A rational finite element basis (Mathematics in Science and Engineering, Volume 114)

A

Rational

Finite Element Basis

ACADEMIC PRESS RAPID MANUSCRIPT REPRODUCTION

This is Volume 114 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.

A

Rational Finite Element Basis Eugene L. Wachspress Knolls Atomic Power Laboratory Schenectady, New York

Academic Press, Inc. NEW YORK SAN FRANCISCO LONDON 1975

A Subsidiary of Harcourt Brace Jovanovich, Publishers

COPYRIGHT 1975, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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

Library of Congress Cataloging in PublicationData

Wachspress,Eugene L A rational finite element basis. (Mathematics in science and engineering ; Bibliography: p. Includes index. 1. Finite element method. I. Title. TA347.F5W3 519.4 75-12594 ISBN 0-12-728950-X

PRINTED IN 'THE UNITED STATES OF AMERICA

11.

Series.

Eo my parents,

Jean and Sidney Wachspress

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CONTENTS xi xiii

PREFACE THEOREMS AND LEMMAS

Chapter 1 . PATCHWORK APPROXIMATION IN NUMERICAL ANALYSIS

1.1 1.2 1.3 1.4 1.5 1.6 1.7

Wedges and Pyramids Definitions and Notation Continuity Patchwork Approximation Spaces and Convergence Wedge Properties Isoparametric Coordinates Generalizations to Sides of Higher Order and to ThreeDimensional Elements 1.8 Remarks and References

1 6

15 18 23 24 28 30

Chapter 2. THE QUADRILATERAL 2.1 Inadequacy of Polynomials

2.2 2.3 2.4 2.5 2.6

Rational Wedges Areal Coordinates as Limits of Rational Wedges An Example of Quadrilateral Wedges Projective Coordinates Polygons?

32 33 37 39

40

49

Chapter 3. RATIONAL WEDGES FOR SELEmED POLYCONS

3.1 3.2 3.3 3.4

The 3-Con of Order Four The CCon of Order Five The Pentagon Some Elementary Congruences

52 63 69 70 vii

CONTENTS

3.5 Wedges for 3-Cons of Orders Five and Six 3.6 Two-sided Elements 3.7 Related Studies

75 83 87

Chapter 4. ALGEBRAIC GEOMETRY FOUNDATIONS

4.1 4.2 4.3 4.4 4.5 4.6 4.7

Motivation Homogeneous Coordinates and the Projective Plane Intersection of Plane Curves The Fundamental Congruence Theorem Associated Points Resolution of Singularities Remarks and References

88 90 92 101 109 112 124

Chapter 5. RATIONAL WEDGE CONSTRUCTION FOR POLYCONS AND POLYPOLS

5.1 5.2 5.3 5.4 5.5 5.6

Polycon Wedge Construction Verification of Polycon Wedge Properties The Case of the Vanishing Denominator Polypols and Deficit Intersection Points Polyp01 Wedge Numerators and Adjunct Intersection Points Illustrative Polycubes

126 142 147 162 167 172

Chapter 6. APPROXIMATION OF HIGHER DEGREE

6.1 6.2 6.3 6.4 6.5 6.6 6.7

Data Fitting Degree Two Approximation Degree Three and Higher Degree Approximation Intermediate Approximation Higher Degree Approximation on Polypols A Concise Algebraic Geometry Analysis Algebrais Reticulation

177 179 189 194 197 199 20 5

Chapter 7. THREE-DIMENSIONAL APPROXIMATION

7.1 7.2 7.3 7.4 7.5 7.6 7.7

Definitions and Background Triangular Prisms and Hexahedra Polyhedra Polycondra The Adjoint of a Well-Set Polypoldron Polypoldra Nodes and Adjacent Factors for Degree k Approximation Attainment of Degree k Approximation viii

206 212 22 1 223 232 240 243

CONTENTS

Chapter 8. A RATIONAL SOLUTION TO AN IRRATIONAL PROBLEM

8.1 8.2 8.3 8.4 8.5

245 249 253 255 2 59

Irrational Wedges The Method of Descent Wedges for an Ill-Set Polycon Nonconvex Quadrilaterals Remarks

Chapter 9. FINITE ELEMENT DISCRETIZATION

9.1 9.2 9.3 9.4 9.5 9.6 9.7

Introductory Remarks Some Simple Quadrature Formulas Consistent Quadrature and the Patch Test Triangle Averaging Mosaic Discretization A Discrete Laplacian for Quadrilaterals Harmonious Discretization

26 1 263 273 279 282 288 296

Chapter ZO. TWO-LEVELCOMPUTATION

10.1 10.2 10.3 10.4

Recapitulation Synthesis Coarse Mesh Rebalancing Concluding Remarks

314 315 319 32 1 322

REFERENCES

327

INDEX

ix

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PREFACE

Popularity of the finite element method is such that an astute lecturer or author may increase his audience by choice of a title like “Finite Elements and -,’ inserting his topic in the blank space, no matter how remotely connected with finite element methods. The title of this book is, nevertheless, precisely the subject of this book. Fundamental to any finite element computation is the definition of an approximation space over a collection of elements. A basis function is associated with each element node so that the approximation within the element is determined by the nodal values. Polynomial basis functions have been widely used, and convergence theory for continuous patchwork polynomial approximation has been developed to a high degree of mathematical sophistication. Elements over which polynomial basis functions apply, within restrictions imposed for rigorous theoretical foundations, are extremely limited. In two space dimensions, for example, triangles and parallelograms are admissible. Isoparametric coordinates enable use of a larger class of three- and four-sided elements that may have parabolic as well as straight sides. Despite limitations on element geometry, polynomial and isoparametric basis functions seem adequate for finite element computations of current concern. Why then do we seek alternatives? As computer capacity expands, computational sophistication grows, and desire for greater precision increases, we may no longer be content with approximate representation of curved boundaries by isoparametric parabolas or with the restriction to three- and four-sided elements. Not many years ago the straight-sided triangle was the all-purpose element. Now isoparametric elements are considered indispensable for some purposes. Tomorrow, we may well demand even greater flexibility. The basis functions described in this book are rational in two senses. They are rational functions (ratios of polynomials), and they are constructed from geometric properties of the elements in a rational (logical) manner. The word basis also has a dual meaning. Besides providing a function basis for polynomial approximation over elements, the theory establishes a logical basis (foundation) for finite element computation. Much of the convergence theory developed for polynomial approximation applies to the rational approximation.

xi

PREFACE

One fascinating aspect of the analysis is the coordination of geometric and algebraic arguments to exploit the interrelationship of element geometry and basis functions. As one proceeds through the successive stages of the development to increasingly more complex elements, the geometric simplicity of the basis function construction becomes more striking, and one suspects that the theory is not an invention but rather the discovery of a natural phenomenon. For finite element computation, one must evaluate within prescribed tolerances and constraints the integrals of certain products of basis functions and their derivatives of various orders over each element. Such integrals play a crucial role in discretization of continuous problems, and errors in their numerical approximation can have a deleterious effect on accuracy of computed solutions. Although construction of basis functions for complex elements is a fascinating mathematical diversion, practical use of these functions depends on our ability to evaluate the integrals within the prescribed tolerances. Chapter 9 deals with this problem. The technique of mosaic discretization described in Section 9.5 is the key to finite element integration with rational basis functions, and this device facilitates application of the theory developed in Chapters 1-8 to finite element computation over algebraically reticulated regions. This analysis was initiated in Dundee, Scotland, while the author was a visiting fellow, participating in a one-year symposium on numerical analysis sponsored by the Science Research Council of Great Britain. This was made possible by a leave of absence granted by the Knolls Atomic Power Laboratory for which the author is most grateful. The author is particularly thankful for the encouragement offered by Professor A.R. Mitchell of the University of Dundee. Discussions with Professor Mitchell motivated this entire investigation. We are also indebted to Professor R. Bellman for his editorial review of an early draft and for incorporating this work in the Mathematics in Science and Engineering series. The material in Chapter 2 on the general quadrilateral was reported at the Dundee Conference of Applications of Numerical Analysis held in April, 197 1 . Most of the analysis was done in Schenectady,,shortly after the author returned from Dundee. Some of the concepts that remained in an amorphous state in Dundee crystallized during the Schenectady winter. Although some of the early work on elements with curved sides was reported in an article in the Journal of the Institute for Mathematics Applications, most of the analysis has not been published previously. A summary of the algebraic geometry foundations appeared in the Proceedings of the 1973 Dundee Conference on Numerical Solution of Differential Equations. We are indebted to Heinrich Guggenheimer (Polytechnic Institute of New York) for first steering us toward the theory of divisors that provide the theoretical foundations for much of this work and to David Brudnoy (Knolls Atomic Power Laboratory) for his critique of the first few chapters. In a work so rich in geometric concepts, we are especially grateful to Albert D. Comley for the exceptionally well-done illustrations. Eugene L. Wachspress Schenectady, New York March 1 , 1975 xii

THEOREMS AND LEMMAS Theorem

Page

Theorem

Page

1.1 1.2 3.1 3.2 3.3 3.4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

8 21 70 72 73 74 98 102 111 112 113 114 117

4.8 4.9 4.10 4.1 1 4.12 4.13 5.1 5.2 7.1 7.2 7.3 7.4 7.5

117 118 119 121 122 124 150 153 210 210 210 210 213

Noether’s Theorem: Page 10 1 Lemma

Page

2.1 3.1 4.1

34 56 99

xiii

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Chapter 1

PATCHWORK APPROXIMATION IN NUMERICAL ANALYSIS

1.1 WEDGES AND PYRAMIDS The numerical solution to a problem is often expressed in terms of an approximation U(5) to the true solution u(x) for 5 in some prescribed region may be defined and U may be chosen as D. A norm the function which best approximates u over some approximation space A in the sense that U(5) = U where

II*11

(x;~,)

Questions of existence and uniqueness of -0 a ' and of convergence of H to zero as the dimension of A is increased somehow, are considered in many research papers and texts. This is a central problem in approximation theory. A common technique is to use the approximation space n A =

u(x;a) =

1

i=l

a.W. (x) 1 1 -

(1.2)

where the Wi are known basis €unctions and the ai 1

PATCHWORK APPROXIMATION

are combining coefficients. Some of these coefficients are chosen to satisfy prescribed boundary conditions, and the remaining ai are obtained by solving the norm minimization problem of Eq. (1.1). When a Ritz-Galerkin type formulation is applied, this often requires evaluation of matrix elements of the form

I

(1.3) Lm[Wi(x) I *LmI[Wj( 5 )I dx I D where Lm and L m l are problem dependent linear operators such as the identity or the gradient. Numerical solution of (1.1) is facilitated by choice of the Wi so that many of the';b vanish. One then determines -0 a by solving a sparse system of linear algebraic equations. The "patchwork" approximation characteristic of the finite element method is generated from basis functions each of which is nonvanishing only over a small subregion of D. The situation is clear in one dimension. Let D be the real interval [a,b] and let xi be a prescribed point set on D: a = x 1 < x2 < - * * < n-1 < xn = b. We may define

by' =

x

-

x. x. < x c xi+l 1 =

'

Then the combining coefficients are the values of U ( x ) on the points xi. We note that U(x) E C[a,bl Alternatively, we could choose U(x) to be piecewise cubic with value and derivative as the nodal parameters. This leads to a "spline" approximation with U(x) in C1 [a,b]. Higher-order derivatives and

.

2

RATIONAL FINITE ELEMENT BASIS

polynomials may be introduced. In each case, the approximation may be expressed in terms of basis functions that are nonzero only over small subintervals of la,bl. Approximation (1.4) can be represented, for example, by U(x) = y=lUiWi(x), where and Wi is the hat function (Fig. 1.1) : Ui = U(xi)

w.1 (x) =

0,

- (x -

= (x

-

F i g . 1.1.

x < xi-l and x > xi+lf XiJ/(Xi - xi-l) , xi-l = < x 5 - Xi'

-

(1.5)

xi+l)/(Xi xi+l), x. < x c xi+l 1 =

The h a t function.

In two dimensions the situation is more complex, - E: Cp(D) with p > 0. Our analespecially when U(x) ysis is restricted to p = 0: U ( 5 ) E C(D). Two patchwork approximations are widely used for twodimensional problems: (i) Triangles. Domain D is partitioned into a network of nonoverlapping triangles and U is a function that is continuous over D, linear within each triangle, and is uniquely defined within a triangle by its values at the triangle vertices. Areal coordinates (Section 2.5) provide a natural basis for approximation within a triangle.

'3

PATCHWORK APPROXIMATION

Fig.

1.2.

Triangle basis functions.

For the triangle in Fig. 1.2,

Let L1(xly) be the linear form (polynomial of degree one) which vanishes on side (2;3) of the triangle. Then we have W1(xty) = Ll(xty)/L1(xl,yl). This "wedge" basis function is associated with node 1 of the triangle. The other wedges are defined similarly. The wedges for the triangles which share vertex i piece together to form a "pyramid" function with value unity at i. This function is continuous over the triangles which share vertex i and vanishes along the triangle sides opposite i (Fig. 1.3).

Fig. (a)

1.3.

Wedge and pyramid

Wedge a t i = 3 ;

(b)

4

functions.

Pyramid

a t i = 3.

RATIONAL FINITE ELEMENT BASIS

Over the collection of triangles the approximation is U(X,Y) = CUiPi(X,Y) i

.

(1.7)

Only wedges of triangles with both i and j as vertices contribute to bij in Eq. (1.3). It is apparent that U(x,y) is continuous over the entire domain. (ii) Rectangles. Domain D may be partitioned into a collection of nonoverlapping rectangles. The wedges are bilinear within each rectangle. Referring to Fig. (1.4):

Fig,

1.4.

Rectangular element.

We note that each bilinear wedge is linear on each side of the rectangle so that the approximation

f

UiWi(x,y) , which is in general bilinear i=l within the rectangle, is linear on each side. This bilinear approximation is also adequate for the parallelogram shown in Fig. 1.5.

U(x,y) =

Fig.

1.5.

Parallelogram element. 5

PATCHWORK APPROXIMATION

Let L1(x,y) = 0 on side (2;3) and let L2(x,y) = 0 on side ( 3 ; 4 ) . Then It will be shown in the next section that W1 is linear on each of the parallelogram sides. The other wedges are defined similarly. Reduction of the interior function behavior to linearity on each side is essential for continuity of the composite approximation. The value of the approximation on any side must depend only on values at the two vertices of that side. Higher degree approximation is achieved by introduction of more nodes. This will be examined in Chapter 6. For the present, we consider only patchwork approximation which is linear on each side of the elements. 1.2

DEFINITIONS AND NOTATION A polynomial of degree n in x and y over the

complex field is of the form r+s 5 n brsx rYs1 I Pn(X#Y) =

>:

(1.10) r,s=O, 1,2,. where the ars are members of the complex field, IK, and there is at least one nonzero coefficient for which r+s = n. We usually designate polynomials by capital letters (P in this instance) with the degrees indicated by subscripts. A polynomial may be identified by a superscript. Sometimes the symbol Pr det notes a generic polynomial constructed from certain data. In such cases the subscript t denotes the maximal degree of the polynomial. Subscripts and super-

..

6

RATIONAL FINITE ELEMENT BASIS

scripts are suppressed when not needed. The set of points on which Pn(x#y) = 0 is a plane algebraic curve of order n. A polynomial is irreducible if and only if it cannot be factored into a product of polynomials of lower positive degrees. The curve of each irreducible factor of a polynomial is a simple component of the curve of the polynomial, and is called nondegenerate or irreducible. Curves Of order one are lines and of order two are conics. Properties of algebraic curves are analyzed in the branch of mathematics known as algebraic geometry. A review of pertinent topics in algebraic geometry is presented in Chapter 4 . Concepts required for the first three chapters of this monograph are discussed in this section. Let Px and P denote the partial derivatives of Y polynomial P with respect to x and yI respectively. A simple point of curve P is a point where either Px or P is nonzero. When both partials vanish,the Y point is said to be a singular point. An intersection point of two curves is a point common to both curves. Two curves which do not have a common component intersect at a finite number of points. This set of points is denoted by the symbol P.Q for curves P and Q. If point p is a simple point on each of two curves that do not have a common tangent at p, then p is called a "simple intersection point" and the curves are said to "intersect transversally" at p Otherwise, p is a "multiple intersection point". The theory of intersections of curves is a primary topic of algebraic geometry and will be discussed 7

PATCHWORK APPROXIMATION

more fully in Chapter 4 . Two polynomials which differ only in normalization have the same curve and are said to be "equivalent". When we speak of polynomial uniqueness we do not distinguish between members of an equivalence class. Whenever a polynomial is constructed from points on which it vanishes, it is assumed that some specific (though arbitrary) normalization is imposed. The value of polynomial P at point p is denoted by P(P) If for given polynomials P,Q,R hhere is a b in k such that P = bQ at all points on curve R, we say that P is "congruent" to Q modulo R and write P E Q mod R. The fundamental theorem of algebra that a polynomial of degree n in x has exactly n zeros, counting multiplicities, is often used in approximation theory. Chebyshev minimax theory, for example, abounds in theorems proved by showing that some polynomial of maximal degree n vanishes at n+l points and thus must be the zero polynomial. The following theorem f o r polynomials in two variables is less definitive but is particularly useful in this analysis.

-

THEOREM 1.1. Let Q be a polynomial in x and y which is a product of distinct irreducible factors and let P be a polynomial which is not identically zero. If P Z 0 mod Q, then Q(x,y) must be a factor of P(X,Y). Proof of this theorem may be found in higheralgebra texts (e.g. Bocher, 1907).

8

RATIONAL FINITE ELEMENT BASIS

We shall often speak of geometric properties of elements. Following Walker (1962, p.35),we assert that an algebraic condition connecting coordinates of points of a space defines a geometric property of these points if satisfaction of the condition does not depend on the coordinate system used. In a broader sense, any property of figures in a space which can be defined without reference to coordinate systems is a geometric property. The geometry of a space consists of the relationships between the geometric properties of figures in that space. In our study of basis functions for two-dimensional elements we will first consider elements in the real plane bounded by segments of irreducible alegraic curves. We designate as "polyconsll a particular subset of these algebraic elements which are treated in depth in the first few chapters. A polycon is a closed figure in the real plane bounded by segments of lines and conics. The polynomials which define these segments have real coefficients. It is a polygon when all the boundary segments are lines. The intersection points of adjacent segments are called vertices. A polycon is well set if and only if the boundary curves intersect transversally at the vertices and the extensions of the boundary segments do not intersect the polycon. (Both branches of a hyperbolic boundary curve are considered in the extension.) A polycon that is not well set is said to be ill set. We note that a polygon is well set if and only if it is convex. A figure with vertices pl, p2, ... is sometimes Thus, we may refer to denoted by [p,,p,, I. triangle [1,2,31

.

...

9

PATCHWORK APPFIOX M A T I ON

Examples of w e l l - s e t i n Fig. 1.6.

CONVEX POLYGON

and i l l - s e t polycons are shown

POLYCONS HYPERBOLA

--@f

NONCONVEX POLYG 0 N /

\

(b> Fig.

1.6.

( a ) Well-set a n d

( b ) ill-set p o l y c o n s .

P o i n t s a t which t h e e x t e n s i o n s of boundary segments intersect are c a l l e d " e x t e r i o r i n t e r s e c t i o n p o i n t s " (EIP). The o r d e r of a polycon i s t h e o r d e r of i t s boundary c u r v e . Thus i f t h e polycon i s bounded by r c o n i c and s l i n e a r segments it i s of o r d e r m = 2r+s. A polycon w i t h n v e r t i c e s i s c a l l e d an n- con I t is advantageous t o a d o p t f o r t h i s development a notation t h a t explicitly displays interrelations h i p s of p o i n t s , c u r v e s and polynomials. L e t IP1,P2, . . . I be a set of p o i n t s t h a t l i e on a c u r v e of o r d e r s. W e d e n o t e t h e c u r v e and a member of t h e e q p i v a l e n c e c l a s s of polynomials of d e g r e e s which

.

10

RATIONAL FINITE ELEMENT BASIS

vanish everywhere on the curve by (P1;P2;...)S' In general, the curve may be identified by supplemental information. In some cases the points themselves determine a unique curve of the indicated order. For example, a straight line is determined by any two of its points so that (when p,# p2) a unique line is given by (p1;p211. The subscript one for a line is suppressed. The value of any polynomial or ratio of polynomials at point p is denoted by a vertica1 line with a p subscript. For example, (r;q)2(3is the value at point 3 of a quadratic function containing points r and q. This value depends on supplementary data which defines the quadratic function (r;q12. Similarly, [(l;2)2(3;41/(l;5)l l 8 is the value of the indicated rational function at point 8 . Nodes are defined for polycon analysis. All vertices are nodes. Additional nodes may be introduced on boundary segments and interior to elements. Certain sides are said to be "opposite" a node and other sides are said to be "adjacent" to a node. All of the polycon sides are opposite any interior node. A side node (which is not a vertex) lies on its adjacent side, and the remaining sides are opposite the side node. A vertex is at a point of intersection of its adjacent sides, and the remaining sides are opposite the vertex. All element nodes are either vertices, side nodes, or interior nodes. Each polycon side is either opposite or adjacent to any given node.

11

PATCHWORK APPROXIMATION

This notation is illustrated in Fig. 1.7.

3u I

2

Fig.

1.7.

A 5-con

o f o r d e r seven.

In general, polynomial normalization is arbitrary. If a linear form (polynomial of degree one) is normalized so that the sum of the squares of the coefficients of x and y is equal to unity, then the absolute value of the linear form evaluated at point p is the distance of p from the line on which the linear form vanishes. When the line contains no interior point of the polycon being studied, the signs of the coefficients can be chosen so that the linear form is positive within the polycon. We use this normalization for linear forms. When the curve of a polynomial of any degree contains no interior point of a polycon, the polynomial may be normalized to be positive within the polycon. We use no specific normalization, however, for polynomials of degrees higher than one.

12

RATIONAL FINITE ELEMENT BASIS

As a further illustration of the notation, we consider the triangle and parallelogram of Figs. 1.2 and 1.5 represented now in Figs. 1.8 and 1.9, respectively.

Fig.

1.8.

Triangle wedges.

The triangle basis functions are:

Fig.

1.9.

Parallelogram wedges.

The parallelogram basis functions f o r nodes 1 and 2 are : W1(x,y) = (2;3)(3;4)/[ (2;3)(3;4)Ill (1.12) W2(x,y) = (3;4)(4;1)/[ (3;4)(4;l)l21. In the discussion following Eq. (1.9) we alluded to a proof that the parallelogram wedges are linear on each of the sides of the element. This proof will now be given for W1. That the other wedges are also linear on the sides follows from symmetry. Side (3;4) is parallel to side (1;2). Therefore, (3;4) = (3;4)1, on side (1;2). [The distance between the Referring to (1.121, we have sides is (3;4)I,.] W1 = (2;3)/[2;3) I,] on side (1;2). In the congruence 13

PATCHWORK APPROXIMATI ON

notation: W1

Z

(2;3) mod (1;2)

.

(1.13)

Similarly, since (2;3) is parallel to (1;4): W1 z (3;4) mod (4;l)

.

(1.14)

The construction of basis functions for algebraic elements will be analyzed in depth. Rational basis functions will often be examined. Some of the polynomial factors appearing in these wedge functions are the irreducible polynomials in terms of which element boundary components are defined. This is the case for all the factors which appear in Eqs. (1.11) and (1.12). In general, there are other factors determined from curves which must be constructed. These factors will be denoted by capital letters with subscripts which are the maximal degrees of the factors for the class of elements being considered. It will be shown, for example, that the maximal degree of the denominator polynomial for a quadrilateral wedge is one. Thus this factor’is denoted by Q1(x,y) in the quadrilateral analysis. When the quadrilateral is a parallelogram, however, the denominator polynomial is chosen as unity (degree zero). The general wedge notation is illustrated by the basis function associated with node 1 in Fig. 1.7. It will be shown that this wedge is of the form

where kl is a normalization constant and polynomials R1 and Q4 are determined by a specified construction. 14

R A T I O N A L FINITE ELEMENT BASIS

1.3

CONTINUITY

Suppose we restrict the elements to polygons. is For i = 1,2,...,n wedge W.(x,y) 1 (a) continuous over the polygon, (b) normalized to unity at vertex i, (c) linear on the two sides adjacent to vertex i, and (d) equal to zero on the sides opposite vertex i. It follows that over the n-gon the function

c n

U(X,Y) =

UiWi(XIY)

i=l has nodal values Ui and is linear on each side of the element. The patchwork approximation is thus conkinuous. Before generalizing to conic sides, we must define "linearity" on a curved side. The geometric configuration and the function behavior should not be confused, despite the intimate interrelationship. Function f is linear on curve P if there are any constants a(P) , b(P) , c(P) for which f(x,y) [a(P)x + b(P)y +c(P)] mod P . A linear form has only two degsees of freedom on a straight line (since x and y are linearly dependent on the line.) Hence, linearity of the patchwork approximation on a straight line side together with fitting of vertex values ensures continuity across the side. On any curved side of a polycon,however, a linear form has three degrees of freedom. Continuity is achieved by fitting the function value at

15

PATCHWORK APPROXIMATION

another point on each conic side. This point is called a "side node". Thus a polycon with r conic and s linear sides has 2r+s nodes. This is equal to the order of the polycon. We will develop the theory for construction of wedge functions for these nodes. It is preferable to partibion the domain of interest into elements such that no node is a vertex of one element and a side node of another element. In certain situations, such "hybrid" nodes are introduced. For example, referring to Fig. 1.10, where the element size is reduced along line (2;3), we observe that node 1 is hybrid. For continuity along side (2;3), we restrict the value of U1 so that the approximation is linear between vertices 2 and 3 : Node 1 is not a node of U1 = (aUZ + bU3)/(a + b) element m. In general, nodal values are restricted to ensure continuity in the presence of hybrid nodes.

.

3

mFig.

A h y b r i d node.

1.10.

A more complicated situation is shown in Fig. 1.11 in which node 1 is a side node of element m and a vertex node for elements p and q.

2

Fig.

1.11.

H y b r i d nodes on a c u r v e d side.

16

RATIO NAL FINITE ELEMENT BASIS

Points 4 and 5 are side nodes for p and q,respectively, but these points are not nodes of element m. The restricted values at these points are expressed in terms of the wedges for element m: 3

-

i=1

i=l

3

CI

We will be concerned primarily with patchwork approximation over collections of well-set polycons. Rational wedge basis functions (rather than polynomials) will be needed for all but a few special elements such as triangles and parallelograms. In Chapter 8, a theory will be developed for constructing basis functions for ill-set elements. Irrational wedge functions are usually required for an ill-set element. An alternative means for treating ill-set elements is provided by rational functions with restricted nodes. In bhe accompanying diagrams, we could use the linear wedges of (1.11) for triangle [1,2,31 to approximate U(x,y) within the ill-set element by

c 3

U(X,Y) =

i=l

17

UiWi(X'Y)

.

PATCHWORK APPROXIMATI ON

This would yield a unique value at restricted node 4 for any choice of ul, u2, u3. Moreover, U(x,y) would be linear on each side of the ill-set element. 1.4

PATCHWORK APPROXIMATION SPACES AND CONVERGENCE

A sequence of patchwork approximations may be defined by successive refinement of the elements. Each approximation is characterized by a length h which goes to zero as the number of elements is increased. We may choose h, for example, as the maximum chord length within the elements. A central pnoblem in convergence analysis is to bound (from above) the error 11 Uh-u [IA by an expression of the form chS Ilull,, where c is a constant, the A- and Bnorms are meaningful in the sense that they give useful error measures, and s is as large as possible for the prescribed scheme. We demand that our wedge basis functions be regular (infinitely differentiable) within their associated polycons. Two properties of approximation by linear combination of these wedges play a significant role in convergence analysis: (1) order of continuity across polycon boundaries, and (2) degree of polynomial for which the wedges form a basis within each polycon. Although application is broader, this is illustrated by the Ritz-Galerkin analysis of the finite element method. The most commonly used functionals for partial differential equations (PDEs) of order 2t admit finite element approximation spaces contained in Ct-’ (R) Ct (R), where the subscript p P denotes piecewise continuity. This space can be

n

18

RATIO NAL FINITE ELEMENT BASIS

generalized somewhat, but this restriction is appropriate for our polycon network where discontinuities occur along cufves rather than at isblated points. Piecewise continuity of all derivatives is inherent in patchwork approximation with regular wedges. Order of continuity over the composite region R is limited by continuity across polycon boundaries. Having restricted ourselves to C0 , it would appear that application of our wedge basis is limited to PDEs of order at most two. This is not the case. Alternative functionals may be found for which C0 approximation suffices for higher degree equations. There are various approaches to this problem, each having been subjected to extensive analysis. One method inwolves addition to the functional of weighted integrals of discontinuities of derivatives along element interfaces. Another method involves reformulation of the PDE to coupled equations of degrees less than three in more than one unknown function. Each function is approximated by a C0 patchwork function and a variational principle which admits these Co approximations as trial functions is devised. In any event, we observe that the order of continuity across element boundaries is of crucial concern. We consider only Co continuity in this monograph. A comprehensive description of Ritz-Galerkin convergence analysis for finite element approximation is given by Strang and Fix (1973). We merely wish to indicate here the importance of continuity and of 19

PATCHWORK APPROXIMATION

degree polynomials for which the wedges form a basis within each polycon. Regarding the latter, the approximation space is said to be of "degree k - 1" within a polycon if the wedge functions provide a basis for all polynomials of degree less than k. It has been shown(Strang and Fix, 1973) that for a wide class of problems a finite element space of degree k - 1 over each element achieves approximation of order hk to an arbitrary smooth function and of order hk-s to its derivatives of order s. A wedge basis is of degree one over a polycon if 2r+s i=1

c

2r+s XiWi(X,Y) = x

,

(1.16b)

i=l

i=l We require that our wedges satisfy these equations. The approximation over a triangle with the wedges of (1.6) is linear and is uniquely determined Eqs by the three non-collinear vertex values. (1.16) are obviously satisfied for the triangle. Let ui be the value of linear function u at vertex i of a parallelogram. Then the wedges in (1.12)

.

yield the approximation U(x,y) =

4

uiWi(x,y) over i=l the parallelogram. By construction, U - u vanishes on the element boundary. By Theorem 1.1, the four 20

RATIONAL FINITE ELEMENT BASIS

linear forms which vanish on the parallelogram sides must all be factors of the (at most) quadratic function U - u. This can be true only if U u is the zero polynomial. This proves that we have achieved degree one approximation. The following theorem illustrates a consistency between continuity across polycon boundaries and attainment of degree one approximation.

-

THEOREM 1.2. Three vertex nodes are insufficient for continuous patchwork approximation of degree one with a triangular element having a conic side. Proof. The triangle (which need not be well set) may be oriented as in Fig. 1.12 with no loss in generality. Actually, although it is common practice to call this element a triangle, in our notation it is more appropriately called a 3-COn of order four.

4

Y

2

I Fig.

1.12.

X

A 3 - c o n of o r d e r f o u r .

attempt at degree one approximation with only the three vertex nodes yields

An

i

YiWi(X*Y) = y i=1 so that W2(x,y) = y/y2,

~ x i w i ( x , y )= x i=l 21

PATCHWORK APPROXIMATION

so that x2y/y2 + x 3W 3 (x,y) = x and W3(X,Y) = Thereeore,

3

1

i=l

x x2y -x3y2

"3

.

Wi(x,y) = 1 yields

W1(x,y) = 1

-

X x2-x3 + -

x3

Continuity across (2;3) possible only if

x3y2

(1.17)

for arbitrary ui is

3

Iu(x,y) =

1

uiWi(x,y)1 mod (2;312

i=l

does not depend on u3. Hence, W 1 must vanish on The three vertex (2;3)2 . This contradicts (1.17) nodes are adequate only when (2;312 degenerates to a straight line.

.

It will be shown that degree one approximation can be achieved when a side node is introduced on (2;312 and another wedge is associated with this node. All four wedges differ from the triangle linear wedges. This introduction of a node on side (2;312 is consistent with the continuity requirement described in Section 1.3. There is a lower bound on the number of basis functions required for continuous degree one patchwork approximation. When this lower bound is achieved, we have a "minimal basis". Uniqueness of a minimal rational basis and the number of functions in this basis are yet to be determined. We have already demonstrated,however, that at least m wedges 22

RATIONAL FINITE ELEMENT BASIS

are required for a polycon of order m. 1.5

WEDGE PROPERTIES.

We now summarize properties thus far required of the wedge basis functions to achieve continuous patchwork degree one approximation over a collection of well-set polycons :

(1) There is a node at each vertex and on each conic side. For each node there is an associated wedge within each polycon containing the node. ( 2 ) Wedge Wi(x,y) associated with node i is normalized to unity at node i. ( 3 ) Wedge Wi is linear on sides adjacent to i. Wedge Wi vanishes on sides opposite node i and (4) at all nodes j for which j # i. (5) The wedges associated with a polycon form a basis for linear functions over the polycon. For a polycon with r conic and s linear sides, there must be at least 2r+s nodes. For these to suffice, we must have:

c

2r+s 2r+s Wi(X,Y) = 1, XiWi(X,Y) = i=1 i=l

1

XI

(1.18)

and

i=l Each wedge function and all its derivatives are continuous within the polycon for which the wedge is a basis function. A function with this property is said to be "regular". (6)

23

PATCHWORK APPROXIMATION

Polynomial wedges satisfying these conditions exist (and are well known) for triangles and parallelograms. We will demonhtrate that for any well-set polycon rational wedges which have these properties can be found by a definitive construction of surprising simplicity. The rational wedge basis functions are not defined at points where the denominator polynomials vanish. Eq. (1.18) applies over the polycon where, by property (6), the wedges are well defined. Much of the analysis applies to polynomials with coefficients in the complex field. Boundaries of algebraic elements such as polycons are defined by irreducible polynomials with real coefficients. We cannot restrict the analysis to polynomials over the reals because this field is not algebraically closed. Points of intersection of curves of polynomials with only real coefficients are in general points with complex coordinates which may reduce in specific cases to real coordinates. These intersection points are of vital concern in this development, and for this reason we perform our analysis with the complex coefficient field. Constructed rational basis functions always have only real coefficients. We are concerned with approximation of functions of real variables. The rational basis functions (wedges) are always real functions of these real variables. 1.6

ISOPARAMETRIC COORDINATES

We digress to describe an ingenious procedure for circumventing difficulties associated with triangles having curved sides and with I-COnS which 24

RATIONAL FINITE ELEMENT BASIS

are not parallelograms. This is the method of isoparametric coordinates developed by Irons (1966),' Ergatoudis (1966), and Zienkiewicz (1967). Isoparametric coordinates do not provide wedges of the type cited in Section 1.5. They allow us, however, to approximate 3-cons and 4-cons by polycons having linear and parabolic sides for which a basis is found in terms of a new local coordinate system. This approach is adequate for many problems and is particularly well suited for finite element application. Certain shapes and choice of side nodes result in coordinate transformations which are not oneto-one and are therefore prohibited. Several papers have been published on "forbidden shapes" in the finite element method (Jordan,l970; Mitchell et al., 1971). The geometric results of Jordan (1970) are especially useful. These forbidden shapes have their counterpart in our restriction to well-set polycons. In the isoparametric formulation, a node is introduced on each side (linear or curved) in addition to the vertex nodes. The (physical) curved sides are replaced by parabolas passing through the vertices and side nodes as shown in Fig. 1.13. TACTUAL

F i g . 1.13

.

The 3 - c o n : a c t u a l a n d model.

25

PATCHWORK APPROX !MATI ON

The local coordinate system is completely defined by the location of the six 3-con or eight 4-con nodes. These coordinates are p , q, r for the 3-con and c , n for the 4-conl as illustrated in Figs. 1.14 and 1.15. For Fig. 1.14:

w1

=

W3 = r(2r-1)

~ ( 2 p - 1 ) ~W2 = q(2q-11, W 5 = 4qr,

w4 = 4Pql

W6 = 4rp

.

P P=

Fig. 1.14.

I s o p a r a m e t r i c c o o r d i n a t e s for a 3-con ( p + q + r = 1).

For Fig. 1.15: w1

w3

=

=

-

(1-n)(1-E.1 (1+5+ll)

-

( l + n ) (1+5)(1-5-n)

4

*

4

w2

w4

26

=

-

=

- ( 1 + 5 ) ( 1 7 )( l - C + r l )

( l + r l ) (1-5) (l+<-Tl)

4 4

RATIONAL FINITE ELEMENT BASIS

Fig.

1.15.

I s o p a r a r n e t r i c c o o r d i n a t e s for a 4 - c o n .

