~

~(5.6)-

obtained from consideration of (5.1) and (4.6). 6. Invariant Imbedding

In the method presented above, it is required to store all the quantities R(n) and r(n) and to perform two computational sweeps. In return, the values of w, at every node are obtained. Here we present a direct method to determine u i j for specified values of i and j , which appears very efficient with respect to time and storage if only a few nodal values are required. Suppose that we wish to determine the nodal value u i j . By superposition we may write u i j = s(i,j, n)u(n)

+ W ,n),

(5.7)

where s(i,j, n) = (sk(i,j , n)), k = 1, . . . , M , , is a row vector of dimension M , , and t ( i , j , n) is a scalar. Equation (5.7) expresses the fact that there exists a linear relationship between u i j and the nodal values, say along column n, i.e., u(n). We assume now that j 2 n and, while keeping i and j fixed, we consider the domain of length N - (n + l), namely, uij = s(i,j, n - l)u(n - 1)

+ l(i,j, n - 1).

(5.8)

l

~

~

92

4 PARTIAL DIFFERENTIAL EQUATIONS I

+

Substitutingu(n) = R(n - l)u(n - 1) r(n - 1) in (5.7), subtracting (5.7) from (5.Q and collecting terms in u(n - l), we finally find the recursive equations s(i,j, n - 1)

= s(i,j, n)R(n -

t(i,j, n - 1) = t(i,j , n)

I),

+ s(i,j , n)r(n - I),

n <j,

(5.9)

n < j,

(5.10)

for the quantities s and t , subject to the initial conditions (5.11) (5.12) respectively, obtained from consideration of (5.7) at n =j.In Eq. (5.1 l), hki is the Kronecker delta. The method consists in the recursive, backward computation of R(n) and r(n) using (5.2) and (5.3) starting at n = N - 1 with (5.4) and (5.5). At n =j, we adjoin Eqs. (5.9)-(5.12) and continue the recursive computation of all the quantities R(n), r(n), s(i,j , n ) and t(i, j , n ) , until n = 1. We finally obtain u i j by employing (5.7) at n = 1, i.e., uij = s(i,j , l)u(l)

+ t ( i , j , I),

(5.13)

where u(1) is given by (5.6) 6 . Discussion

The choice of a certain strip pattern may be determined by a number of considerations associated with the specific problem, such as the type and quantity of information required from the solution, the topological properties of the domain, questions of computational efficiency, etc. It is beyond the purpose of this section to deal systematically with all those aspects. Here we only wish to present a few examples to illustrate some possible applications of the present methodology. When the finite element partition is given, the number of possible strip configurations that may be designed on that partition will depend on the topology of the domain and the finite element mesh. For example, in a simply connected plane domain we distinguish at least RZ different configurations, where R is the number of boundary nodes. Two such possible configurations are presented in Fig. 4-2 (a and b) for a triangular mesh in a given region. When the finite element mesh is not given in advance, there is considerably more freedom in selecting a convenient strip configuration that recognizes some of the special features of the problem. In the examples presented in Fig. 4-2 (c-h), we have subdivided threedomains in two different ways each, to illustrate a few possible configurations. It is interesting to note that the

6. DISCUSSION

93

F&. 4-2

present method handles in a uniform fashion all the cases presented here or any other constructed along similar lines. In particular, there is no operational distinction between constant and variable strip dimensionality. Hence, the strip pattern may be entirely selected to reflect special features of the problem and not to simplify the analytical aspects of the method. In some problems involving mechanical applications, for example, the design of the strip pattern might be dictated by the configuration of the actual layers employed in the construction of structures such as dams, tunnels, etc. In other instances, the same role may be played by an a priori qualitative knowledge of the stress or flow pattern of some of the relevant variables of the problem. We finally wish to discuss some of the merits of this method. The combination of invariant imbedding and finite element ideas accentuates the

94

4

PARTIAL DIFFERENTIAL EQUATIONS I

advantages of both approaches. In fact, the present method inherits the versatility of finite element techniques that handle intricate geometries and general boundary conditions in an elegant, uniform fashion. On the other hand, the use of invariant imbedding is accompanied by several advantages. First, it provides a clear physical meaning to each computational step in the solution of the equations. This, together with the geometric flavor attached to the strip method, enriches the information content of the solution, in comparison with ordinary finite element procedures. Second, the method of Section 5b enables the selection of those nodes on which the solution is desired, with a storage requirement nearly proportional to the number of nodal values to be computed. This, an advantage not shared by standard finite element procedures, increases the analyst's control over the type and quantity of information to be extracted from the solution. Third, the numerical algorithms derived in Sections 5a and b are exponentially stablewith respect to initial conditions, a consequence of the positive definite nature of the stiffness matrices and the intrinsic stability of Riccatilike algorithms. Finally, we note that the present method admits immediate generalizations. We could, for example, add one element (rather than one strip) at a time, furnishing additional flexibility to the method. In this case, the algorithms developed in Section 5 retain their validity if a proper reinterpretation of the meaning of the stiffnessmatrices (Kij(n))of the strips is performed.. We leave this as a research exercise to the reader. EXERCISES 1. Discuss the stability of the quantities R(n) and r(n) given by Eqs. (5.2)-(5.5), with respect to initial conditions. See the paper

D. Rappaport and L. M.Silverman, Structure and Stability of Discrete Time Optimal Systems, IEEE Transactionson Automatic Control AC-16 (3), June 1971, pp. 227-233. 2. Study the solution of the equation

over an arbitrary irregular domain using the method presented above. Assume k(x,y ) > 0.

7. Semidiscrethation. The Elastic Plate A method frequently employed in the numerical solution of partial differential equations consists of the discretization G f the operator with respect to one variable only, thus reducing the problem to the solution of a finite system of ordinary differentialequations subject to two-point-value conditions. '

95

7. SEMIDISCRETIZATION. THE ELASTIC PLATE

This is the basis of the so-called method of the lines. (See the Bibliography at the end of the chapter.) The combination of semidiscretization with ideas of invariant imbedding leads to initial-value formulations of a novel type offering features of theoretical and numerical interest. We shall illustrate the application of the method with the solution of the biharmonic equation in the context of elementary thin plate theory. The discussion is restricted to rectangular domains.

Fig. 4-3

The equations of equilibrium of an elastic plate in terms of the usual mechanical variables (see Fig. 4-3) are given by

mxy= -my,

=

a2w

D ( l - v ) -, axay

a%, ax2

--

+

a2mxy a2m 2-ax + aY

ay

= - ~ z9

where D = Eh3/[12(l- v’)] is the stiffness of the plate with thickness h, modulus of elasticity E, and Poisson’s ratio v. When D is a constant, we may eliminate m,, m y , mxy, q,, and qy from the six equations of (7.1) and obtain the classical partial differential equation

i.e., the biharmonic equation in terms of the displacement w. Introducing

(7.3) representing the combination of the shear forces qx and the contribution of the twisting moments mxyin the direction of the shear, and the slope cp, given by cpx =

awlax,

(7.4)

96

4 PARTIAL DIFFERENTIAL EQUATIONS I

we can combine (7.1), (7.3), and (7.4) to derive the following first-order system in x: aw _ -

ax

cpx,

Our task now is to reduce (7.5) to a system of ordinary differential equations by semidiscretizing with respect to y . To simplify the notation we shall assume D = 1. Clearly, the present method affords consideration of variable stiffness D. We consider the rectangular domain enclosed by x = y = 0, x = L, and y = H. To simplify the present analysis further we assume edges y = 0 and y = H to be either clamped or simply supported. The partial differential operators with respect to y will be replaced by the following difference ones:

+ wj+,)/h2, =( w ~ - 4Wj-1 ~ + 6wi - 4Wj+l + wi+2)/h4,

(azw/ay2),,i h = (wi-l (a4w/ay4),, i h

-

2wj

(7.6) (7.7)

where w i= w(x,ih) and h, the distance between

“

(7.8)

lines,” is chosen to satisfy

h

=H/(N+

(7.9)

l),

where N , the number of “lines,” is an integer. We introduce now vectors 4x1, 44, t(x), --(XI, and ~ ( x given ) by

u(x) = (W,ih)) = (wi(x)), u(x) = ((d/dx)w(x,ih)) = du/dx,

t(x) = (t,(x,

w,

m(x) = (m,(x, ih)),

(7.10)

P(X) = ( P A X ,ih)),

and the N x N matrices K = (l/h2)T,

(7.1 1)

L = K Z+ (i/h4)~,

(7.12)

and

97

8. A STIFFNESS MATRIX

where T is the tridiagonal matrix defined by (16.6) of Chapter 1 and F = (fii) is an N x N matrix whose elements are zero except fil and f N Ngiven by if edge y = 0 is simply supported if edge y = 0 is clamped

0 = (2 0 = (2

*"

if edge y = H is simply supported if edge y = H is clamped.

(7.13)

Using the discretization scheme given by (7.6)-(7.7) and the vector-matrix notation introduced by (7.10)-(7.13), Eqs. (7.5), where D = 1, reduce to the 4N-dimensional system of ordinary differential equations = p -L*M - vKm, v' = - m + VKM, m' = - t - K"u, u' = u,

t'

(7.14)

where L" = L - v 2 K 2 ,

K * = 2(1 - v)K.

(7.15)

EXERCISES 1. Derive Eqs. (7.5). 2. Semidiscretize (7.5) assuming D variable in both the x and y direction. 3. Show that urn- ~ K u -t " LU=P

is the semidiscrete version of the biharmonic equation (7.2) for D u, f , and m from Eqs. (7.14).

(a) 3

1. Hint: Eliminate

4. Show that the semidiscrete versions of m, and tx given by Eqs. (7.1) and (7.3), respectively (assuming D = I), are

m

=

-u"

+ VKU,

t = urn- ( 2 - v)Kv.

5. Write the semidiscrete versions of the remaining quantities, i.e., m,. , m,,

(b)

, qx, and qy

given by (7.1).

8. A Stiffness Matrix Consider now the boundary-value problem of integrating the system (7.14) subject to the end conditions t(0) = m(0) = 0,

M(L) = U(L) = 0.

(8.1)

In the invariant imbedding approach, we consider the following family of problems: Integrate (7.14) subject to t(0) = m(0) = 0 ~ ( l=) C, ~ ( l =) d,

0 I 1 I L,

(8.2)

98

4 PARTIAL DIFFERENTIAL EQUATIONS I

in which c and d are arbitrary N-dimensional vectors. Clearly, this solution reduces to the solution of (7.14) and (8.1), when c = d = 0, and I = L. In the familiar fashion we consider the missing boundary conditions t(l) and m(Z) as the linear combinations t ( l ) = R11U)c + Rl,(Od + C%(O, m(l) = R21(l)c R22(1)d + ~12(Z).

(8.3)

+

Proceeding as usual, we find that R i j and a i , i = 1, 2, satisfy the Riccati differential equations

subject to the initial conditions Ri,(O) = R12(O) = R21(0) = R22(O) = 0, a,(O) = 4 0 ) = 0.

(8.5)

We leave the derivation of (8.4)-(8.5) as an exercise for the reader. Introducing the 2N x 2N matrices

),

A=( O I vK 0

B = (O 0

O),

C=

( y :*),

R=

-I

and the following 2N-dimensional vectors:

Eqs. (7.14), (8.2), and (8.3) reduce to z' = A z

+ By,

y' = - c z - ATy

and

Y(0) = 0,

+ p,

z ( l )=

();

(R1l

'I2),

R,,

8 2 2

99

9. PRESCRIBED DISPLACEMENTS A T OPPOSITE EDGES

respectively. Equations (8.4) and (8.5) can be written in compact form R ' = - C - RA - A'R - RBR, a' = -A'u - RBu + p,

(8.10)

and R(0) = 0,

~ ( 0= ) 0,

(8.11)

respectively, as can be proved by inspection. Equations (8.10) can also be obtained by a direct analysis involving (8.8) and (8.9). Clearly, R ( f )satisfying the matrix Riccati equation (8.10) subject to initial conditions (8.11) is a stiffness matrix. It is a symmetric matrix as can be directly verified using (8.10) and (8.11). This symmetry reflects Betti's law obtained via invariant imbedding. Due to this property, R I 2 = RZ, in Eqs. (8.3). Therefore we can delete one of the differential equations involving the symmetric terms in Eqs. (8.4). EXERCISES 1. Show that R given by (8.10) and (8.11) is R < 0. Hint:

Use the fact that C > 0.

2. Using a fully discretized biharmonic operator, derive the discrete analog of Eqs. (8.10). See the book by Angel and Bellman listed in the Bibliography.

9. Prescribed Displacements at Opposite Edges It is worthwhile to observe that whatever conditions we consider at x = 0, the differential equations (8.4) or (8.10) remain unchanged. This is due to the fact that the edge conditions at x = 0 are only used to define the initial conditions for the Riccati equations (8.4) or (8.10). In this section we shall consider the edge conditions

i.e., we prescribe the displacements at both edges x = 0 and x = 1. In this case Eqs. (8.9) and (8.10) still hold, but (8.11) must be modified. In fact, on physical grounds it is clear that when the displacements are prescribed on opposite edges it is no longer true that the matrix R vanishes as I tends to zero, i.e., the stiffness matrix R is not defined for a clamped plate of length zero. To remove this difficulty we define initial conditions for the Riccati equations at x = A, a point near the origin where the stiffness matrix R can be computed in some convenient manner. This can be done in a number of ways.

100

4 PARTIAL DIFFERENTIAL EQUATIONS I

For example, we can consider the plate of length A as being divided into N uncoupled clamped beams. The stiffness matrix is then found to be

A, vector u(A) is given by and, assuming that the load is constant in 0 I x I (9.3) Initial conditions for R in a neighborhood of x = 0 could also be obtained by a number of asymptotic procedures. The reader interested in this problem can consult the Bibliography at the end of the chapter.

10. A Flexibility Matrix We have shown in Section 8 that R in the equation y = RZ

(10.1)

is a stiffness matrix that satisfies the Riccati equation R'= -C-RA-ATR-RBR.

(10.2)

Similarly, by considering the equation z

=zy,

(10.3)

we can derive a Riccati equation for the flexibility matrix Z. There is no need to use a perturbation procedure here since Z enjoys the property Z

= R-'.

(10.4)

By differentiating (10.4), and making use of (l0.2), we obtain z' = - R - ~ R ' R - ~ =

- Z ( - C - RA - ATR - RBR)Z AZ + Z A ~ ZCZ,

=B

+

+

(10.5)

a Riccati equation for Z which in expanded form reads

+ +

+

+

Z ; , = Z12 ZTz Z,,L*Z,, Z l ZK*ZT2, K 2 1 2 K*Z22, Z ; , =Z22 V Z ~ ~Z,,L*2,2 Z;z = - I + v K Z , + ~ vZT, K + ZT2L*Z12 Z2, K*Z22,

+

+

(10.6)

+

where (10.7)

and where K , K*, and L* have the meaning given in Section 7.

101

1 1 . NUMERICAL EXAMPLE

EXERCISES 1. Show that instead of the approximate value given for R(A) by Eq. (9.2). we could use the exact one given by R(A) = Z - ’ ( A ) ,

where Z is the solution of (10.5) subject to Z(0) = 0. 2. Derive a differential equation for p in the equation z= z y

+ p.

11. Numerical Example We present here a numerical example to show the feasibility and accuracy of the method. In Table 4-1 we present a comparison of the results obtained using invariant imbedding and those appearing in Timoshenko’s book on plates and shells, in the computation of rectangular plates with two opposite TABLE 4-1 Rectangular Plate with Two Opposite Edges Simply Supported and Two Other Edges Clamped (Uniform Distributed Load, v = 0.3)

tY I-,-I x = a/2, y = b/2

Invariant Timoshenko . imbedding 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.00192 0.0025 1 0.00319 0.00388 0.00460 0.00531 0.00603 0.00668 0.00732 0.00790 0.00844

0.00191 0.00252 0.00319 0.00389 0.00461 0.00532 0.00602 0.00669 0.00732 0.00791 0.00846

Invariant Invariant Timoshenko imbedding imbedding -0.0697 -0.0787 -0.0868 -0.0938 -0.0998 -0.1049 -0.1090 -0.1122 -0.1152 -0.1174 -0,1191

-0.0696 -0.0785 -0.0865 -0.0935 -0.0996 -0.1047 -0.1089 -0.1123 -0.1150 -0.1 172 -0.1190

0.0332 0.0371 0.0400 0.0426 0.0448 0.0460 0.0469 0.0475 0.0477 0.0476 0.0474

0.0331 0.0368 0.0399 0.0425 0.0445 0.0459 0.0468 0.0474 0.0476 0.0476 0.0474

102

4

PARTIAL DIFFERENTIAL EQUATIONS I

edges simply supported and the two other edges clamped, subject to uniform load. The differential equations were integrated using a Runge-Kutta scheme with a step size of 0.01. In this example N = 1 1 and A for the initial conditions (9.2) and (9.3) was taken to be equal to 0.1. The moments at the middle of the plate were computed by backward integration using a two-sweep method such as that outlined in Section 3-9. 12. Buckling Problems. Critical Lengths

As an example of the application of the method to buckling problems, we calculate the critical length of a rectangular plate simply supported along y = 0 and y = 1 and clamped along x = 0 and x = L , subject to in-plane compressive forces n,(y) acting on edges x = 0 and x = L. Under the effect of in-plane forces n,, the differential equations (7.1) remain unaltered except for the last one, the equilibrium equation in the z direction, which should read (12.1) The additional term n, azw/ax2in Eq. (12.1) clearly denotes the influence of the curvature a2w/ax2in the equilibrium of normal loads. The corresponding system (7.5) now reads

-aw_ ax

CPX

,

aZw

I 8 C P-X - - m, - v 2,

ax

D

am, - - -t,

ax

aY

a D -. 8% + 2(1 - V ) aY

aY

Semidiscretizing (12.2) under the assumption that D of ordinary differential equations

u‘ = u, u’ = -m vKu, i’ = p - L*u - vKm, m’ = - f - K** 0,

= 1 leads to the system

+

(12.3)

NOTES, COMMENTS, AND BIBLIOGRAPHY

103

where i is the effective shear force given by

-

t=t-Np,

m is the diagonal matrix of the compressive forces given by m diag(n,(ih)), =

(12.4)

(12.5)

and K** is the matrix

K**

= 2(1 -

v)K - W.

(12.6)

We leave the details of the derivation of (12.2) and (12.3) to the reader. We observe that Eqs. (7.14) and (12.3) are formally the same. Therefore, the solution of (7.14) provided by the Riccati system (8.3) and (8.4) may still be used provided that t is substituted by t and K* by K**.With these substitutions in mind we integrate the system (8.4) subject to appropriate initial conditions and find the value of L for which the matrix R = (Rij) ceases to be negative definite. This is the smallest critical length of the plate. Recalling the physical meaning of R, a sufficient condition for criticality is that any element of the matrix R becomes zero. This simplifies the numerical procedure considerably. We now present the following numerical example. The critical force nXcri, of a square plate of length one, simply supported along edges parallel to x and clamped along edges parallel to y , with constant thickness and Poisson ratio v = 0.25, is given by nxcrit= 6.74n2,

(12.7)

according to the Handbook of Engineering Mechanics (see Bibliography at the end of the chapter). We have used this value to compute matrix K** given by Eq. (12.6) and subsequently to compute R(x) by means of Eqs. (8.4). The initial conditions for Eqs. (8.4) were computed at A = 0.1. The differential equations were integrated with a Runge-Kutta scheme using a step size of 0.01. The integration was interrupted when the first element of the stiffness matrix R changed its sign. This procedure yielded a value L erit = 1. No further investigation was done to calculate more precisely where the change of sign took place, which obviously occurred in the interval 1 5 LcritI 1.01. In this computation N = 9. NOTES, COMMENTS, AND BIBLIOGRAPHY

1. The numerical solution of partial differential equations by invariant imbedding procedures is presently a fertile and active area of research. In this and the next chapter we make a short incursion into this rapidly evolving

104

4 PARTIAL DIFFERENTIAL EQUATIONS I

field to show some of the most important features of the methods. The treatment of the Poisson equation presented in Sections 2 to 6 , follows very closely the paper N. Distkfano, Invariant Imbedding and the Solution of Finite Element Equations, J. Math. Anal. Appl. (to appear).

2 . The reader interested in Ritz-Galerkin or more general weighted residuals methods should consult B. A. Finlayson, “The Method of Weighted Residuals and Variational Principles.” Academic Press, New York, 1972.

In a variational context, the parameter ?, appearing in Eq. (2.8) is known as a Courant parameter. See R. Courant, Variational Methods for the Solution of Problems of Equilibrium and Vibrations, Bull. Amer. Math. SOC.49 (1943), 1-23.

and T. Butler and A. V. Martin, On a Method of Courant for Minimizing Functionals, J. Math. Phys. 41 (1962), 291-299.

3. The finite element method, an ingenious extension of Ritz-Galerkin techniques well suited for digital computers, has gained considerable support among engineers and applied mathematicians dealing with the numerical solution of partial differential equations. In addition to the pioneering work by Courant (see the book J. L. Synge, “The Hypercircle in Mathematical Physics.” Cambridge Univ. Press, London and New York, 1957.

and the original paper by Courant cited in Section 2), the method was independently formulated and developed by structural engineers interested in the solution of problems of elasticity with very complicated boundaries. We note in this connection the following papers: M. J. Turner, R . W. Clough, H. C. Martin and L. J. Topp, Stiffness and Deflection Analysis of Complex Structures, J . Aeronaut. Sci. 23 (9) (1956), 805-823; R. W. Clough, The Finite Element in Plane Stress Analysis, Proc. 2nd ASCE Conference on Electronic Computation, Pittsburg, Pa., Sept., 1960; J. H. Argyris, Energy Theorems and Structural Analysis, Butterworths Scientific Publications, London, 1960 (Reprinted from Aircr. Engng., 1954-1955).

The method, with many engineering applications, is presented in 0. C. Zienkiewicz, “The Finite Element Method in Engineering Science.” McGraw-Hill, New York, 1971.

NOTES, COMMENTS, AND BIBLIOGRAPHY

105

The mathematical foundations of the method, with particular emphasis on approximation aspects, may be found in : A. K. Aziz (ed.), “The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations.” Academic Press, New York, 1972. G. Strang and G. Fix, “An Analysis of the Finite Element Method.” Prentice-Hall, Englewood Cliffs, New Jersey, 1973.

4-6. The use of invariant imbedding in finite element equations as formulated here was originally presented in the reference by DistCfano cited in Section 1. For an extensive treatment of partial differential equations using finite differences and invariant imbedding, see the book E. Angel and R. Bellman, “Dynamic Programming and Partial Differential Equations.” Academic Press, New York, 1972.

See also N. Distkfano and A. Jain, A One Sweep Method for Linear Elliptic Equations over Irregular Domains, J . Optimization Theory Appl. (to appear).

7. The method ‘of lines, or line method, is a very versatile semidiscrete method. See,for example, S.A. Mikhlin, “Variational Methods in Mathematical Physics.” Pergamon, Oxford, 1964.

8-10. The results given here for the biharmonic equation were first given in N. Disttfano and J. Schujman, Numerical Solution of Boundary Value Problems in Structural Mechanics by Reduction to an Initial Value Formulation, Earthquake Engrg. Res. Center, Rep. No. 69-4, Univ. of California at Berkeley, March 1969; N. DistCfano and J. Schujrnan, Solutions of Elastic Plates by Invariant Imbedding, Proc. Amer. SOC.Civil Engrg. Engrg. Mech. Div., EM5, October 1971.

11-12. For additional examples and details on the numerical solution of the matrix Riccati equations see N. DistCfano, Dynamic Programming and the Solution of the Biharmonic Equation, Internat. J. Numerical Methods Engrg. 3 (1971), 199-213; P. Nelson, A Comparative Study of Invariant Imbedding and Superposition, Internat. J. Comput. Math. Section B, 3 (1972), 195-207.

Chapter 5

Partial Differential Equations I1

1. Introduction

Searching for appropriate numerical schemes, in the last chapter we applied invariant imbedding ideas to suitable discrete or semidiscrete versions of the partial differential operator. It should be kept in mind, however, that a reduction in the transcendentality of the operator is a convenient device but not an essential one for the application of invariant imbedding arguments to boundary-value problems. In fact, a number of significant results in the theory of partial differential equations can be obtained by application of invariant imbedding ideas without resorting to any prior discretization of the operator. Historically, the first result in this direction appears to be due to Hadamard in his celebrated equation for the variation of the Green’s function of the potential equation in terms of a certain variation of the contour. More recently, it has been shown that Hadamard’s result can be derived and generalized using standard arguments in dynamic programming. This is of interest from both a theoretical and a numerical point of view. We note that the same results can be derived from invariant imbedding considerations. The usual argument consists in considering a one-parameter family of regions, monotone under inclusion, and deriving a Cauchy system in terms of such a parameter. Clearly, 106

2. FORMULATION O F THE PROBLEM

107

a considerable simplification in the formulation occurs when the contour lines are simple in relation to the coordinate system employed. In order to illustrate the ideas involved in this method, we consider again the biharmonic equation in the context of elastic rectangular plates. We take a, the length of the plate in the x direction, to be the pertinent imbedding parameter and derive an appropriate Cauchy system in terms of integrodifferential equations. This is discussed in Sections 2-6. In the last two sections, 7 and 8, we change our course to discuss a new problem, namely, the influence of Poisson’s ratio in the stress distribution of the plate. This, a perturbation problem in elasticity investigated mostly by Westergaard using classical arguments in potential theory, is reformulated here as an initial-value problem in terms of integrodifferential equations. In this fashion the theory of invariant imbedding furnishes a new Cauchy system that can be regarded as a functional perturbation with respect to Poisson’s ratio. The vehicle for this derivation is the construction of an appropriate system of Fredholm integral equations. As usual, a Bibliography with comments is provided at the end of the chapter. 2. Formulation of the Problem

Let w be the deflection of a plate satisfying the biharmonic equation

v4w = w,,,, + 2wx,,, + w y y y y = 0, (2.1) where the subscripts indicate partial differentiation in the rectangular domain 0 Ix I a, 0 5 y I 1. We consider that edges x = 0, y = 0, and y = 1 are rigidly clamped, i.e., w(0, y ) = w(x, 0) = w(x, 1) = 0, (2.2) w,(O, Y ) = wy(x,0 ) = w,(x, 1) = 0, and that the edge x = a is subject to prescribed moments m(y) normal to the edge, and to vertical forces n(y). In terms of displacements the conditions on the edge x = a read w x x ( a , Y ) = m(y), (2.3) w,,, + 2wxyy(a,Y ) = 4.Y). (2.4) Note that for simplicity, and without any loss of generality, we are assuming that v, the Poisson ratio of the material, is zero. We discuss an interesting solution for the case v # 0 in Section 7 and subsequent sections. The problem mentioned above is considered a fundamental problem for the plate in the context of invariant imbedding. We obtain the solution of this problem by superposition of the solutions of the two auxiliary problems defined below. Let be a parameter such that OIa11.

(2.5)

108

5

PARTIAL DIFFERENTIAL EQUATIONS I1

We consider two functions u(x, y , a) and u(x, y , a) satisfying the biharmonic equations

v4u = 0 ,

v4v = 0,

(2.6)

subject to u(0, y , a) = u(x, 0, a) = u(x, 1, a) = 0,

ux(0, y , 4 = u,(x, 0,a) = uy(x, 1 , 4 = 0 , u(0, y , a) = v(x, 0, a) = u(x, 1, a) = 0,

(2.7)

ox@,y , a) = uy(x,0 , a) = uy(x, 1, a) = 0.

If 6(y) denotes the Dirac delta function, we additionally require at x conditions uxx(a, y , 4 = S(Y - a), uxxx(a, y , a> + 2uxyy(a,Y , a) = 0, and uxx(a, y , 4 = 0 , uxxx(a,y , 4 + 2uxy,(a, y , 0)= -6bJ - 4.

=a

the

(2.8)

(2.9)

In physical terms, u represents the deflection of a plate clamped at x = y = 0 and y = 1, and subject to a concentrated force P = 1 applied on the free edge x = a at the point y = a. Similarly, u is the deflection of a plate equally supported but loaded with a concentrated unitary moment at the pointy = a of the free edge x = a. Clearly, u and u are special Green’s functions for the original problem. By superposition, w satisfying Eqs. (2.1)-(2.4) can be expressed as W(X,

y) = -

J

1

V(X,

+ S,u(x, y , a)m(a) da. 1

y , a)n(a) do

0

(2.10)

Our purpose is to derive initial-value problems for u and u , and then use (2.10) to obtain the desired solution. We will regard u and u as being functions not only of x, y , and a but also of the length of the plate a. Thus, we write u = u(x, y , a, a),

v = u(x, y , a, a).

(2.1 1)

To facilitate the use of functions of several variables, the following notation is introduced. We will let u i and u i ,i = 1, 2, 3, 4, denote differentiation of u and u with respect to the ith variable. Thus with this notation u satisfies the biharmonic equation Ullll

and at x

=a

+ 2u1122 + u 2 2 2 2 = 0 ,

(2.12)

we have ulll(a,

%,(a, y , a, 4 = 6 ( y - 4, y , a, a> + 2u,z,(a, y , a, a> = 0.

(2.13)

109

3. INITIAL-VALUE FORMULATION

Similar equations can be written for u. Our next task is to investigate how the solutions for u and u vary as the length of the plate is increased. Although we employ some formal manipulations with delta functions, it is clear that the results can be easily justified via the theory of distributions under the smoothing conditions usually adopted in the theory of thin elastic plates and generally required for numerical computations.

3. Initial-Value Formulation We start by differentiating Eqs. (2.6)-(2.9) with respect to the imbedding parameter a. Assuming all necessary conditions for the interchangeability of the order of differentiation, we find that the new functions u3 and u3 satisfy v4u3 = 0,

v4u3 = 0,

(3.1)

subject to U,(O,

Y , a7 a) = U3(X, 0, a, 0)= U3(X, 1, a7

g) = 0,

[u3(0,Y , 4 0)11 = [u3(X, 0, a, 0112 = [U3(X, 1, a, a112 = 0,

u,(O, Y , a, a) = u3(X, 07 a, 0)= u3(X, 1, a7 0)= 07 [u3(0, Y , a, u)]i = [u3(X7 0, a, 0)]2 = [u3(X, 1, a, .)I2 along the edges x = 0, y = 0, and y to a in Eqs. (2.8) and (2.9) yields

and

=

1. At x

=a

(3.2)

= 0,

differentiation with respect

110

5

PARTIAL DIFFERENTIAL EQUATIONS I1

Comparing the problems for u and v with those for u3 and v 3 , we see that the problems are similar except for the forcing functions at x = a. Applying superposition as in (2.10) we have 1

u3(x,Y , a,

4 = 2 J 0 4x7 Y , a,v)u122(a, ?, 44 d?

and

Let us define two new functions by

@, Y , a,4 = u,(x, Y , a,4,

$(x, Y , a7

0) =

q ( x , Y , a, 4. (3.9)

Differentiation of (3.7) and (3.8) with respect to x yields 1

e3(x,

Y9

e(x?Y ,

a,a) =

?)e22(u7

?7

a,

d?

0

(3.10) and

Using the definitions of 0 and $,(3.7) and (3.8) become

14x7 Y , 1

u3(x,Y , 4 4 = 2

and

0

4 ?)022G7, 'I,a,4 4

111

3. INITIAL-VALUE FORMULATION

These are our fundamental imbedding relationships.We shall regard these four equations as initial-value problems for all a 2 x , x being fixed. In order to define more precisely our Cauchy system, let us introduce the functions ~ ( aY,, 4 = e(a, Y , a, 4 = u&, Y , a, 4, qta, Y , 4 = $(a, Y , a,4 = q(a, Y , a, 4,

rta, Y , 4 = u(a, Y , a, 4, s t 4 Y , 4 = u(a, Y , a, a). Differentiation of p with respect to a yields

(3.14)

pl(a, Y , a) = O,(a, YY a, 4 + 83(a, Y , a, (0 (3.15) = u11 ( ~Y,, a, 0 ) + e3(a,Y , a, 0). If we use (2.13) to replace ull and (3.10) evaluated at x = a to replace 8,, we find that p satisfies 1

Pl(%Y , 4 = StY - 4 + 2 s p(a, Y , rtIPzz(a9 0

r 3

44 (3.16)

In a similar manner we find the following equations for q, r, and s:

(3.17)

From (2.7) we find the initial conditions P(0, Y , 4 = do, Y,0)= 4 0 , Y , 0 ) = 40,Y , 4 = 0.

(3.20)

The following auxiliary boundary conditions were derived from consideration of Eqs. (2.7)-(2.9) : p(a, 094 = P(a, 1 , a ) = p(a, Y , 0) = p(a, Y , 1) = 0, d a , 0 9 4 = qta, 1,o) = q(a, Y , 0) = d a , YY 1) = 0, r(a, 0, o) = r(a, 1, o) = r(a, y, 0) = r(a, y , 1) = 0, s(a, 0,6)= s(a, 1, Q) = s(a, y , 0) = s(a, y, 1) = 0.

(3.21)

112

5

PARTIAL DIFFERENTI/,L EQUATIONS I1

To find u(x, y , a, a) and u(x, y , a, CJ)for a given x , we proceed as follows: Starting with (3.20), we solve Eqs. (3.16)-(3.19), subject to (3.21), from a = 0 to a = x . Then we adjoin Eqs. (3.10)-(3.13) which can be rewritten in view of (3.14) as 1

03(x, Y , a , 4 = 2 J

0

w, Y , a,

?)Pz2(a, ?,

44

1

- Jo $(x7 Y , a, ? ) r 2 2 2 2 ( a * ?, a> 4,

1(/3(x,Y , a , 4 = e ( x , Y , a, 0 ) + 2 J

1

0

w, Y , a, 1)922(a, ?, 4 4

1

- Jo $ ( X ? Y , a9

?)sz222(a,

?>0 ) d%

(3.22)

1

%(X, Y , Q, 4

=2

J u(x, Y , a,

?)P22@,

0

?,

44

I, 1

-

u(x, Y , a9 v)r2222(a, ?, 4 d?, 1

h(X,

Y , a, a) = 4 x 3 Y , a, a) + 2 J 4x9 Y , a , 1)922(a, ?, a> d? 0

1

- Jo u(x, Y , a, ?)S2222(% ?, 6)4. These equations are subject to the initial conditions

w,y , x , 4 =

Y , a), u(x,y , x , a) = 4x9 y , a), P(X,

y , x, 4 = d x , y , 4, u(x, y , x, 4 = 4x3 y , a),

%4

(3.23)

and auxiliary conditions similar to those in (3.21) which follow directly from (2.7)-(2.9). The entire fundamental system has now been described as an initial-value problem in terms of the imbedding parameter a. In order to obtain the solution of the original problem, we use superposition as given by Eq. (2.10). We have therefore accomplished a reduction of the original boundaryvalue problem in terms of the biharmonic equation, to a Cauchy system in terms of integrodifferential equations subject to initial values. We should prove now the converse, i.e., that every solution of the Cauchy system is also a solution of the original boundary-value problem. The reader should try to prove this as an exercise. He can compare his findings with the results in the Bibliography given at the end of the chapter.

113

4. RECIPROCITY RELATIONS

4. Reciprocity Relations

The well-known reciprocity relations

p(a, y, 6) = p(u, 6,y),

s(a, Y, a) = s(a, 0 , Y ) ,

d a , Y,4 = r(a, 6,Y ) ,

(4.1) a special case of the theorem of reciprocity of Betti, can be derived directly from the results obtained in Section 3. In order to prove this we can proceed as follows: Setting

.>

4%y , 4 = p(a, Y , 4 - p(a, OY Y), P(a, Y , 4 = 4 7 ,Y , - s(a, 6,r),

(4-2)

x u ,y, 4 = qta, Y , a) - rta, Q, Y), we seek appropriate differential equations for a,8, and A. This is not difficult to do, by differentiating (4.2) with respect to a and by making appropriate use of the integrodifferential system obtained in Section 3 for the functions p , q, r, and s. We leave as an exercise to the reader to prove that a,p, and A in Eqs. (4.2) satisfy the equations

and

subject to the homogeneous initial conditions

4 0 ,y , a) = 0,

B(0, y , 0 ) = 0,

40, y , a) = 0,

(4.6)

114

5 PARTIAL DIFFERENTIAL EQUATIONS I1

obtained by consideration of (3.20) and (4.1). Clearly 4 7 ,

y , a> = 0,

A h , y , a) = 0

/?(a, y , 0 ) = 0,

(4.7)

is a solution of the homogeneous system (4.3)-(4.6). Since this system is linear, (4.7) is the unique solution, proving (4.1), our original assertion. 5. Integral Riccati Transformations Instead of using the two auxiliary boundary-value problems (2.6)-(2.9) and then applying ideas of invariant imbedding, we could have treated the present problem by directly pursuing the idea of “ missing” boundary conditions, so successfully employed in the previous chapters. The two formulations are equivalent, as should be expected. We illustrate this idea by considering the “missing” functions w(a, y ) and w,(a, y ) at the edge x = a in the example of Section 2, given as linear functionals of the “data” functions n(a) and m(a), i.e., 1 0 1

1

wl(a, v) = - d a , Y , ah(.) 0

+ S0 r(a, Y , a)m(a) da, 1

w(a, V ) = - J” $(a, Y , a>n(a>da

da +

1

1

(5.1)

p(a, y , a)m(a)da.

0

The linear transformation defined by (5.1) generalizes in a natural fashion the Riccati transformation operated with matrices in previous chapters. The kernel of the transformation, i.e., functions p , q, r, and s, is expected to satisfy initial-valued functional equations. We show this in the following. Differentiation of (5.1) with respect to a yields

-1

1

w,(a, V >=

0 1

+ S,

1

w,l(a, Y> = -

1

J” $(a, Y , a)na(a)do Y , a)m(a>do + J” r(a, Y , a)ma(a>do,

sl(a, Y , a>n(o> da -

0

1

0 1

J” qI(a,Y , a)n(a>da - J q(a, Y , a)na(a>do 0

where na and ma are given by

0

(5.2)

115

5. INTEGRAL RICCATI TRANSFORMATIONS

as it follows from differentiating (2.3) and (2.4) with respect to a and making appropriate use of (2.1). Recalling w(a, y ) and w,(a, y ) given by (5.1), Eqs. (5.3) can also be written na(a> =

J’

1

0

~ 2 2 2 2 ( a 0, , ‘IM(’I)d

1

m,(4 = 4 . )

+ 2 J-0 4 2 2 @

‘~ -

J

1

0

4,

r 2 2 2 2 ( a , 0 , ‘I>~(‘I>

(5.4)

1

0,

M r ) 4 - 2 J0 P 2 2 ( G

6 9

v>m(‘I)4

Substitution of w,(a, y) given by (5.1), w,,(a, y ) given by (2.3), and n,,

ma given by (5.4), into (5.2) yields

1

+ P ( 4 Y , 0 ) + 2 J0 A4 Y , 1 ) 4 2 2 ( 4 v, 4 d‘I

) 44I

- Jolda,Y , v ) s 2 2 2 2 ( 4 ‘I,0 ) 4

do = 0,

where the delta function appearing in the second equation has been used in order to include m(a) under the integral sign. Equations (3.16)-(3.19) follow directly from (5.5) since Eqs. (5.5) must hold for all m(o) and n(a). It is finally noted that the initial conditions (3.20) and additional conditions (3.21) follow immediately from consideration of (2.2) and (5.1). We have therefore proved that p, q, r, and s, the kernel functions of the integral Riccati transformations (5. l), satisfy the Cauchy system derived in Section 3, exhibiting the equivalence of both methods. Along similar lines we could prove that functions u(x, y, a, a), ~ ( xy,, a, a), O(x, y, a, a), and +(x, y, a, 0 ) in the equations 1

Nx,Y ) = w ~ ( xV, ) =

1

- J’0 4 x , Y , a, 444 do + J’0 u(x, Y , a, a)m(a) da,

J’

1

- Jib, Y , a, a)n(a) do + 0

1W, y , a, a)m(a) do, 1

0

116

5 PARTIAL DIFFERENTIAL EQUATIONS I1

satisfy the Cauchy system (3.22)-(3.23), and the associated set of additional conditions similar to (3.21). We leave this derivation as an exercise for the reader. 6. Computational Aspects

At this point it is convenient to consider possible numerical schemes for the solution of our equations. The Cauchy systems formulated in previous sections can be solved in principle using various quadrature methods. Use of a discretization scheme in the y direction is one possible path of action. We choose this path to illustrate how our equations can be easily reduced to algorithms suitable for solution with a digital computer. Suppose we are interested in the computation of the displacement at the free edge, x = a, of a rectangular plate produced by the action of moments m(y) and vertical forces n(y) along this edge. These quantities, w(a, y) and wl(a, y), are given by Eqs. (5.1), which we reproduce here: 1

5,

w(a, Y ) = - Nu, Y, >.(n). w1(a, Y) =

-I’

do +

1

1

J 44 Y , a)m(a) da, 0

1

4(a,Y , M a )

0

+ J-0 p(a, Y , 46d do,

(6.1)

to facilitate the task of the reader. Subdividing the edge x = a into N equidistant points so that yi=iA,

aj=jA,

i , j = 1 , 2 ,..., N,

(6.2)

and ( N + 1)A = 1, we can integrate (6.1) over the interval [(i - $)A, (i + +)A] and write the results in the form

where u, v , n, and m are N-dimensional column vectors whose ith components are the average intensity of the corresponding continuous quantities w(a, y), w,(a, y), n(y), and m(y) along the interval [(i - $)A, (i + +)A]; e.g., if di) denotes the ith component of vector u, we shall have

117

7. INFLUENCE OF POISSON’S RATIO

The N x N matrices S, R, Q, and P are similarly given by integration of the functions s, r, q, and p . For example, we have 1

R=A’

(1“ 1 i+1/2)A

(j+1/2)A

(i- I / Z ) A

(j- I / Z ) A

r ( a , y , o)d’do),

i , j = 1,2, ..., N. (6.6)

The composed matrix (6.7)

a flexibility matrix, can be shown to be symmetric, a consequence of the reciprocity relations given in Section 4. We can now easily find ordinary differential equations for the quantities in Eq. (6.7) by differentiating R given by (6.6) and corresponding equations for S, Q, and P , under the integral sign and making use of Eqs. (3.16)-(3.19). Application of a convenient quadrature formula yields the system of ordinary equations

+ RT f RTKR + SLS, R, = P + RKP iSLR, Pa = I + PKP -t RrLR, S,

=R

S(0) = 0, R(0) = 0, (6.8) P(0) = 0, where the initial conditions follow directly from (3.20). The matrices K and L depend on the discretization of the second and fourth derivatives in the integrands of (3.16)-(3.19), the auxiliary conditions (3.21), and the particular quadrature formula used. If the trapezoidal rule is used to approximate integrals and central differences are used to replace derivatives, then the equations in (6.8) reduce exactly to Eqs. (10.6) of Chapter 4 where Poisson’s ratio v should be set equal to zero, taking due account of the notation and different sign convention for the shear forces.

7. Influence of Poisson’s Ratio In the example carried out in this chapter, we assumed for simplicity that v, Poisson’s ratio of the material, was zero. Here we release this restriction and assume that v can take any value in the interval 0 Iv I The upper

+.

limit corresponds to the condition of incompressibility of the material. To take into account the influence of Poisson’s ratio, we consider the same problem discussed in Section 2 and introduce the function of three variables u(x, y , v ) such that

v4u = U l l l l + 2u1122 + u 2 2 2 2

= 0,

(7.1)

and u(0, y , v ) = u(x, 0, v ) = u(x, 1, v ) = 0 ,

Ul(0, Y , v )

= uz(x, 0, v ) = u,(x, 1, v ) = 0 ,

(7.2)

118

5 PARTIAL DIFFERENTIAL EQUATIONS I1

in the rectangular domain 0 I x Ia, 0 I yI 1, and such that u11

Ulll

+ vu22 = W ) ,

(7.3)

+ (2 - v>u12z = n w .

The reader familiar with elementary plate theory is aware that u represents the deflection of a rectangular plate clamped at the edges x = y = 0 and y = 1, subject to forces n(y)and moments m(y)at the free boundary x = a, and made of a material whose Poisson’s ratio is v. Therefore w defined in Section 2 is related to u by the equation w(x, u) = 4 x 7 Y , 0). (7.4) Clearly, we could apply the method developed in Section 3 or that of Section 5 to the solution of the present boundary-value problem defined by Eqs. (7.1)-(7.3). We encourage the reader to pursue one of these paths. Our approach, however, is different. In fact, since v is the variable of interest, we seek the construction of a Cauchy system in terms of v and use the available resultsfor v = 0, as the pertinent initial conditions. Toillustrate the main ideas of the method, we restrict ourselves to the derivation of an initial-value problem for the deflection u(a, y , v) and the slope u,(a, y , v) at the free edge x = a. It is first shown that these quantities satisfy a system of Fredholm integral equations and then we use invariant imbedding to operate the reduction to a Cauchy system. Let v(x, y , v) denote the difference v(x, y , v) = @, y , v) - w(x,

v) = 4%y , v) - u(x, y, 0).

Clearly, v satisfies the boundary-value problem v4v = 0 v(0, y , v) = v ( x , 0, v) = v ( x , 1, v) = 0 V , ( O , y, v) = u2(x, 0, v) = V , ( X , 1, v) = 0 v11

Vlll

(7.5)

(7.6)

= -VU22

+ 2 V m = vu122.

This system is identical to Eqs. (2.1)-(2.4), where the moment m(y)and vertical forces n(y) at the free edge are substituted by the values - vuzz and vulzz, respectively. Therefore, the functions v(a, y , v) = u(a, y , v) - w(a, y ) and vl(a, y , v) = ul(a, y , v) - wl(a, y ) can be represented by superposition:

I

1

u(a, Y , v) = w(a, Y ) - v

s(a, y , c)u122(a,c, v) do

0

- v Jolr(a, Y , c>u,,(a, 0, v) do, (7.7)

1

UI(~,Y , V) = wi(a,

U) -

v j da,Y , c)u1n(a, 0

0 , V)

119

8. INVARIANT IMBEDDING SOLUTION

a system of integrodifferentialequations of the Fredholm type for the quantities of interest, u(a, y, v) and u,(a, y, v). Introducing the following notation

(7.9) and A=

i),

(7.10)

and recalling that w(a, y ) = u(a, y, 0) and that w,(a, y) = u,(a, y, 0), Eqs. (7.7) can be written in the compact form

Note that in this notation the reciprocity relations (4.1) reduce to

XCa,y, 4 = XT(40 , u).

(7.12)

Equation (7.1 I), an integrodifferential equation, can be easily transformed into an ordinary second-order Fredholm integral equation by the simple device of integrating twice by parts, thus obtaining

where conditions (3.21) and (7.2) have been used to eliminate the boundary terms appearing in the integration by parts. Our problem next is to solve Eq. (7.11) or, alternatively, (7.13), assuming that the problem has already been solved for v = 0, i.e., that z(a, y, 0) and X(a,y, a) are known. 8. Invariant Imbedding Solution

Our aim is to reduce (7.11) or (7.13) to a Cauchy system. This can be accomplished in a number of ways, using ideas of invariant imbedding. We could, for example, start our analysis by using the Cauchy system for the resolvent kernel of the integral equations. The imbedding parameter would be in this case the length of the interval of integration. In this fashion we could derive an initial-value problem in terms of the width of the plate, i.e., the length in the y direction. Several types of Fredholm integral equations can be treated along these lines. (See the Bibliography at the end of the chapter for references pertaining to the aforementioned items.)

120

5

PARTIAL DIFFERENTIAL EQUATIONS I1

However, as already noted in Section 7, out task is to use v as the basic imbedding parameter. Surprisingly, this can be easily done. The resulting Cauchy system will be an initial-value problem with respect to v, a convenient parametrization for studying the influence of v in the stress distributions. To this end, we shall use some results on integral equations due to R. Kalaba, which are limiting forms of the classical resolvent equations in functional analysis. We can write the solution of (7.13) in terms of the resolvent kernel R(y, o,v), a 2 x 2 matrix function, as

where for simplicity we have omitted the dependence of R on the length of the plate a. We will now find initial-value problems for both z and R as functions of v. Differentiating(7.13) with respect to v, we have 1

zv(a,Y , v) = -Jo Xz2(a, Y , o ) M a , u, v) do - v

Xzz(a,y , o)Azv(a, o,v) do,

Jol

(8.2) or, using (7.13) to eliminate the first integral in (8.2), 1

zv(a,Y , v) = (l/v)[z(a, y , v) - z(a, Y , 011 -

X d a , Y , o)Azv(a, 6, v) do.

(8.3) Once again we can employ the resolvent kernel to write the solution of Eq. (8.3) in the form zv(a, Y , v) = (l/v)[z(a, Y , v) - 4% Y , 0)l "1

Substitution of z(a, y , v) given by (8.1) into (8.4) yields

an initial-value problem for z(a, y , v) with respect to v since we assumed to know z(a, y , 0), the initial condition of (8.5). In order to solve (8.5), however, we need to know the resolvent R. To this end we now construct an initialvalue problem for R. Substitution of z(a, y , v) given by (8.1) into (8.5) yields the equation 1

zv(a,Y , v)

=

J w,vMa, u,o>do 0,

0

121

NOTES, COMMENTS. A N D BIBLIOGRAPHY

On the other hand, differentiating (8.1) with respect to v, “1

Comparison of (8.6) and (8.7) yields [since z(a,a, 0) is arbitrary] R,(y,

0,v) =

J k Y ,

4,v)R(4,0,v) d4,

0

(8.8)

the desired initial-value problem for the resolvent R in terms of v. Since the kernels R and X,, A must satisfy the classical equation for the resolvent

R(Y,a, v) = - X d a , Y ,

-v

1‘x2,(a, 0

Y , WW, 0,v) dt,

(8.9)

the initial condition for (8.8) will be N Y , 620) = - Xzz(a, Y , O M .

(8.10)

Equation (8.5) subject to the obvious initial condition z(a, a, 0), assumed to be given, and Eq. (8.8) subject to initial condition (8.10) comprise the desired Cauchy system in terms of Poisson’s ratio v. NOTES, COMMENTS, AND BIBLIOGRAPHY

1. For the variation of Green’s function with the contour see J. Hadamard, Memoire sur le problkme d’ analyse relatif i I’equilibre des plaques elastiques encastrees. Memoires present& par divers savants a 1’Academie des Sciences de I’Institut de France, Vol. XXXIII (1908); R. Bellman and H. Osborn, Dynamic Programming and the Variation of Green’s Functions, J . Math. Mech. 7, No. 1 (1958), 81-86; V. Volterra, “Theory of Functionals.” Dover, New York, 1959; M. M. Schiffer, Boundary-Value Problems in Elliptic Partial Differential Equations, in “Modern Mathematics for the Engineer” (E. F. Beckenbach, ed.). McGraw-Hill, New York, 1956 (paperback).

2-4. Here we follow E. Angel, N. Distkfano, and A. Jain, Invariant Imbedding and the Reduction of BoundaryValue Problems of Thin Plate Theory to Cauchy Foimulations, Internut. J . Engrg. Sci. 9 (197 l), 933-945.

A proof of the converse, i.e., that every solution of the nonlinear Cauchy problem satisfies the original linear boundary-value problem, is given in E. Angel and N. Distefano, Equivalence of a Cauchy System and a Class of Boundary-Value Problems in Thin Plate Theory, J. Engrg. Math. 6 , No. 2 (1972), 117-123.

122

5

PARTIAL DIFFERENTJAL EQUATIONS I1

Application of this method to the solution of thin shells of revolution may be found in N. Distkfano and A. Jain, A Cauchy System in the Linear Theory of Thin Shells of Revolution, Proc. SECTAM VII March 1974.

7. The influence of Poisson’s ratio has been studied using classical arguments by Westergaard. See H. M. Westergaard, ‘‘Theory of Elasticity and Plasticity.” Harvard Univ. Press, Cambridge, Massachusetts, 1952.

8. The results presented here were first given in E. Angel and N. DistCfano, Invariant Imbedding and the Effects of Changes of Poisson’s Ratio in Thin Plate Theory, Internat. J. Engrg. Sci. 10 (1972), 401408.

For a derivation of a Cauchy system for the resolvent see R. Bellman, Functional Equations in the Theory of Dynamic Programming-VIII: A Partial Differential Equation for the Fredholm Resolvent, Proc. Amer. Math. SOC.8 (1957). 435440.

For a connection between matrix Riccati equations and Fredholm resolvents see A. Schumitzky, On the Equivalence between Matrix Riccati Equations and Fredholrn Resolvents, J. Comput. System Sci. 2, No. 1 (1968), 76-87.

The solution of Fredholm integral equations using invariant imbedding ideas has been extensively studied by R. Kalaba. For applications of a class of integral equations of interest in radiative transfer and other branches of physics, see R. Kalaba, Reduction of Matrix Integral Equations with Displacement Kernels on the Half-Line to Cauchy Systems, in “ Invariant Imbedding” (R. Bellman and E. Denman, eds.), Chapter 4, Lecture Notes in Operations Research and Mathematical Systems No. 52. Springer-Verlag, Berlin and New York, 1970.

For applications of the theory of dual integral equations in potential theory in connection with the problem of thermal diffusion in a cylinder see R. Kalaba and E. Ruspini, Invariant Imbedding and Potential Theory, InterMr. J. Engrg. Sci.7 (1969), 1091-1101.

The method of this section is based on a result first given in J. Casti and R. Kalaba, On the Equivalence between a Cauchy System and Fredholm Integral Equations, USC-EE Tech. Rep. 70-21 (March 1970).

which in turn may be considered a limiting form of some classical results for the resolvent. See, for example, A. N. Kolmogorov and S.V. Fomin, “Elements of the Theory of Functions and Functional Analysis,” Vol. 1, pp. 110-112. Graylock Press, Rochester, New York, 1957.

Chapter 6

Dynamic Programming

1. Introduction

In this chapter we wish to present some of the fundamental ideas and techniques of dynamic programming, the counterpart of invariant imbedding in the realm of variational processes, as it applies to a number of basic problems in structural mechanics. The origins of dynamic programming lie in the theory of multistage decision processes. The advent of the digital computer and the needs of modern engineering analysis have led in a natural way to a look at the calculus of variations as a multistage decision process, thus enlarging in an almost unlimited fashion the scope of the theory. There are several important sources of variational problems that fall within the scope of this book. In the first place we have the problems of equilibrium in mechanics. In fact, an alternative way to formulate models in mechanics is through the use of variational principles: A quantity, energy, written as a functional of the variables of the system, reaches a minimum value when it corresponds to the configuration of equilibrium of the system. Teleological principles, seeking conditions for equilibrium, are at present well understood and embodied in the developments of the calculus of 123

124

6

DYNAMIC PROGRAMMING

variations, an important branch of analysis. Philosophical as well as scientificrivalry between variational and nonvariational models has reduced at present to only prosaic considerations and disputes over alternative computational advantages. On the other hand, the control of engineering processes on one hand, and the problems arising in the emerging theory of optimal structural design on the other, lead naturally to variational formulations. Last but not least we must consider the important class of inverse problems that can be formulated as optimization problems. In this class we recognize, among others, the problems of system identification, which is of paramount importance in modern nonlinear mechanics. In this chapter we concentrate mainly on the first class of problems mentioned in this introduction, i.e., on those problems arising from consideration of variational principles in mechanics. The inclusion of a problem on system identification at the end of the chapter is an exception to this rule, a decision taken mainly on methodological grounds. A systematic treatment of system identification and structural optimization problems finds its place in specific chapters of this book.

2. ALinearString

To illustrate the basic ideas of the theory, let us consider the example of the linear string supported on elastic springs discussed in Chapter 1. We consider the potential energy ei stored during the deformation of a generic link between the nodes i and i + 1. This energy, assuming for simplicity that there is no external load acting on node i, is formed by the energy needed to produce the displacement uiof the spring plus that needed to elongate the string between nodes iand i + 1. Under the hypotheses of small deformations, ei can be written ei = +(kiu: lSwiz), (2.1) where wi is the slope of the string to the right of node i. Clearly, the functional

+

N- 1

E(ui) =

+ C (ki uiZ+ ISwi’) +

+kN uN2,

i=O

(2.2)

where ui and wi are related by the continuity condition Ui+l = ui

+ lw,,

(2.3)

represents the potential energy of the system formed by N links and in the absence of external loads. We assume the ends of the string to be fixed and, in particular, u N = 0,

uo = c.

(2.4)

125

3. THE PRINCIPLE OF OPTIMALITY

We shall use an imbedding procedure to solve our problem. To this end we define N-1

fn(un) = min

41 (kiui2+ /swi2), n

(2.5)

where the minimization can be performed with respect to either ui or w isince both quantities are related by (2.3). Equation (2.5) exhibits our fundamental imbedding: We are considering the minimum potential energy of the family of strings of variable lengths ( N - n)Z, with n varying from n = N - 1 up to n = 0, and subject to a displacement u, at the end node n. Clearly, fn(u,)is a function of n and u,, but may also be interpreted as a functional of the sequence u,,, ~ , , + 1 , ... , uN-1 (or w , , w,,+~, . . . , w ~ - ~ Our ) . next task is to find a functional equation relating the quantitiesS,(u,,). To this end, choosing w, to be the decision variable, we can write

= min[HknunZ

+ lswn2) +fn+i(un+l)I,

W"

a result that on account of (2.3) yields fn(un) = min[f(k,un2

+ /swn2)

+fn+l(un

+ /wn)I,

(2.7)

W"

a functional equation for the minimum potential energy, subject to the initial condition (2.8) fN-i(uN-1) = f(kN-1 + slM-1,

~ in the equation an expression that was obtained by substituting w ~ =-uN-l/i fN-i(uN-1)

= f(kN-iui-1

+ /swi-i).