An alternative formulation with a ninth point has been used for the +Con, but this is of no consequence in this discussion. We note that 8

1

6

Wi(C,q)' = 1 and

i=l

1

Wi(P,qrr) = 1.

i=l

The mapping from isoparametric to (x,y) coordinates is defined to assure degree one approximation: 6

x =

1

6

8

8

i-1

i=l

xiwi(p,q,r), y = Cyiwi(P,qtr) i=l i=l for the 3-con and

(1.19a)

for the 4-con. The Jacobians o f the transformations are 2 x 2 matrices obtained from these equations. Forbidden shapes are those which result in a zero value for the determinant of the Jacobian somewhere in the n-con. Continuity across n-con boundaries is established in Chapter 3 . (This result was of course known to the originators of isoparametric coord27

PATCHWORK APPROXIMATION

hates.) The inverse mapping from (x,y) to isoparametric coordinates is not known, in general, but is fortunately not needed for most finite element application in which the entire computation is formulated in terms of the local isoparametric coordinates (Irons, 1966; Strang and Fix, 1973; Zienkiewicz, 1971). The construction in Fig. 1.16, communicated to the author by D.B. MacMillan (of the Knolls Atomic Power Laboratory,) yields the slope of the symmetry axis of the unique isoparametric parabola through three points. The parabola is easily determined from the slope of its symqetry axis and the three points. Line ( 3 ; 4 ) in Fig. 1.16 is parallel to the symmetry axis.

CONSTRUCTED FROM NODES I & 2 Fig.

1.16. MacMilfan's c o n s t r c c t i o n of t h e isoparametric parabola symmetry a x i s slope.

1.7 GENERALIZATIONS TO SIDES OF HIGHER ORDER AND TO THREE DIMENSIONAL ELEMENTS Much of the theory was initially developed for polycons. It soon became apparent that generalizations to figures with sides of higher orders and to elements in higher dimensional spaces could be achieved. In two dimensions, polycons afford a modeling flexibility which seems adequate for prac-

28

RATIONAL FINITE ELEMENT BASIS

tical purposes. Nevertheless, the extension to sides of higher orders retains much of the geometric structure of the polycon analysis and enhances the theory. One possible application of higher-order curves is to problems for which adjoining boundary segments of adjacent elements must have a common tangent at their common vertex. We define a "polypol" as a closed planar figure bounded by algebraic curves. The definition of well set for polycons is extended to polypols with an additional condition. A simple curve of order greater than two can have singular points (as defined in Section 1.2) and no singular point of a boundary component of a well-set polypol can fall on the polypol boundary. The extensions of boundary segments may have singular points. The order of a polypol is the sum of the orders of its sides. This is equal to the order of its boundary. An n-sided polypol is called an 'In-pol". Let side Pi of a given n-pol be of order si. Let r and

max s i i

(1.20a)

n m z x si i=l

(1.20b)

3

The value of r classifies the element: r=l is a polygon, r=2 is a polycon, and r>2 is a polypol. The value of m is the order of the element. Generalization to higher dimensions requires delicate considerations. In Chapter 7 we develop a theory for construction of basis functions for three-

29

PATCHWORK APPROXIMATION

dimensional algebraic elements. Patchwork degree one approximation over tetrahedra and parallelepipeds is easily achieved with conventional polynomial wedges. The first generalization with rational basis functions in three dimensions was achieved by Wait (1971) who examined convex hexahedra. A hexahedron is a distorted parallelepiped with quadrilateral rather than rectangular faces in the general case. When we analyze more general elements than polyhedra, it becomes apparent that an edge of order greater than one implies that there is at least one element face which is not a planar figure. Thus a byproduct of the three-dimensional analysis is a theory for approximation over non planar surfaces. Extensions to higher-dimensional spaces are indicated by the analysis. These extensions are noted but not pursued in any depth. 1.8

RJ3MARKS AND FEFERENCES

An excellent reference book on the theory of finite element methods is that of Strang and Fix (1973). Applications oriented texts containing some of the underlying theory and written more for engineers are Zienkiewicz and Cheung (1967) and Zienkiewicz (1971). The role of pyramids and wedges in patchwork approximation and some of the geometric implications is described by Synge (1957). Some of the definitions and symbols introduced in this chapter are new. These include "polyconsn, "polypols", "adjacent" and "opposite" sides, and "well-set polyconstt The wedge properties enumerated in Section 1.5

.

30

RATIONAL FINITE ELEMENT BASIS

have been implicit in the finite element literature, but they have not previously been brought so sharply into focus. Recognition of the importance of these properties is a starting point for construction of finite element basis functions. Isoparametric coordinates are an ingenious alternative for a useful class of elements. Analysis by Ciarlet and Raviart (1972 a,b) provides a theoretical basis for application. We do not attempt to assess relative merits of rational and isoparametric bases, but distinguishing characteristics are discussed. A staggering volume of finite element papers has appeared in journals over the past two decades. The few texts already mentioned provide an adequate background for this treatise, and extensive bibliographies may be found in these references. Having indicated the role of patchwork approximation in numerical analysis and having described some of our objectives, we now direct our attention to the first element to which this theory was applied: the general quadrilateral. It was the study of the quadrilateral, motivated by discussions with Professor A. R. Mitchell while the author was a visiting fellow at the University of Dundee, which opened this entire line of analysis.

31

Chapter 2

THE QUADRILATERAL

I t i s a f o u r - g o n c o n c l u s i o n t h a t the r e c t a n g l e i s a p a r a - g o n of v i r t u e .

2.1

INADEQUACY OF POLYNOMIALS

Wedge basis functions with the properties enumerated in Section 1.5 have been known for many years for the triangle and the parallelogram. When we seek corresponding wedges for a quadrilateral which is not a parallelogram, we encounter insurmountable difficulties with polynomials. Referring to Fig. 2.1,

Fig.

2.1.

A quadrilateral.

we consider first

It is apparent that W 1 of Eq. (2.1) is linear on (4;l) only when (2;3) is parallel to (4;l) and on 32

R A T I O N A L FINITE ELEMENT BASIS

(1;2) only when (3;4) is parallel to (1;2). Both of these conditions are met only for a parallelogram. Property (3) in Section 1.5 is thus violated. Property ( 4 ) implies that both (2;3) and (3;4) must appear as factors in any wedge for vertex 1. Introduction of other factors will only increase the degree of variation on the adjacent sides. No polynomial basis satisfying the conditions of Section 1.5 exists for the quadilateral, except for the special case of a parallelogram. 2.2

RATIONAL WEDGES

Polynomials have many properties that are beneficial in numerical application. They are easily evaluated, continuous, may be differentiated and integrated readily any number of times, and there are many results in approximation theory concerning polynomials. Rational functions, over regions bounded away from curves along which the denominators of the rational functions vanish, share many of these beneficial properties. In fact, only integration is significantly more tedious. Having demonstrated the inadequacy of polynomials as quadrilateral wedges, we seek rational functions which satisfy the properties of Section 1.5. The simpleqt form for the numerator of W1 is (2;3)(3;4) [property (4)]. Thus the rational function of least degree in numerator and denominator which can be a candidate for this wedge is

We therefore seek a linear form Q1 such that 33

THE QUADRILATERAL

(a) Q f 0 within the quadrilateral, and 1 (b) (2;3)(3;4)/Q1 is linear on both (4;l) and (1;21. Property (a) has far-reaching consequences: we must broaden our vision and look ouside the quadrilateral. Referring to Eqs. (1.11) and (1.121, we observe that all the linear forms appearing in the triangle and parallelogram wedges were determined by the sides of these figures. This simple observation is crucial in our search for a rational basis for approximation over quadrilaterals. We shall soon see that the quadrilateral itself reaches out to give us the desired linear form. To understand this somewhat cryptic statement, we must first prove the following lemma: LEMMA 2.1. If three lines intersect at a point, then the ratio of linear forms which vanish on any two of these lines is constant on the third line.

Proof. Referring to Fig. 2.2, we note that for all points j on line (k;b), (a;k)I j/sin A

=

and

(c;k)1 ./sin B 3

[(a;k)/(c;k)l I j = sin A/sin B

Fig. 2.2.

T h r e e l i n e s m e e t i n g at a p o i n t .

34

RATIONAL FINITE ELEMENT BASIS

is independent of j. Moreover, sign [(a;k)/(c;k)]lj does not change along line (b;k). The ratio is not defined at point k. The signs of both linear forms change as point j moves through point k. We are now able to determine the denominator of the wedge in (2.2). We choose Q1 so that (2;3)/Q1 is constant on side (4;l) and so that (3;4)/Q1 is constant on side (1;2). By Lemma 2.1, the first requirement is met if lines (2;3), (4;1), and Q1 have a common point of intersection and the second requirement is met if lines (3;4), (1;2), and Q, have a common point of intersection. If we define points 5 = (2;3) (1;4) and 6 = (1;2)- ( 3 ; 4 ) , we find (Fig. 2.3) that Q1(x,y) = (5;6) is the unique line which meets both requirements. We note in passing that Lemma 2.1 is a very simple case of a powerful algebraic geometry theorem which will be discussed in Chapter 4.

-

3

Fig.

2.3.

The e x t e r i o r d i a g o n a l .

For any convex quadrilateral, line Q1 has no point in the quadrilateral. In the language of the geometer (Coxeter, 1961), Q1 is the "exterior diagonal" of the "complete quadrilateral". It is clear from Fig. 2.3 that the quadrilateral does indeed "reach out to give us the desired linear form" for 35

THE QUADRILATERAL

the denominator of the wedges. Having found this candidate for W1, we quickly ascertain that consistent candidates for all four wedges are: Wl(XrY) = k1(2;3) (3;4)/Q1(XtY)~

(2.3a)

W2(XIY) = k2(3;4) (4;1)/QI(X,Y)

(2.3b)

I

W3(X,Y) = k3(4;1)(1;2)/Q1(XtY) I and W4(x,y) = k4(l;2) (2;3)/Q1(x,y). [The ki are chosen so that Wi(xi,yi) = 1.1

(2.3~) (2.3d)

As the quadrilateral is deformed into a parallelogram, the exterior diagonal moves to infinity and the associated linear form becomes more nearly constant within the quadrilateral. We therefore let Q1(x,y) = 1 for a parallelogram to obtain the standard wedges of Eq. (1.12). For a trapezoid, we define Q, as in Fig. 2.4. The trapezoid exterior

F i g . 2.4.

T h e trapezoid exterior diagonal.

diagonal is parallel to the parallel sides and passes through the intersection point of the other two sides. We note that the exterior diagonal is uniquely defined as the line that intersects the sides of the quadrilateral at all the exterior intersection points of these sides and at no other points. We have yet to establish property (5); the other 36

RATIONAL FINITE ELEMENT BASIS

five properties in Section 1 . 5 are obviously satisfied. Let u(x,y) be a linear function with values ui at the quadrilateral vertices. Then g(x,y) = U(X,Y)

-

c UiWi(X,Y) i=1 4

vanishes on the perimeter of the quadrilateral by virtue of properties ( 2 ) - ( 4 ) . There must be a P2 such that g(x,y) = P2 (x,y)/Q,(x,y), where P2 is zero on the quartic perimeter. By Theorem 1.1, this is possible only if P2 is the zero polynomial. Hence, property (5) is established and our candidates in Eq. ( 2 . 3 ) are rational basis functions that satisfy all the conditions in Section 1.5. A quadrilateral wedge is sketched in Fig. 2 . 5 . Each wedge is linear along any line through either of the points (1;2).(3;4) or (1;4) (2;3), and the construction lines indicate how this property is used in sketching the wedge surface.

-

Fig.

2.3

2.5.

Quadrilateral wedge W

1'

A m A L COORDINATES AS LIMITS OF RATIONAL WEDGES

As one of the interior angles of a quadrilateral is increased to IT,the four rational wedges approach functions, only one of which in this ill-set limit 37

THE QUADRILATERAL

is continuous. Two linear combinations of the other three discontinuous limit functions can be found which are continuous. The three continuous functions thereby obtained are the triangle basis functions (that is, the areal coordinates for the triangle). Referring to Fig. 2.6, we let vertex 4 approach point 5 on side (1;3) of triangle [1,2,31. Let s be As point 4 defined as the ratio (2;3)15/(2;3) approaches point 5 along line (4;5), linear forms (4;1), (3;4) and Q1 all approach linear form (1;3).

Il.

A quadrilateral

Fig. 2 . 6 .

into a triangle.

-

(1;3)

degenerating

(1;2)(2;3)

= 0.

(1;3)

(1;2)(2;3)

For (x,y) on line (1;3), W4 (x,y) approaches the piecewise linear function that vanishes at vertices 1 and 3 and is equal to unity at point 5. For all

38

RATIO NAL FINITE ELEMENT BASIS

lim W1 (x,y) = lim

(2;3)(3;4)

Q1 (XrY)

(2;3) --

For (x,y) on (1;3), W1(x,y) approaches the piecewise linear function that is zero between points 3 and 5 and increases to unity at vertex 1. Although W1 and W4 are not continuous in the limit, we observe that

= W1(x,y) for [1,2,3].

Similarly,

It is thus shown that the discontinuous limit functions of the quadrilateral wedges may be combined to yield the continuous linear basis functions for the limiting triangle. Areal coordinates are a degenerate form of rational quadrilateral wedges. 2.4

AN EXAMPLE OF QUADRILATERAL WEDGES

By way of illustration, we determine the wedges for a sample quadrilateral. Referring to Fig. 2.7, we have (4;l) = y, (1;2) = (2y - 3 x ) / m ,

-

8 y ) / m , (3;4) = (4 - 2~ (2;3) = (5 + 2~ Q1 = (20 + 8~ 17y)/m,

-

-

y)/&‘,

and the rational basis functions for degree one approximation over the quadrilateral are:

39

THE QUADRILATERAL

Fig. 2 . 7 .

W1(x,y) = (5

+ 2~

6 y ( 3 ~-

A s a m p l e quadrilateral.

-

- y)/3(20

W2(x,y) = 20y(4

2~

W3(x,y) =

2y)/(20

+

+

2~

W4(x,y) = 2(3~

-

-

+

8y) (4 - 2 ~ y)/(20

2y) ( 5

8~

-

-

8~

+

8~

-

1 7 ~ ) ~

-

1 7 ~ ) ~

1 7 ~ ) ~

8~)/3(20

+

8~

-

17~).

We verify linearity of W1 on side (1;2) where y = 7x and (4 3x/2: 20 + 8x 17y = 5(4 2) 2x y) = 7x mod (li2). Hence, Wl(x,y) = 1 + (4 - 2) %

-

-

-

5

mod (1;Z). 2.5

-

-

5

PROJECTIVE COORDINATES*

One of the pleasing aspects of the development of rational bases is the interrelationship between the geometry of the elements and the algebra. Application of fundamental projective geometry concepts gives insight into the nature of approximation over quadrilaterals. In this connection, Coxeter's (1961) "Introduction to Geometry" is invaluable. We quote *This section may be skipped on a first reading. Results obtained here are referred to in Chapter 9. 40

RATIONAL FINITE ELEMENT BASIS

a few definitions from it: "If four points in a plane are joined in pairs by six distinct lines, they are called the vertices of a complete quadrangle, and the lines are its six sides. Two sides are said to be opposite if they have no common vertex. Any point of intersection of two opposite sides is called a diagonal point ( p .19) 'I A concise description of projective coordinates may be found on pp.234-237 of Coxeter's book. A statement contained therein which is indicative of the power of projective coordinates for examining the quadrilateral is as follows: "Just as in affine geometry, all triangles are alike, so & projective geometry all quadrangles are alike (p.235)." The homogeneous coordinates (k,m,n) linear in x and y which assume the values (a,O,O), (O,b,O) and (O,O,c) at the vertices of a triangle, where a, b, and c are arbitrary, are called the "barycentric" coordinates of the triangle. The point (ga,gb,gc) for any g # 0 is identioal to the point (a,b,c), this being a characteristic of any set of homogeneous coordinates. Barycentric coordinates normalized to k+m+n = 1 are called "areal" coordinates. These are the values of the triangle wedges: (k,m,n) = (W1,w2,W,)

.

-

To obtain a system of projective coordinates, we first select four points, no three of which are collinear. We then choose three of these points as vertices of a "triangle of reference". The projective coordinates of these three points are equal to their barycentric coordinates. The fourth point is the "unit" point with projective coordinates defined 41

THE QUADRILATERAL

as (l,l,l). The barycentric coordinates of the unit point are determined uniquely (up to a common multiplier, of course) by the location of the other three points. Thus the unit point has barycentric coordinates (k4,m4,n4), and if we denote the projective coordinates by (p,q,r) we have the coordinate relationship: (k4p, m4q, n4r) = (k,m,n).

(2.4)

From Exercise 2 on p. 237 Of Coxeter's (1961) work we obtain the following result: If the four vertices which define a complete quadrangle are given the projective coordinates (1,+1,+1), then the triangle of reference for this system of projective coordinates is the triangle determined by the three diagonal points of the quadrangle. (This is called the "diagonal triangle".) The quadrangle with this coordinate system is shown in Fig. 2.8.

Fig. 2 . 8 .

Projective coordinate system (p,q,r) for a quadrangle.

In barycentric coordinates, the equation of line ( 1 1 2 ) is

42

RATIONAL FINITE ELEMENT BASIS

k

m

n

det

(2.5)

According to the principle of duality (which is a basic principle of projective geometry), all theorems remain valid after a consistent interchange of the words "point" and line". The coordinates of Ly*L: are the coefficients and er in of en, e q' e

P

e

4

We observe that p = 0 on (5;6). For points not on line (5;6) we may choose the constant C in ( 2 . 7 ) so that p = 1. Thus for point 9 = (4;l)-(6;7), we have

[ :p -4 f]= e

det and we choose

C =

-

e

e

-

(eP+e91

1 to obtain (p,q,r)g = (l,l,O).

43

THE QUADRILATERAL

In like manner we obtain 8 = (1,-l,O), 10 = (l,O,-1) and 11 = (l,O,l). Normalization to p = 1 off line ( 5 ; 6 ) yields a (q,r) coordinate system which may be compared with the isoparametric coordinates described in Section 1.6. In both systems the quadrilateral in (x,y) is transformed to a square in the new coordinates. This facilitates numerical integrations occurring in Ritz-Galerkin computations (finite element, etc 1 When we use the rational basis we obtain an explicit dependence of the approximation on x and y. When isoparametric coordinates are used, we usually have only the functional dependence on the isoparametric coordinates, from which we can obtain corresponding x and y values. To evaluate integrals in the projective coordinates and relate them to integrals in (x,y), we must find the Jacobian of the transformation. It is convenient to determine the Jacobian as the product of two Jacobians: J = J1J2, where J1 is for the transformation from (x,y) to the baryeentric coordinates of the diagonal-point-triangle of the quadrilateral and where J2 is for the transformation from barycentric to projective coordinates. The absolute value of the determinant of J1 is twice the area of the triangle of reference:

..

J1

[]

1

1

1

= 1 ' 7 x6 x5]

7'

y6 5'

[]

(2.8)

The Jacobian relating the barycentric and projective coordinates is obtained by the following proce-

44

RATIONAL FINITE ELEMENT BASIS

dure, well known to geometers, described to me by Professor W. Edge of the University of Edinburgh. For nonzero p,qrr,s:

Therefore, (2.10) and s / p = k4 or l/p = k 4 / s ,

s/q = m4 or l/q = m4/s,

s/r = n4 or l/r = n4/s. Hence, (2.11)

The barycentric coordinates in ( 2 . 8 ) are actually the normalized (areal) coordinates : k + m + n = l . It follows from

J2

-

q 1: k

that

s

=

m n

,

(k4p + m4q + n4r).

Normalization to p = 1 gives:

45

(2.12)

THE QUADRILATERAL

It is not difficult to prove that for our convex quadrilateral, k4>l, m4<0, n4<0, and that within the quadrilateral s>l. Hence, the absolute value of the Thus the absolute determinant of J2 is k4m4n4/s3 value of the Jacobian of the transformation from (x,y) to (q,r) coordinates is

.

I det J I

= 2K567k4m4n4/(k4+rn4q+n4r)

, (2.14)

where K567 is the area of the triangle of reference whose vertices are the diagonal points of the quadrangle. The wedges include the following linear forms as factors: (1;2) = c2 (n4k+k4n), 4 (2;3) = c3(m4k+k4m), (2.15)

( 4 ; l ) = c (m k-k4m),

1

(5;6) = c5k.

(3;4) = c4(n4k-k4n),

The C I S are normalizing constants. tion to (q,r) is

The transforma-

k = k41/s, m = qm4/s, n = rn4/s.

(2.16)

Substituting (2.16) into (2.15), we obtain (4;l) = dl(l-q)/s,

(1;2) = d2(l+r)/s,

(2;3) = d3(l+q)/s, (3;4) = d4(l-r)/s,

(2.17)

(5;6) = d5/s,

where the d’s are normalizing constants which may be obtained by normalization directly in the (q,r) coor46

R A T I O N A L FINITE ELEMENT BASIS

dinates

Substituting (2.17) into Eq. (2.3 we obtain: s(qilri) (1 + qiq) (1 + rir) 4 s (q,r)

‘i q,r) =

.

(2.18)

for i = 1,2,3,4. The projective transformation is bilinear. It is interesting to note that the form of the quadrilateral wedges is an invariant of the projective transformation. The exterior diagonal of the quadrilateral is moved to the horizon (infinity) so that the quadrilateral becomes a square. The basis functions do not transform into bilinear functions. Each wedge function remains a bilinear over a linear function. Integrals of products of basis functions and of their derivatives play a crucial role in finite element application. Although we defer extensive consideration of integration to Chapter 9, we will now derive expressions for the integrals of quadrilateral wedge basis functions over their quadrilaterals to illustrate the value of the projective coordinates. We obtain from (2.14) and (2.18) :

wi

Wi(x,y) dx dy

f

=

-

I/

dq dr

1

det

J(S) [Wi(q,r)

2k4m4n4K567 s (qi,ri) 4

I’

F q dr

-1 -1

(2.19) (l+qiq)(l+rir) 4 (k4+m4q+n4r)

for i = 1,2,3,4. The integral on the right-hand side of (2.19) is one of a class of integrals which may be evaluated in closed form through use of the following recursion formulas, obtained by integration by parts: 47

THE QUADRILATERAL

We define

f(s,t) by

f(s*t) = Then

f(0,s-t) =

J:

(k

(1 + qIS

+ mq + nr)t

dq (k

+

m

+ nr)l+t-s

-

(k

-

o

m + nr)l+t-s

s - t) for s - t > 1, (2.20a) 1 k+m+nr = - In for s-t = 1.(2.20b) m k-m+nr

+

m(l

For s > t > 0 , f(s,t) = 2'/m(l

-

t) (k

+

m

+ sf (s-l,t-l)/m(t

+ -

nr)

t-1

1).

(2.21)

We use the symbol [q,r] to denote k4 + m 4q + n4rt and define q' = qiq and r' = r.r* noting that 1 qi = r2 i = 1. she wedge integrals are for i = 1,2,3,4:

(1 + 9') (1 + r')

wi = C r-1 I 1-1 d q l dr'

Iqiq',rir'l 4

* (2.22)

where C is the coefficient before the integral in Eq. (2.19). Thus if we define I by

then

(2.24)

Noting the relationship between w4 and the other wi given by Eqs. (2.22) and (2.23), we apply the recursion formulas in (2.20) and (2.21) to (2.22) to obtain the quadrilateral wedge integrals:

48

RATIONAL FINITE ELEMENT BASIS

wi

--

k4K567 12m4n4

c + 4 qi rim4n4

f

ril In

[-qi,ril [qi,-ril [qi,ril [-qi,-ril

[qi,ril

-

1

[-qi,ril

-

(2.25) 1 [qi,-ril

Integrals over quadrilaterals are considered in greater detail in Chapter 9. Having constructed basis functions for degree one approximation over quadrilaterals, we now direct out attention to generalizations. 2.6

POLYGONS?

When the quadrilateral wedges were discovered, a natural extension to convex polygons with any number of sides seemed to exist (Wachspress,l971). Unfortunately, this "natural" extension was the wrong one. Nevertheless, we go through this early analysis here to show why it falls short of the mark. We seek a wedge of the form

...

(n-1;n) W1 = kl (2;3)(3;4) L1 L2. ..p-3

.

(2.26)

The n-2 linear forms in the numerator are the forms which vanish on the n-2 sides of the n-gon opposite vertex 1. The n-3 linear forms in the denominator are chosen so that: 1 ( a ) the right slant ratios (2;3)/L , 2 (3;4)/L I are constant on adjacent side (nil), and 1 (b) the left slant ratios (3;4)/L , (4;5)/L 2 , are constant on adjacent side (1;2).

...

...

49

THE QUADRltATERAL

The exterior diagonals of a convex polygon are defined as the exterior diagonals of all quadrilaterals formed from the sides of the polygon. Each of these quadrilaterals contains the polygon. Hence, these diagonals are indeed exterior to the polygon. The linear forms in the denominator are appropriate combinations of the polygon exterior diagonals, these combinations being different for each wedge. Wedges constructed in this manner satisfy all but property (5) in Section 1.5. They may be renormalized to yield degree zero approximation over the polygon, but degree one has not been achieved with these wedges. This wedge construction is illustrated for the pentagon in Fig. 2.9. 8

9 - 7 10

6

To demonstrate that property (5) in Section 1.5 is v i o l a t e d , we prove t h a t the sum of these wedges

50

RATIONAL FINITE ELEMENT BASIS

does not equal unity.

We define N6(x,y) by

To prove that N6 is not the zero polynomial, we examine this function at point (6;7)*(8;9)where the contributions to N 6 from all terms other than the term with k3 vanish. Thus at (6;7)*(8;9),we have N6(X,Y)

I

-

(6;7)* (8;9) k3(4;5) (5;l)(1;2)(7;8)(9;lO)(10;6)

1

(6;7) (8;9)

None of the factors can vanish. Therefore, these wedges do not even achieve degree zero, much less degree one approximation. Normalization to unity at the vertices and linearity along the sides does ensure vanishing of the numerator on the perimeter of the polygon. We define

Then, Vi = Wi on the perimeter and Vi(x,y) = 1 in the polygon. The Vi thus provide a basis for degree zero approximation over the polygon. This may be adequate for some limited application. We shall demonstrate, however, in the next chapter, after some preliminary analysis of 3-cons and 4-cons with one conic side, how degree one approximation may be achieved by an entirely different generalization of the quadrilateral wedge construction.

51

Chapter 3

RATIONAL WEDGES FOR SELECTED POLYCONS

3.1

THE 3-CON OF ORDER FOUR

We attempt to apply concepts introduced in Chapter 2 to the element in Fig. 3.1, in which side (2;312 is an arc of the unit circle.

2N1-x-y 2

2

=O

LY=0 Fig. 3 . 1 .

A 3-COn

of order f o u r .

Motivation for generalization of the quadrilateral construction is stimulated by examination of the wedge associated with vertex 1 of this element. The simplest rational function that vanishes on (2;312 is fl(xty) = kl(l

-

x2

- Y2 )/Q,(x,Y)

I

(3.1)

and we seek Q1 to make this function linear on sides (1;2) and (3;l) of the 3-con. Let Q1 = a + bx + cy. 52

R A T I O N A L FINITE ELEMENT BASIS

On (3;1), y=O and fl(x,y)

E

[(l

-

x2)/(a + bx)] mod (3;l)

.

(3.2)

For fl to be linear on (3;l) with fl(x3,y3) = 0 , we must have a = b. On side (1;2), x = 0 and fl(x,y)

f

[(l

-

2

Y )/(a + cy)l mod (1;2)

.

(3.3)

For fl to be linear on (1;2) and f1(x2,y2) = 0 , we must set a = c. When a = b = c: fl(x,y)

S

3

(1 - x) mod (3;l) (1 - y ) mod (li2).

Thus Q, is the line on which 1 + x candidate f o r W1 is the function fl(x,y) = (1 - y2

-

x2 )/(1 + x

+

+

(3.4) y = 0 , and a

y).

(3.5)

We now seek some geometric significance to the denominator in Eq. (3.5). Extending the sides of our +con, we are pleased to observe (Fig. 3.2)

Fig. 3.2.

Geometric s i g n i f i c a n c e of t h e d e n o m i n a t o r of f l ( x , y ) .

that, just as for the quadrilateral, the linear form in the denominator is determined by the exterior intersection points of the extended sides of the 53

RATIONAL WEDGES FOR SELECTED POLYCONS

polycon. Is this merely coincidental, or is the quadrilateral wedge construction a special case of a far more general procedure? We shall show that the latter is true. Before we are through we shall have demonstrated a remarkable connection between the geometric configuration and the algebraic form of the wedge basis functions over a broad class of figures. For any well-set polycon with prescribed side nodes on its conic sides, the wedge basis functions are uniquely determined by the multiple points of the boundary curve. A detailed description of wedge construction for well-set polycons will be given in Chapter 5. In this chapter we will derive a few theorems that are especially useful for verifying properties of wedge basis functions associated with low order polycons. Our primary concern in this chapter is with the extension to curved sides of some of the analysis introduced in Chapter 2. We explore further the relationship between the algebraic form of the rational basis functions and the geometric properties of the polycons. The following qualitative description of the basis functions is intended only to indicate how the wedge functions whose properties are examined in this chapter were obtained. The precise construction recipe is given in Chapter 5. The denominator polynomial, common to all wedges of any given element, is uniquely determined by the element exterior intersection points (EIP). The numerator of wedge Wi associated with node i is a product of an "opposite" and an "adjacent" factor. The opposite factor is the polynomial of least

54

RATIONAL FINITE ELEMENT BASIS

degree that vanishes on all polycon sides opposite node i. For i a side node, the adjacent factor is unity. For vertex node i, the adjacent factor is the unique polynomial of a certain maximal degree that vanishes on a curve determined by the side nodes on sides adjacent to i and by the points at which these adjacent sides intersect. Lemma 2.1 in Section 2 . 2 led to construction of the quadrilateral wedges. We require a more general result for application to conic sides. Having this objective, we first define a "triple point" as a point common to three specified curves: (xo,yo) is a triple point of curves P, Q, R if and only if P(xo,yo) = Q(xO,yO) = R(xO,yO) = 0. The linear form that vanishes on a line may be used to eliminate x or y on the line. If L1 = ax + by + c with a # 0, we may substitute -(by + c)/a for x on line L1. If a = 0, then b # 0 and we may substitute -c/b for y on the line. Let v be the retained variable and let Pn and Qm be polynomials of degrees n and m, respectively. Then 1 Pn (x,y) E Pn(v) mod L1 , (3.6a) l. (3.6b) and Qm(x,y) Qm(v) mod L1, where n (3.6~) PA(V) (v - Vj) j=1 and m (3.6d) (v - wj) Qi(v)

=n =n

.

j=1

Let (xo,yo) be a triple point of Pn, Om, and L1. Let vo = v(xo,yo). Then vo is a zero of both 55

RATIONAL WEDGES FOR SELECTED POLYCONS

1 1 pn (v) and Qm(v).

We order the roots v and w of j j these polynomials so that 1 (v) = (v Pn

and 1v) Q,(

= (v

-

n-1

vO)

(3.7a) j=1

m- 1 v0) T T - ( v j=1

-

w.) 3

.

(3.7b)

We obtain from ( 3 . 7 ) : n-1

j=1

Furthermore, if curves Pn, Q , triple points, then n-s T ( v j=l

-

-

-yy(v-

and L1 have s distinct

Vj)

(3.9) w 3. )

j-1

We have proved the following generalization of Lemma 2.1: LEMMA 3.1. Let the subscripts on P and Q be the degrees of these polynomials. Let Pn, Q , and L1 have s distinct triple points. Then (3.10)

56

RATIO NAL FINITE ELEMENT BASIS

where polynomials P1 and Q1 are derived from Pn and Qm by elimination of x or y on line L1. We suppress the superscripts henceforth. The "reduced" polynomials Pn-s and ,,Q may have common quadratic factors, but these do not yet concern us. Lemma 3.1 enables verification of the following wedge construction for any well-set 3-con of order four. Let node 4 be specified on the conic side of the 3-2011 shown in Fig. 3.3, for which sides (3;1), (1;2), and (2;312 are given. In this case. ( 2 ; 3 1 2 is an ellipse that intersects (1;2) at 2 and A and intersects (1;3) at 3 and B. We now prove that the conditions enumerated in Section 1.5 are satisfied by the rational wedge i '\ I functions: \

.' '---.,/-

A

Fig. 3 . 3 .

A 3-con o f order f o u r .

Properties (l), ( 2 1 , ( 4 1 , and (6)in Section 1.5 are obviously satisfied by these functions. We must show that they are linear on the sides and that degree one approximation is attained over the 3-con.

57

RATIONAL WEDGES FOR SELECTED POLYCONS

Point B is a triple point for (3;1), ( 2 ; 3 ) 2 , and (A;B). Hence, ( 2 ; 3 I 2 / ( A ; B ) is linear on (3;l). Point A is a triple point for (1;2), ( 2 ; 3 ) 2, and (A;B). Hence, ( 2 ; 3 I 2 / ( A ; B ) is linear on (1;2), and we have shown that W1 is linear on the sides adjacent to node 1. W1 vanishes on opposite side (2;312, and we include zero as a particular linear function. Point A is a triple point for (1;2), ( 4 ; A ) , and (A;B) so that ( 4 ; A ) / ( A ; B ) is constant on (1;2). Point B is a triple point for (3;1), ( 4 ; B ) , and ( A ; B ) . Hence, ( 4 ; B ) / ( A ; B ) is constant on ( 3 ; l ) . This establishes linearity of W2 and W 3 on (1;2) and (3;l); W4 vanishes on these two sides. We have shown that the four wedges are linear on the linear sides. Before proving linearity of the wedges on the conic side we establish property (5). Let u(x,y) be a linear function. Then there is a polyncmial P2 for which 4

(3.12) i=l The wedges are linear on the linear sides. Hence, P2 must vanish on (3;l) and (1;2) as well as at node 4 . B y Theorem 1.1, this is possible only if P2 is the zero polynomial, and we have proved that the wedges are a basis for degree one approximation over t.he 3-COn [property ( 5 ) 1. Since W1 is zero on (2;312, Eqs. (1.18) reduce on ( 2 ; 3 1 2 (except at points A and B where W1 is not defined) to

58

R A T I O N A L F I N I T E ELEMENT BASIS

w2 (X,Y) +

W3(x,y) + W4(X,Y) = 1

x2W2(x,y) + X3W3(X,Y) + X4Wq(&Y) y2W2(x,y) + Y3W3(X,Y)

+

= x

(3.13)

Y4Wp.Y) = Y

The matrix M is defined as I1 M =

1

1 ’

x2

x3

x4

, y2

y3

y4,

.

(3.14)

This matrix is nonsingular since points 2,3, and 4 do not lie on a straight line. (A line intersects a nondegenerate conic in at most two points.) From (3.13) we obtain

(3.15)

Therefore, I

bJ

(3.16)

The wedges are thus linear on (2;312. Of course, these wedges are not linear interior to the element. Equation (3.13) applies only on (2;312 where W1 = 0. We have proved that the wedges in (3.11) satisfy the conditions in Section 1.5. Just as for the quadrilateral, we must consider special cases where one or both of EIP A and B are not in the finite plane. In any event, (A;B) is the unique line which meets the 3-con sides extended only at their E I P . When 59

RATIONAL WEDGES FOR SELECTED POLYCONS

b o t h A and B a r e a t i n f i n i t y , w e d e f i n e (A;B) = 1 j u s t as f o r t h e p a r a l l e l o g r a m l i m i t of t h e q u a d r i l a t e r a l . A s i l l u s t r a t e d i n Fig. 3 . 4 , i n t h i s case l i n e s ( 4 ; A ) and ( 4 ; B ) m e e t (1;2) and ( 3 ; 1 ) , r e s p e c t i v e l y , a t t h e E I P a t i n f i n i t y . Thus ( 4 ; A ) is p a r a l l e l t o (1;2) and ( 4 ; B ) is p a r a l l e l t o ( 3 ; l ) .

(4;B)7

l!3

- I-k---I--X

Fig. 3 . 4 .

P o i n t s A and B a t i n f i n i t y ; Q1 = / A ; B J = 1.

Y

I,

Fig.

3.5.

Point B a t i n f i n i t y .

When only B i s a t i n f i n i t y , w e have t h e cons t r u c t i o n of F i g . 3.5 w i t h ( 4 ; B ) p a r a l l e l t o ( 3 ; l ) . 60

RATIONAL FINITE ELEMENT BASIS

Intersections at infinity are handled more elegantly in Chapter 4 through the use of homogeneous coordinates and the projective rather than the affine plane. The relevance of the well-set requirement is illustrated in Fig. 3.6, where it is seen that line ( A ; B ) is no longer exterior to the 3-con. Wedges with ( A ; B ) as a denominator are not regular over the 3-con. The construction fails. 2

Fig. 3 . 6 .

An i l l - s e t 3 - c o n .

The sample well-set 3-COn given in Fig. 3.7 will now be examined in detail to illustrate wedge construction.

[(26+8)-2&Y

Fig.

3.7.

-(5+2&)X

A sample

=O]

3-COn

1 of o r d e r four.

The wedges for the 3-COn in Fig. 3.7 are given in Eqs. (3.17): 61

RATIONAL WEDGES FOR SELECTED POLYCONS

-

4 W1 ( X ,Y)

=

W2(X,Y)

=

W+Y)

=

W4(X,Y)

=

x2

-

4y

2

+

y[2a

-

8

(3.17al

I

2 + x + 6 y

-

2fiy

(5

+

26)xl

2 + x + 6 y (x

+

-

y

-

+ x

1) ( 2

2J5y)

2 + x + 6 y

4(J5

+

l)y(x

+

-

y

,

1)

2 + x + 6 y

,

(3.17b)

(3.17~)

(3.17d)

W e have a l r e a d y proved t h a t t h e wedges i n (3.17)

are l i n e a r on t h e s i d e s o f t h e 3-COn. To d e m o n s t r a t e t h i s p r o p e r t y w e v e r i f y l i n e a r i t y of W3 on ( 2 ; 3 ) 2 : We must show t h a t W3(x,y) : ( 2 ; 4 ) mod (2;312, o r t h a t (2

+

x

+

6y) [y

(X

+

y

-

1) ( 2

+

(1

+ x

-

Substituting 4 ( 1 2(1

+

m y 2 x"l

-

-

-

$)x

11

2 J s y ) mod ( 2 ; 3 1 2

(3.18)

-

y ) for x2 i n (3.181, w e o b t a i n

-

4y

2

+

- n) +

2(1

-

(7

-

45) + 3Js)yl

(3.19)

on t h e l e f t hand s i d e , and -2(2

+ 2

JS)yZ

+

2(1

+

+ x [ l + (1 -

m y

+

2my1

(3.20)

on t h e r i g h t hand s i d e . W e f i n d t h a t t h e e x p r e s s i o n i n (3.19) i s j u s t (1 - fi) t i m e s t h e e x p r e s s i o n i n (3.201, and t h i s v e r i f i e s t h e congruence of ( 3 . 1 8 ) .

62

RATIONAL FINITE ELEMENT BASIS

3.2

THE 4-CON OF ORDER FIVE

The sides of the 4-con of order five in Fig. 3 . 8 have been extended to display intersection points A , B, C, D, and E. We defer treatment of cases where there are fewer than five distinct EIP in the finite plane.

Fig.

3.8.

A 4-COn

o f order f i v e .

In seeking a candidate for wedge W1, we first note that this function must vanish on ( 2 ; 3 ) (3;412, a curve of order three. For all elements thus far examined the denominator curve common to all wedges of a given polycon was of maximal order one less than the wedge numerators. Furthermore, the denominator curve in each case intersected the polycon sides (including their extensions) at and only at the EIP. If this pattern recurs, there should be a unique conic through A, B, C, D, and E that is the denominator curve for the wedges of the 4-con in Fig. 3 . 8 . This will now be verified. A linear form has three degrees of freedom. A line is determined, however, by only two of its points. The fact that a line has one fewer degree of freedom than its linear form is obviously due to the fact that the linear form can be determined only by its normalization at some point off the line. 63

RATIONAL WEDGES FOR SELECTED POLYCONS

The general quadratic in x and y has six degrees of freedom; a conic has five degrees of freedom. It is easily shown (Coxeter, 1961) that five points, no four of which are collinear, determine a unique conic. If any three of the five points are collinear, the conic degenerates into two lines. If no three are collinear, the conic is irreducible. Existence of a unique (A;B;C;D;E)2 is crucial to our development. Inspection of Fig. 3 . 8 reveals that no three of the EIP are collinear. For example, if A, C, and E were collinear, then A would be on line (C;E) = ( 2 ; 3 ) . Since A is on (1;2), this would imply that A = vertex 2, contrary to the well-set hypothesis. A more sophisticated and rigorous proof of the existence of a unique conic through the EIP will be given in Chapter 5 after a discussion in Chapter 4 of algebraic geometry foundations. We denote the unique conic (A;B;C;D;El2 by Q2. A weak form of Bezout's theorem (Theorem 4.1) is that curves of orders n and m having no common component can intersect in at most mn distinct points. We use this theorem to prove that Q2 cannot touch the polycon boundary. Referring to Fig. 3.8, we see that C is in curve Q 2 and not in the irreducible conic ( 3 ; 4 1 2 . Hence, (3;412 and Q 2 intersect only at A, B, D, and E. Therefore, Q 2 cannot touch side (3;412 of the 4-con. Points 3 and 4 are not in Q 2 so that neither (2;3) nor (1;4) can be a component of Q2. Since Q 2 intersects these lines at C, E and C, D, respectively, the denominator curve cannot touch either side ( 2 ; 3 ) or (1;4) of the 4-con. If (1;2) were a component of Q2, then vertex 2 would

64

RATIONAL FINITE ELEMENT BASIS

be in (2;3)*Q2. This has already been ruled out. [(2;3)-Q2 = C, E.1 Hence, (1;2) is not a component of Q2 and the denominator curve cannot touch side ( 1 ; 2 ) of the 4-COn. We have proved that the unique conic (A;B;C;D;E12 does not touch the 4-con boundary. Let Q2(x,y) = a + bx + cy + dx2 + ey2 + fxy, where a, b, c, d, and e are to be determined. Let the coordinate origin be chosen on the 4-COn boundary to assure Q2(0,0) # 0, and normalize Q2 by setting a = 1. The other coefficients may then be found by solving the linear system: > I 2 2 "A YA "A YA X ~ Y b ~ 2 2 X~ YB X~ YB X ~ Y ~ (3.21) I

\ "E

YE

2 X~

\

2

YE X ~ ,Y ,f~,

Since no four of the E I P are collinear, Q2 is unique and the coefficient matrix in (3.21) must be nonsingular. For i = 1,2,. . . , 5 , let the ki in (3.22) be normalizing constants. The wedge function construction recipe previously sketched (and described in detail in Chapter 5 ) yields: Wl(x,y) = k1(2;3) (3;4I2/(A;B;C;D;El2

(3.22a)

W2(x,y) = k2(1;4) (3;4I2/(A;B;C;D;El2

(3.22b)

W3(x,y) = k 3 ( 1 ; 2 ) (1;4)(5;E)/(A;B;C;D;E)2

( 3.22~)

W4(x,y) = k4 (1;2)(2;3)(5;D)/(A;B;C;D;E)

(3.22d)

.

(3.22e) W5(x,y) = k5(4;1) (1;2)(2;3)/(A;B;C;D;EI2 That these wedges have the properties enumerated in Section 1 . 5 will now be proved. 65

RATIONAL WEDGES FOR SELECTED POLYCONS

Properties (1), ( 2 ) , and (4) are obviously satisfied. We first consider linearity of the wedges on the sides of the &Con. To establish linearity on (4;1), we consider points on (4;l) common to numerator and denominator curves. For example, for W1 curve (2;3)(3;4)2 of the numerator intersects curve (A;B;C;D;E)2 of the denominator on line (4;l) at distinct points C and D in Fig. 3 . 8 . By Lemma 3.1, these two curves have two common roots when parametrized on line (4;l). Therefore, W 1 is linear on side (4;l). On ( 1 ; 2 ) , curve (3;412 intersects Q2 at A and B. Hence, (3;4I2/Q2 is constant on (1;2) It may be verified by and W1 f (2;3) mod ( 1 ; 2 ) . similar application of Lemma 3.1 that all five wedges are linear on the straight sides of the polycon. We continue as in the 3-con analysis. For any linear function u(x,y), there is a P3 such that u(x,Y)

-

5

1

uiWi(x,y) = P3/Q2

I

(3.23)

i=1

where P3 vanishes at node 5 and on (4;1), (1;2), and (2;3). This is possible if and only if P3 is the zero polynomial. The wedges in (3.22) are a basis for degree one approximation over the 4-con [property (5)]. Only W3, W4, and W5 are nonzero on (3;4)2* The 3-con analysis leading to (3.16) applies here and establishes linearity of the wedges on the conic side. Properties (1)- (5) have been verified. We now establish regularity [property ( 6 1 1 . It has already been shown that Q2 intersects the boundary only at the EIP. To have a point interior to the 4-c0n, the conic Q2 must have an interior 66

RATIONAL FINITE ELEMENT BASIS

closed branch in addition to the branch passing thru the EIP. Only a hyperbola has more than one branch, anda. hyperbola has no closed branch. Hence, Q2 cannot have a point interior to the 4-con and property (6) has been established. The term "EIP deficiency" has been used to denote cases where there are fewer E I P than the maximum possible for a polycon of specified order. These deficiencies are removed when we (1) pass from the affine plane into the projective plane, and ( 2 ) account for multiplicity of intersections. The general procedure for constructing and analyzing wedges in the presence of EIP deficiency is described in detail in Chapters 4 and 5 . To illustrate the effect of EIP deficiency, we consider two cases qualitatively at this time. First, let sides (4;l) and (2;3) in Fig. 3 . 8 intersect on conic (3;412. Then points C, D, and E coallesce to a point which we designate as C. A unique Q 2 is obtained when we demand that this curve have intersection of multiplicity two with each of sides (4;1), (2;3), and (3;412 at point C and also contain points A and B. The only conic satisfying these requirements is the degenerate conic Q, = (A;C)(B;C). It may be shown that this curve yields wedges that have the properties enumerated in Section 1.5. EIP deficiency is treated in this fashion in general. A unique denominator is always obtained. As a further illustration that involves some new features, we replace the 4-con in Fig. 3.8 by that in Fig. 3.9, in which sides (1;4) and ( 2 ; 3 ) are parallel so that point C falls at infinity

.

61

RATIONAL WEDGES FOR SELECTED WLYCONS

Fig. 3 . 9 .

A

4 - c o n w i t h p a r a l l e l sides.

Denominator conic Q2 intersects ( 4 ; l ) only at D in the finite plane and ( 2 ; 3 ) only at E in the finite plane. Expressing Q2(x,y) as a general quadratic with coefficients to be determined and eliminating y on line ( 4 ; l ) by use of the linear relationship between x and y on the line, we obtain Q,(x,y)

E (a'

+ b'x + c'x2 ) mod ( 4 ; l ) .

(3.24)

For (4;1)-Q2 to have only one point in the finite plane, c' must be zero. This is the fifth condition on the coefficients of Q 2 . The conic may be identified by geometric considerations. It must be either a hyperbola with one asymptote parallel to (2;3) or, when the 4-con has further symmetry, a parabola with axis of symmetry parallel to (2;3). The more probable hyperbolic curve is illustrated in Fig. 3.10.

G

F' Fig. 3 - 1 0 .

L i n e (F;G) i s one of t h e a s y m p t o t e s of t h e h y p e r b o l i c denominator curve.

Q2 for t h e 4-con of F i g .

68

3.9.

RATIONAL F I N I T E ELEMENT BASIS

From (3.22a) we obtain Wl(x,y) = k1(2;3) (3;4I2/Q2, where kl normalizes W1 to unity at vertex 1. We will verify that this wedge is linear on (4;l) to illustrate the role of the parallel sides. Since (2;3) is parallel to (4;1), the factor (2;3) in the numerator is constant on (4;l). Having set c' = 0 in (3.24), we have constructed Q 2 to be linear on (4;l). Since D is a triple point for (3;4)2, (1;4), and Q 2 , polynomials Q2 and (3;4)2 have a common linear factor when parametrized on line ( 4 ; l ) . It follows that W1 is linear on ( 4 ; l ) : (2;3) E 1 mod (4;l) and (3;4I2/Q2 E L1 mod (4;l) yields 3.3

W1(x,y) E L1 mod (4;l).

(3.25)

THE PENTAGON

We return briefly to the pentagon, recalling the failure of the wedges constructed in Section 2.5 to form a basis for degree one approximation. Lemma 3.1 and theQ4-conanalysis in Section 3.2 provide a better introduction for investigation of the pentagon. There are five exterior intersection points when no sides are parallel. Deficiency is handled in the usual manner. The five EIP determine a unique conic which is the appropriate curve of the denominator in all five pentagon wedges. Proof of property ( 5 ) follows the usual reasoning and will not be repeated here. Repetition of previous arguments suffices to verify all properties in Section 1.5 for pentagon wedge functions of the form:

69

RATIONAL WEDGES FOR SELECTEO POLYCONS

W1(x,y) = k1(2;3) (3;4)(4;5)/(A;B;C;D;EI2

(3.26)

for the pentagon displayed in Fig. 3.11. In the next section we continue our quest for further generalizations.

C

c---

Fig.

3.4

3.11.

The pentagon.

SOME ELEMENTARY CONGRUENCES

The congruences in Lemmas 2.1 and 3.1 facilitated verification of the properties required of the wedge functions. These congruences were of the form P1 and

3

Q1 mod R1

P2 G Q2

mod R1

(3.27)

.

(3.28)

Both congruences app,lyalong a line. We now consider congruences on conics, starting with proof of the following theorem: THEOREM 3.1.

and let A , B, C, on P 2 . Then any plete quadrangle congruent to any

Let P 2 be a nondegenerate conic and D be any four distinct points pair of opposite sides of the comgenerated by these four points is other pair of opposite sides mod P2.

The theorem asserts that in Figs. 3.12 and 3.13 the following congruences apply:

70

RATIONAL FINITE ELEMENT BASIS

(B;D) (A;C)

( A ; D ) (B;C) mod P, L

(Ail31 (C;D) mod P2

@~

.

(3.29)

B

A

C

D p2

Congruences Fig. 3.13. hyperbola. on a

Fig. 3.12. Congruences on a n e l l i p s e .

Proof. A quadratic in x and y has six degrees of freedom for which a convenient basis is 1, x, y, x2, y2, and xy. This is in fact the basis in terms of which a quadratic function of x and y is defined. On conic P2 the equation P2(x,y) = 0 may be used to obtain a basis with only five elements for quadratic functions over the restricted domain of P2. Thus a quadratic function has only five degrees of freedom on any given conic. Let Q2 and R2 be conics, either or both of which may degenerate into a product of two lines. Suppose Q2 and R2 intersect P2 at the same four distinct points. This accounts for four Of the five degrees of freedom of polynomials Q2(x,y) and R2(x,y) on conic P2. We consider only the case where P2 is relatively prime to Q2 and R2. At any fifth point on P2 neither Q, nor R2 can vanish (by Bezout's theorem) and we may find a constant, c, Such that Q2(x,y) = cR2(x,y) at this point.

71

RATIONAL WEDGES FOR SELECTED POLYCONS

This exhausts the fifth degree of freedom: polynomial Q2(x,y) - cR2(x,y) = 0 at five distinct points on conic P 2 and must therefore be zero everywhere on P2. This may be expressed as the congruence Q2(x,y)

R2(x,y) mod P2

.

(3.30)

Each of the linear form products in ( 3 . 2 9 ) is a quadratic function which vanishes at A , B, C , and D. We have proved the theorem. We have in fact proved the following more general theorem: THEOREM 3.2. Let P 2 , Q 2 , and R2 be three conics that have no common component. Let the four distinct points A , B , C , and D be on all three conics. Then P2 z Q2 mod R 2 , and the three conics may be interchanged in any manner in this congruence. REMARK 3.1. Given any four distinct points A, B , C, and D on conic R2, we may choose P2 and Q, as any of the line pairs: (A;B) ( C ; D ) , (A;D) ( B ; C ) , (A;C) (B;D)

.

REMARK 3.2.

If conics Q, and R2 intersect at distinct points A , B, C , and D, then Q, is congruent mod R2 to any of the line pairs in Remark 3.1 and R2 is congruent mod Q2 to any of these line pairs. Referring to Fig. 3.14, (A;D)

(B;C)

f

(A;C) (B;D)

(A;B) (C;D)

mod Q 2

Q2 mod p2

mod P2,Q2 (3.31)

This theorem and related analysis has its roots in algebraic geometry. The elegant structure of this theory will be brought more sharply into focus in Chapter 4 .

72

RATIONAL FINITE ELEMENT BASIS

\

Fig. 3 . 1 4 .

I n t e r s e c t i o n o f conics.

In this chapter we establish a few simple theorems and examine selected polycons of low order. This less erudite approach motivated the search for the mathematical foundations described in Chapter 4 and led to generalizations which will be described subsequently. The assertion that two conics having no common component intersect in four points requires clarification. The four points are obtained by solving a quartic equation. The roots are not necessarily real and need not be distinct. Some may occur at infinity, in which case the quartic equation degenerates to a lower-degree equation. Nevertheless, the values of the four roots including the roots at infinity are crucial to this analysis. Theorem 3.2 applies when the intersection points are distinct. The case of two circles, for which two of the intersection points are at infinity (see Fig. 3.15) warrants a separate theorem: THEOREM 3 . 3 .

points A and B. P2

Let circles P 2 and Q, intersect at Then

(A;B) mod Q, and Q 2

5

73

(A;B) mod P2.

(3.32)

RATIONAL WEDGES FOR SELECTED POLYCONS

A

Fig. 3 . 1 5 . C i r c l e s i n t e r s e c t i n g a t A and B . ( a ) Real i n t e r s e c t i o n s ; (b) Complex i n t e r s e c t i o n s .

Proof. Without l o s s in generality, we may choose 2 2 2 P 2 = x + y2 - 1 and Q, = (x - a)’ + (y - b) - r Evaluating polynomial Q2 on conic P2, we have

.

Q,L mod P,

-= -;ax

-

-

-

aI2 + (1 x ) -~ 2by + b2 2x - 2by + a2 + 1 + b2 - r2

(x

.

- r2

(3.33)

The right-hand side of (3.33) is linear and must vanish at A and B. Hence, Q2 Z (A;B) mod P2. The roles of P 2 and Q2 may be interchanged, and the theorem is thus proved. We shall have need for one more congruence theorem for our study of 3-COn wedges: THEOREM 3.4.

Let P3, R3, and Q, have no common component. If these three curves have any six distinct points in common, then P3 Z R3 mod Q 2 . Proof. A cubic has ten degrees of freedom. The relationship Q2(x,y) = 0 reduces this to seven on Q2. For example, suppose we use Q = 0 to express y2 in terms of the basis 1, x, y, x , xy on conic Q 2 .

2

74

RATIO NAL FINITE ELEMENT BASIS

2 3 Then a b a s i s f o r c u b i c s on Q2 i s 1, x , y, x , xy, x , and x 2 y . S i n c e P3 and R3 v a n i s h s i m u l t a n e o u s l y a t s i x p o i n t s on Q 2 , o n l y one of t h e seven d e g r e e s of freedom i s n o t accounted f o r . There must b e a cons t a n t c f o r which P 3 ( x , y ) = c R 3 ( x , y ) on Q 2 , and t h e theorem i s proved.

3.5

WEDGES FOR 3-CONS OF ORDERS FIVE AND SIX

W e f i r s t c o n s i d e r a 3-con w i t h one l i n e a r and t w o c i r c u l a r a r c s as i t s boundary. T h i s element is i l l u s t r a t e d i n F i g . 3.16 i n which c u r v e s (1;212, ( 2 ; 3 ) 2, (4;5;B) 2 , and (A;B;C) are c i r c l e s . Note t h a t t h e l a t t e r two c i r c l e s a r e u n i q u e l y determined by t h e t h r e e i n d i c a t e d p o i n t s w h i l e t h e f i r s t t w o circles are t h e p r e s c r i b e d 3-COn s i d e s .

F i g . 3.16.

3-cOn of o r d e r f i v e w i t h t w o circular b o u n d a r y arcs.

A

75

RATIONAL WEDGES FOR SELECTED POLYCONS

To verify linearity of W1 on the sides, we note that A is a triple point for (3;1), (2;3)2, and (A;B;Cl2so that (2;312 and (A;B;C)2 have a common factor on (3;l). Similarly, from triple point C we find that (4;C) and (A;B;C12 have a different common factor on (3;l). Therefore, W1 L1 mod ( 3 ; l ) . By Theorem 3.3, (2;3)2 E (2;B) mod (1;2)2 and (A;B;C12 = (B;C) mod (1;212. Thus

=

W1(x,y)

5

(4;C)(2;B)/(B;C) mod (1;2)2'

(3.35)

Applying Theorem 3.1 to quadrangle 14 ,C,2 ,Bl , we 2. Substiobtain (4;C)(2;B) (4;2)(B;C) mod tuting this result into (3.351, we obtain Wl(x,y) z (4;2)(B;C)/(B;C) mod (1;212 E (4;2) mod (1;212. (3.36) Linearity of W3 on (3;l) and on (2;312 follows by construction symmetry. To establish linearity of W2 on ( 1 ; 2 ) 2 , we first apply Theorem 3.3 to obtain (4;5;Bj2 E (4;B) mod (1;212 and hence W2(x,y)

=

(1;C)(4;B)/(B,C) mod (1;212.

(3.37)

Applying Theorem 3.1 to quadrangle [l,C,4,BI, we obtain (1;C)(4;B) (1;4)(B;C) mod ( 1 ; 2 1 2 . By substituting this into (3.371, we find that

=

W2(x,y)

=

(1;4) mod (1;2)2.

(3.38)

Linearity of W on (2;3)2 follows from symmetry. 2 To establish linearity of W4 on (1;212, we consider Wq(x,y) = k4(3;1) (2;3I2/(A;B;Cl2 E (1;C)(2;B)/(B;C) mod (1;2)2 .

(3.39)

Using quadrangle [1,2,B,Cl and Theorem 3.1 to obtain (1;2) (B;C) mod (1;212 and the identity (1;C)(2;B) 76

RATIONAL FINITE ELEMENT BASIS

substituting this into (3.391, we obtain w4(x,y)

=

(1;2) mod (1;212.

(3.40)

Linearity of W5 follows by construction symmetry. Property (3) in Section 1.5 is thus verified for the wedges given in (3.34). For any linear u(x,y), there is a P3 such that u(x,y) - cizluiWi(x,y) = P3 (x,y)/ (A;B;C) vanishes on the perimeter of the 3-con, which is of order five. P3 must be the zero polynomial, and degree one approximation [property ( 5 ) l is established. The simplicity of this verification illustrates the value of the congruence theorems in analysis of rational wedge basis functions. We could have used these theorems to derive the wedges for the 3-2011 of order four in Section 3.1, also. For example, in Fig. 3.3: (3;l)(4;A) 5 (3;4)(A;B) mod (2;312 by Theorem 3.1 so that W2 in Eq. (3.11) satisfies W~(X,Y)= k2(3;1) (4;A)/(A;B) E (3;4)(A;B)/(A;B) mod (2;3)2 (3;4) mod (2;312

.

This proof of linearity is more concise than that given in Section 3.1, but the earlier proof was instructive. I B

A

(2;312

C E

(1;3)2

Note t h a t

/

Fig

E

.

and

(C;4;6)2

ar

3.17. A 3-COn w i t h t h r e e c i r c l e a r c s : 2 ) 2 # ( 2 ; 3 1 2 , and (3;1j2.

77

RATIONAL WEDGES FOR SELECTED POLYCONS

Wedges for vertex node 1 and side node 4 of the 3-2011 with three circle arcs given in Fig. 3.17 are and

W1 (x,Y) = kl (2;3)2 (C;4;6 ) 2/ (A;B;C)2 ,

(3.41a)

W4(x,y) = k4(1;3)2(2;3)2/(A;B;C)2.

(3.41b)

The remaining wedges are defined similarly. We omit verification involving trivial application of Theorems 3.1 and 3.3.

Fig.

3.18.

A

3-Con

of order six.

We now consider the 3-COn of order s i x in Fig. 3.18 with conic sides that are in general not circle arcs. The wedges associated with vertex 1 and side node 4 will be displayed and then verified. The other four wedges may be constructed and verified in similar fashion. The 3-con EIP are defined by (1;2)2-(2;3)2 = 2, B, D, and E, (2;3)2-(1:3)2 = 3, A, F, and G, (1;3)2-(1;2)2 = 1, C, H, and I.

(3.42)

Some of the EIP may not be in the finite real plane. It will be proved in Section 5.1 that in any event these EIP determine a unique cubic, which we denote 78

R A T I O N A L F I N I T E ELEMENT BASIS

as Q3. Polynomial Q3 is the denominator for all six wedges. (Although we refer to Q3 as cubic in the general case, it is of maximal degree three. We have already seen that when the three sides are arcs of circles Q3 reduces to a conic.) The adjacent factor for side node 4 is unity. The adjacent factor for vertex node 1 vanishes on the conic determined by 4, 6, C, H, and I, as indicated by the qualitative discussion in Section 3.1. That these points determine a unique conic is proved in Section 5.1. At this time, we are interested primarily in showing how the congruence theorems facilitate verification of the desired wedge properties. We consider and

W1 (x,y) = kl(2;3)2 (4;W;H;I)2/Q3,

( 3.43a)

w4(X,Y) = k4(1;3)2(2;3)2/93*

( 3.43b)

These wedges obviously satisfy properties (11, (21, and ( 4 ) . As in the previous cases, proof of linearity on the sides requires the most analysis. Since (2;3)2. (1;2)2 = 2, B, D, and E, application of Theorem 3.2 yields (2;312 E (2;B) (D;E) mod (1;2)2.

(3.44)

Since (4;6;C;H;I)2-(1;2)2 = 4, C, H, and I, we have by Theorem 3.2: (4;6;C;H;I)

(4;C)(H;I) mod (1;2)2.

(3.45)

Since Q3.(1;2)2 = B, C, D, E, €3, and I, we have by Theorem 3.4, which was introduced in Section 3.4 specifically for this application, Q3(x,y)

f

(B;C)(D;E)(Hi11 mod (1;212.

79

(3.46)

RATIONAL WEDGES FOR SELECTED POLYCONS

By substituting (3.44)-(3.46) W1(X1Y)

into (3.43) I we obtain

=

(2;B) (D;E) (4;C) (H;I)

s

(2;B)(4;C)/(B;C) mod (1;212.

mod (1;2)2

(B;C)(D;E)(Hi11

By Theorem 3.2, (2;B)(4:C) 3 (2;4)(B;C) mod (1;212. Hence, Wl(x,y) : (2;4)(B;C)/(B;C) mod (1;212 = (2;4) mod (1;2)2. Linearity of W1 on (1;3)2 follows from symmetry considerations. The above proof may be repeated for side (1;312 as an exercise. This will not be done here. Proof of linearity of W4 on (1;212 is analogous: (1;3)2 : (1;C)( H ; I ) mod (1;212 and (2;312 Z (2;B) (D;E) mod (1;2)2. Therefore, W4(X,Y)

5

(1;C) (H;I)(2;B) (D;E) (B;C)(D;E)(H;I)

=

mod (1;212

(1;C)(2;B)/(B;C) mod (1;212.

Property (3) is thus verified for W1 and W4. To prove that the wedges are a basis for degree one approximation over the +con, we observe that for any linear u(x,y) there is a P4 such that 6 u(xtY) C u ~ w ~ ( x , Y=) p4/~31 (3.47) i=l

-

and that P4 must vanish on the 3-con perimeter of order six. By Theorem 1.1, P4 is the zero polynomial. Property (5)'is thus verified. We defer discussion of regularity [property (6)1 until Section 5 . 3 . To prove that Q3 does not vanish anywhere in the 3-con we must introduce some algebraic geometry concepts and derive new theorems. We will then prove that these wedges are regular.

80

RATIONAL FINITE ELEMENT BASIS

One may compute wedges for specific 3-cons for a better understanding of the procedure; this is a good pasttime for a rainy day. One may verify the following, for example: Let (3;1)2 = y - x2 , (1;212 = x - y2, and (2;3)2 = 1 - x 2 - y2 Let a = ( A - 1)/2 and b = (16 + 1)/2. Then the cubic denominator is Q3(x,y) = 1 + &(x + y) + 3 ( a - 6 )(x2 + y2) + ( 2 - ~1)xy - ~;(x + y3) a(x 2y + xy2 1 . Another example, for which we give more details is the 3-COn of order five in Fig. 3.19 for which (3;l) = y, (2;312 = 1 - x2 - y2, and 2 2 (1;212 = (x/2 - 1) + y - 1.

.

Fig. 3 . 1 9 .

A s a m p l e 3-COn

of o r d e r f i v e .

The two complex points in (1;2)2-(2;3)2are D = (-2,ni) and E = (-2, -fii). Let Q2 = (A;B;c;D;E)~= alx + a2y + a3x2 + a4y2 + a5xy 1. The coefficients may be determined by solving the linear system obtained by equating Q2 to zero at the EIP:

-

81

RATIONAL WEDGES FOR SELECTED POLYCONS

4

0

16

0

0

2/3

-6/3

4/9

5/9

-26/9

-1

0

1

0

0

-2

6 i

4

-3

-2J5i

-2

-J5i

4

-3

2J5i

al a2 a3 a4

a5 This reducible system is easily solved to yield al = -3/4, a2 = &/2, a3 = 1/4, a4 = 1/2, a5 = &/4. The denominator of the wedges is therefore

+ (1/4)x2 + (1/2)y2 + (&/4)xy Q2 = -f3/4)x + (&/2)y - 1. The wedges will, of course, have numerators dependent on the choice of the conic side nodes: W1(x4y) = k1(4;C) (2;3I2/Q2, W2 = k2(3;1) (4;5;B;D;EI2/Q2, W3 = k3(5;A) (1;2)2/Q2t W4 = k4(3;1) (2;3I2/Q2, and W5 = k5(3;1) (1;2I2/Q2. In the last paragraph in Section 1.5, it was asserted that a l l wedge polynomial factors have only real coefficients. Further insight regarding this assertion is gained by considering the construction of Q, for this last example. The EIP led to a system of linear equations. This system can be written as A& = w,, where a is the vector whose elements are the polynomial coefficients and where w, has components all equal to unity. (In general, w, has zero components corresponding to equations of E I P at infinity.) The elements in row j of A are functions of coordinates (x ,y.) of the j-th E I P . If either coordinate 1 1 is not real, there is another row, say j', €or which - (x,14yjl) = (xj,yj), the bars denoting complex conjugates. If we interchange all such row pairs, 82

RATIO NAL FINITE ELEMENT BASIS

w

= that has the identical we obtain the system solution, 5. Since w_ is real, the conjugate of the are nonsinguSince A and first system is = w_. lar, the latter two systems have the unique real solution 4 = 5.

xi

3.6

TWO-SIDED ELEMENTS

Congruences on lines and conics provide a basis for construction of rational wedges for diverse elements. We now turn to one of the simplest types of polycons, the 2-conI considering first the 2-con o f order three. Curved sides may be introduced in practice to conform to boundaries of regions. For example, in Fig. 3.20 elements I, 11, and I11 each have a curved side along the boundary. It has been shown

7

BOUNDARY OF DOMAIN

Fig.

3.20.

Curved element s i d e s on a boundary.

(Strang and Fix, 1973) that ciently smooth" boundary by Fig. 3.21 does not decrease over a wide range of finite

F i g . 3.21.

replacement of a "suffistraight lines as in the order of convergence element computations.

Segmented boundary.

83

RATIONAL WEDGES FOR SELECTED POLYCONS

Asymptotic theory may not, however, be applicable until the elements are smaller than needed for prescribed accuracy. Thus finer subdivisions may be required than when conforming to the boundary as in Fig. 3.20. The curved sides introduce a few (perhaps minor) inconveniences such as complication of the discretization of a boundary value problem. These inconveniences may be reduced by an alternative representation that will now be examined. We consider the segments in Fig. 3.21 between the boundary and the triangle sides as distinct elements. This may be accomplished by either connecting vertex nodes on the boundary in Fig. 3.20 or by introducing side nodes on the boundary in Fig. 3.21. The resulting elements are shown in Fig. 3.22a. The straightsided elements need not be triangles, and a rectangular element is displayed in Fig. 3.22b.

Fig.

3.22.

T h e s e g m e n t element.

We now show how to construct bases for degree one approximation over segments. There are no E I P so that the wedges are polynomials. Referring to Fig. 3.23, we discover that the wedges are in fact identical to the wedges for triangle [1,2,3].

84

RATIONAL FINITE ELEMENT BASIS

We have W1 = k1(2;3),

Fig.

3.23.

W2 = k2(1;3), W3 = k,(1;2).

(3.48)

A 2-con of o r d e r t h r e e .

The boundary curve enters into a finite element computation in that the domain of integration is the segment rather than the triangle. This is discussed in more detail in Chapter 9. The curved side could be interior to the region of interest (perhaps at a material interface), in which case continuity on this side is assured by the usual argument. Other two-sided elements may be considered. Material defects, fissures, and boundary layers can lead to an assortment of regions for which 2-cons are appropriate.

Fig.

3.24.

The l u n e : A 2-con of o r d e r f o u r .

In Fig. 3.24 we have a lune bounded by two inter-

85

RATIONAL WEDGES FOR SELECTED POLYCONS

secting circle arcs. cation of the wedges

Theorem 3.3 facilitates verifi-

W1 = k1(2;3;4)21 W2 = k2(1;4;3I2 W3 = k3(1;2;4)2,

(3.49)

W4 = k4(1;2;3j2.

Here, the curves of a l l the quadratic functions are the circles through the indicated points. The most general 2-con is shown in Fig. 3.25. 3

. Fig.

.25.

,

T h e g e n e r a l 2-con of

Ber f

ur.

At least one of the two sides is not a circle arc and neither is straight. This usually results in two distinct EIP and rational rather than polynomial wedges. Theorems 3.1 and 3.2 provide the basis for verification of the wedges

(3.50) Wq(xty) = k4(1;2;3)2/(A;B) Our analysis of a variety of selected polycons indicates a close connection between element geometry and basis function construction. The theorems used in this chapter are too fundamental to be new, and the study suggests far deeper concepts. These are

86

RATIONAL F I N I T E ELEMENT BASIS

found in the classical theory of algebraic geometry, and this will be pursued in depth in Chapter 4 . 3.7

RELATED STUDIES

For a number of years there has been widespread interest in finite elements with curved sides. Isoparametric elements with parabolic sides were introduced around 1966 (Ergatoudis, 1966; Irons, 1966) and have been quite popular since that time. The first published description of a curved element in real coordinates was by McLeod and Mitchell (1972), with whom the author exchanged ideas during the formative years of this analysis. McLeod's thesis (1972) contains many interesting ideas. In a more recent paper, McLeod and Mitchell (1975) suggest a procedure for locating nodes on curved sides. The author's first published account of the material presented in this chapter was in 1973. Although the material presented here is self-contained, it is recommended that the reader who is interested in related concepts read the McLeod and Mitchell papers.

87

Chapter 4

ALGEBRAIC GEOMETRY FOUNDATIONS

4.1

MOTIVATION

In our study of selected polycons in Chapter 3 we introduced several concepts that require further investigation. Denominator polynomials were constructed from EIP of polycon boundaries. This raises several questions: When does a set of points on a curve of specified maximal order determine that curve? How do we allow for deficiency in intersection points caused by either intersection at infinity or coallescing of points that are in the general case distinct? Given polynomials P and Q and curve R, what are necessary and sufficient conditions for the existence of some b in the field of complex numbers such that P - bQ = 0 everywhere on R? How may we generalize to polypols? Can we find basis functions for higher-degree approximation by similar techniques? Is there some unifying theory that will facilitate extension to higher-dimensional elements? The theorems already proved do not suffice for the contemplated generalizations. Nevertheless, it

88

RATIONAL FINITE ELEMENT BASIS

is evident that theorems of this sort play a crucial role in the theory. If F(x,y) = N 1/Q 1 and G(x,y) = N 2/Q 2 are functions in elements with interface curve s, and if the patchwork approximation equal to F in the one element and G in the other element is continuous across S , then we must have N1/Q I -= N 2 /Q 2 mod S or N1Q2 5 N 2Q 1 mod S . Proof of such a relationship requires analysis of intersections of the various curves of the factors with boundary curve S. Our task is greatly simplified when all the points of intersection of the element boundary components are distinct and in the affine (finite) plane. This is too restrictive. Theoretical foundations for construction and analysis of rational wedges for any degree approximation over algebraic elements in two and three dimensions are found in algebraic geometry. Much of this theory is concerned with multiple and tangential intersection of curves. When we consider higherorder boundary components, we find that the irreducible curve of an element side may have branches that intersect at singular points of the curve. Difficulties in analysis are compounded when other components pass through such singular points or (even worse) are tangent to any of the branches at a singular point. The structure of the intersection of curves can be quite complicated. For a general theory of rational wedge construction, we must be able to resolve difficulties encountered at singularities. The classical theory of algebraic geometry provides precisely what is needed for wedge construction and analysis. In practice, more sophisticated aspects of the theory rarely occur. 89

ALGEBRAIC GEOMETRY FOUNDATlONS

It is not appropriate for us to develop the algebraic geometry foundations here. Several excellent texts are available. Walker's (1962) "Algebraic Curves" is particularly pertinent. We will only define some of the terms, introduce a few salient concepts, and state with no attempt at proofs a few of the key theorems. A reader who is willing to accept these theorems will find this monograph reasonably self-contained. The reader who has the time and the enthusiasm to do so is advised to study the algebraic geometry foundations in an algebraic geometry text. 4.2

HOMOGENEOUS COORDINATES AND THE PROJECTIVE PLANE

Any two distinct lines meet at a point. When two lines are parallel, their intersection point is at infinity. It is hard to visualize intersection at infinity and even harder to distinguish points from one another at infinity. The situation is resolved by introducing a homogenizing coordinate w and passing from the affine to the projective plane. Each point in the projective plane has coordinates (w,x,y). Any polynomial of degree n in x and y is homogenized by multiplying-each term of degree m aijxiyj in affine by w"-~: Pn(x,y) = i+j 2 n coordinates yields Pn(w,x,y) = < n i+j = in projective coordinates. Consider the curve in the projective plane on which Pn(w,x,y) = 0. For any nonzero g, n Thus the point (w,x,Y) Pn(gw,gx,gy) = g P,(w,x,y). is identical to the point (gw,gx,gy). Point (x,y) in the affine plane has projective coordinates (l,x,y).