3. The Principle of Optimality The functional equation (2.7) was obtained using some properties of the minimum operation, as indicated in Eq. (2.6). We can derive (2.7) in a more direct fashion. Consider a typical member of the imbedding family, e.g., the string comprised between nodes n and N , in the equilibrium configuration.

126

6 DYNAMIC PROGRAMMING

Fig. 6-1

See Fig. 6-1. We shall use an asterisk to differentiate the slopes associated with an equilibrium configuration from those merely satisfying kinematic constraints. Thus, since the energy is an additive quantity, we have by definition

+ ~ S W , +L+l(un *~) + lwn*),

(3.1) i.e., we consider the potential energy of the string (n,N ) in its equilibrium configuration to be equal to the potential energy stored in the link (n,n + l), plus the potential energy of the remaining string (n + 1, N ) which is clearly subject to the displacement u,,+l = u,, + Iw,* at node n + 1 . Now, from the principle of minimum potential energy, if instead of w,*, we use any admissible w,,in (3.1), we must have fn(un)

= $(knun2

L(UJ 5 H k n U n 2 + lSwn2) + L + l ( u n + lwn),

(3.2) where the equality sign holds only for w,, = w,*. Clearly, (2.7) follows immediately from (3.2). The considerations that precede are a particular version of the principle of optimality by Bellman. See any of the books given as references of Section 1 , at the end of the chapter. We return again to this point in Section 25. 4. An Analytical Solution

The results of the last section can be considerably advanced, providing an elegant solution of the string, if we use calculus to find the minimizing slope wn* at each stage of the optimization process. In fact, the condition for a minimum in (2.7) is ISwn

+ (a/awn)fn+l(un + lwn) = 0,

(4.11

which on account of (2.3) reduces to

Clearly, since Sw,,is the vertical component of the string tension, the condition of optimality (4.2) of the Bellman-Hamilton-Jacobi equation (2.7) is an expression of the theorem of Castigliano.

127

EXERCISES

Now, it is easy to show by an inductive reasoning on (2.7) and (2.8) that fn(un)is a quadratic function in its argument. Hence if we write fXUA

= tRn un2,

(4.3)

Eq. ‘(2.7)reduces to R,u,Z

= k,~,’

+ ISW,’ + R,+~(u,,+ Iw,,)~,

(4.4) where w,,in (4.4) is the value that minimizes in (2.7), i.e., w, satisfies the equation SW,, R,+l(u, + ZW,,) = 0, (4.5) from which we derive

+

Instead of directly eliminating w, between (4.4) and (4.6), we take an intermediate step that simplifies the algebra. In fact, we combine (4.4) and (4.5) to rewrite (4.4) in the form

which, on account of (4.6), and after collecting terms in un2,leads to

a backward recursive equation for R, subject to the initial condition RN-1 = kN-1

+ Sll,

(4.8)

obtained from consideration of (2.8) and (4.3). Finally we wish to show that R,, given by (4.7)-(4.8) and 2, given by Eqs. (8.6)-(8.7) of Chapter 1 are related in a simple manner. All we have to do is to compare Z,, with RN-”because of the opposite “time” directions in both cases. Very simple manipulations show that

RN-,,= k,,

+ SZ,, .

(4.9)

EXERCISES 1. Consider k, constant and equal to k > 0. Show that, as n tends to infinity, the initial slope of the string is related to the initial deflection uo by means of the equation

wo=--

R S + 1R ‘O’

where R is the positive solution of

R2- k R - k(S/l)= 0 .

128

6

DYNAMIC PROGRAMMING

2. The potential energy of a string of N links subject to loads p , on the nodes is given by

where u1 and w 1are related by the equation u l +

and

= u1

+ Iw, . Set

M u ) = R.uZ + r,u + s..

Find recurrence relations for R., r., and s.. What are the initial conditions of those relations if U N = O? 3. Show that the Riccati equation R'

=k

- ( l/S ) R 2,

where k = liml,o (kJl), is the limiting form, as I tends to zero, of the differenceequation (4.7).

5. Constraints

The simplicity of the analytical solution derived in the previous sections is largely due to the use of differential calculus to minimize at each stage of the process. This procedure, however, will not be legal in general because we shall be bound by local constraints imposed on the displacements and slopes such as to render the expression to be minimized, nondifferentiable. In general, the structure of the analytical solution of the constrained case will be much more complex than that of the unconstrained one. Paradoxically, the direct numerical solution turns out to be easier in the constrained case than in the unconstrained one. We defer a discussion of the numerical solution to the next section where more general cases are discussed.

6. A Nonlinear String If we assume that the springs are nonlinear, the potential energy stored by them will no longer be a quadratic functional. For example, we can think of a spring whose force-displacement law is given by F, = k,u, + 2rnun3. The potential energy stored in such a spring will be equal to t(k, u,' + r, un4). In general we shall consider, instead of (2.5), the nonquadratic functional N- 1

f,(un)

= min

31 (hi(ui) + lswi2) + hN(uN), n

where hi(ui) is the potential energy stored in the spring i.

(6.1)

6.

129

A NONLINEAR STRING

More generally, we can think of the potential energy stored in the spring i and the link i, i + 1 as a function of the type gi(ui, wi , So),where So is the tension of the string at a given reference state. In this case we consider N-1 f.(un)

= min

C gi(ui

7

wi 9

n

so) + C P ( ~ N ) ,

(6.2)

where p(uN) is the potential energy stored in the last spring. By following the arguments of Section 3 step by step, we readily obtain the functional equation

where v is a dummy variable and where for simplicity we omitted So in gn. The end condition reads fN(u)

(6.4)

= q(u)*

In general, the determination of the structure of the solution of (6.3) and (6.4) by analytical methods can be a matter of considerable difficulty, particularly when u,, and w,, are subject to local constraints. Thus one sees the interest in concentrating here on numerical procedures. To illustrate the essential ideas we shall consider local constraints on the slopes and displacements of the type O
Iw,,~

rd,,.

(6.5)

We consider that for a fixed n, only a discrete set of values can be taken by u,, and w,,. For example,

where u, and /3, are such that a, 6 = c,, and /3, y a discretized version of (6.3) will read

. With these restrictions,

= d,,

(6.7)

where i

= 0, 1,2,

..., u,, and wheref, is subject to the end condition f N ( i 6 ) = q(i6).

(6.8)

We can see that the problem has reduced to the minimization of a sequence ofN - 1 functions of one variable, a problemwithin even modest computational capabilities,if reasonable values of a,,and B,, are prescribed. The procedure to follow at a typical computation stage is very simple. At the stage n we assume to have available in rapid access storage, the values of the functionf,,, at the points 0, 6,26, . . . ,~ , , + ~which 6 , were computed in the previous stage. Now,

130

6

DYNAMIC PROGRAMMING

for each fixed value of i, we minimize the right-hand member of (6.7) by using, for example, an exhaustive enumeration procedure, and retain the values of j that minimize. We shall denote these values byj(i). In order to evaluatef,,, at the points i6 Qy, we make use of the stored values off,, at the points id, together with some interpolation procedure, if needed. We repeat the procedure for each admissible value of i and proceed to store the values of the new functionf, . One of the typical difficulties arising in this method is that for some i 5 a,, and j 5 j,, the argument i6 l j y may fall outside the range for which the function is defined. What we can do in this case is to restrict the set of values of the admissible slopes further. For example, in our present case we would consider only values o f j that satisfy the original constraint I j l 5 fin and the additional one

+

+

0
+ Qy ~ a , + , d ,

0I i
(6.9)

+ 3242) + 3(u52 +&u54),

(7.1)

7. A Routing Problem

As an example we consider the functional 4

J(uJ =

+c

( U i Z +&Ui4

i=O

where ui and ui are related by the continuity equation Ui+l =

ui

+

ui,

(7.2)

subject to the particular initial condition 245

= 3.

(7.3)

In addition to the fixed end condition (7.3) we consider the following constraints on the displacements u i , 0 5 ~ ~ 5 5 i,= O , 1 , 2 , 4 ,

0

5243

52,

(7.4)

and the following constraint on the slopes u i ,

l u i l r l , i = O , l , ..., 4.

(7.5)

We discretize ui and u i in uniform increments of one, i.e., 6 = y = 1. With this discretization in mind and taking into account (7.4), the displacements ui are restricted to take the values

ui={0,1,2,3,4,5}, u3

= (0, 1921,

i=O,l,2,4,

(7.6)

131

7. A ROUTING PROBLEM

and the slopes the values u i = { - 1 , 0 , 11,

i = O , 1,2,3,4.

(7.7)

We observe that consideration of the continuity equation (7.2) in combination with the constraints (7.6) and (7.7) introduces further constraints on the displacements. In Fig. 6-2 we represent with a large dot all possible states of the nodes, i.e., the displacements ui ,and with dotted lines, all possible slopes between adjacent nodes. So posed, the problem can be treated as an optimal routing problem, namely, for any given admissible state u o , find the trajectory connecting feasible states associated with permissible slopes so as to minimize the functional (7.1). We define as usual

f.(un) = min

+

4

(u: +&u: n

+ 3u:) + 5.31,

(7.8)

and construct the functional equation

where un has been replaced by i = 0, 1, .. .,according to (7.0, vi by j = - 1, 0, 1 according to (7.7), and where g(i,j) is the potential energy stored in a generic link given by g(i,j) = +(i2 + h i 4

+ 3j’).

(7.10)

The minimum potential energy h(i)satisfies the initial condition f5(3) = 5.31.

(7.11)

We solve the function equations (7.9) and (7.1 1) directly on the graph of Fig. 6-2. We start by considering the last link and compute

+

f4(2) = g(2, - 1) 5.31 = 8.97 f4(3) = g(3,O) + 5.31 = 10.62.

(7.12)

The values of f4(2) and f4(3) are recorded in the circles located above the large dots representing the nodes. The arrows indicate the optimum slope; in the present case there is only one possible slope for each node at n = 4. Then we consider the chain with two links and compute

f3(1)

= g(1,

f3(2) = min

- 1) +f4(2)

= 2.01

+ 8.97 = 10.98,

g(2,O) +$4(2) = 2.16 +8.97 = 11.13

- 1) +f4(3)

= 3.66

I

+ 10.62 = 14.28

= 11.13 (U = 0).

(7.13)

In I1

c

P C

ro I1

c

133

8. A CONTINUOUS ANALOG

The values of f3( 1) and f3(2) are again recorded in circles above the nodes, with the arrow that indicates the slope u that minimizes. Proceeding in the same fashion, adding one link at a time, we finally compute all thef,(i) in a sequential manner, by minimizing functions of one variable at each stage of the process. At the same time the arrows associated with the minimum potential energy functionsprovide the optimal trajectories, i.e., the elasticas with the same endpoint us = 3. For example, the elastica with the fixed ends uo = 3 and us = 3 is the trajectory obtained following the arrows from the node uo = 3. This is the dashdot line in the figure. Another example is provided by the dashed line in the figure. This, the elastica starting at uo = 5 , has been clearly affected by the constraint on the deflection at n = 3. The reader could convince himself that this is the case by solving the unconstrained case. Or, is there a quicker way to prove that by removing the constraint u3 I 3 , the elastica will change? Finally we note that if the displacement at n = 0 is not prescribed, the associated elastica will be that for which the potential energy is minimum. In the present case the minimum offo(i) occurs at i = 0 withfo(0) = 12.48 and an associated elastica indicated by the dotted line. 8. A Continuous Analog

We consider here the minimization of the functional L

g(u, u'x) dx,

J(u) = 0

which may be considered to represent the potential energy of a general nonhomogeneous, nonlinear string of length L. By analogy with the discrete case, we define f ( u , x) = min

lL

g(u, u', z) dz,

(8.2)

where the minimization may be performed with respect to either u or u'. Again, we choose u' as the pertinent decision variable. Proceeding as in Section 3, we decompose the interval [x, L] in the interval [x, x A] and [x + A, L] and apply the additive property of the integrals

+

x+ A

f(u, x)

= min [Jx UP

L

+ s,,,g(u, u', z ) dz]

[jxx+A g(u, u', z) dz +

- min U'(2) SE[X, x+

g(u, u', z ) dz

A]

s+,.,

L

min u, (2)

zs[x+ A, Ll

- min [/xx+Ag(u,u', z) dz + f ( u ( x + A), x + A)]. u' ZE[X,

(2)

x + A]

1

u', z) dz

(8.3)

134

6 DYNAMIC PROGRAMMING

We can now write, under appropriate smoothing conditions, x+ A

5,

9(u, uf, 4 dz = g(u, u’, x)A

+ @I,

(8.4)

and U(X

+ A) = U ( X ) + u’(x)A + o(A),

(8.5)

where the term o(A) is such that limA+oo(A)/A = 0. We can now substitute (8.4) and (8.5) into (8.3) obtaining

f(u, x) = min[g(u, w, x)A +f(u

+wA, x + A) + o(A)],

(8.6)

W

a functional equation similar to (6.3) where the minimization is performed with respect to w = u’(x), a dummy function in the present case. The limiting form of (8.6) as A + 0 can be constructed easily. Assuming the existence and boundedness of the necessary derivatives, we can write

f(u

+ wA,x + A) = f ( u , X) +fX(u, x)A + wfU(u,x)A + o(A),

(8.7)

where u stands for u(x) and where a subscript indicates partial derivative. Substitution of (8.7) into (8.6) yields

- U u , x) = min[g(u, w,x ) + wfXu, w, x)l,

(8.8)

W

a partial differential equation for the minimum potential energy. When the displacement u(L) is not prescribed, function f(u, x) must satisfy the end condition f ( u ( J 3 , L )= 0. (8.9) When, on the contrary, we prescribe u(L),u’(x) will become unbounded as x -+L and therefore the function f(u, x) will not in general be defined at x = L. In this case we look for an initial condition at a point x in the neighborhood of L. This can be done easily using several asymptotic expansions forf(u, x). The simplest one can be obtained by substituting u‘(x), given by

L - A 5 x -< L,

u’(x) = [u(L)- u(L - A)]/A,

valid for A

(8.10)

< I , into Eq. (8.2), i.e., L

f(u(L- A), L - A) = L-A

g(u5 u’, z) dz,

(8.1 1)

where u’ is given by (8.10). Finally we could write

f(u, X) 2 ( L - X)g(u, (d - u)/(L- x), x), the desired asymptotic result, where d stands for u(L).

L - x Q 1,

(8.12)

135

9. EULER EQUATIONS

EXERCISE Show that the problem of mimimizing the functional

where u and u are related through the differential equation u’

= h(u,

u, x),

u(0) = u g ,

can be reduced to the integration of the partial differential equation

-fx(u, x ) = min[g(u,

”

0,

XI + Mu, u, XMU, 41,

where f ( u , x ) = min J’:L&

0,

5) d5,

subject to the intial condition f(u,. L ) = 0.

9. Euler Equations We consider here the minimization of the functional

I

L

J(u) =

0

g(u, u’, x ) dx

(9.1)

by classical variational methods. We assume u prescribed at both ends ~ ( 0= ) C, u(L) = d. (9.2) This is known as the simplest problem in the calculus of variations. The function g(u, u’, x ) is assumed to be continuously differentiable up to the second order with respect to any of the three arguments u, u’, and x. In addition, u’ is assumed to be piecewise continuous. This condition may be considerably weakened, but the present restriction is enough for our purposes. We consider now a variation of the function u constructed in the following way: Let q(x) be an admissible function defined in the closed interval [0, L], and vanishing at the endpoints of the interval. For sufficiently small absolute values of a parameter E , the functions

U=u+&q (9.3) lie in the neighborhood of u. We say that 6u = eq is a variation of the function u. We now compute J at U = u + eq, i.e.,

+ E V ) = S g(u + ~ q U‘, + EV’, X) dx. L

J(u

(9.4)

0

A necessary condition for a minimum is given by (d/dE)J(u

+ &q)

Je=O

= 0.

(9.5)

136

6 DYNAMIC PROGRAMMING

Consequently, differentiating under the integral sign in (9.3), we obtain JOL(9J + 9.. ?') dx = 0,

(9.6)

an equation that, integrated by parts, and recalling that q vanishes at the endpoints, yields

valid for all admissible trial functions q. From the fundamental lemma of the calculus of variations it follows that

(dPxlg,. - 9, = 0, (9.8) a second-order ordinary differential equation subject to the two-point conditions (9.2). In expanded form (9.8) reads g,.,. u"

+ g,., u' +

gurx

- 9" = 0.

(9.9)

In order to be able to solve (9.9) for the higher derivative we must have

gu,,, # 0,

(9.10)

an inequality known as the Legendre condition. We observe that in integrating (9.6) by parts in order to obtain (9.7), we are implicitly assuming the existence of the second derivative of u, a fact that has not been postulated. Fortunately, under our present assumption regarding u and g, the existence of U" can be readily proved, validating in this fashion the Euler equations (9.8). The reader can find references on this aspect at the end of the chapter. Our next task is to show the connection between dynamic programming and the classical calculus of variations. Specificallywe wish to prove that u in f(u, x) which satisfies (8.8), satisfies the Euler equation (9.8). This is very easy to do. In fact, assuming that the minimization in (8.8) can be carried out by differential calculus, instead of (8.8) we consider the system

f, = -9 - UYU, 0 =9.* + f u ,

(9.11) (9.12)

where the second equation furnishes the condition for a minimum. Now differentiating(9.12) with respect to x and (9.11) with respect to u, we obtain

-9u - U I f U U , d d d 0 = -9,. f, = -g,. dx dx dx

fxu =

+fuuu)+f,,, from which (9.8) follows by equating f,, and f,, . +-

(9.13)

137

10. CONVEXITY

RARRCIS~S 1. Show that when g(u, u', x) = tu"

+ h(u),the Euler equation is u"

= h(u).

2. Show that if u(L) is not prescribed, the natural condition is u'(L) = 0.

10. Convexity

We consider again the functional

1

L

J(u) =

g(u, u', X) dx.

(10.1)

0

By expanding the integrand in a Taylor's series with a remainder after two terms, around an admissible function u, we can compute J(u

+ u) = J(u) + SJ(u) + f 6 2 J ( U ) ,

(10.2)

where u is an admissible variation, 6 J is the first variation given by ,L

6J(u) = J (ug, 0

+ u'g,,,)dx,

(10.3)

and where S*J(u), the second variation, is given by L

S2J(u) =

J0 (v2S,, + ~ u u ' ~ , , , +. u ' ~ ~ ~dx, ~ , . )

(10.4)

where an overbar indicates that the derivatives have been calculated along intermediate curves U. For example,

+ U ,u' + U', x), where U is a function that lies between u and u + u. g,,

= guu(u

(10.5)

When u = u* is an extremal, i.e., when it satisfies the boundary conditions (9.2) and the Euler equation (9.8), the first variation vanishes identically for every admissible trial function u. This can be verified by integrating by parts in (10.3). Therefore J(u*

+ u) - J(u*) = #J(u*),

(10.6)

and a necessary condition for a minimum is that the second variation S2J(u*) be nonnegative for all admissible U lying in the neighborhood of an extremal u*. This is a local convexity property that can be reduced, after an adroit treatment, to the elegant Legendre condition gu,,, 2 0,

where the second derivative is evaluated along the extremal.

(10.7)

138

6 DYNAMIC PROGRAMMING

A more general result is obtained if the second variation given by the quadratic functional (10.4) is nonnegative for all admissible functions u and u. This, a global convexity property, is a sufficient condition for a minimum. The minimum obtained in this fashion is an absolute minimum. If the strict case a2J > O holds, i.e., if J i s strictly convex for all admissible u and v , we can additionally guarantee the uniqueness of the minimum, or in other words, we can establish the uniqueness of the Euler equation. This can be proved readily by contradiction. Clearly, a sufficientcondition for a minimum is that the Hessian matrix H, associated with the quadratic form under the integral sign in (10.4), i.e., (10.8) be nonnegative definite. If, furthermore, H > 0, the Euler equation yields the unique solution of the minimization problem. It is clear that conditions for anabsolute minimum are far more demanding and difficult to establish than those associated with local aspects. In general, we shall always be able to assert the existence and uniqueness of the solution of the minimization problem, or equivalently, of the Euler equations in a local sense, i.e., for small interval L and small IIu(x) - u(O)((.Results of this type can therefore be used to establish appropriate conditions and bounds for the existence and uniqueness of f ( u , x) satisfying the partial differential equation (8.8) subject to appropriate initial conditions. A direct treatment of questions of existence and uniqueness in Eq. (8.8) without resorting to results in the classical theory is possible, but is beyond our present scope. The reader can consult the references at the end of the chapter. We return partially to this subject in the next chapter, in connection with some topics of quasilinearization. EXERCISES I . The potential energy of a beam resting on a nonlinear foundation characterized by the force-displacement relation F = U + ~ X U ~

(a)

is given by the functional J(u) = I I L ( E h N z 2pu

+ u2 + hu4) dx.

(b)

0

Assuming the end x = 0 to be rigidly clamped, i.e., u(0) = u’(0) = 0,

(4

show that the Euler equation associated with the minimization of the functional (b) is

(d2/dx2)(EZdzu/dr2) +u

+ 2 h 3= p ,

subject to (c) and the natural conditions

u”(L)= (EZU”(L))’= 0.

(dl

139

11, NUMERICAL ASPECTS

2.

show that if X 2 0 ,the Euler equation (d) subiect to the two-point boundary conditions (c) and (e) admits one and only one solution for all L 2 Q.

3. If < 0, we cannot ensure in general uniqueness of the solution. However, if Ihl 4 1 and IpI Q 1, we expect to enjoy uniqueness. A number of bounds on h and p can be determined for this purpose. For example, show that if a function u(x) satisfies (c), (d), (e), and in addition the inequality

IU I

< (1/6h)”*,

(f)

then it is the unique solution. 4. Can you construct bounds on IpI and

1x1 that are independent of u?

5. Suppose that for a given h < 0 and p , the problem admits more than one solution. What physical interpretation of this fact would you give in the present case?

11. Numerical Aspects

We have reduced the solution of our original unconstrained minimization problem to the solution of the partial differential equation (8.8) which can be written in the form f = - g - wfu (11.1) 5

where w = u‘ is given by the optimality condition 9w

Additionally, when the end x to the initial condition

=L

+f”= 0.

(11.2)

is free, the minimum function f is subject

f (4L ) = 0,

(11.3)

whileiftheend x = Lisfixed, with u(L) = d, we must use (8.12) as the pertinent initial condition. In either case the problem consists of the solution of the partial differential system (1 1. I)-( 1 1.2) subject to appropriate initial conditions. This can be done in a number of ways. Since (1 1.1) and (1 1.2) comprise a quasilinear system that can be written in the form fx

= -9

+ wgw,

fu

=

-9w7

(11.4)

we can apply the theory of characteristics and reduce the problem to that of integrating ordinary differential equations along the characteristics. This approach may be shown to be equivalent to that of integrating the associated Euler equations written in canonical form. We must expect, therefore, to be confronted with two-point conditions, a situation that we systematically try to avoid. In other instances, a particular structure of the function g might suggest a particular direction of attack. This is true, for example, when g is given in the form g(u, u’, x ) = +u’2 h(u). (11.5)

+

140

6 DYNAMIC PROGRAMMING

Equation (1 1.5) can be interpreted as the potential energy per unit length of a string subject to small displacements, lying on a nonlinear foundation. In this case Eq. (1 1.2) yields w = u ' = -f U ) (1 1.6) again a manifestation of the theorem of complementary energy, this time in a nonlinear context. Using (1 1.6), Eq. (1 1.1) reduces to

f, = - h(u) + *f"Z.

(11.7)

When h(u) is a polynomial, one is tempted to try expansions of the type

f (u, x) = ~ ~ ( x ) u+' R , ( x ) u ~+ R , ( x ) u ~+ . . .

(11.8)

In other cases h(u) may be of the form h(u, E )

= +u2

+ Eh,(U),

(I 1.9)

where E is a small parameter and h,(u) is a polynomial of order higher than 2. In this case we may attempt expansions in terms of the parameter E , i.e.,

f ( u , x, 4 =fo(u, x ) + Efi(U, x> + EZf2(U, x) + . - * .

(1 1.10)

If the nonlinear effect, given by the term affected by E , is relatively small, we can reasonably expect, on physical grounds, that the series (11.10) possesses a nonzero radius of convergence. Exercises and references on these analytical techniques can be found at the end of the chapter. A direct numerical approach by using difference schemes on (1 1.1) and (11.2) is also possible. In this case however, there is much to gain if instead of the system (1 1 .I)-(I 1.2) we treat directly the original equation (8.8), or even better, the discretized equation (8.6) from which (8.8) originated. The reasons for this are several. In the first place Eq. (8.6) reflects the physical nature of the problem directly since it has been constructed using physical entities such as energy. We can therefore expect to inherit, in the process of our calculations, the benefits of the inherent stability of the physical quantities involved. On the other hand, we note that in a nondissimulated effort to exhibit analyticity, in Sections 8-10 we have carefully and systematically avoided consideration of local constraints on the slopes and deflections, a fact of the utmost importance in dealing with realistic processes. But we know that when local constraints are to be considered, the analytical structure that we began to build in the preceding three paragraphs will almost completely collapse, confronting us with a new highly intricate problem, for whose solution we cannot unfortunately offer any uniform theoretical treatment at present. On the contrary, a solution based on the discrete scheme (8.6) will recognize the existence of local constraints, which in addition will generally facilitate the solution by naturally delimitating the range of existence of the variables.

141

12. DISCUSSION

In order to illustrate the basic ideas, we write (8.6) without the o(A) term

f(u, x)

= min[g(u, w,

+

x)A f (u

+ wA, x + A)].

(1 1.1 1)

W

Assuming the right end at x = L free, the end condition

f(u, L) = 0

(I 1.12)

must be imposed. Making (1 1.13) and g(u, w, n

(11.14)

w = g&, w),

(11.11) and (11.12) reduce to

L(4 = min[g,(u, w) +fn+1(u + w 4 1 , W

f N W = 0,

(11.15)

a recursive system for the f.(u) that has been discussed in some detail in Sections 6 and 7. We finally note that the solution of Eq. (8.8) can be tackled via the use of the method of successive approximations. We delay this matter until Chapter 7, where it is treated in connection with the problem of approximation in policy space. 12. Discussion

In the preceding sections we discussed the solution of the optimization problem L

min Jo g(u, u', x ) dx,

u(0) = c,

(12.1)

Y

where the other end x = L can either be fixed or free, by means of two utterly different approaches. On the one hand, we have seen how the classical calculus of variations inexorably leads us to two-point boundary-value problems in terms of ordinary differential equations. On the other hand, the dynamic programming formulation resulted in quasilinear partial differential equations subject to initial conditions.

142

6

DYNAMIC PROGRAMMING

This striking structural difference in the resulting functional equations is a consequence of the different imbedding used in the formulation of the problem. We clarify these concepts in the following. In fact, the solution space in the classical formulation is the set of all continuous functions u(x) satisfying the boundary conditions. Once the problem has been posed in this way, we must select that (or those) admissible function(s) u(x) for which the functional (12.1) is a minimum. To solve this global problem we then resort to a local formulation and in fact we derive the Euler equations that provide necessary conditions for local optimality. In the dynamic programming formulation we regard the minimization of the functional (12.1) as a multistage decision process and consider at each stage the value of the slope u'(x) that minimizes the remainder of the process associated with the admissible family of u(x) at the same stage x . In other words, for a given x we compute the value of u' associated with every possible u. In mathematical terms what we do here is to determine the function u'(u); in the terminology of control theory, we are determining at each stage of the process, the decision in terms of the state of the system, a feedback rule. The sequence of decisions constitutes a policy, and in fact, the dynamic programming approach to problems in the calculus of variations is to determine optimum policies in a sequential fashion, a fact rich in consequences. Since at each stage we minimize with respect to every possible decision and for every admissible state, the solution resulting from this formulation is an absolute minimum. Therefore a local formulation at each stage leads to a global solution. It is in this duality, u(t) versus u'(u), or local versus global, that the difference between the two theories lies. The implications of this duality in a computation context are obvious. Clearly, in dealing with the Euler equations we are compelled to reformulate the problem in some adroit fashion if our intention is to solve the problem on a digital computer. In fact, there is no standard procedure to solve nonlinear boundary-value problems numerically. The dynamic programming approach, on the other hand, does not require a special reformulation because the functional equation itself leads in a natural fashion to a recursive computational scheme of the optimum policies and associated optimum value function. 13. Partial Differential Equations

In this and subsequent sections we wish to discuss the application of dynamic programming ideas to the solution of partial differential equations. The natural device in this context is the use of variational formulations. Consider, for example, the Dirichlet functional

!Iv

(w:

+ :w

- 2pw) dx

dy,

(13.1)

143

13. PARTIAL DIFFERENTIAL EQUATIONS

where w is a function with piecewise continuous first derivatives in the domain V, taking prescribed values along the boundary S, i.e.,

(13.2)

on S.

w(x,y) = wo

We know that the function w that minimizes in (1 3.1) and satisfies boundary conditions (13.2) is the (unique) solution of Poisson's equation

(13.3) w,, + W'yy = P, subject to (13.2). It is often convenient to replace the constrained minimization problem (13.1)-(13.2), by an unconstrained one. This may be done by incorporating the constraint (13.2) into the functional (13. l), via the introduction of a Courant parameter A, i.e., by constructing the functional J(w) =

JJV (w,' + :w

- 2pw)

dx dy + A

[ (w - w,#

ds.

(13.4)

S

As expected, it is possible to prove that w(x, y , A) that minimizes in (13.4) will approach the minimum of (13.1) subject to (13.2), as A 00. In order to solve this minimization problem, we discretize the integrand of (13.4) using finite element procedures. To that end we use the same finite element partition, strip configuration, and notation as introduced in Sections 3 and 4 of Chapter 4. In this fashion, by making w = c i N i ' u iin Eq. (13.4), we are led, after appropriate grouping of terms, to the discrete functional --f

N-

J(u(l), ~ ( 2 1 *, * * u ( W ) = 9

1

C1 En(u(n>,u(n + I)),

(13.5)

n=

where u(n) is the vector defined by Eq. (4.1) of Chapter 4 and En is the energy stored in the strip (n,n l), given by the quadratic form

+

E,(u(n), u(n + 1)) = (44, K,,(n)u(n)) + (u(n + I), K z , ( M n -t 1)) + 2(u(n), K,,(n)u(n + 1)) + 2(u(n),p,(n))

+ 2(4n + l),~z(n))-

(13.6)

In (13.6), matrices Kij(n) and vectors pi(n) retain the same meaning as in Section 4 of Chapter 4. Introducing N- 1

f.(u(n)) = min

C

Ei(u(i),u(i

u(n+l) i = n

+ I)),

(1 3.7)

application of standard arguments in dynamic programming leads to the functional equation

whose solution may be written in the form

144

6

DYNAMIC PROGRAMMING

In (13.9), Q(n), q(n),and s(n) are quantities to be determined. Proceeding as in Section 4, we find

EXERCISES 1. Work out the details of the derivations of Eqs. (13.10)-(13.12). 2. Show that the quantities Q(n) and q(n) are related to R(n) and r(n) defined in Section 5 of Chapter 4 by

Qb)= Kl1(n)+ K I ~Wn), )

q(n) = p l ( n )

+ K12(n)r(n)

14. The Biharmonic Equation

We illustrate now the application of dynamic programming to problems of higher order using semidiscrete operators, by considering again the bending of a thin rectangular plate enclosed by x = y = 0, x = L , and y = H . We know that the configuration of equilibrium of such a plate is associated with the minimization of the functional J(w) =

+ lJ[(w,,

+ w,,,)~ - 2(1 - v)(w,,

wyy- wqy) - 2pw] dx dy,

(14.1)

where w is the deflection, p is the load per unit area, and v is Poisson's ratio. We assume that the edges y = 0, y = H are rigidly clamped, i.e., w = W" = 0,

(14.2)

while the values of w(0, y ) and those of the derivatives w,(O, y ) along x = 0 are prescribed. Additional conditions will be specified later on the remaining edge x = L.

145

14. THE BIHARMONIC EQUATION

Now, considering the semidiscretization scheme w,,(x. ih) z ( l/h)(wi - wi-1), wyy(x,ih) z ( ~ / h ~ ) ( w-~2wi -~

(14.3)

+ wi+l),

where w i= w(x, ih), and using the matrix-vector notation introduced in Section 4-7, we can readily transform (14.1) into the approximate functional L

J(u) =

+J [(u”,u“) - ~ ( K u ”u,) + (Lu, u) + 2( 1 - v)((Ku”,U) + (Ku’, u’))- 2(P, u)]dx, 0

(14.4)

in terms of the displacement vector u given by (14.5)

and where the matrices K and L have the meaning given by Eqs. (7.11) and (7.12) of Chapter 4, respectively. The mesh size in the y direction, h, is related to N by H = ( N + I)h. It is convenient to introduce here the vector forces m and t defined by Eqs. (7.14) of Chapter 4, i.e.,

m = -ufr

+ vKu

(14.6)

and t

= urrr- (2 -

(14.7)

v)Ku’.

Using m given by (14.6), (14.4) reduces to

S, [(m,m) + (L*u, u) + (K*u’, u’) L

J(u) = f

- 2(P, u)] dx,

(14.8)

where L* = L - v2K2 and K* = 2(1 - v)K have been found previously in Eqs.(7.15)ofChapter4.InEqs.(14.4)and(14.8),P=(p(x,ih)),i= 1, . . . N . We define as usual

where (14.10)

u’ = u,

and where m is related to u by means of (14.6). Now considering the decomposition L

minJx U

=

min

min

m(t)

u(<)

x c < ~ x + Ax t A s < < L

lxL =

L

min

min Jxx+A u(<) ‘x+A m(<) xs<jx+A x+As<sL

(14.1 1)

146

6 DYNAMIC PROGRAMMING

based on the additive property of the integrals and an elementary property of the minimum function, we can write

f(u, u, x) = min[+((m, m) + (L*u, u) + (K*u, u) - 2(P, u))A m(x)

+f ( u ( x + A), u(x + A), x + A)] + 4 A ) .

(14.12)

Assuming the existence of the necessary derivatives and recalling that, with the help of (14.6), u' = U" can be written in the form

u" = - m

+ VKU,

( 14.13)

it is easy to show that

-fx(u, u, x)

= min[+((m, rn) m

+ (fu

1

+ (L*u, u) + (K*v, u) - 2(P, u))

u> - ( f u

9

4 + v(fu

7

w,

(14.14)

where f, stands for aJax and f,,and f, are the gradients fu = (af/au,),

f , = (af/au,),

i = 1 , 2,

,

. . , N,

( 14.15)

where uiand u iare the components of the vectors u and u, respectively, is the limiting form of (14.12) when A 4 0 .We leave this derivation as an exercise for the reader. The minimum in (14.14) is attained at m =fv,

(14.16)

a value that substituted into (14.14) yields

fx

= -$(L*u, U) - $(K*u, 0)

+ $(fu

,fu) - (f,,,V ) - ~ ( f ,Ku) , + (P,u), (14.17)

a partial differential equation for f subject to appropriate initial conditions which are discussed below. First we wish to find a meaning for the gradient fu and to construct appropriate equations for fu and f,. We do this in the next section. 15. Theorem of Castigliano

We compute f,, by taking the gradient with respect to namely,

21

in (14.17),

-K*u + f , " f , - f u - f u u v - vf,vKu = - K * v + s , , ( f u - V W -fu - f u l J u , where with the symbol fms we denote the Jacobian matrix

(15.1)

...,N.

(1 5.2)

fxu

=

faS = (a2f/aai ag,),

i , j = i,2,

I47

16. EULER EQUATIONS

We now compute m'

= dm/dx from

m'

(14.16):

=fux

+fuuv

=f,x

+fuuv -fu,(m - VKU).

+fUUV'

(1 5.3)

Adding (1 5.1) and ( I 5.3), taking into account that m =f , , we obtain

m'

=

-K*u -f U .

(1 5.4)

Recalling (14.6), (14.7), Eq. (1 5.4) yields

f" = t ,

(15.5)

the desired result which, together with (14.16),expresses Castigliano's theorem. 16. Euler Equations

We now take the gradient with respect to u in (14.17):

+P

fxu =

-L*U

+f,,f, -fu,,v -vfv,,Ku

=

-L*u

+ P +fuu(f,- VKU)-fuu(v- VKf,),

- vKfu

(16.1)

and differentiate (1 5.5) with respect to x : t'

=fux

+fuuv

+fuuv'

=fux

+fuuo

-fuu(m - V K 4 .

(16.2)

Combining (16.1) and (16.2) we readily obtain t' = -L*u - vKm

+ P,

(16.3)

a differential equation for t . Writing (15.4) in the form

m'

=

-K*u' - t ,

(16.4)

and eliminating m and t from among (14.16), (16.3), and (16.4), taking into account that L* = L - v 2 K Zand K* = 2(1 - v)K, we finally arrive at

+

u'" - ~ K u " LU = P,

(16.5)

the Euler equation associated with the minimization of (14.8). It is remarkable that (16.5) coincides with Eq.(a), p. 97, Chapter 4, a differential equation obtained by semidiscretization of the biharmonic operator. The similarity is not casual, but the complete coincidence is a consequence of the same discretization scheme employed.

148

6 DYNAMIC PROGRAMMING

17. Natural Boundary Conditions If we do not prescribe the deformations at the boundary x = L , the minimum potential energy of a plate of length zero will be zero. Thus

(17.1) f(u, u, L) = 0. Since (17.1) is valid whatever values of u and u we consider, we pick those values for whichfis a minimum, i.e., we must have

f,(u,0,L ) = 0, f"@,u, L ) = 0, which, on account of (14.16) and (15.5) yields

( I 7.2)

m(L) = t(L) = 0, the natural boundary conditions of the classical formulation.

(17.3)

18. Solution of (14.17). Betti's Theorem

Sincefis the minimum of a quadratic functional, it is intuitively clear that it is a quadratic function of its arguments; i.e.,fmust be of the form

f ( u , U, X) = +[(R,iu, U) + 2(R12 U, U) where the associated matrix

+ (R22 U, u)] + a,u +

u

+

S,

(18.1)

(18.2) is positive definite as a consequence of the positive character of the potential energy. Using (15.5) and (14.16) we can derive expressions for the forces t and m,namely, t =fu = Rllu + R,,u M', (18.3) m =f, = RT,u R2,u a,.

+

+ +

The symmetry of R given by (1 8.2) reflects in (18.3) the reciprocity theorem of Betti, a result derived from dynamic programming considerations. Substitutingf given by (1 8.1) into the partial differential equation (14.17), and collecting terms of equal power, we find that the N x N matrices R , , , R , , , and R,, , N-dimensional vectors a', a,, and the scalar s, satisfy the Ricatti differential equations R;, = -L* + R , ZRT2 - vKRT, - vR,, K , R;2 = R12 R22 - R,, - vKR,,, R;2 = -K* - R , , - RT, iR:, , (18.4) ~ ( 1= ' P R12 ~ ( 2 vKc~,,

+

~ ( 2= ' R2, ~ 1 2- ~ 1 ,

s' = Ha2 9 a,),

149

19. A STATICALLY INDETERMINATE STRUCTURE

subject to the initial conditions R*I(L) = R12W = R 2 m = 0, a,(L) = a,(L) = 0, s(L) = 0,

(18.5)

derived from consideration of (17.1) and (18.3). This result had been previously derived in Eqs. (8.4) and (8.5) of Chapter 4 using invariant imbedding. In comparing both results, the reader should recall that R,, = R l l and take into account the difference introduced by the change of origin that subjects the Riccati equations derived from dynamic programming to conditions at x = L rather than at x = 0 of the invariant imbedding formulation. EXERCISES 1. The potential energy of a beam on elastic foundation is given by

+

J(u) = 4 ~oL(Elu"2 ku2 - 2pu) dx,

(a)

where El is the stiffness, k is the coefficient of the foundation, and p is the load. Consider the end x = 0 clamped, i.e., u(0) = u'(0) = 0, and let f ( u , u'x) be the minimum potential energy of a beam of length L - x , i.e.,

+

f ( u , u'x) = min f ~ L ( E l u " 2 ku2 - 2pu) dx. u

x

(b)

Show that f satisfies the quasilinear partial differential equation f x = (1/2EI)f$

-f . ~ '- BkU2

+ PU,

f ( U , V,

L) = 0.

(C)

Hint: Use u"(x) as the decision variable. 2. Set f ( u , u', x ) = tr,u2

+ r2 uu' + 4r2 u" + s I u+ s2 u' + s3.

(d)

Using (c), find the differential equations and corresponding initial conditions at x = L for the quantities r l , r 2 , r g ,s,, s2, and s3. 3. Show that fu* =

i.e.,

-EW,

fu

=

-(E~u")',

and f. are the moment and shear force of the beam, respectively.

4. Show that if k = 0, the differential equations for rl , rs , r 3 , s,, s 2 , and s3 can be solved

by quadratures.

19. A Statically Indeterminate Structure

A system of N + 1 pinned elastic bars, not necessarily linearly elastic, such as indicated in Fig. 6-3a, is a convenient device to illustrate the application of energy principles in structures. We shall treat this problem using dynamic programming. Let u ibe a unit vector in the direction of the ith bar.

150

6

DYNAMIC PROGRAMMING

(b)

(a)

Fig. 6-3. (a) Complete structural system. (b) Imbedded structural system.

If Ai is a scalar quantity denoting the intensity of the force along such a bar, the equation of equilibrium can be written N+ 1

P

=

1 Aiui,

(19.1)

i= 1

where p is a vector denoting the external force acting on the structure. Now, if cpi(Ai) denotes the complementary energy at the ith bar, we know that the set of forces {Ai} that minimize the functional

(1 9.2) and satisfy the conditions of equilibrium (1 9. l), corresponds to kinematically admissible displacements. In other words, they furnish the solution(s) of the structural system. To solve this problem we consider the structure that results from elimination of the first n - 1 bars, replacing them with the equivalent resultant

c

n- 1

qn = -

AiUi,

(19.3)

i= 1

as indicated in Fig. 6-3b. In this fashion, the equilibrium of the remaining bars is undisturbed. This is an imbedding procedure. We denote by pn the vector given by n- 1

Pn = P -

1 A i u i = P + 4.5

(19.4)

i= t

the resultant force acting on the reduced system. Clearly, Pn+l = P n -

A n ~ n .

(19.5)

We now definef,(pn), a scalar function of vector pn, to be the complementary energy of the reduced system, i.e.,

(19.6) where the Ai’s are quantities constrained by the equilibrium equation (19.5).

19. A STATICALLY INDETERMINATE STRUCTURE

151

In the familiar way we now derive the pertinent functional equation forf.(pn), namely, f n ( P n ) = min[Vn(An) + f . + ~ ( ~ n + ~ > l , (19.7) 1,

or, on account of (19.5), f n ( p n ) = min[Vn(&)

+f.+ l(pn - A n un)l.

(19.8)

A,

We can easily find an initial condition for (19.8) by considering the reduced system with only two bars (Nand N + 1) if it is a plane system, or with only three bars ( N - 1, N , and N + 1) if it is a spatial structure. This is a statically determinate problem. Suppose the structure is plane. The equation of equilibrium (19.9) PN = &uN + A~+iuN+i furnishes two linear equations for the computation of AN and AN+1, values that substituted in (19.6) for n = N yield (19.10) fN&) = vN(&) + v N + I(&+ I), the desired initial condition. Clearly, AN and in (19.10) depend on the two components of vector pN. Equations (19.8) and (19.10) may be used to obtain a direct numerical solution of the problem. We could use here a procedure similar to that employed in Section 6 in dealing with a nonlinear string. We note, however, that in the present case we must deal with scalar functions of a vector rather than with functions of a scalar variable. For example, if the structure is plane, i.e., the bars are coplanar, the minimum energy function f,(p,) will be a function of two variables, the two components of the vector in the plane. This means that if we discretize each component of the vector in, say, M points, we need M 2 values to representf,(p,) in that grid, rather than the M values that would be required iff, were a function of a scalar quantity. If, for example, M = 100, a plane structure would demand the storage of lo4 quantities as compared with the (102)3= lo6 required for a spatial system. If we now refine the mesh, i.e., if we increase M , the limit of currently available computers will be reached. This, a severe limitation to the direct, nalve application of dynamic programming, is eloquently called after Bellman, the curse of dimensionality. There are several ways t o overcome this difficulty. The reader will find references at the end of the chapter. We finally observe that under very weak conditions we could establish the existence and uniqueness of the minimum complementary energy function f,(p,) given by (19.8) and (19.10), subject to possible additional constraints on Ai. This, in turn, can be used to establish the existence of a set of forces Ai,i = 1,2, . . . , N 1, that satisfy equilibrium and compatibility. The uniqueness of such a set demands, however, stronger conditions, such as, for example, the convexity of functions qn(An).

+

152

6

DYNAMIC PROGRAMMING

20. The Theorem of Complementary Energy

When no local constraints on forces or elongations are considered, and under appropriate differentiability and convexity conditions on the complementary energy functions cpi(Ai), we may in principle minimize at each stage of the computational process by differentiation. Therefore, under these conditions, Eq. (19.8) is equivalent to the system f,(Pn)

+f,+,(Pn

= cPn(An)

0

= dVn/dJn

- Anun),

+ a ! + 1laan .

(20.1)

(20.2)

Using the fact that the derivative of the complementary energy of bar n with respect to the axial force is the elongation A, of the bar under such a force, i.e., An

(20.3)

= dVn/dAn >

Eq. (20.2) reduces to An

=

(20.4)

-af,+llaAn.

Using (19.5) we can express (20.4) in the form An = (un gradf,+l), 9

(20.5)

where the gradient is taken with respect to P,,+~,the external load. Equations (20.4) and (20.5) are expressions of the theorem of complementary energy. EXERCISE Show that grad fi(p) is the component of the displacement of the vertex with respect to the direction of the force p of the structure with N 1 bars.

+

21. Linearity

If the force-displacement relationship of each bar is linear, the complementary energy functions 'piwill be quadratic in li, i.e., cp. = 3a.1.2 f I )

(21.1)

ai = l,& A i

(21.2)

where

are coefficients involving the lengths I { , Young's modulus Ei,and crosssectionalareas A i. When this is the case, and no constraints are imposed on ,Ii, we can considerably advance the analyticaltreatment of Eqs. (20.1) and (20.2), and in fact we can provide a remarkably simple solution to this problem.

153

21. LINEARITY

By induction, or otherwise, we can prove thatf,(pn) is a quadratic function, i.e.,

L(PJ = f(RnPn> Pn),

(21.3)

where Rn is a 2 x 2 symmetric matrix to be determined. Substitution of (21.3) into (20.1) and (20.2) where ‘pi is given by (21.1) yields

+ (Rn+1 ( P n - An un), Pn -

(RnPn pn) = a n A n 2 9

An

un)

(21.4)

and an& -(Pn

(21.5)

Rn+,un) = O ,

-Anun,

respectively. From (21.5) we can easily compute

We now rewrite (21.4) in the expanded form (Rn P n 9 Pn) = an A,2

+ (&+

l(Pn

- An un), pn)

- A n u n , Rn+lun),

-

(21.7)

an equation that on account of (21.5) and (21.6) reduces to (Rn Pn > Pn> = (Rn + 1P n 9

9

Pn)

- ( l/Pn)(Rn+ 1un P J 2 , 9

(21.8)

where Pn is given by Pn

= an

+ (Rn+lUn,4.

(21.9)

Now, let Rn+,be the symmetric 2 x 2 matrix rl(n r2(n

+ 1) + 1)

r2(n r3(n

+ 1) + 1)

(21.9‘)

and let l,, l, be the components of the unit vector un. Introducing the quantities a, = r1(n a2 = rz(n

+ I)/, + rz(n + l)lZ, + 1)1, + + 1)1,,

(21.10)

r3(n

and the matrix (21.11) the reader can easily verify that the quantity (R,+,un, p,)’ which appears in (21.8) can be written as the scalar product (Rn+,un ,Pn)’ = (An+lPn9 PA*

(21.12)

154

DYNAMIC PROGRAMMING

6

Substituting (21.12) into (21.8) immediately yields (21.13)

Rn = Rn+1 - (l/finMn+I,

a backward recursive equation for R , . We find an initial condition for Rn by considering the structure with only two bars. Since now the system is statically determined, we can easily find the flexibility matrix K and write the strain energy in the form J N ( P N ) = +(@N

9

(21.14)

PN),

from which the desired initial condition RN = K

(2 1.I 5)

follows. Now by combining (19.5) and (21.6) we obtain Pn + 1 = Pn

- (1 IPn)(Rn+ Isn

9

Pn)Un

(21.1 6)

9

a forward recursive relation for p N subject to the obvious initial condition P1

( 2 1.17)

=P.

The forces An follow from (19.5), i.e., I n

= ( P n - Pn-

(21.18)

1, ~ n > .

EXERCISES 1. If the structure is spatial, the solution method outlined in this section remains formally the same. The matrix R., however, will be 3 x 3 instead of 2 x 2. Find the elements of the matrix A n + l introduced in (21.12), for the spatial case.

2. Show that R. is a flexibility matrix.

3. Show that the elements ri(n) of the flexibility matrix R, may be recurrently computed by using the relations rl(n) = r l ( n 3- 1) -

+ + 2r1(n+ I)r& + l ) Z 1 + rz2(n+ 1 ) f Z 2 + + 1)lIz + 2r2(n + 1)11lz + r3(n+ 1)lz2 r l ( n + l ) r z ( n + 1)ZI2 + rl(n + h ( n + 1)11

r 1 2 ( n 1)Il2 a. rl(n

12

f2

155

22. AN IDENTIFICATION PROBLEM

22. An IdentificationProblem We consider once more the two-point-value problem u(0) = u(L) = 0, u" - k(x)u =p(x), (22.1) governing the deflection u(x) of an elastic wire stretched with a unit tension, held fixed at the ends, loaded with a force p(x), and resting on elastic foundation k(x). Ordinarily,p and k are known and we are asked to determine u so as to satisfy (22.1). So posed, this is a direct problem. There is, however, a more general class of inverse problems, known as system identification problems, that can be formulated. The objective of this new class of problems is to determine part of the structure of the system, say the foundation function k(x), from suitable experimental information. Although system identification is a topic which is thoroughly discussed in future chapters, we decided mainly on methodological grounds that the present example belongs here. In what follows we shall assume that the deflectionfunction u(x) associated to a given load p(x) has been measured. On the basis of this information we wish to determine the foundation function k(x) > 0, in some optimal fashion. In this chapter we perform the identification of the foundation by means of the technique of segmental approximation in the sense of Bellman and Roth. See the Bibliography at the end of the chapter. We make no assumptions regarding the analytical structure of the unknown foundation function k(x). We pursue this path in connection with a quasilinear approach in a later chapter. In lieu of that approach we consider here the total length L of the string divided into N subintervals [xi-l, xi], such that 0 = x,, c x1 < . c x i < . . c xN = L, and consider a sequence of numbers ki > 0 which are sought to represent k(x) optimally within each subinterval, such as indicated in Fig. 6-4.

-

Xi-]

Xi Fig. 6-4

Xi+t

156

6

DYNAMIC PROGRAMMING

Let Ei(xi-l,x i , k i ) be a measure of the error, under suitable norm, which results from the use of ki instead of k ( x ) in the subinterval [ x i - l , x i ] . We define the optimal k i , within each subinterval, as the one that minimizes Ei . Let q i ( x i - l , x i ) = min x i ,k i ) (22.2) ki

be the minimum error associated with the subinterval [ x i - l ,x i ] . For a given number N of subintervals, there still remains the flexibility of choosing the subinterval endpoints in such a way as to minimize the total error measure given by N

CPN(L)=

1 qi(xi-

i= 1

1,

xi),

(22.3)

and thus optimally approximate k ( x ) over its entire domain by a number of segmental constant values. Dynamic programming provides an elegant solution to the minimization of (22.3), i.e., to the optimal determination of the segment endpoints. Let

f"(4=

min XI.XZ...

. ,X n - 1

CP"W

= the minimum total error measure obtained

by optimally picking the end points of n segments over the interval [0, X I .

(22.4)

Then, by the principle of optimality of dynamic programming, or otherwise, we construct the functional equation

f,(x) = min

[qn(~7

XI

+fn-~(~)l,

(22.5)

o
where n = 2, 3,

. . . , N , subject to the initial condition L(X)= CPI(O7 4.

(22.6)

The functional equation (22.5) subject to (22.6) can be used to computef,(l). The value of y that minimizes at stage i will be denoted by x i . The sequence x i , i = 1, 2, . . . , N - 1, furnishes the abscissas of the optimal partition of the segment [0, L ] . The values of kifor each subinterval follow from (22.2). 23. The Error Measure An error measure Ei , given by the functional Ei(X i - 1 , xi ,ki) =

j

xi

p(x)(u"- ki 24 - p)' dx,

xi-I

may be adopted, where p(x) is a weighting factor.

(23.1)

157

24. A NUMERICAL EXAMPLE

Thus, for a given subinterval [ x i - l ,x i ] ,we can readily compute the optimal ki*. By differentiating with respect to k iin (23.1) we obtain

ki* = (jx:-lpu"(u - p ) d x ) / ( j xix -i I p u z d x ) ,

(23.2)

and therefore Xi

p"(" - ki*u -p)'

dx.

(23.3)

The functions u(x) and p(x) are assumed to be known as experimental data values at a number of points in the interval [0, L].To evaluate the expressions (23.2) and (23.3) numerically, we must conveniently smooth the data, particularly in view of the second derivative of u in both equations. This can be done in a number of ways using spline functions and other interpolation procedures. The reader interested in this aspect should consult the literature at the end of the chapter. EXERCISES Discuss the determination of k ( x ) in the equation

where h(x) and q(x) are given functions in the interval [0, L ] . This is a uniaxial version of an important inverse problem in hydrology.

A meaningful nonlinear version of (a) is furnished by the equation

which represents the "unconfined" case. How does the appearance of a nonlinear term modify the identification procedure?

24. A Numerical Example

One of the most relevant features of the identification algorithm presented above is that it does not require special analytical properties on the unknown function k(x). Another important feature is that every stage of the computation has a clear physical meaning. In fact, the first stage, i.e., the initial condition of the functional equation, provides the best fit with one segment, the second stage furnishes the best fit with two segments, and so on.

158

6

DYNAMIC PROGRAMMING

To illustrate the basic ideas we present here some numerical results obtained from application of the method to a particular example. With a load function 5

p(x)=

1 (6 - i)cos inx,

(24.1)

i=l

and the foundation function

r2

if 0 s x 1 0 . 4 k(x)= 4 if 0.4 XI <0.6 (24.2) 5(1 - x) if 0.6 XI 51.0, we integrated Eq. (22.1) using invariant imbedding and an Adams-Moulton integration scheme with step size A = 0.02. The values of u at x =j/25, j = 1,2, . . . ,25, obtained in this fashion were used for identification purposes. The results for 1, 3, 5, and 7 segments in [0, 11 are shown in Fig. 6-5. 4-

--

I

--3r5 7

0

0.2

0.6

0.4

0.8

1.0

X

Fig. 6-5. Identification of Winkler’s foundation function k(x) by segmental approxlmation. (After Distefano and Nagy [19711.)

25. Discussion

In this chapter we presented a number of carefully selected topics to illustrate some of the fundamental ideas and methods of dynamic programming. In all the cases we have constructed the pertinent functional equation guided by intuitive notions on the process to be optimized. In defining the optimum value function the notion of imbedding has played the central role, while in obtaining the functional relationship we have been constantly relying on a basic structural property of the optimum policies that can be elegantly expressed in the following principle by R. Bellman.

NOTES, COMMENTS, AND BIBLIOGRAPHY

159

Principle of Optimality An optimal policy has the property that whatever the initial state and the initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision. NOTES, COMMENTS, AND BIBLIOGRAPHY

1. Some of the recommended books on dynamic programming, which cover basic as well as general aspects of the theory, are R. Bellman, ‘‘Dynamic Programming.” Princeton Univ. Press, Princeton, New Jersey, 1957; R.Bellman, “Adaptive Control Process: A Guided Tour.” Princeton Univ. Press, Princeton, New Jersey, 1961; R. Bellman and S. Dreyfus, “Applied Dynamic Programming.” Princeton Univ. Press, Princeton, New Jersey, 1962.

2-5. The linear string provides an example of a larger class of optimization and control processes, namely those having linear dynamics and quadratic functionals. For an extended treatment of this important subject both from a classical and a dynamic programming point of view, see R. Bellman, “Introduction to the Mathematical Theory of Control Processes,” Vol. I, Linear Equations and Quadratic Criteria. Academic Press, New York, 1968.

6. The nonlinear string offers one of the simplest examples of nonlinear control processes in the realm of equilibrium problems in mechanics. Most of the rest of this book is devoted to the treatment of mechanical processes from a modern control theory point of view. See R. Bellman, “Introduction to the Mathematical Theory of Control Processes,” Vol. 11, Nonlinear Processes. Academic Press, New York, 1971.

7. For an elementary treatment of general routing problems see R. Bellman and K. Cooke, “Algorithms, Graphs and Computers.” Academic Press, New York, 1970.