90

RATIO NAL FINITE ELEMENT BASIS

Points with projective coordinate w = 0 define the absolute line in the projective plane, which is the mapping of points at infinity in the affine plane. Consider the affine line on which ax + by + c = 0. If b = 0, the line is vertical. If b f 3 , the line has slope -a/b. In homogeneous coordinates this is the line on which ax + by + cw = 0 in the projective plane. This line intersects the absolute line at point (0,-b,a). If b = 0, this is the point (O,O,l), and all vertical lines pass through this point. If b # 0, this is the point (O,l,-a/b). All lines with the same slope intersect the absolute line at a common point. The real plane may be mapped into the projective plane if one defines a system of projective coordinates as was done in Section 2.5 for the analysis of quadrilateral wedges. The system is defined by assigning to four points, no three of which are collinear, the projective coordinates (l,O,O), (O,l,O), (O,O,l), and (l,l,l). The first three points are the vertices of the triangle of reference and the last is the unity point. This determines uniquely (up to a common nonzero multiplier, of course) the projective coordinates of all points in the plane. Sketches of curves in the projective plane are shown in Fig. 4.2b (parallel lines), Fig. 4.4b ( a hyperbola), and Fig. 4 . 7 . The transformation of the cubic sketched in the affine plane in Fig. 4 . 6 to the projective plane (Fig. 4 . 7 ) is particularly illuminating. Projective coordinates (xo,x1,x2) are used in Fig. 4 . 7 , where the absolute line is xo = 0. Since (0,0,1) is a triple point in the intersection

91

ALGEBRAIC GEOMETRY FOUNDATIONS

of the cubic with the absolute line, curve Q in Fig. 4.7 has an inflection point at (0,0,1) and is tangent to the absolute line at this point.

4.3

I N T E R S E C T I O N OF PLANE CURVES A thorough study of the intersection of plane

curves leads to analysis beyond our scope. To introduce this subject, we will describe some material in Fulton's (1969) text, "Algebraic Curves", in which many terms are defined and the "intersection number" is characterized. We are concerned with polynomials over the com- plex field: F E K[x,y]. If F = ri(Fi)ei where the Fi are the irreducible factors of F, we say that the Fi are the components of curve F and ei is the Two polynomials that multiplicity of component Fi have no common factor of degree greater than zero are relatively prime. Component F~ is simple if ei = 1; otherwise it is multiple. The partial derivatives of F with respect to x and y evaluated at Point p in curve F is point p are Fx(p) and F (p) Y a simple point of F if either or both of these derivatives is nonzero. When both derivatives are zero, the curve does not have a unique well-defined tangent at p, which is then a singular or multiple point of F. A curve with only simple points is called nonsingular. A l l lines and nondegenerate conics are nonsingular. Subtleties of the algebraic geometry analysis arise primarily in connection with multiple points. We now define the multiplicity of curve F at p. If p is not on the absolute line, we translate

.

92

R A T I O N A L F I N I T E ELEMENT BASIS

the affine coordinate origin so that p = (1,0,0) and write polynomial F of degree n in affine coordinates + Fn, (that is, with w = 1) as F = FS + Fs+l + where Fs # 0 and F is a homogeneous form of degree j j in x and y. Then s is the multiplicity of F at p and we denote this by m (F) = s . If p is on the P absolute line, then either the x or y value at p is nonzero. Say x ( p ) # 0. We then choose coordinates so that p = (O,l,O), set y = 1, and determine the multiplicity of F in the (w,x)-plane. It can be shown that multiplicity is independent of which coordinate is set equal to unity to define an affine plane. (See p.104 in Fulton.) We note that p E F if and only if m (F) > 0 and P that p is a simple point of F if and only if m (F) = P 1. If m (F) = 2 , then p is a double point; if P mp(F) = 3 , then p is a triple point, etc. Point p is a simple point of F if and only if p belongs to just one component of F, this being a simple component, and p is a.simple point of the component. Plane curves intersect properly at a common point if they do not have a common component passing through the point. The intersection number of curves F and G at p is defined by Fulton by seven properties, some of which, as observed by Fulton, are redundant. It is denoted by the symbol I(p,F-G) in our analysis. where we use the dot.) (Fulton uses the symbol The seven properties which de5ine a unique intersection number are: * a *

n

(1) I (p,F-G)is a nonnegative integer for any F, G, and p such that F and G intersect properly at 93

ALGEBRAIC GEOMETRY FOUNDATIONS

p.

1f F and G do not intersect properly at p and p

is in both F and G, then I(p,F*G) = (2) I(p,F-G) = 0 if and only if p f! F-G. The intersection number depends only on the components of F and G that pass through point p . ( 3 ) If T is a projective change in coordinates and T(p) = q, then I(q,FT-GT) = I(p,F-G). ( 4 ) I(p,F*G) = I(pIG*F). 03.

Curves F and G intersect transversally at p if p is a simple point of F and of G, and if F and G do not have a common tangent line at p . The intersection number must be unity at such a point. More generally, we require: ( 5 ) I(p,F*G) m (F)m ( G ) with equality if and P P only if F and G have no tangent lines in common at p .

The intersection numbers should add when we take the unions of curves: (Fi)ri and G = (Gj)Sj, (6) If F = i i, r.s .I(p,Fi-Gj) (We use then I(p,F-G) = 1 2 subscripts identifying Fulton's notation here with polynomials and superscripts equal to exponents.) ( 7 ) For arbitrary F, I(p,F-G) = I(p,F. [G+AFI) for any A such that deg A = deg G - deg F.

nj .

We now describe how Sylvester's dialytic method (Bocher, 1907; Hodge and Pedoe, 1968; Macaulay, 1916; Muir, 1960) yields the intersection points and the corresponding intersection numbers. We define t i=0 94

RATIONAL FINITE ELEMENT BASIS

(4.lb) and the bigradient matrix of Pt and Qs:

co

t+l columns al

0

a .

s-1 columns

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

....

al...........

.................................. 0

0

0

.*.

a.

al............

........ 0 bo bl ......-b b 0 .... s-1 s .................................. 0 0 0 ...... bo bl

bo bl

b2

*.*-.*

bS

s

a t

0

rows

.(4.2) t rows

S

The resultant of Pt and Qs is the determinant of the bigradient: (4 3 ) This resultant vanishes if and only if Pt and Q S have a common zero. Any polynomial in n variables may be expressed as a homogeneous polynomial in n + 1 variables by introducing a variable of homogeneity xo in the following manner: In E q s . (4.4), il + i2 + - * - + in 5 t. The polynomial expressed in affine coordinates in (4.4a) is homogenized in (4.4b).

95

ALGEBRAIC GEOMETRY FOUNDATiONS

Pt(xO,xl,...xn) =

(4.4a)

1

ai1

( t-il-

nxo

-*-in)il in x1 *.*x n' (4.4b)

We note that for any nonzero r the point with homogeneous coordinates (po,pl,...,pn) is the same When xo # 0, we may as the point (rpo,rpl,...,rpn). normalize this coordinate to unity. The process of expressing a polynomial in several variables as a polynomial in a subset of these variables with the remaining variables appearing in the coefficients is known as specialization (Hodge and Pedoe, 1968, Vol. 1, p. 22). Let Pt and Q, be relatively prime. We introduce the variable of homogeneity w and specialize Pt(w,x,y) and Qs(w,x,y) to the form of Eqs. (4.1), having chosen the (x,y) coordinate system to assure aObO # 0. Then coefficients ai and bi are homogeneous polynomials of degree i in w and x. The point ( O , O , O ) is a trivial root of all homogeneous polynomials and is not counted when considering intersections of curves. The resultant of Pt and Q, is in general a homogeneous polynomial of degree st in w and x that vanishes at st nontrivial points. These may be grouped into points not on the absolute line (w = 1) and points on the absolute line (w = 0). Corresponding to each root there is at least one value of y for which both P (w,x,y) and Qs(w,x,y) vanish. t

96

RATIONAL FINITE ELEMENT BASIS

The number of solutions (points) in the general case distinct that come into coincidence for a particular solution, say p, in the particular example considered is equal to I(p,Pt-Qs). Ambiguities may arise when two or more intersection points have the same y value. Such difficulties may be removed by a preliminary rotation of coordinates. The curves of two relatively prime polynomials intersect in a finite number of points so there is always a choice of coordinates that yields distinct y values for distinct points. A practical alternative is to specialize in terms of x to resolve ambiguities uncovered by the y specialization. This is illustrat2 w2 and ed by the following example: P = x + y2 2 2 Specializing in x, we obtain w Q = x + 4y2 4 Res [P(x) ,Q(x) ] = y , and the four intersection We know points coallesce to (l,l,O) and ( l , - l , O ) . that the sum of the intersection numbers of these points is equal to four, but we do not yet know which values to assign to the two points. Specializing in 2 2 terms of y , we obtain Res [P(y),Q(y) I = (w2 - x Setting w = 0, we obtain only the trivial root (O,O,O). Setting w = 1, we have (1 x2I2 = (1 - x) 2 (1 + x)2 , and there are double roots at x = 1 and x = -1. This establishes that I((l,l,O),P-Q) = 2 and I((l,-l,O),P-Q) = 2 . If we had recognized that P and Q have common tangents at ( l , l , O ) and at (1,-l,O), we could have obtained this result directly from property (5) of the intersection numbers. In this example affine coordinates would have sufficed. None of the intersection points are on the absolute line.

-

-

.

.

-

97

ALGEBRAIC GEOMETRY FOUNDATIONS

The value of projective coordinates is illustrated by the calculation of the intersection points of the circles on which P = x2 + -y2 w2 and Q = [x - (w/2)I2 + y2 - w2 vanish. We have

-

1

0

(x2 - w2)

0

1

0

1

O[(x-;)2

0

1

-

0

0

(x2 - w 2 1

- w 21 [(+I2

0

-

w21

I

-

(w/4) 3 L . The four intersection points are (lI1/4,*m/4) and (O,l,?i). The points on the absolute line are called the "cyclic points at infinity" and are common to all circles. Two circles meet in only two points in the affine plane but in four points in the projective plane. A mote general result (see Fulton, pp. 112115 f o r a proof) is the famous theorem: W ' I X

(Bezout's theorem). Let Fm and THEOREM 4.1. Gn be projective plane curves of orders m and n, respectively. Let Fm and Gn have no common components. Then I ( P , F ~ * G ~=)mn , P where the summation is over all points in the projective plane [not counting point ( O , O , O ) j .

C

The intersection cycle of projective plane curves Fm and Gn with no common component is defined as (4.5)

98

RATIONAL FINITE ELEMENT BASIS

The summation in (4.5) is symbolic. The intersection cycle is a set of points in which p appears I(p,Fm*Gn) times. The number of elements in this set is called the degree or order of FmoGn' and Bezout's theorem asserts that the order of Fm-Gn is mn. Another property of intersection numbers proved by Fulton (p. 8 2 ) is: LEMMA 4.1.

If p is a simple point on F, then

I (p,F-[G + HI) 2 - Min[I(p,F.G), I(p,F*H)1 .

(4.6)

REMARK 4 . 1 . The need for p to be a simple point is illustrated by the following case, in which ( 4 . 6 ) does not hold at multiple point p: F = y 2 - x ( x - - 12) , G = y + x - 1 , H = y - x + l ,

p = (1,O); I(p,F*G) = I(p,F-H) = 3 but I(p,F*[G + HI) = 2. Affine coordinates suffice here since the intersection is not on the absolute line. The lemma may be generalized if we distinguish intersections with different branches of F at p. Let F1 = y + &(x - 1) and F2 = y - &(x - 1). These are not polynomials in x and y. Nevertheless, line y - x + 1 is tangent to F1 at (1,O) while y + x -1 is tangent to F2 at (1,O). If we could extend the concept of intersection numbers to include nonalgebraic curves, we might define I((l,O),F 1 *G) = 2, I((l,O),F 1 -H) = 1, I((l,0),F2.G) = 1, and I((l,0),F2-H) = 2. Applying ( 4 . 6 ) to each branch of F, we would then have - 1 SO that 1 and I(p,F2-[G + HI) 2 I(p,F1* [G + HI) I ( P , F ~ F [G ~ .+ HI) = I(~,F* [G + H I ) 2 - 2 at multiple point p = (1,O) of curve F. The theory that places this generalization on a rigorous footing will be 99

ALGEBRAIC GEOMETRY FOUNDATIONS

given i n Section 4.6.

REMARK 4 . 2 . The lemma a p p l i e s when p i s a multThis i s i p l e p o i n t of G and a s i m p l e p o i n t of H . 2 2 2 i l l u s t r a t e d by F = x + y - 1, G = y2 (x 1) , Referring t o Fig. 4 . 1 , H = y2 (x - 1) a t p = ( 1 , O ) . w e o b s e r v e t h a t p i s a double p o i n t i n G - F and i n H-F. W e v e r i f y by d i r e c t computation t h a t for any

-

-

-

F H

Fig.

4.1.

Mixed double contact.

+

real c o n s t a n t s c1 and c 2 , I ( p , F . [clG c2HI) 2 2. Normalizing t o c1 + c2 = 1 when t h e sum i s n o t z e r o , w e o b t a i n c1G + c 2 H P = y2 - c1 (x-1) - ( l - c l ) (x-1) P,(p) = 0 and P x ( p ) = ( c l 1). I f c1 = 1, P reduces t o G and t h e r e i s n o t h i n g t o be proved. If c1 # 1, Px # 0 and P has a v e r t i c a l t a n g e n t a t p. Since F a l s o has a vertical tangent a t p, t h i s i s a double p o i n t i n F - P ( a t l e a s t ) . If on t h e o t h e r hand c1 + c 2 = 0 , we have P = -cl (x-1) + c l ( x - l ) = c l ( l - x ) ( x - 2 ) . Component x-1 has a v e r t i c a l t a n g e n t a t p and we a g a i n have a double p o i n t i n P . F .

-

I n t e r s e c t i o n a t i n f i n i t y p o s e s no new problems. I n t e r s e c t i o n i s a p r o j e c t i v e i n v a r i a n t and a l l t h a t w e have s a i d a p p l i e s i n t h e p r o j e c t i v e p l a n e . A polynomial of z e r o d e g r e e maps i n t o t h e a b s o l u t e l i n e . AS a l i n e moves away from a f i g u r e , i t s l i n e a r f o r m approaches a c o n s t a n t v a l u e o v e r t h e f i g u r e . 100

.

RATIONAL FINITE ELEMENT BASIS

4.4

THE FUNDAMENTAL CONGRUENCE THEOREM

Having discussed some of the algebraic geometry background, we now direct our attention to a fundamental theorem relating to construction and analysis of polycon wedge basis functions. This is a special case of Max Noether's fundamental theorem. Consider the Euclidean algorithm for integers. Let {f) denote the divisors of integer f, counting multiplicities, and let fog denote {f).{g). For distinct integers f, g, and h there are integers a and b such that af + bg = h if and only if the greatest common divisor of f and g divides h. This last condition is equivalent to hof 2 gof. This generalizes to polynomials in one variable over the complex field if we replace the concept of divisors by that of polynomial roots. We define f 0 g as the set of roots common to f (XI and g(x), counting multiplicities. Noether's theorem is concerned with the generalization of this result to polynomials in two variables. There are several obscure points. If we attempt to use the intersection cycle of f and g as f(x,y) 0 g(x,y), the theorem holds only for a restricted class of intersections. Noether gave the necessary conditions relating to intersection cycles for the proper generalization: Max Noether's Fundamental Theorem. Let F, G, and H be projective plane curves, where F and G do not have a common component. Then there is an equation H = AF + BG (with A and B forms of degrees deg H - deg F and deg H - deg G, respectively) if and only if Noether's conditions are satisfied for 101

ALGEBRAIC GEOMETRY FOUNDATIONS

every p in the intersection of F and G. The theorem is meaningless until Noether's conditions are defined. At this point, we remark that these conditions are satisfied at a simple point p on F if I(p,H.F) 2 I(p,G-F). More subtle criteria apply when p is a multiple point on F, and we shall elaborate on this subsequently. In Noether's theorem, let deg H = deg G. Then deg B = deg H - deg G = 0 and B is an element in the polynomial coefficient ring, the complex field in our study. The symbol b is used instead of B for this case: H = A??

+ bG.

(4.7)

In our congruence notation, the above yields H : G mod F.

(4.8)

Suppose I(p,H-F) = I(p,G-F) for all p on F. Then the intersection cycles defined in ( 4 . 5 ) satisfy G*F = H-F

.

(4.9)

If, in addition, all these intersection points are simple points of F, Noether's conditions are satisfied. This brings us to the fundamental theorem for construction and verification of rational wedge basis functions for polycons: THEOREM 4 . 2

(The fundamental polycon theorem). Let Qs be an irreducible polynomial of degree s. Let P and R be relatively prime to Q s with Let all the points in these intersecP.Qs = R-0,. Then tion cycles be simple points of 9,. P

=

R mod

102

Q

S'

(4.10)

RATIONAL FINITE ELEMENT BASIS

Proof. Since the theorem is a derivative of Noether's theorem and Noether's conditions are satisfied, a proof is not needed. Further insight is gained, however, by proving the theorem for this special case, and we give a concise proof. By Bezout's theorem (Theorem 4.11, the order of F O G is the product of the degrees of F and G. Hence, deg P = deg R, and denoting this degree by t we note that the order of P.Q S (= R-Q,) is st. There must be a point, say po, on curve QS at which neither P nor R vanishes so that we can find c1 and c2 such that: S = c 1P + c 2 R (4.11) vanishes at p0' Hence, po E S-Q,. Since P-Qs contains only simple points of Qs, we obtain from Lemma 4.1: I(P,Qs*S) 2 I(PrQs*P) = I(P,Qs*R)

(4.12)

for all p E P-Q,. Therefore, P-Qs is contained in SsQ,, and the st elements of P-Q, together with po yield at least 1 + st elements in S-Q,. Since S is of maximal degree t, S and Q cannot be relatively prime (Theorem 4.1). Since Q is irreducible, there must be a T such that S = TQ, and it follows that 0 mod Q, or c P + cZR : 0 mod Q, which is equivS 1 alent to P = R mod Q. In the projective plane the absolute line is of order one. This line transforms into a constant (order zero) in the affine plane. Hence, it is possible for the theorem to apply to cases where P and R are not of the same order in the affine plane. This and other aspects of the theorem will be 103

ALGEBRAIC GEOMETRY FOUNDATIONS

clarified by example. EXAMPLE 4 . 1

(See Fig. 4.2).

P = x, R = y-x,

P Y =o Fig.

4.2.

N

I n t e r s e c t i o n of t h r e e l i n e s .

and Q = y: x y - x mod y. This is Lemma 2.1. Parallel lines illustrate the value of projective coordinates in treating intersections at infinity. P = x and R = 1 - x meet on the absolute line ( M ; N ) . Thus P. ( M ; N ) = R. ( M ; N ) and P E R mod ( M ; N ) . In the affine plane this yields x (I - x) mod 1, a useless congruence since 1 # 0 in the affine plane. We note that P * R = ( M ; N ) - R so that P E ( M ; N ) mod R, and this yields the more meaningful affine plane congruence: x 5 1 mod (1 - x ) . The wedges in Chapter 2 for trapezoids and parallelograms are no different from the general quadrilateral wedges. They may be obtained by recognizing the role of the absolute line in establishing congruences. EXAMPLE 4 . 2 .

Intersecting conics are shown in Fig. 4.3. We have P 2 - Q 2 = (A;D)(B;C)*Q2 so that P2 : ( A ; D ) ( B ; C ) mod Q2. This is Theorem 3.2. If P2 and Q2 are circles, then two of the four intersection points (say A and D) are on the absolute line and P(x,y) (B;C) mod Q 2 in the affine plane. 104

RATIONAL FINITE ELEMENT BASIS

This is Theorem 3.3.

Fig. 4.3.

Intersecting conics.

EXAMPLE 4.3. In Fig. 4.4a, we have P = 1 - x 4 y, and Q, = xy 1. Homogenizing, we obtain w, R = y - w 4 and Q, = xy w2 in the projective plane (Fig. 4.4b). Specializing P and Q, we

R = 1 P = x

-

-

-

Y

XY- I =o

Fig. 4.4.

Intersecting curves, (a) Affine p l a n e ; (b) P r o j e c t i v e p l a n e . 105

A LGEBRAlC GEOMETRY FOUNDATIONS

-

wL and P ( x ) = x - w , both of which The bigradient is

obtain Q ( x ) = yx are linear in x.

For w = 1, the resultant vanishes at y = 1, corresponding to intersection point (w,x,y) = (l,l,l). When w = 0, we obtain the nontrivial intersection point (O,O,l). Similar analysis of R and Q yields R - Q = (l,l,l), (O,l,O). Thus P - Q # R - Q , as shown in Fig. 4.4b: P - Q = A , M and R * Q = A, N. Let T be the tangent to Q, at A. Then ( M ; N ) T - Q 2 = (A;M) ( A ; N ) Equality of these intersection .Q2 = M, N , A , A . cycles yields (M;N)T : (A;M) (A;N) mod Q 2 , or (X

+ y

-

2)

(X - 1) ( y

5

-

1) mod (XY - 1).

Another congruence is obtained from P - Q 2 = P . ( M ; N ) R = A , M : (M;N)R = Q, mod P , or (y

-

1)

5

(xy

-

1) mod (x

-

1).

These last two congruences illustrate how the transformation of the absolute line into a constant in the affine plane results in polynomials of different degrees being congruent on a curve. EXAMPLE 4 . 4 . Let P 2 = x - 1 - y2 and let (A;B) = 1 + x - y. Define the cubic P 3 by P 3 = (2 + x)y(l - x - y) = (2 + x) (A;C)(B;C) and the quadratic Q 2 = 1 - x2 - y2 (Fig. 4 . 5 ) . We specialize P 2 and Q, to polynomials in x , noting that P2 is quadratic in x and y but only linear in x. Thus, in Eq. (4.2) we have t = 1 and s = 2 in the bigradient of P, and Q,:

106

RATIONAL FINITE ELEMENT BASIS

B

Fig. 4 . 5 .

Curves f o r example 4 . 4 .

R e s ( P , Q ) = y2 - 1 + (1 + with a double root at y = 0 and simple roots at y = G i and y = -6i. The double root is at (l,O), which is point C in Fig. 4.5, while the other intersection points are (-2,543i). Designating the latter two as D and E, we have (D;E) = 2 + x. Curves P 2 and Q, have vertical tangents at double point C . We note that (A;B)P2.Q2 = P 3 - Q2 = A, B, c , C, D, E. Hence, (A;B)P2 P 3 mod Q2, or

(l+x-y)(x-1-y2 ) -= (2+x)y(l-x-y) mod (1-x2-y2 ) . This may be verified by direct expansion. Two features of this example are particularly interesting. First, points D and E are in the complex plane. Second, curve P2 has a well-defined tangent at point C, a double point in P2-Q2, whereas C is a singular point on curve P3. The two branches of P 3 intersect Q, transversally at C.

107

ALGEBRAIC GEOMETRY FOUNDAT IONS

EXAMPLE 4 . 5 . We now illustrate the theorem for a case where intersection occurs at a singular point of an irreducible curve. Let P = y2 - x(x - 1)2 , R = (1 - x - y)3 , and Q = 1 - x + y. Referring to Remark 4.1 after Lemma 4.1, we recall that P.Q = R - Q = (l,O), (l,O), (1,O). Therefore, y2

-

x(x

-

1)2

(1

-

x

-

y)3 mod (1 - x

+

y).

The theorem applies when p is a multiple point of P or R, but Noether's conditions are more obscure when p is a multiple point of Q. For example, let 2 F = y2 - x(x - 1) , G = 1 - x - y, and H = 1 - x + y . Then G - F = H-F = ( l , O ) * (l,O)-* (1,O). But point ( 1 , O ) is not a simple point of F. Noether's conditions, not yet given for multiple points of F, are not satisfied here. It may be verified that G j! H mod F. Lemma 4.1 was used in our proof of Theorem 4.2. We discussed in Remark 4.1 some of the considerations at a multiple point. This discussion suggests what Noether's conditions should imply. If F has m distinct tangents at point p of multiplicity m, then p is called an ordinary multiple point of F. Let Ir(p,F*G) be the intersection number defined somehow for the intersection of G with branch r of F. Noether's condition should imply that

...,

Ir (p,F*H) Ir(p,F.G), r = 1, 2, m, at any ordinary multiple point of multiplicity m. The intersection cycle does not distinguish the Ir at p. The more general set in which this distinction is made is known as the divisor of G on F. This is considered in greater depth in Section 4 . 6 .

108

R A T I O N A L F I N I T E ELEM E N T BASIS

4.5

ASSOCIATED POINTS

The number of points on an algebraic curve that determine the curve may be obtained by counting degrees of freedom. Muir (1960, Vol. 1, pp. 11-12) cites Cramer's thesis in 1750 as an early source of the well-known theorem "that the equation of a curve of the n-th degree is determinable when (n/2)(n + 3 ) points of the curve are known." This requires qualification. The number of terms in the general polynomial of degree n in x and y is n+2C2, where the symbol nCr denotes the binomial coefficient and is defined to be zero when n < r. The number of degrees of freedom in curve Pn is equal to n+2C2 - 1 = (n/2)(n + 31, which is Cramer's result. To show that conditions must be imposed on these points we consider a cubic P3 that does not pass through the origin. In homogeneous coordinates, ~,(w,x,y) = aOw3 + alw2 x

+

0 . -

+ agy3 = o on curve

P3. Dividing through by the nonzero ao, we have nine coefficient ratios corresponding to nine degrees of freedom. Given any nine points on the curve, (wi,xi,yi) for i = 1, 2, 9, from which these ratios are to be determined, we define 2 2 3 I Yi) T =- (wixi, wiyi, v.

...,

-1

Let V be the matrix of order nine whose rows are the T vectors v.. Then the coefficient ratios satisfy the -1 linear system V a = -3,which has a unique solution when det V # 0. A set of J points is said to be 109

ALGEBRAIC GEOMETRY FOUNDATIONS

.

"cubic independent" when the v. for j = 1,2# . . # 9 -1 are linearly independent. Thus det V # 0 if and only if the nine points are cubic independent; Two relatively prime cubics intersect at nine points. These nine points must be cubic dependent. It follows that if two relatively prime cubics intersect at eight cubic independent points, then there is a unique associated ninth point of intersection which together with the other eight points forms a cubic dependent set. (This is given as Proposition ( 3 ) on p. 124 of Fulton.) Uniqueness of the ninth point is established by noting that if there were more than one the two relatively prime cubics would meet at more than nine points, contrary to Theorem 4.1. A simple illustration is provided by the nine intersection points of three parallel lines with another set of three parallel lines, not parallel to the first set. Any eight of these intersection points determine the ninth point. This is related to a class of projective geometry theorems discussed by Coxeter (1961, p . 259). Similar conditions apply when values of slopes of tangents to curves are specified along with points on the curves. Any nine conditions are cubic independent only if they determine a unique cubic curve. The argument is easily extended to curves of any order. The dimension of the space of polynomials of maximal degree t in x and y on a curve of order s is t+2C2 - t-s+2C2. The dimension of the space Vt of curves of maximal order t on curve PS of order s is equal to dim Vt (mod Ps) = t+2C2 110

-

t-s+2c2

-

1.

When t L - s , this reduces to

By Theorem 4.1, the intersection cycle of curves Rt and Ps has st elements. If no proper subset of a specific set of elements forces the remaining elements of this set to lie in the cycle, then the members of the set are said to be independent elements of the cycle. Thus any st - (s-1)(s-2)/2 elements of Pt.Rs (for t s) determine the remaining (s-1)(s-2)/2 associated elements. This is the content of the following theorem: THEOREM 4.3.

If Pt and Q, are relatively prime and t s, then st - (s-1)(s-2)/2 independent elements of Pt -Qs uniquely determine the remaining (s-1)(s-2)/2 elements of Pt*Qs.

The associated elements described here are related to the residual sets discussed in Chapter VI of Walker (1962). There are s(s+3)/2 degrees of freedom in a curve of order s and the number of elements in Pt=Qs exceeds this when st > s(s+3)/2 or when 2t - 3 > s. For a given Pt with 2t - 3 > s, we note that s(s+3)/2 s-independent points in Pt*Qs determine curve Qs and the remaining intersection points. For example, let Pg = y - x3 and let ( 0 , O ) and (1,l) be given on P3-Q1. Then Q, = y - x, and the third intersection point must be (-l,-l). Here t = 3, s = 1, 2t - 3 = 3 > s = 1. Thus s(s+3)/2 = 2 inter,section points determine line Q1 and the other st s(s+3)/2 = 1 intersection point, (-l,-l).

-

111

ALGEBRAIC GEOMETRY FOUNDATIONS

On pages 108-112 of Fulton's work, he discusses linear systems of curves. Curves of order d form a projective space of dimension d(d + 3)/2. The dimension of the space is equal to the number of degrees of freedom in the general curve of given maximal order. If we put conditions on the curve, the subset of curves satisfying these conditions is a linear subvariety and is called a linear system of curves. Fulton proves:

...,

Let pl, p2, pn be points in THEOREM 4 . 4 . the projective plane and let rl, r2, rn be nonnegative integers. Let V ( d ; rlpl, r2p2, rnpn) be the set of curves of maximal order d such that for curve F in V: m (F) 2 ri ; i = 1,2,...,n. Then Pi (1) V is a linear subspace of curves of order d with n dim V 1, d(d+3)/2 ri(ri+1)/2, and i=l (2) If d (Zy=l ri) - 1, then equality holds above.

...,

...,

1

We note that a linear system of dimension R is determined by any R + 1 of its independent curves. Thus there is a unique curve in a linear system of dimension zero. 4.6

RESOLUTION OF SINGULARITIES

Analysis of multiple points of curves is of major concern in algebraic geometry. We have seen in Chapter 3 the importance of the multiple points of polycon boundaries in the construction of wedge functions. Although Fulton's work contains the 112

RATIONAL FINITE ELEMENT BASIS

necessary theory, we shall follow the development of Walker (1962, Chap. 111, Sec. 7). The following summary of some of this theory indicates the content of the crucial theorems, but one should read Walker for a better appreciation of the subtleties in the analysis. The relations yo = x1x2, y1 = xox2, and y2 = x0x1 between coordinates xi in projective plane S and yi in projective plane S' define the quadratic transformation denoted by T of S into S ' . Each point of S with the exception of the fundamental points (1,0,0), (0,1,0), and (0,0,1) is transformed into a unique point in S'. Any nonfundamental point on the line xi = 0 is transformed into the point yi = 1, Yj = Yk = 0 (i, j, and k all different). The three lines xi = 0 are called irregular lines of the transformation. If F(x) = 0 on a curve in S , the transform of its points will satisfy G(y) = 0 in S ' . Curve G is called the algebraic transform of F. Let F(x) have no irregular line as a carnponent. If G(y) = H(y)F' (y), where H is a product of powers of the yi and F' is not divisible by any yi, we say that F' is the transform of F by T. When given a curve in S, it is not clear from the definition of S' how one determines the algebraic transform. This is the content of Walker's theorem 111.7.1: THEOREM 4 . 5 .

If F' is the transform of F by T, then F is the transform of F ' by T' where T' designates the transformation from S' to S defined by xi - yjyk (i, j, and k all different). With a finite number of exceptions the points of F and F' are in one-to-one correspondence, and the components of 113

ALGEBRAIC GEOMETRY FOUNDATIONS

F and F ' a l s o c o r r e s p o n d . According t o t h i s theorem, w e need o n l y s u b s t i t u t e y . y f o r xk t o f i n d t h e t r a n s f o r m of a c u r v e . l j 2 (xl + xo)xl. The a l g e For example, l e t Q = xox; b r a i c t r a n s f o r m of Q i s

-

2 D i v i d i n g T ( Q ) by yoy2, w e o b t a i n t h e t r a n s f o r m of Q by T: 3 2 Q' = Y 1 Y 2 ( Y O + Y,)

-

The a l g e b r a i c t r a n s f o r m of Q' back i n t o S i s 2 2 T ' ( Q ' ) = x3x3 - x x x ( X + xo). 0 2 0 1 2 1 2 D i v i d i n g T ' ( Q ' ) by x o x 2 , w e o b t a i n t h e t r a n s f o r m o f Q' by T ' :

Q,"=

2

xox2

-

2

x 1( x1

+

xo).

i s equal t o Q.

As a s s e r t e d i n t h e theorem, Q"

W e n e x t c o n s i d e r Walker's theorem 111.7.2: L e t F be a curve o f o r d e r n

with an r - f o l d (ri 2 0) fundamental p o i n t x = xk = 0 i j ( i , j , and k a l l d i f f e r e n t ) , no t a n g e n t t o t h e c u r v e a t a fundamental p o i n t b e i n g an i r r e g u l a r l i n e . Then (1) The a l g e b r a i c t r a n s f o r m G of F h a s t h e l i n e yi = 0 as an r i - f o l d component and so F' is of o r d e r 2n ri THEOREM 4 . 6 .

1

.

( 2 ) T h e r e is a one-to-one correspondence, p r e s e r v i n g m u l t i p l i c i t i e s , between t h e t a n g e n t s of F - 0 and t h e nonfundamental i n t e r s e c t i o n s at xj = Xk of F w i t h xi = 0.

-

r j - rk a t y j = yk ( 3 ) F' h a s m u l t i p l i c i t y n 0 , t h e t a n g e n t s b e i n g d i s t i n c t from t h e i r r e g u l a r 114

RATI ONAL FINITE ELEMENT BASIS

lines and corresponding to the nonfundamental intersections of F with xi = 0. For illustrative purposes, we will consider the curves shown in Figs. 4 . 6 , 4 . 7 , and 4 . 8 in the affine, S,and S ' planes, respectively. In these figures, Q = y2 - x2 (x + 1) and F = (x + y)Q is a curve of order four with a nonordinary singular point of multiplicity three at the origin.

Q X

Y=O

Fig. 4 . 6 . y

2

-

x

2

C u r v e F = (y + x ) Q w h e r e Q = ( x + l), s h o w n i n t h e a f f i n e p l a n e .

We are concerned only with the characteristics of the transform that are determined by the behavior of the curve at fundamental point x1 = x2 = 0. Parts (1) and (2) of the theorem then apply with i = 0, independent of the tangents of the curve at the other two fundamental points. For example, let 2 2 F = (x, + x2) [xox2 - (xo + xl)xl]. No tangent of F at (1,0,0) is an irregular line. If this were not the case, we would rotate coordinates before finding the algebraic transform. We have

= yoy2(y1 3 + Y2) l Y 3,

-

Y,2(Y,

115

+

Y$'

ALGEBRAIC GEOMETRY FOUNDATIONS

and the factor of y o3 confirms (1) in the theorem for this function, being that fundamental point (1,0,0) is a 3-fold point of F.

Fig. 4 . 7 .

F = (xl

f

x z ) Q i n the S p l a n e .

According to (2), there should be two nonfundamental intersection points of F' with yo = 0. Point (O,l,-1) should be a double point and (O,l,l) should be a simple point in Fig. 4 . 8 . We have 2 3 F' = ( ~ +1 y2) [ Y , Y2(Yo + Y1)It and

-

116

R A T I O N A L F I N I T E E L E M E N T BASIS

FIRST NEIGHBORHOOD OF A TYg=oA

("'

Fig.

C u r v e F'

4.8. F'

=

(Y,

*

i n t h e S' p l a n e , where 3 2 Y 2 ) [Y, .- Y 2 ( Y o + Y1)l

-

Thus yO'F' = (0,0,1) (Orlt-l), (0,1,-1), (0,1,1)r confirming part ( 2 ) of the theorem for this function. Theorems 4.7-4.9 should be clear without illustrative examples. The proofs are given by Walker. THEOREM 4 . 7

(Walker's theorem 111.7.3). An rfold point of F not on an irregular line is transformed into an r-fold point of F', and the tangents of these two points correspond in multiplicities. By a THEOmM 4 . 8 (Walker's theorem 111.7.4). finite succession of quadratic transformations any

117

A LG EERA IC GEOMETRY FOUNDATIONS

irreducible curve can be transformed into one having only ordinary singularities. (Recall that the ri tangents are distinct at an ri-fold ordinary singular point.) Walker shows that Theorem 4 . 8 applies to any curve with distinct components, and we state this as a separate theorem: THEOREM 4 . 9 .

By a finite succession of quadratic transformations any algebraic curve with distinct components can be transformed into one having only ordinary singularities. We now consider a few concepts discussed in various chapters of the books by Walker and Fulton that are particularly relevant to the construction of basis functions for polycons and polypols. Let rl, r2, be the multiplicities of all the singular points of an irreducible curve F of order n.

...

is a nonnegative number. If F has only ordinary singularities, then the genus of F is defined as equal to p. Quadratic transformations are members of the more general class of birational transformations (Walker, Chap. V, sec. 4 ) . If curve F' is in birational correspondence with curve F, then F and F' are said to be birationally equivalent. By Theorem 4 . 8 , any irreducible curve is birationally equivalent to a curve with no singularities other than 118

RATIONAL FINITE ELEMENT BASIS

ordinary multiple points. The invariants of transformations are of fundamental significance in the development of any theory. The genus of a curve is a birational invariant. We now describe another invariant. The irreducible algebraic curve on which F(x,y) = 0 is defined as rational (Walker, chap. 111, sec. 5 ) if there exist two rational functions u(s) and v(s) in K ( s ) , where K is the field o f the polynomial coefficients, such that (1) f o r all but a finite set of s o in K, (u(so),v(so)) is a point of F, and

with a finite number of exceptions, for every point (xo,yo)of F there is a unique s o in K such that xo = u(so) and yo = v(so). (2)

Rationality of a curve is a birational invariant. Walker (p. 187) proves: THEOREM 4.10.

Rationality and genus zero are

equivalent. The genus of curve F with only ordinary singularities is defined by Eq. (4.13). We have not explicitly defined the genus of an irreducible curve having nonordinary singularities, even though the definition is implicit in our statement that genus is a birational invariant. By Theorem 4 . 8 , any irreducible curve is birationally equivalent to a curve having only ordinary singular points. The genus of an irreducible curve having nonordinary singularities is equal to the genus of any birationally equivalent curve to which (4.13) applies.

119

ALGEBRAIC GEOMETRY FOUNDATIONS

To illustrate the subtleties involved, Walker cites an example of an irreducible rational curve for which p = 2: F(x,y) = (x2 - y)' - y3 vanishes on a curve whose only singularity is a nonordinary double point at the origin. That F is rational is seen by choosing u ( s ) = (s2 - l ) / s 3 and Curve F is birationally v(s) = ( s 2 - 1)2 /s 4 equivalent to a curve having only ordinary singular points for which p = 0. The concepts of neighboring points and neighborhoods (Walker, Chap. 111, sec. 7.6) of a point on a curve is introduced to facilitate resolution of singularities at nonordinary multiple points. Let p be an r-fold point on curve F. We choose p as the fundamental point x = xk = 0 of a quadratic transj formation such that no tangent at p of curve F coincides with an irregular line. The transform of p is the set of nonfundamental points pi, pi, ...,pi on line yi of multiplicities ri, r;, ri on F'. Each p' corresponds to a distinct tangent of F at p j (Theorem 4.6). We say that in the first neighborhood of p, F has points pi, pi, pi of multiplicities ri, r;, ri. This process may be repeated for each p! to yield pi;, pj2,...,pjk. ' I I t of 3 multiplicities r!' r;i,...,r''jk.. These poiAts are 11' said to be in the second neighbdrhood of p. Thus neighborhood n of p is defined in terms of transformed points of neighborhood n - 1. If p is an ordinary multiple point of F, then all the p! are simple points of F' (rj = 1). A cusp 3 is a nonordinary double point with one simple point in its first neighborhood. The curve on which

.

...,

...,

...,

120

RATIO NAL FlNlTE ELEMENT BASIS

x4 + x2y2 - y2 = 0 has a nonordinary multiple point of order two at the origin with an ordinary double point in its first neighborhood. This type of singularity is called a tacnode. It can be shown (van der Waerden, 1939) that the analysis of a singularity in terms of neighborhoods is independent of the specific transformations chosen for the resolution. Referring to Fig. 4.7, we observe that for the illus(F) = 3 . The first neighbortrated curve m (1,0,0) hood of (1,0,0) is the nonfundamental points on line y o in Fig. 4 . 8 . In the first neighborhood of (1,0,0) curve F has point (O,l,l) of multiplicity one and point (O,l,-1) of multiplicity two. The second neighborhood of (1,0,0) in Fig. 4 . 7 is the union of the first neighborhoods of (O,l,l) and (O,l,-l) in Fig. 4 . 8 . This will consist of-three simple points obtained by transformations of F' about (O,l,l) and (0,1,-1).

Our next theorem is a stronger form of Walker's theorem 111.7.5 alluded to be Walker in his proof of the theorem. The number g defined in this theorem is considered to be one of the most important invariants of an irreducible curve.

...

THEOREM 4.11. Let rl, r2, be the multiplicities of all the singular points (including the neighboring ones) of an irreducible curve F of order n, and let g be defined by g(F) = (n - 1)(n - 2 ) / 2

-

c

ri(ri

[

-

1)]/2.

i Then g is a nonnegative number that is invariant under quadratic transformations. 121

ALGEBRAIC GEOMETRY FOUNDATIONS

This number is actually invariant under the more general class of birational transformations (Walker, Chap. V, sec. 4 ) . This invariant is called the genus of F, and this is a consistent generalization of the genus defined in Eq. (4.13) for curves having only ordinary singular points. Exercise 3 on p. 66 of Walker's book may be generalized to give a result of sufficient importance to warrant statement as a separate theorem:

...,

THEOREM 4.12.

Let gl, g2, gn be the genera of the n distinct irreducible components of curve F of order m having multiplicities rl, r2, at all the singular points (including neighbors) of F. Then n (m - 1) (m - 2) ri(ri - 1) + n - l = gj2 2 j=1 i

...

c

1

=En

We define the genus of an n-pol as the sum of the genera of its sides: gn-pol j=1 gj A Polypol is said to be a rational polyp01 if and only if its genus is zero. All polycons are rational. In Section 4.3, we alluded to a generalization of the intersection cycle which differentiates intersections with different branches of a curve at a point. This generalization is developed in the theory of divisors (Fulton, Chap. 8 ) or cycles (Walker, Chap. VI). A detailed description of this theory would take us too far afield. The analysis deals with local power series parametrizations of branches of curves. We shall present here an intuitive description in terms of neighboring points.

122

R A T I O N A L F I N I T E ELEMENT BASIS

Let point p and its neighbors on F be denoted by pk. Let ri be the multiplicity of pi pl, p2, on F and si the multiplicity of pi on G. The divisor of G on F (and of F on G), denoted by F 0 G, is the symbolic series F 0 G D(p,F.G), where D(p,F.G) = (r.s,)pi. We note that risi = I(p,F-G), but that the divisor contains more information than the intersection cycle. In our example in Section 4.3 (remark 4.1) with regard to the intersection of two lines with a cubic at its singular point, we noted that for Q = y2 - x(x - 1)2 , R = y + x - 1, and S = y x + 1, R - Q = S - Q = 3A. (A is the point (1,O) .) For these curves, the divisors are R 0 Q = 2A + A; and S 0 Q = 2A i A;, where A; and A; are distinct points in the first neighborhood of A, as sketched in Fig. 4.9.

...,

=Ip

xi

xi

-

Fig. 4 . 9 .

N e i g h b o r h o o d s of A

( s e e Figs.

4.6-4.8).

In our application of Noether's theorem to the theory of construction of basis functions, the element sides play the role of Q, in Eq.(4.10). Lines and nondegenerate conics are nonsingular. Hence, Theorem 4.2 suffices for polycon analysis. 123

ALGEBRAIC GEOMETRY FOUNDATIONS

When we generalize to higher-order sides, the singular points of the sides enter into the analysis, and Theorem 4 . 2 is no longer adequate. This last section on resolution of singularities was motivated primarily by the need for more powerful theorems for analysis of complexities accompanying the generalization from polycons to polypols. Much of the material in this section is required only when a side of order greater than two occurs. The appropriate generalization of Theorem 4 . 2 may be found in the work of Walker as Theorem VI.2.2 and in the work of Fulton as Corollary 2 on p. 189. We state this crucial theorem without proof. Our previous discussion relating to the information contained in divisors beyond that in intersection cycles should provide a qualitative basis for acceptance of the theorem, proof of which may be found in the referenced texts: THEOREM 4.13 (The fundamental congruence theorem). Let Q be an irreducible curve and let curves P and R not have Q as a component. If P 0 Q = R 0 Q, then P 3 R mod Q. The simplicity of the statement of this theorem bears testimony to the elegance of the underlying algebraic geometry theory. 4.7

REMARKS AND REFERENCES

Algebraic geometry is rarely used by numerical analysts. It is hoped that this application to one of the frontier areas in numerical analysis will revitalize interest in this fascinating mathematics discipline as a tool for applied mathematicians. 124

We have referred extensively to Fulton (1969) and to Walker (1962) in this chapter. The work of Carr (1970) contains many useful formulas, and Muir (1960) traces the history of analysis of the bigradient matrix. Other books on algebraic geometry include those by Hodge and Pedoe (19681, Macaulay (19161, and Verdina (1971). Although van der Waerden (1950) treats some pertinent topics in his "Modern Algebra", VOl. 2 , a better reference for the classical treatment of algebraic geometry is his German text (19391, "Einfuhrung in der Algebraische Geometrie". A more extensive bibliography may be found in the referenced texts. The intimate relationship between element geometry and the algebraic form of the basis functions exploited in the analysis in Chapters 2 and 3 motivated use of Theorems 4.2 and 4.13 in generalizations to a wide class of elements. The analysis summarized in Section 4.6 is particularly relevant to the construction of basis functions for polypols.

125

Chapter 5

RATIONAL WEDGE CONSTRUCTION FOR POLYCONS AND POLYPOLS

5.1

POLYCON WEDGE CONSTRUCTION

Each polycon wedge is of the form Wi(x,y) = kiPiRi/Q, where ki normalizes the polycon wedge to unity at node (xi,yi) and the three polynomials are determined so that the properties enumerated in Section 1.5 are achieved. We first elaborate on the construction mentioned in Section 3.1 and then prove that this construction yields the required properties. Polynomial Pi in the numerator is called the opposite factor. It is the product of the linear and quadratic forms which vanish on the sides opposite node i. Polynomial Ri, the other factor in the numerator, is called the adjacent factor. It is unity for all side nodes and for all vertex nodes at the intersection of two linear sides. For these nodes, Pi is of degree m - 2 when the polycon is of order m. A vertex node at the intersection of a linear and a conic side has an opposite factor of degree m - 3 . The adjacent factor is the linear form that vanishes on (A;B) in Fig. 5.1.

126

RATIONAL FINITE ELEMENT BASIS

Fig.

/ L i n e a r a d j a c e n t f a c t o r (A;B).

5.1.

If the EIP of the sides adjacent to node i is at infinity (that is, if point B is on the absolute line in the projective plane), then line (A;B) is parallel to the linear side at vertex i. A vertex node at the intersection of two conic sides has an opposite factor of degree m - 4 . The adjacent factor is the quadratic function that vanishes on A , B, C, D, and E in Fig. 5.2. These five

Fig. 5.2.

Q u a d r a t i c a d j a c e n t f a c t o r (A,B,C,D,E)

2'

points are the two nodes adjacent to i and the three EIP of the sides adjacent to vertex i. When any of these points coallesce, the curve of the adjacent factor is determined by the intersection cycle of the conic sides: Ri-S1 = s1.s2

-

i + B, and Ri.S2 = S1.s2 127

-

i

+

A.

RATIONAL WEDGE CONSTR UCTlON

To prove that this yields a unique adjacent factor, we use an argument that recurs in rational wedge analysis. We first note that a conic has five degrees of freedom so that we may always find at least one Ri having the required intersection cycles with S 1 and S 2 Since Ri cannot contain vertex i, we may normalize any two candidates, say Ril and Ri2, to unity at i. Then Ril - Ri2 =- 0 mod S1S 2 by repeated Ri2 is application of Theorem 4 . 2 , and since Ril of maximal degree two while S1S2 is a quartic with only simple components we may apply Theorem 1.1 to prove that Ri1 - Ri2 must be the zero polynomial. We observe that in any case PiRi is a unique polynomial of degree m - 2. A denominator factor of maximal degree m - 3 is constructed in the following manner: Let Pi be the polynomial of minimal degree that vanishes on boundary segment (i;i+l) k(i) of a well-set polycon of order m = s + 2r, where s is the number of linear sides, r is the number of conic sides, and the symbol ( [s+rl ;[s+r+ll ) (s+r) is defined to be ( [ ~ + r I ; l ) ~ ( ~ + ~The ) . value of k(i) is one for a linear side and two f o r a conic side. Let

.

-

and

Eij

= pi.pJ,

(5.1)

The number of elements in Ei is s - 1 + 2r if Pi is linear and 2s + 4 (r - 1) if Pi is conic. The polycon vertices at the ends of segment Pi are two of the elements in Ei' Let Eij and Ei be the sets in (5.1) and (5.2) with the vertices excluded.

128

RATIONAL FINITE ELEMENT BASIS

The number of elements in Ei is equal to s + 2r - 3 = m - 3 when Pi is linear and 2 ( s + 2r) - 6 = 2 (m - 3 ) when Pi is conic. over all nodes, we obtain

i=l = (s

+ 2r) (m -

3 ) = m(m

-

3) ,

Summing

(5.3)

where O(Ei) is the order of (number of elements in) set Ei. Since all elements of Pi-PJ appear in both Ei and E there are at most m(m - 3)/2 distinct j* EIP in the projective plane. This is precisely the dimension of the space af curves of order m - 3. We will show that there is a unique curve of maximal order m - 3, say Qm-3, for which i = 1, 2,

..., r + s .

(5.4)

It will then be shown that Qm-3 is the denominator factor in the polycon wedge functions. This construction yields denominator polynomials identical to those already found for the polycons examined in Chapters 2 and 3 . An important feature of (5.4) is that multiple points and points on the absolute line are automatically taken into account. There is no "EIP deficiency". We have yet to demonstrate that Eq. ( 5 . 4 ) exhausts the appropriate number of degrees of freedom in the presence of multiple EIP. A qualitative description of a few simple cases will first be given to help provide an intuitive grasp of the handling of multiple points. This will be followed by a more rigorous treatment, drawing directly on algebraic geometry theorems. 129

RATIONAL WEDGE CONSTRUCTION

Boundary curves are shown in Figs. 5.3a-d in a neighborhood of exterior intersection point j.

Fig. 5.3.

B o u n d a r y curves a t EIP j.

The conditions imposed on the denominator curves from intersection at point j are: For Fig. 5.3a, Q(j) = 0, exhausting one degree of freedom; For Fig. 5.3b, Q(j) = 0 and Z-VQI = 0, exhaustj ing two degrees of freedom; For Fig. 5.3c, Qm-3 mod L1 = Qm-3(t) and Qm-3(t) = (t - t.1 2Qm-$t) accounts for two degrees 3 of freedom and assures Qm-3 mod L2 = (u uj)Qm_,(u). Another degree of freedom is used to yield Qm,4 (u) = (u - uj)Qm+ (u), and this assures that j is a double point (at least) in Qm-3-CZ. A total of three degrees of freedom is accounted for; For Fig. 5.3d, Qm-3 mod L2 = Qm-3(~), and 2 Q,-,(v) = ( v - v.) Qm-,(v) accounts for two degrees 3 of freedom while assuring Qm-3 mod L1 =

-

(u - U~)Q,_~(U). Another two degrees of freedom are used to assure Qm-4(u) = (u u.)2Qm-6(u). This

-

130

I

RATIONAL FINITE ELEMENT BASIS

makes j at least a triple point in Qm-3-C2. A total of four degrees of freedom have been exhausted by the,requirementsat j. For these illustrative cases and in all cases, the number of conditions imposed by intersections at j is equal to the number of EIP, in the general case distinct, which have coallesced to j in the particular case. A more precise algebraic geometry treatment will now be presented. We first consider the subset of polycons for which all EIP are at transverse intersections. That is, all multiple points of the polycon boundary curve are ordinary singular points. Suppose 1 + ri sides intersect transversally at pi. Then pi is an ordinary (1 + ri)-fold multiple point of the boundary curve. Let V be the space of curves of maximal order d having a point of multiplicity not less than ri at pi. In the case where the 1 + ri sides meet at distinct points, there are ri(l + ri)/2 EIP from exterior intersections of these sides with one another. These EIP coallesce to pi. The dimension of V for the case with distinct points is:

d(d

+

3)/2

-

(5.5)

ri(ri + 1)/2.

At most one degree of freedom is used to place each point on a curve in V. Referring to Theorem 4.4, we observe that the requirement that pi be of multiplicity not less than ri yields the same bound on the dimension of V. Thus coallescing of transverse intersection points does not affect the lower bound 13 1

RATIONAL WEDGE CONSTRUCTION

on the dimension of V. Let C be the curve of the polynomial of least degree that vanishes on the boundary of a well--setpolycon of order m. In Theorem 4 . 4 , let pl, p2, pn be the EIP of the polycon. Since we have assumed transverse intersections, these EIP are all ordinary multiple points of C. We have m ( C ) = r. + 1 polycon sides intersecting at pi Let V1be the space of curves in which the Pi denominator curve must lie. We demand that m (F) 2 Pi r and that in Theorem 4.4 d = m - 3 . We i have shown that when all the points are distinct there are d (d + 3)/ 2 = m(m - 3)/2 EIP, and that this yields a lower bound of zero on the dimension of V. We have also shown that for each ri > 1 the term ri(ri + 1)/2 subtracted from the bound on dim V accounts for precisely the number of points which have coallesced to pi. Therefore, dim V 0 in any case, and there is always at least one denominator curve that satisfies the stated conditions. The assumption of transverse intersection of the sides was made only to demonstrate the applicability of Theorem 4.4. Appropriate bookkeeping at nonordinary multiple points yields the same result. This has already been shown for a few examples, and we will examine this in greater depth after having proved that this construction yields a unique denominat.or curve. Let Q1 and Q2 be two curves in V and let Ps be the curve of side s of the polycon. By construction, Q1*Ps = Q2 .Ps . By Theorem 4.2,

...,

Q1

Q2

mod Ps.

132

(5.6)

R A T I O N A L FINITE ELEMENT BASIS

Curves Q1and Q2 may be normalized so that Q1 = Q2 at vertex 2, where sides P1 and P2 intersect. Then 2 Q1 - Q2 E 0 mod PIP2. Hence, Q1 = Q at vertex 3 , where sides P2 and P3 intersect. It follows that 0 mod P1P2P 3 Proceeding around the bounQ1 Q2 dary in this fashion, we obtain Q1 - Q 2 f 0 mod Pm, where Pm is the curve of order m defined by the polycon boundary. This boundary curve is a product of simple irreducible components. By Theorem 1.1, 2 Q1 - Q , a polynomial of maximal degree m - 3, can vanish everywhere on Pm only if Q1 - Q2 is the zero polynomial. Thus the denominator curve is unique. An example of multiple EIP (already discussed in Section 3.2) is the 4-con of order five for which C, D, and E in Fig. 3.10 coallesce to point C as shown in Fig. 5.4. Quadratic (A;C)(B;C) is the only

.

-

Q 24:f- A

\

\

\

,*--A/’

\

Fig. 5.4.

A multiple

,

I---

exterior intersection point.

polynomial of maximal degree two whose curve intersects (3;4)2 at A and B and is such that I(C,Q*(4;l)) = I(C,Q. (2;3)) = I(C,Q. (3;4)2) = 2. We have shown that there is a unique Qm-3 for which Qm-3-Pi contains Ei for all i. We have yet to prove that Qm-3*Pi = Ei. Suppose there were one or more additional points on side j. Then (m - 3 ) deg Pj < O(Qm,3- PI) ’ and, by Thearem 4.1, 133

RATIONAL WEDGE CONSTRUCTION

and Pj would have a common component. Since Qm- 3 would have PJ as a component. Vertex j + 1 (where vertex r + s + 1 is vertex 1) would then have to be in Qm-3. Continuing around the boundary in this fashion, we would find that Pm must be a component of Qm-3. This is impossible. Hence, Qm-3.P3 cannot contain any points in addition to E This is true for j = 1, 2, r + s. j' We return now to consideration of nonordinary EIP. Eq. (5.4) was used to define Qm-3 in terms of the polycon EIP. An alternative and more general definition will now be given. Let Cm be the boundary curve of a well-set polycon of order m. Let pl, p2, be the singular points of Cm including neighbors of multiplicities rI, r2, . Let V be the space of curves of maximal order m - 3 having multiplicity not less than ri - 1 at each pi that is not a polycon vertex. Theorem 4.4 applies in the transformed space corresponding to each neighborhood. The maximum number of degrees of freedom of V exhausted by requiring that pi have multiplicity not less than ri - 1 is r.(ri-l)/2. At each vertex, 1 ri = 2 and ri(ri - 1)/2 = 1. Having excluded the n vertex double points, we obtain Q

PT- is irreducible,

...,

...

dim

v

...

2 m(m

-

3)/2

-

L

v

[

all i

ri(ri

-

11/21 + n. (5.7)

It was observed after the statement of Theorem 4.12 that all polycons are rational and are thus of genus zero. We obtain from Theorem 4.12 with g(Cm) = 0: ri(ri-1)/2 = (m-1)(m-2)/2 + n - 1. i (5.8)

C

134

RATIONAL FINITE ELEMENT BASIS

Substituting (5.8) into (5.7) , we obtain dim V - m(m

-

3)/2

+ 1 - (m -1)(m -

2)/2 = 0.

This establishes the existence of at least one curve in V. To prove uniqueness, we note that (5.6) and the argument following (5.6) applies for any two elements in V. Hence, V must have dimension zero, and we choose Qm-3 as the unique curve in V. The construction is summarized as follows: be t h e m u l t i p l i c i t i e s of a l l t h e n o n i vertex singular points p i n c l u d i n g n e i g h b o r s , of i' boundary curve C of a w e l l - s e t p o l y c o n of o r d e r m . Let r

Then Qm-3

rn i s t h e u n i q u e c u r v e of m a x i m a l o r d e r m

w i t h m u l t i p l i c i t y n o t less than r

i

-

-

3

1 a t each p i .

Let Fm be an irreducible curve of order m and let p range over all points for which m (F ) 2 2. P m Any curve P for which m (P) m (F,) - 1 is called P P an adjoint of Fm (Walker, 1962). Adjoints of order m - 3 are of particular interest and are called special adjoints. The dimension of the space of special adjoints is at least equal to g - 1, where g is the genus of curve Fm. Boundary curve Cm of an n-con of order m is a product of n irreducible components and is thus reducible. Adjoints are defined for irreducible curves. We have just proved, however, that there is a unique Qm-3 such that mp(Qm-3) = > mp ( Cm ) - 1 if we exclude vertices. Thus Qm-3 is related to Cm in a manner similar to a specia1 adjoint. We therefore call Qm-3 the polycon adjoint curve and refer to the wedge denominators as adjoints. These polycon adjoints are of crucial importance in the theory of continuous patchwork approximation with rational basis functions. 135

RATIONAL WEDGE CONSTRUCTION

We have described three levels of polycon adjoint construction. The first level was for distinct EIP and required little more than Cramer's result that m(m - 3 ) / 2 points determine a curve of order m - 3 . The second level allowed ordinary multiple points as EIP, and the construction required the greater sophistication of intersection number theory and dimensionality of subspaces of curves satisfying specified conditions. In the third level, we allowed nonordinary multiple points, and for this generalization we used the additional concept of neighborhoods. These levels of complexity persist in the analysis. By proceeding from level to level, we have tried to clarify subtleties that might otherwise have been obscure. This approach is followed throughout this work. To illustrate the role of neighborhoods and application of quadratic transformations to polycon adjoint construction, we consider the 4-con of order six shown in Fig. 5.5. The vertices in this figure

AY

( I -x

2

--Y

--X

Fig. 5.5.

A 4-con

with a nonordinary E I P .

136

RATIONAL FINITE ELEMENT BASIS

are 1 = (0,-1/2), 2 = (O,l), 3 = (l,O), and a/2] ,-1/2) The EIP in the real plane 4 = ( [l are A = (J?j2,-1/2) , B = ([1 + J7/21,-1/2) , C = (-/F/2,-1/2) , and D = (O,-l). The two remaining EIP are the cyclic points where the circles meet on the absolute line. In homogeneous coordinates, these are E = (O,l,i) and F = (O,l,-i). Point D is a nonordinary singular point of the boundary curve. The perturbation displayed in Fig. 5.6 indicates how four points have coallesced to D. The coordinate

.

-

PERTURBED (1;2)

Fig. 5 . 6 .

7

A perturbation a t point D.

origin is on side (1;2) s o that Q3(0,0) # 0 and we may express Q3 in homogeneous coordinates as ~,(w,x,y) = w3 + w 2 (alx + a2y) + w(a3x2 + a4y2 a xy) + (a6x3 + a7x2y + a8xy2 + a9y 31 . 5

+

(5.9)

Conditions on Q3 from EIP A, B, and C are

-

Q,- mod (1;4) E (h/2 - x ) (h/2 + x ) (1 + &/2 x) = 3(1 + & / 2 ) / 4 - 3~/4 (5.10) -(1 + Js/2)x2 + x 3 * where we have set w = 1 since A, B, and C are in the 137

RATIONAL WEDGE CONSTRUCTION

affine plane. Substituting 1 for w and -1/2 for y in (5,9), we obtain Q3 mod (1;4)

-

-

(1 a2/2 + a4/4 ag/8) + (al a5/2 + a8/4)x + (a3 a7/2)x2 + a6x3

-

(5.11)

.

Comparing (5.10) and (5.11), we observe that a6 # 0. Multiplying (5.10) by a6 and equating like powers of x in (5.10) and (5.111, we obtain the three conditions on Q3 from EIP A, B, and C: 3(1

+

fi/2)a6/4 = 1

-

a3 -

-3a6/4 = al -(1

+

J3/2)a6 =

-

ag/8, (5.12a)

+ a8/4,

(5.12b)

+ a4/4

a2/2 a5/2

(5.12~)

a+.

The conditions from points E and F are: %(Q3)

=

1, or Q3(0,1,i) = a6

mF (Q3) = 1, or Q3(0,1,-i)= as

+

-

a7 i a7 i

-

a8 - a 9i = O ; a8 + a9 i = 0.

These equations yield a 6 - a8 = O a7 a9 = 0.

-

and

(5.13a) (5.13b)

The remaining four conditions on Q3 are at D. We first determine these conditions without recourse to a quadratic transformation. The boundary curve has a triple point at D. Hence, Q must have a double point at D. The conditions Q(l,O,-1) = 0, Qx(l,O,-l) = 0, and Q (l,O,-1) = 0 yield

1

-

Y

a2 4- a4 al

and

a2

-

-

-

a9 = 0,

(5.14a)

as + a8 = 0,

(5.14b)

2a4

+

138

3a9 = 0.

(5.14~)

RATIONAL FINITE ELEMENT BASIS

Curves ( 3 ; 4) and (1;2) have a common tangent along the y-axis at D. The remaining condition on Qg is that at least one of its branches must have a vertical tangent at D. If D were not a singular point of Q, this could be accomplished by setting Q (D) = Y 0. This is not appropriate here since Q,(D) and Q (D) have already been set to zero in (5.14) to Y make D a double point of Q. To obtain the required condition without a quadratic transformation, we let Q = PR, where P and R are branches of Q at D that need not be polynomials. Then P ( D ) = 0 and R ( D ) = 0. We have Q = P R + PR and Qyy = Py y R + P R + 2 P R Y Y Y YY Y Y' I f either P or R has a vertical At D, Qyy = 2 P R Y Y tangent at D, then Qyy(D) = 0 . This is the ninth condition: Q (l,O,-l) = 2a4 - 6ag = 0 , or YY a4 - 3ag = 0 . (5.15)

.

Now let us see how (5.14) and (5.15) may be obtained through the use of a quadratic transformation to resolve the singularity at D . We first transform the origin to D:

w' = w,

x ' = x,

and y ' = y

+

w.

(5.16a)

Theorem 4 . 6 cannot yet be invoked since the tangent conunon to (1;2) and ( 3 ~ 4 is ) ~ the irregular line x' = 0 . We therefore define x2 = y', and x1 = x' xo = w ' , and obtain from (5.9) :

139

+ y'

(5.16b)

RATIONAL WEDGE CONSTRUCTION

Q3 (xo,x1,x2) = x3 0

+

+

+

+

x2 0 [a1 (x1-x2

xO[a3(x1-x2)2

+

+ a2 (x2-xO)I 2 a ( x -x

4

2

0

(5.17)

a (x -x l 3 + a7(x1-x2) 2 (x2-x0) 6 1 2 a (x -x (x2-x0l2 + ag(x2-xO)3 8 1 2

.

The algebraic transform of Q as described in Section 4.6 is

agYl 2 (Y,-Y,) 3

(5.18)

The requirement that D be a double point of Q, is equivalent to the condition that yo be a component of T(Q3) of order t w o [see (1) in Theorem 4.61. Expressing T ( Q 3 ) as a polynomial in yo, we obtain the linear terms which must vanish for all (yo,y1,y2):

+

3 2 y0y1y 2 (a2 - a1

-

2a4

+

a5

-

a8

+ 3ag).

We obtain (5.14a) from the first term and (5.14b) from the second term. Substituting (5.14b) into the last term, we obtain (5.14~). This verifies the equivalence asserted in part (1) of Theorem 4.6. 2 after imposing (5.141, we obDividing T(Q3) by yo tain the transform of Q3 by T: 140

RATIONAL WEDGE CONSTRUCTION

+ Yo(

.'.

1,

where the terms that vanish when yo = 0 are not needed for the analysis, and are therefore not displayed in (5.19). Line (1;2) and circle (3;412 have a common tangent at D. In the first neighborhood of D (the nonfundamental points on the line yo), curves (1;2) and (3;4) have transforms that intersect at (O,l,l) The corresponding condition on Q; is that Q;(O,l,l) = 0. Referring to (5.191, we have from this condition: a4 3ag = 0. This is Eq. (5.15), the previously derived ninth condition. For this example, the quadratic transformation was more tedious than setting Q (D) = 0. For more YY complex singularities, however, the transformation provides a rigorous recipe even when the structure of the nonordinary singularity is not clear. In any event, the neighborhood analysis is essential for a precise mathematical exposition. Adjoint curve Qm-3 of any well-set polycon is uniquely defined by the polycon boundary multiple points. The construction may be generalized to ill-set elements. When the vertices are ordinary boundary curve double points, the construction is unchanged. If any vertex is a nonordinary double point or is of higher-order multiplicity, it is possible to define a unique Qm- 3 by removing an appropriate constraint on Qm-3 at each vertex.

.

-

141

RATIONAL WEDGE CONSTRUCTION

We shall have occasion to deal with such ill-set polycons in connection with analysis of wedge regularity in Section 5.3. The adjoint of any polycon of order m is a curve on which a polynomial of maximal degree m - 3 with real coefficients vanishes. That the coefficients are real is a consequence of the construction from multiple points of boundary curves of polynomials having only real coefficients. The analysis at the end of Section 3.5 applies to both the adjoint polynomial and the adjacent factors. Thus the wedge basis functions are rational functions of x and y over the reals. 5.2

VERIFICATION OF POLYCON WEDGE PROPERTIES

By construction, none of the factors for the wedge at node i can vanish at i. Therefore, we can choose a ki to normalize Wi to unity at i. The opposite factor vanishes on all sides opposite i and the adjacent factors vanish at side nodes adjacent to i. Therefore, 1, j = i

w.1 ( x3. , y 3. )

=

{

O,J#i

is assured. We must show that Wi is linear on sides adjacent to i. We first consider a side node. Referring to Fig. 5.7 and denoting the product of

a

L

Fig.

5.7.

S i d e node i .

142

RATIONAL FINITE ELEMENT BASIS

the forms that vanish on the sides opposite i by Pi, we have Wi = kiPi/Qm-3. (The adjacent factor of a side node is unity.) Let (j;i;k)2 be the conic on which node i lies. Then P1-(j;i;k)z = {EIP on (j;i;k)2}

+ {j,kl,

Qm,3*(j;i;k)z = {EIP on (j;i.;kl2l, and (j;k) (-j;i;k)z = {j,k}. Hence, (j;k)Qm,3- (j;i;k12 = Pi-(j;i;k)2. By Theorem 4.2, (j;k)Qm-3 = Pi mod (j;i;kI2, or Wi = kiPi/Qm-3 = (j;k) mod (j;i;k12, and we have established linearity of wedges associated with side nodes on their adjacent (conic) sides. Three vertex node cases are shown in Fig. 5.8: Fig. 5.8a shows vertex i at the intersection of two linear sides. We have

+ {j), , and

(i;j).pi = EEIP on (i;j)) Qm-3-(i;J)= (EIP on (i;j)

(m;j)-(i;j)= {jl, where point m is any point not on line (i;j) (m;j)Q,-,. (i;j) = Pl-(i;j) and by Theorem 4 . 2 : P1 = (m;j)Qm-3 mod (i;]), or i Wi = kiP /Qm-3 5 (m;j) mod (i;j).

.

Thus

Fig. 5.8b shows vertex i at the intersection of a linear side and a conic side. We have Ri = (m;A) and Ri (i;j) = {A), Qm-3-(i;j) = {EIP on (i;j)1, (j;m)*(i;j)= {j), 143

RATIONAL WEDG E CONSTRUCTION

= { E I P on ( i ; ] ) + } (j)

and P i - ( i ; J )

-

{A}.

[ W e n o t e t h a t t h e E I P on ( i ; j ) i n c l u d e p o i n t A and

t h a t t h e symbol -{A) i n t h e above e q u a t i o n d e n o t e s t h e removal of p o i n t A. I n t e r s e c t i o n c y c l e s c a n b e d e f i n e d t o i n c l u d e n e g a t i v e p o i n t s , and such c y c l e s are called v i r t u a l c y c l e s . W e are n o t concerned w i t h a v i r t u a l c y c l e i n t h i s case.]

dPi ..

L

Fig.

5.8.

/

Vertex nodes. ( a ) Line-line; (b) l i n e - c o n i c ; (c) conic-conic.

i i.

F o r case (b) w e t h u s have P R

and PiRi

m

(i;j) = (j;m)Qm-3-(i;j)

: (j;m)Qm-3 mod ( i . ; ] ) ,o r Wi

= kip

i i R

5

144

(J;m) mod ( i ; j ) .

R A T I O N A L FINITE ELEMENT BASIS

Denoting the conic side adjacent to node i in Fig. 5.8b by (i;m;k)2, we have Pi-(i;m;k)2 = {EIP on (i;m;kI2) + {k) - (A),

-

R1- (i;m;k) = (m;A) (i;m;k) = {m,A}, Qm-3*(i;m;k)2 = {EIP on (i;m;k12), and

.

(k;m) (i;m;k) = {k,m). Thus PiRi. (i;m;k) = (k;m)Qm,3. (i;m;k) and application of Theorem 4 . 2 yields Wi = kiPiRi/Qm-3 = (kim) mod (i;m;k12. Throughout this monograph it is understood that the congruences apply only where W i is defined. It will be shown that Qm-3 # 0 over the polycon so that the congruences apply within the regions of interest. Fig. 5 . 8 ~shows vertex i at the intersection of two conic sides. We have the adjacent factor Ri = (m;n;A;B;CI2, the unique conic through the five indicated points, and Ri -(i;m;jI2 = {m,A,B,C), i P .(i;m;jl2 = {EIP on (i;m;jl21 -{A,B,c), Qm-3-(i;m;j)2 = {EIP on (i;rn;]I2},

+ {jl and

(j;m)*(i;m;j)2 = {m,jl. Thus PiRi. (i;m;jI2 = (j;m)Qm-3*(i;m;j)2 and i i Wi = kiP R /Qm-3 = (j;m) mod (i;m;j)2. The same analysis applies with n and k replaci.ngm and j on the other adjacent side. Linearity has been established on the adjacent sides for a l l cases. 145

RATIONAL WEDGE CONSTRUCTION

We now direct our attention to property (5) in Section 1.5: we must show that the wedges form a basis for polynomials of maximal degree one over the polycon. Let Cm be the polycon boundary curve and let u(x,y) be any polynomial of maximal degree one with value ui at node i. Then U(X,Y)

-

f

uiWi(x,y) E 0 mod Cm

i=l

f

and multiplying by Qrn-3,we have rn . . u(x,~)Q~-~(x,y) kiuiP1R1 =- 0 mod Cm.

1

i=1 The left-hand side of this congruence is a polynomial of maximal degree m - 2. Curve Cm is a product of distinct irreducible components and is of order m. By Theorem 1.1, the left-hand side must be the zero polynomial. For all (x,y) not on curve Qm-3 I U(X,Y) =

f

UiWi(X,Y).

i=1 This completes verification of properties (1) (5) of Section 1.5 for the constructed wedges. We have yet to establish property (61, regularity of the wedges over their polycons. We must prove that Qm-3 does not vanish at any point of the well-set polycon bounded by curve Cm.