9-10. There are a number of excellent books in the calculus of variations. For an introduction we recommend 1. M. Gelfand and S. V. Fomin, ‘‘Calculus of Variations.” Prentice-Hall, Englewood Cliffs, New Jersey, 1963.

For the study of optimal control processes from the point of view of the classical calculus of variations, see L. S. Pontryagin et al., “The Mathematical Theory of Optimal Processes.” Pergamon, Oxford, 1964; M. Hestenes, “ Calculus of Variations and Optimal Control Theory.” Wiley, New York, 1966.

160

6

DYNAMIC PROGRAMMING

For the connections between dynamic programming and the classical calculus of variations, see S. Dreyfus, “Dynamic Programming and the Calculus of Variations.” Academic Press, New York, 1965.

11. There are many books devoted to the numerical treatment of functional equations arising from control processes. For example, P. Dyer and S. R. McReynolds, “The Computation and Theory of Optimal Control.” Academic Press, New York, 1970; D. H. Jacobson and D. Q. Mayne, “Differential Dynamic Programming.” American Elsevier, New York, 1970; R. E. Larson, ‘‘ State Increment Dynamic Programming.” American Elsevier, New York, 1968.

12. An interesting reading on modern control theory can be found in R. Bellman and R. Kalaba, “Dynamic Programming and Modern Control Theory.” Academic Press, New York, 1965.

13. For a classical treatment of partial differential equations in a variational framework, see R. Courant and D. Hilbert, ‘‘ Methods of Mathematical Physics,” Vol.1 .Wiley (Interscience), New York, 1953.

Here we follow in part N. Distefano and A. Samartin, “ A Dynamic Programming Approach to the Formulation and Solution of Finite Element Equations” (to appear).

14-18. Here we follow part of N. DistCfano, Dynamic Programming and the Solution of the Biharmonic Equation, Internat. J. Numer. Methods Engrg. 3 (1971), 199-213.

19-21. The material of these sections is new. 22-24. Here we follow part of N. Distefano and D. Nagy, Parametric and Adaptive, Nonparametric System Identification Procedures in Structural Mechanics, Proc. 1971 Summer Comput. Simulation, Boston, Massachusetts (July 1971).

Chapter ?

Quasilinearization

1. Introduction

Quasilinearization is a theory devoted to the study of nonlinear equations from the properties reflected by a family of associated linear ones. The idea is very simple. Let T(u, u) be a function of the real variables u and u with the condition that T is linear in u. We assume now that T(u,Y) has a minimum with respect to u E S, where S is a given set, and denote that minimum by F(u),i.e., F(u) = min T(u, u). LIES

We now say that (1.1) is a quasilinear representation of the generally nonlinear function F(u) and expect to derive properties of F from the properties of the linear family T. In this chapter we wish to present the method of successive approximations in the framework of the theory of quasilinearization. The advantages are multiple: Quasilinearization offers a unified device to study the properties of the approximating sequence and to establish appropriate intervals of convergence. 161

162

7

QUASILINEARIZATION

We motivate the present analysis with an introductory section on approximation in policy space, a method at the very basis of dynamic programming. It is followed by a section on quasilinearization as it applies to the study of convex operators. In Sections 4-6 we present a detailed treatment of a nonlinear Fredholm integral equation. In Sections 7-9 we apply the results to some nonlinear differential equations, including a discussion on numerical aspects. Section 10 discusses the possibility of obtaining lower and upper bounds while the rest of the chapter is devoted to the more involved problem of the solution of large-dimension nonlinear systems starting, as usual, with the derivation of the Newton-Raphson-Kantorovich method. In Sections 13-1 5 we discuss problems of dimensionality and some procedures to overcome them. The chapter ends with a parametric design problem. 2. Approximation in Policy Space In discussing the numerical solution of the partial differential equation (8.8) in Chapter 6 , we intentionally avoided the application of the method of successive approximations, an unavoidable instrument of nonlinear analysis. The purpose of this omission was to present this tool in the framework of quasilinearization, a powerful technique whose origin lies in some monotone properties of the functional operators of dynamic programming. We shall in fact concentrate here on a procedure known as approximation in policy space. The reader will find a number of references in the Bibliography at the end of the chapter. To illustrate the basic ideas we consider the partial differential equation

-f, = min[g(u, u’, x ) + uIJ&,

x>J,

S(u, L) = 0,

(2.1)

UP

which is associated with the minimization of the functional

f ( u , x ) = min Y

s:

g(u, u’, 5) d t ,

u(0) = c.

We start the process by guessing an initial policy; i.e., we pick the decision variable u’ in terms of the state u. Let u’ = wo(u)

(2.3)

be the guessed policy. We compute the first approximationf, of the optimum value function using the equation (2.4)

163

2. APPROXIMATION IN POLICY SPACE

where the function u(5) appearing in the integrand is uniquely determined by integration of du(5)ld5 = wo(u(5>),

5 2 x,

u ( x ) = u,

(2.5)

a family of initial-value problems in terms of every possible state u at x . Clearly,fo in (2.4) satisfies the equation -

a ax

X) = g(u,

a

+ w o ( ~ -fo(u, ) au

wo(u), X )

fo(~,L) = 0,

x),

(2.6)

where the symbols of partial derivatives were explicitly used in order to avoid confusion with the subscripts indicating the order of the approximation. To improve the initial policy wo(u), we compute the new function w(u) using the equation wl(u)

= arg

[

min g(u, w ,x) W

+w

(2.7)

With wl(u) determined in this fashion we proceed to compute the second approximation for the optimum value function, i.e.,

fl(%4 = J

L

g ( u ( 0 , w,(u(t>>96) dt9

(2.8)

X

where, as before, u(5) is given by du(Old5

= w,(u>,

(2.9)

u(x) = u,

for each value of u. Here again fl satisfies a partial differential equation,

a

- -f1(u, X I = g(u, w,(u), x)

ax

a

+ w1(4 -auf1(u, 4,

f1(u, L) = 0.

(2.10)

We wish to compare now the values of two successive approximationsf, andf'. To this end we observe that (2.7) implies g(u, w ,x)

a

where equality occurs for w Eq. (2.1 1) yields

Making

a

+ w -gpo(u, 4 2 g(u, w1,x) + w1 -fo(u, au = wl.

x),

(2.11)

Setting w = w o and considering (2.6),

164

7

QUASILINEARIZATION

subtraction of (2.10) from (2.12) yields azlax

+ w1 azlau I 0,

(2.14)

a functional inequality for the quantity z . Recalling (2.9), we observe that the left-hand side of (2.14) represents the total derivative of z(u, x ) along an arbitrary function u(x). Therefore, since z(u, L ) = 0, Eq. (2.14) implies

z(u, x ) > 0,

x E [O, L),

(2.15)

i.e., fo 2fl. Continuing the process in the same fashion, we generate the monotone sequence fo 2 fi 2

*. *

2f,2

* * .

2f

(2.16)

that presumably converges toward the solution of (2.1). We call this monotone approximation. It is important to keep in mind that although we have not as yet proved convergence, the present method affords a procedure to improve systematically any estimate of the original minimization problem. The problem of convergence is more involved and requires further assumptions in general. We do not enter into that aspect here. The illustration of the main properties of the sequence generated by the approximation in policy space has been the main objective of this section. EXERCISE Using the preceding arguments, prove the uniqueness of the solution of (2.1),

3. Quasilinearization

The results of the preceding section, in the realm of dynamic programming, highlight the interest in writing equations of the form

u = F(u),

(3.1)

in terms of the minimum (or maximum) operation, i.e., u = min T(u, v ) , U

where T is a function or functional associated with F and where v may be regarded as the policy function of a fictitious optimization process. When T(u, v ) is linear in u we call (3.2) a quasilinear representation of (3.1). Clearly, for any u in (3.2) we have u IT(u, 0).

(3.3)

165

3. QUASILINEARIZATION

Now let w be the solution of the linear equation w

= T(w, v).

(3.4)

If we can prove by comparison of (3.3) and (3.4) that u

I w,

(3.5)

then any choice of v in (3.4) will automatically provide an upper bound w(v) on the function u. Let z be defined as z=w-u;

(3.6)

then subtracting (3.3) from (3.4) and recalling the assumption that T is linear in u, we have

z 2 T(z, v). (3.7) If, from (3.7), we can infer that z 2 0, property ( 3 . 9 , i.e., u Iw, will hold. This is a positivity property of the operator (.) - T(*,0). We are interested in establishing conditions under which this property holds because, as the reader might recognize, this enables us to determine upper bounds and to improve estimates in a monotone fashion systematically. These ideas are at the very basis of the theory of quasilinearization as it applies to the study of functional equations. In particular, an important development of this theory is obtained if we restrict attention to nonlinear operators exhibiting some kind of convexity (concavity) because they afford a very convenient quasilinear representation in terms of the maximum (minimum) operation. To illustrate the ideas we consider here the scalar function S(x).We say that S(x)is convex (concave) in the interval [a, b] if we have S”(x) 2 0 (S”(x) 5 0 ) in that interval. Strict convexity (strict concavity) implies the strict inequality, i.e., S”(x) > 0 (S”(x) < 0). By expanding S(x) in Taylor’s series, or otherwise, we can easily show that, in the domain of convexity, S(x) can be represented by S(x) = max[S(y) Y

+ (x - Y)S’(Y)l,

(3.8)

the required quasilinear representation. See Fig. 7-1. Similarly, if S(x) is concave, S(x) = min[S(y) Y

+ (x - y)S’(y)].

(3.9)

Equation (3.8) or (3.9) may be used to implement a number of successive approximation procedures and, by imposing some additional conditions on S, we may ensure monotone approximation. We deal with problems of this sort in the next few sections, in connection with a number of nonlinear operators of interest in our applications.

166

7 QUASILINEARIZATION

S(X’f

EXERCISES

1. Show that r z = max(2rs - s’). 5

2. Let S(x) = S(x, , x 2 , . . ., nw) be a strictly convex function of x = ( x i ) , for all x. Show that S(x) = max[S(y)

+ (x

-Y.

grad S(y))l,

(b)

Y

and that the unique maximum is attained at y

= x.

4. A Nonlinear Fredholm Operator

We wish to apply the ideas exposed in Section 3 to the study of the nonlinear integral equation

where g is such that g(s, 0) # 0. Under very mild conditions on g, and using fixed-point theorems, or otherwise, we can in general assert that, for L sufficiently small, a solution to (4.1) exists and is unique. For our subsequent purposes, however, the following result will be enough: Theorem (Hammerstein) If

(a) K ( x , y ) is a positive (in the operator sense), symmetric L2 kernel, i.e.,

J’s K(x, y)u(x)u(y) dx dy > 0

1’sK Z

dx dy < 00,

for all u(x),

(4.2) K ( x , V ) = K(y, x);

167

4. A NONLINEAR FREDHOLM OPERATOR

(b) function g(s, u) is continuous and satisfies uniformly a Lipschitz condition of the form Ig(s,

- g(s,

41

az Iu - U I 3

(4.3)

where a’ is a positive constant such that a2 < 1,

(4.4)

where ,I1 > 0 is the smallest characteristic value of the linear equation

then a solution to (4.1)exists and is unique. The construction of a convergent sequence of functions ul, u 2 , .. . ,u,,, . . ., could now be obtained using the classical Picard’s successive approximation scheme, namely

with uo = 0, for example. To construct a solution, however, we pursue a different path. In effect, we are interested in the application of the general ideas on quasilinearization discussed in Section 3. The reasons for this preference are various, but it is certainly worth mentioning the slow rate of convergence of Picard’s algorithm. First, we shall try to determine conditions for monotone behavior. To thisend, in addition to (4.2)-(4.5), we require g to be strictly convex and twice differentiable, i.e., a2gpv2 = gvu(s,v ) > 0. (4.7) Therefore, we can write g(s,

4 = max[g(s, 4 + (u - v)g,(s, 41,

(4.8)

0

where gu stands for ag/av. Substituting (4.8)into (4.1), L

= maxJo

W,

s)[g(s, v ) + (u - v)g,(s, v ) ds, ~

(4.9)

U

or equivalently,

where the equality sign holds only for u = u. Clearly, (4.9)and (4.10)hold for any function v defined in the interval of existence of u.

168

7

QUASILINEARIZATION

We now consider the linear equation in w, L

w(x> =

J K(x, s)[g(s,

0)

0

+ (w - v)g,(s, 4 1 ds,

(4.1 1)

and proceed to compare u satisfying (4.10) with w given by (4.11). To this end we form the difference z=u-w, (4.12) which satisfies the functional inequality

obtained by subtracting (4.1 1) from (4.10). Now, the determination of lower bounds by this method relies on the nonnegativity of the operator, L

L(z) = Z ( X ) - J K(x, s)g,(s,

U)Z(S)

ds.

(4.14)

0

In fact, if L(z) 2 0 implies z 2 0, then w We observe that (4.13) is equivalent to

u, the desired monotone property.

(4.15)

L(4 =f(x>,

wheref(x) 2 0. In some cases we can ensure the nonnegativity of the integral operator. Consider, for example, > gu > 0 and that the kernel K(x, y), in addition to satisfying conditions (4.2), is pointwise nonnegative, i.e., K(x, y ) 2- 0, x, y E [0, L]. Under these restrictions, z 2 0, as follows from consideration of the Neumann series z(x> =

J

L

0

K(x, ylgu(y, W ( s > ds

+ JoL JLK(x, YMY,s)g,(y, U)g,(s, v > f ( s )dy ds +

* *

.,

(4.1 6)

obtained by iteration in (4.15). When gu is of a more general nature, the pointwise nonnegativity of the kernel K(x, y ) will not be enough in general to guarantee the nonnegativity of the operator L(z), as simple counterexamples show. In some applications it will be enough to require a bound of the type suPlg"l

<&.

(4.17)

See, for example, the case presented in Section 8. In this case, when L2 2 A,, it is clear that the monotone property holds with the maximum interval of validity in the sense that the conditions for existence and uniqueness of the original nonlinear operator are included in the conditions for monotone behavior.

169

5. SUCCESSIVE APPROXIMATIONS-MONOTONICITY

5. Successive Approximations-Monotonicity In the previous section we have seen how to construct a lower bound. Here we wish to construct an approximating sequence of functions. Fortunately, Eq. (4.9) is well suited to implement various successive approximation schemes. If, for example, we consider an initial arbitrary function u = uo , and disregard the maximum operation indicated in Eq. (4.9), then clearly the solution of the resulting linear equation (4.9) will yield a new approximation u1 which substituted back in place of v yields another approximation uz , and so on. This process of using the last approximation to compute the new one in (4.9) turns out to be the Newton-Raphson scheme in function space. The successive approximations are generated by the following recursive system of linear integral equations :

Jb

L

un+ 1

=

K(X, s)[g(s, un> + (un+ 1

- Un)gu,,(s,

UJI

n=

ds,

0 9 1 ,

..

. ?

(5.1)

where uo is some suitable initial choice. Equation (5.1) follows from (4.9) by making u = u , + ~and setting u, in place of v. As noted above, this process coincides with the Newton-Raphson scheme in function space. In fact, to use the Newton-Raphson method we would expand function g appearing under the integral sign of the original nonlinear functional equation (4.1) in a Taylor’s series around a known function u, , i.e., g(s, u ) = g(s, un)

+ (u - un)gu,,(s,

un) + . * * .

(5.2)

Retaining only linear terms, substitution of (5.2) into (4.1) shows the coincidence with ( 5 . 1 ) if we denote the current value of u by u,+l. This method appears to have been employed first by Kantorovich to solve functional equations. For that reason it usually bears the name Newton-RaphsonKantorovich. We are now interested in the convergence and some monotone properties of the sequence u l , uz , . . . , generated using (5.1). Assuming the nonnegativity of the linear operator, we can readily establish that

n = l , 2 ,...,

(5.3) i.e., any element of the approximating sequence is a lower bound to the solution. Now we want to prove that uru,,

u , , + ~2 u , ,

n = 1 , 2 , ....

(5.4)

To this end, consider Eq. (5.1). We can write

Jb

L

L

un+ 1

=

K(x, S)[g(s, un) + (un+1

- un)gu,,(s, un)l dS IJ0 K(x, s)g(s, u,+ 1) ds, (5.5)

170

7

QUASILINEARIZATION

since u,,,, is the function that maximizes the integrand. Considering u , + ~ given by (5.1), i.e., L

un

+2

=

S, WX*

S)[g(s, un + 1)

+ (un + 2 - un + l)gu,,+

I(s,

un

+ 111 ds,

(5.6)

and subtracting (5.5) from (5.6), we obtain un+z

-un+,

2

I

K(X,S)Su,,,(S,un+l)(un+z

(5.7)

-un+I)ds,

a familiar functional inequality that, under the appropriate restrictions, implies un+z

(5.8)

2 un+t,

the desired result. Combining (5.3) and (5.8) we have, inductively,

’

the desired property. 6. Quadratic Convergence

Convergence can be proved using standard techniques, but in addition we expect to have quadratic convergence since we are actually dealing with the Newton-Raphson method in function space. By quadratic convergence we mean that IIun+i

- uII IIIun - uIIz . M

holds, where the norm indicated by

(6.1)

11.11 is defined by

and M is some positive constant. To investigate a property of this kind we consider an expansion of g(s, u) around un using a remainder in the Taylor’s expansion, i.e., .ds, u) = g ( ~un) ,

+ (u - un)Su.(S,

un)

+ f(u - uJ2see(s, 8),

(6.3)

where gee stands for (a2g(s,8)/802 and 8 is a function lying between u and u,, , i.e., u(x) I e(x) s

U,(X),

x E 10, LI.

(6.4)

171

7. GREEN’S FUNCTION

a constant independent of n. EXERCISE

Consider Picard’s scheme of successive approximations in Eq. (4.1), i.e., U”+ 1 =

joLK(x,S M S , U.(S)) dS.

.

Show that the sequence ut , u 2 , . . ,converges uniformly toward the unique solution u such that maxIu.+l - u I S k m a x I u n - u I , X

X

where k is a constant independent of n.

7. Green’s Function

The results of Sections 4-6 are of interest on their own, for so many problems in nonlinear mechanics can be cast in integral form. But, in addition, they offer a convenient way of discussing a class of nonlinear boundary-value problems in terms of differential operators. This can be done by introducing appropriate Green’s functions. We recall here the basic ideas of this important tool in mathematics and mechanics.

172

7 QUASILINEARIZATION

Consider the linear differential equation u" = p ( x )

(7.1)

and the boundary conditions u(0) = u(L) = 0.

Instead of treating (7.1)-(7.2)directly we wish to proceed by superposition. To this end let G(x, y ) be the solution of the associated differential equation uw= 6(x

-y),

x, y E 10, L1,

(7.3)

where 6(x - y ) is Dirac's delta, subject to boundary conditions (7.2). We easily find G(x, r>= (x/L)o,- L )

(7.4)

if x I y , and G(x, y ) = GO., x) if y I x. We can now use superposition and write the solution of (7.1)-(7.2) in the form

Similarly, if we consider the nonlinear equation uW= e",

subject to boundary conditions (7.2), we can write (7.7) where G is the same function given by (7.4). In general, the nonlinear differential equation

L ( 4 = g(u, XI,

x E 10, LI,

(7.8)

subject to boundary conditions

&(ON

= S(U(O),

(7.9)

where L, R, and S are linear differential operators and g(u, x) is a nonlinear function of u such that g(0, x) # 0, can be transformed into the nonlinear integral equation L

.(X)

=

J G(x, Y)g(u(Y),Y>4,

(7.10)

0

where G(x, y ) is the Green's function associated with the differential operators

L, R, and S.

173

8. BEAM ON NONLINEAR FOUNDATION

We now observe that according to the results obtained in the last sections, the nature of the approximating sequence u l , u 2 , . . . , given by the solution of the linear differential equations subject to two-point conditions U U n

+ 1) - gu,(uu x)un + 1 = d u n X ) - gu,(un x)un 7

7

3

3

(7.11)

= s(un+1(L)),

R(un+

obtained by substituting the expression dun+

11

X ) = dun >

+ gu,(un

9

x)(~n + 1 - un)

(7.12)

into ( 7 . Q depends on the positivity of the Green’s function G(x,y), some convexity on g(u, x ) , and some additional properties opportunely discussed. In general we expect to satisfy these conditions in a local sense rather than unconditionally. In the next section we illustrate the ideas with an example. EXERCISES 1. Derive Eq. (7.4).

2. Show that the solution of u” = g(u, u’),

u(0) = u(1)

= 0,

is given by the system of coupled nonlinear integral equations

where G is given by (7.4). 3. Prove that the Green’s function associated with the linear equation u”

+ a(x)u = 0,

u(0) = u(L) = 0,

is nonnegative provided that a ( x ) < n2/Lz.For an elegant proof based on a variational approach see

R. Bellman, On the Nonnegativity of Green’s Functions, Boll. Un.Mat. I d . 12 (1957), 411413.

8. Beam on Nonlinear Foundation

We consider a beam resting on a nonlinear foundation characterized by the force-displacement relation F =u

+ h(u),

(8.1)

where h(u) is a given nonlinear function of u. To simplify we consider the stiffness of the beam to be constant and, in particular, EZ = 1. The ends of the beam will be considered hinged. Under these conditions it is well known that the displacement u satisfies the fourth-order differential equation d4u/dx4

+ u + h(u) = p ( x ) ,

(8.2)

174

7 QUASILINEARIZATION

where p is the load per unit length, subject to the conditions at the ends u(0) = u(L) = u”(0)= u”(L)= 0.

(8.3) Using the Newton-Raphson method we consider, instead of (8.2)-(8.3), the associated linear systems

+ + h’(un))un+ = AX)+ unh’(un) - h ( u A

d4un+ l/dx4 ( 1

1

un+,(O) = U,+l(L) = uf+,(O) = U f + l ( L ) = 0,

(8.4)

where the first element of the sequence uo is assumed to be given. Before we enter into problems of numerical feasibility we wish to examine the convergence of the sequence ulr u 2 , . . . . To this end we consider the Green’s function associated with the linear operator u”’ and boundary conditions (8.3), given by

G(x,y ) = (1/6L)x(y- L)(x2 - 2Ly

+y2)

(8.5)

if y I x , and G(x, y ) = G(y, x ) (Maxwell) if x Iy. The associated integral equation is given by JO

where

(8.7) is a loading term. Clearly G 2 0 in the domain of definition. We compute now the first characteristic value of the kernel G : L

~ ( x) J. J G ( x ,y ) ~ ( ydy)

= 0,

(8.8)

0

or, equivalently, in differential form, u’”

- lu

= 0,

u(0) = u(L)= u”(0)

= u”(L)= 0 ,

(8.9)

obtaining

A, = (7@)4.

(8.10)

We leave the derivation of this result as an exercise to the reader. Recalling the results of previous sections we know that the monotone behavior of the successive approximations depends on the nonnegativity of a certain associate linear operator which, in the present case, turns out to be

175

8. BEAM ON NONLINEAR FOUNDATION

or, in differential form z'"

+ (1 + ~ ' ( u ) )=zf ( x ) ,

~ ( 0=) ~ " ( 0=) z(L) = z"(L)= 0, (8.12)

+

where z = u - w and such that if u h(u) is concave (convex), we must have f 2 0 ( f S 0 ) . The reader can easily verify this assertion. Let us assume for now that u + h(u) is concave; i.e., h" I 0. We wish to show now that if there are two positive numbers pl and pz such that yz 2 1

+ h'(u) 2 -y,

(8.13)

and such that the operators L

UYl)

= Y l - P l S G(x, ElYl(5) d5 0

=f

(8.14)

and (8.15)

are nonnegative in the sense that, for a l l f 2 0, Eqs. (8.14) and (8.15) imply that y, 2 0 and y 2 2 0, respectively, then z in (8.11) or (8.12) is nonnegative. To

prove this assertion we make yz

-q = 1

+ h'(u),

(8.16)

where q is nonnegative if (8.13) is taken into account. Substituting 1 given by (8.16) into (8.11) we obtain, after some manipulations,

+ h'

where (8.18)

and where K(x, y ) is the resolvent kernel of (8.15) which is pointwise nonnegative in 0 5 ( x , y ) I L by virtue of the postulated nonnegativity of (8.15). Now, the Neumann series

obtained by iteration in (8.17), converges provided that q satisfies the characteristic value condition SUP

4 < a. ,

(8.20)

176

7

QUASILINEARIZATION

where A,, is the smaller characteristic value of K(x, y ) , a condition that is clearly verified if inf(1

+ h‘) > -

7 ~ ~ 1 ~ ~ .

(8.21)

X

The required nonnegativity of z follows immediately from the nonnegativity of K and o! in (8.19). It remains now to provide estimates for pl and p 2 in (8.13). It is easy to see that a lower bound on p1 is

>

-7c4/~4;

(8.22)

i.e., if pl satisfies (8.22), then y1 in (8.14) is nonnegative. This result follows from considerations of the Neumann series in the operator equation (8.14). We leave this verification as an exercise for the reader. The lower bound for -pl given by (8.22) is the best possible in the sense that for pL1> n4/L4 the solution of (8.14), with f = sin(n/l)x, is negative. A conservative estimate on the upper bound pt of 1 + h‘ may be shown to be 7c4/L4. This estimate is not the best possible? but it is certainly enough for our purposes. In fact, now the condition for monotone behavior may be compactly written sup 11

+ h’I

< n4/L4,

(8.23)

which indicates that it holds with the maximum interval of validity since (8.23) is a necessary condition for the existence and uniqueness of the original nonlinear problem (8.2)-( 8.3). We may now combine the results derived above in a single statement: Theorem If the nonlinear representation of the foundation is such that

h” < 0

(h” > O),

(8.24)

and sup1 1

+ h’l

< n4L4,

(8.25)

then the sequence u, generated by the solution of the linear boundary-value problem (8.4) will monotonically and quadratically converge from below (above) to u, the unique solution of the original nonlinear two-point-value problem (8.2)-( 8.3). The greatest upper bound on p2 may be shown to be 4h4, where h is the smallest positive root of the equation tan h = tanh h. See the papers by J. Schroder given in the Bibliography.

177

9. NUMERICAL EXAMPLE

EXERCISES

1. Prove that the eigenvalues of (8.8) are positive and, in particular, show that the first one is given by (8.10). 2. Derive Eqs. (8.11) and (8.12). 3. Consider an elastic plate resting on a nonlinear foundation characterized by the function g(u), and simply supported at the boundary aB. The deflection u satisfies the nonlinear boundary-value problem V4u = Q(u),

U ( P ) = azu(P)/an2= 0,

P E aB,

(a)

where 11 denotes the normal to the contour aB and V 4 = ( 8 ’ / a x 2 + a2/ay3)2is the biharmonic operator. Let G(x, y, q, 5) be the Green’s function of this operator for the same boundary conditions of (a). Show that

where B is the domain enclosed by aB, and prove the symmetry of the Green’s function, i.e., C k Y , 7, 5) = G(7, 6, x , Y ) . 4. Let H(x, y, q, f ) be the Green’s function associated with the second-order partial differential equation and boundary conditions v z u = a2uiaxZ

+ a 2 u p y = au,

U ( P ) = 0,

P E aB,

(C)

where a > 0. Prove, by invoking only elementary results in differential calculus, that HrO. 5. Let G be the Green’s function of the Laplace operator Q’u and boundary conditions u = 0 on a contour as, and let Go be the Green’s function of the same operator under

the same boundary conditions but on a contour aBo c aB. Using Hadamard’s equation for the variation of Green’s function, or otherwise, prove the following monotonic property of G: Go IG.

(d)

6. Can the results of Exercises 4 and 5 be proved for the biharmonic operator?

9. Numerical Example

We rewrite Eq. (8.4) in the more convenient form

It is clear that if uo is known and available in fast storage, we can readily proceed to compute ui using some of the methods outlined in Chapter 2 or 3, because (9.1) is an equation linear in u,,,. In turn, we use u1 to compute u 2 , and so on. By proceeding in this way we require that un be available in fast storage when we compute un+1. In problems such as the one we are discussing

178

7 QUASILINEARIZATION

here this demand of storage will be easily satisfied even with small computers, but when dealing with higher-dimensional problems we might encounter serious difficulties associated with the storage of the previous approximation. We study variations of this method and an alternative quasilinear approach later in this chapter, in connection with the solution of nonlinear systems. In order to illustrate the efficiency, accuracy, and other features of the method, we present here a numerical example involving the fourth-order equation

+ + h(u) = (1 + 7c4)sin nx + h(sin nx),

d4u/dx4 u

(94

subject to boundary conditions u(0) = u”(0) = u(1) = u”(1) = 0.

(9.3)

The solution of (9.2)-(9.3) is u = sin nx,

(9.4)

for any continuous function h satisfying (8.25). For numerical purposes we shall consider

h(u) = cedu,

(9.5)

+

where c and d are constants such that 11 cdedU(< 7c4. Now, recalling the results of the last section we are able to assert that when c > 0 (c < 0), the approximating sequence derived from quasilinearization will monotonically converge from above (below) to the unique solution u = sin 7cx. In the present case Eq. (9.1) reduces to d4u,+ l/dx4

+ (1 + cd exp(du,))u,+

= q(x)

- c exp(du,)(l - du,),

%+l(O) = u,”+,(O) = % I + l ( l )= 4+1(1) = 0,

(9.6)

where q(x) is given by q(x) = (1

+ n4)sin nx + c exp(d sin nx).

(9.7)

The solution of Eq. (9.6) was numerically obtained for two sets of values of c and d as follows: c=5.0,

d=2.0,

and

c=-0.1,

d=1.0.

(9 * 8)

In both cases the initial approximation chosen to start the process was uo = 0. Equation (9.6) was solved using the method of superposition of principal solutions numerically obtained by integration of the associated scheme and various step sizes for comparison purposes. The results are displayed in Tables 7-1 and 7-2, where only the value of the successive approximations at x = 0.5, ie., u,(0.5), is presented. The exact value in both cases is u(0.5) = 1.0.

179

10. LOWER AND UPPER BOUNDS

TABLE 7-1 Newton-Raphson-Kantorovich, c = 5.0,d = 2.0 h = 0.010

u. uo(x)

E [0, 11 1.156831 1.008793 1 .OOO025 1.OOOO00 1.00OOOO

0.0 for x

~~(0.5) ~~(0.5) udo.5) ~q(0.5) uS(0.5)

h = 0.005 0.0 for x E [0, 11

1.1568311 1.0087937 1 .oooO255 1.0000000 1.0000000

h = 0.002 0.0 for x

E [0, 11 1.156831167 1.008793789 1.oooO25558

1.000000000

1.000000000

h = 0.001 0.0 for x E [0, 11 1.156831 1671 1 .0087937892 l.oooO255582 1.0000000002 1.0000000000

TABLE 7-2 Newton-Raphson-Kantorovich, c = -0.1, d = 1.0

u.

h = 0.010

0.0 for x E [O,11 u0(x) Ui(O.5) 0.999404 ~~(0.5) 1 .OOOo00 ~~(0.5) 1.000000

h = 0.005 E [0, 11 0.99940433 1 .00OOO000 1 .oOOOoooo

0.0 for x

h = 0.002

h = 0.001

0.0 for x E [0, 11

0.0 for x E [O, 11

0.999404329 0.999999999 1.000000000

0.9994043290 0.9999999996 1.0000000000

The computations were performed for four step sizes, namely h = 0.010, 0.005, 0.002, and 0.001, using an Adams-Moulton scheme on a CDC 6400 computer. For each step size, the number of accurate figures was determined separately using a similar equation whose exact solution was known. Only the figures estimated to be exact are shown in the tables. 10. Lower and Upper Bounds In the preceding sections we have seen that under appropriate conditions, quasilinearization enables us to construct lower (upper) bounds of the solution. If we could have an equally effective method of constructing upper (lower) bounds, we could greatly improve our knowledge of the solution. Unfortunately this is a problem that is considerably more difficult to solve. For some problems deriving from a variational principle, Bellman has shown how to construct lower and upper bounds using classical concepts in duality. See the exercises below and the Bibliography at the end of the chapter. Here we give some ideas on another systematic approach. We illustrate the ideas by considering a problem involving a continuous function of one variable. Let a < b be real numbers,f(a)f(b) < 0, and f ( x ) = 0.

(10.1)

180

7 QUASILINEARIZATION

We consider an expansion off in linear terms, i.e., f(Xn>

+ (x,+ 1 - xn>f'(x,>= 09

(1 0.2)

from which the classical Newton-Raphson equation x,+,

= x,-

f(x,) f '(x,)'

(10.3)

follows. If x1 = a and .f'(x)f"(x> < 0,

(10.4)

we can easily prove that the sequence x, converges monotonically from below to r, the unique root of (10.1) in the interval of consideration. Therefore x1 < x z < - ~ - < x , < * ~ ~ < r .

(10.5)

Now instead of (10.3) we consider the sequence y , constructed on the regula falsi concept, in the following manner: (10.6)

i.e., using the slope of the secant given by [ f ( x , ) - f ( y , ) ] / ( x , - y,) instead of the derivative f'(x,). Still assuming that (10.4) holds, we can readily prove that the sequence yn converges monotonically to r from above, i.e., r<


+ . -

< y z < y,.

Clearly, iff '(x)f"(x)> 0, as shown in Fig. 7-2, we use y , obtaining y1 < y , * * *<

(10.7) =a

and x1 = b,

~ , < ~ ~ ~ < I < ~ ~ ~ < X , < ~ ~ (10.8) ~ < X ~ < X ~

Fig. 7-2

181

11. NONLINEAR SYSTEMS

This procedure, which automatically furnishes lower and upper bounds, can be generalized to function spaces, making possible in this fashion the determination of upper and lower bounds of functional equations. The reader can find references at the end of the chapter. EXERCISES 1. The equilibrium configurations of elastic beams on nonlinear foundations with hinged ends are characterized by the minimum of the functional

min 4 u

J:

(EZu"*

+ cp(u) - 2pu) dx,

u(0) = u(L) = 0,

(a)

where ~ ( u is) the potential energy stored by the nonlinear foundation and where the remaining conditions for hinged ends are furnished by the natural conditions of the variational problem, i.e., by u"(0) = u"(L) = 0. Assume cp(u) convex and write

where g(u) = cp'(u). Substituting (b) into (a) and assuming that the min-max operations are interchangeable, i.e., min max U

"

I

= max

min

V

Y

J,

show that min 4 I

J:

+ cp(u) - 2pu] dx = max + JoL[EIW"~+ ( w - u)g(u) - 2pw + ~ ( v ) dx, ]

[EZun2

v

(c)

where w is the solution of the differential equation

(EZW")" = p - 4g(u),

w(0) = w"(0) = w(L) = w"(L) = 0.

2. Using (c) and appropriate trial functions, determine convenient lower and upper bounds for the variational problem. 3. Show that the max-min operations in Exercise 1 are interchangeable.

11. Nonlinear Systems

The scalar case studied in the previous sections of this chapter admits a natural generalization involving higher-dimensional quantities. Possibly the most significant application of quasilinearization lies in its ability to solve large systems of functional equations of various kinds subject to two-point or multipoint boundary conditions, in a very efficient and accurate manner. We shall see in the remaining chapters of this book how a large class of significant problems in the area of design, optimum design, system identification, and analysis can be cast in the form of generally nonlinear functional equations in terms of differential, differencedifferential, integrodifferential, etc., operators, subject to boundary conditions.

182

7 QUASILINEARIZATION

In the rest of this chapter we deal with some examples of application to the areas of analysis and design. To begin with, we consider here a mechanical system described by the following nonlinear system of first-order differential equations :

du/dx = g(u),

(11.1)

where u and g are N-dimensional vectors given by

subject to the two-point boundary conditions ui(0)= c i , ui(L)=di,

i = 1, 2, . . . , k, i = k + l , k + 2 ,..., N.

(11.3)

It should be noted at this point that the present method can handle far more general boundary conditions than those given by (1 1.3), including nonlinear ones. We restrict ourselves to (11.3) in an attempt to clarify the central ideas of the method. Now, assuming that g(u) is convex in some region, we consider the quasilinear representation (11.4) where v is an N-dimensional vector

(11.5)

and J(v) is the Jacobian matrix defined by J(v) = dgi/duj,

i, j

=

1, 2,

...,N.

(11.6)

Clearly, (1 1.4) is the multidimensional version of the scalar representation (3.8). Substituting (11.4) into (11.1) and proceeding as in the scalar case, we consider the sequence of linear boundary-value problems

dd""/dx = g(u") + J(u")(u"+' - u"),

(1 1.7)

where u"+l is subject to boundary conditions (1 1.3). In (1 1.7) we use a superscript to indicate the order of the element in the approximating sequence,

12.

183

THE LINEAR ASSOCIATED PROBLEM

and it should not be taken as an exponent. We use this notation in the multidimensional case to avoid confusion with subscripts which indicate the component of the vector. We recognize that (11.7) is the Newton-Raphson method which can be obtained by substituting the linear multidimensional Taylor's expansion of g(u) around u", i.e.,

+ J(u")(u"+' - u"),

g(u) = du")

(11.8)

in (11.1). We observe that conditions to ensure monotone approximation and convergence of U" given by (11.7)-(11.3) are far more difficult to obtain than in the scalar case. When, instead of boundary conditions such as (11.3), we deal with initial conditions, then a necessary and sufficient condition for monotonicity is that the Jacobian matrix be nonnegative (or nonpositive) in the sense that dgi/duj 2 0

ag,/au, 5 0,

or

i, j = 1, 2, ... , N .

(11.9)

A few other results are available. The reader should consult the Bibliography at the end of the chapter. 12. The Linear Associated Problem

The method hinges on the solution of a sequence of linear boundaryvalue problems (1 1.7)-(11.3), where the first element of the approximating sequence uo is assumed to be given. We can readily solve this system by using some of the methods of Chapters 2 and 3. In order to illustrate the application of the method, we discuss here the solution of (11.7)-(11.3) by means of the method of superposition of linearly independent solutions. In Sections 14 and 15 we show how invariant imbedding is the natural tool to use in conjunction with an alternative version of quasilinearization. To this end we write (11.7) in the more convenient form dd'+'/dx = J(u")u"+' + w",

(12.1)

where w" is the vector given by

w" = g(u7

- J(u")U".

(12.2)

Introducing the matrix differential equation dUn+'/dx = J(u")U"+',

U(0)= I,

(12.3)

in terms of the fundamental matrix Un+'(x),we can readily obtain the solution of (12.1) by superposition, i.e., Un+l

=

un + l un + l (0) + u;+1,

(12.4)

184

7 QUASILINEARIZATION

where u"+'(O), the initial condition, is a vector incompletely known and uF+' is a particular solution of (12.1) subject to the initial condition u;+'(O) = 0, i.e., d$+'/dX = J(u")u;+' w", u;+'(O) = 0. (12.5)

+

We integrate the N 2 differential equations (12.3) and the N differential equations (12.5), both subject to initial values, in the forward direction by means of a numerical procedure such as the Runge-Kutta or the AdamsMoulton scheme. When we arrive at x = L we can formulate the following algebraic problem : u;;'(L) +

u;;w

,

which furnishes the missing boundary conditions, thus completely determining vector u" '. +

13. Computational Aspects. Storage The method of successive approximations as exposed in the last section requires the storage of the last approximation u" in order to compute the new one u"+'. If N is the dimension of the vector u and M is the number of points where the function u" has to be known, then N M is the number that sets the limits of the method in connection with any given digital computer. To remove some of the difficulties associated with the problem of storage, we need to modify the procedure. Fortunately this can be done. What we do is to generate the function rather than store it. To do this all we have to store are the initial conditions u'(0) of the vectors ur, r = 1, 2, . .. , n, quantities that become available right after the solution of the algebraic equations. In effect, in this way we are able to integrate the set of simultaneous differential equations du'/dX = J(u')u' + W O du2/dx = J(u')uZ + w1 (13.1)

du"/dx = J(u"-')u" + w"-', subject to known initial conditions u'(O), r = 1, 2, . . . , n. We are implicitly assuming that the initial approximation uo is a vector whose components are functions easy to store such as constants or exponentials. We are clearly trading memory for the time needed to integrate the additional system of nN differential equations (13.1) per iteration. In a final analysis this method

185

15. A MATRIX RICCATI EQUATION

+

requires the integration, at each iteration, of the N 2 N equations (12.3) and (12.5) plus the adjunct nN equations (13.1). This makes a total of R(N2 N ) + $NR(R - 1) equations in R iterations. Since the convergence of this process is quadratic, when convergent at all, we shall not need too many iterations, say R < 10 or so. The present procedure of adjoining differential equations has been extensively used by Bellman and Kalaba to bypass some of the limitations of the Newton-Raphson scheme of successive approximations. In the next section we discuss a procedure that further reduces the computational effort while preserving the same convergency characteristics.

+

14. An Alternative QuasilinearizationProcedure

If, in the quasilinear representation (1 1.4), we use the previous approximation un in place of u, the maximizing function, we are led to the classical Newton-Raphson-Kantorovich scheme (1 1.7). This is not the only way to generate a sequence of approximating functions from the first linear problem associated with (1 1.I), and in fact it might be advantageous to proceed otherwise. For example, we may consider the sequence W" generated by the sequence of linear boundary-value problems

dW"+'/dx = g(v") + J(u")(w"+~ - u"), i = 1 , 2 , . . ., k, wl"(0) = c i , w;+'(L) = di, i = k + 1, k + 2, ..., N , where the function v" is the solution of the initial-value problem du"/dx = g(v"),

~ " ( 0=) ~ " ( 0 ) .

(14.1)

(14.2)

The sequences of functions W" and U" exhibit properties of convergency and monotonicity which are similar to those of u" generated with the classical Newton-Raphson-Kantorovich method and are exempted from some of the limitations. In fact, here we neither have to store the previous approximating function u" in order to compute u"", nor do we have to adjoin an increasingly larger system of equations at each iteration. The reader can easily verify that the total number of differential equations to integrate in this method amounts to N ( N - k + 2) k per iteration, when the integration scheme proposed in the next section is used.

+

15. A Matrix Riccati Equation If convergence occurs, then the sequence u" tends to u, the solution of the original boundary-value problem (1 1.1) and (1 1.3). Then it is interesting to note that the main role played by the associated linear boundary-value problem (14.1) is to make possible the determination of w"+l(O),the initial condition

186

7

QUASILINEARIZATION

of Eq. (14.2). Therefore, the determination of the initial vectors becomes the central part of the method. This can be done in a number of ways, taking advantage of the linearity of the equations. In general we can use the method of superposition of the principal solutions or, alternatively, we can use invariant imbedding. We choose the latter to illustrate the application. To this end, we rewrite (14.1) in the more convenient form

+ , dyldx = J3 z + J4y + yo, +

dZ/dX = J ~ Z J2y

ZO

~(0= ) C,

(15.1)

y(L) = d,

where z and y are the k- and ( N - k)-dimensional vectors, respectively, given by

J,, J , , J 3 , and J4 are the partitions of the matrix J by the kth row and column, vectors zo and yo are obtained from an appropriate partition of vector g(u") - J(u")u", and where c and d are self-evident, recalling the boundary conditions in (14.1). Considering now the interval [x, L],in view of the linearity of the problem we can write Y ( 4 = R(x)z(x) + S(XlY(L) + a(x>,

(15.3)

where the matrices R and S a n d the vector a are of the appropriate dimension. Now, operating as usual in invariant imbedding, i.e., differentiating (15.3) with respect to x , substituting y ( x ) back, employing (15.1) in the resulting equation, and collecting terms in z ( x ) and y(L), we finally obtain dR/dx = J3 + 54 R - RJ, - RJ2 R , dSldX = (54 - RJ,)S, daldx = (J4 - RJ2)a + yo - Rzo ,

R(L) = 0, s(L) = IN-k,

(15.4)

a(L) = 0,

a system of Riccati differential equations for R,S, and a, where is the identity matrix of dimension N - k . Clearly, the system (15.4) must be integrated in the backward direction. The "missing" conditions at x = 0 are furnished by ~ ( 0= ) R(0)c

+ S(0)d + a(O),

(15.5)

and the vector w""(0) is finally given by (15.6)

187

16. A PARAMETRIC DESIGN PROBLEM

16. A Parametric Design Problem

The theory of design is a prolific source of nonlinear boundary-value problems. In general the problem consists of the determination of one or more design parameters such as to satisfy a number of carefully chosen design specifications. We illustrate the application of quasilinearization to problems of this type with a simple design problem involving a circular, simply supported plate subject to a given axisymmetric load q(r), where r is the radius. The depth h of the plate is assumed to be variable according to the law

h(r) = ae-br,

(16.1)

where a and b are two positive constants to be determined such that, for a fixed volume of material, the deflection u(0) at the center of the plate reaches a prescribed value c. To formulate this problem in mathematical terms, we consider the differential equations of bending for a simply supported circular plate of radius R given by

u(R)= 0, u(0) = 0,

dm 1-v E 1 -=-m-h3v - 2qr, dr r 12r2

(16.2)

m(R) = 0,

where u, v, and m are the deflection, slope, and radial bending moment, respectively, and where E and v are Young’s modulus and Poisson’s ratio, respectively. The design specifications are given by the equations u(0) = c,

(16.3)

and

hr dr = V,.

272

(1 6.4)

JoR

Our objective is to find u, u, m, a, and b such that Eqs. (16.1)-(16.4) are satisfied. This is clearly a nonlinear boundary-value problem that can be solved by quasilinearization.To this end we shall consider a and b as auxiliary variables satisfying the differential equations daldr = 0,

dbldr = 0,

(16.5)

188

7 QUASILINEARIZATION

which express the obvious fact that both a and b are constants. It is also convenient to homogenize the structure of the nonlinear problem by reducing the constraints given by (16.1) and (16.4) to differential equations. In this fashion, Eq. (16.1) reduces to

dhldr = -bh,

h(0) = a(O),

(16.6)

and, by introducing the auxiliary variable 4 r ) = f h S d4, 0

Eq. (16.4) reduces to

dz/dr = hr,

z(0) = 0,

z(R) = V0/2n.

(16.8)

Now, Eqs. (16.2), (16.3), (16.5), (16.6), and (16.8) are a system of nonlinear equations in u, v, m, a, b, h, and z subject to two-point conditions, similar to that studied in Section 11 and subsequent ones. Application of quasilinearization leads to the system

dun+ 1 -dr - 'n+l 36(1 - v2)mn 36(1 - v2)mn don + 1 - - V- V n + 1 - 12(1 - v 2 ) -hn+l + mn+t +

dr

r

Ehn3

Eh/

Eh2

9

subject to the two-point-value conditions

(1 6.10) zn + 1(0) = 0,

We can now solve the linear system (16.9)-(16.10) by a number of methods. We leave the discussion of this problem as an exercise to the reader. We note that if instead of (16.3) we wish to prescribe the deflection at a point ro < R, i.e., if 4 r o ) = c, (16.11)

NOTES, COMMENTS, AND BIBLIOGRAPHY

189

nothing is changed except that now we have three-point-value rather than two-point-value conditions. Again, we leave the discussion of this problem as an exercise to the reader. EXERCISES 1. Discuss suitable initial approximations for the solution of this parametric design problem. 2. Solve the linear system (16.9)-(16.10) by invariant imbedding.

3. Extend the solution of Exercise 2 to the case where, instead of 4 0 ) = c, we consider (16.11). 4. Formulate the solution of this problem by using the alternative quasilinearization procedure discussed in Section 14.

5. How would you study numerically the dependence of the parameters a and b on the quantity u(0) = c? Could you construct differential equations for the quantities a and b

in terms of c ? 6. Instead of h given by (16.1) consider

where @< are given functions and ci are unknown constants to be determined so as to render minimum the quantity u2(0) while satisfying the constraint (16.4) imposed on the

volume. Discuss the solution of this version of the problem by quasilinearization.

NOTES, COMMENTS, AND BIBLIOGRAPHY

1. QuusiZinearizution i s a theory with roots in dynamic programming. The following are some of the books available on applications to various areas : R. Bellman and R. Kalaba, “ Quasilinearization and Nonlinear Boundary-Value Problems.” American Elsevier, New York, 1965; E. S. h. Quasilinearization and Invariant Imbedding.” Academic Press, New York, 1968;

.‘

J. R. Radbill and G. A. McCue, “ Quasilinearization and Nonlinear Problems in Fluid and Orbital Mechanics.” American Elsevier, New York, 1970.

2. The theory of dynamic programming suggests a type of approximation known as approximation in policy space, with no counterpart in classical analysis. In this section we follow R. Bellman, “ Dynamic Programming.” Princeton Univ. Press, Princeton, New Jersey, 1957.

190

7

QUASILINEARIZATION

3. The intimate connection of quasilinearization to the theories of monotone operators and functional inequalities is significant. See E. F. Beckenbach and R. Bellman, “Inequalities.” Springer-Verlag, Berlin and New York, 1965;

L.Collatz, Functional Analysis and Numerical Mathematics.” Academic Press, New York, “

1966.

4. A comprehensive treatment of nonlinear Fredholm integral equations using fixed-point theorems is given by

W.Pogorzelski, Integral Equations and Their Applications,” Vol. I. Pergamon, Oxford, “

1966.

The present version of Hammerstein’s theorem is given in F. G. Tricomi, “Integral Equations.” Wiley (Interscience), New York, 1957.

The same book is highly recommended as an introduction to integral equations in a classical context. 5-6. When the last approximation is used to compute the new one, quasilinearization overlaps Newton-Raphson’s method in function space, a method initially developed by Kantorovich. See the book by L. Collatz quoted in Section 3 above. The theory of quasilinearization permits increasing the domain of validity of the Newton-Raphson-Kantorovich method and furnishes a convenient framework for establishing various properties of the approximating sequence. The problem of monotone behavior and convergence has been studied by Kalaba : R. Kalaba, On Nonlinear Differential Equations, the Maximum Operation, and Monotone Convergence, J. Math. Mech. 8 (1959), 519-574.

Some results for nonlinear Volterra operators in connection with some problems on nonlinear creep of columns are given in N. Disttfano and J. Sackman, Quasilinearization and the Computation of the Deflections of Metal Structures Undergoing Creep, in ‘‘Creep in Structures” (I. Hult, ed.). SpringerVerlag, Berlin and New York, 1972.

7. A comprehensive treatment of Green’s function and associated SturmLiouville problems can be found in C. Lanczos, “ Linear Differential Operators.” Van Nostrand Reinhold, Princeton, New Jersey, 1961.

For the connection to the solution of nonlinear problems see H. T. Davis, “Introduction to Nonlinear Differential and Integral Equations.” U S A . At. Energy Comm., 1961.

NOTES, COMMENTS, AND BIBLIOGRAPHY

191

8-9. Here we follow in part N. Distbfano and R. Todeschini, A Quasilinearization Approach to the Solution of Elastic Beams on Nonlinear Foundation (to appear).

An approach to the study of the positive solutions of general linear fourthorder differential equations may be found in J. Schroder, Randwertaufgaben vierter Ordnung mit positiver Greenscher Funktion, Math. Z. 90 (1965), 429-440; J. Schrader, "Zusammenhangende Mengen inverspositiver Differentialoperatoren vierter Ordnung," Math. 2 . 9 6 (1967), 89-110.

10. The determination of upper and lower bounds for certain classes of variational problems has been given in a note: R.Bellman, Quasilinearization and Upper and Lower Bounds for Variational Problems, Quart. AppI. Math. XIV, NO.4 (1962), 349-350. Similar ideas are exposed in the book Quasilinearization by the same author. The extension of the method of secants to function space can be found in V. h d a , A Remark to Quasilinearization, J. Math. Anal. Appf. 23 (19681, 130-138. General considerations and abundant bibliography on this important subject can be found in the book by Collatz referred to in the literature of Section 3. 11-13. Here we give some general ideas associated with the solution of nonlinear systems via quasilinearization. In the rest of the book the reader will find extensive application of these ideas and will witness the development of new procedures to cope with specific problems.

14. This method was used in N. DistBfano, Quasilinearization and the Solution of Nonlinear Design Problems in Structures Undergoing Creep Deformations, Internat. J. Solids Structures 8 (1972), 215-225. See also SESM Tech. Rep. 70-14, Berkeley (August 1970).

It was thereafter used in connection with a number of problems on identification of viscoelastic materials and optimum design. 15. For an application to system identification problems in structures see N. DistBfano and D. Nagy, Parametric and Adaptive, Non-Parametric System Identification Procedures in Structural Mechanics, Proc. 1971 Summer Comput. Simulation Con& , Boston, Massachusetts I (1971), 688-695.

Miscellaneous References on the theory of positive operators see the book M. A. Krasnosel'skiI, '' Positive Solutions of Operator Equations-* (transIated into Enslish). Noordhoff. Groningen, The Netherlands, 1964. For an extended reading

Chapter 8

Elements of Optimal Structural Design 1. Introduction Engineering design, blend of art and science, is presently undergoing a rapid transformation in its supporting methodologies. Structural design is not immune to this trend. A theory of optimal design is emerging from a judicious combination of structural mechanics and the modern mathematical theory of processes. In this book we expose some of the fundamental ideas and associated methodologies in the field. In particular, this chapter is devoted to a carefully chosen class of problems that can be treated with elementary methods. These last include a modest amount in the theory of Lagrange multipliers and convex sets and, occasionally, some discrete dynamic programming in addition to elementary principles in structural mechanics. Interest in this class of problems stems from both methodological and practical considerations. In the first place, they provide the substance for the development of a theory of optimal structural design with its associated advantages (easy transmission of knowledge, generalization of results, etc.). In the second place, they provide an excellent framework for the formulation (modeling) of " realistic " structural design problems to be ultimately solved by direct programming techniques. 192

3. ELASTIC TRUSSES

193

In the next chapter we present some selected topics in modern controI theory as a complementary methodological approach to a more involved class of structural design problems. Finally, in Chapters 10 and 11, some applications of the control theory approach are presented, completing in this fashion our involvement with structural design problems.

2. Objective Functions and Constraints Problems of optimal design, when formulated in mathematical terms, consist of an objective function, a design vector, and a number of appropriate side constraints. To every design vector there is an associated design conjiguration or just a design. The set of all possible designs is called the design space. The closure of the design space, a sufficient condition for the design problem to be well posed, is usually performed, if necessary, by the incorporation of appropriate constraints. An optimal design is the element of the design space for which the objective function achieves an extremum. We shall restrict ourselves to minimum volume design; i.e., our objective function will be the vglume, weight, or a similar functional of the structure. The constraints will usually be incorporated in the form of a compliance, i.e., a measure of the degree of deformability of the structure. Additional (or alternative) constraints, such as maximum stresses and strains, are also possible. For a given set of loads acting on the structure, popular candidates for a compliance are 1. the work of deformation of the given loads on the displacements of their points of application, 2. a deflection at an arbitrarily prescribed point of the structure. When there is a single concentrated load acting on the structure and we consider the deflection in the direction of the applied load, both quantities (1) and (2) coincide. A less commonly used, but not less important constraint, is 3. the maximum displacement.

The difficulty with (3) is the lack of analyticity usually associated with it.

3. Elastic Trusses We consider an elastic truss with a given layout of N bars of length l i , i = 1,2, . ..,N , subject to a single load P applied a t a specified node. We shall discuss the minimum volume design of such a truss subject to a

194

8

ELEMENTS OF OPTIMAL STRUCTURAL DESIGN

constraint on the work of deformation, a quantity that for a fixed value of P qualifies as a statical compliance of the truss. If Si denotes the axial force, A , the cross-sectional area, and E the modulus of elasticity of the ith bar, then the volume of the truss is given by N

V= CliAi, i= 1

(3.1)

and the work of deformation by

where u is the displacement of the truss at the point of application and in the direction of the load P . We form the functional

where 2 is a positive Lagrange multiplier. We know that conditions under which J is stationary, i.e., the first variation of J (with respect to small, arbitrary variations 6 A i in the design) vanishes, are necessary conditions for a minimum of (3.1) subject to the constraint (3.2). Performing the variations in the usual fashion, we find

where 6 S i ,the system of forces in the bars induced by a variation in the design, is a state of self-stress because no changes in the external loads have occurred. Therefore, the last term in (3.4), representing the virtual work of 6Si along a kinematically admissible set of deformations, drops out and (3.4) reduces to 1 N ( I i - A S ) 6 A i = O .

(3.5) i= 1

Clearly, when the truss is statically determinate, the last term in (3.4) vanishes because 6S, = 0. Since 6 A i are arbitrary quantities, (3.5) is satisfied if and only if oi2= E l l ,

(3.6) where oi = S i / A i is the stress at the ith bar. Equation (3.6) is a necessary condition for optimality, expressing the fact that the stresses have the same absolute value in each bar of the truss. This is usually known as a fully stressed design. Equation (3.6) also expresses that the strains E~ = oi/E, in absolute value, must be the same for all the bars of the truss. This, a compatibility constraint, will not be generally satisfied in a statically indeterminate truss.

195

4. SUFFICIENCY. PRAGERS DEVICE

Thus the optimal truss subject to a single loading condition will be, in general, statically determinate. The value of 1, the Lagrange multiplier, should be chosen to satisfy the constraint (3.2) which, in terms of the optimal volume Y*,reads

v* = APU.

(3.7)

It remains now to investigate the sufficiency of the optimality condition (3.6). This is what we do in the next section.