-

146

RATIONAL

5.3

F INlTE ELEMENT BASIS

THE CASE OF THE VANISHING DENOMINATOR

The denominator polynomial for a well-set polycon of order m is the unique polynomial of maximal degree m-3 determined from the multiple points of the polycon boundary curve. The wedge functions are regular within their polycon if and only if adjoint polynomial Qm-3 does not vanish within the polycon. Regularity is easily proved for convex polygon wedges. All adjacent factors are unity for a polygon. Let N i (x,y) = kiPi ( x , y ) and let Qm-3 be normalized so that it is positive on boundary curve Cm of the polycon. This normalization is always possible since Qm-3 is constructed to intersect Cm only at EIP. In establishing property (51, we proved that

[We may choose u(x,y) = 1 in property (5). I For a convex polygon, N i is positive interior to the polygon for all i. Hence, the sum in (5.20) is positive over the polygon. The situation is more obscure for well-set polycons. When m = 5, Q is positive on the boundary and the conic Q can have no closed branch or isolated point interior to the polycon. Regularity is thus established for any well-set polycon of order less than six. Analysis of wedges for well-set 3-cons of order six reveals some of the difficulties encountered in establishing positivity within a polycon of its adjoint curve. Consider, for example, the 3-con in Fig. 5.9. Without loss of generality, 147

RATIONAL WEDGE CONSTRUCTION

F i g . 5.9.

3-con

[ 1 , 2 , 3 ] of o r d e r s i x .

we may assume that EIP A, B, and C are in the affine plane. A projective transformation could be used to bring any of these points "in from infinity". The adjoint is a geometric property and as such is a projective invariant. An isolated point of a curve is a multiple point. In Fig. 5 . 9 , suppose adjoint Q3 has either an isolated point at 4 or a closed branch encircling 4 interior to the 3-con. If 0 over the 3-con, then there must be a point Q3 like 4. Let L be any line not contained in Q 3 . Then O(L-Q3) = 3 . Our assumption of the existence of point 4 precludes the possibility of Q, having another closed loop for any line connecting the two loops would intersect Q3 in four places. In general, a cubic curve can have at most one closed loop. Points B and C in Fig. 5.9 must lie on an open branch of Q 3 . For the illustrated 3-con4 line (A;4) intersects the polycon boundary and cannot be a component of Q 3 . Line (A;4) also intersects the open branch of Q3 containing B and C. Therefore, O((A;4).Q3) 24. This is not possible. There can be no such point as 4 , and this well-set 3-2011 of order six has an adjoint that is positive over the polycon. 148

RATIONAL FINITE ELEMENT BASIS

Now suppose one of the 3-COn sides is concave as shown in Fig. 5.10. If we postulate the existence

7

OPEN BRANCHOF Q3 CLOSED BRANCH OF Q3,

F i g . 5.10.

A well-set

3-COn

with a concave s i d e .

of an isolated point at 4 or a loop of Q3 encircling 4 , we cannot arrive at a contradiction by the same reasoning a s that applied to the convex 3-COn of Fig. 5 . 9 . The possibility of an adjoint of the type labelled Q, in Fig. 5.10 motivated search for additional theoretical tools with which to establish regularity. (Curve Q3 in Fig. 5.10 is not the true adjoint curve. It is only a hypothetical curve that will eventually be ruled out.) We now prove a remarkable theorem, indispensable in regularity analysis, connecting adjoints of three related polycons. A schematic drawing of the three polycons is given in Fig. 5.11.

PL

F i g . 5.11.

Polycons T 149

1

,

2

T , and T

3

.

RATIONAL WEDGE CONSTRUCTION

THEOREM 5.1,

Let T

1

, T 2 , and

T

3

denote t h r e e L e t t h e polycon h P j , Pk, and P

polycons, n o t n e c e s s a r i l y w e l l set. i b o u n d a r i e s be segments o f c u r v e s P , such t h a t t h e boundary of 1 T : [ i h j ] is P i P h P j , a b b r e v i a t e d as PihJ, 2 k h j khj T : [khj] i s P P P , abbreviated a s P , 3 i h k T : [ i h k ] i s PiPhPk, a b b r e v i a t e d as P

.

L e t t h e c o r r e s p o n d i n g a d j o i n t s b e d h j , Qkhj, and

Then t h e r e are real numbers a , b , and c , n o t a l l o f which a r e z e r o , such t h a t

Qihk.

apiQkhj

+

bpkQihj

+

cpjQihk = 0.

(5.21)

i

REMARK 5.1. I n F i g . 5.11, P = ( 1 ; 4 I L , r. pJ = ( 2 ; 5 ) u , Pk = ( 3 ; 6 ) v , and Ph i s t h e p r o d u c t of

hl = P (1;2;3)wq,

ph2

= (4;5;6),_:

h

Ph = P

4

h '

P'

h

'.

The

o r d e r of P i s w = w1 + w 2 . Polycon T3 i s t h e union of T1 and T2 formed by removing boundary c u r v e P j

.

Any of t h e c u r v e s may d e g e n e r a t e For example, when nodes 1, 2 , and 3

REMARK 5.2.

t o a point.

h

coallesce t o one p o i n t , s a y 2 , c u r v e P 1 i s p o i n t 2.

W e d e f i n e polynomial P hl a s u n i t y i n t h i s case, and note t h a t p o i n t 2 i s a v e r t e x a t w h i c h P i , P I , and k P intersect. Proof of Theorem 5.1. W e r e s o l v e a l l nonordin a r y s i n g u l a r i t i e s of c u r v e PihJk by a sequence o f The symbol q u a d r a t i c t r a n s f o r m a t i o n s (Theorem 4 . 9 ) .

1;

d e n o t e s summation o v e r a l l p o i n t s p i n t h e t r a n s f o r m e d s p a c e , e x c l u d i n g t h e n p o i n t s associated w i t h t h e vertices of e a c h n-con. L e t ms d e n o t e t h e P m u l t i p l i c i t y of c u r v e pS a t p o i n t p. 150

RATIONAL FINITE ELEMENT BASIS

By c o n s t r u c t i o n :

Q

ihk o-pk P

Moreover, p i 0 pk =

(5.22b)

P

1';; C'

(rn m ) p

+ vertices

(mjmk)p P P

+

i n pi

P

and pj

0

pk =

P

Any v e r t e x i n Pi

ph'

0

pk.

(5.23b)

i s also i n PI

0 Pk and con-

Such vertices occur o n l y when one o r b o t h

versely. of

0 Pk

vertices i n ~j

pk, (5.23a)

0

and Ph2 i s u n i t y . PiQkhj

0

From (5.22) and ( 5 . 2 3 ) :

pk = p jQ i h k

pk.

(5.24)

Curve Pk i s a p r o d u c t of d i s t i n c t i r r e d u c i b l e comp o n e n t s , and a l l t h e f a c t o r s have o n l y r e a l c o e f f i cients.

Theorem 4.13 assures t h e e x i s t e n c e of a

real b ' such t h a t piqkhj

- blpjQihk

mod

k

.

(5.25)

By Theorem 1.1, t h e r e i s a polynomial A f o r which p i Q k h j - btPjQihk = A p k (5.26)

.

As i n d i c a t e d i n Remark 5 . 1 , t h e l e f t - h a n d s i d e of (5.26) i s of maximal d e g r e e ( t + w + u + v - 3 ) . T h i s i s e q u a l t o t h e d e g r e e of Pk p l u s t h e maximal d e g r e e of Q i h j

A = Qihj.

and t h e r e f o r e s u g g e s t s t h a t p e r h a p s

I f t h i s w e r e t r u e , t h e n (5.26) c o u l d be

w r i t t e n i n t h e m o r e symmetric form of

151

(5.21).

That

RATIONAL WEDGE CONSTRUCTION

this is the case will now be shown. We recall that Qihj is uniquely determined by the requirement that (5.27) where p excludes the n vertex double points of the bounding curve of polycon [ihj], or the equivalent n conditions for ill-set vertices. Since the maximal order of A is equal to that of QihJ, we need only demonstrate that i 1) (5.28) m (A) 2 (mp + m + mJ P P P By Lemma 4.1, to prove that A = c'Qihj for some c'

-

.

for any curves F and G, m (F + G ) > minim (F),m (GI1.

-

P

P

P

(5.29)

Also, the property m (FG) = m (F) + m (G) is an P P P obvious consequence of the definition of multiplicity. Thus the polynomials on the left-hand side of (5.26) satisfy 1

+ +

+ (mi +

-

(mk

P

mh

+

-

mJ 1)I (5.30a) P and mh + mk 1) (5.30b) mp(P J Qihk) 2 mJ - P P P P From (5.29) and (5.301, we obtain for the left side of (5.26): j ihk mi mh i khj mJ + mk 1. mp(P Q b'P Q 1 = P P P P (5.31) On the right-hand side of (5.261, we have m (APk ) = m (A) + m (Pk ) = m (A) + rnk Therefore, P P P P P' i (5.32) m (A) 2 mp + mh +.mJ 1, P P P and the theorem is proved. i mp(P iQkhj1 2 mp

,

P

+

+

-

152

-

-

RATIO NAL FINITE ELEMENT BASIS

A p a r t i c u l a r configuration warrants a separate theorem. L e t t h e t h r e e polycons i n t h e theorem a l l be w e l l s e t . Then each component of t h e bounding c u r v e of e a c h polycon does n o t change s i g n w i t h i n i t s polycon. L e t Pi and Pk b e normalized t o be posi t i v e i n t e r i o r t o [ i h k l . L e t P’ be normalized t o be p o s i t i v e i n t e r i o r t o [ k h j ] and n e g a t i v e i n t e r i o r t o [ihj] Suppose t h a t

.

(1) Qihk > 0 o v e r [ i h k l , and ( 2 ) QihJ > 0 over [ i h k l .

Eq. (5.10) may be w r i t t e n i n t h e form piakhI = d p j Q i h k + eP k i h j

.

(5.33)

A d j o i n t Q k h j i s p o s i t i v e on t h e boundary of w e l l set polycon [ k h j ] . Thus a t p o i n t s i n F i g . 5.11, P J = 0 and t h e o t h e r polynomial f a c t o r s i n ( 5 . 3 3 ) a r e p o s i t i v e . Hence, e > 0 . A t p o i n t q i n F i g . 5 . 1 1 , Pk = 0 and t h e other polynomial f a c t o r s i n (5.33) a r e p o s i t i v e . Hence, d > 0 . I t f o l l o w s t h a t t h e r i g h t - h a n d s i d e of (5.33) i s p o s i t i v e i n t e r i o r t o

. .

[khjl W e have normalized Pi t o be p o s i t i v e over [khj] Hence, Q k h j must a l s o be p o s i t i v e i n [ k h j ] W e have proved t h e f o l l o w i n g theorem: THEOREM 5.2.

L e t T1 = [ i h j !

T

2

.

= [ k h j ] , and

T3 = [ i h k l be w e l l - s e t polycons such t h a t T1

+

T2 =

3 a s shown i n Fig. 5.11. I f Qihk and Qihj are 2 b o t h p o s i t i v e i n T , t h e n Qk h j > o i n T 2 T

.

T h i s theorem p r o v i d e s a b a s i s f o r e s t a b l i s h i n g

r e g u l a r i t y of wedges o v e r a v a r i e t y of w e l l - s e t polycons. Let T d e n o t e a w e l l - s e t polycon of order t. W e a t t e m p t t o e s t a b l i s h t h a t a d j o i n t Q ( T ) i s p o s i t i v e o v e r T by r e l a t i n g Q ( T ) t o t h e a d j o i n t of 153

RATIONAL WEOGE CONSTRUCTION

a polycon of lower order, using Theorems 5.1 and 5 . 2 . For any specific T , this process may be repeated until we arrive at a polycon for which a direct proof of positivity is known. An inductive proof of regularity is suggested. Let t be the least order for which there is postulated the existence of a wellset polycon, say T, for which Q ( T ) % 0 over T. We have already shown that t > 5. We embed T in a polycon of lower order than t in a manner which leads to a contradiction on application of Theorems 5.1 and 5.2. Unfortunately, it has not yet been shown that this embedding is always possible even though all particular polycons thus far considered have been amenable to this procedure. Several examples will now be treated to illustrate the versatility of the theorems. EXAMPLE 5 . 1 (Convex polygons). We have already proved regularity for convex polygons. An alternative proof, using Theorem 5.2, will now be given. Referring to Fig. 5.12, triangle T1 is erected as shown on side Pj of the polygon. Q(T1) = 1 is positive over T and T1 In applying Theorem 5.2, we 'choose polygon T as T2 in the theorem. According to the theorem, Q(T1 + T2 ) > 0 in T2 yields Q(T 2 ) > 0 in T2 . The order of T1 e T2 is one less than that of T2 This reduction in order may be repeated until a polygon of order five is generated, and the adjoint of this polygon is known to be positive over the element. This illustrates the induction argument in its purest form.

.

.

154

R A T I O N A L F I N I T E ELEMENT BASIS

P L =I

Fig.

5.12.

A c o n v e x polygon.

In the following examples, we will not give the detailed analysis. Polycons Tl, T2 , and T3 will be identified in each example. The crucial part of the argument in each case is the proof that Q(T 1 ) does not vanish on T1 + T2

.

EXAMPLE 5.2 (A 3-COn of order six with a con-

cave side). The 3-COn in Fig. 5.10 that motivated the search for Theorems 5.1 and 5.2 is treated by trivial application of Theorem 5.2 (Fig. 5.13).

‘\qL

I/

/

/

Fig. 5 . 1 3 .

A nonconvex

/‘. 3 - c o n of o r d e r s i x .

EXAMPLE 5 . 3 (A convex n-con with a linear side). It is easily shown that a convex n-con with a linear side can be embedded in a lower-order element for

155

RATIONAL WEDGE CONSTRUCTION

the inductive proof of regularity when elements 1 2 T + T in the configurations shown in Fig. 5.14 are well set. The possibility of the kind of interfer-

INTERFERENCE7

(C) F i g . 5.14.

Convex n-cons

w i t h linear sides.

ence shown in Fig. 5.14d is one of the factors respons i b l e for our failure to prove regularity in general. Triangle [1,2,31 could not be chosen as T1 in Fig. 5.14d because T1 + T2 would then be ill set. EXAMPLE 5.4 (A convex 3-con of order six).

We consider first the 3-con in Fig. 5.15a, in which vertex 3 is a triple point of the boundary curve. A perturbation of side (1;212 as shown in Fig. 5.15b yields a well-set element. ESP A and B of this well156

R A T I O N A L FINITE ELEMENT BASIS

‘,2

Fig. 5 . 1 5 .

P e r t u r b a t i o n o f an i l l - s e t

3-con.

set element coallesce to vertex 3 in Fig. 5.15a. In the ill-set limit, adjoint Q(T) is tangent to ( 1 ; 2 1 2 at vertex 3 as sketched in Fig. 5.15a. Thus Q(T) is positive over T with vertex 3 removed and is equal to zero at 3. The well-set 3-con of order six shown in Fig. 5.9 and redrawn in Fig. 5.16 may be treated Polycon T1 + T2 in with Theorems 5.1 and 5 . 2 . Fig. 5.16 is similar to the element shown in Fig. Adjoint curves Q(T 1 ) and Q(T1 + T2 1 are 5.15. sketched in Fig. 5.16. The 3-con labelled TI in this figure is bounded by (2;3)2(1;3)2(C;2). Adjoint Q(T 1) = ( B ; D ; E ) 2 is sketched. The other two points that determine this adjoint are the complex elements of the cycle (1;312 0 (2;3)2. We observe

157

RATIONAL WEDGE CONSTRUCTION

4

‘+T2)

F i g . 5.16.

A c o n v e x 3-con

of order s i x .

.

that Q(T 1) > 0 over T1 + T2 + T 3 We then note that T1 + T2 is an ill-set 3-con of the type shown in Fig. 5.15 and that its adjoint is positive over T1 + T2 + T4 with point C (at which the adjoint vanishes) excluded. Application of Theorem 5.1 yields Q(T) > 0 over T1 + T2 + T4

.

For a given 3-c0n, this construction is not always possible, but alternatives have been found for all cases thus far examined. Lack of a definitive construction for the general convex 3-COn of order six is another illustration of the elusiveness of a general proof of regularity f o r well-set polycons. EXAMPLE 5.5.

Consider the section of a ring shown in Fig. 5.17a. Polycon T may be embedded in T + T1. We have Q(T1) = 1 and Q(T + T1) > 0 in T. By Theorem 5.2, Q(T) > 0 in T. This approach fails for polycon T in Fig. 5.17b, for in this case T + T1 is ill set. Referring to Fig. 5.17c, we note that even though T1 is ill set Theorem 5.1 yields aPkQ(T1)

+ b(l;2)Q(T) + cPj(A;B) 158

= 0.

RATIONAL FINITE ELEMENT BASIS

(C 1 Fig.

5.17.

A s e c t i o n of a r i n g .

Here, (A;B) = Q(T + TI) is normalized to be positive in T. A l s o , Q(T1) must be an ellipse or a circle k and is normalized to be positive in T. P , (1;2), and Pj are also normalized to be positive interior to T. There are real numbers d and e such that (1;2)Q(T) = dPkQ(T') + e(A;B)PJ. At vertex 3 , d = (1;2)Q(T)/PkQ(T') establishes d > 0. At vertex 4 , e = (1;2)Q(T)/Pj(A;B) establishes e > 0. It follows from the positivity of the polynomial factors interior to T that Q(T) > 0 interior to T. By construction, Q(T) > 0 on the boundary of T.

159

RATIONAL WEDGE CONSTRUCTION

EXAMPLE 5.6 ( A 4-con with interfering hyperbolas). In Fig. 5.18, hyperbolic arcs (1;212 and (3;412 have branches that interfere with lines (1;2) and (3;4)

(1;2)2

(3;412 F i g . 5.18.

I n t e r f e r i n g h y p e r b o l i c arcs.

when we use the approach of Example 5.2. Regularity of the wedges €or this element is established by two applications of Theorem 5.1. The various boundary segments are indicated in Fig. 5.19. In Fig. 5.19a, Q(T 1 ) = 1 and Q(T 1 + T2 ) > 0 over convex quadrilateral T1 + T2. By Theorem 5.1, there are real d and e for which pipJk = dpJQik + epkQ ij (5.34)

.

We normalize Pi and Pk to be positive interior to T. BY considering (5.34) at points p and q in Fig. 5.19a, we establish positivity of d and e. Even though T2 is ill set, we have proved that Q ( T 2 ) = Qjk > 0 on T. A negative value is possible only in the shaded region of T3 in Fig. 5.19a. The second application of Theorem 5.1 is illustrated in Fig. 5.19b. Again, Q ( T1) = 1. Polycon T1 + T2 in Fig. 5.19b is polycon T2 in Fig. 5.19a. We have already demonstrated that Q(T1 + T 2 ) in Fig. 5.19b is positive on T. For the boundary 160

RATIO NAL FINITE ELEMENT BASIS

F i g . 5.19.

A p p l i c a t i o n of T h e o r e m 5.1.

curves defined in Fig. 5.19b, the values of d and e in Eq. (5.34) are again positive. It follows that Q(T) > 0 over polycon T. Establishing regularity of wedges over specified elements through use of Theorems 5.1 and 5.2 is an entertaining pasttime, and one should examine a few exotic elements to become convinced that there is a reasonable basis for the assumption that the constructed wedges are indeed regular for all well-set polycons. Although Theorems 5.1 and 5.2 provide a convenient means for verification of wedge regularity for specific polycons, a general proof seems to 161

RATIONAL WEDGE CONSTRUCTION

require resolution of difficult topological problems. Nevertheless, the remarkable relationship in Theorem 5.1 is evidence of the deep algebraic and geometric connection between the elements and their rational wedge basis functions. That Max Noether's fundamental theorem and modern algebraic geometry can be used effectively in analysis of a problem of central concern to numerical analysts and applied mathematicians is most gratifying. 5.4

POLYPOLS

AND DEFICIT INTERSECTION POINTS

The polypol was defined in Section 1.7 as the generalization of the polycon to an algebraic element with sides of any order. Although polygons and polycons are polypols, we reserve the more general name for figures with at least one side of order greater than two. A figure with sides of maximum order three is called a polycube. Polycons provide a versatile representation for patchwork approximation and the need for greater freedom of element boundary curves is limited. It has already been observed that triangles, parallelograms and isoparametric elements satisfy most practical needs. Generalization of polycon wedge construction to polypols was motivated more by a desire to illustrate the scope 0.f the theory than by a need for polypols in practice. Associated points (Theorem 4 . 3 ) are significant, wedge uniqueness is lost, and regularity is more difficult to establish in polypol analysis. Nevertheless, there are satisfying aspects of the generalization that justify exposition. We consider first the construction of the denominator polynomial, common to all wedges of a given polypol. 162

RATIONAL FINITE ELEMENT BASIS

The denominator polynomial is of maximal degree m - 3 for a polypol of order m. Let PE be the curve of order k that defines side s, and let Qm- be the polypol adjoint (denominator) curve. Let Pmmk be the boundary curve with component !P removed. Thus polynomial Pt-k vanishes on all polypol sides other than s . We note that O(PE*Pz-k) = k(m k) and that when we subtract the two vertices there are k(m - k) - 2 EIP in pSk - 'm-k' We next note that O(PE-Q,3) = k(m - 3) and that this exceeds the number of EIP in PE.Pi-k by k(m 3 ) -k 2 - k(m - k) = (k 1) (k - 2 ) elements. This excess is nonzero only when k > 2 and for this reason did not enter into polycon analysis. It is clearly impossible to construct a polypol adjoint curve of order m - 3 that intersects the polypol sides only at the EIP. It will be shown that this leads to some arbitrariness in the wedge construction. We may choose (k - 1)(k - 2 ) / 2 points on the extension of each side of order k which together with the k(m - k) - 2 EIP on the side form an (m - 3 ) independent point set. We demand that the adjoint intersect the side at these points. The appended points are called deficit intersection points (DIP). A unique Qmm3 may be constructed once the DIP have S 3 2 k. Curves Pk and been chosen. Suppose m Qmw3 are relatively prime. By Theorem 4.3, [k(m - 3) - (k - 1)(k - 2 ) / 2 ] (m - 3)-independent points in PE-Qm-sdetermine the remaining (k - 1)(k - 2 ) / 2 points uniquely. Let f be the number of EIP plus DIP on side PE:

2

-

-

-

-

163

RATIONAL WEDGE CONSTRUCTION

- k) k(m - 3)

f = k(m =

-

2

+

(k

-

-

1)(k - 2)/2 1)(k - 2)/2

(k

(5.35)

.

We have assumed that the DIP have been chosen so that these f points are (m 3)-independent. Hence, Theorem 4.3 applies and these points determine the S .Qm-3' Suppose, on the other remaining points in Pk 3 < k. Then either hand, that k > 2 but that m m - k = 1 or m - k = 2. As observed in the discussion following Theorem 4.3, there are at most m(m - 3)/2 (m - 3)-independent points in the intersection set and these points determine curve Q m-3 ' We have for the two possibilities:

-

-

- k = 1. f = (m - 1)(m 3) = m(m - 3)/2, and ( 2 ) m - k = 2. f = (m - 2 ) (m 3) = m(m - 3)/2. (1) m

-

-

(m

-

2) (m

-

3)/2

-

-

(m

-

3) (m

-

4)/2

-

-

Thus in all cases the addition of (k 1)(k 2)/2 DIP to the EIP on side s of order k yields just the right number of conditions for determining Qm-3 OPE. We have yet to show that there are sufficient degrees of freedom in Qm- 3 to enable simultaneous satisfaction of these conditions on all polypol sides. The total number of EIP and DIP is

f

f

k ( s ) Im - k(s)1/2 = [m k2(s)]/2 s=l s=l points of intersection of the polypol sides (counting multiplicities) - n polypol vertices +

164

RATIONAL FINITE ELEMENT BASIS

n

n

s=l

s= 1

+ n DIP for a total of m /2 - 3m/2 = m(m - 3)/2 points. This is precisely the number of conditions required to determine a curve of order m - 3. That these points are (m - 3)-independent and determine a unique curve may be proved by the polycon argument. This consideration of degrees of freedom of Q,-3 to establish existence and uniqueness of the adjoint (once the DIP have been chosen) parallels the preliminary polycon adjoint discussion in Section 5.1 which was subsequently made more precise by applicaA similar application of altion of Theorem 4 . 4 . gebraic geometry is possible here, and coallescing of EIP may be treated satisfactorily within such a framework. Even so, the polypol construction is less definitive than the polycon construction. Aside from the arbitrariness of the DIP, there is the possibility of EIP or associated intersection points falling at singular points of polypol sides. Theorem 4.2 would than not apply. Furthermore, wedge regularity is achieved only if the adjoint is nonzero over the polypol, and we have no assurance that the associated points fall on extensions of sides rather than on polypol boundary segments. In any event, regularity must still be established. If these questions can be resolved, it is likely that any one of a family of adjoints can be found for a given polypol. There is a class of polypols for which one choice has great merit, and 2

165

RATIONAL WEDGE CONSTRUCTION

for which the entire polycon theory generalizes in a straightforward manner. This preferred class is that of well-set rational polypols. We have Pacitly used Eq. (5.4) to define Qm-3. This accounts for all singularities on polypol boundary curve Cm at intersections of different components of Cm but does not account for the singular points of the components themselves. The "third level" definition of Qme3 in Section 5.1 includes component singularities. .The number of conditions imposed by singularities of a component of order k and genus zero is by Eq. (4.13) equal to (k-1) (k-2)/2, and this eliminates the need for DIP. We note that if the component were not rational its genus would be greater than zero and the fewer singular points would have to be supplemented by deficit points. A component double point is in effect its own associated point. Now that we have included component singular points on curve Qm-3, we can no longer use Theorem 4.2. The analysis in Section 4 . 6 was presented in anticipation of just this situation. By using the divisor of Qm-3 on Cm instead of the intersection cycle in the development in Sections 5.2 and 5 . 3 and by using Theorem 4.13 rather than Theorem 4.2, one may establish the existence of a unique adjoint for any well-set rational polypol. Regularity Theorems 5.1 and 5.2 are valid in this context, for these theorems may be proved with divisors replacing intersection cycles. For any polypol of genus greater than zero, we may retain the component singular point conditions and introduce gs D I P on side s of genus gs. One 166

RATIONAL FINITE ELEMENT BASIS

convenient choice is a gS -fold DIP at infinity. This may be introduced after satisfying all other conditions on a side by setting to zero the gs terms of highest degree in the resultant of Ps and Qm-3 in affine coordinates. Supplementary checks on regularity are required. 5.5

POLYPOL WEDGE NUMERATORS AND ADJUNCT INTERSECTION POINTS

Polypol wedge numerators are products of opposite and adjacent polynomials. The opposite factors are defined as for polycons. The adjacent factors require further study. A linear form has three degrees of freedom on a side of order greater than one. A side node is therefore required on each polyp01 side of order greater than one. Let i be the side node on side PE, and let p and q be the adjacent vertices as shown in Fig. 5.20. Wedge Wi must be

Fig.

5.20.

Polypol s i d e node i.

linear on Ps and vanish at vertices p and q. Hence, Wi E (p;q) mod Ps. Continuity of the patchwork approximation across Ps is assured only if function values at the vertices and side node determine a unique linear form on the side. Values of a linear form at any three noncollinear points on a curve determine the form uniquely on that curve. Thus 167

RATtONAL WEDGE CONSTRUCTION

side node i must be chosen off line (p;q). This requirement did not appear in the polycon analysis S where line (p;q) could not intersect conic side P2 at any point other than vertices p and q. Adjacent factor Ri of wedge Wi is the polynomial of maximal degree k - 2 that is the a d j o i n t of t h e 2-pol of o r d e r k

f

1 b o u n d e d b y s e g m e n t s Ps a n d

If Ps is not rational the DIP on this side of the 2-pol are chosen as those of the polypol. This defines a unique Ri. Further insight is gained if we consider in greater detail the situation where all points in (p;q) 0 ! P are distinct and do not fall on Qm-3. The points in this divisor, excluding vertices p and q, are called the adjunct intersection points associated with side node i. There are k 2 adjunct points i on R The singular points of PE impose (k - 1) ( k 2)/2 - gs conditions on Ri and the DIP impose another g, conditions for a total of ( p ; q ) w i t h vertices p a n d q .

.

k

-

2

+

-

-

[(k

-

1) (k

-

2)/2

-

gsl

+

gs = ( k - 2 ) ( k + 1 ) / 2

conditions. At least one curve of maximal order k - 2 satisfies these conditions. We have shown previously that coallescing of intersection points does not alter the number of conditions imposed. We have already observed that Ri is unique. The role of associated points and the importance of Theorem 4.3 becomes apparent when we investigate linearity of Wi on Ps. Let SIPs denote the singular points of 'P in Q,-~ 0 Ps, counting neighbors and multiplicities, and let DIPs denote the deficit points. For the chosen DIP, there are gs associated points in Qm-3 0 Ps that are in general not multiple 168

RATIONAL FINITE ELEMENT BASIS

We denote these

points of polypol boundary curve Cm. associated points by AIPs. Then i Pm-k

0

S

Pk = p

+

g

+

Qm-3

0

S

Pk

-

SIPs

-

DIPs

-

AIPs. (5.36)

By construction, <-2

0

S Pz 2 (p;q) 0 Pk

-

p

-

q

+

SIPs

Adding (5.36) and (5.37), we obtain i Ri S 'm-k k-2 0 p! (P;q)Q,,3 pk

-

+

DIPs. (5.37)

AIPs-

(5.38)

The order m of the polypol must be greater than the i order k of side Ps. If m = k + 1, then (p;q) = P i i and Qm-3 = Ri so that (p;q)Qm-3 = P R and W, = k,PiRi/Qm-3 : (piq) mod Ps. 1

1

In this simple case Wi = k'(p;q) for some k'. 2 2 - k. Moreover, I f m > k + 1, then m 4, = < (k 1) (k - 2)/2. We may apply Theorem 4.3 to establish that the right-hand side of (5.38) contains sufficient elements to ensure

-

-

Ps = (p;q)Qmm3 0 PS. i i = Then Theorem 4.13 yields P R - (p;q)Qm,3 mod Ps and we have proved linearity of Wi on Ps. We now consider the adjacent factor at a vertex node, examining only the case where both adjacent

PiRi

0

sides are of order greater than unity and using the nomenclature in Fig. 5.21. Let the orders of adjacent sides P1 and P2 be kl and k2, respectively. Then adjacent factor Ri is of maximal degree kl + 2 k2 - 2, and the divisors of Ri with P1 and P satisfy the following conditions:

169

RATIONAL WEDGE CONSTR UCTlON

Fig. 5 . 2 1 .

R~

0

P1 2

R~ o p 2 2

A p o l y p 0 1 w i t h v e r t e x i.

DIP^ + SIP^ + DIP^ + SIP^ +

(p;jl)P2 (q;j2)P1

0

' P

0

p2

-

p q

-

i, i.

Adding the above expressions, we obtain R~ 0 p1p2=

DIP^

+

+ DIP^ +

(q;j2) o p2

SIP^ + SIP^

+

2~'

0

p2

+

(p;jl) 0 P1

-p-q-

2i.

(5.39) By adding the orders of the sets on the right-hand side of (5.39) and applying Theorem 4.3, we ascertain that there is at least one Ri of maximal degree kl + k2 - 2 that satisfies (5.39). To establish uniqueness, we note that ( 5 . 3 8 ) and Theorem 4.3 give Ril 0 p1p2 = Ri2 0 p 1p 2 for any two adjacent factor candidates. Theorem 4.13 may then be applied to prove uniqueness. Linearity of Wi on the sides adjacent to node i can also be established by the usual argument. The construction assures PiRi o p l = (p;jl)Qm-3 0 Pl and PiRi 0 P2 - (q;j2)QmY30 P2 Theorem 4.13 may now be applied to establish linearity, and we dispense with the details.

.

170

R A T I O N A L FINITE ELEMENT BASIS

The adjacent factor is constructed from five types of points: (1) side nodes, (2) singular points of adjacent sides, ( 3 ) deficit points on adjacent sides, ( 4 ) adjunct points on adjacent sides, and (5) points other than the vertex in P1 0 P2. Care must be exercised when any of these points coallesce. The construction is unique for the general rational polypol and for any polypol once the DIP have been selected. Wedge Wi is normalized to unity at i, it is linear on sides adjacent to i, and Wi vanishes on all sides opposite i. Regularity may be verified with the aid of Theorems 5.1 and 5.2 for specific elements, although a general proof of regularity for well-set polypols has not been found. Degree one approximation is achieved; the polycon proof goes over to polypols without modification. The polycon theory generalizes to well-set rational polypols with regard to construction and verification of a unique rational wedge basis for degree one approximation. Although the wedges for nonrational polypols depend on arbitrary choice of DIP, the construction is reasonably well founded. A few illustrative elements will now be examined. As the elements increase in complexity, the algebra of numerical studies grows. For this reason, the examples are quite restricted. Studies should be conducted with the aid of a The geometric aspects of polypol analycomputer. sis suggest fascinating graphic displays. The algebraic geometry involved in adjoint construction introduces challenging computer programming. Algorithms have not yet been developed for this task. 171

RATIONAL WEDGE CONSTRUCTION

5.6

ILLUSTRATIVE POLYCUBES

We consider first the polycube in Fig. 5.22

Fig.

5.22.

A polycube of order f i v e .

-

2 3 bounded by curve C5 on which x(w y ) (w y - x ) = 0. We have used projective coordinates (w,x,y) here to facilitate examination of the triple point on the absolute line where side (1;2) intersects a cusp of the cubic side. Aside from the vertices, C5 has double points at (l,xA,l) and ( l , x B , l ) , where xA and xB are the roots of 1 + x + x2 = 0. These two points together with vertex 3 are the elements in (2;3) 0 (3;4;113. The unique adjoint of this rational polyp01 is the curve Q 2 satisfying:

2 2 * rn(l,XA,lfQ2)L1' rn(l,xg,l) (Q,) 2 w2 + a wx + a x2 + a3wy + a4y2 + a5xy. 1 2

m (0,0,1) (Q2)

Let Q =

1.

Then m (o,o,l) (Q) 2 2 yields a3 = a4 = a = 0 . The 52 = 1 + x + other two conditions yield 1 + alx + a2x x2 or al = a2 = 1. Hence, Q2 = 1 + x + x2 .

172

RATIO NAL FINITE ELEMENT BASIS

Construction of wedge W4 illustrates computation of adjacent factors. The opposite factor in W4 is, of course, p4 = x(l y) . The adjacent factor is the adjoint curve of the polyp01 with vertices at 1 Curve C4 and 3 bounded by C4 = (y - x) (w2y - x 3 ) has a double point at (l,-l,-l) and a cusp at (O,O,l). The line through the two double points is the adjacent factor for node 4 : R4 = 1 + x. The wedge is normalized to unity at node 4 :

-

.

W4(x,y) = k4x(l

+

x) (1 - y)/(l

+ x2 + x) .

Construction of the other wedges follows this pattern. For example, adjacent factor R1 in W1 has 1 1 multiplicities: m 2 2 , m (W4*X4*Y4)(R 2 1, (0* 0 r l ) ( R For any choice of node 4 and m (w rX *YF)(RI) 1. 1. on the Eubfc side, these five conditions yield a unique R1 (We recall that, by Theorem 4.4, locating a double point on a curve imposes at most three conditions,) If we chose not to use the singular point as the DIP for the cubic side, we would obtain a different set of wedges. Suppose, for example, we select (l,-l,-l) as a DIP in place of (O,O,l). The adjoint 2 is then P2(x,y) = 1 + (3x/2) + x - (xy/2). The associated point of intersection of curve P2 with the cubic side is then point (1,2,8). The adjacent factor is linear form 2 + 3x - y whose curve is (Our choice of DIP tangent to the cubic at (l,-l,-l). happens to be one of the points of intersection of (1;3) with the cubic.) Wedge W 4 is now

.

W4(xty) = k4x(l

-

y ) (2

+ 3x - y)/P,(x,y).

The enthusiastic reader may construct and verify the 173

RATIONAL WEDGE CONSTRUCTION

wedge basis for degree one approximation over this polycube The polycube of order six in Fig. 5.23 is an ex-

.

BRANCH 2 OF (I:3)zL Fig. 5.23..