4. Sufficiency. Prager’s Device We use here an ingenious method extensively used by Prager to establish optimality conditions. To this end we rewrite (3.2) in terms of the bar elongations di = Si Ii/EAi,i.e.,

In order to find a condition for minimum volume we compare trusses of the family 9, whose elements are designs with the same compliance Pu. Let {Ai}and be two arbitrary designs of the same family. Thus

{ai}

where the 6i are the elongations associated with the design (Ai). Clearly, the elongations si are kinematically admissible for the truss with design {Ai}. Therefore, if we introduce the total potential energy function

the theorem of minimum potential energy demands that

where ii is the displacement associated with the design have Pu = Pii,(4.4) reduces to

{Ai}.Since

we

196

8

ELEMENTS OF OPTIMAL STRUCTURAL DESIGN

which, on account of (4.2), yields

c N

O i 2 ( A i li

- A i li) 2 0 .

i= 1

Thus, if(3.6) holds, (4.6) shows that { A i } is the minimum volume design. establishing the sufficiency of (3.6). 5. Discussion When several loads P , , P 2 , . . . , Pk act simultaneously on the truss, instead of (3.2) we consider (5.1) i.e., the work of the external loads P i along the displacements uj of their points of application. This extension does not modify Eq. (3.6), which still remains as the necessary and sufficient condition for optimality, but in the present case the constraint (5.1) loses the geometrical flavor attached to (3.2), i.e., when only one load is considered. We observe that the optimization problem presented in Section 3 admits the dual formulation: Minimize the compliance W given by (3.2) [or (5.1)] for a fixed value of P [or Pi] and the volume V . It is finally noted that if the truss is statically determinate, Si are independent of A i , and therefore the sufficiency of Eq. (3.6) follows immediately from the convexity of the functional (3.3).

6. Dual Loading If instead of a single loading condition we consider two (or more) loading systems P ( l ) and P ( 2 ) acting in a nonsimultaneous manner in a truss with a given layout, we may formulate the following optimal design problem: N

min C A i l i ,

AizO i=l

(6.1)

subject to the constraints

where 8;j) denotes the elongation of the ith bar due to thejth loading condition, j = 1, 2, and where u l , u2 denote the displacement, at the point of application, of P ( l ) and P ( 2 ) ,respectively, and where c1 and c2 are two arbitrary constants.

7. KUHN-TUCKER CONDITIONS

197

If we consider the two constraints (6.2) separately, and assume that strict equality holds in (6.2), the argument of Section 4 shows that the following inequalities N

C a!’’2(2iili - A i li) 2 0 , i= 1 N

C ai2’,(AiZi - Ai li) 2 0,

(6.3)

i= 1

where ap),j = 1, 2, is the stress at the ith bar due to thejth loading system acting on the truss with design { A i } , and where {Ai} is an arbitrary design, must hold simultaneously. Multiplying the first inequality of (6.3) by Al > 0 and the second by A2 > 0 and adding together, we obtain N

an inequality that shows as a sufficient condition for optimality. To investigate necessity, we form the functional

where 1, and 1, are nonnegative Lagrange multipliers. Now, conditions that ensure J to be stationary, i.e., that the first variation of J is zero, are necessary conditions for a minimum of (6.1) subject to the constraints (6.2) assumed to be active, i.e., when strict equality holds in (6.2). Recalling that variations in the internal forces induced by variations in the design originate only states of self-stresses in the structure, a standard variational computation in (6.6) leads to (6.5) where I , = X,/E and I , = &/E. This shows (6.5) to be a necessary and sufficient condition for optimality provided that strict equality holds in (6.2). In the next section we release this assumption, completing our investigation on the present subject. 7. Kuhn-Tucker Conditions

When in (6.2) the strict equality is substituted by the inequality, the corresponding condition on the Lagrange multipliers J j is that either Xi = 0 or > 0 and uj - c j = 0, j = 1,2. Therefore

xj

198

8

ELEMENTS OF OPTIMAL STRUCTURAL DESIGN

where the overbars were dropped and where

are necessary conditions for a minimum of the volume given by (6.1) subject to the constraints (6.2). That they are also sufficient follows, from consideration of the first part of Section 6. If the truss is statically determinate, it is clear that sufficiency follows directly from the convexity of the Lagrangian functional (6.6). Equations (7.1) are a particular version of the so-called Kuhn-Tucker conditions, a central result of nonlinear programming. 8. Prescribed Displacements

A popular behavioral constraint in structural design is a displacement along a prescribed direction at a location other than the point of application of the load. Consider, for example, an elastic truss of a given layout, subject to external loading and to a prescribed displacement u at a given node k in a specified direction v. Invoking the principle of virtual work we can write

where Si and Si are the axial forces in the ith bar of the truss due to external loading and to a unit load acting at the node k in the direction v, respectively. As usual, we construct the functional i= 1

EA

where A is a Lagrange multiplier, and seek for conditions under which 6J = 0. Performing the variations in (8.2), we readily find that

laici= 1, (8.3) where ui = S i / A i and rFi = S i / A i , is a necessary condition for optimality. Now, if we specify the deflection u to be positive, I will be positive and sign Si= sign Si , as implied by (8.3). Under dual loading we consider the displacement constraints

(8.4)

9. DUAL LOADING IN STATICALLY DETERMINATE ELASTIC TRUSSES

199

where SY) and sY) are the corresponding quantities associated with the loading systemj = 1,2, and form the functional

where ,Il and A2 are nonnegative Lagrange multiplers. The Kuhn-Tucker conditions of this problem read

AlfJyc?y) + A 2 c q ) $ 2 ) I , 2 0, A2 2 0,

A1@1

= 1,

- c1) = 0,

(8.7)

- c2) = 0,

where u1 and u2 are given by the equalities in (8.5). As usual, there is a case that demands little effort to prove sufficiency. In fact, when the quantities SV) and i = 1, 2, . . . , N ;j = 1,2, are independent of A , , the functional J is readily shown to be convex in l / A i . This observation is enough to ensure the sufficiency of the necessary conditions (8.7) when the truss is statically determinate. We note that in previous applications we have systematically invoked convexity of the Lagrangian with respect to the A , . In the present case, however, this cannot be guaranteed by direct inspection because we do not know the signs of SI” with respect to those of $”.

sy),

EXERCISE Consider the optimization problem associated with the functional (8.6). How are the Kuhn-Tucker conditions (8.7) modified by the addition of the constraints ( s , / A , ) 2i a o 2

(4

on the stresses of fhe bars?

9. Dual Loading in Statically Determinate Elastic Trusses Subject to Displacement Constraints It is clear that apart from some carefully chosen cases where the KuhnTucker conditions may be employed to determine optimal designs in a rather simple fashion, the problem of optimal design for two loading conditions under displacement constraints poses in general a complex nonlinear programming problem whose treatment is beyond the elementary framework chosen for this chapter. There is, however, a class of structures that may be solved by elementary methods. The class in question is that of statically determinate trusses and the method is that of dynamic programming.

200

ELEMENTS OF OPTIMAL STRUCTURAL DESIGN

8

To illustrate the ideas, consider a statically determinate elastic truss of a given layout of N bars subject to a dual system of loads P ( l ) and P(’),and let x and y be the displacements of the truss at given nodes and along given and P(’), respectively. Using the principle directions, under the effect of P(’) of virtual work, we may write N

N

where

mi = , S ~ ’ ) ~ ~ ’ ) l i / EPi , = S!Z’S~Z’[i/E,

(9.2)

are known quantities since the truss is statically determinate. As usual, a superscript in parentheses denotes the loading system. Additionally, and with no loss in generality, we specify the quantities x and y to be nonnegative. We may now formulate the following optimization problem : N

min

2li,ii,

(9.3)

Ai>Oi=l

subject to the displacement constraints X S C ,

y ~ d ,

(9.4)

where c and d a r e two prescribed quantities, and to the stress constraint (9.5)

ISiIlAi 5 bi,

where oi > 0 is the allowable stress in the ith bar. In order to solve this problem by dynamic programming, we introduce the minimum volume function N

f N ( c ,d) = min

1li A i , i=

(9.6)

1

where the minimization is sought over the set of permissible cross-sectional areas. Now, application of the principle of optimality leads to

f~(c,d ) = min [IN A N + f ~ - i ( c- “ / A N d - PN/AN)I, 7

A N E d~

(9.7)

where d N is the set of all AN such that

AN 2 @N/c,

AN

2 PNId,

AN

2

I sN I l b N

9

(9.8)

or, equivalently, such that AN

2 max(@N/c,P N I d ,

I sN I / O N > .

(9.9)

The functional equation (9.7) is subject to the initial condition

I

fl(G d) = 4 maX(~1IGPlld, Sl 1/01).

(9.10)

201

10. BEAM OPTIMIZATION

Equation (9.7) subject to (9.10) is a straightforward two-dimensional dynamic programming problem in N stages. The advantages associated with the use of the functional equation are numerous. Among others, we mention that the present method yields the absolute minimum even when other local minimums exist. On the other hand, it allows the incorporation,

in a natural fashion, of local constraints of the type i = 1 , 2 ,..., N ,

gi(Ai,S!j))
j=l,2,

(9.11)

such as (9.9, or those o ccu r r i n g w h e n b u ck l i ng is t a k e n into account. It

also handles naturally discrete, as opposed to continuous, design variables suchas A i. As a matter of fact, theincorporation of localconstraints simplifies, rather than complicates, the solution of the functional equation by reducing the size of the state and design space. We finally note that the present problem may be reduced to the solution of a one-dimensional functional equation to the expense of introducing a Lagrange multiplier. See paper by DistCfano and Rath in the Bibliography. EXERCISES 1. Discuss the use of a Lagrange multiplier to reduce the present problem to a one-dimensional functional equation. 2. Discuss the incorporation of buckling as a constraint.

10. Beam Optimization

Consider a beam of length L subject to a single concentrated load P applied at a specified point 0 < x1 < L. The deflection u, at x1 in the direction of P , satisfies the equation

Pu

(Ijcr)m2dx,

=

(10.1)

JoL

where a,given by a = EI,

(10.2)

is the stiffness of the cross section and m is the bending moment such that the curvature U" = -m/a satisfies the compatibility conditions at specified points of the beam. We shall adopt (10.1) as a measure of the compliance of the beam. The volume V may be written in terms of a by means of (10.3) where g(a, x ) is a function that depends on the geometry of the cross section.

202

8 ELEMENTS OF OPTIMAL STRUCTURAL DESIGN

For example, for solid rectangular beams of constant width b and variable height h, g = (12b2u/E)”3.Now we can formulate the following optimization problem: Minimize the volume (10.3) subject to the constraint (10.1). We introduce the functional

1 L

J(a) =

M a , x>

0

+ 1(1/.)m’>

dx,

(10.4)

where 1 is a positive Lagrange multiplier. Performing a small, arbitrary change in design 6a, we compute the first variation L

6J =

(g,(u,

L

x) - A.(m2/u2))6udx

0

+ 2 1 l (m/a)Smdx,

(10.5)

0

where ga = dg/du, and where 6m, the variation of bending moment induced by a variation of design, is a state of self-stress because no changes in the external loads have occurred. Thus, by the principle of virtual work, the second integral in (10.5) drops out and the condition for J to be stationary reduces to L

6J = J” (9. - 1(m2/a2))6udx = 0.

(10.6)

0

Since the variations in design 6a are arbitrary, the fundamental lemma of the calculus of variations demands that A(m/Q>Z= 9.: 9

(10.7)

a necessary condition for optimality. If g is convex in u and the beam is statically determinate, (10.7) is also a sufficient condition for global optimality, as it follows from the convexity of the Lagrangian (10.4). Except for this, and some other carefully chosen examples, we must content ourselves with conditions that ensure local as opposed to global optimality. We finally observe that (10.7) was derived under the assumption that a is continuously differentiable. This, a too restrictive condition, may generally be released, as indicated in the following sections. EXERCISES 1. Show that for a solid rectangular beam of fixed depth h and variable width 6, the crosssectional area g appearing in (10.3) is given by g = 12cr/Ehz,

(a)

i.e., V is a linear functional of the stiffness a. 2. Show that when g is given by (a), (10.7) is a necessary and sufficient condition of optimality.

203

11. SANDWICH BEAMS

11. Sandwich Beams We consider here the class of sandwich beams, i.e., those whose cross section is indicated in Fig. 8-1 where h is assumed constant. If E denotes the modulus of elasticity and A the cross-sectional area of the covering sheets, we have a = EAh'. Therefore the cross section A in terms of the stiffness a is given by g = a/Eh' and (10.3) will then read

1

L

V = (1/Eh')

a dx,

(11.1)

0

i.e., the volume of the beam is a linear functional of the stiffness. The necessary condition (10.7) derived in the last section now reads

v(rn/a)'

= 1,

(11.2)

where v = Eh'L is a positive constant.

Fig. 8-1

We shall now use Prager's method to show that (11.2) is also a sufficient condition for optimality. To this end we write the compliance given by (10.1) in terms of the displacements u(x), i.e., P6 =

IoL dx. CLU"'

(11.3)

We now consider two designs a and 8 of the same family, i.e., such that P6

= P8,

(11.4)

or, alternatively, such that (11.5) where ii and 8 are the quantities associated with the design 8. In other words, we shall compare designs having the same overall compliance. By the theorem of minimum potential energy we must have JoLEiin2 dx - 2P8 I

loL8um2dx - 2P6,

(11.6)

204

8 ELEMENTS OF OPTIMAL STRUCTURAL DESIGN

because u and 6 are kinematically admissible displacements for the beam with design ti. Using (11.4) and (11.5) in (11.6) we obtain L

Jociu"'

L

dx I

J', ZU"'

dx,

(11.7)

an inequality which shows that u"' = constant

(11.8)

implies

joLa dx 5 JoLtidx, i.e., that CI is the lightest possible design. Therefore (11.8) is a sufficient condition for optimality. Substitution of U" by its equivalent value - m / a in (11.8) and comparison with (1 1.2) proves the desired property. 12. Discussion

We may rewrite (1 1.8) in the form vu"' = 1, or alternatively in terms of m and a, v(m/a)' = 1,

(12.1) (12.2)

where v is an arbitrary positive constant that may be chosen so as to satisfy the deflection constraint given by (1 1.3). Since for a sandwich beam the curvature mla is proportional to 0 , the stress in the flanges or covering sheets, then (12.2) shows that an optimal sandwich beam is also a fully stressed design. We note that Eq. (12.1), subject to appropriate boundary conditions at x = 0 and x = L,furnishes in principle the elastica of the beam within an arbitrary positive constant v. This observation shows that the shape of the elastica depends only on the conditions at the ends, not on the load distribution. The actual determination of the optimal design depends on the degree of indeterminacy of the beam. If the beam is statically determinate, m may be evaluated independently of GI and the optimal design follows from (12.2) where the constant v is to be chosen so as to satisfy the constraint (10.1). When the beam is statically indeterminate, we must integrate (12.1), a nonlinear second-order differential equation, subject to appropriate boundary conditions, This integration furnishes the elastica and consequently the points of inflection of the elastica, for which m and a are zero. This reduces the problem to a statically determinate one for which the determination of the optimal GI is immediate.

205

13. EXAMPLE

We may finally note that the results of Sections 10 and 11 still hold if instead of a single load P, we consider several concentrated loads PI, P 2 , . . . , Pk acting simultaneously at various points, or a continuously distributed load p ( x ) . In these cases, the compliance given by (10.1) must be replaced by k

C pjuj= j= 1

(l/c()m2dx

(12.3)

JoL

or L

~(x)u(x) dx = J (l/a)m2 dx,

(12.4)

0

respectively. When this is the case, the compliance ceases to exhibit the clear geometrical meaning of a deflection that exists when only one concentrated load acts on the beam, to become a more abstract mathematical entity, i.e., a linear functional of the displacement field. EXERCISES 1. Consider a column of the sandwich type subject to an axial load P. If u(x) is the lateral deflection, o! the cross-sectional stiffness, and 6 the shortening of the column due to bending, it is well known that the buckling load is characterized by the fact that, under a lateral perturbation, the internal work equates the value of the external work, a fact that in formulas reads

Using (a) as a constraint, and the method outlined in Section 11, show that u’I2 = constant

(b)

is a necessary and sufficient condition for minimum volume. (See the paper by Prager and Taylor in the Bibliography.)

2. Find the deflected shape of the column of problem 1 for the cases: (a) simply supported ends, and (b) clamped ends.

13. Example To illustrate the use of the optimality conditions developed for sandwich beams, we present here a representative example involving the optimal design of a clamped-simply supported beam as indicated in Fig. 8-2a and b. The differential equation of the elastica follows from the optimality condition (12. l), namely, U” = *A,

(13.1)

206

8

ELEMENTS OF OPTIMAL STRUCTURAL DESIGN

P

P

MOMENTS

DESIGNS

h (b)

(a) Fig. 8-2

where 1 is an arbitrary positive quantity. Since the beam is simply supported at one end and clamped at the other, it will have an inflection point at an unknown distance z from the clamped end. See Fig. 8-2. We determine the elastica by integrating (13.1) subject to the boundary conditions u(0) = u'(0) = 0, U ( Z + ) = u(z-),

U ' ( Z + ) = u'(z-),

u(L) = 0 ,

clamped end, continuity of u at z , continuity of u' at z , support at x = L.

(13.2a) (13.2b) (1 3 . 2 ~ ) (1 3.2d)

Assuming a downward deflection of the portion 0 I x 5 z , i.e., 1 > 0, we can integrate (13.1) subject to (13.2a) and obtain u = 31x2,

0 5 x I z,

(13.3)

i.e., the elastica in the first portion of the beam. Using (13.2b) and (13.2~) together with (13.3) we can easily construct the initial conditions for the elastics in the second portion of the beam, namely, U ( Z + ) = +1z2,

U'(Z+) = I z .

(13.4)

Recalling that the curvature changes its sign in the second portion of the beam, the differential equation (13.1) now reads U'' =

Integration of (13.5)-(13.4) yields

-2.

(13.5)

207

14. PIECEWISE CONTINUOUS STIFFNESS

Finally, imposing condition (13.2d) on (I 3.6), we find 2

= (1 - (2)”2/2)L,

(13.7)

the location of the inflection point. Now if we assume the beam loaded as indicated in Fig. 8-2a, i.e., with a concentrated load P at x = L/2, the reader will have no difficulty in verifying that the maximum moments m, and mz are given by

m, = -2[(2)1’2 - 1](~L/4), m2 = [2 - (2)’”](PL/4).

(13.8)

Therefore the optimal design given by (12.2) will be as indicated in Fig. 8-2a where the curvature 2 should be chosen ‘so as to satisfy the deflection condition ~(15.12)= 6, i.e.,

A=

86/L2 1 - 4[3 - 2(2)”2]’

(13.9)

a condition that follows from (13.6). We assume now that the concentrated load P acts at a point x1 Iz, such as in Fig. 8-2b. In this case the bending moments are = - P ( x , - x),

m(x)

0 5 x 5 xi, x1 5 x I L.

E 0,

( 13.10)

Consequently, the optimal beam will be the cantilever in the portion 0 I x I XI. 14. Piecewise Continuous Stiffness

The derivations of the preceding sections have been made under the implicit assumption that a, the cross-sectional stiffness, is a continuous function of x . In most practical applications, however, this assumption must be relaxed. We consider here one of such cases, namely the case of segmentwise constant cross-sectional stiffness in a sandwich structure. In this instance, Eq. (1 1.3) must be replaced by N

P6 =

a i l x ‘ u f 2dx, i=1

(14.1)

XI-,

where N is the number of segments of given lengths li = X ~ - X ~ along - ~ which the stiffness ai is constant. Accordingly, the volume is given by N

v = (l/Eh2)1 Uili. i= 1

(14.2)

208

8

ELEMENTS OF OPTIMAL STRUCTURAL DESIGN

Now, using an argument similar to that employed in Section 11, we can easily prove that (14.3) where Xi

pi2 = ( l / l i ) s

u;’ dx

(14.4)

xi-1

is a sufficient condition for optimality. Using classical arguments in the calculus of variations one may easily prove that (14.3) is also a necessary condition. In other words, a necessary and sufficient condition for optimality is that the mean square curvature (14.4) be constant at every segment i = 1 , 2) . . . )N . EXERCISE

Prove that (14.3) is a necessary and sufficient condition for optimality.

15. Prescribed Deflections In this section we discuss minimum volume design of sandwich beams with prescribed displacements given at fixed locations. When only one load is acting on the beam and the displacement is prescribed at the point of application of the load, the present case reduces to prescribing the work of deformation, a problem extensively studied in previous sections. We consider a sandwich beam of length L and cross-sectional stiffness a(x), subject to external loading that originates bending moments m(x). The deflection u(x) at a prescribed point 0 < x1 < L is given by L

(mG/a)d x ,

u(xl)=

(15.1)

0

according to the principle of virtual work, where iE is the bending moment due to a unit load applied at x l . The volume to be minimized is proportional to the quantity fk a d x . Thus if (15.1) is prescribed, we form as usual the functional

J= where that

loL(a + AmEi/a)dx,

(15.2)

A is a Lagrange multiplier. Familiar variational considerations show A(mrn/a’) = 1

(15.3)

209

15. PRESCRIBED DEFLECTIONS

is a necessary condition for optimality. If we now specify u(xl) > 0, A will be positive and (15.3) demands mE > 0.

(15.4)

When a is restricted to belong to a closed domain d , then instead of Eq. (15.3) we use aopt= arg min(a a€&

+ AmE/tl>,

(15.3)’

whose derivation will be deferred until Chapter 10. When the beam is statically determinate, (15.3) and (15.4) are explicit optimality conditions. In fact, in this case m and Z are known quantities and aOptmay be evaluated using (15.5)

tl =

an equation that follows from (15.3). The value of A should be determined so as to enforce the deflection constraint (15.1). When the beam is statically indeterminate, m and Ei depend on the unknown design a and therefore Eqs. (15.3) and (15.4) must be regarded as implicit optimality conditions. In simple cases they provide the solution of the problem if a modicum of nonlinear algebra is allowed, as the example of the next section illustrates. Finally, when we consider dual (or multiple) loading, the pertinent optimization problem reads L

min Jo a dx,

(15.6)

a

subject to

( I 5.7) and L m(2)&2)

UZ(X2) =

Jo dx < a

~ 2 ,

(15.8)

where m(’)and m(’) are the bending moments due to both loading systems, moments produced by a unit load applied at x1 and x 2 ,respectively, and c1 and c2 are two arbitrary numbers. By analogy with the developments of our previous sections, we suspect that

dl) and ?d2) are the virtual

Alm(l)jj.-p)

+ A z m ( 2 ) j j p ) = t12,

(15.9)

where I , and I , are nonnegative multipliers, is a necessary condition for optimality. This is in fact so, but we leave the proof as an exercise to the reader, who might also write the complete set of Kuhn-Tucker conditions associated with this problem.

210

8

ELEMENTS OF OPTIMAL STRUCTURAL DESIGN

16. An Example

To illustrate the method outlined in Section 15, in particular the use of the optimality condition given by (1 5.3), we consider a sandwich beam clamped at both ends x = 0 and x = 2L, and subject to a uniformly distributed load of intensity q. The problem is to minimize the volume subject to a constraint on the deflection at the center of the span x = L, namely u(L) = 6.

(16.1)

Since the beam is symmetric and clamped at both ends, we know that there will be an inflection point located at a distance z (to be determined) from both ends. The optimality condition (15.3) yields a = A’/’(mm)’/’.

(16.2)

The values of the quantities m and iii are not known in advance but we know that m(z) = m(2L - z) = 0 because z is the inflection point of the elastica, and that Z(z) = iii(2L - z) = 0, because otherwise the convexity condition (15.4) would be violated. Therefore m and E are given by m = - (z - x)(2L - z - x)q/2, Fl = - ( z - x)/2,

(16.3)

for x IL. See Fig. 8-3. The value of z will be obtained from the condition of minimum volume of the beam, which on account of (16.2) reads (16.4)

Substituting m and iii given by (16.3) in (16.4), and performing the operations, we finally derive the following nonlinear algebraic equation

(512 - 224(2)’l2)z3- (1473 - 672(2)’12)Lz2 (1395 - 672(2)’/2)L2~ - (433 - 224(2)’l2)L3= 0,

+

(16.5)

whose unique real solution z = 0.502215 furnishes, with four exact decimal places, the position of the inflection point. The deflection u(x) may be readily obtained by integration of the differential equation of the elastica u” = -mi. which, on account of (16.2) and (16.3), reads

- x)”2, = -(q/A)”2(2L - z - x y ,

u“ = (q/A)”2(2L - z

O S X I Z ,

2.4’’

z < x I L,

(16.6)

subject to the initial conditions u(0) = u’(0) = 0,

(16.7)

21 1

EXERCISES

-----

0 0.2 0.4 0.6 0.8

1.0

1.2

1.4

1.6

1.8 2.0

Fig. 8-3

and the continuity conditions u ( z - ) = U(Z+),

u’(L)= 0.

(16.8)

Equations (16.6)-(16.8) can be integrated easily, furnishing the elastica of the optimal beam. We choose 2 appearing in the equations so as to enforce the deflection condition (16.1). EXERCISES 1. Write the analytical expressions for the optimal design. 2. Integrate the system (16.6)-(16.8). 3. Show that making =

2q’/z [(16(2)”’ - 7)(L- z ) ~ / ’ - (4L - 7z)(2L - z ) ~ ’ ’ ] 156

in the equations for the deflection u(x), the constraint (16.1) is satisfied.

212

8 ELEMENTS OF OPTIMAL STRUCTURAL DESIGN

17. Rotating Disks We present here an elementary version of the problem of the optimal profile for an elastic rotating disk, a problem that is investigated thoroughly in a subsequent chapter. In this section we restrict ourselves to outlining how dynamic programming can handle, in a rather straightforward manner, a generally nonlinear discrete version of the problem. We consider the disk formed by the assemblage of N concentric rings of thickness H i , i = 1 , 2, . . . , N , as indicated in Fig. 8-4.

Fig. 8-4

We introduce now the state variables U i and Si,the radial displacements and the radial traction, respectively, at the interface i between the (i - 1)th and ith rings. It is assumed that

ui, = ui-, si, = si-,

(17.1)

i.e., there is continuity of displacements and radial tractions at the interfaces. We are justified in assuming U i and Sionly i dependent, in view of the axisymmetry of the problem. By using a number of approximate methods, or otherwise, we can always construct relationships of the type U i - l = Fi(U i , S i , Hi), Si-l

=

Gi(Ui, S i , Hi),

(17.2)

which relate the values of the radial tractions and displacements at the two faces of the ring i. Equations (17.2) include the effect of the inertia forces due to rotation and are not restricted to be linear in U i and Si ; i.e., they can be constructed assuming nonlinear material behavior and/or large displacements. Assuming the transverse thickness H i of the rings to be constant within each ring, the volume of the disk with n elements (rings) will be given by n

v,= C d i H i , i= 1

(17.3)

213

EXERCISES

where di = nei(Ri+ R i - l ) is the annular area between the radius Ri-l and R i. (See Fig. 8-4.) We can now formulate an optimization problem by requiring the volume of the disk to be a minimum for prescribed values of the radial tractions and displacements at the outer radius and possibly subject to additional constraints on Sior H i , or a combination of both. To this end we define f , ( U f l ,S,,) = minimum volume of the disk of n elements, subject to radial displacements U,, and (17.4) radial tractions S,, . By the principle of optimality, or otherwise, we can write f,,(U,,, S,,) = min[d,,Kl

+fn-1(Un-1,

Sn-1)1,

(17.5)

H,

or, on account of (17.2), (17.6)

a functional equation for the minimum volume function f,,(U,,, S,,), subject to the initial condition (17.7) fl(U, S ) = D(U,S ) , where D(U,S ) is the volume of a circular disk of uniform thickness, in terms of radial forces and radial displacements at the outer edge and subject to possible constraints on stresses or thickness. For example, if the material is isotropic and follows Hooke’s law, D is given by f l ( U , S ) = D(U, S ) = 4(1 - v)neI3S/(4EU- (1 - v)po2e13), (17.8)

where v is Poisson’s ratio, E is Young’s modulus, p is the density of the material, and w is the angular velocity of the disk. Other expressions for f l ( U , S ) may be found if different assumptions are made. Clearly, Eqs. (17.6) and (17.7) can be solved using the standard numerical procedures for two-dimensional dynamic programming. They are particularly valuable when discussing various nonlinear versions of the problem. When the problem is linear, i.e., when Eqs. (17.2) are linear in U iand S i , there is no need to implement a two-dimensional functional equation such as (17.6). In fact, the linearity of the problem allows the reduction of the two-dimensional equation (17.6) to a one-dimensional functional equation in terms of the displacements U i . See the paper by D. Nagy listed in the Bibliography. EXERCISES

1. Using the well-known formulas for thick-walled cylinders, derive the specific form for Eqs. (17.2) in the linear case. 2. Derive Eq. (17.8).

214

8

ELEMENTS OF OPTIMAL STRUCTURAL DESIGN

NOTES, COMMENTS, AND BIBLIOGRAPHY

1. Two main review articles on structural optimization are available: Z. Wasiutynski and A. Brandt, The Present State of Knowledge in the Field of Optimum Design of Structures, Appl. Mech. Rev. 16, No. 5 (1963), 341-350. C. Y. Sheu and W. Prager, Recent Developments in Optimal Structural Design, Appl. Mech. Rev. 21, No. 10 (1968).

See also W. Prager, Optimization of Structural Design, J. Optimization Theory Appl. 6, No. 1 (1970), 1-21.

2. In addition to some elementary principles in mechanics, the main analytical tools of the following sections are based on some notions on convexity and Lagrange multipliers. The reader who wishes to refresh his knowledge in this area may consult R. Bellman and S. Dreyfus, “Applied Dynamic Programming,” Chapter 11. Princeton Univ. Press, Princeton, New Jersey, 1962; M. R. Hestenes, “Calculus of Variations and Optimal Control Theory,” Chapter 1. Wiley, New York, 1966; D. G. Luenberger, “Introduction to Linear and Nonlinear Programming.” AddisonWesley, Reading, Massachusetts, 1973.

3. Equation (3.6) is a particular case of a more general result. In fact, adopting the work of deformation of the given loads on their points of application as a measure of compliance of the structure, it is possible to prove that among all the elastic structures with the same compliance, one with minimum volume is characterized by the constancy, over the whole body, of the specific strain energy. This, a necessary condition for optimality, was presented and extensively used by Wasiutyriski et al. in a series of articles, starting in 1939, on optimal structural design. See the references of this author in both review articles quoted above.

4. The use of minimum principles to construct sufficient conditions for optimality in structural design has been extensively employed by W. Prager et al. For a number of applications involving elastic structures, see W. Prager, Optimality Criteria Derived from Classical Extremum Principles, in “An Introduction to Structural Optimization” (M. Z. Cohn, ed.), Studies Series No. 1, pp. 165-178. Univ. of Waterloo, Waterloo, Canada, 1969; W. Prager and J. E. Taylor, Problems of Optimal Structural Design, Trans. A S M E (March 1968), 102-106; C. Y. Sheu and W. Prager, Minimum-Weight Design with Piecewise Constant Specific Stiffness, J. Optimization Theory Appl. 2, No. 3 (1968), 179-186;

NOTES, COMMENTS, AND BIBLIOGRAPHY

215

W. Prager and R. T. Shield, Optimal Design of Multi-Purpose Structures, Internat. J. Solids Structures 4 (1968), 469-475; C. Y. Sheu, Elastic Minimum-Weight Design for Specified Fundamental Frequency, Internat. J. Solids Structures 4 (1968), 953-958.

See also the survey paper by Prager in J.O.T.A. quoted above. 6-8. The present results are a combination of Prager’s method to establish sufficiency and the Kuhn-Tucker conditions of nonlinear programming. The reader interested in expanding his notions in the theory of multipliers may consult J. Abadie (ed.), “Nonlinear Programming.” North-Holland Publ., Amsterdam, 1967.

in addition to the original paper by H. W. Kuhn and A. W. Tucker, Nonlinear Programming, in Proc. Berkeley Symp. Math. Stat. andProb., 2nd, Berkeley (1951), 481-492.

9. We follow here N. Distefano and A. Rath, A Dynamic Programming Approach to the Optimization of Elastic Trusses, J. Optimization Theory Appl. (issue dedicated to W. Prager, to appear Dec. 1974).

For a treatment of the same problem using the Kuhn-Tucker conditions, including some thoroughly developed examples, see J. M. Chern and W. Prager, Minimum-Weight Design of Statically Determinate Trusses Subject to Multiple Constraints, Internat. J. Solids Structures 7 (1971), 931-940.

10. The subject of beam optimization is considerably expanded in Chapter 10. In this and some of the following sections we present some fundamental results involving mainly sandwich beams. They require some elementary notions on the variations of continuous functionals. The reader may consult Chapter 1 of I. M. Gelfand and S. V. Fomin, “ Calculus of Variations.” Prentice-Hall, Englewood Cliffs, New Jersey, 1963.

11-12. The results presented here were first reported in the paper by Prager and Taylor quoted above. 14. The results presented here were first given in the paper by Sheu and Prager quoted above. 15. See J. B. Martin, Optimal Design of Elastic Structures for Multipurpose Loading, J. Optimizu-

tion Theory Appl. 6, No. 1 (1970), 2 2 4 0 ; J. B. Martin, The Optimal Design of Beams and Frames with Compliance Constraints,

Internat. J. Solids Structures 7 (1971), 63-81.

216

8

ELEMENTS OF OPTIMAL STRUCTURAL DESIGN

16. This example was presented, for comparative purposes, in N. Distkfano and R. Todeschini, Invariant Imbedding and Optimum Beam Design with Displacement Constraints, Znternat. J . Solid Structures 8 (1972), 1073-1087.

17. These ideas were exposed by the author at the 1972 National Structural ASCE Meeting in Cleveland. The linear disk is thoroughly treated by discrete dynamic programming in D. A. Nagy, Some System Identification and Optimization Procedures in Structural Mechanics, Ph. D . Dissertation, Univ. of California, Berkeley, California (December 1971).

Miscellaneous References In this chapter we have not attempted to use any of the numerical methods of nonlinear programming. For those interested in such applications, the following references may be well suited: Symp. on Structural Optimization AGARD Conf. Proceeding No. 36 (1970);

An Introduction to Structural Optimization. Seven Special Lectures at the Univ. of Waterloo (1968) (M. Z. Cohn, ed.). Univ. of Waterloo, Waterloo, Canada, 1969; R. L. Fox, “ Optimization Methods for Engineering Design.” Addison-Wesley, Reading, Massachusetts, 1971.

We have not attempted a discussion on any aspect of the theory of layout. The reader will find some references in the survey paper by Prager in J.O.T.A. An interesting reading is also found in the book H. L. Cox, “The Design of Structures of Least Weight.” Pergamon, Oxford, 1965.

See also the articles P. Pedersen, On the Minimum Mass Layout of Trusses, in the AGARD Conf. Proc. quoted above; AGARD Conf. Pre-Print No. 123, Symp. Structural Optimization, 2nd (1973); A. C. Palmer and D. J. Sheppard, Optimizing the Shape of Pin-Jointed Structures, Proc. Znst. Civil Engrs 47 (1970), 363-376.

Chapter 9

Control Theory

1. Introduction

A first fundamental problem in the study of dynamical systems lies in the construction of suitable mathematical equations to simulate the evolution of the process. This, a descriptive stage in the study of the system, is associated with modeling and identification, two notions at the very basis of modern engineering science. A further major step is associated with the notion of control. The main ideas behind this concept are very simple and intuitive. We usually visualize a system with a block diagram such as that indicated below: INPUT

SYSTEM

OUTPUT 4

If, as time proceeds, the system does not perform satisfactorily, i.e., if the output does not fulfill our expectations in a certain sense, we then perform some modifications in the system to alter its behavior. When this is done, it is said that we exert control on the system. An ingenious way to exert control consists in using the observed output to determine the amount of control to be applied so as to correct the evolution of the system in a desired fashion. This, a feedback control scheme, is illustrated by the block diagram that follows. 217

218

9 CONTROL THEORY

INPUT

-~

OUTPUT v

SYSTEM

-L.

1' I

-

t

c l CONTROL

and it is called an open-loop control system as opposed to feedback control which is usually known as closed-loop control. This terminology is selfexplanatory. Conceptually and practically both control procedures are utterly different. Feedback control relies on the observation of the output and in fact provides the control as a function of the state of the system. As opposed to this concept, an open-loop control system relies on an a priori computation of the control which should then be provided as a function of time in order to be implemented. When the performance of the system is associated with the minimum of a certain quantity (functional), we talk of optimal control. The conceptual and theoretical framework of optimal feedback control processes lies in the theory of dynamic programming while the classical calculus of variations plays that role for optimal open-loop control processes. The duality of both mathematical theories underlies the duality between the two control processes. It is only when we extend the notion of control to stochastic systems that the duality between feedback and open-loop control ceases to hold. The reason for this is that in the stochastic case, as opposed to the deterministic one, we can generate not only two, but a family of control processes according to the manner in which the states are observed. We do not discuss this important topic here. The reader may consult some of the books given as references for Section 10. In this chapter we present some results in optimal control theory that will be required in the applications throughout the remainder of the book, The underlying theory is dynamic programming and the systems to be studied are characterized by dynamical equations in terms of differential (or difference)

219

2. DYNAMICAL EQUATIONS

equations. In the first part of the chapter (Sections 5-13), we refer constantly to dynamical systems evolving in real time and associated with initial conditions. This is preferable from a methodological point of view. In the remaining sections of the chapter, however, we enlarge the notion of dynamical systems to encompass a number of processes that do not occur in real time and that are generally associated with boundary conditions. This permits us to introduce design and optimal design as dynamical processes evolving in a fictitious time. A number of examples from the theory of viscoelasticity, structural design, etc., are presented to illustrate the points of theory. 2. Dynamical Equations

The equations employed to describe the evolution of a dynamical process are usually called the dynamical equations of the process. They can be written in terms of differential, integral, and integrodifferential equations, and so on. Differential models are frequently encountered. If the vector

denotes the state of the system at time t, the generally nonlinear ordinary differential equation du/dt = h ( ~ ) , ~ ( 0 = ) C, (2.2) where h is a vector function and c is the initial condition, describes the dynamics of the system. Equation (2.2) is implicitly stating that the rate of change in the state of the system depends on the current value of the state and not on the past history of the process. In other instances we are interested in exhibiting explicitly the dependence of the current state of the system with the past history of the process. This leads naturally to dynamical equations in terms of Volterra integral equations such as

z(t) =

f k(z(t),t

-7)

dz,

-m

where z(t), the state of the system, is an M-dimensional vector and k is a generally nonlinear vector function. Or, we may consider integrodifferencedifferential models of the type dz/dt = z(t - tl)

+

It

-m

where t , is a discrete time lag, and so on.

k ( z ( t ) ,t - t) dt,

(2.4)

220

9

CONTROL THEORY

It is clear that some functional equations express more eloquently than others some special features of the system we are interested in stressing. However, we must be very careful in associating those features with intrinsic properties of the real system we are trying to describe. For example, the notion of hereditary processes, so indissolubly attached to Volterra integral equations, may be equally well expressed by means of systems of ordinary differential equations at the expense of a convenient increase in the‘dimension of the state vector. (See the exercises below.) Similar considerations are valid in connection with other classical processes such as diffusion, transport, and propagation of disturbances. In other words, a physical process may be alternatively represented by utterly different functional equations. Thus, preference for one among various alternative dynamical equations should be found on analytical or computational grounds rather than by invoking apparent qualities of structure. EXERCISES 1. The “dynamics” of a linear mechanical oscillator with variable mass m starting from a configuration at rest at t = 0 is given by

+

d/dt(mdu/dt) ku = f ( t ) ,

u(0) = u’(0) = 0,

(a)

where u is the displacement, k is the spring constant, and f is the external excitation. Show that (a) is equivalent to the first-order system duldt = vlm, dvldt =f - ku,

u(0) = 0,

~(0= ) 0,

where v = m duldt is the momentum. 2. The stress-strain relationship u - E of a linear viscoelastic material in uniaxial tension is given by

where E is Young’s modulus and f ( t ) is a function that depends on the viscous properties of the material. Assuming that f ( t ) is given by the sum of exponentials N

f(d = 2 CI exp(X1t ) , 1= 1

show that (c) reduces to the “equation of state” N E=

ulE+ C c i z i , I= 1

where the z!’s are auxiliary (hidden) variables that satisfy the dynamical system dzl/dt=X,zl

+ u,

zf(0)=O,

i= l,2,

..., N.

(f)

22 1

4. CONSTRAINTS

3. Control Variables. Objective Functionals In order to exert control, instead of the dynamical equation (2.2), i.e., du/dt = h(u),

u(0) = C,

(3.1)

we consider the more general family

where v , a vector of dimension K , is called the control vector. Clearly, we require that g(u, 0) = h(u); i.e., in the absence of control influences we recover the original system (3.1). This, an imbedding procedure, must be implemented with care, based on the observation and knowledge of the system to be controlled. This aspect of the problem is far beyond the purpose of this chapter and is not discussed here. As we see in the next sections, there are generally a number of explicit requirements (constraints) that a control vector v must fulfill. These requirements, generally dictated by engineering considerations, make up the family of admissible controls. For each control vector in this family, Eq. (3.2) provides an associated admissible trajectory. In order to choose a control vector, we introduce a positive quantity J, the cost of the process, in the form of the functional J = s ’ l ( u , v ) dt 0

+ @(u(T),T),

(3.3)

where both l and @ are scalar functions of the corresponding vector quantities. The terminal time Tmay be specified in advance or its determination may be part of the problem. The quantity J , representing the cost of the process, is generally known as the objective or criterion functional. Our purpose is to find an admissible control vector that minimizes the criterion functional (3.3). But before we formulate the problem in more precise terms, let us first discuss the principal types of constraints met in practice. This is done in the next section. 4. Constraints A natural way to introduce a constraint in a control problem is by limiting the amount of resources. This generally leads to global constraints of the type (4.1)

222

9 CONTROL THEORY

where q and Q are specified scalar or vector quantities. We can also think of terminal constraints such as W T ) , T = 0,

(4.2)

where $ may also be a scalar or vector function. In addition to (4.1) and (4.2), the following are constraints of a local nature: For example, S(u(t), t ) 50,

0 I t I T,

(4.3)

G(u(t), t ) I0,

0 I t IT,

(4.4)

0 It IT.

(4.5)

or or H(u(r), u(t), t ) 5 0,

5. A Control Problem To begin, we formulate a relatively simple control problem, namely: Find the control vector u such that the criterion functional given by T

J= 0

l(u, u) dt

+ Cg(u(T),T),

where u and u are related by the dynamical equations du/dt = g(u, u),

u(0) = c,

(5.2)

is a minimum, and such that the terminal constraint +(u(T), T ) = 0

(5.3)

is satisfied. Here we assume that u, u, and $ are N, K, and L I N dimensional, respectively. We observe that the present control problem includes as a particular case, and generalizes in several directions, the simplest problem in the calculus of variations treated in Chapter 6. In the next section we use dynamic programming to construct a fundamental partial differential equation associated with the present control problem. 6. A Fundamental Equation

We define the minimum value function

223

6 . A FUNDAMENTAL EQUATION

Proceeding as usual, we consider

S,

[J)t+d' +

T

min

=

u(C)

C E [I,

min U(C)

C E [t,

TI

min u(C)

s,:~],

(6.2)

CE[t+A?,Tl

?+At]

a decomposition that follows from the additive properties of integrals and from an elementary property of the minimum function. Recalling that by definition, i.e., by (6.1), we have f(u(t

T

+ At), t + tA) =

min o(C)

< E [t+At,

l(u, 0) d t

[lt

TI

+ A?

+ @(u(T),TI]

(6.3)

application of (6.2) in (6.1) yields

[i

t+Ai

f (u, r) =

min u(€)

Z(u, u) dc

1

+f (u(t + At), r + At) .

t E [t, t + A i l

(6.4)

We now make the approximations

S,

f+At

Z(u, u) d< = Z(u(t), v(t))At

+ o(At)

(6.5)

and

u(t + At) = ~ ( t +) u'(t)At + o(At), which substituted into (6.4) yield

f (u, r)

= min[Z(u, v

u)At + f ( u + u'At

+ o(At), t + At)] + o(Ar),

(6.6) (6.7)

where u and u are now evaluated at t . Note that disregarding the o(At) terms, Eq. (6.7) furnishes a discrete, approximate equation for the quantity f ( u , t). Now, if the second derivatives a2fl(aui au,), where u i ,i = 1,2, . . . , N , are the components of vector u, exist and are bounded, the limit of (6.7), as At -+ 0, is given by -ft(u, t ) = mi"u, 0) + (fu dl, (6.8) 9

U

where f, stands for aflat and f, for grad f, i.e.,

Equation (6.8), usually known as the Bellman-Hamilton-Jacobi equation ofthe process, is a quasilinear partial differential equation for the optimum value function f subject to the initial condition (6.10) f (W),T ) = @(u(T), TI, where u(T) is constrained by the terminal condition given by (5.3). Clearly, (6.10) follows from consideration of (6. l), by making t = T .

224

9

CONTROL THEORY

7. The Optimality Condition The optimal control is the vector u* that minimizes in (6.8), i.e., u* = arg min[I(u, u) v

+ (f,,

g)].

If we use the optimal control u* in (6.8), the minimization operation may be dropped, yielding 2) = &u,

-ft(u,

+ (f

*)

,s),

(7.2)

a differential equation subject to f(u,

r>= @ ( K TI,

(7.3)

and additional terminal constraint (5.3). By the way in which the fundamental equation (6.8) was constructed, it is clear that (7.1)-(7.3) are necessary and sufficient conditions for optimality. We leave the proof of this property, which is straightforward, as an exercise to the reader. When functions 1 and g are continuously differentiable in ti, and no local constraints are imposed on v , then a necessary condition for a minimum in (7.1) is given by the K conditions

Lj+ ( L ,g,,>

= 0,

j

=

1 , 2 , . . ., K .

(7.4)

EXERCISES 1. Let min jtTh(u,u) dt = f ( u , u', t ) ,

"

where u and u are related by means of the dynamical equation U" = g(u,

u', u, I ) ,

u(0) = C,

~ ' ( 0 )= d.

Show that fsatisfies --ft

= min[h

"

+f.u' + gf.,],

f ( u , u', T ) = 0.

2. Derive the pertinent optimality condition for the previous exercise.

8. Optimal Loading of a Creeping Bar Axially Stretched In order to show the application of the partial differential equation derived in the last section, we present an example involving the optimal loading history of a metallic bar undergoing nonlinear creep deformations.

8. OPTIMAL LOADING OF A CREEPING BAR AXIALLY STRETCHED

225

Suppose that a metallic bar at constant temperature is subject to an axial stress a(t) and undergoes creep strains p ( t ) . We introduce a dissipation function

where an overdot indicates the derivative, as the work of the stress along the creep strains. The creep rate is assumed to follow a strain-hardening law given by

ti = g(P,0). It is required to find a loading history a([), t

P(0)= 0,

E

(8.2) (0, T ) , for which

(8.3)

p m = c,

and which minimizes the dissipation function (8.1). We introduce the minimum value function f ( p , t ) = minJ:aj d

dr,

t

< T,

(8.4)

representing the minimum value of the dissipation function in a process of duration t . Note that for convenience we have performed the imbedding in (8.4) using the upper limit of the integral rather than the lower one used in the last section. If due account is taken of this fact, we may apply the method outlined in the previous section and obtain f d P , 0 = min(ad - f p ( P , t ) i ) ) 0

= min(o

- f p ( P , t ) ) g ( p , a),

(8.5)

d

a partial differential equation for the minimum dissipation function. For small t , i.e., t 5 A 4 1, we may assume p ( t ) E (c/A)t, which, substituted into (8.4), furnishes A

f(c,4 z

0

ag(cl/A,

44

(8.6)

an approximate initial condition forf(c, t ) , where a is given by the solution of the equation c/A = g(c/A, a). We could employ now a number of numerical procedures to solve the initial-value problem (8.5)-(8.6). However, we do not pursue this path here, in favor of an analytical treatment of the problem. To this end we shall asume that the strain-hardening law (8.2) is of the type ppa = a*,

(8.7)

226

9 CONTROL THEORY

where a 2 0 and m > 0 are two constants of the material. Using (8.7) in (8.5) we obtain f t ( p , t ) = p - " min[a"+' -f p ( p ,t)o"]. (8.8) 0

The optimality condition follows by taking derivatives in (8.8), namely,

m

a=-f,. m+l

Substituting (8.9) into (8.8), this last equation reduces to f :-

(")

m+l

-p-"' m m + l

fa+'

(8.10)

Equation (8.10) may be solved by separation of variables. In fact, making f(Pt t> = r(t)@(p),

(8.11)

and substituting into (8. lo), we obtain (8.12)

where 0'stands for &/dp and i

= drldt. By

requiring that

@ = p-"[@']m+l,

(8.13)

a condition that is satisfied by taking (8.14)

Eq. (8.12) reduces to (8.15)

Integrating (8.15) we obtain -(m

+l)/m (8.16)

Using (8.14) and (8.16) in (8.1 1) we finally obtain

(8.17) the desired minimum dissipation function. Using (8.9) and (8.17) we may evaluate the optimal loading policy, given by (8.18)

9. LINEAR DYNAMICS AND QUADRATIC CRITERION

227

a function that substituted into (8.7) yields (8.19) a differential equation for the optimal creep strain. Integration of (8.19) using conditions (8.3) furnishes Popt

= c(tlr)

(rn+l)/(rn+l +u)

(8.20)

EXERCISES 1. If, instead of (8.2), we consider the more general equation of state

P = d p , q,

4, 4 = 0,q, 4,

(a)

in terms of a “ hidden” variable q, show that the minimum dissipation function

9. Linear Dynamics and Quadratic Criterion

In general, the Bellman-Hamilton-Jacobi equation of the process given by Eq. (6.8) cannot be explicitly integrated. Thus, one sees the interest in finding classes of problems for which the special structure of the equations permits a further advancement in the analytical treatment, before numerical methods are applied. The most important class of this type is that for which the criterion is a quadratic functional and the dynamical equations are linear. Problems in this class are important on their own, as is seen in the forthcoming sections, but also because they are a natural step in the solution of more general nonlinear processes via the method of successive approximations. We consider here the following case: Minimize J given by the quadratic functional

where u and u are both vectors of dimension N related by the linear dynamical equations dU/dt = CU

+ Du,

~ ( 0=) C.

(9.2)

228

9

CONTROL THEORY

In (9.1) and (9.2), A , B, C, and D are N x N generally time-dependent matrices. It is assumed that A 2 0 and B > 0. Introducing the minimum value function ~ ( ut ), = min u

41

T

[ ( ~ uu) ,

r

+ (BU,V>I dt,

(9.3)

and following the method outlined in Section 6 , we derive the partial differential equation

-f,= min[*(du, u) + +(Bv, v) + (fu, Cu + Dv)],

(9.4)

u

subject to the initial condition at t

= T given

by

f ( u ( T ) , 71= 0.

(9.5)

In view of the structure of the problem we seek a solution to (9.4) and (9.5) of the form

f ( u , t ) = +(u, Ru),

R(t) 2 0 ,

(9.6)

i.e., a quadratic, nonnegative-definite form, for the minimum value functionf. Substituting (9.6) into (9.4) -+(u, R'u) = min[+(Au, u) u

+ +(Bv, v) + (Ru, Cu + Dv)].

(9.7)

If no further constraints on u or u are imposed, we can minimize in (9.7) by taking derivatives, namely, BV

+ DTRu = 0,

(9.8)

where, as usual, the superscript T indicates transpose. From (9.8) we obtain u=-B-~D~RU.

(9.9)

Making use of (9.8) and (9.9) to eliminate v in (9.7) we arrive at

- t ( ~R'u) , = *(Au, u) + $(u,RCU)+ +(CTRu,U) - +(RDB-'DTRu, u), (9.10) an equation which must hold for all u. Therefore R'

=

- A - RC - C T R +- RTR,

(9.1 1)

a Riccati differential equation subject to R ( T ) = 0,

(9.12)

r = DB-ID=.

(9.13)

where

229

10. SMOOTHING, FILTERING, AND PREDICTION

The optimal control follows from (9.9) where the matrix R is given by the backward solution of (9.11)-(9.12). The optimal trajectories may be evaluated by forward integration of (9.2), where v has been substituted by (9.9), i.e., du/dt = (C - TR)u,

~ ( 0= ) C.

(9.14)

EXERCISES 1. Instead of (9.3) consider

where F is a nonnegative-definite matrix. Show that

v = -B-'D'Ru, where R is given by (9.11) subject to R(T) = F, 2. Instead of (9.2) consider the more general dynamics

+ DV + p ,

duldt = CU

~(0) = C,

where p is a given N-dimensional vector. Set f(u, t ) = m u , u)

+ (r, u) + s.

(el

Find the corresponding equations for matrix R, vector r , and scalar s. 3. Assuming that v is a vector of dimension M # Nand B, D , are matrices of the necessary order, rederive the equations of Section 9.

4. Show that under the assumptions made for the matrices A, B, C, and D, the Riccati equations (9.1 1)-(9.12)are numerically stable. 5. Assuming that A, B, C,and D are constant matrices, are the conditions of this section enough to ensure an asymptotic control as T + m ? In other words, does a positive-

definite solution of the matrix Riccati equation A

+ R , C + CTR, - R , DB-'D'R,

=0

(f)

exist?

10. Smoothing, Filtering, and Prediction

As an application of the method developed in the preceding section for linear problems with quadratic criterion, we present here a linear deterministic version of a filtering problem. Suppose that noisy observations are made on a physical system whose evolution is sought to be described by a certain functional equation, during a

230

9 CONTROL THEORY

period of duration T. Given the observations in [0, TI and the dynamical equations of the process, we wish (a) to determine the best (in a sense made precise later) estimate of the state at time t ; (b) to upgrade this estimate as more observations become available. If t < T, the problem is a smoothing or interpolation problem; if t = T, it is a filtering problem; and if t > T, we talk of a prediction or extrapolation problem. Clearly, these are three aspects of the same problem, to be studied in a unified manner. The mathematical model constructed to solve this problem is called a sequential estimator. Only the scalar version of the problem is presented here. The matrix-vector extension is straightforward and is left as an exercise to the reader. The process to be estimated is assumed to be described by the linear equation du/dt = a(t)u

+ p ( t ) + v,

(10.1)

where u is the state, p is a known forcing function, a(t) is a given function, and v , the dynamical error, is an unknown function, assumed small in norm, which accounts for imperfections in the model, errors in the forcing term, small external disturbances, and so on. The observations of the process are given in the form w(t> = bu(t) + ?(t>,

(10.2)

where b is a known coefficient and q ( t ) is the observational error. To any estimate u in the interval [0, TI,we associate a cost functional given by T

J

=

4 0 [k(t)(w- b

+

~ I(t)v2] ) ~ dt

+ +m(u(O)- c)',

(10.3)

where u and v are related by the dynamical equation (lO.l), k and I are two given positive functions, m is a given positive number, and c is an estimate of the initial condition. The functional J represents a measure of the merits of any given estimate. The smaller it is, the better the estimate is. Our objective is to choose the dynamical error v and the state u which minimize J given by (10.3). This is clearly an optimal control problem of the type treated in the last section. We define the minimum value function f ( u , )'2 = min J. U

(10.4)

23 1

10. SMOOTHING, FILTERING, AND PREDICTION

Application of the principle of optimality in the form

f (u, T ) = min +

JLI

(10.5)

leads to the partial differential equation fT(u, T ) = min[-fk(w - bu)’ v

+ +lu’

-f.(u, T)(au + p

+ u)],

(10.6)

subject to the initial condition f ( u , 0) = +m(u - c)’.

(10.7)

The optimal control u is obtained by minimizing in (10.6), i.e., = (1IOLI3

(10.8)

a quantity that substituted back into (10.6) yields ST(^, T ) = 4k(w - bu)’

-f.(~, T)(au + P ) - (1/20LI2(~,T),

(10.9)

an equation whose solution is sought in the form

f (4r ) = t r ( m ’ + 4 T ) U + q(73.

(10.10)

Substitution of (10.10) into (10.9) and (10.7) yields, after collecting terms in powers of u,

dr/dT= kb’ - 2ar - (I/l)r’, ds/dT = - ( r / l + a)s - pr - kwb,

r(0) = m, s(0) = mc,

dq/dT = +kw2 - (1 121)s’ - PS,

4(0) = 3mc2,

(10.11)

a system of Riccati equations subject to initial conditions. The optimal control follows from (10.8) and (lO.lO), i.e.,

+

u = (I/Z)(ru s).

(10.12)

To evaluate the optimal filter, i.e., the best estimate of u(t) at t = we proceed as follows. Since our original minimization is not constrained at t = T, we must determine u ( T ) so as to minimize the cost function. This leads to

u(T) = arg minf(u, T ) , U

which, on account of (lO.lO), yields

an expression for the optimal filter in terms of s and r

( 10.13)

232

CONTROL THEORY

9

We can derive an alternative system of differential equations for the optimal filter. To this end we construct a differential equation for u(T)using the expression (10.14). Taking derivatives in (10.14) and eliminating the resulting quantities r' and s' using (lO.ll), we arrive at

duldT = au + p

+ Fkb(bu - w),

u(0)

= c,

(10.15)

where 7 = l/r. Equation (10.15) shows that the optimal filter satisfies the original dynamical equation (10.1) driven by a feedback term proportional to the difference bu - w. The quantity f = l/r, where r is given by integration of the first equation of (10.1 l), may be obtained as a solution of the Riccati equation di-/dT= 111+ 2aT - kb2F2, 7(O) = l/m, (10.16) as the reader may easily verify by direct substitution in (10.11). This observation is important in the multidimensional case when r is a matrix, not a scalar as in the present case, because it avoids the inversion of matrix r. The reader familiar with statistical estimation theory will recognize (10.15) and (10.16) as the dual (deterministic) formulation of the KalmanBucy filter. When the observations are interrupted at a time T = TI, we make k(T) = 0, T 2 T I , in (10.15) and continue the integration. This provides the optimal extrapolation. The treatment of the optimal interpolator is deferred until Chapter 12. EXERCISE

Discuss the multidimensional version of the filtering problem.

11. Euler-Lagrange Equations In this section we show how to transform the fundamental partial differential equation of the process, Eq. (6.8), subject to initial conditions (6. lo), into a set of generally nonlinear ordinary differential equations subject to boundary conditions. The aim of this reduction is twofold. In the first place in so proceeding we make connection with the classical treatment of the present class of control problems and, in particular, with Pontryagin's extremum principles. In the second place, our scope is to offer an alternative route, based on the necessary conditions, for the numerical solution of the fundamental partial differential equation. We consider here the partial differential equation -ft(u, t ) = 1(u, 0) + (f.(u, 0,g(u, O N ,

(11.1)

233

11. EULER-LAGRANGE EQUATIONS

subject to the initial condition

f ( u , r> = @(u,TI,

(11.2)

*(% T> = 0.

(11.3)

and the terminal constraint The control vector in (1 1.1) satisfies the optimality condition 1,

+

(11.4)

= 0.

( f , , d U

Here we assume, as usual, that u, u, and $ are N , K , and L I N dimensional, respectively. We wish now to construct a differential equation for the vectorf,. To this end we compute the total derivative off, along an optimal trajectory, i.e.,

(44-6=ft, + L u g 3

(11.5)

where f,, is the N x N Jacobian matrix

f,, = (a2f/a,, a,,),

i, j

=

1, 2, . .. , N .

(11.6)

Differentiating (1 1.1) with respect to u we obtain -fu, =

1,

+flus

+S u f , +

UUIlU

+

(fUY

d”1,

(11.7)

where guand u, are the Jacobian matrices 9,= (dgilau,),

i,j

1,2, . .., N ,

(11.8)

and i = 1, 2,

u, = (aui/dUj),

. . . ,k ,

j = 1, 2,

. . .,N ,

(11.9)

respectively. Recalling (1 1.4), Eq. (1 1.7) reduces to -fut

=I, +flus

+Sufu,

(11.10)

a characteristic equation along an optimal trajectory. Eliminating f,,=ft, between (1 1.5) and (1 1.10) we finally obtain (d/dt)S, =

-9,L

(1 1.11)

- 1,s

the desired ordinary differential equation for the quantity f,. Differentiating (1 1.2) with respect to u, we find -6(4 = cD,(u, TI,

(11.12)

an end condition for& where u(T) is constrained to satisfy (11.3). Customarily, a constraint such as (1 1.3) is adjoined in the form fu(u, T ) = @,(%

T> +

*Id

v,

(11.13)

where v is a multiplier vector to be determined by requirements of optimality.

234

9 CONTROL THEORY

Summarizing, the original dynamical equation of the process, Eq. (5.2), plus the auxiliary differential equations (1 1.1l), (1 1.13), and the optimality condition (1 1.4) constitute a coupled, nonlinear system of ordinary differential equations subject to two-point-value conditions. By the way this system was constructed, it provides a necessary, although, in general, not sufficient condition for the solution of our original control problem. In the next section we recast this system in a more convenient form and discuss some aspects of its application to the effective solution of control problems. EXERCISE

Instead of (5.2) consider the nonautonomous system duldt = g(u, U, t ) , u(0) = 0, (a) and derive the necessary conditions. Hint: Reduce (a) to an autonomous system by introducing the ( N 1)th-state variable uN+ = t and associated dynamical equation d++i/dt = 1.

+

12. The Minimum Principle

In order to exhibit their basic features, the necessary conditions derived in the preceding section may be written conveniently in the form du/dt = g(u, u), dA/dt = - HJu, v),

u(0) = c,

A(T) = Q(u(T),T ) + $" V , where the N-dimensional vector l ( t ) is given by

(12.1)

(12.2) (12.3) In the terminology of the calculus of variations, A is a multiplier vector and H is the Hamiltonian of the process. The optimal control vector u appearing in (12.1) is implicitly given by the optimality condition (11.4), which, in the present notation, reads

H, = 0,

(12.4)

or by the more general equation u = arg min H.

(12.5) When H is continuously differentiable, (12.4) follows from (12.5) by differentiation. In general the presence of constraints on u, or v, or both u and u, prevents the use of (12.4), and therefore the minimization indicated in (12.5) must be performed numerically or by using methods other than differential calculus.

235

EXERCISES

Equations (12.1) and (12.9, a system of 2N generally nonlinear ordinary differential equations subject to boundary conditions, are a particular version of Pontryagin's minimum principle for autonomous systems. They have been derived here under the assumption of differentiability of H . For a proof that releases this assumption, the reader is urged to consult the Bibliography at the end of the chapter. We must observe that in the course of reducing Eqs. (6.8) and (6.10) to the ordinary system (12.1) and (12.5), we have lost one of the most conspicuous and significant properties of the fundamental partial differential equation of the process, namely, its character of being both a necessary and sufficient condition for optimality. In fact, (12.1) and (12.5) are necessary but, generally, not sufficient conditions. In addition they are boundary valued, a fact that prevents the use of routine methods of integration. The interest in the necessary conditions becomes apparent in connection with quasilinearization and invariant imbedding in the forthcoming chapters. EXERCISES

1. Consider the following control problem: min f v

joT((Cu,u ) + (Du,

u)) dr,

(a)

where the N-dimensional vectors u and u are related by the dynamical equations duldt = Au

+ Bu,

~ ( 0= ) c.

(b)

Show that the necessary conditions may be written in terms of the linear two-pohtvalue problem duldt = AU - BD-'BTh, dhldt= -CU - ATh,

~ ( 0=) C,

h(T)= 0.

(C)

2. Show that the optimal control u in Eqs. (a) and (b) is given in terms of the costate h by means of u=-D-'B~.

(4

3. Derive the necessary conditions for the optimal filter studied in Section 10.

4. The simplest problem in the calculus of variations studied in Chapter 6 may be reformulated as follows :

min J:g(u,

0,

x) d

~,

V

where u and u are related by the dynamical equation

Derive the necessary conditions using the minimum principle.

236

9 CONTROL THEORY

13. Optimal Loading Policy for a Voigt Model

We further illustrate here the use of dynamic programming and the minimum principle by solving some optimization problems in the theory of viscoelasticity. We consider a Voigt model, i.e., one for which the stress (a)-strain ( E ) relationship is given by the linear first-order differential equation d

+ (1/T)E = (l/q)a,

(13.1)

where T, the retardation time, and q-, the viscosity coefficient, are positive ) 0, it is required to find the strain constants of the material. Assuming ~ ( 0 = rate history i(t), t E [0, TI, for which the work functional W=

I

T

ad dt

(13.2)

0

is a minimum, subject to the elongation constraint E(T)= e T . This problem is equivalent to T (0’

minJo

+ (~/T)Eu) dt,

(13.3)

c

where &(O) = 0,

d = 0,

& ( T )= & T ,

(13.4)

as the reader may verify easily. We apply here the minimum principle and form the Hamiltonian

H

= U’

+ (l/T)&U+ h,

(13.5)

where 2 is a multiplier that satisfies the differential equation

I

=

- H E = -(l/ T)V.

(13.6)

Comparison of (13.4) and (13.6) yields 1 = - (l/z)B, or, after integration,

A = -(l/T)& + K ,

(1 3.7)

where K is a constant to be determined. The optimal strain rate v follows from the optimality condition vopt = arg min H

=

-+(A

+ (I/T)E),

(13.8)

an equation that combined with (13.7) yields vOpt= -3K.

(13.9)

231

14. OPTIMAL DESIGN AS A CONTROL PROCESS

Integrating (13.4), where u is given by (1 3.9), we finally obtain E

( 13.1 0)

= (t/T)ET.

The optimal rate I: is thus a constant equal to eJT. Compare Eq. (13.10) with (8.20). EXERCISES 1. Show that (13.4). (13.6), and (13.8) are sufficient conditions for optimality.

2. Show that min W = q(l/T+ &(1/7))cT2.

(a)

3. Solve the present problem by dynamic programming.

14. Optimal Design as a Control Process

In the problems formulated so far, for a given admissible control vector u, the dynamical equations uniquely ’determine a trajectory u by initial

conditions specified at a certain point. This is most typical of time-evolutionary systems. However, in a number of important applications, the dynamical equations describing the process are determined by boundary conditions. This prevents in general the construction of the functional equation of dynamic programming in terms of the usual imbedding, i.e., in terms of the data vector, because there is not a single point where the state is completely known. This situation forces us to enlarge the imbedding family so as to include the missing components of the state, assuming that they are known. When this family of problems has been solved, we recover the original problem by requiring that the “ missing” components of the state at the origin, say, be chosen so as to optimize the minimum value function of the auxiliary problem. In order to illustrate how these ideas can be applied, we consider here an example involving a nonlinear string held fixed at both ends x = x l and x = x 2 , whose deflection u(x) satisfies the nonlinear differential equation and boundary conditions

u”(4

= g(u, v),

4%)= a,,

4x2)

= a2,

(14.1)

where a1 and a, are the deflections imposed at xi and x 2 , respectively, and where u(x) is a design variable to be chosen so as to minimize a criterion functional of the form x2

J = f h(u, U) dx. XI

(14.2)

238

9

CONTROL THEORY

It is assumed that for every admissible design u(x), Eq. (14.1) admits a solution which is unique. Now, in order to apply the usual imbedding procedure of dynamic programming, we should introduce the minimum value function x?.

. f ( a l , a 2 ,xl)

= min v

J,, M u , 4 dx,

(14.3)

in terms of the data vector (al, a2). Although we may be able to prove the existence and uniqueness of function f ( a , , u 2 , xl) given by (14.3), it is not apparent how to construct a functional equation for this quantity that results in a useful computational device. For this reason, instead of (14.1) and (14.2) we shall consider the following larger family of problems: Minimize the functional

J

u) dx,

= f?(w,

(14.4)

where w and u are related by the dynamical equation w" =

m,4,

(14.5)

subject to boundary conditions W(X1) = a,,

4x2)

= a2,

w ' ( x , ) = c,

(14.6)

where c is an imbedding variable. We now define (14.7)

s ( a l ,c, x, u2) =

where w and u are related by (14.5)-(14.6). The principle of optimality now readily furnishes

-Jf/dx

= min(h(a,, u) V

+ (d]/8u,)c + (df/dc)g(al,u)),

(14.8)

a partial differential equation for the auxiliary minimum value function f, subject to the initial condition (14.9) We can see now that the required minimum value function f(a,, u 2 , x,) defined by (14.3) is a member of the larger familyj(u,, c, x, az) and, in particular,

f(%

a2

9

x1)

= m i n m , c, x1,az). C

(14.10)

239

16. THE NECESSARY CONDITIONS

15. Discussion It is clear from the preceding analysis, that in order to solve, via dynamic programming, control problems with boundary-valued differential constraints, we need to solve the larger class of problems resulting from incorporating the missing boundary conditions at one end, as known variables of the problem. The original problem is subsequently recovered as a particular element of the larger family by requiring the minimum value function of the auxiliary problem to be a minimum with respect to all possible values of the missing boundary conditions. This, an imbedding procedure, is repeatedly employed in Chapters 10 and 11 in connection with structural optimization problems. 16. The Necessary Conditions

It is of interest to derive the necessary conditions of the process defined in the last section. We apply the method outlined in Section 11. To this end it is convenient to rewrite (14.8) in the form -fx

= h(w,

4

w' + f w d w ,

+fW

4

(16.1)

wheref =f(w, w',x , az)and where u, the optimal design, satisfies the optimality condition

h, + f w t g , = 0.

(16.2)

Taking partials with respect to w and w' in (16.1) we obtain the characteristic equations -fwx -fW,,

=hw = h,

+ h" dv/dw + f w w w' + .fww,g+ .fw,(s,+ S" dvldw), (16.3) dv/dw' W' .fw fwjW*g fw,guduldw',

+

fWtw

+ +

+

which, along an optimal trajectory, reduce to -fwx -.fwpw

=hw =f w t w

+fwww' + j w t w g+fw,.gw,

w' + f w

+

fWf,.,.

(16.4)

g,

a system obtained using the optimality condition (16.2) in (16.3). Now we compute the total derivatives of the quantities f w and f w , , namely

(d/dx)fw= f x w +fwww' +fW#,S, (d/dx)f,, =f x w * +.fww, w' + fW*,* 9.

(16.5)

Substituting f x w and f x w , given by (16.4) into (16.5), we finally arrive at

dfwldx = - h w - TwfSw Y

dfwddx =

-fw

9

(16.6)

240

9 CONTROL THEORY

a system of ordinary differential equations for the multiplier functions f w andfwt. Recalling (14.10) we must have (16.7)

fw,(w(x,),w’(x,>,x i , a21 = 0,

and, since a, in the initial condition (14.9) is arbitrary, any variation in this quantity does not modify the value off evaluated at x2 . Therefore df/da, = 0, or (16.8) fwf,(w(xz), w’(x,), xz w ( x 2 ) ) = 0. 9

At this point we can interchange w for u. Introducing the notation

44 = f4

(16.9)

3

the necessary conditions are given by 24’‘

= g(u, u),

4x1) = a19

I” = h, + Ag,,

A(x,) = 0,

4 x 2 ) = a2 A’(xz)= 0,

9

( 16.10)

a system of ordinary differential equations coupled by the optimal design function u ( x ) given by Eq. (16.2), which in the present notation reads

h”

+ Ag” = 0,

(16.11)

or, more generally, by u = arg min(h

+ Ag).

(1 6.12)

The multiplier rule of the minimum principle emerges clear from consideration of Eqs. (16.10) and (16.12). NOTES, COMMENTS, A N D BIBLIOGRAPHY

I . Introductory to modern control theory is R. Bellman, “Introduction to the Mathematical Theory of Control Processes,” Vols. I and 11. Academic Press, New York, 1967 and 1971, respectively.

2-4. Very stimulating reading on modeling and control is found in R. Bellman, “Adaptive Control Processes : A Guided Tour.” Princeton Univ. Press, Princeton, New York, 1961.

5-7. The present method of treating problems in the calculus of variations, an extension of the procedure outlined in Chapter 6, was first announced by Bellman in a communication to the Nat. Acad. Sci., Vol. 39 (1953). See also R. Bellman, “Dynamic Programming.” Princeton Univ. Press, Princeton, New Jersey, 1957.

NOTES, COMMENTS, AND BIBLIOGRAPHY

24 1

The solution of the fundamental partial differential equation is a challenging research area. In addition to the books quoted above see P. Dyer and S. R. McReynolds, “The Computation and Theory of Optimal Control.” Academic Press, New York, 1970; R. Larson, “ State Increment Dynamic Programming.” American Elsevier, New York, 1968; D. H. Jacobson and D. Q. Mayne, “ Differential Dynamic Programming.” American Elsevier, New York, 1970.

8. Here we follow N. Distkfano, The Optimal Control of Some Viscoelastic Processes (to appear).

A treatment of constitutive equations for creep of metals may be found, with many applications, in Yu. N. Rabotnov, “Creep Problems in Structural Members.” North-Holland Publ., Amsterdam, 1969. 9. The reduction of linear quadratic control problems to the solution of Riccati equations was first given by R. Bellman, On a Class of Variational Problems, Quart. Appl. Math. 14 (1957).

For a detailed, rigorous treatment, see Volume I of the book by Bellman quoted above. 10. Filtering has occupied much of the modern literature in control theory. Known also asprediction and estimation theory, it was first formulated by Kolmogorov and Wiener independently in the early 1940s within the framework of the theory of random processes. See N. Wiener, “Time Series.” M.I.T. Press Paperback Ed., Cambridge, Massachusetts, 1966,

for the classical theory and historical remarks. This book contains a very clear appendix by Levinson, on Wiener’s filtering theory.

A significant reformulation of the filtering problem took place some 20 years later. In effect, in the early 1960s Kalman and Bucy, in addition to releasing some of the limitations of the classical theory (extension to finite interval of observation and consideration of generally nonstationary time series), reformulated the problem in terms of algorithms suitable for digital computation, allowing for a widespread use of the procedures and considerably extending the conceptual framework of the theory. For references, theory, and applications see the book R. S. Bucy and P. D. Joseph, “Filtering for Stochastic Processes with Applications to Guidance.” Wiley (Interscience), New York, 1968.

242

9 CONTROL THEORY

A conceptual framework of the Kalman-Bucy theory of filtering is the theory of dynamic programming, via the duality between estimation and control theory. This duality was first pointed out in R. E. Kalman and R. S. Bucy, New Results in Linear Filtering and Prediction Theory, Trans. ASME Ser. D, J. Basic Engrg. 83 (1961), 95-107.

The interest in the duality lies in the possibility of formulating an entirely deterministic and independent theory of filtering and prediction based on the results of control theory. In the present section we follow that line. For extensions to nonlinear filtering see R. Bellman, H. H. Kagiwada, R.Kalaba, and R. Sridhar, Invariant Imbedding and Nonlinear Filtering Theory, J. Astronaut. Sci.13, No. 3, (1966), 110-115; H. H. Kagiwada, R. Kalaba, A. Schumitzky, and R. Sridhar, Invariant Imbedding and Sequential Interpolating Filters for Nonlinear Processes,RAND Memorandum RM-5507PR. Santa Monica (Nov. 1967).

The connection between the classical Wiener-Kolmogorov theory and the Kalman-Bucy theory may be found in the theory of invariant imbedding and, in particular, in the duality between the Wiener-Hopf equation and a certain matrix Riccati differential equation. See A. Schumitzky, On the Equivalence between Matrix Riccati Equations and Fredholm Resolvents, J. Comput. Syst. Sci. 2 (1968). 76-87.

Additional readings in estimation and control theory may be found in the books M. Aoki, “Optimization of Stochastic Systems.” Academic Press, New York, 1967; K. J. Astrom, ‘‘ Introduction to Stochastic Control Theory.” Academic Press, New York, 1970.

Interest in the present ideas on practical problems in solid mechanics and structures becomes apparent in Chapter 12 in connection with identification and control. 11. The connections between dynamic programming and the classical calculus of variations have been studied in great detail in S. Dreyfus, “Dynamic Programming and the Calculus of Variations.” Academic

Press,

New York, 1965.

12. See L. S. Pontryagin, V. G. Bol’tanskii, R. S. Gamkrelidze, and E. F. Mishenko, “The Mathematical Theory of Optimal Processes.” Macmillan, New York, 1964.

For a treatment along classical lines, see M.R.Hestenes, “Calculus of Variations and Optimal Control Theory.” Wiley, New York, 1966.

NOTES, COMMENTS, AND BIBLIOGRAPHY

243

An elegant and very accessible treatment of the theory of fields and related applications to problems involving propagation of disturbances and optimal control theory may be found in I. M.Gelfand and S. V. Fomin, “ Calculus of Variations.” Prentice-Hall, Englewood Cliffs, New Jersey, 1963.

13. Here we follow the paper by Distkfano quoted in Section 8. The problem of determining optimal loading policies for viscoelastic bodies generalizesa problem that has recently attracted some interest in the literature. See S. Breuer, Lower Bounds on Work in Linear Viscoelasticity, Quart. Appl. Math. 27 (1969), 139-146; S. Breuer, The Minimizing Strain-Rate History and the Resulting Greatest Lower Bound on Work in Linear Viscoelasticity, Angew. Math. Mech. 49 (1969), 209-213;

A. R. S. Ponter, “The Derivation of Energy Theorems for Creep Constitutive Relationships, in “Creep in Structures” (J. Hult, ed.). Springer-Verlag,Berlin and New York, 1970.

14-16. Here we follow N. Distkfano, An Optimum Control Theory Approach to Structural Design (to appear).

Chapter I 0

Optimal Beam Design

1. Introduction

In this and the following chapter we present a unified methodology for the analysis of a class of optimal design problems. The main idea of these two chapters is to treat, within the same conceptual framework, the seemingly related problems of formulating the model, deriving the optimality conditions, and constructing suitable numerical algorithms. In so doing we are naturally enlarging the scope and results of Chapter 8. Here we treat elastic beams and in the next chapter rotating disks. In the first part of this chapter we present a dynamic programming version of the max-min problem for minimum-weight beam design that results from constraining the value of the potential energy. The thorough treatment of this specific problem should be taken more as an example of the use of dynamic programming procedures to a larger class of problems in structural optimization rather than as an isolated case of theoretical importance. Interest in a dynamic programming approach to structural optimization stems from both practical and theoretical considerations. Some of these aspects are already apparent to the reader. 244

2. MAX-MIN OPTIMIZATION PROBLEM

245

The problem is formulated in Section 2, while in Section 3 the basic dynamic programming equations are obtained. In Section 5 the classical EulerLagrange equations for the beam are derived from the fundamental BellmanHamilton-Jacobi equation. It is shown that the conditions of optimality associated with the minimum operation are expressions of the theorem of Castigliano, an expected result repeatedly obtained. An analytical solution for the optimum cantilever laying on elastic foundation is found in Section 4, while a method of successive approximations consisting in a stable, twosweep iterative procedure is developed in Section 6. In order to illustrate the theory, numerical examples are presented in Section 7. In the second part of this chapter, we use the calculus of variations in the fashion of Pontryagin’s minimum principle to derive optimality conditions for a beam under elastic foundation subject to a displacement constraint. Again, this problem is presented as representative of a larger class of problems in structural mechanics. Using ideas of invariant imbedding and the method of successive approximations, the pertinent nonlinear mixed boundary-value problem is reduced to a two-sweep iterative procedure in terms of a system of Riccati differential equations subject to initial values and exhibiting favorable numerical stability properties. Two examples are finally presented to illustrate the application and accuracy of the method. I. DYNAMIC PROGRAMMING 2. Max-Min Optimization Problem

We consider a beam of stiffness a(x) = EZ, where E is Young’s modulus and I is the moment of inertia, and length L, laying on a Winkler foundation k(x), subject to distributed forcesp(x) and a concentrated force P and moment M at the end x = L. The beam is subject to prescribed deformations, deflection w(O), and slope w’(O), at the origin x = 0. We restrict attention to beams of the sandwich type, so that the volume to be minimized is a quantity proportional to the integral jt a dx. The strain energy is given by the functional

where a prime indicates derivative. We define the potential energy V(w, a) = U(w, a) - W(w,a),

where

(2.2)

246

10 OPTIMAL BEAM DESIGN

is the virtual work of the distributed forces p ( x ) and the concentrated forces P and it4 at x = L, over kinematically admissible displacements. By the theorem of minimum potential energy, the function w* that minimizes V in (2.2) and satisfies the boundary conditions is the “true ” displacement function of the beam with a given design a(x). The value of the minimum is min V(w,a) = -+W(w*, a).

(2.4)

W

Clearly, for a given set of forcesp, P,and M , the virtual work W(w*, a) along the true displacement field can be used as a measure of compliance. Therefore, if our aim is to minimize the volume a dx for a fixed value of the compliance W(w*, a), we can formulate the minimization problem:

1

min[JoLa dx - 2p min(U(w, a) - W(w,a)) , a

W

(2.5)

a min-min problem in the deflection w and the design a, where p is a Lagrange multiplier used to incorporate the constraint. Alternatively, if for a given system of forces we wish to determine a design a for which the compliance W(w*,a) is a minimum among all possible designs with the same volume, we formulate the following problem :

or, on account of (2.2) and (2.4),

a noninterchangeable max-min problem in a and w, respectively. We assume to belong to A, the set of all admissible designs. Clearly, the two problems (2.5) and (2.6) are equivalent and L = l/p. In the following we shall use the formulation given by (2.6). 01

EXERCISE Formulate mathematically the following ma& min, problems: (a) maximum buckling load of an axially loaded prismatic bar for a specified volume; (b) largest minimum frequency of a beam for a specified volume. [In both problems assume the volume to be given by V = j!j g(a)dx, whereg is a specified function of the cross-sectionalstiffness a.]

3. Dynamic Programming Instead of a beam of length (0,L), we consider the family of beams of lengths (2,L), where 0 Iz I L, subject to prescribed displacements

w(x)I,=, = 421,

w‘(x)[,=, = 421,

(3.1)

247

3. DYNAMIC PROGRAMMING

at x = z. Recalling Eqs. (2.1), (2.3), and (2.6), we define the optimum value function f(u, u, z ) = max min a.zA

w

[+ (CLW"'+ kw2 L Jz

- 2pw - h)dx

1

- Pw(L) - Mw'(L) , (3.2)

where V ( Z ) = du(z)/dz.

(3.3)

To obtain a differential equation for the optimum value function, we proceed as usual and find the following functional equation

[

f ( u , u, z ) = max min 6 V - 1 5 dE C[z, ) zw(C) +Az]

Jzz+Az + f ( u ( z + Az), u dx

v(z

1

+ Az), z + Az)

,

(3.4)

where

$1

z+Az

6 ~ =

z

(awn2 + kw2 - 2pw) dx

(3.5)

is the potential energy of the slice of beam between z and z + Az. Equation (3.4) together with ( 3 . 9 , which may be easily justified on mechanical grounds, is a local expression of the principle of optimality. Introducing the curvature K

= d2w/dxzl x Z z ,

(3.6)

standard arguments in dynamic programming show that, as Az+O, Eqs. (3.4) and (3.5) yield

fz

= min

max[ - ( u K 2

a~

+ ku2 - 2pu - h ) / 2 -fu

2,

-f,K],

(3.7)

the Bellman-Hamilton-Jacobi equation for the optimum value function, where f,stands for aflaa, a = z, u, u, andf(u, u, z ) is subject to the initial condition f ( u , u, L) = Pu

+ Mu,

(3* 8)

an equation that follows directly from consideration of Eq. (3.2). Minimizing with respect to K in Eq. (3.7) we obtain K =

(3.9)

-fv/u,

that substituted back into Eq. (3.7) yields

fz

= minv;2/2a a

-fuv - ku2/2

+ p u + Au/2].

(3.10)

248

10 OPTIMAL BEAM DESIGN

If the design a is not subject to constraints, we can minimize in (3.10) by taking derivatives, obtaining in this fashion

f,’la’

(3.11)

=I,

an equation that together with (3.9) yields the optimality condition

I

(3.12)

= K2.

Combining Eqs. (3.10) and (3.1 1) we obtain

f, + uf, + ku2/2 - PU

=f:/a,

(3.13)

=A f:,

(3.14)

or, eliminating a using Eq. (3.1 l),

+ ~f.+ ku2/2 - PU)’

(fi

a nonlinear partial differential equation for the optimum value function subject to the initial condition (3.8). EXERCISES 1. It is known that the buckling load P of a prismatic axially loaded bar is given by

P = min w (f fawe2 dx)

/

(4 J:w”

dx) ,

i.e., by the minimum of a Rayleigh quotient. Show that the optimization problem subject to JOLg(a)dx = V,

max P, (I

where V, the volume, is a prescribed quantity and g is a given function of a denoting the cross-sectional area of the bar, may be formulated as max min 4 a

w

:j(awl2- h,w’, -

(4

2h2g(a))dx,

where h, and h, are two constant Lagrange multipliers. 2. In Exercise 1, assume the column clamped at x = 0 and free at x

f(u, u, x)

= max min 0

1

K

where u(x) = w(x), u(x) = u’(x), and differential equation

f

s:

K =

(aK2

-A,

= L.

u2 - 2hg(a)) dx,

Set (4

u”(x), and show that f satisfies the partial

-fx(u, u, x) = max(-(1/2a)fv2 - fhl u2 - A d a ) +f.u), 01

f(u, u, L ) = 0.

(4

249

4. CONSTANT CURVATURE

4. Constant Curvature

If the curvature of the beam does not change the sign, then Eq. (3.14) admits a remarkably simple solution. In fact, assuming K ( Z ) > 0 for 0 Iz < L, Eq. (3.14) reduces to f,

+ Ofu + ku2/2 - p~ = A 1 J 2 f , ,

(4.1)

which admits a solution of the form

f

= rtu2

+ 2r2 uv + r3 v2 + r4 u + r5 v + r 6 ,

(4.2) where r i , i = 1, . . . , 6, are functions of z that satisfy the following initialvalue system :

dr,/dz = -k/2, dr2/dz= - r l ,

r,(L) = 0, r2(L)= 0,

dr4/dz = p + 2A'I2r2, r4(L) = P, dr,/dz = -r4 + 2A'I2r3, r,(L) = M ,

(4.3)

dr,/dz = - 2 r 2 , r3(L)= 0 , dr6/dz = A1l2r5, r 6 ( L ) = O, obtained upon substitution off given by (4.2) into (4.1) and collecting terms in powers of u and v. The initial conditions follow from consideration of Eq. (3.8). Clearly, Eqs. (4.3) admit a solution in quadratures. For example, assume p and k to be constant in 0 I z I L and consider the initial conditions v(0) = 0.

u(0) = 0,

(4.4)

Integration of (4.3) yields

rt = k t / 2 , r2 = kt2/4, r3 = kt3/6, r4 = P - p t - kA'I2t3/6, r5 = M + P t - p t 2 / 2 - kA'I2t4/8, r6 = - M t - P t 2 / 2 p t 3 / 2+ kA'l2t5/40,

(4.5)

+

where t = L - z. On the other hand, integration of (3.12), where by (3.6) subject to the initial conditions (4.4), yields

u(z) = A'/2z2/2, v(z) = A'/2z. The optimal design is given by Eq. (3.1 l), i.e., 01

=fu/A'12

= (2r2u

K

is given (4.6)

+ 2r3 v + r5)/A'J2,

which on account of Eqs. (4.5) and (4.6) yields 01

=

[M

+ PLS - P L ~ ~ ~ / ~+][2( / A1 ' /9)'~ + 85( 1 - 5)/3 - f2][2kL4/8,

(4.7) where [ =
250

10 OPTIMAL BEAM DESIGN

EXERCISES 1 . Consider problem 2 of Section 3. Assumingg(a) = kal/”,where k is a positive constant and n is a positive integer, solve Eq. (e). 2. If n = 1, show that the curvature of the column, in its deformed position, is a constant.

3. Show that when n = 1, the optimum design is proportional to Lz - x z .

5. Euler-Lagrange Equations

We illustrate the procedure outlined in the last chapter, and derive the pertinent necessary conditions of the present optimization problem. Along an optimal design aOpt,Eq. (3.10) holds without the minimum operation, i.e., f z =f,”/2a -fu v - ku2/2+ PU

+ 2~12,

(5.1)

where u satisfies (3.11). On account of (3.11) we can write (5.1) in the form fz

= ACC- f u v - ku2/2

+ PU.

(5.2)

We compute now the total derivatives o f f , and f u with respect to z, i.e., (dldzlfu = f z u +fuuv +fuuv’, (dldzls, = f z u + f u u v + f U U V ’ ,

(5.3)

and also the partial derivatives f , , and fvz from Eq. (5.2), i.e., fuz

= 2% - f u u v - ku

Lz =

-Luv

+ P,

(5.4)

-fu,

which substituted into (5.3) yields (dlWfI4 = P - ku (dldzlf, =

-fu

+

-fufuula,

+ 2%-Lfvul4

(5.5)

where v‘ has been substituted by its equivalent dvldz

=

-fv/U,

(5.6)

obtained recalling (3.3), (3.6), and (3.9). Now, taking partial derivatives with respect to u and v in Eq. (3.11) we have

2% -f v fuula = 0, La” -f u f o v l a = 0, respectively, so that Eqs. (5.5) reduce to (dldz)S, = P - ku, ( 4 W f V

=

-fu

Y

(5.7)

6 . DESIGN CONSTRAINTS. SUCCESSIVE APPROXIMATIONS

25 1

the equations of equilibrium of the beam which together with Eqs. (3.3) and (5.6) are the Euler-Lagrange equations of the original variational problem (2.6). They hold, of course, for any u other than the optimal one given by (5.9) an equation derived directly from (3.11). In Eq. (5.8) werecognizef,, to be the shear force andf, the bending moment of the beam. In fact, the equations (5.10)

f" = -uw" obtained from (3.6) and (3.9), and

f"= -f,' = (olw")',

(5.1 1)

obtained from (5.8) and (5.10), are local expressions of the theorem of Castigliano obtained via dynamic programming. Elimination of fu between (5.8) and (5.1 1) yields (dZ/dx2)(ud2wldxZ)

+ kw = p ,

(5.12)

the classical equation of the beam where u(z) in Eq. (5.8) has been substituted by w ( 4 . EXERCISE Using the method outlined in Chapter 9, discuss the end conditions on the quantities

f. and fu . 6. Design Constraints. Successive Approximations There are several reasons why we wish to solve the problem of optimum beam design using a method of successive approximations. In fact, closedform solutions of the type obtained in Section 4 are the exception rather than the rule. In general we must resort to Eq. (3.14) or to the more complicated ones which are obtained if the hypotheses of beams of sandwich type are relaxed; and although Eq. (3.14) is subject to initial conditions, its numerical solution is not a routine matter in view of the singularities associated with the inflection points of the beam, i.e., where the curvature changes the sign. In addition to this we observe that Eq. (3.14) was derived assuming that there are no constraints on the design, a condition seldom met in practice. It is therefore desirable that the development of successive approximation schemes suitable for the solution of various classes of problems be derived in a unified fashion.

252

10 OPTIMAL BEAM DESIGN

We start by considering Eq. (3.10). For a fixed design oro this equation holds without the minimum operation, i.e.,

f, =fv2/2u0 -f , v

- ku2/2

+ PU + Ia0/2,

(6.1) subject to the initial condition (3.8). Equation (6.1) admits a solution of the form

where si,

f = s1u2 + 2s, uv + s3 v2 + s4 u + s5 v + ss , i = 1, . . . , 6, are functions of z given by ds4/dz= 2s, s5/ao + p (d) ds,/dz= 2s, s5/ao - s4 (e) dS6/dZ= s52/2ao+ h 0 / 2 (f)

dsl/dz= 2szz/u0- k/2 (a) ds2/dz= 2s2s3/a0- s1 (b) ds3/dz= 2s3’/a0 - 2s, (c)

(6.2)

(6.3)

a system of Riccati differential equations subject to the initial conditions S,(L) = S z ( L ) = s3(L) = Sg(L)

s4(L) = P,

= 0,

S,(L) = M ,

(6.4)

obtained from consideration of Eqs. (3.8) and (6.2). Functions u and v can be obtained from Eqs. (3.3) and (5.6), i.e.,

du/dz= V , dvldz = -(2s2 u + 2s3 v

+ s,)/ao,

(6.5)

a coupled system of differential equations subject to the initial conditions

u(0) = uo ,

v(0) = 00,

(6.6)

where uo and vo are prescribed values of the deflection and the slope at z = 0, respectively. Now we are in a position to implement a successive approximation scheme for the computation of the optimum structure. In fact, for a given estimate uo of the design, we integrate Eqs. (6.3a)-(6.3e), and obtain si,i = 1, . . . , 5, which substituted into (6.5)-(6.6) yield functions u and u by forward integration. Using those quantities we can therefore compute a’ = arg min[2(s2 u asA

+ s3 v + s5/2)’/a + la/2],

(6.7)

an equation of optimality derived directly from (3.10) and (6.2), which yields an improved estimate for the design. The quantity I appearing in (6.7) can be computed easily by making use, at each iteration of the process, of Eq. (3.1 1) which, on account of (6.2), reads

1 = (s5(o)/a0(o))2 at z

= 0.

This process is repeated until convergence is achieved.

(6.8)

253

7. NUMERICAL EXAMPLE

EXERCISE How would you proceed to satisfy, at each iteration of the process, the condition

Pu(L)

+ MU@)+

p u dx

(a)

= C,

JoL

where P,M, and c are prescribed quantities? Hint: Instead of (6.8), compute h to satisfy (a).

7. Numerical Example To show the feasibility and accuracy of the method developed in Section 6, we present a numerical example involving a cantilever beam on elastic foundation subject to a load P and a moment M such that the rotation at the free end x = L is zero. This last condition is incorporated for the sole purpose of simplifying the analytical solution. In fact, the exact solution for the optimum design of the present beam configuration and loading condition can be found easily by application of the results in Section 3. From Eq. (3.12) we have K =

+A”’,

(7.1)

i.e., the curvature in absolute value is a constant along the entire beam. Therefore, the elastica consists of two arcs of a parabola with the same curvature intersecting at an inflection point xo = L/2 and satisfying the geometric boundary conditions

w(0) = w’(0) = w’(L) = 0, w’(x0-) = w‘(x, +). By requiring that the slope of the two parabolas be equal at xo, the shear force V at the inflection point can be found easily be integration of all the forces acting perpendicularly to the beam, obtaining in this fashion V = P - 5kL3A“’/48.

(7.3)

Using V furnished by (7.3) we can now easily compute the moments acting along the entire beam, namely,

m(5) = (PL/2)(1 - 25)[1 - (23 - 25 - 45’ - 8t3)>lB],

5 5 3,

(7.4)

3I 5I 1,

(7.5)

05

and

m(5) = (PL/2)(25 - 1)[1

- (21

+ 105 - 285’ + 8t3)/P],

where

p = 192P/kL3A1I2.

(7.6)

254

10 OPTIMAL BEAM DESIGN

In particular, the moment M necessary to ensure the horizontality of the elastica at 5 = 1 is given by Eq. (7.5), namely, M

= m(1) = (PL/2)( 1

- 21/P).

(7.7)

The optimum design follows from Eq. (3.1 I), i.e., a = m(5)/A'I2.

Therefore &(<) = (1

0 < 5 < 3,

(7.8)

+ 105 - 28t2 + 8y3)/P], 3 I 5 I1 ,

(7.9)

- 25)[1 - (23 - 2< - 4 t 2 - 853)/p],

and

Cr(5)

= (25

- 1)[1 - (21

where Cr is the dimensionless quantity given by Cr = ( 2 F / P L ) u *

(7.10)

Clearly, P = 23 is the smallest value with physical meaning that this parameter cantake. The longitudinal stresses in the flanges, in absolute value, are constant over the entire beam and their value is given by

I o I = A1/'Eh/2,

(7.1 1)

where E is the modulus of elasticity and h is the height of the beam, i.e., the distance between the flanges. Hence, for a prescribed value of the stress u, A l l 2 can be computed from (7.1 1) and its value substituted into (7.6) to compute the dimensionless quantity P. The optimum dimensionless design ti, for several values of the parameter P, is presented in Fig. 10-1. X

h

I .o

0.8

dko,6 ,I

IU

2 04 ' cn W

0.2 0 0

0.1

0.2

0.3

0.4

0.5

[=

xL

0.6

0.7

0.8

0.9 I 3

Fig. 10-1. Optimal beam designs, where a = El, p = 192P/kCL3, C = 2u/Eh, and h is the height of the beam. (After Distkfano [1972].)

TABLE 10-1 Comparison of Exact and Computed Designs &ompvtcd

5

(constraint: Emin= 0.005)

L a c *

Initial estimate

1st iteration

2nd iteration

3rd iteration

5th iteration

7th iteration

9th iteration

11th iteration

0.77718 0.47106 0.16139 0.16090 0.50843

0.77199 0.46722 0.15887 0.16220 0.50880

0.89000

0.89000

0.77055 0.46615 0.15817 0.16265 0.50890 0.89000

0.77015 0.46586 0.15798 0.16267 0.50893 0.89000

12th iteration

~

0.00 0.20 0.40 0.60 0.80 1.0

0.77000 0.46574 0.15790 0.16270 0.50894 0.89000

1.0 1.0 1.0 1.0 1.0 1.0

0.89812 0.55810 0.21538 0.13570 0.50212 0.89000

0.82566 0.50691 0.18487 0.14876 0.50495 0.89000

0.79773 0.48631 0.17144 0.15566 0.50692 0.89000

0.77008 0.46580 0.15794 0.16268 0.50894 0.89000

256

10 OPTIMAL BEAM DESIGN

The solution of the same problem, using the method of successive approximations developed in Section 6, can be obtained as follows. At a generic iteration, using the available design a', we integrate Eqs. (6.3a)-(6.3e) in the backward direction, obtaining the values of si,i = 1, . . . , 5, which substituted into (6.5)-(6.6) yield functions u and v by forward integration. Using these quantities we compute an improved design a' by making use of Eq. (6.7). In the numerical example presented in Table 10-1 and corresponding to fi = 100, we started the process by considering a uniform design a' = I . The integration was performed using an Adams-Moulton scheme with step size H = L/200. It is noticed that the singular behavior of the derivatives in Eqs. (6.3), in the neighborhood of the inflection point, prevents in general the routine use of standard methods of numerical integration. A number of asymptotic solutions near the singularity can be obtained and the corresponding results can be matched to the numerical solution furnished by the Adams-Moulton integration scheme, in the neighborhood of the singularity. In the present case it was found that the simple device of imposing a constraint on the design of the form c12 0.005 worked very well and avoided the need of complicating the routine of integration. The results in Table 10-1 show the accuracy of the method.

11. INVARIANT IMBEDDING 8. Alternative Formulation

In order to incorporate displacement constraints and additional local restrictions on the design and the state variables, the remainder of this chapter is devoted to an alternative formulation of the beam-optimization problem. In view of the different imbedding used here, we shall modify the notation slightly. To illustrate the procedures we consider a beam of finite length L, laying on elastic foundation with coefficient k(x) and subject to external forces q(x) that might eventually contain a singularity due to a concentrated load. If u(x) denotes the deflection, v(x) the slope, m(x) the bending moment, and t(x) the shear, the constitutive equations of a Euler-Bernoulli beam are given by duldx = v, dvldx = - (1/a)m, (8.1) and the equilibrium equations by

dmldx

=

-t,

dtldx

= q - ku,

(8.2)

where

u

= EZ

(8.3)

257

8. ALTERNATIVE FORMULATION

is the stiffness, I is the moment of inertia, and E is Young’s modulus. We choose the stiffness a as the design variable, and assume that the crosssectional area A of the beam is given by the formula

where g is a function that depends on the particular geometry of the cross section. For example, for rectangular beams with constant width b and variable height h, g = (12bZa/E)’/3.The volume of the beam is given by .L

Considering u, v , m, and t subject to an appropriate set of boundary conditions, we can now formulate the following minimum volume problem for a prescribed displacement u1 at x = x l , and additional inequality constraints: Minimize the quantity V given by ( 8 . 5 ) subject to the conditions

and possibly to a number of additional constraints cpi(m,t,a,x)
i=1,2

,....

(8.8)

We assume that this optimization problem is well posed; i.e., a solution exists, is unique, and is continuous with respect to boundary conditions and constraints. The study of classes of constraints under which a given optimization problem is well posed is one of great interest and importance, but it is beyond the scope of this chapter whose main purpose is the discussion of numerical computational procedures. The derivation of the necessary conditions for this problem may be done using dynamic programming as outlined in Chapter 9. We leave this derivation as an exercise to the reader. Here we proceed directly, using the formalism of the minimum principle. In order to incorporate the constraint on the deflection u, instead of (8.6) we consider the equivalent integral expression

where 6 is the Dirac delta. Now we form the Hamiltonian

258

10 OPTIMAL BEAM DESIGN

where 1and I l to 1, are Lagrange multipliers used to incorporate the constraints (8.1), (8.2), and (8.9). It is well known that the multipliers satisfy the adjoint differential equations

(8.1 1)

From the first equation of (8.11) we see that 1is a constant, to be chosen to enforce condition (8.6). Comparison of (8.1) and (8.2) with Eqs. (8.11) shows that the Lagrange multipliers I , to I , are the forces and displacements of the beam subject to a virtual concentrated load of intensity I at x = xi. More precisely,

I,

= AU,

2,

= 16,

12

=

-26,

I1 =

-If,

(8.12)

where U , 6, E , and t are the displacements and forces due to a unit virtual force at-xl. Minimization of (810), taking into account (8.12), yields the optimality condition aOpt= arg min[l(mE/a)

+ g ( 4 41,

(8.13)

a

where clop, denotes the optimum design. EXERCISES 1. Let f(u, u, t , rn, x ) be the minimum volume function given by

f(u, u, t , m, x ) = min a

S:

g(a,x ) dx,

(a)

subject to the differential constraints (8.1) and (8.2) and to the deflection constraint (8.6). Show that f satisfies the partial differential equation

-f = min[g(a, x ) + puIS(x - X I ) +f.u m

- (l/a)fum +fh- ku) - A t ] ,

(b)

where p is a constant Lagrange multiplier. 2. Using the method outlined in Chapter 9, derive ordinary differential equations for f., fu, fc, and fm . How are these quantities related to the adjoint variables X I to satisfying (8.1l ) ? 3. How would you include consideration of constraints such as:

Ilongitudinal stress I < u,, ,

I slope I

=L

< uL ?

9. AN ALTERNATIVE DERIVATION OF THE OPTIMALITY CONDITION

259

4. Consider a sandwich beam as indicated in Fig. 8-1. Let E = F(U)

be the (nonlinear) stress-strain law of the flanges. Therefore the curvature K will be given bY K = -(1/2h)[F(m/Ah) - F(-m/Ah)]. (4 If, additionally, we consider a nonlinear elastic foundation characterized by the forcedisplacement relation p=ku-pu', 820, (6) the differential equations of the beam will be given by duldx = U, = - ( l / X ) [ F ( m / A h ) - F(-m/Ah)],

du/&

dm/& = -t, dtldx = q - ku

(0

+ Pu'.

Assuming F to be continuously differentiable with respect to A, derive the pertinent necessary conditions for the following optimization problem: min /'A dx, A20

0

subject to the deflection constraint (8.6) and the differential constraints (f). Use A as the pertinent design variable.

9. An Alternative Derivation of the Optimality Condition Instead of incorporating the differential constraints (8.1) and (8.2) using the Lagrange multipliers I1 to I,, we can proceed in the following way: In place of (8.9) we use the integral representation "L

+

u1 = ~o(mfi/cc kuii) dx,

(9.1)

furnished by the theorem of virtual work, where Z and i7 are the moment and the displacement, respectively, of the beam subject to a virtual load applied at x1 and in the direction of the prescribed displacement. Combining (8.5) and (9.1) we form the Hamiltonian 231 = g(a, x )

+ I(rnE/a + h a ) ,

(9.2)

where 1 2 0 is a Lagrange multiplier that satisfies the equation

dI/ax = - aH1laul = 0.

(9.3) Hence I is a positive constant which is to be chosen to satisfy (8.6). Clearly, minimization of (9.2) yields (8.13), as expected.

260

10 OPTIMAL BEAM DESIGN

EXERCISES 1. Derive the optirnality condition from Eq. (b) of Exercise 1, Section 8.

2. Assuming a not subject to constraints andg(a, x ) = k ~ ( ' /derive ~ , the pertinent condition of optirnality starting from (8.13).

10. Optimum Cantilever under Prescribed Displacement

To illustrate the method and the problems associated with the numerical computations, we consider the case of a cantilever beam laying on elastic foundation and subject to a prescribed displacement at x = xl. The boundary conditions of Eqs. (8.1) and (8.2) are u(0) = 0,

~ ( 0=) 0,

m(L) = M , t(L) = T.

(10.1)

For simplicity we rewrite Eqs. (8.11) in terms of U, V , E,and i given by

dU/dx = i5, dV/dx - (1/a)E, 1

dE/dx = - i , di/dx = 6(x - xl) - ku,

(10.2)

subject to the homogeneous boundary conditions

U(0) = V(0) = 0,

E(L) = t(L)= 0.

(10.3)

So formulated, the solution of the optimum beam reduces to the task of integrating Eqs. (8.1), (8.2), and (10.2) subject to boundary conditions (10.1) and (10.3), respectively. These two systems of equations are coupled together through the optimality condition (8.13). In a number of important cases in the applications we possess an explicit representation for the minimum operation in (8.13). This simplifies in some sense the solution of the system. In any case, this nonlinear boundary-value problem can be integrated using a quasilinearization scheme. (See the exercises below.) We do not pursue this path here, i.e., a direct treatment of the nonlinear boundary-value problem, in favor of the implementation of a simple, first-order, stable iterative method based on ideas of invariant imbedding. This is done in the following section. EXERCISES 1. Assuming u, u, t , m subject to boundary conditions (10.1) show thatfin Eq. (b), Exercise 1, Section 8, satisfies the conditions

fmb, u, t , m,0) = 0, fXu, u, t, m,0) = 0,

A h , 0, t , m,0 = 0, f d u , u, I, m,L) = 0.

(a)

2. Write the boundary conditions for the adjoint variables 8,6, i, and 177,when the beam is (a) clamped-clamped, (b) clamped-hinged, (c) hinged-hinged.

26 1

11. INVARIANT IMBEDDING

3. Derive the conditions on f satisfying the partial differential equation (b) in Exercise 1, Section 8, for the cases indicated above.

4. Implement a quasilinearizationscheme for the solution of Eqs. (S.l), (8.2), (10.1)-(10.3), and (8.13). 5. Implement a quasilinearizationscheme for the solution of Exercise 5, Section 8.

11. Invariant Imbedding

For a given nominal design c1, we consider the uncoupled linear boundaryvalue systems given by Eqs. (8.1), (8.2), (lO.l), and (10.2), (10.3), respectively. First we consider system (8.1)-(8.2). Instead of boundary conditions (10.1) we set m(L) = M , u(X) = w, (11.1) t(L)= T, v(X) = Z, i.e., we consider the families of beams of length L - X subject to arbitrary displacements w and z at the end x = X . This is an “imbedding” procedure that in the fashion of invariant imbedding affords the property of reducing the computation of the original boundary-value problem (8. l), (8.2), (lO.l), to a stable, two-sweep procedure. We seek solutions of the form

t ( X ) = r,(X)w + r12(X)z + S l W , m ( X ) = rzl(X)w r,(X)z s2(X).

+

+

(11.2)

Differentiation of (1 1.2) with respect to X and elimination of the derivatives

w’ = u’(X) and z’ = v’(X) using Eqs. (8.1) and (8.2) evaluated at x = X and further elimination of t(X) and m ( X ) using (1 1.2) yields (rl’

+ k - r12(1/cOr21)w+ (G2+ rl - rl2(l/a)r2>z+ (sl’ - q - rlz(l/4sz) = 0,

and

(rL + rl - r 2 ( 1 / 4 r 2 d w + h’+ r12 + r21 - r,(l/4rz)z

+ (s,’ + s1 - r2( l/u)s,)

= 0,

a system of equations that must be valid for any w and z. Therefore

+

rl’ = - k rl2(1/c1)rZ1, r2’ = -rlz - r21 + r2(1/u)r2, (11.3) r i 2 = -rl + rl2(1/4r2 , s1’ = q + r12U/a)s2, s2’ = -sJ + r2(l/u)s2, ril = -rl r2(1/u)r21, a system of Riccati equations subject to the initial conditions at X = L,

+

r,(L) = rI2(L)= r21(L)= r,(L) = 0,

s1(L) = T,

s2(L)= M ,

(11.4)

262

10 OPTIMAL BEAM DESIGN

obtained upon consideration of Eqs. (1 1.1) and (1 1.2). Consideration of the second and third equation in (1 1.3) and corresponding initial conditions readily yields r12 = r21, (11.5) an expression of Maxwell’s theorem derived from invariant imbedding. Therefore Eqs. (1 1.3) and (1 1.4) reduce to

+ (l/a)r;,, = -r, + (1/.)rlZ~Zr = -2r1, + (1/a)rZ2, 81’ = q + (l/a)r12sz -k

r,’ r;, r,‘

=

s‘,

= -sl

r,(L) = 0, r,,(L) = 0, r,(L) = 0,

(11.6)

s,(L) = T, s,(L) = M .

9

+ (l/a)rzs,,

Substitution of m(X) given by the second equation of (11.2), into Eqs. (8.1) evaluated at x = X and due consideration of Eqs. (10.1) yield u(0) = 0, du/dX = V , (11.7) dv/dX = -(l/a)(r,, u + r2 v + s2), v ( 0 ) = 0, an initial-value problem in the forward direction for the deflection u and slope v of the beam, where the quantities r , , , r , , and s, are given by the integration of (1 1.6) in the backward direction. Substitution of u and v given by (1 1.7) into (1 1.2), taking into account that w = u ( X ) and z = v ( X ) , finally yields the remaining state variables of the beam, i.e., the moment m and the shear force t in the interval 0 I X 5 L . We can treat the virtual system (10.2) in a similar fashion. We consider U, 6, iii, and t satisfying Eqs. (10.2) to be subject to the boundary conditions

U(X)= w,

E(L) = 0, t(L)= 0,

q x ) = 5,

(11.8)

and write for t ( X ) and % ( X ) ,the missing boundary conditions of the imbedded beam of length L - X , equations similar to (1 1.2), i.e.,

f ( X )= T;,(X)W+ f,,(X)Z + S,(X), E ( X ) = T‘,l(X)W r,(X)z S,(X).

+

+

(11.9)

Carrying out the same perturbation analysis done to Eqs. (1 1.2), on Eqs. (1 1.9), we obtain

+ (l/~)?;,, i ; , = - r, + (l/a)r,, r,, 7,’ = -2r12 + (l/a)F22, F,’= -k

+

6,’ = 6 ( X - x1) (l/a)?,, 6,‘ = - 6 , ( l / a ) 7 , s,,

+

f,(L) = 0, = 0, F,(L) = 0, S,(L) = 0, S,(L) = 0.

F12(L)

s, ,

(11.10)

263

12. A TWO-SWEEP ITERATIVE PROCEDURE

Similarly, we can write dU/dX = O, dO/dX = -(l/a)(F12 U + i 2 ij + iz),

U(0) = 0,

(1 1.11)

O(0) = 0,

an initial-value problem for U and ij obtained after substitution of E,given by the second equation of (1 1.9), into the first two equations of (10.2) evaluated at x = X . Finally, t and E can be obtained from (1 1.9) recalling that W = U(X) and Z = tj(X). EXERCISES 1. If, instead of (ll.l), the boundary conditions were u(X) = w, m(X)= M,

u(L)= z, t(L) = T,

how would the initial conditions of (11.6) change? 2. Repeat the previous analysis for the conditions u ( X ) = w, v ( X ) = 2,

u(L) = 0, u(L) = 0.

In this case, some of the r's in (11.3) are not defined at x = L. Derive approximate asymptotic expressions for those quantities at x = L - A, A < 1.

12. A Two-Sweep Iterative Procedure

Using the results of the preceding section we can now implement a simple iterative method for the solution of the optimization problem. To this end let dn)denote the value of the quantity a at the nth iteration. Then at a generic iteration (n + 1) we use the currently available design a(") to integrate (11.6) in the backward direction and subsequently (1 1.7) in the forward direction, computing in turn m("+')given by (1 1.2). Similarly, we compute the values of E'"+ by using the two-sweep process given by (1 1.9)-(11.11). The improved value of the design a("+') follows from the optimality condition 1)

= arg min[l(n+

')m("+1 )/a) + da,41,

(12.1)

a

where the new estimate of 1= A("+') needs to be taken so as to ensure an appropriate convergence of the sequence u(")(xl) to the prescribed value ul. This can be done in a number of ways. In general we shall use formulas of the type

A("+1) = F(1("), u(")(xl),u("-l)(xl),...).

(12.2)

264

10 OPTIMAL BEAM DESIGN

(12.3)

The procedure is repeated until convergence is achieved. Initially we need an a priori estimate of the design a(’). In the absence of special information, a uniform design ~(‘’(x) = constant is usually taken as the initial design. A numerical example in Section 15 illustrates the application of the method. EXERCISES 1. Consider a circular plate of radius a, thickness h, Young’s modulus E, and Poisson’s ratio v. If u denotes the deflection and u and m the radial slope and bending moment, respectively, the equations of equilibrium of the plate undergoing bending may be written u‘

=u

u’ = -(v/r)u - ( l / a ) m m‘ = -((I - v)/r)m- ((1

(a) - v2)/r’)au - Jqr,

where r is the radial coordinate, q(r) is the load, and the stiffness a is given by a = Eh3/(12(l

- v’)).

(b)

Assuming one of the following boundary conditions: Clamped Simply supported

u(a) = 0,

u(0)

u(a) = 0,

u(0) = 0,

= u(a) = 0,

m(a) = 0,

(4

and using the method outlined in this and foregoing sections, solve the following optimization problem: min 1;ra1l3 dr,

subject to the displacement constraint u(0) = 1.

(d)

Clearly, the integral in Eq. (d) is a quantity proportional to the volume of the plate. 2. A circular footing with outer radius a is carrying a column of radius 6, with axial load P. Assuming thin-plate theory and Winkler foundation, the associated boundaryvalue problem for the bending of the footing is given by P

u’ = u,

u(b) = 0,

- ( l / a ) m - (v/r)u, m‘= -((I - v)/r)m - ((1 - v2)/r2)au- ikur,

u(a) = 0,

U’ =

I

2

a

I

and the additional equilibrium condition P = nb’ku(b)

+ 2njbakurdr,

m(a) = 0,

(4

265

14. STABILITY CONSIDERATIONS

where k is the coefficient of the foundation. Derive the optimality conditions and implement a two-sweep method for the solution of the following optimization problem: max P,

(g)

subject to

dr = C, where C is a quantity proportional to the volume of the plate, and to the displacement constraint 0 Iu(b) 5 c. 3. In Exercise 2 above, include a constraint on the absolute values of the shear and radial stresses.