A polycube

of o r d e r six.

ample of a nonrational polypol. Cubic side (1;313 has no singular point. We will construct the wedge associated with side node 4. We define the E I P A, B, H by: (1;2)-(2;3)2 = 2, A,

...,

(1;2)-(1;3)3 = 1, B, c , (2;312. (1;313 = 3, D, E t F, G, H. We choose deficit point I at infinity so that (1;313 and adjoint Q3 intersect on the absolute line at point I. We note that Q3 = ( A ; B ; . ..;I) 3 and that the eight E I P and DIP determine a unique associated ninth point (Theorem 4.3 with t = s = 3), which we denote by J, in the intersection cycle (1;3)3.Q3. Adjunct point K and deficit point I determine adjacent factor ( K ; I ) for node 4 . The opposite factor is P 4 = (1;2) (2;3)2. 3hus the wedge is w4 (x8y) = k4 (K;I) (1;2)(2;3)2/Q3'

To verify that W4 is linear on the cubic side, we J)+{l, 3, K] note that (1;3)Q3-(1;313 =fB, C, and that (K;I)(1;2)(2;3)2*(1;3)3 2 { K , d + 8 , B , C)+

...,

174

RATIONAL FINITE ELEMENT BASIS

9,

H). The last two intersection cycles have 11 points in common. Theorem 4.3 with t = 4 and s = 3 assures us that these 11 points determine a unique 12th point in the cycles. This is point J. Hence, both cycles are the same and W4 is indeed linear on the cubic side. We now illustrate application of regularity theorems 5.1 and 5.2 to polycubes. In Fig. 5.24, the D,...,

Fig. 5 . 2 4 .

A

2-pol of o r d e r f i v e .

2-pol of order five has a quadratic denominator polynomial that cannot vanish interior to polycube T. This is verified by application of Theorem 5.2 to T and T1 Line (A;B) is parallel to the y-axis. We have Q(T + T1) = 1 > 0 in T + T1, and Q(T 1) = (A;B) > 0 in T. Therefore, Q(T) > 0 in T. In Fig. 5.25, the cubic side has a cusp at B. The preceding argument applies when B is chosen as the DIP. If instead the DIP is chosen at infinity, then ( A ; B ' ) is parallel to the y-axis. The regularity argument remains valid. We must take care to associate DIP with curves rather than with polypols when we relate adjoints for application of Theorems

.

175

RATIONAL WEDGE CONSTRUCTION

Fig.

5.25.

Another 2-pol

o f order f i v e .

5.1 and 5.2. The DIP f o r curves common to two polypols in the analysis must be the same for the poly-

pols. For rational polypols, use of the singular points of the components as the DIP eliminates all ambiguity. No well-set rational polyp01 has been found for which this choice does not yield regular wedges.

176

Chapter 6

APPROXIMATION OF HIGHER DEGREE

6.1

DATA FITTING

The m wedge basis functions for a polycon of order m and products of these functions may be used to fit discrete data on polycon sides while maintaining continuity across polycon boundaries. The side between vertices i and i + 1 in Fig. 6.1 is a conic

F i g . 6.1.

D a t a f i t t i n g on a p o l y c o n s i d e .

boundary component with side node p. Wedges Wi, W P' and Wi+l provide a basis for fitting data on this side. The number of basis functions must equal the number of data points. Referring to Sections 1.3 and 3 . 4 , we recall that on a conic side a linear function has three degrees of freedom and a quadratic function has five degrees of freedom. For each successive degree polynomial there are two more degrees of freedom. In terms of the wedges, a basis 177

APPROXIMATION OF HIGHER DEGREE

for fitting polynomials on the side in Fig. 6.1 is: 2

= P },...(6.1) {Wi4Wi+14W l l IW.W 4WpWi+l}, IWiWpWi+llW.W P 1 P where the first set suffices for linear fitting, the second set is appended for quadratic fitting, the third for cubic fitting, etc. Wedges Wi, W P' and Wi+l differ in the two polycons that share this boundary component. These functions, however, coincide on the boundary. If the basis in Eq.(6.1) is used in approximating data over both polycons, then continuity will be achieved across this boundary. The continuations into the respective polycons will of course differ in general. In Fig. 6.lb, for example, the six data points indicated by X may be fit by the five functions in the first two sets in (6.1) and any one of the functions in the third set. Although this approach is possible, there is arbitrariness in choice of a basis. Haphazard fitting of data lacks motivation, and a far more meaningful alternative will be described in depth in this chapter. A polycon of order m has one wedge associated with each of its m nodes that form a basis for degree one approximation over the polycon. We seek a judicious choice of additional nodes to retain continuity of the composite approximation while increasing the degree to two, three, or as high as we wish. In the next section, we show how a basis may be constructed for continuous degree two approximation. This leads to a general formulation for any degree approximation over any well-set polypol.

178

RATIONAL FINITE ELEMENT BASIS

6.2

DEGREE TWO APPROXIMATION

We first consider the triangle, €or which areal coordinates provide a polynomial (and hence rational) basis for degree one approximation. Introducing a node on each side, we have six data points that determine a unique quadratic function. A minimal basis for degree two approximation over the triangle in Fig. 6.2 is provided by functions of the type dis-

Fig. 6.2.

The s i x - n o d e

triangle.

played in Eq.(6.2):

(6.2)

where the k. are the usual normalizing factors. 1 The other four Wi are chosen similarly. For any quadratic u(x,y), there is a P2(x,y) such that u(x,y) i=1 uiWi(x,y) = Pz(x,y) vanishes on the triangle perimeter of order three. Thus P2 must be the zero polynomial and degree two approximation is established. This is one of the most commonly used degree two elements in finite element computation (Zienkiewicz and Cheung, 1967). A similar basis for the quadrilateral in Fig. 6.3 is provided by functi.cns of the type given in Eq.(6.3) for nodes 1 and 5.

l6

179

APPROXIMATION OF HIGHER DEGREE

F i g . 6.3

Degree t w o approximation over a quadrilateral.

(6.3)

These basis functions are quadratic on the perimeter and cubic/(A;B) interior to the quadrilateral. Hence, there is a P 3 such that for any quadratic u, uiWi(xfy) = P3(x,y)/(A;B) vanishes u(x,y) on the perimeter of order four. It follows that P3 must be the zero polynomial and degree two approximation is achieved. A n eight node isoparametric element has been used in finite element computation. This element has degree two accuracy in the local (isoparametric) coordinates but is only of degree one in x and y. This will now be clarified by example. The quadrilateral in F i g . 6.4 w a s cho-

1Xzl

F i g . 6.4.

A sample quadrilateral.

180

RATIONAL FINITE ELEMENT BASIS

sen to allow simple reversion of the isoparametric transformation. Referring to Fig. 1.15 and Eq. (1.19) in Section 1.6, we find that

5

= x and 0 = 2y/(x

+

3).

(6.4)

If the isoparametric approximation were of degree 2 yiWi(x,y) with two, then we would have y2 = the isoparametric W i of Fig. 1.15. This is not the case :

c 8

2 YiWi(X’Y) =

+ w1 + w2 + T(WS 9

W*)

+

4 (W3

+ W4)

i=l

2(5 2Y ( 3

+ +

3x) XI2

-

1

4

x2

It may be verified that this does reduce to y2 on the boundary. The error at the origin is 1/4. If the quadrilateral dimensions are reduced by a factor 2 of h, then the error at the origin is h / 4 . Thus 2 the quadratic error is O(h ) , and this is characteristic of degree one approximation. The rational basis functions of the type shown in Eq.(6.3) required to examine the approximation to y2 are given in Eq. (6.6). W 5 and W 7 are not needed since y5 = y7 = 0.

181

APPROXIMATION OF HIGHER DEGREE

W3(X,Y)

=

-

3(1 +

X) (X

w8 ( x , y ) = w6 ( x , - y )

3

+

2y) 11

16(3

+

x)

+

-

-

x

(2y/3)1 I

-

S u b s t i t u t i n g t h e s e f u n c t i o n s i n t o (6.51, w e o b t a i n 8

1

Y;wi(x,Y)

=

-

i=l

+ = y

3 ( 1-X) 12 (x+3) ( l + x ) 8 (3+x) 2 9 (1-x ) 3(l+x)

2

-

4

r-

1

-

x

3 + x

+

4 (3+x)

2(1

+

(8y2/3) 1

I 2 (x+3) (1-x)

x)

3 + x

-

-

I ' Y

8~*/31

2

.

W e need n o t v e r i f y t h a t xy and x 2 a r e a l s o o b t a i n e d

w i t h no error: d e g r e e t w o approximation i s a s s u r e d

by t h e t h e o r y . W e now d i r e c t o u r a t t e n t i o n t o d e g r e e t w o approxi m a t i o n o v e r polycons, s t a r t i n g w i t h t h e s i m p l e 3-COn of o r d e r f o u r shown i n F i g . 6.5. Three nodes are a d e q u a t e f o r f i t t i n g a q u a d r a t i c f u n c t i o n on a s t r a i g h t l i n e , b u t a q u a d r a t i c f u n c t i o n has f i v e d e g r e e s of freedom on a c o n i c s i d e . Thus w e have used t h e l e a s t p o s s i b l e number of nodes i n F i g . 6.5.

182

RATIONAL FINITE ELEMENT BASIS

Fig. 6.5. N o d e s for a p p r o x i m a t i o n over a 3-con of o r d e r f o u r ; ( a ) d e g r e e o n e , ( b ) d e g r e e t w o .

The six node isoparametric triangle cannot be of degree two when any of its sides is not linear in the x,y plane. For example, consider points (O,O), (1/2,1/4), and (1,l) on the parabola for which 2 y - x = 0. Both xy and y(1 + 2y)/3 have the same values at the three points, but these functions are not equal on the parabola. Three nodes just do not determine unique quadratic variation on a conic. The denominator for the degree one rational basis functions of the 3-COn in Fig. 6.5 is (A;B). We seek a degree two basis analogous to that previously found for the quadrilateral [Eq. (6.3) 3 . It will be shown that the following functions are appropriate: W1(X,y) = k1(2;3)2(5;6)/(A;B)

(6.7a)

W2(X,Y) = k2(3;1) (4;B;6;7;8)2/(A;B)

(6.7b)

W3(x,y) = k3(l;2) (A;5;4;7;8I2/(A;B),

(6.7~)

W ~ ( X , Y ) = k4(7;8) (3;1) (1;2)/(A;B),

(6.7d)

W5(x,y) = k5(l;2) (2;3I2/(A;B),

(6.7e)

183

APPROXIMATION OF HIGHER DEGREE

w6(x,y) = k6(1i3) (2;3)2/(A;B) ,

(6.7f)

W7(x,y) = k7(3i1) ( 1 i 2 ) (4;8)/(A;B)

(6.7g)

W8(x,y) = k8(3;1) (1;2) (4;7)/(A;B).

(6.7h)

Before verifying these functions, we use the argument in Section 5.1 to prove that the 'adjacent 1 2 factors in W2 and W3 are unique. Let R and R be two conics containing points 4 , B, 6, 7, and 8 . 2 Since R 1 . ( 1 ; 2 ) = R 2 - (1;2) = {6,B}, neither R1 nor R 2 contains vertex 2 so that polynomials R1 and R may be normalized to unity at 2 . By Theorem 4.2, R1 R2 E 0 mod (li2). Similarly,

-

R1- (2;3) = R2- (2;3)2 = {B,4,7,8),

so that R1 - R2 5 0 mod (2;312. Thus, R1 - R2 is a polynomial of maximal degree two that vanishes on a curve of order three having distinct irreducible 2 components. By Theorem 1.1, R1 - R must be the zero polynomial. Similar analysis may be used to show that the adjacent factor in W3 is unique. If we refer to Fig. 3.3 and Eq.(3.11), we see that the functions are related to the 3-con wedges. A comparison of (6.7) with (3.11) reveals that all but W2 and W3 in (6.7) are obviously quadratic on the 3-COn sides. It is less obvious but also true that W2 and W3 are quadratic on the 3-con sides. This will now be demonstrated for W2; the same analysis applies to W3. We first note that (4;B;6;7;8l2*( 1 ; 2 ) = (6;7)(A;B) (1;2) = {6,B}. By Theorem 4 . 2 , (4;B;6;7;8I2 z (6;7)(A;B) mod (1;2), and hence, W2

=

k2(3;1) (4;B;6;7;8)2/(A;B)

184

(3;l)(6;7) mod (1;2).

RATIO NAL FINITE ELEMENT BASIS

Similarly, (3;l)(4;B;6;7;8)2-(2;3)2 =

-

(3;7)(4;8)( A ; B ) (2;312 = {4,B,7,8,At3}. By Theorem (3;l)(4;B;6;7;8I2 Z (3;7)(4;8)(A;B) mod (2;3)2, and therefore 4.2,

W2 = k2(3;1) (4;B;6;7;8I2/(A;B)

(3;7)(4;8) mod (2;312.

It is thus established that W2 is quadratic on each side adjacent to node 2. For any quadratic u(x,y), there is a P3 for which 8 uiWi(x,y) = P3 (x,y)/(A;B) vanishes on U(X,Y) the boundary of order four. By Theorem 1.1, P3 is the zero polynomial, and we have shown that the functions in (6.7) achieve degree two approximation. An instructive example is illustrated in Fig. 6.6.

Fig.

6.6.

A

s a m p l e 3-COn o f o r d e r f o u r .

Point 6 has been chosen so that the isoparametric arc (2;3;612 is the true 3-COn side: (2;3;6)2 = y x2 + 1. This is best seen by referring to MacMillan's construction (Fig. 1.16). The isoparametric coordinates obtained from Fig. 1.15 are:

-

p = (1 + y

-

x2 )/(I

+

x), q

=

x , r = -y/(l

+ XI. (6.8)

Point 6 in Fig. 6.6 corresponds to point 5 in Fig. 1.15. We note that for Fig. 6.6, W6(x,y) = 4qr = -4xy/(l + x) and that the isoparametric approxima185

APPROXIMATION OF HIGHER DEGREE

tion to xy is U(x,y) = 3xy/2(1 + x)

.

(6.9)

The rational wedge basis function associated with node 6 for degree one approximation over t is 3-con is (6.10) W6(x,y) = -4xy/(l + XI, and this is identical to the isoparametric basis function associated with this node. Thus 6.9) is the degree one rational approximation to x y . Degree two rational approximation requires two more nodes on (2;3)2. We choose points 7 = (1/4,-15/16) and 8 = (3/4,-7/16). The three basis functions that affect the approximation of xy are: W6(x,y) = 64xy[y

-

x + (19/16)1/(1

+ x), (6.11a

~), W7(x,y) = - 1 2 8 ~ ~ [ ~ - ( 5 ~ / 4 ) + ( 1 1 / 8 ) ] / 3 ( 1 +(6.11b ~). W8(x,y) = - 1 2 8 ~ ~ [ ~ - ( 3 ~ / 4 ) + ( 9 / 8 ) 1 / 3 ( 1 +(6.11~

We obtain from these functions, the xy approximation: 5x + -111 (x,y) = xy[lO(y - 4 8 19 3x 9 - 24(y - x + i-g) + 14(y - 4 + g)]/(1 + X) = xy[(-

= xy(1

50 4 +

24

- 442) x +

+ x)/(l + x)

(110 - 12.19 + 14*9)/81/(1 + X)

= xy,

as predicted by theory. The eight node rational approximation is of degree two over this 3-con. This result generalizes to all well-set polycons. Let the wedges for degree one approximation over a well-set polycon be Wi (i = 1, 2, m). Let the basis functions for degree two approximation

...,

186

RATIONAL FINITE ELEMENT BASIS

...,

over the same polycon be Vi (i = 1, 2, 2nd , where the first m of these Vi are associated with the same nodes as the corresponding w TO increase the i' degree of approximation from one to two, we add one side node on each linear side and two side nodes on each conic side. The boundary curve multiple points remain fixed. Hence, the adjoint curve does not change. The opposite factors of Vi and Wi f o r i 5 m are also the same. Let sides Pi and Pi+' of orders si and s ~ + ~ , respectively, intersect at vertex i. The adjacent factor in Wi vanishes on the si + si+l 2 adjacent side nodes and on the sisi+l - 1 EIP in Piepi+'. As shown in Section 5.1, these points determine a unique adjacent curve of order si + si+l - 2 . For degree two approximation, we introduce an additional si + si+l side nodes on the sides adjacent to vertex i, giving a total of

-

t(si,si+l) =

(Si

+

+ si+l

(Si

-

2)

+ (sisi+l - 1)

+ si+l)

points (in the general case) on which the adjacent factor must vanish. We observe that

and that a polynomial of degree one more that that required for degree one approximation suffices for the adjacent factor when degree two approximation is demanded. We omit the proof, which follows the usual pattern, that the adjacent curve is unique and of order 'i+l - 1. The opposite factor is of degree i ' m - s i - s i+l' Hence, the numerator of each Vi +

187

APPROXIMATION OF HIGHER DEGREE

associated with a vertex node is of degree m - 1. We now consider basis functions associated with side nodes. Let j be a side node on a conic side for degree one approximation. Basis function V for j degree two approximation is equal to W times the j linear form determined by the two nodes added on the conic side containing j to achieve degree two. The basis functions associated with these added nodes are each equal to W times the linear form determined j by the other two side nodes on the side containing j . Thus each numerator factor is one degree higher than the degree one numerator. All the Vi have numerators of degree m - 1. Each V. has quadratic variation on each polycon 1 side. For any quadratic u with nodal values ui there is a P 1 such that u(x,y) mPm-l (x,y)/Qm-3 (x,y) vanishes on the perimeter of order m. By Theorem 1.1, 'm- 1 is the zero polynomial, and we have proved that degree two approximation is achieved. In proving that the m degree one basis functions replicated polynomials through degree one, we applied Theorem 1.1 to a polynomial of degree m - 2 on a boundary of order m. In this degree two analysis the polynomial that vanishes on the perimeter of order m is of maximal degree m - 1. This portends trouble when we attempt generalization of the argument to establish higher degree approximation. A polynomial of degree greater than m - 1 can vanish on a curve of order m without being the zero polynomial. This anticipated difficulty will be resolved in the next section in a most satisfactory manner.

188

APPROXIMATION OF HIGHER DEGREE

6.3

DEGREE THREE AND H I G H E R DEGR?3E APPROXIMATION

A s t u d y of d e g r e e s of freedom shows t h a t w e req u i r e f o u r p o i n t s on a l i n e and seven p o i n t s on a c o n i c t o d e t e r m i n e a polynomial of d e g r e e t h r e e i n x and y . W e therefore r e q u i r e t w o side nodes on each l i n e a r s i d e and f i v e s i d e nodes on e a c h c o n i c s i d e t o achieve interelement c o n t i n u i t y with degree three v a r i a t i o n a l o n g boundary components. The v a l u e s € o r t ( s i , s i + l ) i n Eq.(6.12) i n c r e a s e f o r d e g r e e three approximation t o : t(1,l)= 4 , t ( 1 , 2 ) = t ( 2 , l )

= 8, t ( 2 , 2 ) = 13.(6.13)

I n each case there i s one fewer p o i n t t h a n t h e numb e r needed t o d e f i n e a unique a d j a c e n t c u r v e one o r d e r h i g h e r t h a n t h e d e g r e e two a d j a c e n t c u r v e . T h i s seems t o pose a s e r i o u s problem. T h i s problem p e r s i s t s when w e a t t e m p t t o d e t e r m i n e side node adj a c e n t f a c t o r s . On each l i n e a r s i d e , t h e a d j a c e n t c u r v e for one of t h e s i d e nodes must c o n t a i n o n l y t h e o t h e r s i d e node on t h a t l i n e a r s i d e . On each c o n i c s i d e , t h e a d j a c e n t c u r v e must c o n t a i n t h e o t h e r f o u r nodes on t h e c o n i c s i d e . There i s always one p o i n t less t h a n t h e number r e q u i r e d t o d e f i n e a unique c u r v e . Suppose t h i s problem w e r e r e s o l v e d somehow. Then each b a s i s f u n c t i o n Vi would be of t h e form i Vi(x,y) = F,(x,Y)/Q,,~ ( x , y ) , and f o r any c u b i c u there would be a P i such t h a t 3m i=l

1 where Pm v a n i s h e s on t h e polycon boundary c u r v e o f 189

APPROXIMATION OF HIGHER DEGREE

1 order m. This is possible without Pm being the zero 1 could be a constant times polynomial; in fact, Pm the polynomial whose curve is the polycon boundary. The difficulty anticipated at the end of Section 6.2 has materialized to yield a second dilemma. B o t h problems a r e r e s o l v e d b y i n t r o d u c t i o n of a node i n t e r i o r t o t h e p o l y c o n .

The basis function associated with the interior node is V3m+l(x,y) - k3m+lPm (x,Y)/Qm-3 (x,y), where Pm is the polynomial of degree m that vanishes on the polycon boundary curve. The adjacent curves of all other nodes are now required to contain the interior node. This additional condition leads to unique adjacent factors in all cases. The summation in Eq. ( 6 -14) is now over 3m + 1 nodes, and the resulting polynomial of degree m, 1 vanishes on boundary curve Pm and at the interPm, 1 is the zero ior node. This is possible only if Pm 1 vanishes on the boundary folpolynomial! That Pm lows from the cubic variation of the basis functions on the boundary components (on which they are nonzero) together with our choice of the proper number of nodes to determine unique cubics on each side. The points which determine the adjacent factors for degree three approximation are shown in Fig. 6 . 7 for vertices and in Fig. 6.8 for side nodes. We illustrate the role of the interior point in the uniqueness argument first encountered in Section 5.1 by proving that the adjacent factor of the basis function associated with vertex i in Fig. 6.7b is i2 uniquely determined by points 1-9. Let RS’ and R3 be two candidates. The sides adjacent to vertex i are (i;l) and S. We normalize Ri’ and Ri2 to the 190

RATIONAL FINITE ELEMENT BASIS

F i g . 6.7. Points t h a t determine adjacent f a c t o r for d e g r e e t h r e e a p p r o x i m a t i o n w i t h v e r t e x i a t t h e i n t e r s e c t i o n of ( a ) l i n e a r s i d e s , ( b ) a l i n e a r and a conic s i d e , and ( c ) two conic s i d e s .

R

i

Points t h a t determine a d j a c e n t f a c t o r Fig. 6.8. f o r d e g r e e three a p p r o x i m a t i o n a t s i d e n o d e i o n ( a ) a l i n e a r s i d e and ( b ) a conic s i d e .

Ri

191

APPROXIMATION OF HIGHER DEGREE

same value at node i and obtain by the usual argument: Ri2 = Ril - 0 mod (i;l)S. By Theorem 1.1, this is Iti2 = c(i;l)S for some conpossible only if Ril stant c. Both Ril and Ri2 vanish at interior node 9, and this node is contained in neither curve (i;l) nor S. Hence, c = 0 and we have established uniqueness of adjacent factor Ri This analysis generalizes to approximation of any degree. As the degree increases and the appropriate number of nodes is introduced on the boundary curve, the number of boundary conditions lacking for unique determination of adjacent factors of the required degrees is precisely equal to the number of interior nodes that must be introduced to yield a basis. Some aspects of this generalization that may not be clear are clarified by analysis of degree four approximation. Significant considerations will be discussed without detailed analysis. Each basis function has a numerator of degree m + 1 and is congruent to a polynomial of degree four on each of its adjacent sides. Let Cm be the boundary curve. Then for any u(x,y) of maximal degree four,

-

-

.

U(X,Y)

-

1

u.V. 1 1(x,Y) = Pm+,(x,y)/Qm_,(x,y)

all nodes vanishes on Cm, and there must be a linear L such 1 that (6-15) pm+l (X,Y) = L1 (X,Y)C,(X,Y)

-

If Pm+l is to be the zero polynomial, we must account for the three degrees of freedom in L1. This is done by introducing three noncollinear interior nodes as shown in Fig. 6.9. The basis functions associated with these interior nodes are given in Eq.(6.16). 192

RATIONAL FINITE ELEMENT BASIS

Fig.

v4m+l

6.9.

-

T h r e e i n t e r i o r nodes for d e g r e e

f o u r approximation.

k4m+l ( 4m+2 ,4m+31 Cm ( x ,Y 1 /Qm-

,

( x y 1 I ( 6 .16a1

The basis functions for the 4m boundary nodes all vanish on the interior nodes. Thus L1 in Eq.(6.15) vanishes at these nodes and must be the zero polynomial. In general, for approximation of degree k there are km boundary nodes and (k - 1) (k - 2)/2 interior nodes that do not lie on any curve of order less than k - 4. The wedge numerators are of degree m + k - 3. Application of higher degree approximation is limited. Improved accuracy is counterbalanced by growth in complexity. Low degree approximation within each element is more characteristic of finite element approximation. The role of interior nodes in finite element computation is discussed in greater depth in Chapter 9 . These nodes are effectively removed by "static condensation".

193

APPROXIMATION OF HIGHER DEGREE

6.4

INTERMEDIATE APPROXIMATION

Side nodes were chosen in Sections 6.2 and 6.3 to yield a prescribed degree approximation. We return now to the less formal data-fitting problem mentioned in Section 6.1 where nodes were determined from considerations other than degree of approximation. Rather than choosing products of degree one basis functions f o r interpolation as was done in Section 6.1, we may retain the adjoint polynomial as the only term in the denominator and construct appropriate numerators. These numerators are not always determined uniquely by the parameters, and the technique will now be illustrated with four examples. EXAMPLE 6.1.

Fig. 6.10.

In Fig. 6.10, (4;5) is parallel to

A f o u r node t r i a n g l e .

(1;3) and (4;6) is parallel to (1;2). We define W1 = k1(2;3), W2 = k2(1;3)(4;6) W3 = k3(1;2) (4;5), and W4 = k4(1;2)(1;3). This provides a basis for degree one approximation satisfying the conditions in Section 1.5 and allowing quadratic variation on side (2;3). EXAMPLE 6.2. For the element shown in Fig. 6.11, (4;6) is parallel to (1;2) and conic side (2;5;312 is a circle. Let (4;5;A)2 be the circle defined by points 4, 5, and A. A basis for degree one approximation with quadratic variation on side (1;3) is 194

RATIO NAL FINITE ELEMENT BASIS

W1 = k1(2;5;3)2(4;6)/(A;B), W3 = k3(1;2) (4;5;AI2/(A;B)

W 2 = k2(1;3) (5;B)/(A;B),

, W4

= k4 (1;2)(2;5;3I2/(A;B),

and W5 = k5 (1;2) (1;3)/(A;B).

Fig. 6.11.

A f i v e node 3-con o f o r d e r f o u r .

EXAMPLE 6.3. W e s e e k a q u a d r a t i c f a c t o r f o r t h e numerator of W3 associated w i t h node 3 i n F i g . 6.12 whose c u r v e w i l l c o n t a i n 4 , 5, 6, and A b u t w i l l i n t e r s e c t ( 1 ; 3 ) o n l y a t A. W e have i n e f f e c t a s i d e

\ F i g . 6.12.

A s i x n o d e 3-con of o r d e r f o u r .

node a t i n f i n i t y on ( 1 ; 3 ) t h a t w e d e n o t e a s 7. Then i f w e choose W3 = k3(1;2) (4;5;6;A;7)2/(A;B), w e have (4;5;6;A;7I2 : (4;5) (A;6) m o d (2;3)2 and (1;2) (A;6)

=

(A;B) (2;6) mod (2;312 so t h a t

(A;B) (2;6)(4;5)/(A;B) : (2;6)(4;s) m o d (2;312. F u n c t i o n s of t h i s t y p e y i e l d d e g r e e one approximat i o n w i t h q u a d r a t i c v a r i a t i o n on t h e c o n i c s i d e .

W3 Z

195

APPROXIMATION OF HIGHER DEGREE

EXAMPLE 6.4. For this last example (Fig. 6.131, the basis is not unique. This arbitrariness occurs whenever there is an even number of side nodes on a conic side. In this example, there are too many nodes for a linear fit and too few nodes for a quadratic fit on the conic side. To maintain continuity

Fig.

6.13.

Two s i d e n o d e s on a c o n i c s i d e .

across the conic side, w e must use consistent approximations within the contiguous elements. If we select W 3 = k3(1;2) (A;4) (4;5)/(A;B), then W3 (2;4)( 4 ; 5 ) mod (2;3)2. If we choose instead W3 = k3(1;2) (A;5) (4;5)/(A;B), then we find that W3 : (2;5) ( 4 ; 5 ) mod (2;312. In general, one must exercise some ingenuity in fitting specified data.

A side node is required on each curved side for degree one approximation over any well-set element. Let a curved side be deformed continuously into the line connecting its vertices. We examine the basis functions as the curve approaches the line, where in the limit the side node is superfluous. It is apparent from the property Wi(x ,y.) - 6ij that the 1 1 three basis functions on the side in question do not become linearly dependent when this side degenerates into the straight line. on the contrary, they form 196

RATIONAL FINITE ELEMENT BASIS

a basis for quadratic fitting on the linear side. Consider, for example, the element shown in Fig. 3 . 3 . As the conic side becomes linear, adjoint (A;B) moves toward infinity. Lines (4;A) and (4;B) approach the lines through 4 parallel to (1;2) and (1;3), respectively. In the limit we obtain the wedges for intermediate approximation with quadratic fitting on (2;3) and linear fitting on (1;2) and (1;3) shown in Fig. 6.10. In general, when a side degenerates to a curve on which the dimension of the space of polynomials of concern is less than the number of nodes on the side (including the vertices), the wedge functions form a basis for a class of polynomials including those of concern. Thus t h e e x t r a nodes i n t r o d u c e n e i t h e r degeneracy of b a s i s f u n c t i o n s nor c o m p u t a t i o n a l s e n s i t i v i t y .

6.5

HIGHER DEGREE APPROXIMATION ON POLYPOLS

The theory for higher degree approximation over polycons generalizes to polypols. A qualitative discussion given in this section is made more precise in Section 6.6. Each time we increase the degree of approximation by one, more side nodes are introduced. More interior nodes are also added for approximation higher than two. The nodes demanded for degree k approximation always determine unique basis functions. This is illustrated by analysis of a vertex at the intersection of sides of orders kl and k2. The adjacent factor is of degree kl + k2 2 for degree one approximation. A side of order s requires 2 s - 1 side nodes for degree two approximation. Thus for degree two approximation the vertex adjacent curve must contain

-

197

APPROXIMATION OF HIGHER DEGREE

+ k2

-

1) side nodes. The number of EIP and DIP conditions is not altered when we advance .from degree one to degree two approximation. We thus introduce kl + k2 additional side nodes to obtain as the total number of conditions on the adjacent curve : 2(kl

(kl

+

k2

-

2)(kl

+

k2

+

1)/2 + kl + k, =

This is the dimension of the space of curves of order kl + k2 - 1 and a curve of this maximal order can always be constructed to satisfy the conditions. It can be shown that there is only one such curve. Note that the order o f this adjacent factor has increased by one on passing from degree one to degree two approximation. Some of the points of intersection of the adjacent curve with the adjacent sides are adjunct points. For example, in Fig. 6.14 degree two approxCUBIC SIDE

T h e c u r v a t u r e of ( A ; B ; C ; D ; j ) 2 i n (b) i s g r o s s l y exagg e r a t e d t o d i s t i n g u i s h t h i s c o n i c from t h e c u b i c s i d e F i g . 6 . 1 4 . B d j o i n t p o i n t s and s i d e n o d e s ; ( a ) d e g r e e o n e , and (b) d e g r e e t w o .

imation on a cubic side requires four side nodes. These together with the two vertices on the cubic 198

RATIONAL FINITE ELEMENT BASIS

side account for the six degrees of freedom of a quadratic function on this side. The conic adjacent curve associated with vertex i intersects the cubic curve at six points, including the four side nodes and vertex j. The sixth intersection point is adjunct point E. Just as nodes j and A determine B for degree one approximation, points j, A, B, C, and D determine E for degree two approximation. 6.6

A CONCISE ALGEBRAIC GEOMETRY ANALYSIS

We have analyzed the construction of rational wedges for polypols, proceeding from degree one approximation over simple elements to arbitrarily high degree approximation over well-set polypols. While building up the theory gradually, we have attempted to describe the algebraic mechanism of wedge construction and to motivate successive stages of the development. A more concise treatment (Wachspress, 1974) based on the algebraic geometry foundations outlined in Chapter 4 will now be given. We confine this more sophisticated discussion to construction of rational basis functions for wellset rational polypols. Let boundary curve Cm have singular points (including neighbors) pi of multiplicities ri. The definition of well set assures r(v.) = 2 for the 3 n vertices (j = 1,2,. ..,n). Let V be the linear system of curves of maximal order m 3 having multiplicity ri-1 at each nonvertex pi. Each of the n irredpcible components of the rational polyp01 boundary curve is of genus zero. A lower bound on the dimension of V is given by Theorem 4 . 4 . On substi-

-

199

APPROXIMATION OF HIGHER DEGREE

tuting Theorem 4.12 into Theorem 4.4 with all the = 0, we find that dim V 2 - 0. Let Q1 and Q2 be gj any two elements in V. Then

(6.18)

By Theorem 4.13, there is a b such that Q1 - bQ2 = 0 on Cm. By Theorem 1.1, Q1 - bQ 2 must therefore contain each irreducible component of Cm. This is possible only if Q1 - bQ 2 is the zero polynomial. Thus the dimension of V is zero and the unique Qm-3 is the polypol adjoint which is the denominator for all basis functions, regardless of the degree approximation. In general, wedge numerators of degree m + k - 3 are constructed to achieve degree k approximation over a polypol of order m. For any u of maximal degree k, U(X,Y)

-

c

U.W. 1 1 (X,y) = Nm+k-3/Qm-3'

(6.19)

all nodes On each component of the polypol boundary each wedge is either zero or of degree k. Enough nodes are placed on each polypol side to assure Nm+k-3 = 0 on boundary Cm. For k < 3, this suffices to establish that Nm+k-3 is the zero polynomial. For k L - 3, there could be a Gk-3 such that Nm+k-3 (x,Y) = Gk-3(x,y)Cm(xiY)

-

Interior nodes are introduced to yield degree k approximation when k 2 3. Since any k(k - 3)/2 200

(6.20)

R A T I O N A L F I N I T E E L E M E N T BASIS

p o i n t s can be l o c a t e d on a c u r v e of maximal o r d e r k - 3, w e choose [ k ( k - 31/21 f 1 = ( k - 1)( k - 2 ) / 2 i n t e r i o r p o i n t s t h a t do n o t a l l l i e on any c u r v e of o r d e r z(k -

-

3).

The wedge a s s o c i a t e d w i t h each i n -

t e r i o r node j i s = k . C s -j 3 / Q m - 3 i (6.21) j 7 m i s t h e unique c u r v e c o n t a i n i n g a l l i n t e r -

W

where $-3

i o r nodes o t h e r t h a n j . v a n i s h e s on boundary Cm.

Each of t h e s e f u n c t i o n s The a d j a c e n t f a c t o r i n t h e

numerator of each wedge a s s o c i a t e d w i t h a boundary node i s c o n s t r u c t e d t o v a n i s h on a l l t h e i n t e r i o r = 0 on Cm and a l s o on a T h i s a s s u r e s Nm+k-3 s e t of i n t e r i o r nodes c o n t a i n e d i n no curve of max-

nodes.

i m a l o r d e r k - 3 . Thus Gk-3 i n Eq.(6.20) is t h e z e r o polynomial and d e g r e e k approximation i s e s t a b lished. W e now c o n s i d e r t h e boundary-node wedges i n det a i l . A s s o c i a t e d w i t h boundary node q i s wedge (6.22)

O p p o s i t e f a c t o r Fq v a n i s h e s on a l l s i d e s o p p o s i t e q and remains t h e same f o r any d e g r e e approximation. (Of c o u r s e , a d d i t i o n a l nodes a r e i n t r o d u c e d a s t h e d e g r e e of approximation i s i n c r e a s e d . )

We now con-

s i d e r a d j a c e n t f a c t o r Rq f o r q on s i d e Pi Let t' d ( t , k ) be t h e dimension of polynomials of d e g r e e k on a c u r v e of o r d e r t ( S e c t i o n 4.5). chosen

SO

S i d e nodes are

t h a t any c u r v e Hk determined by a l l b u t

one of t h e [ d ( t , k ) - 21 s i d e nodes and t h e 2 v e r t e x i nodes on P t does n o t c o n t a i n t h e d e l e t e d s i d e node. T h i s a s s u r e s unique d e t e r m i n a t i o n of a polynomial of d e g r e e k on Pi i n t e r m s of d ( t , k ) nodal v a l u e s . 20 1

APPROXIMATION OF HIGHER DEGREE

Let HZ denote the curve obtained by deleting side node q. Then O(HZ 0 Pt) = kt and all of these points (including neighbors and multiplicities) are uniquely determined by the d(t,k) - 1 nodes on Pi, excluding q. The adjacent factor Rq is constructed so that i By Theorem 4.13, this is true if and Wq 5 Hq mod P k only if Rq is the curve of maximal order t + k - 3 for which

.

*m-3~qk

...

0

= F:-~R~

: P

0

P;.

(6.23)

are the multiplicities of a l l the If ril# ri2 * i singular points pilt pi2, of P (including neighbors) and if vi-l and vi are the vertices of boundary component Pi , then by construction: Qm-3

...