13. Inertia Forces When the weight of the beam is to be taken into account, q in Eqs. (8.2), (8.10), and (11.6) must be replaced by 4=p

+ YA,

(13.1)

where p is the external forces, y is the specific weight of the beam, and A is the cross-sectional area given by Eq. (8.4). Therefore the optimality condition (8.13) must be substituted by uOpt= arg min[A(m%/u)

+ (1 + y/lU)g(u,XI],

(13.2)

a

and Eq. (13.1) must be taken into account when integrating Eqs. (11.6). Similarly, other mass forces can be taken into account. EXERCISE

In Exercise 1 of Section 12, incorporate the weight of the plate in the formulation of the optimization problem.

14. Stability Considerations

In order to prove the stability of the method, we should show that u and u given by (1 1.6) and ( 1 1.7) are stable with respect to small errors introduced in the computational process. To this end it is enough to prove the stability of the differential system (1 1.6) and, in particular, of the quantities rl, r12, and r 2 . We rewrite the first three equations of (1 1.6) in the matrix form R' = - A - BR - RBT + RCR,

where

R(L) = 0,

(14.1)

266

10 OPTIMAL BEAM DESIGN

It is easy to show, by recalling the results of Section 3-12, that the stability of R in the backward direction will depend on the stability of the differential equation dz/dx = ( B - RC)Z

(14.3)

in the forward direction, which in turn follows from consideration of the Liapunov function V ( Z ( X )= ) (z(x), R - ’ ( L - x)z(x))

(14.4)

which clearly exists, is bounded, and is positive definite for x > 0, since R-’ is a flexibility matrix. We leave the details of the demonstrations to the reader. EXERCISES 1 . What are matrices A , B, C, and D in (14.2) when, instead of (8.1), we use the constitutive equations

+

a l l t + a12m, u’=a21t+a22rn,

u’= u

where ax2 = a z l ? 2. When we use Eq. (a) above and assume that the matrix (a,,) is positive definite, do we still enjoy the stability of the Riccati equation (14.1)? 3. Prove the numerical stability of all the quantities involved in Sections 11 and 12.

15. Numerical Examples a. Cantilever on Elastic Foundation

We consider a beam of the sandwich type such as that whose cross section is indicated in Fig. 8-1. In this case if 2h is a fixed quantity denoting the distance between the covering sheets and A/2 is the area of each one of the sheets, we have tl = EZ = EAh’, and g in (8.4) reduces to g(a, X) = UlEh’.

(15.1)

The volume is therefore proportional to a, i.e., L

V = ( l / E h 2 ) I adx.

(15.2)

0

In addition we shall take x1 = L and q = P6(x - L ) ; i.e., we prescribe the displacement u1 in correspondence with a concentrated load applied at the free end of the cantilever. The reason to consider this simplified example is because under the present assumptions we can afford a closed-form solution

267

15. NUMERICAL EXAMPLES

of the problem that can be used to compare the accuracy of the numerical procedures. In fact, in our present case we have m = P E and Eq. (8.13) reduces to

+ E),

aOpt = arg min(p(m2/a) U

(15.3)

where p = (EhZ/P)Ais a constant to be determined such that u(L) = ul. Minimizing by taking derivatives, Eq. (15.3) reduces to =p

(15.4)

2 .

Combining (8.1) and (1 5.4) we obtain (u")2

= lip.

(15.5)

A solution that satisfies the nonlinear differential equation (15.5) and the boundary conditions u(0) = u'(0) = 0 and u(L) = u1 is given by (15.6)

u(x) = U 1 X 2 / L Z .

Equation (15.6) is valid if sign m = sign u", a condition that is clearly fulfilled in our example. The curvature is given by lu"l = (l/p)l'Z

= 2u,/L2.

(15.7)

Combining (15.4) and (15.7), aopt= LZm/2ul,

(15.8)

where the bending moment m can be computed readily by direct integration, namely, m = P ( L - x) -

1°C.- x)ku

dz

0

= P L ( ~- rl) - kU1L2(3- 4q

+ 14)/12,

(15.9)

where we put q = x/L. In a similar fashion we can derive for the shear force t the expression t =~

[-i4 j ( i - 191,

(15.10)

where

j= ku1L/12P.

(15.11)

Combination of (15.8) and (15.9) yields Lop, = 1 - q - p(3 - 41

+ q4),

(15.12)

where c1 is a dimensionless design given by

L =2u,a/P~~.

(1 5.1 3)

268

10 OPTIMAL BEAM DESIGN

0.8

0.6

a OPT. 0.4

0.2

0.0

0.2

0.4 0.6 r] =x/L

0.8

I .o

Fig. 10-2. Optimal beam designs, where EOp,= (2ul/PL3)aOp, and (After Distefano and Todeschini [1972].)

= kLul/12P.

Clearly, the range of p for which clopt is positive is 0 Ip IQ. In Fig. 10-2, Eopt given by (15.12) is presented for several values of p. For purposes of comparison, the same example was solved using the procedure outlined in Section 12. At a generic iteration we integrate Eqs. (11.6) in the backward direction using thecurrently available estimate for the design. There is no need to integrate Eqs. (1 1.10) since in this particular case we have m = PE. Subsequently, using the values of the r's previously computed, we integrate (1 1.7) in the forward direction. In turn we calculate a new, upgraded estimate of the design by means of &+I) = (/p+l) l / z m c n + l , ( 15.14) 1 9

an expression for u("+') that follows from (12.1) by taking derivatives and making m = PE and g = a/EhZ. In Eq. (15.14), p ( " + l ) ,given by p(n+l)

= (Eh2/p)A(n+l),

(15.15)

is a constant at each iteration that must be chosen to ensure the convergence of the sequence u(")(x,) to the prescribed value ul. We have taken in the present case p(n+l) = p(")u(")(xl)/ul.

(1 5.16)

The procedure is continued until the following convergence criterion Ek=max(lEkI, IEk+l[,IEk+21)

(1 5.17)

269

15. NUMERICAL EXAMPLES

where k --

1 - ,$k)/ki'k

+ 1)

(1 5.18)

is fulfilled. All the integrations were performed numerically using an AdamsMoulton scheme with step size 0.005 on a CDC 6400 computer. The resulting values for three different values of and at four equidistant sections of the cantilever are presented in Table 10-2. It is seen that in order to reach the same accuracy, the number of iterations increases for increasing values of the design parameter p. On the other hand, it is of interest to note that the convergence of the process is of an oscillatory type. This can be appreciated in Fig. 10-3 where the relative error &k versus the number of iterations k for two different values of has been plotted. Methods to improve the convergence properties of the process can be devised for the present problem but we do not enter into that discussion here. 0.02

0.01

-0.001

-0.01

-0.02 Fig. 10-3. Convergence of approximations where (After Distkfano and Todeschini [1972].)

E~

= 1 - 6,ck)/GckI ) and +

7 = 0.0.

N

4

0

TABLE 10-2 Comparison of Exact and Computed Designs. Cantilever Beam on Elastic Foundation"

1 12

0.00 0.25 0.50 0.75

0.75000 0.58301 0.41146 0.22363

2.0 2.0 2.0 2.0

0.74922(8) 0.58246(8) 0.41 114(8) 0.22351(8)

0.74983(9) 0.58290(9) 0.41 14Q(9) 0.22362(9)

0.82 x 0.75 x 0.64 x 0.48 x

10-3(8) 10-3(8) 10-3(8) 10-3(8)

0.78 x 0.55 x 0.32 x 0.12 x

10-3(8) 10-3(8) 10-3(8) 10-3(8)

1 6

0.00 0.25

0.5oooO

0.75

0.41602 0.32292 0.19727

2.0 2.0 2.0 2.0

0.5oooS(15) 0.41604(15) 0.32315(14) 0.19732(14)

0.49979(16) 0.41584(16) 0.32288(15) 0.19721(15)

10-3(15) 10-3(15) 10-3(14) 10-3(14)

0.25000 0.24902 0.23438 0.17090

2.0 2.0 2.0 2.0

0.25026(28) 0.24907(27) 0.23432(26) 0.17084(25)

0.25040(29) 0.24927(28) 0.23450(27) 0.17097(26)

10-3(15) 10-3(15) 10-3(14) 10-3(14) 10-3(28) 10-3(27) 10-3(26) 10-3(25)

0.09 x 0.02 x 0.23 x 0.05 x

0.00 0.25 0.50 0.75

0.60 x 0.48 x 0.83 x 0.55 x 0.59 x 0.80 x 0.80 x 0.74 x

0.26 x 0.05 x 0.06 x 0.06 x

10-3(28) 10-3(27) 10-3(26) 10-3(25)

0.50

1

7

1

0

0

4

5

r W

Numbers in parentheses denote the number of iterations.

K? n

27 1

15. NUMERICAL EXAMPLES

A further application of the method in connection with piecewise constant design is presented below. The design is taken in the form

c N

u=

(15.19)

C i H ( X - Xi),

i=l

where H is the usual unit step function. The problem consists in determining the cross-sectional stiffness ci and the lengths x i which minimize the volume of the beam. The values of ci are constrained to be one of any possible combinations of a given set of values a l , a 2 , . . . , a M . This is a version of a piecewise constant optimum design problem that occurs when the flanges of the beam must be constructed using a number of available sections a l , u 2 , . . . , aM. Substituting a given by (15.19) into the optimality condition (15.3), we can solve this problem by minimizing with respect to all possible values of ci. In the present example we have considered a1 = u2 = ... = us = 0.125. Therefore ci = 0.125j(i) where the possible values of j are 1, . . ., 6. The results are shown in Fig. 10-4 where a comparison with the unconstrained solution for B = & is possible. LO\

I

I

I

I

I

\

0.6

-

\

I

I

I

-

- \ \ 0.8 \up 0 \

I

-

\

\

\

212

10 OPTIMAL BEAM DESIGN

b. Clamped Beam under Uniformly Distributed Load An optimal design is characterized by in&

> 0,

(15.20)

a convexity condition that follows from consideration of the optimality condition (8.13). During the computation of the successive approximations following the procedure developed in Section 12, Eq. (1 5.20) might be violated leading to an indeterminacy in Eq. (8.13), unless additional information is furnished. This can be done in a number of ways. For example, we can require a positive lower bound for the design E, i.e., (15.21)

Ci26>0,

where 6 is usually taken to be the order of magnitude of the step of integration of the equations. This problem did not occur in our previous example involving the cantilever on elastic foundation, because in that case, Eq. (15.20) was identically satisfied since m = PE. The purpose of the present example is to show that Eq. (1 5.21) is enough to bypass the difficulties created by a possible violation of (15.20) during the first iterations of the process. To this end we consider a sandwich-clamped beam of length 2L subject to uniformly distributed load q. It is required that the deflection at the middle be u l . The exact optimal solution can be obtained without difficulties. In fact, assuming E to be continuously differentiable, the exact solution of this problem has been given in Section 8-16. The optimal design obtained using the method of successive approximations may be compared with the exact one, given in Chapter 8, in Fig. 10-5 and Table 10-3. In this example, the dimensionless design c( = (10u1/qL4)ccand the dimensionless quantities q = xlL and qo = X[L were introduced. The procedure used to TABLE 10-3

Comparison of Exact and Computed Designs. Clamped-Clamped Beam under Uniformly Distributed Load cy(rlj

C(O’(7lj

7

exact

initial approximation

0.00 0.20 0.40 0.60 0.80 1.00

0.75560 0.42326 0.13168 0.11387 0.30576 0.43171

0.5 0.5 0.5 0.5 0.5 0.5

0.58926 0.74725 0.30957 0.39508 0.05271 0.08591 0.13437 0.17484 0.30277 0.37943 0.41667 0.5 1566

0.72994 0.39632 0.10335 0.14341 0.33713 0.46570

0.74267 0.74948 0.40962 0.41649 0.11719 0.12414 0.12904 0.12198 0.32215 0.31492 0.44985 0.44228

NOTES, COMMENTS, AND BIBLIOGRAPHY

I

*

O

*

213

0' .0

'OPT.

q=x/L

Fig. 10-5. Comparison of successive approximations with optimal design in a clamped beam under uniformly distributed load, where GOpt= ( ~ O U ~ / ~ L and ~ ) LT~ Y= ~ ~0.50224. , (After DistCfano and Todeschini [1972].)

compute the solution using the method of successive approximations is similar to that presented in Section 12, making k = 0, and where the Riccati equations (1 l .6) and (1 l . 10) were subject to appropriate initial conditions in order to account for the difference in boundary conditions of the present beam. NOTES, COMMENTS, AND BIBLIOGRAPHY

2-7. In Part I of this chapter we present a dynamic programming treatment of the min-max problem which results from constraining the work of deformation of the given external loads on their own displacements. Here we follow closely N. DistCfano, Dynamic Programming and a Max-Min Problem in the Theory of Structures, J. Franklin Inst. 294, No. 5 (1972), 339-350.

In addition to the work by Wasiutyriski quoted in Chapter 8, the criterion of the work of deformation has been applied to some problems involving elastic plates. For a theoretical treatment of elastic plates, prestressed or not, according to the design criterion based on the work of deformation, see W. Dzieniszewski, Optimum Design of Plates of Variable Thickness for Minimum Potential Energy, Bull. Academie Polon. Sci., Sir.Sci. Tech. 13, Nc. 6 (1965).

274

10 OPTIMAL BEAM DESIGN

A numerical application of this design criterion to circular plates under uniform pressure is found in N. C . Huang, Optimal Design of Elastic Structures for Maximum Stiffness, Internat. J. Solids Structures 4 (1968), 689-700.

8-15. In Part I1 of this chapter, we follow closely N. Distkfano and R. Todeschini, Invariant Imbedding and Optimum Beam Design with Displacement Constraints, Internat. J . Solids Structures 8 (1972), 1073-1087.

Deflections at specified points of the structure are natural design constraints in most engineering applications. See R. L. Barnett, Minimum-Weight Design of Beams for Deflection, Proc. Amer. SOC.Civil Engrs. Engrg. Mech. Div. 87 (1961), 75-109.

As noted by Barnett, the problem resulting from prescribing the deflection of a beam at a single point might not be well posed. The indeterminacy is generally released, however, by incorporating additional constraints. For an example of this, see G. A. Dupuis, Optimal Design of Statically Determinate Beams Subject to Displacement and Stress Constraints, Div. of Engrg., Brown Univ., Rep. F33615-1826/1 (July 1970).

The first attempt of a systematic treatment of statically determinate beams using the methods of the calculus of variations appears to have been done by E. J. Haug, Jr. and P. G. Kirmser, Minimum Weight Design of Beams with Inequality Constraints on Stress and Deflection, J . Appl. Mech. 34 (Dec. 1967), 999-1004.

The problem of body forces in beams has been treated by invariant imbedding in the paper by Disttfano and Todeschini quoted above. See also R. L. Barnett, Minimum Deflection Design of a Uniformly Accelerating Cantilever Beam, J . Appl. Mech. 30 (19631, 466-467; J. M. Chern, Optimal Structural Design for Given Deflections in Presence of Body Forces, Internat. J . Solid Structures 7 (1971), 373-382.

In this paper the author derives the pertinent necessary condition from the principle of mutual potential energy introduced by Shield and Prager (see reference below) to treat a class of optimal design problems. In the paper by Chern, the author points out the inadequacy of the necessary condition used by Barnett to treat the problem of body forces. Additional references involving beams with deflection constraints are N. C. Huang, Optimal Design of Elastic Beams for Minimum-Maximum Deflection, Trans. ASME (Dec. 1971). 1078-1081 ;

R. T. Shield and W. Prager, Optimal Structural Design for Given Deflection, 2. Angew. Math. P h r ~21 . (1970), 513-523.

NOTES, COMMENTS, AND BIBLIOGRAPHY

275

Miscellaneous References Although not treated in this chapter, the problem of minimum-weight beam design with prescribed eigenvalues is intimately related to the problem discussed here. See, for example, J. B. Keller, The Shape of the Strongest Column, Arch. Rational Mech. Anal. 5 , Number 4 (1960), 275-285; J. B. Keller and F. I. Niordson, The Tallest Column, J. Math. Mech. 16, No. 5 (1966), 433446; C. Y . Sheu, Elastic Minimum-Weight Design for Specified Fundamental Frequency, Internat. J. Solids Structures 4 (1968), 953-958, 1968; N. C. Huang and C. Y . Sheu, Optimal Design of an Elastic Column of Thin-Walled Cross Section, J. Appl. Mech. 35 (1968), 285-288; B. L. Karihaloo and F. I. Niordson, Optimum Design of Vibrating Cantilevers, The Danish Center for Appl. Math. and Mech., Rep. No. 15 (May 1971).

See also J. L. Armand, Applications of Optimal Control Theory to Structural OptimiLation: Analytical and Numerical Approach, preprint of the IUTAM Symp. Optimization Structural Design, Polish Acad. Sci., Warsaw, Poland, Aug. 21-25, 1973.

Optimal Design of Rotating Disks

1. Introduction

In this chapter we treat minimum weight design of axisymmetric rotating disks subjectto appropriate constraints via dynamic programming and invariant imbedding. There are several advantages in so proceeding. In the first place the present method affords consideration of a number of integral and terminal constraints as well as inequality design constraints (piecewise thickness, etc.) within the same computational framework. Second, the algorithms for numerical computation that are derived in a natural fashionfrom the present theoretical formulation are initial valued as opposed to the two-point boundary-value problems occurring along classical formulations. And finally, this method handles the constraints on the radial displacement in a direct manner rather than in the parametric fashion afforded by other existing procedures. The first type of constraint considered here is the potential energy of the disk, a quantity extensively used by Wasiutyriski as a measure of rigidity in optimum structural design. See the Bibliography in Chapter 8. We derive the pertinent conditions of optimality from the Bellman-Hamilton-Jacobi equation associated with the corresponding max-min problem. When the thickness of the optimum disk is not subject to bounds or local constraints, 276

277

2. THE PROBLEM

the formulation in terms of the minimum potential energy affords a remarkably simple analytical solution in terms of a first-order linear partial differential equation subject to initial conditions. When the assumption of continuous differentiability is released, a method of successive approximations consisting of a two-sweep iterative scheme in terms of stable Riccati equations can be used. In the second part of this chapter the results previously presented are generalized by consideration of more general forms of compliances for the disk. In particular, the case in which the constraints are the radial displacements and the tractions at the edge of the disk is discussed in detail as an example of application of the present method. Invariant imbedding procedures are employed to implement a stable two-sweep iterative scheme starting from the necessary conditions. Finally, numerical examples are presented to show the accuracy and feasibility of the method and to compare optimum designs under various constraints. The insensitivity of the minimum volume with respect to various design constraints is stressed.

I. MINIMUM POTENTIAL ENERGY

2. The Problem We consider a thin elastic axisymmetric disk rotating at angular velocity w about the axis of symmetry and subject to constant radial tractions T per

unit length of the edge r = R. We are interested in formulating a minimum weight design problem by the specification of a proper set of constraints. This can be done by prescribing the value of a compliance, i.e., a quantity denoting the degree of deformability of the structure. Among several well-qualified candidates for a measure of a compliance we mention the work done by the centrifugal forces and the edge tractions, a quantity directly related to the potential energy of the disk. In the first part of this chapter we adopt this measure of compliance. In the second part, Section 7, we generalize the results by allowing for more general forms of compliances. In order to make the formulation of the optimization problem precise we proceed to define the quantities of interest. We assume a totally symmetric configuration; i.e., the density p of the material and the thickness h of the disk are functions of r only. Therefore the radial displacement u, the strains E , , E ~ and , the stresses crr , crg , will be functions of r alone. The strain energy will thus be given by

I0 R

U = +2n

(E,

crr

+ E~ a,)hr dr,

(2.1)

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11 OPTIMAL DESIGN OF ROTATING DISKS

or, on account of the stress-strain-displacement relations, = du/dr = (a,

E,

E@ = u/r

- va,)/E,

= (a, - va,)/E,

by

E R U(u, h) = +27c - (u'~ u2/rz 1- V2Jb

+

+ 2vuu'/r)hr dr,

(2.3)

a functional in terms of u and h only, where E and v denoteyoung's modulus and Poisson's ratio, respectively, and where a prime indicates derivative with respect to r. We define the potential energy V ( U , h) =

U(u,h) - W(u,h),

(2.4)

where R

W(u,h) = 2 n s pW2uhr2dr

+ 2nRTu(R)

(2.5)

0

is the virtual work of body forces and edge tractions T. By the theorem of minimum potential energy, the function u = u* that minimizes V in (2.4) and satisfies appropriate end conditions is the true displacement function of the elastic disk associated with a given design h. The value of this minimum is min V(u,h ) = -+W(u*, h).

(2.6)

U

Clearly, the quantity on the right-hand side of Eq. (2.6) represents the work done by the centrifugal forces and edge tractions along the true displacement field u*(r),with the minus sign. We shall use this quantity W(u*, h) as a measure of the compliance of the disk. Therefore, if for a given system of forces our objective is to minimize the weight 271 st phr dr for a constant value of the compliance W(u*,h), we can formulate the minimization problem R

min [2n Jo phr dr

+ p W(u*, h)] ,

h

or using Eqs. (2.4) and (2.6):

1.

min b n JoRphrdr - 2 p min( U(u, h) - W(u, h)) h

U

(2.7)

a min-min problem in h and u, where p is a Lagrange multiplier. Alternatively, if for a given system of forces we wish to obtain h for which the compliance W(u*, h) is a minimum among all possible designs with the same weight, we formulate the following problem : h

279

3. DYNAMIC PROGRAMMING

which combined with (2.4) and (2.6) yields R

U(u, h) - W(u,h) - A x / phr dr h

u

(2.8)

0

a max-min problem in h and u, respectively. Obviously the two problems (2.7) and (2.8) are equivalent and mulation given by Eq. (2.8).

A = l/p. In the following we use the for-

3. Dynamic Programming Instead of a plain disk, we consider the family of optimum annular disks with fixed external radius R subject to tractions T at the external edge and given radial displacements u(R), and with an inner edge of variable radius x subject to radial displacements u(x). Referring to Eqs. (2.3), (2.5), and (2.8) we define the optimum value function

['

+

+

f ( u ( x ) ,x ) = max min - -JXR(ut2 u2/r2 2vuu'/r)hr'dr h u 21-v2

-

R

IxR

po2uhr2 dr - RTu(R) -

1

phr dr , X

(3.1)

where the constant 2x has been dropped. Clearly, we recover our original problem by setting x = 0, and u(0) = 0; i.e., our problem is a particular member of the family of problems defined by Eq. (3.1). Using standard arguments in dynamic programming we obtain the following functional equation for the optimum value function :

f ( u ( x ) ,x ) = max min[BY - AJXxiAphrdr + f ( u ( x

+ A), x + A)

u'

h

1

, (3.2)

where

+ u2/r2+ 2vuu'/r)hr dr - ~xxiApw2uhr2dr (3.3) is the potential energy stored in the annular disk between x and x + A. As (u"

A --t 0, Eqs. (3.2) and (3.3) yield, in the usual fashion, -fx(u, x ) = max min[(d2 + u2/x2 + 2vuu'/x)Ehx/2(1 - v2) h

u'

- pW2UhX2- Aphx

+ fU(u, x)u'],

(3.4)

the Bellman-Hamilton-Jacobi equation of the process, where = afaE,

= u, x ,

(3.5)

280

11

OPTIMAL DESIGN OF ROTATING DISKS

and where f (u, x) is subject to the initial condition

f(u,R ) = - RTu,

(3.6)

obtained by consideration of Eq. (3.1). Clearly, the max-min operations in (3.4) are not interchangeable. Minimizing first with respect to u’ we obtain, by taking derivatives in (3.4) with respect to u’,

+

(u’ VU/X)EhX/(l - v’) +f, = 0.

Recalling that

6,in

(3.7)

terms of displacements reads or = (u’

+ vu/x)E/(l - v’),

(3.8)

f, in Eq. (3.7) reduces to

f, = -urhx,

(3.9)

i.e., the gradient of the minimum potential energy with respect to the displacement is the stress function of the disk with the minus sign, a consequence of the theorem of complementary energy derived from dynamic programming arguments. Elimination of u‘ between (3.4) and (3.7) yields, after some algebraic manipulations,

f, = min[(G’(u, x, A)h +f,’/h)(l - v2)/2Ex + vfuu/x],

(3.10)

C(U,X, A) = {[2p(J. + W’UX) - Eu’/x’]Ex’/( 1 - v’)}’’~

(3.1 1)

h

where

Now, if h is not subject to constraints, a restriction that is released in Section 5, the minimum operation appearing in (3.10) can be carried out explicitly, yielding.

(L/h)’ = G 2 ( U , x, 4,

(3.12)

the optimality condition for the design h in terms of the stress function f,, the displacement u, and the constant Lagrange multiplier 1.Since x = 0 is an isotropic point, we have ~ ~ (=0~ ) ~ (=0e0), Therefore, the limit of (3.12) as x + 0 yields

p A = Ee0’/(1 - v)

= (1

- v)a,’/E 2 0,

(3.13)

an expression for the Lagrange multiplier in terms of the stress or strain at the origin. Elimination of h between (3.10) and (3.12) yields

(f,- vf,~/x>’= (G(u, X, A)fu(l - v’)/Ex)’,

(3.14)

28 1

EXERCISES

a nonlinear partial differential equation for the optimum value function f. Equation (3.12) yields f"

=

Only the minus sign corresponds in (3.15) since h 2 0 and f , I 0 in 0 Ix IR. Hence Eq. (3.14) reduces to f,

= (-

C ( U ,X,

(3.15)

f h C ( u , x , A). =

-a,(x)xh(x)

A)( 1 - ?)/EX + V U / X ) S , ,

(3.16)

a linear partial differential equation subject to the initial condition f (u,R) = - RTu(R).

(3.17)

It can be shown, via the theory of characteristics, or otherwise, that the initial-value problem given by the linear partial differential equations (3.16)(3.17) is equivalent to the following nonlinear system of ordinary differential equations: df,/dx

= [V/X

- ( P ~ -~u/x)/G(u, x ~ X, A)lf,,

(3.18)

+ G(u, X, A)( 1 - ?)/EX,

(3.19)

du/dx = - V U / X

subject to the two-point boundary conditions u(0) = 0,

(3.20)

S,(R) = -RT.

(3.21)

u(R) = u1

(3.22)

and

If the end condition

is to be enforced, we adjoin the differential equation dA/dx = 0

(3.23)

on the Lagrange multiplier A. Equations (3.18)-(3.23) are the Euler equations of our original variational problem (2.8). EXERCISES 1. Prove that i f f satisfies the partial differential equations (3.16)-(3.17), it also satisfies the two-point-valueproblem (3.18)-(3.21). 2. Derive a partial differential equation for the minimization problem indicated by (2.7). 3. Discuss the formulation of the present optimization problem for an annular rotating disk subject to prescribed radial displacements at the inner edge.

282

11

OPTIMAL DESIGN OF ROTATING DISKS

4. Numerical Aspects The results of the preceding sections show that the solution of our optimization problem can be reduced to the solution of the linear, partial differential equation (3.16), subject to the initial condition (3.17). This is a simple result in that, for a fixed value of I , Eqs. (3.16)-(3.17) can be solved directly by using one of the several available numerical methods for initial-value problems. Alternatively, it can be integrated using the theory of characteristics, a procedure that in the present case yields the system of ordinary, nonlinear differential equations (3.18), (3.19), and (3.23) subject to the two-point boundary conditions (3.20)-(3.22). It is worth noting that if instead of enforcing the value of the radial displacement u(R) we prescribe the stress or strain at the origin, I can be obtained from (3.13), u can be computed in the forward direction using (3.19)-(3.20), andf, can be computed subsequently in the backward direction using (3.18) and (3.21). Finally, h and cr are computed from Eqs. (3.9) and (3.12). This appears to be a remarkably simple solution of the original optimization problem. Clearly, integration of Eq. (3.19) in the forward direction by numerical methods requires an asymptotic expansion around the origin since u/x is not defined at that point. This can be easily done in a number of ways as will be shown in the applications. EXERCISES

1. How would you construct a difference operator for the direct numerical solution of (3.16)-(3.17) starting from (3.2) and (3.3)? 2. Discuss the solution of (3.18)-(3.23) by quasilinearization. 3. Find an asymptotic expression for u(x), x e 1, in (3.19)-(3.20). Compare your results with those given in the next section.

5. Design Constraints. Successive Approximations The analytical results obtained in Section 3 owe their existence to the explicit determination of the optimum design condition (3.12) which in turn is a consequence of the continuous differentiability assumption made on h. In most realistic engineering applications one must impose a number of constraints on the design h. In order to be able to handle constraints of the form

g(h, x ) so,

(5.1)

or more generally, of the type

d h , cr

>

u, X > 5 0,

(5.2)

5. DESIGN CONSTRAINTS. SUCCESSIVE APPROXIMATIONS

283

we shall consider an algorithmic solution of Eq. (3.10) based on a successive approximation scheme, in particular, a version of approximation in policy space. In fact, we guess an initial design h(O) and compute f ( ” ( u , x ) using Eq. (3.10), which we rewrite

fil)= (G’(u, X , A)h‘O’ + (f!,l))’/h(o))(l - v2)/2Ex+ v ~ ! ” u / x f(l)(u, R ) = - RTu,.

(5.3)

This automatically leads to an upper bound on the functionf(u, x ) . We then improve the design by using the optimality condition

h(’) = arg min(G’(u, x , A)h + (fJ1))’/h), h E H*

(5.4)

where H* is the set of all admissible designs, i.e., h E H* if and only if h satisfies Eq. (5.1), and where u is given by the solution of Eq. (3.7) subject to the initial condition u(0) = 0. The resulting function h(’) is substituted back into Eq. (5.3) and the process is repeated again until convergence is achieved. We turn now to the solution of the generic partial differential equation (5.3) where is understood to be the design function computed in the previous iteration. It can be easily proved that (5.3) admits a quadratic solution in u, namely, f ( ” ( u , x ) = ~ ( x+) b(x)u + &(x)u’.

(5.5)

Substitutingf‘” given by (5.5) into (5.3) and equating terms in powers of u, we obtain

da/dx = Apxh‘O) + b’(1 - V ’ ) / ~ E X ~ ( ~ ) ,

(5.6)

db/dx = p o ’ ~ ’ h ( + ~ )bv/x + bc(1 - v’)/Exh(O),

(5.7)

+ ~ V C / X+ ~ ’ ( 1- v’)/Exh(O),

(5.8)

dc/dx = -Eh(O)/x

a system of Riccati equations for the coefficients a , b, and c at a generic iteration, subject to the initial conditions at x = R

u(R) = 0 ,

(5.9)

b(R) = -RT,

(5.10)

c(R) = 0 ,

(5.1 1)

which follow from consideration of Eq. (5.3). The stress function f:’) can be readily computed from ( 5 3 , i.e.,

fJ’)(u,

X) =

b(x) + C ( X ) U .

(5.12)

284

11

OPTIMAL DESIGN OF ROTATING DISKS

Substitutingf,“) given by (5.12) into (5.4), the improved design h(’) is given by

h(’) = arg min(G2(u, x, I ) h + (b + cu)’/h),

(5.13)

hsH*

where u satisfies the linear equation du/dx = - (b

+ CU)( 1 - V ’ ) / E ~ ( ~-) XVU/X,

(5.14)

obtained from (3.7) and (5.12), subject to the initial condition (3.20), and where I in (5.13) is given by

I

= (E/p(l - v)) lim(u/x)’,

(5.15)

x- 0

an equation that follows from (3.13) and the second equation of (2.2). Clearly, (3.20) is insufficient to integrate (5.14) unless the limit u/x as x tends to zero is defined. In addition, Eq. (5.15) requires knowledge of the same limit. This is easy to calculate by assuming an appropriate asymptotic expansion of u around x = 0. For example, in linear terms we have u(x) = u(0)

+ xu’(O),

x small,

which, on account of (3.20) and (5.14), yields lim u(x)/x

=

- (Eh(O)(O)/(I - v)

+ ~( 0 ) )lim b(x)/x,

(5.16)

x+o

x-0

the desired result. Summarizing, given an initial design h ( O ) we integrate Eqs. (5.7) and (5.8) subject to the initial conditions (5.10) and (5.1 I), respectively, until a fixed value of x = x o , xo 4 1, is reached. Then Eq. (5.14) is integrated forward subject to the initial conditions u(xo) = xo lim u(x)/x.

(5.17)

x-0

Simultaneously with this integration we compute the improved design h(’)(x) by means of Eq. (5.13) where I in that equation is given by (5.15). The process continues in the same fashion until convergence is reached. We note that (5.6) has not been used in this process. It may, however, be used as a check of the computations. In fact, since u(0) = 0, Eq. (5.5) yields a(0) =f‘”(O,

0).

(5.18)

As an exercise, the reader should now prove that 241

= u(R) = X(R)(l

- v2)/ERT - Y ( R ) p d / R T ,

(5.19)

6. EQUILIBRIUM AND COMPATABILITY-EULER

EQUATIONS

285

where X and Y satisfy the differential equations

dX/dx = b2/xh(0),

X(0) = 0,

(5.20)

d Y/dx = u ~ “ ’ x ’ ,

Y(0)= 0.

(5.21)

and

Equations (5.20) and (5.21) can be integrated simultaneously with Eq. (5.14), providing through Eq. (5.19) a check in the computation of u(R). EXERCISES 1. Prove that if a, b, and c are a solution of the initial-value problem given by (5.6)-(5.11), thenfgiven by (5.5) satisfies (5.3).

2. Show the numerical stability of a, b, and c given by (5.6)-(5.11).

6. Equilibrium and Compatibility-Euler

Equations

The results of the previous sections have been derived from the fundamental equation (3.10) obtained from dynamic programming considerations. We show now how the equation of equilibrium and compatibility of a rotating disk of fixed design can be derived from the Bellman-Hamilton-Jacobi equation (3.10). In fact, for a fixed design we can drop the min operation in (3.10), i.e.,

f, = (G2(u,X, A)h +fU2/h)(l - v2>/2Ex+ vfUu/x, (6.1) where G’(u, x , A) is given by (3.1 1). We wish now to derive a differential equation for the stress function f.. To this end we compute the derivatives of

f, with respect to x , i.e., dfu/dx =f,uu’

+f,x

*

(6.2)

Now, computing dfx/duusing Eq. (6.1) we obtain

f,, = (PO’X’ - Eu/x)h + v ~ , / x+ ( V U / X + (1 - v’)f,/Exh)f,, .

(6.3)

Comparison of (3.7), (6.2), and (6.3) yields

df,/dX

= vf,/x

- E ~ u / x+ po2XZh,

(6.4)

which we recognize as the equation of equilibrium in terms of the stress function f, = -06,hx and the displacements u. Adjoining Eq. (3.7), an equation of compatibility, to Eq. (6.4), the quantities u andf, remain uniquely determined when appropriate boundary conditions are considered. Elimination of u between (3.7) and (6.4) yields the classical second-order equation for the stress functionf, of a rotating axisymmetric disk.

286

11

OPTIMAL DESIGN OF ROTATING DISKS

EXERCISES 1. Using a procedure similar to that used to derive (7.14) and (7.15), derive an ordinary differential equation for the quantity fx . 2. Use (3.7) and (6.4) to derive the classical formulas for the rotating disk of uniform thickness. 3. A so-called disk of constant strength is characterized by the fact that ur = a@= constant throughout the disk. Using this condition and the Euler equations, find the profile of such

a disk. 4. Integrate the Euler equations assuming h = kr", where k and rn are given numbers.

II. PRESCRIBED DISPLACEMENTS

7. Optimization In our previous treatment of the disk we considered the potential energy as a measure of the degree of deformability of the structure and used this quantity as a constraint for minimum weight design. In engineering applications, however, one is led to consideration of not one but several types of constraints and to comparison of the corresponding results. For this reason we wish to extend the results previously discussed by incorporating, in the computational scheme, a family of design constraints. This can be done using ideas of invariant imbedding. Specifically, we are interested in terminal constraints such as

W ( N ,R) = 0,

(7.1)

R),R) = 0,

(7.2)

F(f"(U,

and integral constraints of the form loRGi(u,fu, h, r ) dr = m i ,

i = 1,2,. .. , N .

(7.3)

Constraints of the type given by (7.1) and (7.2) are incorporated as boundary conditions while the integral constraints (7.3) may be easily handled by Lagrange multipliers. In order to present the general ideas of the method we solve here the minimum weight design problem of a rotating disk subject to prescribed radial traction T and displacements u1 at the outer edge. To this effect we rewrite the equations of equilibrium and compatibility given by (3.7) and (6.4) in the form

dU/dx = - ( V / X ) U - (( 1 - v 2 ) / E h x ) ~ ,

+ ( V / X ) U + po2x2h,

dv/dx = - ( E ~ / x ) u

(7.4) (7.5)

287

7. OPTIMIZATION

where o stands for the stress function S. = -ohx, subject to the boundary conditions u(0) = 0,

(7.6)

u(R) = u1,

(7.7)

v(R)= -RT.

(7.8)

This problem is overdetermined if h is assumed to be known. Clearly, this is not the case because h is precisely the design variable. In order to determine the optimal profile h, we formulate the following optimization problem :

I. phr dr, R

min h

(7.9)

subject to the constraints (7.4)-(7.8). As usual, we introduce the minimum weight function R

g(u, o, x ) = min

(7.10)

phr dr.

h

By the principle of optimality we readily obtain the partial differential equation

[

-gx = min phx h

-

v -u (x X

v

1-vz +Ehx

+ po’x2h)g.],

(7.11)

where g is subject to the end condition (7.12)

g(u, U, R ) = 0. The optimality condition follows from (7.1 l), i.e., 1 -v’ h = arg rnin phx - Ehx

[

X

By proceeding as outlined in Chapter 9, we derive ordinary differential equations for the quantities gu and g o , namely,

Since v is not given at x

= 0,

we determine it by requiring

o(0) = arg min g(u, v, 0).

(7.16)

288

11 OPTIMAL DESIGN OF ROTATING DISKS

(7.17) an initial condition on g o . To determine additional conditions we observe that (7.12) must hold simultaneously with (7.7) and (7.8). Forming the Lagrangian function

L = g(u, U, R) - P~(U - ui) - p 2 ( ~+ RT) = 0,

(7.18)

we can treat u and v as arbitrary and require that u and u be such as to minimize L. Therefore gu(% y,

R) = P1,

g,@, u, R) = p2

*

(7.19)

Now it is clear that pl and p2 are not independent multipliers. In fact, by multiplying (7.18) by a constant factor A, the optimization problem remains unchanged and we can set pi or p 2 , arbitrarily. We choose (7.20)

g,(u, v, R) = l / I ,

the remaining boundary condition for Eqs. (7.14) and (7.15). Introducing the notation

v = -Ag",

U=Ag,,

(7.21)

Eqs. (7.14), (7.15), (7.17), and (7.20) reduce to

dU/dx = -(V/x)U-((l

- v2)/Ehx)V,

dY/dx = - (Eh/x)U + (v/x)V,

U(O) = 0,

(7.22)

U(R) = 1,

(7.23)

and the optimality condition (7.13) to

h = arg min Ipxh +

[

-u +-

(:

Ehx

Eh

u

+ -Xv v + pwZx2h (7.24)

where the constant I , which is to be determined, satisfies the differential equation dA/dx = 0. (7.25) It is clear now that the system (7.4)-(7.8) plus (7.22)-(7.24) is determined and furnishes the necessary conditions for optimality. Comparison of u and 8 with U and V through Eqs. (7.4)-(7.7) and (7.22), (7.23) shows that the adjoint variables U and V represent the displacement and stress function, respectively, of a disk subject to a (virtual) unitary displacement at the outer edge and to no centrifugal forces. Along with this observation it is convenient to consider Eqs. (7.4)-(7.7) and (7.22), (7.23), where h is given by (7.24) and A is a constant to be determined such as to satisfy ( 7 . Q as the pertinent necessary conditions.

289

EXERCISES

This interpretation facilitates the determination of the constant A. In fact, let h(x, A,), u(x, A,), V(x,A,), u(x, A,), and U ( x , A,) denote the values of h, u, V,u, and U, respectively, satisfying Eqs. (7.4)-(7.7) and (7.22), (7.23) evaluated for a particular value A = A,. In general, v(x, A,) obtained in this fashion will not satisfy Eq. (7.8). We form now the quantities

h*

= ph(x, A,),

u* = pu(x, A,),

u* = u(x, A,),

v* = p V ( x , A,),

u* = U(x, A,),

(7.26)

where p

= RT/U(R,A,).

(7.27)

It may be easily proved by inspection that the new quantities defined by (7.26) are a solution of the necessary conditions and that in addition they satisfy the constraint (7.8). Therefore,

h*

= h(x,

A,) = ph(x, A,),

V* = V ( X ,A,).= /LV(X,A,), u* = U(X, A,)

u* = v(x, A,) = pu(x, A,), U* = U(X, A,) = U(X, A,), =

(7.28)

U ( x , A,),

where 2, is the value of A for which Eq. (7.8) is satisfied. When the design is unconstrained, h given by (7.24) can be explicitly given by

h

= [(l - V’>(VV/X’)/(A~E - E’UV/X’

+ ~Eo’xU)]”’,

(7.29)

an expression for h that substituted into Eqs. (7.4)-(7.7) and (7.22), (7.23) yields a nonlinear boundary-value problem in terms of u, u, U , and V which can be treated by quasilinearization and invariant imbedding. Here we shall resort to a successive approximation scheme for the solution of the problem. A simple, first-order procedure consists in the use of a nominal design h(O’ to compute the quantities u, u, U , and I/, which substituted in the optimality condition (7.24), lead to an improved design /z(’), and so on. In the following we develop a stable two-sweep iteration procedure for the implementation of such a successive approximation scheme. EXERCISES 1. Derive the necessary conditions using the minimum principle. 2. Show that gx satisfies the ordinary differential equation dgJdx = agx/ax.

3. Discuss the solution of the necessary conditions by quasilinearization.

290

11 OPTIMAL DESIGN OF ROTATING DISKS

8. Successive Approximations Using ideas of invariant imbedding we consider the family of annular disks with fixed external radius R subject to radial displacements u(R) = u1 and with an inner edge of variable radius x subject to the stress function v(x). In the usual fashion we consider themissing boundary conditions u(x) given by

+ b(x)v(x),

u(x) = a ( x )

(8.1)

i.e., a linear combination of the dara values u(R) = 1 and v ( x ) . Differentiation of (8.1) with respect to x yields u’(x)= a’(x) + b’(x)v(x)+ b(x)v’(x).

(8.2)

Elimination of u ’ , v’, and u from (7.4),(7.5),(8.1), and (8.2)yields, aftercollecting terms in u,

+ Ehba/x - pu2x2hb, dbldx = -2vb/x + EhXb2/X2- (1 - V2)/EhX,

daldx

=

-va/x

(8.3) (8.4)

a set of differential equations for the quantities a and b subject to the initial conditions a(1) = 1,

(8.5)

b(1)

(8.6)

= 0.

Substitution of u given by (8.1) into Eq. (7.5) yields dvldx

=

( v - Ehb)v/x - Eha/x + p u 2 x 2 h ,

(8.7)

subject to the initial condition u(0) = 0 which follows from the requirement that 0, be finite at x = 0. Asymptotic considerations similar to those made in the derivation of Eq. (5.16) in Section 6 show that the limiting value of v / x in Eq. (8.7) can be given by the expression lim v ( x ) / x = -((I x+o

- v)/Eh(O)+ b ( 0 ) )lim a / x . x+o

(8.8)

Thus the numerical integration of (8.7) can be performed using the initial condition u(xo) = xo lim v ( x ) / x , x-0

where xo is a value near the origin. Clearly, Eqs. (7.22) and (7.23) can be solved using the same procedure. We find U , given by U(X) = a ( x )

+ b(x)V(x),

(8.10)

29 1

8. SUCCESSIVE APPROXIMATIONS

where ii satisfies the equation

dii/dx = - vii/x + Ehbi/x,

ii(1) = 1,

(8.11)

and b = b, is given by (8.4) subject to (8.6). Similarly, V is given by the equation

dV/dx = (V - Ehb)V/x - Ehii/x,

(8.12)

subject to the initial condition

(8.13)

V(xo)= xo lim V(x)/x, x-ro

where the limit is given by lim V ( x ) / x= -((1

- v)/Eh(O)+ b(0))lim ii(x)/x.

(8.14)

x-ro

x-0

We are in a position now to implement a stable, two-sweep, successive approximation scheme for the determination of the optimal disk. In fact, starting with an initial estimate for the design h(O),we compute the quantities a, b, and ii by backward integration of Eqs. (8.3)-(8.6) and (8.11) until a prescribed value x = xo 6 1 is reached. At this point we initiate the forward integration of Eqs. (8.7), (8.9) and (8.12), (8.13), yielding the quantities v(x) and V ( x ) , respectively. We then compute the quantity

A(’) = arg min[(lp - EuU/xz + p o 2 x U ) h + ((1 - v2)/Ehxz)vV], (8.15) hEHt

an equation derived directly from (7.24). Finally the quantity

h*

= h‘”RT/v(R)

(8.16)

furnishes an improved estimate of the design h. Similarly, we can define the quantities v* = -RTv(x)/v(R),

and

u* = u,

associated with the improved design h*. Clearly, v*(R) = - RT. The process is continued until convergence is achieved. The constant 1 appearing in Eq. (8.15) can be evaluated using

dgxldx = agx/ax,

(8.17)

an ordinary differential equation for the quantity gx that may be derived using the same procedure employed to derive (7.14) and (7.15). Using (7.1 1) (without the min operation) and (7.21), Eq. (8.17) yields

dH/dx where H

= - lg,

.

+ ( l / x ) H = 21ph + 3pwZxhU,

H(0) = 0,

(8.18)

292

11 OPTIMAL DESIGN OF ROTATING DISKS

Making H

+ 1pH2 ,where H I and H , satisfy the differential equations dH,/dx + HJX = 3p02xhU, Hl(0) = 0, dH,/dx + H Z / x= 2h, H2(0) = 0,

= Hl

it is easy to derive for the quantity I p , the expression

+ ( ( 1 - V2)/Eh)V/X)V - (- Ehu/X + vV/X + p ~ ’ ~ ~ h ) U ] /-( H2). xh

1p = [HI - (vU/X

(8.19)

EXERCISES 1. Discuss the solution of an annular rotating disk whose inner edge of radius a is subject to the condition u(a>

+ pu(a)/a = 0.

2. Derive Eq. (8.19).

9. Numerical Examples To show the feasibility and accuracy of the procedures developed in previous sections, we present some numerical examples. In the computations we have used the following dimensionless variables :

Z = (R/Eu,)cT,

y = ~ w ’ R ~ / E u , , X = x/R,

w = u/u1,

w = u/u1,

S = v/RT,

H = (Eul/RT)h, (9.1) S = VIRT.

The results of the first series of numerical examples are presented in Figs. 1 1 - 1 and 11-2. Equations (3.18)-(3.23), conveniently modified by the change of variables (9. I), have been solved using quasilinearization for five different values of y. The resulting dimensionless designs H and stresses C appear in solid lines in Figs. 1 1 - 1 and 11-2, respectively. The equations were integrated using an Adams-Moulton scheme with a step size of 0.5 x lo-,. When y = 1, three iterations were needed to repeat four significant figures, whereas for y = 0.25, only two iterations were required to achieve the same accuracy. The choice of an initial estimate of the design did not seem to influence the rate of convergence of the process. In fact, the same rate of convergence was observed using H = 1, H = 0.1, and H = 1 - X as initial approximations. Using the values of a,(O) obtained in this fashion, we computed 1given by Eq. (3.13) and then proceeded to integrate Eqs. (3.19) and (3.20).The values of the radial displacements computed by this one-sweep process agreed in seven significant figures with those obtained solving the boundary-value problem (3.18)-(3.23) by using quasilinearization.

9. NUMERICAL EXAMPLES

293

RT I

Fig. 11-1. Thickness of minimum-weight rotating disks for two design criteria: minimum potential energy; - - u, and T prescribed. y = pu2R3/Eu,. (After Distkfano [1972].)

-

The dashed curves in Figs. 11-1 and 11-2 are the thickness and the radial stresses, respectively, of the minimum weight disks subject to prescribed radial displacements and radial tractions at the edge. These results were obtained using the method of successive approximations presented in Section 8. At each iteration the thickness was improved by computing h with (8.15) where il was obtained using (8.19). The integrations were performed using an Adams-Moulton routine with a step size of 0.5 x lo-’. Here, as in the previous example, the choice of the initial approximation does not seem to influence the rate of convergence of the process, but the value of the parameter y does. In fact, for y = 1, six iterations were needed to repeat four significant figures, whereas for y = 0.25 only five iterations were required to reach the same accuracy. In Figs. 11-3 and 11-4 we have plotted the dimensionless thickness and stresses, respectively, of minimum weight disks subject to prescribed radial displacements and tractions at the edge, for y = 1 , and subject to the following additional design conditions: (1) unconstrained design, (2) H I 0 . 9 0 , and

294

11

OPTIMAL DESIGN OF ROTATING DISKS

1.40

I .30

I .20

1.15

0

0.5 r -

1.0

R

Fig. 11- Radial stresses of minimum-weight rotating disks for two L-sign criteria: -minimum potential energy; - - - u1 and T prescribed. (After Distbfano [1972].)

0.61 0

I

0.5

I.

Fig. 11-3. Optimal profiles of rotating disks. Design criterion: u1 and T prescribed. Constraints: (1) none; (2) (Eu,/RT)h50.90; (3) (Eul/RT)h2 0.80. y = ~ w ~ R ~=/ 1.0. E u ~ (After Disthfano 119721.)

295

9. NUMERICAL EXAMPLES

1.55 I.50

-hE U l RT

I .40

1.30

1.20

0

1.0

0.5

-Rr

Fig. 11-4. Radial stresses of minimum-weight rotating disks. Design criterion: u1 and T prescribed. Constraints: (1) none; (2) (Eu,/RT)h5 0.90; (3) (Eul/RT)h2 0.80. y = pu2R3/Eu1= 1.0. (After Distkfano [1972].)

TABLE 11-1 Comparison of Volumes y

Design criteria

~~

0.00

Minimum potential energy (unconstrained) Radial displacement and radial traction (unconstrained) Radial displacement and radial traction constraint: (Eu,/RT)h 5 0 . 9 0 Radial displacement and radial traction constraint: (Eu,/RT)h 2 0.80 Uniform strength (unconstrained)

=p u 2 R 3 / E ~ 1 ~~

0.25

~

~

0.50

0.75

1.00

2.1991 1 2.29971 2.40979 2.53068 2.66394 2.1991 1 2.29814 2.40300 2.51409 2.63184 2.1991 1 2.29814 2.40300 2.51428 2.63894

2.1991 1

-

-

2.52911 2.64352

2.29820 2.40326 2.51472 2.63309

296

11 OPTIMAL DESIGN OF ROTATING DISKS

(3) H 2 0.80. The results presented in Figs. 11-3 and 11-4 were obtained using the method of successive approximations developed in Section 8. The unconstrained solution converged in six iterations, whereas the constrained designs H 5 0.90 and H 2 0.80 needed 10 and 14 iterations, respectively, to repeat four significant figures. In all cases the initial approximation was H = 1. Finally, the volume of the disks designed under various assumptions is presented in Table 11-1. The insensitivity of the volume with the design criteria is noticeable.

10. Conclusions The optimum design of rotating disks under various design criteria has been studied using ideas of dynamic programming and invariant imbedding. In all the cases studied, the methods for numerical computation derived from those theoretical considerations afford numerical stability, an important property if accuracy is required, since in the present problem the stresses are highly sensitive to small changes in the thickness of the disk. This extreme sensitivity is compensated, on the other hand, by the high stability of the volume with respect to the design criterion, a striking feature of rotating disk design. NOTES, COMMENTS, AND BIBLIOGRAPHY

In this chapter we present, with minor changes, the article N. DistCfano, Dynamic Programming and the Optimum Design of Rotating Disks, J. Optimization Theory Appl. 10, No. 2 (1972), 109-128.

For a different computational method see: D. Nagy and N. Distbfano, A Dynamic Programming Approach to Optimal Disc Design. Proc. Internat. Conf, Variational Methods Engrg. organized by Southampton Univ. and Internat. Assoc. for Shell Structures, Southampton, 1972.

In this last paper, taking advantage of the special structure of the problem, the necessary conditions are written as a set of three nonlinear ordinary differential equations subject to two-point boundary-value conditions. A convenient shooting scheme is implemented for the numerical solution. Abundant numerical experimentation and results are presented. For the solution of the rotating disk using discrete dynamic programming, see D. Nagy, Some System Identificationand OptimizationProceduresin Structural Mechanics, Ph. D. Dissertation, Dept. of Civil Engrg., Berkeley, California (1971).

NOTES, COMMENTS, AND BIBLIOGRAPHY

297

A treatment of a discrete nonlinear disk by dynamic programming has been presented in Section 8-17. For a derivation of an optimality condition using the principle of stationary mutual potential energy, see J. M. Chern and W. Prager, Optimal Design of Rotating Disk for Given Radial Displacement of Edge, J . Optimization Theory Appl. 6 , No. 2 (1970), 161-170.

For a discussion of disks of constant strength see the article by Chern and Prager above, or J. P. Den Hartog, “Advanced Strength of Materials.” McGraw-Hill, New York, 1952.

Chapter 12A

Modeling and Identification of Hereditary Processes 1. Introduction

In a number of important applications in various domains of engineering science, we are required to deal with processes whose intimate structure is partially or completely unknown. Yet, we must construct approximate mathematical models of the processes in order to satisfy various technological needs. This is the case, for example, when we are asked to describe the time behavior of viscoelastic materials. The construction of constitutive relations for viscoelastic materials still rests heavily on the phenomenologicalgrounds of the theory of hereditary processes developed by Volterra at the beginning of the century. Thus, one sees the interest in providing a solid methodological framework for this theory in connection with the use of the digital computer. This and the following chapter are intended to be a step in that direction. Although most of our theory is presented with the theory of viscoelasticity in mind, the reader will have no difficulty in applying some of these ideas to problems involving other hereditary processes. Some aspects of the theory of approximation could not be absent from the context of the present chapter. In this fashion some Tauberian results are shown to apply naturally to the study of the asymptotic properties of a class of viscoelastic materials and structures. Similarly, the notion of differential approximation is introduced 298

2. MODELING ASPECTS IN VISCOELASTICITY

299

as the basic ingredient relative to the numerical solution of nonlinear Volterra integral equations. The present chapter ends with an expository presentation of a problem of paramount importance in modern engineering science: the identiJicution problem. In the next chapter we continue with the discussion of this problem in the context of the theory of viscoelasticity. 2. Modeling Aspects in Viscoelasticity

In this section we introduce some notions on the mathematical representation of time-dependent behavior of real materials. To this end we present a brief survey of the various models afforded by the theory of viscoelasticity. The exposition is restricted to a scalar, time-invariant version of the problem. Some extensions and properties are presented subsequently in connection with more general hereditary processes. a.

Diyevential Models

The simplest representation of a linear elastic material is afforded by Hooke’s law a = EE, (2.1) where Q is the stress, E is the strain, and E is Young’s modulus. Equation (2.1) is inappropriate when dealing with materials exhibiting time-dependent behavior, e.g., increase in deformation when subject to a constant stress. In a first approximation this behavior may be accounted for by assuming that the stress is strain-rate dependent. The simplest material in this class is given by Newton’s law a = rls, (2.2) where an overdot indicates a time derivative and q is the viscosity coefficient. The analog models usually associated with (2.1) and (2.2) are the spring and dashpot indicated in Fig. 12A-1.

SPRING

DASHPOT

Fig 12A-1

300

12A MODELING AND IDENTIFICATION OF HEREDITARY PROCESSES

Generalization of Eqs. (2.1) and (2.2) may be accomplished in a number of ways. For example, one can construct a number of analog models based on those given in Fig. 12A-1. Parallel and cascade arrangements of springs and dashpots yield the Voigt and Maxwell models, respectively, as indicated in Fig, 12A-2. These models are associated with the differential equations Voigt: Maxwell :

EE + qf. = o, E8 = (E/q)a + 6.

(2.3)

i

E

y' MAXWELL

VOlGT

Fig.12A-2

Simulation of more complex time behavior may be achieved by means of higher order models. For example, the Burgers model is obtained from a cascade arrangement of a Voigt and Maxwell model, as indicated in Fig. 12A-3. We can easily derive differential equations for higher-order models constructed with springs and dashpots. For example, consider the Burgers

BURGERS

Fig. 12A-3

30 1

2. MODELING ASPECTS IN VISCOELASTICITY

model given in Fig. 12A-3, and introduce the auxiliary (internal) variables and c2 defined by =strain of the dashpot in the Maxwell element of the Burgers model, c2 = strain of the Voigt element in the Burgers model. E~

Using this notation we can write

where and e2 are related to the observable variables E and CT by means of the equation of state E

= o/El

+ c1 + E

~ .

(2.5)

Equations (2.4) and (2.5) uniquely describe the time evolution of the system when appropriate initial conditions are specified. We can also write the constitutive equations of the Burgers model using a single, higher-order equation involving only the two observable variables o and E . To this end we have to eliminate the two internal variables and c2 from the three equations of (2.4) and (2.5), obtaining

a second-order differential equation. Clearly, analog models constructed with springs and dashpots may be justified only on heuristic grounds. They do not attempt to offer any molecular or structural interpretation of the material being studied. We could in fact just as well have considered the differential equation

as the general linear, second-order, constitutive relation, instead of the Burgers model given by (2.6) or any other alternative second-order model derived from springs and dashpots. Extension to order higher than 2 is obvious.

b. Linear Integral Models A linear system may be represented elegantly by using ideas of superposition. Let us see how this procedure applies to the modeling of linear viscoelastic materials. Let E(t, 2) be the E response of the material at time t, due to a unit step stress function o = l ( t - z) starting at time 2 5 t. Function

302

12A

MODELING AND IDENTIFICATION OF HEREDITARY PROCESSES

E(t, 7 ) is usually called the creep function. Here we assume that the material is time invariant; i.e., the response E(t, z) is invariant with respect to an arbitrary time translation h. In symbols, E(t

+ h, z + h) = E(t, 7 ) .

(2.8)

This, a group property, is satisfied if and only if B is a function of the type

q t , 7 ) = C(r - 7 ) .

(2.9)

Now, the response of the material to a general o input starting at time to will be given, by superposition, by (2.10) Integration by parts yields E ( t ) = (1/E)ff(t)-k

f o ( 7 ) f ( t- Z) dt,

(2.11)

to

where we have put 1/E = C(O), f ( t - 7 ) = - ( a / a r ) c ( t - z),

to I 7

< t.

(2.12)

Clearly, in (2.11) the term o/E accounts for the instantaneous response while the integral term exhibits the time-dependent behavior. The lower limit of integration in (2.11) comes from the assumption that o(t) = 0 if t < t o . When this assumption does not hold any longer and the stress history is defined in the whole semiinterval (- 03, t ) , we consider the &-arelation given by E ( t ) = (l/E)o(t)

+ J"

o ( z ) f ( t - 7 ) dz.

(2.13)

-W

Similarly, if we start the analysis using the relaxation function R(t), i.e., the o response to a unit step function E = l(t), we arrive at the 0-E relationship o(t) = EE(t)

+ 1'

~ ( ~ ) r-( 7t ) dt,

(2.14)

-m

where E = R(O), r(t - 7 ) = -(a/az)R(t - z ) ,

-03

< 7 < t.

(2.15)

The reader familiar with the theory of Volterra integral equations will recognize thatfand r in Eqs. (2.13) and (2.14) are reciprocal kernels. See the exercises at the end of the section.

303

2. MODELING ASPECTS IN VISCOELASTICITY

There exists an intimate connection between integral and differential models of constitutive equations for viscoelastic materials. To see this, consider, for example, f i n Eq. (2.11) given as a sum of exponential functions N

f ( t ) = C ci exp( - Ai t ) .

(2.16)

i=l

Substitution off given by (2.16) into (2.11) yields N

E

= (l/E)o

+1

(2.17)

CiZ,(t),

i=l

where the zi are auxiliary variables defined by zi = Jt:o(z) exp[ - A,(t

- z)] dz,

(2.18)

which may be shown, on direct differentiation with respect to t , to satisfy the system of differential equations i i

+

= 6,

AiZ,

Zi(t0)

= 0.

(2.19)

Equations (2.17) and (2.19) are clearly an internal (or hidden) variables representation of a linear viscoelastic material. Note that the Ai are not restricted to be real numbers. Also, note that (2.4) and (2.5) are a special form of (2.11) and (2.13), as expected. The present results show the connection between the internal-variables approach and Volterra's approach, in the linear case. Additional equivalences may be found if we assume thatf, instead of being given by (2.16), satisfies the initial-value problem

c qf'"

N- 1

+f") = 0,

f'"(0) = f i ,

i=l

i = 0, 1, ..., N - 1, (2.20)

where the a, an,d fi are given constants, and where f i ) stands for d%dti. We compute now the ith derivative of E ( ' ) in (2.11) and form the sum a, d i )+ E"). Recalling (2.20), this procedure yields

cy=-:

N

1 ai

c b, #), N

E(i)

=

i=l

i= 1

a,

E

1,

(2.21)

a differential constitutive equation where the bi are constant coefficients that depend on the a, andf, coefficients. See the exercises below. Clearly, (2.21) is the nth-order version of (2.7) derived from the integral constitutive equation. We observe that if the roots of the characteristic polynomial N-1 i=O

ail'+ AN = 0

(2.22)

304

12A MODELING A N D IDENTIFICATION OF HEREDITARY PROCESSES

are equal to the exponentials Ai appearing in (2.16), then the representations given by the Volterra integral equation (2.1 1), the hidden-variables approach (2.17), (2.19), and the Nth-order ordinary differential equation (2.21) are equivalent. c. Nonlinear Models

Here we make a brief incursion into the challenging and almost unexplored domain of modeling functional equations to simulate nonlinear viscoelastic behavior. To this end, we present nonlinear versions of some of the functional operators previously discussed. A straightforward generalization of the hidden-variables formulation given by the linear system (2.17) and (2.19) consists in postulating the existence of an equation of state

0,

(2.23)

4 t ) = g(4t),z(t),

where z(t) is the Nth-dimensional vector whose components zi(t),i = I, 2, . . . , N , are the hidden variables of the process which are assumed to satisfy the differential equations of evolution

dzjdt = h(a, Z, t),

~ ( 0=) 0.

(2.24)

Clearly, g in (2.23) is a scalar function of cr and the vector function z. In turn, h in (2.24) is a vector function of the same quantities. Although much effort has been devoted to attaching specific physical meanings to the hidden variables zi,a process such as (2.23) and (2.24) will be observed generally through the external variables cr and E , the internalvariables vector z merely playing the role of an auxiliary, nonobservable modeling quantity. Some comments and literature on the early work in this field can be found at the end of the chapter. A different modeling approach going back to Volterra himself is based on integral expansions of the form ~ ( t= )

It

cr(z)fi(t - z) dz

-m

1 '

+ -J 2!

-m

J

o(71b(z,)f,(t

-a

1 + . .+*

N!

' * *

- z1, t

- 4 dz, dz,

a(zl) . . . a(z,)f,(t

Jf

- zl,

. . . , t -),z dzl . . . dz,,

-m

(2.25) where fl(t),f2(tl, t,), . . . , are characteristic functions of the material. The rationale behind this model will be better understood in the next sections, in connection with some more general hereditary processes.

305

2. MODELING ASPECTS IN VISCOELASTICITY

By assuming various forms of the kernels fi, fi, . . . ,fN, we can reduce (2.25) to a number of nonlinear differential and integral equations of interest in the applications. To see how this can be done, we consider an example by assuming the kernels to be separable and of the type given by

nf ( t n

f , ( t - zl,. . .,t - z), = a,

Ti),

(2.26)

i= 1

where the a, are given coefficients andf(t) is a given function that contains, in general, a Dirac delta function to account for nonzero instantaneous deformations. Using (2.26), Eq. (2.25) reduces to

c (l/n!)a,[J' N

E(t)

=

n= 1

rJ(z)f(t - 7) d.1,.

(2.27)

-a

We now assume that the sequence y, given by (2.28)

converges toward a function y(z), as N -+ co. Let z be the inverse function of y ; i.e., z = y - l . Then, (2.27) reduces to Z(&) =

1' o ( z ) f ( t

(2.29)

- 7) dz,

-a

a nonlinear integral equation. Making explicit the 6 singularity contained in f, i.e., making f ( t >= (1 l a w ) + f(0,

(2.30)

Eq. (2.29), in expanded form, reads Z(E)

= (l/E)a

+

I'

a(t)f(t - z) dt.

(2.31)

-a

A similar, very convenient nonlinear model is furnished by the classical nonlinear Volterra integral equation & ( t ) = g(a)

+f

h(a(z), t - z) dz,

(2.32)

-W

or the simpler equation E ( t ) = g(4

+

1' h(o)f(t -

dz,

(2.33)

-W

where g and h are nonlinear functions of IT.We return to (2.33) in subsequent sections in connection with the identification problem in nonlinear viscoelasticity.

306

12A

MODELING A N D IDENTIFICATION OF HEREDITARY PROCESSES

d. Diyerence-Dcyerential Models We see here how difference-differential models can be employed as constitutive equations for viscoelastic materials. To this end we restrict the discussion to the linear case, and consider again the a-E relation E ( t ) = (l/E)a(t)

+ It

( ~ ( z ) f (-t z) dz.

(2.34)

- W

When (r = 0 for t < t o , Eq. (2.34) uniquely determines ~ ( t [or ) o(t)] when a(t) [or ~ ( t ) is ] given in the interval [ t o , t ] . In many important applications, however, this assumption does not hold. This is particularly true when the system is permanently evolving in time and, to make things worse, we ignore the 0 and E histories prior to a certain instant when we start observing the system. This situation poses new kinds of challenging problems in viscoelasticity, particularly in connection with a number of problems in biomechanics and geotechnics. To illustrate the ideas, let us consider the case where (T is observed in the interval [ - T,01, T being a given positive quantity. Using this information, our objective is to construct an estimate w(t) which approximates a(t) in [0, t ] in the sense of IJw - 011 I 6(T),

lim 6 = 0,

(2.35)

T-+m

where 11.11 indicates a prescribed norm. Under appropriate conditions on the kernelf and functions 0,E , we can show that w given by the solution of the functional equation

+ jr w(z)f(t

~ ( t=) (l/E)w(t)

- z) dz

(2.36)

t-T

approximates

0

in the sense of (2.35). Assumingfgiven by the expansion

it is not difficult to transform (2.36) into the difference-differential system (2.38) subject to the initial conditions zi(0) =

I

0

-T

w(z) exp(,Iiz) dz,

~ ( t =) ~ ( t ) ,

- T I t 5 0.

i = 1, 2,

. . . ,N ,

(2.39)

307

3. HEREDITARY PROCESSES

Equations (2.38) and (2.39) afford a generalization of the hidden-variables approach to the solution of ongoing viscoelastic processes. They were first introduced in connection with some problems in biomechanics. See the Bibliography at the end of the chapter. An alternative version of this problem is afforded by filtering models. This will be discussed in Section 20 in connection with the identification problem. See also Chapter 9. EXERCISES 1. Find the response of the second-order model given by Eq. (2.7), when subject to the

following inputs u = l(t),

u = l ( t ) sin wt,

E = l(t),

(a)

where l ( t ) denotes the unit step function. 2. Let C ( t )be the response of (2.7) to a unit step function u = l(r). Show that the response of (2.7) to any arbitrary stress history u(t), such that u = 0 for t < 0, may be computed by superposition as follows:

3. Show that ,-I

"I

~ ( ~ )f (T t) dr =

u(t

Jo

Jo

- T ) ~ ( Td)

~ .

(4

4. Let 2(s) be the Laplace transform of a function 4 t h i.e., 2 ( s ) = J.omr(t)e--"dt.

Show thatf(s) and ?(s) in Eqs. (2.13) and (2.14) are related by

+

+

(1/E f(s))(E i ( s ) ) = 1.

5. In Eq. (2.21), the b1 coefficients are related to the al and fi coefficients. Find recursive equations for such a dependence. 6. Let e(t, fo) be the response of (2.21) to a stress function u(t) = l(t - to)& - to), where g ( t ) is an arbitrary function of time. Show that

+ h, t o + h).

&(I, t o ) = &(t

(f)

7. Show that Eq. (f) holds if instead of (2.21) we consider Eq. (2.13) or (2.14).

3. Hereditary Processes

In an effort to include hysteresis effects in the models of electromagnetism and elastic phenomena, Volterra in 1909 introduced corrective terms in the constitutive equations of the classical theories. The corrections introduced by the Italian mathematician consisted in additive quantities that depended on

308

12A MODELING A N D IDENTIFICATION OF HEREDITARY PROCESSES

the whole previous history (functionals) of the pertinent state variables. This, an application of the theory of functionals developed by Volterra himself several years before (1887), was the starting point of the theory of hereditary processes. We illustrate the ideas by considering a hereditary correction to Hooke's law, given by E

= (l/E)a.

(3.1)

A correction to (3.1) in the hereditary sense consists in the addition of a term depending on the whole history of 6,i.e., f

E

= (I/E)o

+ F[a(z)], --OD

where F denotes a functional of postulated, i.e., if

Q

in the interval [-

4Ql + Qz)

=E(Qd

oc), t].

+ &(QZ),

If superposition is

(3.3)

Eq. (3.2) must be written @) = (l/QtJ(O

+ f-aQ ( T ) f ( t , 7) dz,

(3.4)

i.e., as a nonanticipatory linear functional of Q. Iffis assumed to be a function of t - 7 , Eq. (3.4) reduces to (2.1 1). In general, a hereditary process in the scalar quantities u and u depending on t , is given by the functional relationship t

~ ( t=) F[u(z)]. -m

(3-5)

In order to make explicit some structure in the functional relationship (3.5), we expand F i n series of powers, similar to a Taylor series. This is given by the series of multiple integrals

(3.6) Functionsfi,fi, . ..,the kernels of the multiple integral expansion, are known as the functional derivatives of F. A hereditary process may therefore be determined through knowledge of the functional derivatives fi,fi , .... In applications we usually consider truncated forms of (3.6). In this fashion, the construction of constitutive equations for nonlinear viscoelastic materials such as that given by (2.25) can be easily justified.

309

3. HEREDITARY PROCESSES

It should be clear that the theory of hereditary processes is of a phenomenological nature. In fact, it does not pretend to furnish any internal (structural) interpretation of the process to be studied. Its success lies precisely in that it provides adequate guidelines for the modeling of processes whose intimate dynamical structure is not well known or not at all understood. Had the constitutive relations of the process been derived from first principles, in terms of differential equations, the notion of heredity would be unnecessary. However, the need for the solution of practical problems invariably leads to the use of phenomenological descriptions of the real world. Thus, one sees the interest in the theory of hereditary processes as a valuable modeling tool. We note in passing that this last observation is probably more valid at present than it was at the beginning of the century when, using similar arguments, Volterra challenged Painleve’s objection on the importance of hereditary effects in mechanics. The pertinent references on this historical remark may be found in the Bibliography at the end of the chapter. Although the notion of heredity is indissolubly attached to Volterra integral equations, a most natural modeling tool, we must keep in mind that the ultimate structure of the mathematical model should not necessarily be restricted to integral equations. For example, a function z ( t ) satisfying the differential equation

i = g(2, a),

(3.7)

z(0) = 0,

is indeed a functional of a in the interval [0, t ] . Therefore the equation

40 = (1/JvJ(t) +

(3.8)

where z is given by (3.7), may be considered a hereditary correction to Hooke’s law. If, for example, we take g of (3.7) in the form g

=

-az

+ ba,

(3.9)

where a and b are positive constants, then (3.7) and (3.8) reduce to i:

+ a& = (l/E)6 + (b + @)a,

(3.10)

i.e., the standard viscoelastic solid. We leave this derivation as an exercise to the reader. It is interesting to note here that the price of avoiding Volterra integral equations in the constitutive relations of the process is the introduction of auxiliary variables which, as z in (3.7)-(3.8), are generally nonobservable quantities. This comment is of course pertinent in connection with the so-called rate theories in nonlinear viscoelasticity. Along similar reasoning lines, we might find it convenient to model hereditary processes in terms of difference-differential equations such as zi = s(u(t), u(t - 41, U W ,

(3.11)

310

12A MODELING AND IDENTIFICATION OF HEREDITARY PROCESSES

where t, is a time lag, or integrodifferential equations, or by more involved functional equations, such as partial differential equations. The determination of suitable functional structures for modeling purposes is a challenging area of research in modern engineering science. We return to this topic in connection with the identification problem. EXERCISES Derive Eq. (3.10). Construct a series expansion of the type (3.6) to represent the solution of the nonlinear differential equation dvldt

+ v2 = U.

(a)

See

A. A. Wolf, Volterra Kernels. A Generalization of the Convolution Integral for Solving Nonlinear Systems, J. Quunt. Spectrosc. Rudiat. Transfer8 (1968). 495-501. Show that if g in (3.7) is given by g = -A2

+h(d,

(6

where h(m) is an arbitrary nonlinear function of u, then the u-.s relationship given by (3.7) and (3.8) may be written in terms of the nonlinear Volterra integral equation

4. Dissipation of Hereditary Effects Consider a linear hereditary process characterized by the functional relation

The quantity u ( z ) f ( t , z) dz, z I t, may be interpreted as the contribution of the quantity u(z), acting during the small interval (7, z + dz), at the current instant t . The integral term in (4.1) is thus representing the linear superposition of all those effects. This is usually called Bokzmann’s superposition principle. We may restrict the class of functionsf(t, z) in (4.1) by postulating the principle of dissipation of hereditary efects which, in loose terms, expresses that the influence of an input u(z) dz at a future time t > z tends to be negligible for t - z sufficiently large. This is clearly satisfied if lim f ( t , z) = 0, r-+--co

in addition to other regularity conditions.

(4.2)

31 1

5. TIME-INVARIANT PROCESSES

If, in general, we consider a nonlinear functional relationship of the form t

v ( t ) = F[u($l,

(4.3)

-m

we may postulate the dissipation of hereditary effects by requiring that for a bounded function u(t), i.e., 1IuI1 5 M in (- co,t ] , and, for a given tl < t , the variation of v , with respect to a prescribed norm, due to a bounded variation of u in the interval (- co,tl], can be made as small as we please by making t - t , sufficiently large. Clearly, the dissipation of hereditary effects, or fading memory as it is sometimes called, is a stability concept that may be treated within the modern theory of stability. EXERCISES 1. Assume that f in (4.1) is given by f ( t , T) = S(t - T) - (a/a~)[G(~)[l- exp(-h(t - T))]]

(a)

where 6 is Dirac’s delta, h is a positive constant, and G is a positive, monotonically decreasingfunction. Show that the principle of dissipationof hereditary effectsis satisfied.

2. Do the same, assuming that f ( t , T) satisfies the condition

1 f ( t . 7)1 < A(r - TI-’, where u > 1 and A is an aribitrary positive quantity.

5. Time-Invariant Processes

Let u and v be related by a functional relationship t

v ( t ) = F[u(z)]. - W

If, for any positive number h and arbitrary function u, we have t+h

t

P[u(z - h)l = m91, -m

(5.2)

-m

i.e., if F i s invariant with respect to time translations, we say that the process is time invariant. Now, suppose that u and v are periodic functions with the same period w > 0. When the following property t+W

r

F[u(z - w)] = F[u(t)] --m

(5.3)

-m

holds, we say that the conditions of closed cycle are satisfied. In a celebrated memoire, Volterra proved the following theorem, valid for processes satisfying the principle of dissipation of hereditary effects:

312

I2A MODELING AND IDENTIFICATION OF HEREDITARY PROCESSES

If the conditions of closed cycle are satisfied for any period w > 0, the process is time invariant and conversely, if the process is time invariant, the conditions of closed cycle are satisfied for all w > 0.

Assuming the necessary conditions of regularity on F which ensure the existence of the expansion (3.6), Volterra has also proved that for a process to be time invariant, it is necessary and sufficient that the kernels in (3.6) be of the form A(t, 7)

=A0 -

f2(4 z1, 7 2 ) = f 2 0

$3

- TI,

t

- zd,

(5-4)

In particular, linear time-invariant processes are represented by v(t) = ku(t)

+

u(z)f(t - Z) dz, --m

(5.5)

where k is a constant. EXERCISES

1. Prove that if f ( t ) in (5.5) is given by f ( t )= t-' sin t, the principle of dissipation of hereditary effects holds. 2. We know that if the kernel f satisfies the ordinary differential equation (2.20), u and Y in (5.5) satisfy differential equations of the type

Show that the condition for time invariance implies that the aLand b, in Eq. (a) must be constants.

6. Aging Processes

When the conditions for time invariancy discussed in the last section are not fulfilled,then we say that the process is time uarying, In this case the characteristics of the system change with time. An important class of time-varying processes occurring in the applications is that of aging processes. Roughly speaking we say that a time-varying system is undergoing an aging process if the response of the system to a prescribed class of input functions decreases,in a certain sense, with the age of the system. Consider, for example, the linear process

313

7. BOUNDS AND ASYMPTOTIC BEHAVIOR

In order to exhibit aging we may require f(t

+ h, 7 -t h) < f ( t , 7>,

7


(6.2)

for arbitrary h > 0, or the less demanding condition G(t

+ h, t o + h) < G(t, t o ) ,

(6.3)

where

is the response of the system to a unit step function starting at t = to EXERCISE Show that the linear o--Eviscoelastic law

+

- ~ ( t=) ( l / E ( O ) o ( t )

s:,

d ~ ) f ( t 7, ) d7,

where f i s the kernel of the Maslov-Arutiunian law for creep of concrete given by

+ + A / ~ ) y (-l expi-h(t

f ( t , 7 ) = - (a/a~)[l/E(~)(1

- 7)])1,

t >_ 7 2 to > 0,

where A , y, and are positive constants and E ( t ) is a positive, monotonically growing function of 1, is an aging process in the sense of (6.2).

7. Bounds and Asymptotic Behavior Consider the linear, time-invariant process characterized by the Volterra equation u(t) = u(t)

+ A / * u ( t ) f ( t- 7)dt. 0

(7.1)

A meaningful class of problems arises when we are required to derive some properties of function u from some given properties of functions z1 and f. In particular, we are interested in conditions which ensure boundedness and asymptotic properties of u as t -, 00. To this end we make use of a number of available Tauberian results. In order to facilitate the exposition we present some of the most interesting results of this class in theorem form. In subsequent sections we show how these results apply to the solution of a number of problems involving viscoelastic structures.

Theorem 1 (Paley and Wiener) If the limit c = lim v(t) t+m

3 14

12A

MODELING A N D IDENTIFICATION OF HEREDITARY PROCESSES

exists and f is absolutely integrable, i.e.,

exists, then the limit of u ( t ) in Eq. (7.1) is lim u ( t ) = c / (1

+A~

~ ~ dz), f ( r )

(7.4)

1-rW

if and only if

A

e - " f ( t ) dt lom

Re s > 0.

# - 1,

(7.5)

If we require the kernelf to be nonnegative, then the condition expressed by (7.5) is replaced by -i.Imf(t) dt < 1. 0

If u ( t ) does not have a limit but is nonnegative and possesses an upper bound, we may still use the following results. Introducing for convenience the operator (7.7) Eq. (7.1) can be written L,u

= v.

It is not difficult to prove now that if monotone property in the sense that v1

A 5 0, the inverse operator enjoys a 2 L;'v, 2 0.

2 v 2 2 O*L,lV,

Hence, if c now denotes an upper bound of v ( t ) , [0, a),we have

tE

(7.9)

R , where R is the interval

(7.10)

providedfis nonnegative and Eq. (7.6) is satisfied. Further, the solution of (7.8) for arbitrary u can be obtained by superposition of the solution of the following equation L A X = l(t), (7.1 1) where l ( t ) is the unit step function, by means of u( t ) = v(t)x(o)

+ 1/ ' a ( t

- z) dx(z),

0

under conditions for which the integral exists.

(7.12)

315

7. BOUNDS A N D ASYMPTOTIC BEHAVIOR

The foregoing results can be summarized in the following:

Theorem 2 Let c be an upper bound of the nonnegative function u(t), satisfies Eq. (7.6), x ( t ) is the (unique) solution of (7.1 l), and A 5 0. Then u in (7.8) is bounded in R and satisfies Eqs. (7.10) and (7.12).

t

E R ;f ( t ) > 0

When, instead of considering time-invariant processes as described by Eq. (7.1), we must deal with more general processes such as u(t) = u(t)

+ A/'u(z)f(t, 0

(7.13)

z) dt,

the problem may still be tractable by means of a deep Tauberian result due to Pitt. We restrict the kernelf(t, T ) to belong to the class %? of functions which may be approximated to functions of the type k ( t - z ) in the sense that f ( t , T ) E %? [0 I crl I cr I crz ; k(t)] implies f ( t , z)exp[- a(t - z)] and k(t)e-"' belong to L and

1

m

a(t) =

sup a,sa
I f ( t , z) - k(t - ~)Iexp[-o(t- z)] dz,

(7.14)

0

is bounded and lim a(t) = 0.

(7.15)

t+m

Under these restrictions on f ( t , T), Pitt's theorem now reads :

Theorem 3 (Pitt) Suppose f ( t , r ) E %? [0 I crl I cr I crz ; k(t)], u(t) is bounded in R,and that ,

lJome-"'k(t)dt # - I ,

Re s > 0.

Then u(t) in Eq. (7.13) is bounded and there exists f ( t , T ) and u(t) such that ulTG, tER.

(7.16)

r and G depending on (7.17)

When the kernelf(t, T ) is specified to be nonnegative we further require the existence of nonnegative approximating functions k(t) such that the condition given by Eq. (7.16) is replaced by

-i.I"k(t)dt < 1.

(7.18)

0

We note that Theorem 3 embodies Theorems 1 and 2 in generality, but is less precise as far as an estimate for an upper bound is concerned, as expected. If additional properties are prescribed for functionf(t, T ) , sharper estimates may be constructed. See, for example, references of Section 8 and 9 in connection with some problems of creep buckling involving aging materials.

316

12A

MODELING A N D IDENTIFICATION OF HEREDITARY PROCESSES

EXERCISES

1. Prove that under the specified conditions, Eq. (7.9) holds. 2. Prove the validity of the inequality (7.10). 3. Let u be given by

where f satisfies the conditions of Theorem 3 and h is a bounded function having a limit ho as t -+ w . Show that the conditions derived for Theorem 3 still hold on Eq. (a) if h in (7.16) and (7.18) is replaced by A,.

8. Asymptotic Stability of Linear Viscoelastic Structures As an application of the preceding results, we present here an example involving the determination of conditions under which the deflection of a certain viscoelastic structure is bounded as t -+ co. To illustrate the ideas we consider a viscoelastically restrained rigid bar submitted to constant axial and lateral forces P and F, respectively, as shown in Fig. 12A-4, where the

A NONLINEAR VISCOELASTIC SPRING

i POSITION

Fig. 12A-4

restraint element AB is assumed to be a linear, time-invariant viscoelastic spring whose force-displacement relation is given by u(t) = (I/E)H(t)

+f

-OD

H(z)f(t - z)

dt,

(8.1)

317

8. ASYMPTOTIC STABILITY OF LINEAR VISCOELASTIC STRUCTURES

where E is a positive constant and f is a positive function associated with the creep of the spring. Now, at any instant, equilibrium in the bar requires that the force H in the spring A B be given by H = (P/h)u+ F,

(8.2)

where P and F a r e constant forces, both assumed to be applied at t Substituting H given by (8.2) into (8.1), we obtain u(t) - A J ' u ( z ) f ( t - z) dz = Au(t),

= 0.

(8.3)

0

where

A = PE/(P, - P),

P, = Eh,

(8.4)

and

[1/E + J:f(z)

u = (Fh/P)

dz] .

(8.5)

Clearly, if P approaches the critical value P, , the instantaneous displacement u becomes unbounded. Thus P , represents the Euler's load of the present mechanical system. Now, if the creep of the spring is limited, i.e., if

we can apply Theorem 1 to Eq. (8.3) and obtain as the pertinent condition for asymptotic boundedness of the displacements, the inequality

4J < 1,

(8.7)

p < (1/(1 4-EY))P,.

(8.8)

which, on account of (8.4), yields

The asymptotic value of the displacement u, as t + 00, is given by u(c0)

=

1 (E/(l + Ey))h - P

In order to provide a physical meaning to Eqs. (8.8) and (8.9), consider the purely elastic case, i.e., y = 0. In this case, (8.8) and (8.9) reduce to

P < P, = Eh,

(8.10)

to) = (1/(Eh- P))Fh,

(8.1 1)

and

31 8

12A

MODELING A N D IDENTIFICATION OF HEREDITARY PROCESSES

respectively. Comparison of (8.8) and (8.9) with (8.10) and (8.11) shows that the condition for asymptotic stability and the asymptotic value of the displacement in the viscoelastic case may be derived from the purely elastic one by the simple expedient of replacing the elastic modulus E by the reduced (or effective) modulus E/(1 + Ey). This, a generalization of the so-called principle of correspondence in linear viscoelasticity, for consideration of asymptotic stability problems, was presented by the author in a series of notes to the Academy of Lincei (Rome) in 1959 and subsequent years. See the Bibliography at the end of the chapter. EXERCISES 1. Derive Eqs. (8.8) and (8.9). Use Theorem 1 of Section 7. 2. Show that the deflected shape u(x, t ) of an axially loaded bar whose cross section has a moment of inertia f ( x ) and is made from a linear viscoelastic material that obeys the E-u law given by (a) satisfies the integrodifferential equation u"(x, t)f(t - 7)dr =

-Pw"

where E is the instantaneous modulus of elasticity, f is a characteristic function of the material, P is a constant denoting the axial load, and w is an initial imperfection of the bar. In Eq. (b), a prime indicates derivative with respect to x . 3. When f = 0 (purely elastic problem), Eq. (b) reduces to

+

(Efu")" Pu" = -Pw".

(C)

Let P, be the smallest characteristic value of the homogeneous equation associated with arbitrarily given boundary conditions. The value P, is known as the Euler buckling load of the bar. Now, i f f > 0 and

show that, regardless of the shape of the initial imperfection w ( x ) and the boundary conditions of the bar, the deflection function u in Eq. (b) will be asymptotically bounded if and only if p
+ Ey),

(4

and that, as t + co, u will tend to a function u(x) that satisfies the differential equation [EIu"/(l

+ Ey)]"+ Pu" = -Pw",

(f)

subject to appropriate boundary conditions. See N. DistCfano, Sulla Stabilith in Regime Viscoelastico a Comportamento Lineare, Nota I, Rend. Acc. Nuz. Lincei Ser. VIII, 27 (1959), 205-211.

319

9. AGING SPRING

9. Aging Spring

Instead of (8.1), we consider now the linear aging viscoelastic law u(t) = H(t)/E(t)

+ f- m H(z).f(t, T) dz

(9.1)

for the spring AB, wheref(t, z) is assumed to be asymptotically approximated by a positive function k(t - z) in the sense of (7.14) and (7.15), and where E(t) is a positive, monotonically increasing function of time, having the limiting value Em= lirnt+, E(t). Eliminating H between (9.1) and (8.2), and assuming for simplicity that both P and F a r e forces applied simultaneously at f = t o , we obtain

40 - 4 0 fu(z)f(f,

2)

C h = 4 M t , to),

(9.2)

to

where

A(t) = PE(t)/(Pc(t)- PI,

PC(O= E(t)h,

(9.3)

t 2 to.

(9.5)

and

Clearly, instantaneous stability requires P < Pc(t,) IPc(t),

Our problem consists in determining the largest upper bound on P such that u(t) remains bounded as t + a.This can be done using Pitt’s theorem in (9.2). In fact, if by hypothesis f ( t , z) E V [al5 CJ 5 C J ;~ k(t)], P satisfies

( 9 3 , and E(t) is bounded, then the right-hand member of (9.2) is bounded in R. Further, if f ( t , z) E % [0 5 (rl < a Ia2 ; k(t)], then the function A(t)f(t, z) E %‘ [0 Icl < (r s a2 ; Am k(t)], where Am is the limiting value of A(t) as t + co, i.e., Am

= PEmI(Pcm - PI,

(9.6)

where Po = Emh, i.e., Euler’s critical load evaluated using the limiting value of the modulus of elasticity E , . Therefore, Eq. (9.2) satisfiesthe conditions of Pitt’s theorem from which we can easily derive the condition of asymptotic stability in the form

320

12.4

MODELING AND IDENTIFICATION OF HEREDITARY PROCESSES

This is a remarkable result in that it provides the lowest upper bound for the load P which ensures asymptotic boundedness of the displacement u, using the limiting values of the quantities involved, not their respective histories. Before closing this section we note that the integral term appearing in (9.7) admits a physical interpretation that increases further the interest in this condition. Let y ( t , to) be the quantity defined by

i.e., y ( t , to) denotes the viscous (or creep) displacement of the spring AB at time t , when subject to a unit step traction applied at time t o . It is now easy to show that if y ( t , to) tends asymptotically to a limit for t and to large, then Iomk(t)dt = lim y ( t , co); r+m

in other words, the integral appearing in (9.7) denotes the limiting creep displacement of the spring asymptotically aged. This is an important observation because it permits the establishment of estimates (possibly by suitable experiments) on the quantities appearing in (9.7) without resorting to analytical expressions forf(t, 7). EXERCISES 1. Derive Eq. (9.7). 2. Show that the kernel

where A ,

t o , y I , and

X i are constants, may be approximated by the function N

k(t - T) =

C yihr exp[-h,(t

- 711,

I= 1

in the sense of (7.14). 3. Rederive Eqs. (9.2), (9.3), and (9.4) assuming that P and Fare not constants but bounded functions of time. 4. Assume that limr+, P ( t ) = P , exists. Show that Eq. (9.7) should be replaced by

32 1

10. NONCONSTANT LOADING. BOUNDS

10. Nonconstant Loading. Bounds

We return now to consideration of a linear, time-invariant spring, but we shall assume here that P is not a constant force. In this case, the pertinent equation for the displacement u is given by

whereAis given by (8.4) and v by (8.5). Assuming that 0 I PII P s P, I P,, an interesting problem here is the determination of suitable lower and upper bounds for the deflection function u. To study problems of this type, however, we shall introduce the more general (intimately related) equation u(t) - S'd(z)u(~)f(t - z) dz = g(t),

(10.2)

0

where d and g are arbitrary functions satisfying

(10.3)

0 5 A, 5 A(t) I A2

and 0 I g I c, respectively. We can now prove the following theorems: Theorem 4 Let c be an upper bound of the nonnegative function g(t), t E R, and let d(t) satisfy Eq. (10.3). Then, i f f > 0 and 1 2 Jrnf ( t ) dt < 1,

(10.4)

0

u(t) in Eq. (10.2) is nonnegative and bounded in R.

Proof. Consider the sequence (u,} given by uo = 9,

%+, = 9

+ J)(z)f(t

- z)u,(T) dr.

(10.5)

By recalling the conditions of the hypotheses and observing that (10.6)

it can be easily proved that the series ~ ; = = , ( u ~ u,) + ~ converges absolutely and uniformly in R , and that the sequence {u,} converges to the nonnegative bounded function u with an upper bound (10.7)

322

12A

MODELING A N D IDENTIFICATION OF HEREDITARY PROCESSES

We may obtain a more precise upper bound, but we first prove the following: Theorem 5 Let z ( t ) , t E R , be a continuous bounded function, and A(1) and f ( t ) satisfy the conditions of Theorem 4. Then (10.8)

z(t) 2 / ' A ( r ) f ( t - z)z(z) dz, 0

implies z(t) 2 0 in R. Proof. Equation (10.8) may be written z ( t ) - f A ( r ) f ( t - z)z(z) dz = h(t),

(10.9)

0

where h is a nonnegative function of t E R . The nonnegativity of z follows from the nonnegativity of the resolvent kernel k(t) in the equation z(t) = h(t)

+ f 0k ( t - z)h(z) dz,

t E R.

(10.10)

We leave this verification as an exercise to the reader. We can now prove the following:

Theorem 6 Let u(t) satisfy Eq. (10.2), andf, I , g satisfy the conditions of Theorem 4. If x and y are the solutions of the equations

4 0 - A1 S'f(t 0

- z ) x ( 4 dz = d t ) ,

(10.1 1)

Y ( t ) - A2 J'f'l

- zlY(z) dz = g(t),

(10.12)

and 0

respectively, then t E R.

0 I x(t) 5 u(t) I y(t),

Proof.

(10.13)

Using Theorems 4 and 5 we can establish that 0 I y 5 c / (1 - I , jomf(t)d t ) .

(10.14)

Then, to prove that u(t) I y(t), it is enough to prove that z ( t ) = y(t>- 4 0

(10.15)

is positive for t E R. In fact, comparison of Eqs. (10.2), (10.12), and (10.15) yields the following equation for z ( t ) :

z(r) - J ' A ( z ) f ( t - z)z(z) dz = 0

- A)f(t - z)u(z) dt.

(10.16)

323

11. ALMOST PERIODIC LOADING. TIME AVERAGES

Since the right-hand member of (10.16) is positive and bounded, then z is bounded, recalling Theorem 4, and z ( t ) 2 0, t E R, recalling Theorem 5. In very much the same way we can prove the rest of the proposition. We observe that from Theorems 4 and 5 we may establish the following: Lemma Let A(t) andf(t) satisfy the conditions of Theorem 4, and let c be a positive constant. If uI c

+ [>(z)

f ( t - z)u(z) dz,

then 0 I u Iv

Ic/(l - A2Jomf(z)dz),

t E R,

(10.17)

where u(t) is the solution of z,

- /)(z)f(t

- z)u(z) dz = c.

(10.18)

This is one of the ways that the Gronwall-Bellman lemma can be generalized. 11. Almost Periodic Loading. Time Averages

When our knowledge on the loading function P(t) increases, we expect to draw additional information on the asymptotic behavior of the displacement u(t). This is the case when we assume P to be a periodic or an almost periodic function within the limits 0 < PI 5 P IP, . To illustrate the ideas we consider an example. We assume that 1 may be written in the form

A

= [I

+ +(Ol@,

(11.1)

where @ is a constant and is a function that takes the value v > 0 with probability p 1 and the value - v with probability 1 - p i . Substituting 1 given by (11.1) into (10.2) we obtain (11.2) the pertinent equation for the displacements u, where f is assumed to satisfy (8.6) and g to be an almost constant function with limt+mg = 1. Under these conditions, for t large, u in (11.2) oscillates from the value u1

= 1/(1 - (1

+ v)@y),

(11.3)

324

12A MODELING A N D IDENTIFICATION OF HEREDITARY PROCESSES

with probability p l , to the value u2

=

1/(1 - (1 - v)@y),

(11.4)

with probability 1 - p l . Therefore, the time average U will be given by

U=

1 + 1 - (1 + v)@y 1 - (1 - v)@y' -P1

(11.5)

Clearly, the condition for boundedness is given by (1

+ v)@

J-omf(r)dz .< 1.

(11.6)

12. Stochastic Aspects

An interesting class of problems arises when function I in Eq. (10.1) depends on a random function with known probability distribution. In this case it is required to determine the mean and various statistical quantities associated with the stochastic process u. When the randomly fluctuating part of I is small, a standard perturbation procedure leading to a successive approximation scheme in terms of a sequence of linear integral equations for the ensemble averages ( u ) and higher-order quantities may be successfully employed. It is interesting to observe that the smallness of the randomly fluctuating part of I is not enough to ensure good results using perturbation procedures. In fact, in addition to that condition, we should also require (12.1) i.e., the smallness of creep contribution. A technique of truncated hierarchy surmounts this restriction, although to the expense of more involved integral equations for the approximated mean ( u ) . The method of truncated hierarchy is based on the assumption that the random functions I(t,)I(t,) * * * I(tn)I(t) and u ( t ) are statistically independent, i.e.,

< I ( t l ) I ( t J*

*

J(tn)J-(t)u(t)>= ( I ( ~ I ) I ( ~ .. Z 4tn >).I ( t ) ) < u ( t ) ) .

(12.2)

This decomposition permits the construction of a hierarchical system of integral equations whose truncation at a certain order provides a consistent set of equations in the pertinent quantities. This method has yielded accurate results by truncation at low orders. The present topic, however, goes beyond the scope of this chapter and is not pursued further. The reader interested in stochastic aspects should consult the Bibliography at the end of the chapter.

325

14. SYSTEM IDENTIFICATION

13. Nonlinear Models. Generalities Instead of the linear relationship (8.1) for the spring A B in the mechanical model shown in Fig. 12A-4, we may assume the nonlinear functional relationship t

u(t) = G[H(z)].

(13.1)

-w

Substituting H given by (8.2) into (13.1) we obtain a functional equation for the displacement u, namely,

'

u(t) = G [ ~ T ) ~+( fWl, T)

(13.2)

-m

where we have put a = P/h. The problem consists now in studying the properties of u in (13.2) from the properties of the functional G . The nonlinear case affords the appearance of critical times, a phenomenon analogous to the formation of shocks in nonlinear wave propagation. A critical time is the time at which the deflection becomes unbounded. Since in general we know very little about the analytical structure of the functional relationship (13.1), we usually expand G in terms of Volterra integrals such as

By now assuming some properties on the kernelsf,, f z , . . . , we can use perturbation procedures to study short- and long-term stability of u. It is interesting to observe that some of the Tauberian results employed in the linear case may be extended to the study of nonlinear functionals. In addition to this we observe that if u is small enough, the stability of the system will be associated with certain properties of the linear term in the expansion (13.3). Clearly, it is possible to construct a number of PoincarC-Liapunov analogs, using multiple integral expansions. We do not pursue this topic here. The interested reader can find appropriate literature in the Bibliography.

14. System Identification In this and subsequent sections we discuss a number of aspects associated with the identijication problem, i.e., the determination of the structure of a system from suitable experimental input-output data. In the following chapter, we present an application of this problem to the representation of nonlinear viscoelastic materials.

326

12A MODELING AND IDENTIFICATION OF HEREDITARY PROCESSES

System identification clearly belongs to the class of inverse problems and, as such, usually leads to ill-posed formulations. An example will help to clarify this point. We consider a linear, time-invariant system, whose imputoutput relationship is given by the convolution integral u ( t ) = f u ( t - z ) f ( z ) dz.

(14.1)

0

Given the input function u and output v in the interval [0, TI, we are asked to determine the transfer function$ An approach to the solution of this problem consists in regarding (14.1) as a Volterra integral equation of the first kind in the unknown function f ( t ) . Unfortunately, since the integral in (14.1) is a smoothing operation, we must then expect that small errors in u and o will produce arbitrarily large errors in $ This, an undesirable property exhibited by the whole class of linear operators of the first kind, disqualifies this approach as a reliable identification procedure for linear systems. The way to transform a problem like this into a well-posed problem is by the incorporation of appropriate additional information on the function to be determined. For example, we can restrictfto be a smooth function. This, a regularization idea in the sense of Tichonov, may be implemented in a number of ways. We can, for example, use a Galerkin procedure and form the least squares functional (14.2) Assuming f to be given by f(t)

= cl@.,(t)

+ c2 Q 2 ( t )+ . . . + cNQN(t),

(14.3)

we can easily compute the ci in (14.3) by requiring Vin (14.2) to be aminimum. The advantage of this method is that the minimization in (14.2) leads to a linear system of algebraic equations. By picking the mi's in an adroit fashion, the computation of the elements of the associated matrix may be considerably simplified. To this end we can, for example, use Laguerre functions (14.4) and take advantage of their orthonormal properties, among others. We could also use Prony series such as f ( t ) = c1 exp(-All)

+ c2 exp(-A2t) + . . . + c,exp(-A,t),

(14.5)

where the Ai are given real numbers. The disadvantage of this procedure is that it requires the availability of the A i , which is an important part of the structure we are trying to identify. If, on the other hand, the problem is reformulated such as to determine both the ci and the Ai in (14.5), then the

15. FORMULATION OF THE IDENTIFICATION PROBLEM

327

minimization in (14.2) will not? any longer, yield a linear algebraic problem for the unknown quantities. When any of the available nonlinear least squares routines are employed to solve this problem, we face a number of difficulties. But here is precisely where our true story begins. EXERCISES 1. Assuming that the input u is given by u(t) = l ( t ) t ,

and that f i s given by the polynomial

f ( t ) = CI t

+ B c t~Z + . + (l/N!)c.wtN,

(b)

compute the elements of the matrix associated with the minimization in (14.2). 2. Do the same as in Exercise 1, but assuming that f i s given by the Prony series (14.5) when the exponents A , , i = 1,2, ..., N, are known quantities. 3. Discuss the implementation of a method for general input-output data, using Laguerre’s functions. See J. C. I. Dooge, Analysis of Linear Systems by Means of Laguerre Functions, J. SZAM Control Ser. A, 2 (1965), 396-408.

15. Formulation of the Identification Problem The formulation of an identification problem usually proceeds along the following steps : 1. We select an appropriate class of models to describe the dynamics of the process. For example, we may adopt a differential equation such as

dujdt

z = h(u, v, c),

= g(u, v, c),

(15.1)

where u is the state, v is the imput, z is the output, and cis a vector of unknown parameters. Or, we may use an integral model of the Volterra type

z = h(u, 21, c),

where u, v, z, and c have the same meaning as before. 2. Regardless of the structure of the model, in order to determine a suitable vector c, we introduce a criterion function such as V(c)=

J

T

(z - Z)2 dt,

(15.3)

0

where Z stands for the measured value of the output z. 3. For a suitable class of input-output pairs (v; z), we determine c such as to minimize the functional V given by (15.3).

328

12A MODELING A N D IDENTIFICATION OF HEREDITARY PROCESSES

As outlined here, the identification problem has been reduced to an optimization problem in the unknown vector c. This is sometimes referred to as a parametric identification problem, as opposed to nonparametric identification problems where, instead of a vector of unknown constants, we seek the numerical values of one or more functions. It is clear that the adoption of a least squares criterion functional is associated with several advantages, but in general may be substituted by any other error functional, if necessary. We finally mention that the solution of a problem such as that presented here may be performed using various procedures such as those outlined in Chapters 7 and 9. We deal with some applications in Chapter 12B, in connection with the identification problem in viscoelasticity. 16. Identifiability. Hidden Variables Once the identification problem has been formulated as outlined in the previous section, it remains for us to prove that the associated optimization problem is well posed, i.e., that a solution exists and is unique and stable with respect to small perturbations in the data vector. In this connection, the structure of the model selected to describe the dynamics of the process may play a decisive role in the solution of the identification problem. We illustrate this point with a simple example. Consider the linear input-output relation v(t) = f u ( t - T ) ~ ( T dz, )

(16.1)

0

where the unknown transfer function f is assumed to be given by a sum of exponentials f ( t ) = c1 exp(--A,t)

+ c2 exp(-A,t) + + c,exp(-&t),

and attempt to determine the ciand tional

(16.2)

Ai by minimizing the least squares func(16.3)

where u is given by (16.1) and 6 is the measured value of 2’. We can consider here, with no loss in generality, that the input u is the unit step function. This reduces the problem to that of fitting a given function u with a sum of exponentials with unknown amplitudes ci and retardation times l/Ai, by using a least squares criterion. This problem is known to yield highly ill-conditioned matrices, as a result of the intrinsic instability of the exponents Ai as functionals of the function v ( t ) in any interval [0, TI. The reader can find a numerical example illustrating this characteristic property of exponential functions in the book by Lanczos given in the Bibliography at the end of the chapter.

329

17. AN ALTERNATIVE FORMULATION

In our present context, the interest of the preceding observation lies in the fact that the optimization problem given by (16.1)-( 16.3) is clearly equivalent to: Find the parameters c , and exponents A, which minimize (16.3), where v is given by the state equation N

(16.4) where the hidden variables x i satisfy the equations of evolution dx,/dt

+ Aixi = u,

i = 1, 2,

. . . , N.

(16.5)

In other words, the hidden-variables approach appears to yield an incorrectly posed identification problem. Therefore, a good deal of care should be exercised in the formulation of the pertinent problem, in view of the influence of the model structure in the success of the identification. To emphasize the role of structure further, we present an alternative formulation to this problem in the next section. We finally note that here we discussed identifiability in connection with the stability of the unknown vector. The problem of whether such a vector exists at all leads to a very important and interesting class of problems. See the Bibliography at the end of the chapter. 17. An Alternative Formulation Instead of using (16.4) and (16.5) as the pertinent representation of (16.2), we can alternatively use the ordinary differential equation N-1

1a if ( ' ) +f ( N )

= 0,

f")(0)=fi,

i=O,l,

..., N - 1 ,

(17.1)

i=O

where the a, are constants such that the exponents A, in (16.2) are the roots of the characteristic equation

c u,Ai+ AN

N-1

= 0.

(17.2)

0

The relation between the vector of the fi appearing in (17.1) and that of the c, appearing in (16.2) is a linear one, as the reader can easily verify. By adopting (17.1) as the pertinent representation off in Eq. (16.1), the identification problem reads : Find the coefficients a, and initial conditionsf, which minimize (16.3), where u is given by (1 6.1) and f is the solution of the initial-value problem (1 7.1). This problem, mathematically equivalent to that presented in Section 16 in terms of the hidden variables x i , is now correctly posed, by virtue of the stability of the a, coefficients as functionals of v . We exploit properties of this kind when formulating several identification problems in Chapter 12B.

330

12A MODELING AND IDENTIFICATION OF HEREDITARY PROCESSES

EXERCISES

..

1. Letf= (A), i = 0,1, . ,N - 1, be the vector of the initial conditionsfi given in (17.1), and let c = (c0. i = 1,2, ...,N, be the vector of the coefficients c, appearing in (16.2). Prove that the N x N matrix in the equation

f= Ac (a) is nonsingular. 2. Write the expressions for the coefficients ut in (17.2) as functions of the N distinct roots h i , i = l , 2 ,..., N .

18. Differential Approximation of Functions

The use of linear differential equations such as (17.1) to represent an unknown function f in an identification problem is a very convenient device in applications, but it tends to introduce an uncomfortably large number of constants to be determined. In some cases, a priori information on the process leads to functional forms containing a smaller number of constants than that required by a representation of the type (17.1). This is particularly true in the field of linear and nonlinear viscoelasticity where the input-output relationship is generally given by Volterra integral equations of the form (18.1) Here, a representation o f f by differential equations of the type (17.1) is always a convenient parametrization procedure. In many applications, however, some a priori information leads us to functions of the form (18.2) where A, a, and /3 are constants. When this is the case, the identification problem is simpler because there are, in general, fewer constants to identify. However, in dealing with integral equations such as (18.1) the prediction problem, i.e., for given u,J g, and h to compute 0, is easier iffis of the type given by (16.2) or, equivalently, (17.1). This is true because when this is the case, i.e.,fis given by (16.2) or (17,1), we can readily transform an integral equation such as (18.1) into a system of differential equations subject to initial values, with its computational advantages. The procedure for this reduction was outlined in Section 2 in connection with the equivalence of various functional representations, and need not be repeated here. Thus, interest exists in constructing accurate approximations of functions in terms of exponentials such as (16.2). Following Bellman, we can solve this problem by diferential approximation. The idea of the method is as follows: Let f ( t ) be the function to be approximated by f satisfying (17.1).

33 1

EXERCISES

We construct the error functional

1

T

V(ai)=

0

[aof

+ al df/dt+ . . + dNf/dtNI2dt, *

(18.3)

and determine the ai by minimization. This leads to a linear system of algebraic equations A.X+b=O,

(18.4)

where x is the vector (ai), i = 0, 1, .. . , N - 1, A is the matrix whose typical term a i j is given by T

=

J0

f(i)p)

dt,

(18.5)

and b is the vector whose ith component is given by

dt.

b i ( T )= [Tf""'"'

(18.6)

0

The quantities aij and bi may be computed by integrating the initial-value problem

dUii/dt = f ( i Y ( j ) , dbi/dt = f ( i ) f ( N ) ,

U i j ( 0 ) = 0, bi(0) = 0,

(18.7)

until t = T. Simultaneously, we can compute f and its successive derivatives by constructing an appropriate initial-value problem for those quantities. This leads to additional efficiency, iff is of a complex form, Once the a, have been obtained by inversion of (18.4), the approximating function f is given by the differential equation (18.8) subject to the approximated initial conditions f"'(0) = f(i)(O).

(18.9)

At this point, the determination of the ci and the Ai in (16.2), using Eqs. (18.8) and (18.9), follows a routine pattern. We leave this as an exercise to the reader. EXERCISES 1. Construct a pertinent (nonlinear) system of differential equations to compute the function]= t - = and its first N - 1 derivatives.

2. How would you proceed to compute the ci and the Ai appearing in (16.2) from Eqs. (18.8) and (18.9)? Assume that h, is in general a complex number.

332

12A MODELING AND IDENTIFICATION O F HEREDITARY PROCESSES

19. Controlled Inputs. Discussion Although not explicitly stated, we have been tacitly assuming in the preceding sections that the input-output functions are of an arbitrary nature, not restricted to any special form. This generality is particularly valuable when we can exert partial or no control at all on the input function, a situation which occurs frequently in biomechanics, geotechnics, hydrology, and other important engineering applications. When we can arbitrarily prescribe the type of input to apply to the system, a meaningful problem is then the determination of the class of input functions for which the error functional is minimized. In spite of its practical interest, we do not pursue this topic here. In some instances, the use of inputs of certain types simplifies the analytical aspects of the identification problem. Well known are the advantages of frequency analysis in the study of linear systems. Similarly, the use of step functions is sometimes advantageous in the determination of transfer functions. This is the classical procedure to determine the creep and relaxation functions of viscoelastic materials. The situation is not, however, as brilliant as it might appear at first glance. This is due to the fact that real systems are imperfectly linear. Therefore we shall obtain better results if the identification is performed using realistic inputs trying to simulate actual conditions, than if we use special inputs whose only objective is to achieve a simplification in the analytical aspects of the identification procedure. When we deal with nonlinear systems, the advantage of using inputs of prescribed form to simplify the identification process is very limited. In fact, the determination of higher-order kernels in Volterra integral expansions by using multiple-dimensional step functions necessitates so many experiments and tedious data manipulation, that the procedure is made very unattractive. On the other hand, the use of Wiener’s method using white noise inputs does not seem to have any appreciable advantage with respect to a direct identification in space state, using optimization procedures such as that outlined in Section 15.

20. Nonlinear Filtering and Sequential Identification The solution of the class of identification problems formulated in Section 15 can be obtained in a number of ways. In Chapter 12B we employ quasilinearization and other perturbation techniques to solve a version of the identification problem in viscoelasticity. Here we wish to point out the connection of the identification problem and a seemingly unrelated problem:

20. NONLINEAR FILTERING A N D SEQUENTIAL IDENTIFICATION

333

the nonlinear filtering problem. To illustrate the ideas we consider a dynamical system given by the nonlinear differential equation

ti

(20.1)

= g(u),

where the state u is a vector of dimension N . The process is continuously observed through the M-dimensional vector w given by

+ rl,

w =r u

(20.2)

where q denotes the observational errors and r is a rectangular matrix of full rank. On the basis of the observations w in the interval 0 It I T, we wish to determine an optimal estimate of the state u at t = T such as to minimize the quadratic error functionf(u(T), T ) given by T

f(u(~T ) ,) = J (w - r u , w - r u ) dt 0

+ (u(o) - b , A ( ~ ( o )- b)),

(20.3)

where b is the best a priori estimate of u(0) and A is a nonsingular matrix establishing the degree of confidence in such an estimate. We first construct an equation forf(u, T). Using ideas of invariant imbedding we easily find

s,(c, TI = (w - r c , w - TC)- ~

( c TI, , gw),

(20.4)

where for simplicity of notation we made u(T) = c. Clearly, f ( c , T )is subject to the initial condition

f(~, 0)

(C

- b, A(c - b)).

(20.5)

Now, if

e(T) = arg minf(c, T )

(20.6)

C

denotes the optimal filter, or estimate, then L ( e , T ) = 0.

(20.7)

Taking total derivatives in (20.7) we obtain = 0,

(20.8)

deldT = -fL'(e, T)fc,(e, T ) .

(20.9)

fcc(e, T ) de

+fc,(e,

T>dT

from which In order to obtain a more convenient form for the right-hand side of (20.9), we calculatef,, using Eq. (20.4), i.e., fTC(C,

= -2TT(W - r c )

-fcc(c,

T ) g(c) - gc(c)fc(c, TI.

(20.10)

Combining (20.9) and (20.10), and taking into account (20.7), Eq. (20.9) may be written deldT = g(e) Q(T)rT(w- re), (20.11)

+

334

12A MODELING A N D IDENTIFICATION OF HEREDITARY PROCESSES

the equation for the optimal nonlinear filter, where Q is the matrix given by (20.12)

Q ( T ) = 2fZ' (e, T).

In order to integrate (20.11) we need an initial condition for e and the matrix Q(T).As an initial condition we may use the best a priori estimate of u(O), i.e., e(0) = b .

(20.13)

The determination of matrix Q requires, however, further analysis. In fact, to computef,, we requirefgiven by (20.4)-(20.5). In order to simplify the integration of this partial differential equation, we assume that, in the neighborhood of the optimal estimate e, function g(c) behaves linearly, i.e., g(c) = s(e>+ sc(e) (c - el.

(20.14)

Thenf(c, T)in (20.4)-(20.5), admits the representation f(c, T ) = (Rc, c ) + ( r , c )

+ s,

(20.15)

from which Q given by (20.12) follows: Q ( T )= R - ' ( T ) .

(20.16)

Now, in order to obtain a differential equation for R, we substitute (20.14) and (20.15) in (20.4)-(20.5) and collect the quadratic terms in c, obtaining dR/dT = TTT - Rg,(e) - gCT(e)R,

R(0) = A,

(20.17)

or, on account of (20.16), dQ/dT = g,(e)Q

+ QgCT(e)- Q r T r Q ,

Q(0) = A-'.

(20.18)

Equations (20.1l), (20.13), and (20.18) are the desired differential equations for the optimal nonlinear filter. Compare with Eqs. (10.15) and (10.16) of Chapter 9 obtained for the linear filter. We finally observe that the nonlinear filtering approach presented in this section may be readily used as a sequential identifier for processes governed by differential equations of the form dx/dt = h(x, a),

(20.19)

where a is an unknown vector, if we replace (20.19) by the system

ddt (") a =

(4,

and determine an optimal estimate of the augmented state ( x ; a) by the method of this section. The sequential identification performed in this fashion enjoys the advantage of continuously upgrading the estimate of the constant vector as additional information becomes available.

NOTES, COMMENTS, AND BIBLIOGRAPHY

335

NOTES, COMMENTS, AND BIBLIOGRAPHY

2. Here we follow in part N. Distbfano, System identification Problems in Hereditary Biomechanical Processes, Proc. ASLLOMAR Con$ Circuits Syst., 5th Nov. 8-10 (1971), Pacific Grove, California, pp. 248-251.

The connections between linear differential and integral models have been extensively investigated in the context of the theory or viscoelasticity in N. Distbfano, On Alternative Representations of Time Varying Viscoelastic Materials, Internat. J. Solids Structures 6 (1970), 1021-1033.

The evolution of irreversible processes can be described by differential equations in terms of internal or thermodynamical variables. A theory of such processes is due to L. Onsager. For some of the early work see L. Onsager, Reciprocal Relations in Irreversible Proce-sses, Phys. Reu. 37 (1931), 405; 38 (1931);2265.

and for an early variational formulation of those processes, L. Onsager, Fluctuationsand Irreversible Processes, Phys. Rev. 91, No.6 (1953), 1505-1515. For some of the early applications of these ideas to viscoelasticity see M. A. Biot, Theory of Stress-StrainRelations in Anisotropic Viscoelasticity and Relaxation Phenomena, J. Appl. Phys. 25, No. 11 (1954), 1385-1391; M. A. Biot, Variational Principles in Irreversible Thermodynamics with Applications to Viscoelasticity, Phys. Rev. 97, No. 6 (1959,1463-1469.

The notion of internal variables was extensively used by Rabotnov to construct a phenomenological theory of creep of metals. See Yu. N. Rabotnov, Amsterdam, 1969.

“

Creep Problems in Structural Members.” North-Holland Publ.,

The use of difference-differential models to describe ongoing viscoelastic behavior was first presented in K. L. Cooke, N. Disttfano, and B. Kashef, On a Class of Hereditary Processes in Biomechanics, Dept. of Elec. Engrg., Univ. of Southern California, Tech. Rep. No. 72-5, Jan. 1972;Math. Biosci. 16 (1973), 359-373.

where a nonlinear version of the problem is investigated and some numerical experimentation is presented.

3-4. For the early work of V. Volterra in the theory of hereditary processes, see V. Volterra, Sulle Equazioni della Ellettrodinarnica, Rend. Acc. Lincei Ser. 5, 18 (1909). 203-21 1 ; V. Volterra, Sulle Equazioni Integro-Differenziali della Teoria dell’ ElasticitA, Rend. Acc. Lincei Ser. 5, 18 (1909), 295-301.

336

12A MODELING A N D IDENTIFICATION OF HEREDITARY PROCESSES

A more comprehensive exposition of the theory was presented later in V. Volterra, Sur La equations Inttgro-Difftrentielles et Leur Applications, Actu Math. 35 (1912), 295-356.

In connection with Painleve’s objections to hereditary mechanics, see V. Volterra, “ L q o n s Sur Les Fonctions de Lignes,” Chapter XIV. Gauthier-Villars, Paris, 1913.

5. For the conditions of closed-cycle and time-invariant processes, see V. Volterra, Sui Fenomeni Ereditarii, Rend. Acc. Lincei Ser. 5,22 (1913), 529-539.

In English the reader may consult V. Volterra, “Theory of Functionals and of Integral and Integro-Differential Equations.” Dover, New York, 1959.

6. An application of Volterra theory of hereditary processes to the representation of aging viscoelasticmaterials is afforded by the theory of Arutiunian for creep of concrete. See the book N. KH. Arutiunian, “Applications de la Theorie du Fluage.” Eyrolles, Paris, 1957.

7. See R. Paley and N. Wiener, Fourier Transforms in the Complex Domain, Amer. Math. SOC. Colloq. Publ. 19 (1934), 59-63; H. R. Pitts, “Tauberian Theorems.” Oxford Univ. hess, London and New York, 1958.

8-9. The application of Tauberian results to a number of problems in the theory of viscoelasticity appeared in a series of notes : N. Disttfano, Sulla Stabiliti in Regime Viscoelastico a Comportamento Lineare, Nota I, Rend. Acc. Lincei Ser. VIII, 27 (1959), 205-21 1 ; N. Disttfano, Nota 11, Rend Acc. Lincei Ser. VIII, 27 (1959), 256-361 ; N. Disttfano, Ancora Sulla Stabiliti Asintotica delle Deflezioni di una Trave Viscoelastica, Rend. Acc. Lincei Ser. VIII, 35 (1963), 504-508; N. Dist6fano. Sul Comportamento Asintotico di Corpi Viscoelastici a Ereditarieti Invariabile, Afri Accud. Sci. Torino 95 (1960-61), 1-8; N. Disttfano, Sul Comportamento Asintotico di Corpi Viscoelastici nella Teoria delle Coazioni, Atti Accud. Sci. Torino 96 (1961-62), 1-5.

For technical applications of the foregoing results, see N. Distkfano, Redistribution of Stresses in a Continuously Supported Beam, Final Rep. of the 6th Congr. of the Internat. Assoc. for Bridge and Structural Engrg. (1960), 417-428; N. Disttfano, Creep Deflections in Concrete and Reinforced Concrete Columns, Publ. Internat. Assoc. Bridge Structural Engrg. 21 (1961), 37-48;

NOTES, COMMENTS, A N D BIBLIOGRAPHY

337

N. Distefano, Creep-Buckling of Slender Columns, Proc. Amer. SOC.Civil Engrs. J. Structural Div. 91 (1965), 127-150; N. DistCfano and M. Gradowczyk, Creep Behavior of Homogeneous Anisotropic Prismatic Shells, Proc. Z.A.S.S. Symp. Non-Classical Shell Probl. (1963), 409427; N. DistCfano and M. Gradowczyk, The Viscoelastic Theory of Thin Shallow Shells, Zngenieur-Arch. 34 (1965), 304-312.

10-12. Here we follow in part N. DistCfano, A Volterra Integral Equation in the Stability of Some Linear Hereditary Phenomena, J. Math. Anal. Appl. 23, No. 2 (1968), 365-383.

For a comprehensive treatment of stochastic integral equations see the books A. T. Bharucha-Reid, “ Random Integral Equations.” Academic Press, New York, 1972; C. P. Tsokos and W. J. Padgett, “ Random Integral Equations with Applications to Life Sciences and Engineering.” Academic Press, New York, 1974.

13. An application of Volterra’s multiple integral expansion to the stability of some nonlinear viscoelastic systems appeared in a series of articles: N. Disttfano and J. Sackman, On Asymptotic Stability of Nonlinear Hereditary Phenomena, Quat. Appl. Math. 24, No. 2 (1968), 133-141 ; N. DistCfano and J. Sackman, Stability Analysis of a Nonlinear Viscoelastic Column, 2. Angew. Math. Mech. 47, Heft 66 (1967), 349-358; N. Disttfano and J. Sackman, Creep Buckling of a Nonlinear Viscoelastic Beam Column, Internat. J. Solid Structures 4 (1968), 341-354.

14. The solution of inverse problems in engineering and other fields is presently stimulating extensive mathematical research. See R. LattCs and J. L. Lions, “ The Method of Quasi-Reversibility. Applications to Partial Differential Equations.” American Elsevier, New York, 1969; M. M. Laurentiev, “Some Improperly Posed Problems of Mathematical Physics.” Springer-Verlag, Berlin and New York, 1967; M. M. Laurentiev, V. G. Romanov, and V. P. Vasiliev, Multidimensional Inverse Problems for Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, New York, 1970; A. N. Tichonov, On the Solution of Ill-posed Problems and the Method of Regularization, Dok. Akad. Nauk. SSSR 151 (1963), 501-504; G. A. Phillipson, ‘‘ Identification of Distributed Systems.” American Elsevier, New York, 1971.

15. A comprehensive review article in system identification containing 230 references is available: K. J. Astrom and P. Eykhoff, System Identification, Automatica 7 (1971), 123-162.

338

12A

MODELING A N D IDENTIFICATION OF HEREDITARY PROCESSES

See also the book A. P. Sage and J. L. Melsa, “System Identification.” Academic Press, New York, 1971.

16. The difficulties associated with the identification of linear viscoelastic materials described by equations of the form (16.4) and (16.5) were pointed out in the first reference of Section 2. For some conditions on structural identifiability see R. Bellman and K. J. Astrsm, On Structural Identifiability, Math. Biosci. 1 (1969), 329-339.

17. A number of applications of this method are presented in Chapter 12B. 18. This method was first presented in R.Bellman, R. Kalaba, and B. Kotkin, Differential Approximation Applied to the Solution of Convolution Equations, Math. Cornp. 18, No. 87 (1964).

19. For Wiener’s theory, see N. Wiener, “ Nonlinear Problems in Random Theory.” The Technology Press, Cambridge, 1958.

20. Application of nonlinear filtering to the problem of sequential estimation of parameters via invariant imbedding was first presented in the paper by Bellman et al. in the references of Section 9-10. For an application to structural dynamics, see N. DistCfano, On the Identification of a Nonlinear Viscoelastic Spring Under Dynamical Conditions. A Filtering Approach, in ‘‘ Mechanics of Deformable Solids and Structures” (anniversary volume dedicated to Yu.N. Rabotnov). Acad. Sci. USSR (to appear Feb. 1975).

Miscellaneous References

For an application of the theory of functionals to the construction of constitutive equations for viscoelastic materials in the presence of thermodynamic effects, see B. D. Coleman, Thermodynamics of Materials with Memory, Arch. Rat. Mech. Anal. 17 (1964), 1-46;

and also the book A. C. Eringen,

“

Mechanics of Continua.” Wiley, New York, 1967,

where, in addition to many references, the reader will find an account of qualitative theories of materials in an axiomatic framework.

Chapter 12B

The Identification Problem in Viscoelasticity

1. Introduction

Here we apply some of the ideas presented previously in connection with the approximation of nonlinear functionals to the solution of a class of identification problems in viscoelasticity. In this chapter we are interested in various kinds of analytical approximations to an imperfectly known nonlinear functional relationship

given through K, generally independent, experimentally determined functions Oi = oi + tiand Bi = E~ y l i , i = 1, . . . , K , where ti and yli denote errors of measurement. We shall limit ourselves to consideration of the simplest problem in this class involving the (longitudinal) stress and strain on a rod undergoing quasistatic isothermal effects. The general problem consists in the determination of suitable analytical functionals which approximate (1.1) in a sense to be specified later. Although the methods to be presented in this chapter may be successfully applied to a variety of analytical functionals of interest in studies of visco-

+

339

340

12B THE IDENTIFICATION PROBLEM IN VISCOELASTICITY

elasticity, here we shall deal only with a limited family of such functionals, namely those given in terms of Volterra integral equations such as a(t) = h(E(t)) + J ' s c . c T , , s c t

- t>dr.

0

(1 2

The problem consists in the determination of suitable functions h and g which best approximate (1.1). Formulated in this fashion, the identification problem clearly reduces to a best-fit problem in function space. Equation (1.2) has been chosen to exemplify the procedures because in addition to simplicity in structure, it affords a remarkable versatility in the realm of problems in viscoelasticity. The eKamples presented at the end of the chapter illustrate this point. The problem is formulated in Section 2 and the solution is outlined in Section 3. Some numerical aspects are discussed in Sections 4-7, while in 8 and 9 we present some examples involving the identification of two polymeric materials. 2. The Identification Problem

To illustrate the procedure, we shall employ the o-c law given in terms of the nonlinear Volterra integral equation

where A h, and g are (generally) nonlinear functions of their arguments and given parametrically in terms of the constant vectors a = (ao, a , , . . . , u 2 N - l ) , b = (bl, b z , . . . , bR),and c = (c,, c z , . . . , cs), respectively, whose determination is part of the identification problem. To that effect we are given Kpairs of experimentally determined stress and strain histories 5,(t) and &(t), k = 1, 2, . .., K , 2 E [0, T,], respectively. The unknown vectors a, b, and c will be determined by requiring that the quadratic error functional K

J(u, b, C) =

Ti

C q i l0 (ai - ai)' dt, i=l

(2.2)

where Ti is the duration of the ith experiment, q i are suitable weighting factors, and ai is the predicted stress history given by (2.3) be a minimum with respect to all admissible choices of vectors a, b, and c.

34 1

3. QUASILINEARIZATION

When, instead of continuous time histories C i ( t )we onlypossess measurements of the stress output at M given values of time, Eq. (2.2) will be substituted by

a discrete functional where

Cik

=ci(tk)

and

bik

= ai(fk).

3. Quasilinearization Here we outline a method based on quasilinearization to solve the optimization problem formulated in the last section. To this end we define the ( R + S + 2N)-dimensional vector x = (b1, b,, . . . , 6,; cl, c,, . . . , c,; a,, a , , . . . , a Z N - of the unknown quantities and regard ai given by (2.3) as a function of x. Quasilinearizing only with respect to x , we obtain, in the usual way a!"f1) = a?) + (grad a!"),

1,

(3.1)

where (., .) denotes inner product, grad a$")stands for the gradient vector grad

(TI") = (&rI")/dx:!)),

j = 1, 2,

. . . , R + S + 2N,

(3.2)

and AX("+')given by

AX(n+l)= x("+1)-

x(fl)

7

(3.3)

is the correction of vector x at the ( n + 1)th iteration. Clearly, a?) denotes the value of ai in (2.3) evaluated at x = x(").We evaluate grad a!") by differentiating Eq. (2.3), i.e.,

grad a?) =

342

12B THE IDENTIFICATION PROBLEM IN VISCOELASTICITY

where hb,(") = ah/abjn),

j = 1, 2, . . . , R,

(3.5)

gCk(") = ag/acp,

k

= I , 2,

. . . , s,

(3.6)

I

= 0,

and = aflaa?),

1 , .. .,

2 - ~1.

(3.7)

As usual, at each iteration of the numerical process we determine a correction vector Ax("+') such as to render minimum the error functional (2.2). Thus, given by (3.1) into Eq. (2.2), and minimization with resubstitution of spect to Ax("+') by taking derivatives, yields

.In+')

K

1q i s i=l

K

Ti

a?) dt =

(grad oi"),Ax("")grad

1q i i i=l

0

Ti

(cTi - aj"))grad oj")dt,

0

(3.8) a linear system of algebraic equations in the R + S + 2N components of the = (Ab"'", Ad""), Ad"")). This process is correction vector quadratically convergent, if convergent at all. To initiate the computations we are required to guess an initial approximation vector x(O). An appropriate choice of x(O) may considerably speed up the convergence of this process. In general, some a priori knowledge of the class of material to be identified permits the construction of a suitable initial approximation. When no information at all is available, we may take x(O) 0. This is what we have usually done in the many examples whose references are listed in the Bibliography at the end of the chapter. 4. Computational Aspects

+ +

If with (a;)),j = 1, 2 , . . . , R + S + 2N, we denote the (R S 2 N ) x K matrix whose columns are the vectors grad a?), i = 1 , 2 , . . . , K , then Eq. (3.8) may be compactly written A(")Ax(n+ 1 ) = Y (n)9 (4.1) where A(") is the symmetric (R elements a j k are given by K

a!") = ($) Jk

Ti

i=l

N

qil

i=1

j,k

a$)ag) dt,

yi

=

=

1 , 2 , . . .,R

+ S + 2N,

0

and where y'") is the 2N by yy)=

+ S + 2 N ) x ( R + S + 2 N ) matrix

+R +S

(4.2)

vector whose components y y ) are given

Ti

0

whose

.(ifi - o?))a$)dt,

j

=

1 , 2 , . . ., R

+ S + 2N.

(4.3)

343

5. COMPUTATION OF THE DERIVATIVES

The evaluation of the quantities a$) and $) may be conveniently reduced to the solution of an initial-value differential system. The central idea involved here is to reduce the computation of the convolutions and the derivatives appearing in the expressions for of" and grad a?' given by Eqs. (3.1) and (3.4), respectively, to the solution of a system of differential equations subject to initial conditions. This may be achieved by using some ideas of differential approximation. This is discussed in the next two sections. 5. Computation of the Derivatives The derivatives hb,, j = 1,2, ..., R ; g,,, k = 1,2, . . . , S ; and f a , , 1 = 0, 1, . . . , 2 N - 1, appearing in (3.4), may be evaluated easily if functions h, g, andfare explicitly given in terms of vectors b, c, and a. This is the case when we use polynomials of exponentials. For example,

h ( ~b) , = blE + b, eZ + or

; g(E, c) = E exp(c,e); f ( t , a) = a, t - " ' ,

h ( ~b), = blE + bz + . .. ; f ( t , a) = a. exp(-alt2),

g(&,C) = E

+

C~E'

+

* * *

(5.1)

;

(5.2)

etc. In other instances, h, g, or f may be given in differential form, i.e., as the solution of a differential equation. In this case we cannot evaluate the derivatives directly, but we can, in general, by differentiation, construct appropriate differential equations for the pertinent derivatives. We illustrate this point with an example. A case of some interest is provided when the relaxation kernel f satisfies the ordinary differential equation

d N f d t N+

N-1

C a, dfldt' = 0,

(5.3)

i=O

subject to the initial conditions

i = 0, 1, . . . , N - 1. (dfldti),=o= aN+i, Differentiation of (5.3) and (5.4) with respect to a, yields

(5.4)

N-I

dNf,/dtN+-C-ui i=O d x j / d r i= -dif/dtj,

(5.5)

subject to the homogeneous initial conditions i , j = 0, 1, . . . , N dyaf,,/dri = 0, forthequantitiesf,,,j=O, 1, ..., N - 1, and

-

1,

(5.6)

N-l

dNfaN+jdtN +

i=O

a, dYaN+,/dti = 0,

(5.7)

344

12B THE IDENTIFICATION PROBLEM IN VISCOELASTICITY

subject to the initial conditions

i, j = 0, 1, . . . , N - 1, (5.8) d%,+,/dt' = d,,, where dij is the Kronecker symbol, for the quantities fa,, j = N , N + 1, . .. , 2N - 1. Now, integration of the coupled system of differential equations (5.3), ( 5 . 9 , and (5.7), subject to the initial conditions (5.4),(5.Q and (5.8), respectively, furnishes the quantities of interest. Clearly, in order to determine required by Eq. (3.4), we must substitute aj by a y ) in the derivatives LjCn) Eqs. (5.3)-(5.8). It remains now to discuss the computation of the convolutions appearing in (3.4). This is presented in the next section. EXERCISE Show that u = exp(ur *) satisfies the differential equation u' - 2aru = 0,

u(0)= 1,

and construct an appropriate differential equation for the quantity u = au/au.

6. Computation of the Convolutions

It is well known that a direct numerical procedure, i.e., by quadratures, to compute convolutions of the form appearing in Eq. (3.4) is an inefficient process insofar as time and storage is concerned. Thus, there are obvious advantages in treating the problem in an indirect rather than in a direct manner, if this circumvents the need to use quadrature formulas for the computation of the convolutions. We can implement an efficient computational scheme when the kernelsf'") and appearing in the convolutions of Eq. (3.4) are given in differential form of the type discussed in the preceding section. In order to illustrate the ideas, it is enough to consider the computation of the convolution u ( t ) = f v ( r ) f ( t - 7) dt, 0

where f satisfies the linear differential equation (5.3) subject to the initial conditions (5.4). If l j ,j = 1, . . . , N, denote the roots of the characteristic equation

aN +

N- 1

=o,

i=O

which for simplicity we assume real and different, then f satisfying (5.3) and (5.4) may be written N

f( t ) = iC Yi exp(Ji t ) , = 1

(6.3)

7. PREDICTION. INVERSION OF A NONLINEAR VOLTERRA EQUATION

345

where the coefficients y i are determined via the solution of a well-conditioned (nonsingular) linear system of algebraic equations. Now, substitution of (6.3) into (6.1) yields

where the zi's are the functions determined by the solution of the initialvalue system

dZi/dt - R i Z i

= u,

Z i ( 0 ) = 0,

i = 1,

.. . ) N .

We note that the previous derivations were made under the assumption that the roots Ri are different and real. Multiple and imaginary roots can be easily handled with obvious modifications. When the kernelfis not given in the linear differential form (5.3)-(5.4), we employ differential approximation as outlined in Section 12A-18 to reduce the problem to the format used in this section. This is the case whenfis given by equations such as (5.1) or (5.2). Relatively low order for the approximation ( N = 3 or 4) usually leads to excellent accuracy. See the examples in the Bibliography at the end of the chapter. EXERCISE Using differential approximation, find suitable approximations of the form

f-

C ci exp(--hi t ) i=l

for function f = t- ', CL > 0. See the paper N. Distkfano and R. Todeschini, Modeling, Identification and Prediction of a Class of Nonlinear Viscoelastic Materials, (I), lnternat. J. Solids Structures 9 (1973), 805-818.

7. Prediction. Inversion of a Nonlinear Volterra Equation Once a set of constants has been optimally determined using the method outlined in previous sections, we are in a position to discuss a number of problems associated with the prediction of strains and stresses. Consider, for example, the model given by Eq. (2.1). Two basic direct problems immediately arise : (a) Given a certain strain history (b) Given a certain stress history

E,

find the associated stress history a; the corresponding strain E .

6,find

Problem (a) is clearly the simplest of the two. We can readily compute a by direct substitution of E in the model equation (2.1). In order to avoid the computation of the convolutions, we proceed as in the preceding section.

346

12B THE IDENTIFICATION PROBLEM IN VISCOELASTICITY

Suppose that by using differential approximation we find the following exponential approximation for f: N

f=

C y i exp(-Ai

t).

i=l

Substituting (7.1) in (2.1) we write

where the zi satisfy the differential equations dzi/dt + ,Ii zi = g(E),

z,(O) = 0,

i = 1,2, . .. , N .

(7.3)

Integration of the initial system (7.3) furnishes the stress u via Eq. (7.2). In g and h appearing in (7.2) and (7.3) we omitted the dependence of these functions on the material constants, in an effort to simplify the notation. The solution of problem (b), which along classical numerical lines is considerably more complicated than problem (a), in the present formulation requires only very little additional effort. In fact, assuming that function h ( ~appearing ) in (2.1) possesses an inverse, i.e.,

h ( ~=) k e E = H ( k ) ,

(7.4)

then we can invert Eq. (7.2) in the sense of N i= 1

(7.5)

a function that substituted into Eq. (7.3) yields

i = 1,2, . . . , N , a system of nonlinear differential equations subject to initial conditions, for the quantities z i , that substituted in (7.5) furnish the desired value of E . The success of this method, i.e., the inversion of nonlinear integral equations of the Volterra type by reduction to differential systems, is due to the expansion of the kernel in exponential functions and in the possibility of inverting the nonlinear function A(&). If this inversion cannot be analytically performed, i.e., if H in Eq. (7.4) does not afford an analytical expression, we can always perform the inversion of h ( ~in) a numerical fashion. We observe that the solution of problems (a) and (b) requires only the integration of systems of first-order, ordinary differential equations subject to initial conditions, a task for which many standard algorithms and computer

347

8. IDENTIFICATION OF SOLID POLYURETHANE

routines are available. But what is important to note is that this process requires only the storage of the computer program and the quantities currently being computed, an insignificant number in comparison with the storage of the whole set of functions if a quadrature approach to the computation of the convolutions is used. We finally observe that the computation of creep from (several) relaxation tests, and conversely, the determination of the relaxation function from creep tests, is a particular case of the procedure outlined in this section.

8. Identification of Solid Polyurethane from General 0-8 Data Our next step now should be to test the accuracy and stability of the numerical processes involved in the identification method previously developed. However, we shall not pursue studies of this type at this time. The reader interested in these aspects should consult the papers by the author quoted in the Bibliography, where the accuracy and stability of the method were thoroughly examined using synthetic uncorrupted and corrupted data. Here we are interested in the use of models of the class given by (2.1) to predict nonlinear viscoelastic behavior of real materials. To this end we employ experimental data furnished in a paper by J. S. Y. Lay and W. H. Findley. The experiments were performed at constant temperature on a rod of solid polyurethane subject to various axial strains and stresses. We have divided some of the available data into three groups; that is, we have considered the experimental data associated with the following strain histories : Strain Histories I (Relaxation at various

E(f)

4.333 l(t) 6.500 l ( t ) = 8.666 l ( t ) 10.833 l(t)

E

levels)

x 1 0 - ~in./in., x in./in.,

x x

in./in., in./in.,

tI 2, t I 2, tI 2,

t I 2,

Strain History 11 ~ ( t =) 4.333[1(t)

+ l ( t - 2) - l ( t - 3)] x

in&,

tI 4,

(8.2)

Strain History III

~ ( t=) [10.833 l(t) - 4.333 l ( t - 2)] x

in./in.,

t53,

(8.3)

where the time t is given in hours. The first group is clearly formed by the relaxation curves obtained during two-hour experiments at various constant strain levels. The lack of proportionality of 0 in terms of E at various times indicates the nonlinear nature of the process involved.

348

12B THE IDENTIFICATION PROBLEM IN VISCOELASTICITY

To account for such nonlinear behavior Lay and Findley employed a model of the type o(t) = klE

+ k2 eZ + k , E~ + k, f&(T)(t 0

- r)" d T

(8.4) i.e., a multiple integral expansion of the Volterra type, truncated at the third order, with degenerate kernels. Under a constant strain c i , ts in (8.4) reduces to ai(t) = klEi

+ k , ci2 + k, + ( k , q + k , E i 2 + k6 ~~,>[l/(rn+ l)]t"",

(8.5)

an expression used by Lay and Findley to fit the relaxation data associated with the strain histories I, by using a combination of graphical and analytical procedures. The model used by us is given by

where c i , i = 1, . . . , 5, are constants to be determined by using the method outlined in previous sections. In order to exhibit the influence of the amount and type of data used for identification purposes in the predictive ability of the model, we have performed two separate identifications. In the first one we only employed the four strain histories I for the determination of the unknown constants. This is exactly the same data used by Lay and Findley to fit Eq. (8.5). For the second identification, we used concurrently the strain histories I1 and 111. Using the two sets of five parameters determined in both identifications, we subsequently proceeded to predict the stresses associated with the strain histories 1-111, in order to check the predictive ability of the model, since we can compare the predicted results with the experimental ones. The resulting stresses obtained using the parameters determined with the data I are denoted by D1while those obtained using the parameters determined with data I1 and I11 are denoted by D, . For computational purposes, the data were prepared as follows. The strain histories were entered using the analytical expressions given by (8.1)(8.3), where the time is given in hours. The stresses were determined graphically, by directly measuring the corresponding ordinates in appropriately enlarged pictures of the drawings appearing in the paper by Lay and Findley.

349

8. IDENTIFICATION OF SOLID POLYURETHANE

The values o f t for which the ordinates were measured are not equally spaced. Forty-three points in each experiment were measured in the stress histories I, 71 in the stress history 11, and 60 in the stress history 111. The values of the unknown constants c , , c2 , . . . , c5 obtained in the two identifications are given in Table 12B-1. It is noted that both identifications have been performed using the same initial approximation and the same lower and upper permissible limits. To reach the same convergency rate, given by

I (cjn+l)

-Cy)/CS”+y

5 10-5,

j

= 1,.

..,5,

the identification employing data I needed 17 iterations, while the one employing data I1 and I11 needed 22 iterations. TABLE 12B-1 Estimation of Parameters ~

~~

Parameter

Initial approximation

c,[lOz ksi] 0.01 0.01 cZ[1o4ksil c3[102ksi/hr(’-‘5)] -0.0099 0.01 c4[1O21 0.01 C5

Lower permissible limit

Upper permissible limit

0.01 0.01

10.0 0.999

Identification using (u,E ) histories I I1 and I11 (17 iterations) (22 iterations) 5.2358 -0.39263 -0.055363 0.47245 0.75872

5.1141 -0.25744 -0.032759 1.0676 0.74664

The predictive ability of both identifications may be measured by the mean square stress error given by

c R

(l/R)

( o p - o;;’k””)Z

k= 1

and displayed in Table 12B-2. The stress histories predicted by using both sets of parameters are shown in Figs, 12B-1, 12B-2, and 12B-3 together with the experimental results and those obtained by Lay and Findley. We observe that the results obtained using the relaxation curves, denoted by D, in the figures, fit the experimental curves closer than the results obtained by Lay and Findley. In turn, the results denoted by D , , i.e., those obtained by using more general (variable) strain and stress histories, are considerably better than those of D,. The amount and type of data to be used for identification purposes are a delicate matter. In general we tend to use as much data as

350

12B THE IDENTIFICATION PROBLEM IN VISOCELASTICITY

TABLE 12B-2 Mean Square Deviation of o-Histories

Stress-strain history

Mean square stress error [lo-'] obtained using parameters identified with data:

I

I1 and I11

I &=4.333 x 10-3 E = 6.500x 10-3 E = 8.666 x E = 10.833 x I1 ,111

1.19 5.16 13.36 7.16 42.49 38.04

6.36 6.27 47.37 18.38 14.20 13.90

Total mean square error

107.40

106.48

5.3 5.2 -

, , , ,

I

I

I

I

,

,

'= YZ

. ,< s,-.
STRAIN HISTORY1

-t

-

4.7b 4.3'4.2

-

-

-

Ela

I-

in

z

0

3a

I2

0

0.5

I.o

1.5

2.0

TIME (HOURS)

- - -prediction D1, Fig. I2B-I. Model responses to strain histories I: -experiment; using constants determined with stress-strain histories I. (After DistCfano and Todeschini [1973b].)

available. In both identifications (and predictions) used here, the integration of the equations was performed using an Adams-Moulton integration scheme with step size H = 0.005. For more information on numerical matters, the reader should consult the original paper listed in the Bibliography at the end of the chapter.

8. IDENTIFICATION OF SOLID POLYURETHANE

, , , ,

5.3

, , , ,

, , , , z

, , I , , , ,

5.2

1

5.1

5.0

v, Y

4.9

STRAIN HISTORY

3.1

I

LL

k ",

l

LANDF7

Y I

2.7"""

0

I

~

0.5

I

I

I.o

1

I

I

I

I

1

I

I

\EXP ~

I

1.5 TIME (HOURS)

Ill

1

1

1

1

1

1

1

1

1

2.5

2.0

3.0

Fig. 128-2. Model response to strain history 11. D , : prediction using constants determined with stress-strain histories I; D 2: prediction using constants determined with stressstrain histories I1 and 111. (After Distkfano and Todeschini [1973b].)

STRAIN HISTORY II

1.9

-

1.8 0

'

~

'

'

~

0.5

~

~

~

1.0

~

1

1

1

15

1

1

1

1

1

1

2 .o

1

1

2.5

1

,

,

1

30

1

1

1

1

3.5

1

~

,

1

,

4.0

T I M E (HOURS)

Fig. 12B-3. Model response to strain history 111. D , : prediction using constants determined with stress-strain histories I; D 2: prediction using constants determined with stressstrain histories I1 and 111. (After DistCfano and Todeschini [1973b].)

1

~

1

1

,

1

352

12B THE IDENTIFICATION PROBLEM IN VISCOELASTICITY

9. Another Example : Low-Density Polyethylene In this section we use again an input-output model of the family (2.1) to represent mathematically the nonlinear behavior of a viscoelastic material subject to both tension and compression. The main purpose of this additional example is to exhibit once more the exceptional ability of relatively simple models of the family (2.1) to predict the behavior of nonlinear viscoelastic materials subject to general loading histories, and to show the advantages of the present identification method over conventional nonparametric procedures based on the use of multidimensional step inputs. As in the example presented in the preceding section, we also devote a preponderant portion of the numerical experimentation here to exhibit the influence of the type and amount of data employed in the identification of the predictive ability of the resulting model. The experimental data used in this example were obtained by V. V. Neis and J. L. Sackman. These investigators performed a large number of experiments in tension and compression employing more than 80 specimens under isothermal conditions. The data so obtained were conveniently processed and employed to fit the kernel functionsf,, f 2 , and f 3 in a model of the type

+ J:

1;

1~'(rl)a(r2)D(73)h(t

-

t - z2

9

- T 3 ) dzl

dz2 dz3

7

(9.1)

i.e., a multiple Volterra integral expansion truncated at the third order. Here, unlike the model (8.4) used by Lay and Findley, the kernels are not given in parametric form. The reader is urged to consult the original paper by Neis and Sackman regarding the method employed for the determination of the kernels, which is based on an adroit use of multidimensional step inputs. The model employed by us contains seven constants and is given by &(t)= C ~ C T+ c2 o2 + c3o3 +

J:

(c4 D

+ c5 a* + c6 03)(t- T)-"

dr.

(9.2)

From the excellent and abundant experimental data reported by Neis and Sackman, we have chosen the five experiments associated with the following stress input histories: Stress History C1 ( t in seconds) ?l(t)

= 0.4

l ( t ) - 0.7 l ( t - 600) + 0.3 l ( t - 1200),

(9.3)

Stress History o2 ( t in seconds) C z ( t ) = -C1(t),

(9.4)

353

9. ANOTHER EXAMPLE: LOW-DENSITY POLYETHYLENE

Stress History C3 (t in seconds) 1.6224t x 0.5062, 1.3881 - 2.1302t x

0 It I 312, t I 414, 312 I 414 I t I 540, 540 I t I 604, t I 788, 604 I tI 894, 788 I tI 1157, 894 I

+

-0.6433 1.4587t x 0.5062, 2.2269 - 1.9247t x

(9.5)

Stress History if4 (t in seconds) 54=

[

10-3t, 1 - 10-3t, -2 10-3t,

+

0I t s 500, 500 I t s 1500, 1500 5 t 5 2000,

where the stresses are given in ksi. The strain histories E l , E 2 , E 3 , and E4, corresponding to a,, C 2 , 03,and C4 given above, were tabulated from convenient enlargements of the figures appearing in the paper by Neis and Sackman. These stress-strain histories were combined in various ways to perform the determination of the unknown vector c, using the method described in previous sections. Specifically, the following five combinations of experiments were considered Ei+Ez, El E2

E3+E4, Ei+Ez+E3, El +E2 E3 E4,

(9.7)

+ +

+ + E4,

where E,denotes the experiment whose input is the stress history Ci and whose output is the strain history Ei. Table 12B-3 presents the results of the five TABLE 12B-3 Estimation of Parameters

Parameter identification using the following combination of experiments: Parameter Ei

+ E2

1.8297 0.11505 6.3886 1.2093 0.21362 - 2.5973 0.80961

E3

+ E4

0.4099 -0.1131 -1.7018 0.7030 0.1104 0.3121 0.8950

Ei

+ Ez + E3

2.7401 0.15156 0.60482 1.0011 0.25671 -0.61657 0.79929

Ei

+ Ez + E4

2.3709 0.061140 3.0737 1.1467 0.2203 3 - 1.6556 0.79826

Ei

+ Ez + E3 + 2.7625 0.083646 0.30575 0.96353 0.24199 -0.3 1681 0.80017

354

1 2 8 THE IDENTIFICATION PROBLEM I N VISCOELASTICITY

identifications performed using the various combinations of experiments given by (9.7). To exhibit the predictive ability of the various identifications performed, in Table 12B-4 we present the predicted values of the strain history c 3 ( t ) obtained by using the five vectors c appearing in Table 12B-3. In order to display further the predictive ability of the model relative to the type and amount of data employed in the identification, we have computed the mean square deviations N

mean square deviation

= (1/N)

11[ci(k)- li(k)I2,

i = 1,2, 3,4,

(9.8)

k=

of the predicted strain histories E ~ c,2 , I, , and c 4 , with respect to the corresponding experimental values li . These results are presented in Table 12B-5. In analyzing the results displayed in Tables 12B-3, 12B-4, and 12B-5, we observe that the prediction of the various models is consistently good. I t is, however, clear from the results that the prediction of those strain histories TABLE 12B-4 Comparison of Various Predictions of Strain History c3

Time 1,

sec 0.0 100 200 300 360 400 420 500 540 600 700 800 900 lo00 1100 1200 1300 1400 1500

Experimental strain 10’ 2 3 El 0.0 0.733 1.644 2.633 2.917 3.000 2.983 2.256 1.878 1.750 2.289 3.056 3.167 2.261 1.222 0.517 0.406 0.360 0.328

Predicted values of loz E~ obtained by using the following combinations of experiments:

+

E g

0.0 0.750 1.679 2.694 2.848 2.875 2.807 1.978 1.587 1.563 2.290 3.055 3.012 2.058 1.138 0.492 0.394 0.338 0.300

E3

+ E4

0.0 0.729 1.618 2.599 2.897 2.969 2.977 2.214 1.776 1.652 2.276 3.084 3.209 2.297 1.262 0.540 0.429 0.363 0.317

El

+ Ez + 0.0 0.783 1.692 2.651 2.867 2.924 2.898 2.137 1.717 1.659 2.338 3.078 3.138 2.241 1.234 0.549 0.447 0.386 0.343

E3

El

+ + E4 E2

0.0 0.766 1.672 2.629 2.810 2.852 2.809 2.048 1.651 1.611 2.297 3.022 3.035 2.146 1.195 0.529 0.429 0.369 0.328

Et

+ Ez+ En + E4 0.0 0.776 1.677 2.640 2.867 2.928 2.905 2.139 1.716 1.653 2.326 3.076 3.148 2.243 1.234 0.553 0.451 0.389 0.346

355

9. ANOTHER EXAMPLE: LOW-DENSITY POLYETHYLENE

TABLE 12B-5 Mean Square Deviations of &-Histories Strain history

Mean square deviation [lo-*]

EI

Ei

El E2

E3

E4

Total

+ Ez

28.5 21.2 145.2 91.6 286.8

E3 + E4

E~ + E~ + E~

485.4 450.2 14.9 21.2 971.7

48.5 28.1 29.1 67.5 173.2

E~ + E~ + E~ El 30.4 27.6 88.9 62.8 209.7

+

Ez

+

E3

+

E4

47.1 36.8 27.1 52.4 163.4

not used in the determination of the constants is somewhat poorer than the prediction of the strain histories employed in the identification procedure. An exacerbation of this trend is clearly exhibited by the model obtained using the combination of experiments E , + E 4 . A close look at the model constants shows the lack of ability of this model to predict instantaneous responses accurately. In fact, simple calculations show that the response to an instantaneous r~ jump is less than 10 % of the value of the same response predicted by the other models. If, however, the response to a r~ jump is averaged over a short period of time (say 10 sec), then the prediction, although still poorer than with the other models, is indeed remarkably accurate. The reason for this is that the viscous mechanism, represented by theconstant c, inthismodel, has been automatically adjusted to compensate for the lack of ability of the model to predict elastic instantaneous responses. Another remarkable feature of this anomalous model is that when used to predict either c3 or c4, it yields the least mean square error of Table 12B-5. An explanation for the behavior of this model should be found in the fact that the experiments E, and E4 employed in the determination of the constants contain no jumps, making it difficult for the model to determine accurately the parameters responsible for the elastic instantaneous response. This is clearly underlining a basic operational principle in system identification that should be kept in mind when designing the experiments for the determination of the model constants: The predictive ability of a given model, relative to a family of inputs, is improved if in the identification we employ histories qualitatively and quantitatively similar to those whose prediction is sought.

356

12B THE IDENTIFICATION PROBLEM IN VISCOELASTICITY

It is also clear that consideration during the identification process, of any additional information on the model behavior obtained from theoretical or empirical sources, will definitely improve the predictive ability of the model. Finally, the response of the model to the strain histories c l , c 2 , and c3 by using the combination El + E2 + E, for the determination of c was plotted in Figs. 12B-4, 12B-5, and 12B-6, respectively. In order to facilitate the comparison of results, we also plotted the experimental strains and those predicted by the model employed by Neis and Sackman. 10. Discussion In this chapter we have presented a limited, although significant version of the identification problem in viscoelasticity. A more comprehensive view of the problem will be presented in a future volume. The procedures were illustrated using equations of the type (1.2). This equation appears to yield satisfactory results in applications involving a number of polymeric materials.

0.03

0.02

:

U L

-300

v)

TIME (SECI

-z

0.01

STRESS HISTORY I

-

. z

-

z

-

0

a t x

100

200

300

I

400

I

500

I

700

1300 1400

I

800

I

900

I

1000

I

1100

12

Iv)

-

- 0.01

--__- - _

1

-0.02I

Fig. 12B-4. Model response to stress history I :

experimental; - - -prediction;

_ _ _ _ Neis and Sackman. Constants determined using El + E2 + E 3 . (After DistCfano and Todeschini [1974].)

~

357

10. DISCUSSION

0.02

0.01 I

I

I00

200

I

I

300 400

I

I

-

5

a I1I

I-ln -0.01

I

I

700 800 900 loo0 TIME (SEC)

500

1100 1200 1300 1400

-

-

-0.02

Fig. 128-5. Model response to stress history 11: experimental; - - predictions; and Sackman. Constants determined using El E2 E 3 . (After DistCfano and Todeschini [1974].)

+ +

~

- - - _Neis

312

414

540 604

788

894

-

1157

-

TIME (SEC)

Fig. 12B-6. Model response to stress history 111:

experimental; - - prediction:

_ _ _ _ Neis and Sackman. Constants determined using E , + Ea + E 3 .(After DistCfano and Todeschini [1974].)

~

358

12B THE IDENTIFICATION PROBLEM IN VISCOELASTICITY

NOTES, COMMENTS, AND BIBLIOGRAPHY

1 . The first attempt to simulate nonlinear viscoelastic behavior using Volterra integral equations of the type (1.2) appears to have been done as early as 1943 by H. Leaderman, “ Elastic and Creep Properties of Filamentous and Other High Polymers.” The Textile Foundation, Washington, D.C., 1943.

2-7. The method outlined here was originally presented in N. DistCfano, Some Numerical Aspects in the Identification of Nonlinear Viscoelastic Materials, U.S.C., Dept. of Elec. Engrg., Tech. Rep. 71-25. Also in Z. Angew. Math. Mech. 52 (19721, 389-395.

This paper contains abundant numerical experimentation using synthetic corrupted and uncorrupted data to test the stability and accuracy of the identification procedure. Although simpler, the identification problem in linear viscoelasticity is a significant problem in many important applications. Using a slightly different version of the quasilinearization approach presented here, a solution to the identification problem in the realm of linear viscoelasticity was first presented in N. DistCfano, On the Identification Problem in Linear Viscoelasticity, Z. Angew. Math. Mech. 50 (1970), 683-690.

For an application of the foregoing technique to the identification of thermorheologically simple materials, see N. DistCfano and K. Pister, On the Identification Problem for Thermorheologically Simple Materials, Acta Mech. 13 (1972), 179-190.

For application to biomechanics, see N. DistCfano and K. Pister, On Some Modeling and Identification Problems in Biomechanics, J. Biomed. Syst. 1, No. 2 (1970), 32-47; N. DistCfano and K. Pister, On Modeling and Identification in Biophysics with Applications to the Rheology of the Red-Cell Membrane, Conf Engrg. Med. Biol., 23rd Washington, D.C. (Nov. 1970).

Finally, for an application to a problem in structural mechanics see N. DistCfano and D. Nagy, Parametric and Adaptive, Non-Parametric System Identification Procedures in Structural Mechanics, Proc. 1971 Summer Comp. Simulation Conf Boston I (1971), 688-695.

8. The example of this section was extracted from N. Distdfano and R. Todeschini, Modeling, Identification and Prediction of Nonlinear Viscoelastic Materials, (I), Znternat. J. Solids Structures 9 (1973a), 805-818; N. DistCfano and R. Todeschini, Modeling, Identification and Prediction of a Class of Nonlinear Viscoelastic Materials, (II), Znternat. J. Solids Structures 9 (1973b), 1431-1438.

NOTES, COMMENTS, AND BIBLIOGRAPHY

359

The data used in this example were taken from J. S . Y. Lai and W. N. Findley, Stress Relaxation of Nonlinear Viscoelastic Materials Under Uniaxial Strain, Trans. SOC.Rheol. 12:2 (1968), 259-280.

9. The example of this section was extracted from: N. Distefano and R. Todeschini, On the Identification of Nonlinear Viscoelastic Characteristics of a Class of Polymeric Materials, Z. Agnew. Math. Mech. (to appear 1974).

The data used here were taken from V. V. Neis and J. L. Sackman, An Experimental Study of a Nonlinear Material with Memory, Trans. Soc. Rheol. 11 (1967), 307. See also: SESM Technical Report No. 66-9. University of California, Berkeley, California (1966).

Miscellaneous References The following papers. bring attention to some challenging problems relative to the mathematical descriptions of creep of plastics : R. 0. Stafford, On Mathematical Forms for the Material Functions in Nonlinear Viscoelasticity, J. Mech. Phys. Solids 17 (1969), 339-358; F. J. Lockett and S. Turner, Nonlinear Creep of Plastics, J. Mech. Phys. Solids 19 (1971), 201-214.

The structural inability of the Leaderman model to simulate some creep and recovery effects observed in certain plastics was pointed out by I. M. Ward and E. T. Onat, Non-Linear Mechanical Behavior of Oriented Polypropylene J. Mech. Phys. Solids 11 (1963), 217-229.

Index

A Adams-Moulton scheme, 42, 179, 256, 268, 292, 350 Aging processes, 312, 336 Algorithm, 42 Almost periodic loading, 323 Approximating sequence, 169 Approximation in policy space, 162, 189 Approximation, theory of, 43 differential, 330 Asymptotic stability of viscoelastic structures, 316-318

B Bandwidth of stiffness matrices, 88 Beams cantilever on elastic foundation, 60 continuous, 15 generalized nonlinear, 54

imbedded, 56 on nonlinear foundation, 138, 173, 181 optimal design of, 201, 244, 274 sandwich, 203 Beamlike structures, 48 Bellman-Hamilton-Jacobi equation, 126, 223, 247, 279 Betti’s theorem, 77, 113, 148 Biharmonic equation, 95, 105, 107, 144 Biomechanics, modeling and identification in, 358 Boltzmann’s superposition principle, 310 Boundary conditions in elliptic problems, Dirichlet, 85 essential, 85 mixed, 85 natural, 148 Boundary-value problems elliptic, 84 nonlinear, 171, 187 Bounds, 313, 321 lower, 168, 179 upper, 179 361

362

INDEX

Buckling classical, 21 invariant imbedding, 24 optimal design for, 205 of a plate, 102 Burgers model, 300

Convexity of the Lagrangian, 202 Convolution integral, 310, 326 computation of, 344 Courant parameter, 104, 143 Creep, 224, 302 Creep-buckling, 337 Critical length, 25 Critical load, 23, 25

C

D Calculus of variations, 123, 159, 240 fundamental lemma of the, 136, 202 necessary conditions, 137 simplest problem of the, 135, 235 sufficient conditions, 138 Castigliano, theorem of, 126, 146 Cauchy systems, 84, 111, 112, 121, 122 Characteristic value condition, 175 Characteristic value problems, 21, 28, 318 Characteristic values of the kernel, 174 Characteristic vectors, 5 Characteristic, 139 Closed cycle, conditions of, 31 1 Complementary energy, theorem of, 140 152 Compliance, 193, 246 Concavity, 165 Constitutive equations for viscoelastic materials, 338 of a bar, 49 Constraints, 128, 193, 221 design, 251 global, 221 local, 129, 222 Continuous analog, 18, 133 Continuous fraction, 25 Control asymptotic, 229 feedback, 217 open-loop, 218 optimal, 218 stochastic, 242 Controlled inputs, 332 Control of engineering processes, 124 Control problem, 222 Control theory, 43, 159, 217, 240 Control variables, 221 Convergent sequence, 167 Convex operators, 162 Convexity, 137, 165

D’Alembert principle, 26 Decision, 142 Design fully stressed, 194 optimal, 237 optimal structural, 124, 192 theory of, 187 Design constraints, 282 Design space, 193 Design vector, 193 Difference equations, 3, 26, 48, 73, 88, 90 Difference-differential equations, 306, 309, 335 Differential approximation, 330, 338 Differential equations, 29, 46 Riccati (see Riccati differential equations) stiff, 47 systems of, 32 vector matrix, 48 Digital computer, 41, 123 Dimensionality, curse of, 151 Disks, rotating (see Optimal rotating disks) Displacements constraints, 199 Dissipation function, 225 Dual formulation, 246, 278 Duality, 192 Dual loading, 196 Dynamical equations, 219 Dynamical error, 230 Dynamical systems, stability of, 46 Dynamic programming, 123, 159, 162, 189, 199, 213, 215, 241, 245, 279

E Elastic plate (see Plates) Euler-Bernoulli beam, 49 Euler critical load (see Critical load)

363

INDEX

Euler equations, 135, 147, 232, 250, 285 Extrapolation, 230

F Feedback control, 217 Feedback rule, 142 Filter invariant imbedding, 242 Kalman-Bucy, 232, 241 nonlinear, 242, 332, 338 optimal, 232 Wiener, 241 Filtering problem, 229, 241 Filtering theory, 82 Finite differences and invariant imbedding, 105 Finite-dimensional operator, 53 Finite elements, 50, 84, 86, 104, 143, 160 Flexibility matrix, 78, 100, 154 Fourier decomposition, 53 Fredholm integral equations (see Integral equations) Fredholm operator, nonlinear, 166 Fredholm resolvents, 122 Frequencies, natural, 26 Functional derivatives, 308 Functional equation (of dynamic programming), 125, 129, 151, 156, 160 Functional inequalities, 164, 168, 170, 190 Functionals, theory of, 308, 336 Galerkin procedure, 326 Galerkin (weighting functions), 87 Gauss-Newton method in function space, 341 Global constraints, 221 Green’s function, 108, 171, 174, 177, 190 variation of, 106, 121 Group property, 302

H Hamiltonian, 234, 236, 257 Hamilton’s equations, 57 Hammerstein theorem, 166 Hegelian proposition, 41 Hereditary effects, dissipation of, 310 Hereditary mechanics, 336

Hereditary processes, 307 aging, 312 asymptotic behavior, 313 biomechanical, 335 modeling and identification, 298 nonlinear models, 325 stochastic aspects, 324 system identification, 325 time- invariant, 311 Hessian, 138 Hidden (internal) variables, 227, 304, 328

I Identifiability, 328, 338 Identification (see System identification) Identification of low density polyethylene, 352 Identification of solid polyurethane, 347 Identification problem, 155, 339, 340, 358 Imbedding classical, 3 invariant (see Invariant imbedding) notion of, 158 Information content of the solution, 94 Initial-value formulation, 109 Initial-value problems, 29 Initial-value systems, 40 Integral equations (Fredholm), 107, 119, 122, 166, 190 characteristic values, 167 resolvent kernel of, 120, 121 systems of coupled nonlinear, 173 Integral equations (Volterra), 304 hierarchical system of, 324 inversion of nonlinear, 345 reciprocal kernel of, 302 Integral Riccati transformations, 114 Integrodifferential equations, 107, 112, 119, 318 Internal (hidden) variables, 303, 335 Interpolation functions, 87 Invariant imbedding, 1, 9, 12, 26, 54, 81, 82,91, 104, 106, 119, 186,216,256, 261 one-sweep method, 17, 63 perturbation analysis, 56 two-sweep method, 16, 63, 263 Invariant imbedding and finite elements, 93

364

INDEX

Inverse matrix, 31 Irreversible processes, 335 J Jacobian matrix, 41, 146, 182, 233 Jacobi's formula, 36

K Kalman-Bucy filter, 232, 241 Kernel pointwise nonnegative, 168 positive definite, 166 resolvent, 120 reciprocal, 302 Kronecker delta, 92 Kuhn-Tucker conditions, 197, 215 L Lagrange multiplier (see Multipliers), 194, 199, 208,215 Laguerre functions, 326 Legendre condition, 136, 137 Liapunov function, 69, 266 Linear dynamics and quadratic criterion, 159,227, 241 Linear homogeneous system, 34 Linear inhomogeneous systems, 35 Lines, method of the, 95, 105 Lower bounds, 168, 179 best possible, 176 M

Mathematical model, 39 Matrix Riccati equations (see Riccati differential equations) Matrix Riccati transformations, 58, 83, 90 Matrix theory, 29-32 Maximum operation, 165 Max-min optimization problems, 245, 273 Maxwell model, 300 Memory, materials with, 338 Minimum potential energy, theorem of, 203

Minimum principle, 234 Min-max operation, 181 Missing boundary conditions, 58, 98, 114, 186 Mixed boundary-value problems, 82 Monotone approximation, 164 Monotone operators, 190 Monotonicity, 169 Multipliers (see Lagrange multipliers), 234, 258 Multipoint boundary conditions, 181, 189 Multistage decision processes, 123, 142 Mutual potential energy, principle of, 274, 297 N

Necessary conditions (see Euler equations), 239 Neumann series, 168 Newton-Raphson equation, 180 Newton-Raphson-Kantorovich method, 169, 174, 179, 183 Nonlinear boundary-value problems (see Boundary-value problems) Nonlinear mechanics, 171 Nonlinear programming, 198, 216 Nonlinear systems, 181 Nonnegativity, 168 Norm of a matrix, 68 Norm of a vector, 40 Numerical process, 42 stability of, 42, 47 Numerical stability (see stability) 0

Objective functionals in control theory, 221 Objective functions and constraints in structural design, 193 One-sweep method, 17, 63 Optimal beam design (see Beams, optimal design of) filter, 232 loading policy, 224, 236 rotating disks, 212, 276, 296 sandwich beams, examples of, 205, 210, 253, 260,266 structural design, 192, 244

365

INDEX

Optimality condition, 126, 224, 248, 259 Optimality, principle of, 159 Optimum policies, 158

P Partial differential equations, 84, 106, 142 characteristics, 139 of dynamic programming, 134 quasilinear, 58, 141 reduction to a first order system, 50, 96 Patch basis, 87 Physical systems, 39 Piecewise continuous stiffness, 207 Poincare-Liapunov analogs, 325 Polymers, 359 Positive definite kernel, 166 Positive definite matrix, 32 Pontryagin’s extremum principles, 232 Prager’s device, 195 Prediction, 229, 345 Principal solutions, 7 matrix of the, 34 superposition of, 8, 47, 186 Processes dynamical equations of the, 219 hereditary, 220, 298

Q Quadratic convergence, 170 Quadratic form, 31 Quadratic functional, 148 Quasilinearization, 161, 164, 185, 189, 341 Quasilinear representation, 161, 165 ,182

R Raleigh quotient, 248 Reciprocity relation, 16, 113 Reciprocity theorem of Betti, 148 Red-cell, rheology of the, 358 Regula falsi, 180 Regularization, 326, 337 Relaxation, 302 Retardation time, 236 Riccati difference equations, 15

Riccati differential equations, 19, 59, 67, 98, 122, 148, 185, 186, 231, 252, 261, 283 asymptotic behavior of, 21, 229 Riccati transformations (see Matrix Riccati transformations) integral, 114 Ritz-Galerkin methods, 104 Rotating disks (see Optimal rotating disks) Routing problem, 130 Runge-Kutta, 42, 102, 103

S Scattering matrices, 75 rule of composition of, 77 structural, 66 Segmental approximation, 155 Self-adjoint systems, 65, 82 Semidiscrete operators, 144 Semidiscretization, 95, 96 Semigroup relations, 12 Semigroup properties of generalized beams, 54 Sequential estimator, 230 Sequential identification, 332, 338 Shells cylindrical, 50, 80 of revolution, 81, 122 Smoothing, 229 Solution space, 3 Sparce matrix, 88 Spring, aging, 319 Stability asymptotic, 69, 316 exponential, 68 numerical, 8, 11, 67, 265 Stability matrix, 68 Stability theory, 43, 311 State variables, 39 Statically indeterminate structure, 149 Stiffness, 10 Stiffness matrix, 75, 88, 97 Strain hardening law, 225 String, 2 infinite, 20 linear, 124, 159 nonlinear, 128, 159 Strip configuration, 92

INDEX

Strip, stiffness matrix of, 89 Structural design, elements of optimal, 192 Successive approximations, 161, 251, 282 monotone properties of the, 169, 174 Sufficiency of the optimality conditions, 195 Superposition of effects (see Viscoelasticity) Superposition principle (see Boltzmann’s superposition principle) Systems dynamical, 219 stochastic, 218 System identification, 160, 191, 325, 335, 337, 338 operational principle in, 355

T Tauberian theorems, 336 Paley and Wiener, 313 Pitt, 315 Teleological principles, 123 Time averages, 323 Time-invariant process (see Hereditary processes) Topology, 1, 92 Trace, 36 Transfer matrices, 74 rule of composition for, 77 Transmissibility matrix, 69 Trial functions, 86, 136 Truncated hierarchy, method of, 324 Trusses, elastic, 193 Two-point boundary conditions, 85 general, 70 Two-point-value problems, 3, 29, 43 linear, 44 Two-sweep methods, 16, 63, 263

U Upper bounds, 179

V Variational problems, 123 Variation of parameters, 36 Vector-matrix notation, 29-32 Virtual work, principle of, 198 Viscoelasticity, 236, 335 creep and relaxation functions, 302 difference-differential models, 306 differential models, 299 identification problem in, 339, 358, 359 linear integral models, 301 modeling aspects, 299 nonlinear models, 304 principle of correspondence, 318 superposition of effects, 302 Viscoelastic processes, optimal control of, 24 1 Viscosity coefficient, 236 Voigt model, 300 optimal loading policy of, 236 Volterra integral equations (see Integral equations) Volterra integral expansions, 304, 308 Volterra operators, 190

W Weighted residuals, 84, 104 Weighted function, 85 Winkler foundation, 245 Work functional, 196, 236, 246, 278

A 6 G D

4 5 6

F G H 1 J

9 O 1 2 3

~a

l