Pi t = FZ+

+c 0

Pi t

-

{V~-~,V~}

rij (rij

-

(6.24)

l)Pij.

j

We obtain from (6.23) and (6.24) , the requirement that Rq 0 Pi rij(rij - l)pij. This is met if t 2 By Theorem 4.4, this imposes at (R 9 2 rij 1 . m P;; Ad most rij(rij - 1)/2 conditions on Rq. Since Pi is rational (and hence of genus zero), Theorem 4.11 asserts that this is at most (t - 1) (t - 2)/2 conditions. Another tk - 2 conditions are imposed to

zj

lj

(6.25)

Curve Rq must also contain the (k - 1) (k - 2)/2 interior nodes. Let V1 be the space of all curves of maximal order t + k - 3 which satisfies the conditions imposed on Rq. By Theorem 4.4,

202

RATIONAL FINITE ELEMENT BASIS

dim V1 2 - (t + k -(k

-

-

3 ) (t

1)(k

-

+ k)/2

2)/2

-

(tk

(t

-

-

2)

1)(t

-

2)/2 (6.26)

The right-hand side of (6.26) is identically zero. Thus there is at least one Rq in V 1 Uniqueness is easily demonstrated: For any candidate, Rq 0 p1 = Hq 0 Pi Fq 0 Pi. Let Rql and Rq2 %-3 k both be elements of space V1. Then Rql 0 Pi = Rq2 0 Pi and by Theorem 4.13 there is a b f o r which Rql - bRq2 = 0 on Pi. By Theorem 1.1, there is a P such that Rql - bRq2 = Pk-3Pt. Both Rql and 623 R vanish on the interior nodes which do not lie on Pi and have been chosen not to all lie on any curve of order less than k - 2. Hence, Pk-3 must be the zero polynomial, the dimension of V1 is zero, and Rq is a unique curve. This proof parallels an argument that recurs in one-variable Chebyshev minimax theory. We next examine the adjacent factor at vertex 1 and Hk2 be the unique curves of node q = vi' Let Hk order k through the nodes other than vi on boundary components Pi-' and Pi , respectively. Factor Rq t2 is chosen so that pi-l pi-l (6.27a) Fq m-t1-t Rq 0 t, - 'm-3 H1 k t, L L and (6.27b) Rq 0 Pi = Qm-3H2 0 Pi Fq m-tl-t2 t2 t2

.

-

.

Polynomial Rq is of maximal degree tl + t2 + k - 3 . Conditions similar to those described for side nodes are imposed. We have (tl - l)(tl - 2)/2 + (t2 - 1) (t2 - 2)/2 conditions from the singular

203

APPROXIMATION OF HIGHER DEGREE

points on the adjacent sides. From the elements in we obtain another k(tl f t2) - 2 conditions. The interior nodes must lie on curve Rq, and this gives another (k - l)(k - 2)/2 conditions. Finally, there are another tlt2 - 1 conditions, which have no counterpart in the side node analysis, imposed by the i-1 divisor Pt 0 Pi - {vi). L e t V 2 be the space of curves of 'maximai order tl + t 2 + k - 3 that satisfy all these conditions. Then dim V 2 2 - (tl + t2

-

+

k

-

3 ) (tl + t2

+ k)/2

(tl - 1)(tl - 2)/2

-

(k

[k(tl

-

1) (k

-

2)/2

(tlt2 - 1).

-

(t2 - 1) (t,

-

2)/2

+ t2) - 21 (6.28)

The right-hand side of (6.28) is identically zero. Thus V2 contains at least one curve. The usual argument establishes uniqueness: dim V2 = 0. The conditions imposed on the divisors of Rq on sides adjacent to node q ensure degree k variation of the basis function W on these sides. The number and 9 placement of side nodes guarantees continuity of the patchwork approximation across the polyp01 boundary components. This concludes our concise algebraic geometry repetition of the analysis. The development in the first six chapters can be contracted into a few pages of compact analysis, drawing on algebraic geometry foundations.

204

RATIO NAL FINITE ELEMENT BASIS

6.7

ALGEBRAIC RETICULATION

The term triangulation or generalized triangulation is commonly used to denote partitioning of a region for finite element computation. Generalization from the simple three- and four-sided elements to the more varied algebraic elements leads to networks for which the term reticulation seems more appropriate. We therefore refer to any covering of a planar region by nonoverlapping algebraic elements as an algebraic reticulation. Triangulation is a simple form of algebraic reticulation. The unique role of well-set rational algebraic elements in finite element theory is emphasized by referring to a partition containing only such elements as a regular algebraic reticulation. The word "reticulation" brings to mind precisely the kind of network made feasible for finite element computation by the introduction of rational basis functions. It also suggests new areas for application. Whether or not finite element computation with algebraic reticulation will provide a tool for analysis of natural phenomena involving reticulated structures is a fascinating question that suggests paths for research. In any event, we have developed a theory for patchwork interpolation over algebraically reticulated regions.

205

Chapter 7

THREE-DIMENSIONAL APPROXIMATION

7.1

DEFINITIONS AND BACKGROUND A polypoldron (p-p) is a three-dimensional alge-

braic element defined by a set of boundary faces enclosing a simply connected region. The faces intersect on boundary edges that are segments of space curves. These edges meet at the vertices of the element. A p-p is simple when Euler's equation is satisfied: vertices (V)

-

edges (E)

+

faces (F) = 2 .

(7.1)

Face j is a section of the surface on which Pi (x,y,z) = 0 . The subscript t is the order of j 1 the surface and also the degree of Pj (x,y,z) The order of a p-p is F m =

C

tj.

.

(7.2)

j=1

When all the boundary faces are planes, the p-p is a polyhedron. The order of a polyhedron is equal to F. PJ be the boundary surface of a p-p. Let

TTj=1

The p-p is defined as well set if it is simple and 206

RATIO NAL

FINITE ELEMENT BASIS

(1) each vertex is an ordinary triple point of Pm, ( 2 ) excluding vertices, each edge contains only double points of Pm, ( 3 ) excluding edges, each face contains only simple points of Pm, and ( 4 ) the p-p interior contains no point of Pm. Conditions ( 2 ) - { 4 ) are obvious generalizations of the properties of well-set two-dimensional elements. The significance of the first condition will now be explained. The order of a vertex is equal to the number of edges (or faces) that meet at the vertex. Referring to Fig. 7.1, we observe that for any vertex of order greater than three at least one ex-

Fig.

7.1.

A p-p

v e r t e x of o r d e r f o u r .

terior edge of the p-p passes through the vertex: P1 = (1;2;3), P2 = (1;4;5), P = (1;3;4), P4 = (1;2;5) P1.P2 is line (1;6) and this line contains vertex 1. If a p-p is to be well set, all intersections of p-p surfaces other than those that define edges of the p-p must be exterior to the element. For this reason we must exclude vertices of order greater than three. To attain degree k approximation over a Well-Set p-p, we introduce nodes q and associated rational 207

THREE-DIMENSIONAL APPROXIMATION

wedges o f t h e form

where t h e f a c t o r s a r e analogous t o t h o s e of t h e two-dimensional wedges. A p r e c i s e r e c i p e f o r wedge c o n s t r u c t i o n w i l l e v e n t u a l l y be g i v e n . B y way of i n t r o d u c t i o n , w e f i r s t d e s c r i b e wedges f o r a few l o w o r d e r e l e m e n t s h a v i n g o n l y p l a n a r and q u a d r i c f a c e s . W e assume t h a t t h e boundary s u r f a c e s have t h e i r f u l l complement of d i s t i n c t t r i p l e p o i n t s . Higher o r d e r and n o n o r d i n a r y s i n g u l a r i t i e s w i l l be c o n s i d e r e d l a t e r . L e t m be t h e o r d e r of t h e p-p. The a d j o i n t i s t h e unique s u r f a c e of maximal o r d e r m-4 t h a t c o n t a i n s a l l t h e e x t e r i o r t r i p l e p o i n t s and c e r t a i n d e f i c i t p o i n t s y e t t o be d e f i n e d . The opposit;e f a c t o r v a n i s h e s on a l l s u r f a c e s oppos i t e node q . If s u r f a c e s of orders kl, k2, and k3 m e e t a t q, t h e n deg Fq = m kl k2 - k3. I f s u r f a c e s of o r d e r s kl and k2 m e e t a t edge node q , t h e n kl k2. I f q i s a node on a f a c e of deg Fq = m order k , t h e n deg Fq = m k. I f q i s an i n t e r i o r node, t h e n Fq = Pm. The a d j a c e n t f a c t o r Rq i s of d e g r e e such t h a t t h e wedge numerator i s of d e g r e e m - 4 k f o r d e g r e e k approximation. Some of t h e a l g e b r a i c geometry theorems r e l a t i n g t o d i v i s o r s g e n e r a l i z e r e a d i l y from two- t o t h r e e = d i m e n s i o n a l space. The b i n o m i a l c o e f f i c i e n t ):( C i s d e f i n e d t o b e z e r o when n < r . The i d e n t i t y n r

-

-

-

-

-

-

(7.4) nc r = n - I C r + n-1 Cr-1 i s q u i t e u s e f u l . L e t Vt be t h e s p a c e of polynomials i n t h r e e a f f i n e ( f o u r homogeneous) v a r i a b l e s of maxi m a l d e g r e e t . Then 208

RATIONAL FINITE ELEMENT BASIS

dim Vt

-

t+3C3

-

(7.5)

Let Ps be the surface of order s on which Ps(xly,z) = 0. The dimension of Vt on Ps is dim Vt mod Ps = t+3C3 - t-s+3C3' (7.6) Let Ps-Qr be the edge of intersection of surfaces Ps and Qr of orders s and r, respectively. Then dim Vt mod Ps-Q, = t+3C3

When t 2 s

+ r

-

-

t-s+3C3 ' C

t-r+3'3

t-r-s+3 3'

can be written as

3, Eq.(7.7)

dim Vt mod Ps-Qr = sr[t

+

(7 7)

+

2

-

<s+r)/2].

(7 8)

The dimension of the space of surfaces of maximal order t over any variety is one less than dim Vt over that variety. (A variety is a set of points, curves, or surfaces.) There are subtleties that must be taken into consideration. For example, let s = r = t = 2 in Eq.(7.8) to obtain dim V2 mod P2.Qz = 8. There are eight points in the intersection of three relatively prime quadrics. This seems to apply the absurd result that any two quadrics are congruent on any third relatively prime quadric surface. This anomaly is resolved (Coxeter, 1961, p.259) when we note that seven points of general position determine a unique eighth point such that every quadric through the seven passes also through the eighth point. Only seven of the eight points common to three relatively prime quadrics are "quadric-independent". To be congruent on a quadric surface, two polynomials of degree two must vanish on quadric surfaces that

209

THREE-DIMENSIONAL APPROXIMATION

i n t e r s e c t t h e q u a d r i c on which t h e y are congruent a t a n i n t h ( e i g h t h independent) p o i n t . They must t h e n 1 2 have a common edge of i n t e r s e c t i o n : P2*Q2 = P2-Q2 f o r q u a d r i c s P1 and P2 t o be congruent mod Q. A f e w of t h e a l g e b r a i c geometry theorems i n threespace t h a t w e s h a l l have o c c a s i o n t o use a r e : (Bezout). If s u r f a c e s F, G, and H have no common component o r s p a c e c u r v e , t h e n O ( F r 0 Gs 0 H t ) = rst. THEOREM 7 . 1

L e t n e i t h e r P n o r R have i r r e d u c -

THEOREM 7.2.

i b l e s u r f a c e Q a s a component.

I f and only i f

P 0 Q = R 0 Q, t h e r e i s a b f o r which P everywhere on s u r f a c e Q.

bR = 0

I f F, P, and R s h a r e no common

THEOREM 7.3.

component and i f F i s a b such t h a t F

0

-

P 0 R = G 0 P 0 R, t h e n t h e r e bG = 0 on s p a c e curve P9R.

L e t V be t h e l i n e a r series o f

THEOREM 7.4.

algebraic s u r f a c e s of o r d e r s

c i t i e s ri a t p o i n t s pi. d i m V - m-1

-

3

-

L(m -

4 ) with m u l t i p l i -

Then v

-

i

ri+Zc3

The genus of an i r r e d u c i b l e s u r f a c e i s less w e l l d e f i n e d than t h a t of an i r r e d u c i b l e curve. D i f f e r e n t genera have been described i n t h e l i t e r a t u r e , and w e s h a l l u s e one of these throughout. L e t

si = m j

pi

(pi)

.

(7.9)

J

Then t h e genus of i r r e d u c i b l e s u r f a c e Pi i s d e f i n e d as t h e nonnegative number

210

RATIONAL FINITE ELEMENT BASIS

where the summation is over all points, including neighbors. The concept of neighbors will not be developed here. To simplify the analysis, we will assume throughout that p-p surfaces intersect at only ordinary multiple points. Our objective in this chapter is to generalize to three dimensions the basis function construction for two-dimensional algebraic elements. We start with degree one approximation over a tetrahedron with one truncated corner. This is a triangular prism with end planes that need not be parallel. We then consider the hexahedron, this being the three dimensional generalization of the quadrilateral. Higher degree approximation will be examined for the hexahedron. A polyhedron is well set if it is convex and its vertices are all of order three. Wedge construction will be described and analyzed for the general well-set polyhedron. Nonplanar faces result in considerable complexity. For example, quadric surfaces may intersect in fourth-degree space curves that usually do not lie in a plane. A byproduct of generalization to nonplanar faces is a theory for construction of rational wedge bases for approximation over nonplanar surfaces. Construction of basis functions for well-set p-p with surfaces of arbitrary orders is quite intricate. In the last section of this chapter we describe some of the salient aspects of this construction.

211

THREE-DIMENSIONAL APPROXIMATION

7.2

TRIANGULAR PRISMS AND HEXAHEDRA

The tetrahedron is the generalization of the triangle to three dimensions. A tetrahedron has four triangular faces and no exterior triple points. For degree one approximation, the wedge associated with vertex i opposite face Fi is the linear function i i (7.11) Wi(x,y,Z) = F (x,ytz)/F (XitYitzi). The rectangle generalizes to the rectangular parallelepiped with six planar faces, no exterior triple points in finite space, and having trilinear wedges. These elements are well known and have been used extensively in finite element computation. Isoparametric coordinates have been used to extend these elements by allowing curved (isoparametric) surfaces. A simple element for which rational basis functions seem well suited is a tetrahedron with one of its corners truncated as shown in Fig. 7 . 2 . There

Fig.

7.2.

A

triangular prism.

are two faces opposite each vertex, and the product of the linear forms that vanish on these faces is the bilinear wedge numerator associated with the vertex. The element has m = 5 faces and our construction recipe calls for an adjoint of degree not greater than m - 4 = 1. We seek a plane that passes 212

RATIONAL F INlTE ELEMENT BASIS

through a l l t h e e x t e r i o r t r i p l e p o i n t s . The f i v e p l a n a r f a c e s i n t e r s e c t i n 5C3 = 1 0 t r i p l e p o i n t s , s i x of which are v e r t i c e s of t h e element. There a r e f o u r e x t e r i o r t r i p l e p o i n t s ( E T P ) . A p l a n e is determined by any t h r e e n o n c o l l i n e a r p o i n t s . I t a p p e a r s a t f i r s t g l a n c e t h a t t h e c o n s t r u c t i o n b r e a k s down. W e r e g a i n a d e g r e e of freedom, however, through a p p l i c theorem ( F a u l k n e r , 1 9 6 0 , p . l O ) , a t i o n of Desargues' a theorem as i m p o r t a n t i n t h e f o u n d a t i o n s of p r o j e c t i v e geometry as t h e p a r a l l e l p o s t u l a t e i n t h e found a t i o n s of E u c l i d e a n geometry. (Desargues' t h e o r e m ) . I f t w o t r i a n g l e s ( F i g . 7.3) ABC and A ' B ' C ' are such t h a t ( A ; A ' ) , ( B ; B ' ) , and ( C ; C ' ) m e e t i n a p o i n t 0 , t h e n ( B ; C ) meets ( B ' ; C ' ) i n L, ( C ; A ) meets ( C ' ; A ' ) i n M I and (A;B) meets (A';B') i n N where L, M , and N are collinear. THEOREM 7.5

Fig.

7.3.

Desargues'

Proof. For o u r p u r p o s e s , ABC d i f f e r e n t p l a n e s and t h e proof is ( B ; B ' ) and ( C ; C ' ) i n t e r s e c t a t 0 , and C ' l i e on a p l a n e (one of t h e o u r prism) and (B;C) meets ( B ' ; C ' )

213

theorem.

and A ' B ' C ' are i n simplified. Since p o i n t s B, B ' , C , l a t e r a l faces of i n a p o i n t , L.

THREE-DIMENSIONAL APPROXIMATION

Similarly, (C;A) meets (C';A') in M and (A;B) meets (A';B') in N. The three points L, M, and N lie on the line of intersection of planes (A;B;C) and (A';B';C'). The proof is much more subtle when Fig. 7.3 is treated as a planar figure. The ETP of our prism are 0, L, M, and N. If any two of L, M, and N are in a particular plane, then the line containing all three of these points is in that plane. Thus the unique plane through 0 and any two of L, M, and N contains all four ETP. This is the adjoint of the prism. I t w i l l e v e n t u a l l y be d e m o n s t r a t e d t h a t D e s a r ques'

theorem g e n e r a l i z e s t o y i e l d j u s t enough i n -

t e r r e l a t i o n s h i p s among ETP of p - p

to validate the

wedge c o n s t r u c t i o n r e c i p e f o r t h e general w e l l - s e t pol ypol dron

.

The wedge construction recipe in Section 7.1 yields for the prism in Fig. 7 . 3 wedges of the form WA = kA(B;B';C) (A';B';C')/(O;M;N) Although the wedges are uniquely defined, any three noncollinear points in a plane may be used to indicate the factor that vanishes on the plane. Thus (B;B';C) = (0;B;C). Analysis is the same for all six wedges so that we need only verify that WA has the desired properties. We observe that WA vanishes on all faces opposite A, is normalized to unity at A, and is regular over the prism. Let (B;C;X) denote any plane that intersects (A;B;C) on line (B;C). Then

.

-

(A;B;C) (0;B;C)(A';B';C') = { (B;C),(M;N)1 , and

(A;B;C)* (0;M;N)(B;C;X) = { (B;C),(M;N)1 .

214

RATIONAL FINITE ELEMENT BASIS

By Theorem 7.2, ( 0 ; B ; C ) ( A ' ;B' ; C ' )

= =

(0;M;N)

(B;C;X)

(M;N) ( B ; C )

mod (A;B;C) mod ( A ; B ; C ) .

Moreover, Thus WA E (B;C) mod ( A ; B ; C ) . (A;A' ; C ) (0;M;N) = (0;M) and (A;A' ; C ) ( 0 ; B ; C ) (A' ; B ' ; C') = {(C;C'),(A';C')}. Hence,

-

-

WA E ( C ; C ' ) (A' ; C ' ) / ( O ; M )

mod (A;A' ;C).

.

Similarly, WA

(B;B') (A';B')/(O;N)

mod ( A ; B ; B ' ) .

W e have proved t h a t WA reduces on each f a c e a d j a c e n t

t o v e r t e x A t o t h e a p p r o p r i a t e two-variable wedge f o r degree one approximation on t h a t f a c e . A hexahedron h a s s i x f a c e s , each of which i s a

q u a d r i l a t e r a l . A convex hexahedron i s w e l l set. Hexahedron wedges w e r e first d e s c r i b e d by Wait ( 1 9 7 1 ) . The a n a l y s i s can be performed i n a geometric s e t t i n g as w a s done by W a i t , b u t w e s h a l l t a k e a more algebr a i c viewpoint t h a t s u g g e s t s g e n e r a l i z a t i o n s . The s i x p l a n a r f a c e s o f a given hexahedron i n t e r s e c t i n 6C3 = 20 t r i p l e p o i n t s (assumed d i s t i n c t f o r t h e p r e s e n t ) of which e i g h t are vertices. There are 1 2 ETP. The a d j o i n t s u r f a c e i s of order m 4 = 2 and has 5C3 1 = 9 d e g r e e s of freedom. To prove t h a t a q u a d r i c having o n l y 9 d e g r e e s of freedom can be c o n s t r u c t e d t o c o n t a i n a l l 1 2 ETP, w e must examine t h e hexahedron i n g r e a t e r depth. T h e s i x faces i n t e r s e c t i n 6C2 = 1 5 l i n e s , of which 1 2 are edges of t h e hexahedron and 3 are ext e r i o r edges. If t h r e e c o l l i n e a r p o i n t s a r e on a quadric, then the l i n e containing t h e s e p o i n t s i s

-

-

215

THR E E 4 IMENSIONAL APPROXIMATION

in the quadric. This is amplified in Section 7.5. Each of the three exterior edges contains the four ETP at which the other four planes meet this edge of intersection of two planes of nonadjacent faces. Demanding that any three of these four ETP lie on a quadric suffices to place the fourth ETP on that quadric. We discard one ETP on each exterior edge and fit the quadric adjoint surface to the remaining nine ETP. The seeds of a general Desarguean-type theorem are contained in this discussion, and we shall return to this later. Let Pi be the product of the linear forms that 3 vanish on the faces opposite vertex i. Then Wi(x,y,z) = kiP3(X,Y,z)/Q2(x'Y,z) i (7.12) is the wedge associated with vertex i constructed according to the recipe in Section 7.1. We will now show that each wedge reduces to the appropriate quadrilateral wedge on each face of the hexahedron. If P 5 Q mod R, we say that the "projection of P on R is Q". In Fig. 7 . 4 , the hexahedron faces are

Fig. 7 . 4 .

A hexahedron.

216

RATIONAL FINITE ELEMENT BASIS

F1 = (1;2;3;4), F2 = (5;6;7;8), F3 = (1;2;6;5), F4 = (3;4;7;8), F5 = (2;3;6;7), and F6 = (1;4;8;5), where any three of the indicated vertices determine the plane. The analysis is the same €or all eight wedges, and we consider only W3: 2 3 6

W 3 = k3F F F /Q2.

(7.13)

1 1 We observe that F6 : (1;4) mod F , F3 E (1;2) mod F , F2 s (F1 .F 2 ) mod F1 , and that since Q, is a quadratic function that vanishes on line F1-F2: Q, E L1(F 1 * F2 ) mod F 1 for some linear function, L1. We note that Q2(A) = Q2(B) = 0. Thus L1 = (A;B) and Q2

and

E

(A;B) (F1 - F2) mod F1,

W3 E (1;4) (1;2)(F1 - F2 )/(A;B) (F1 - F2 ) mod F1 E (1;4) (1;2)/(A;B) mod F A .

(7.14)

(7.15)

Referring to Eq.(2.3), we see that this is the two1 dimensional wedge for vertex 3 on quadrilateral F The rational approximation over the hexahedron thus projects on each of its faces to the two-dimensional rational approximation in terms of the vertex values for each quadrilateral face. Continuity is thus assured across faces in the composite approximation. The standard argument establishes that degree one approximation is achieved over the hexahedron. For any linear u, u(x,y,z) uiwi(x,y,z) = Pm,,(x,y,z)/Q2(x,y,z) vanishes on the boundary surface of order m. Thus Pm-3 is the zero polynomial. (Theorem 7.1 leads to the generalization of Theorem 1.1 that enables us to arrive at this result.)

.

1

217

THREE-DIMENSIONAL APPROXIMATION

Higher d e g r e e approximation may be a c h i e v e d by the technique described f o r t w o space v a r i a b l e s i n C h a p t e r 6. For d e g r e e t w o approximation w e i n t r o d u c e a n edge node on e a c h edge. T h e t h r e e edge nodes on t h e e d g e s t h a t m e e t a t v e r t e x i d e t e r m i n e t h e unique p l a n e Ri of t h e a d j a c e n t f a c t o r a t i , and t h e d e g r e e two wedge i s 2 ( 21 (7.16) Wi(x,y,z) = ki 3 1 2' For d e g r e e t h r e e approximation, w e r e q u i r e t w o edge nodes on each edge and one f a c e node on each f a c e . For v e r t e x i, t h e a d j a c e n t c u r v e must c o n t a i n t h e two edge nodes on each a d j a c e n t edge and t h e f a c e node on each a d j a c e n t f a c e . T h i s i s a t o t a l of n i n e p o i n t s and s u f f i c e s t o d e t e r m i n e a unique q u a d r i c : i 5C3 - 1 = 9. L e t t h i s q u a d r i c b e denoted by S2. Then (7.17)

i s t h e d e g r e e t h r e e wedge a s s o c i a t e d w i t h v e r t e x i. The a d j a c e n t l i n e a r f a c t o r f o r e a c h d e g r e e t h r e e edge wedge i s u n i q u e l y determined by t h e a d j a c e n t edge node and t h e f a c e nodes on t h e two a d j a c e n t f a c e s . The a d j a c e n t f a c t o r s f o r d e g r e e two edge and d e g r e e t h r e e f a c e nodes a r e a l l u n i t y . C o n t i n u i t y across element faces and a t t a i n m e n t of t h e d e s i r e d d e g r e e approximation i s proved by t h e u s u a l argument. W e n o t e t h a t f o r any c u b i c u , t h e r e i s a P5 f o r which

i v a n i s h e s on t h e boundary of o r d e r s i x . Thus P5 must b e t h e z e r o polynomial. For d e g r e e t h r e e approximat i o n o v e r t h e hexahedron, unique a d j a c e n t f a c t o r s 218

RATIONAL FINITE ELEMENT BASIS

w e r e determined w i t h o u t nodes i n t e r i o r t o t h e element. T h i s i s c o n s i s t e n t w i t h t h e l a c k of a need f o r i n t e r i o r nodes i n e s t a b l i s h i n g a t t a i n m e n t of degree three approximation. Each face does, o f c o u r s e , have i t s node so t h a t t h e two-dimensional p r o j e c t i o n s o f t h e wedges on t o t h e faces a c h i e v e d e g r e e three approxi m a t i o n on each f a c e . When w e advance t o degree f o u r , w e place three edge nodes on each edge and three n o n c o l l i n e a r nodes on each f a c e . The a d j a c e n t c u r v e f o r each vertex wedge must now be a aubic t h a t p a s s e s through t h e 1 8 nodes on a d j a c e n t edges and faces. There are 1 = 1 9 d e g r e e s of freedom i n a cubic. Intro6'3 d u c t i o n of one i n t e r i o r node y i e l d s a unique v e r t e x a d j a c e n t curve of order t h r e e . I t can be shown t h a t a l l a d j a c e n t c u r v e s are uniquely determined. L e t T6 denote t h e p r o d u c t of the l i n e a r forms Fi t h a t vani s h on t h e hexahedron boundary. Then t h e wedge a s s o c i a t e d w i t h i n t e r i o r node j i s

-

(7.19)

For any q u a r t i c u, there is a P 6 such t h a t

i

v a n i s h e s on t h e s i x f a c e s and a t t h e i n t e r i o r node. The i n t e r i o r node is needed t o establish degree f o u r approximation. J u s t as i n t h e two-variable a n a l y s i s , t h e number of nodes r e q u i r e d t o a s s u r e d e g r e e k approximation i s always c o n s i s t e n t w i t h t h e number o f nodes needed f o r a unique d e f i n i t i o n of a d j a c e n t curves. 219

THREE-DIMENSIONAL APPROXIMATION

To illustrate the generality of the construction, we consider the adjacent surface for a vertex basis function when degree k approximation is sought over a convex polyhedron. There are k 1 edge nodes on each of the three adjacent edges and (k - 1) (k 2)/2 face nodes on each of the three adjacent faces. To establish the number of interior nodes, we note first that by Theorem 7 . 4 , the dimension of the space of surfaces of order s is

-

dim Vs =

s+3C3

-

-

(7.21)

1.

We introduce enough interior points so that they can all be contained in no surface of order less than k - 3. If p is the number of interior points, we require P = k-lC3 (7.22) points that do not all lie on any surface of order less than k - 3. Thus the total number of points that should determine the adjacent surface is tk = 3(k - 1) (edge)

+

3(k

-

1)(k (face)

-

2)/2

+

(7.23) (interior) k-1C3.

The right-hand side of (7.23) simplifies to k+2C3 - 1 and, referring to Eq.(7.21), we obtain tk = dim Vk-l.

(7.24)

Hence, there is at least one surface of maximal order k - 1 that contains the tk points. It will be shown in Section 7.6 that there is only one surface. A generalization to n variables is suggested. For degree k approximation in one variable, we require k - 1 nodes between vertices. In two variables, we require k-1 C2 interior points in each well-set 220

R A T I O N A L FINITE ELEMENT BASIS

polypol. I n t h r e e v a r i a b l e s , we r e q u i r e C interk-1 3 I f w e can d e f i n e a i o r nodes i n each well-set p-p. w e l l - s e t element i n n v a r i a b l e s , w e s u s p e c t t h a t C interior d e g r e e k approximation w i l l r e q u i r e k-1 n nodes. 7.3

POLYHEDRA

The wedge a s s o c i a t e d w i t h v e r t e x i f o r d e g r e e one approximation o v e r a well-set polyhedron of o r der m is (7.25)

O p p o s i t e f a c t o r Pi i s t h e p r o d u c t of t h e l i n e a r forms t h a t v a n i s h on t h e f a c e s o p p o s i t e node i. I t w i l l now b e shown t h a t a d j o i n t Qm-4 is u n i q u e l y determined by t h e ETP. W e f i r s t n o t e t h a t a l l t h e vertices of a well-set polyhedron are of o r d e r t h r e e and t h a t t h i s t o g e t h e r w i t h Eq.(7.1) y i e l d s

-

V = 2(m

2) and

E = 3(m

-

(7.26)

2).

Hence, t h e number of ETP i s P = mC3

-

v

= m(m

-

l)(m

-

-

2)/6

2(m

-

2). (7.27) The e x t e r i o r edge formed by any t w o n o n a d j a c e n t planes pierces t h e other planes a t m 2 triple p o i n t s . S i n c e t h e a d j o i n t is of o r d e r m - 4 , w e need

-

-

only place m 3 of t h e s e p o i n t s i n Q,-4 t o assure t h a t t h e e x t e r i o r edge w i l l l i e i n t h e a d j o i n t . ( D e s a r g u e s ' theorem i s a s p e c i a l case of t h i s r e s u l t . ) The number of exterior edges is e q u a l t o q = m(m

-

1)/2

-

3(m

-

21,

and t h e number of ETP t o be p l a c e d i n a d j o i n t

221

(7.28)

s-4

THREE-DIMENSIONAL APPROXIMATION

is, therefore, t = p

-

-

q = m(m l)(m m(m - 1)

-

-

2)/6 - 2(m 3(m - 2).

+

-

2)

It is easily shown that this reduces to t = m-1 c 3

-

(7.29)

1.

Referring to Theorem 7.4, we see that there is at 4 that conleast one surface of maximal order m tains these points. It will be shown in Section 7.5 by arguments similar to those applied in Chapter 5 that the adjoint constructed from these points is unique. Positivity of the numerators in Eq.(7.25) establishes Qm-4 # 0 over the polyhedron. To verify that the wedges in (7.25) have the desired properties, we will first show that these wedges project on each face into the appropriate two-variable wedges. Let F = 0 on a polyhedron face of concern. We seek common factors of Pm-3 mod F and Qm-4 mod F. Suppose F has p edges. Then the m - (p + 1) exteri ior edges on surface F are common factors of Pm-3 and Qmm4 mod F for all nodes i on face F. For each i, cancellation of these common factors leaves a (m - p - 1) = p 2 numerator of degree (m - 3) that vanishes on the p - 2 edges of face F opposite vertex i. This is congruent mod F to the numerator of the two-variable wedge associated with vertex i for the polygonal element. The adjoint is reduced 3 that vanishes at the to a polynomial of degree p points where the exterior edges generated by the p planes defining the boundary of polygon F (Fig. 7.5) pierce plane F. These are just the E I P of the edges

-

-

-

222

-

RATI ONAL F I N I T E ELEMENT BASIS

-

-

(2;3;X) (4;S;X) mod F. 6 7 F.F and F.F are exterior edges.

A = (2;3) (4;5) E

7.5. P r o j e c t i o n of a t h r e e - d i m e n s i o n a l on a p e n t a g o n a l f a c e of a p o l y h e d r o n .

Fig. wedge

of F (the boundary of the polygon) known to determine a unique adjoint curve of order p 3 for the polygon. Thus the projection of Qm-4 onto F after cancellation of common factors in wedge numerators and denominator is the polygon adjoint. Continuity of the patchwork approximation is assured across each of the polyhedron faces. The rational approximation on each face is the two-dimensional interpolation of values at nodes on the face. Proof that degree one approximation is achieved and generalization to degree k approximation introduces no new concepts.

-

7.4

POLYCONDRA

We define a polyquadron as a well-set p-p whose faces are planar and quadric with at least one quadric face. A polyquadron need not have any planar face. Two quadrics intersect in a quartic space curve. We consider in this section the restricted 223

THREE-DIMENSIONAL APPROXIMATION

class of polyquadra t h a t have no q u a r t i c e d g e s , and w e c a l l t h e s e e l e m e n t s polycondra. A polycondron may have e x t e r i o r q u a r t i c edges. Each edge o f t h e element i s on a t l e a s t one of t h e p l a n a r f a c e s . L e t t h e orders of f a c e s a d j a c e n t t o node i b e kl, k 2 , and k3. For an edge node, k3 does n o t app e a r ; o n l y t w o f a c e s m e e t a t an edge node. W e w i l l v e r i f y t h e wedge c a n d i d a t e Wi = k Pi i (7.30) i m-kl-k2-k Rk +k +k -3 /Qm-4' 3 1 2 3 Suppose t h e polycondron h a s s p l a n a r and r q u a d r i c f a c e s . I t i s of o r d e r m = 2 r + s and h a s V = 2 ( r + s - 2 ) v e r t i c e s and E = 3 ( r + s 2 ) edges [by E q . ( 7 . 2 6 ) 1 . The number of boundary t r i p l e points i s

-

g =

sc3 +

2r

c + 4s

s 2

rC2

+ 8

rC3.

(7.31)

The t e r m s i n g are, r e s p e c t i v e l y , t h e i n t e r s e c t i o n s of t h r e e p l a n e s , of t w o p l a n e s and a q u a d r i c , of t w o q u a d r i c s and a p l a n e , and of three q u a d r i c s . The number of e x t e r i o r l i n e a r edges i s eL - sC2 - t h e number of polycondron l i n e a r edges (7.32) and t h e number of e x t e r i o r c o n i c edges i s

eC = rs

-

t h e number of polycondron c o n i c edges (7.33)

so t h a t w e have eL

+

eC = sC2

+

rs

-

3(r

+ s

-

2).

(7.34)

Each e x t e r i o r l i n e a r edge p i e r c e s t h e o t h e r polycondron faces a t s - 2 + 2 r t r i p l e p o i n t s . We n o t e t h a t Qm-4 has m - 3 = s + 2 r 3 d e g r e e s of

-

224

RATIONAL FINITE ELEMENT BASIS

freedom on any line. Thus if Qm-4 = 0 at any s + 2r 3 of the triple points on an exterior linear edge, then surface Qm-4 contains the entire line. We may discard one triple point on each linear exterior edge as a condition on Qm-4. Each exterior conic edge contains 2(s - 1) + 4(r - 1) = 2(s + 2r - 4) + 2 = 2(m 3) ETP. We set s = 1, r = 2 and t = m - 4 in Eq.(7.8) to get 3) 1. dim Vm-4 mod P1-Q2 = 2(m - 2 - 3/2) = 2(m This is the dimension of the space of polynomials of 4 on a conic. Thus if Qm,q = 0 maximal degfree m 3) 1 of the 2(m 3 ) ETP on an at any 2(m exterior conic edge, then the edge must be contained in surface Qm-4 * We may discard one ETP on each exterior conic edge as a condition on Qm-4 The polycondron vertices are included in g of Eq.(7.31) and must also be excluded as conditions on Qm-4. The remaining number of conditions is

-

-

-

-

-

-

-

t = g - eL - eC - V = C + 2r sC2 + 4s rC2 s 3

-

sC2

-

-

rs + (r +

s

-

+

8 rC3

(7.35)

2).

The number of degrees of freedom in adjoint surface Qm-4 is equal to m-1C3 - 1 = 2r+s-1C3 1. By expanding all the binomial coefficients and collecting terms, we discover after some tedious algebraic manipulation that

-

t = 2r+s-1c3 - 1 (7.36) as asserted in Section 7.1! (We have established the existence of a surface of order not greater than 4 that contains all the ETP. Uniqueness is m

-

225

THR EE--DIMENSIONAL APPROXIMATION

proved in a more general setting in Section 7.5.) We now consider the adjacent factors in the wedge numerators. At a vertex where three planes meet, the opposite polynomial is of degree m 3 and the adjacent polynomial is unity. At a vertex where two planes and a quadric meet, the opposite polynomial is of degree m - 4 and the adjacent polynomial must be linear. Referring to Fig. 7.6, we observe that this

-

F i g . 7.6. Points that determine the adjacent p o l y n o m i a l f o r v e r t e x i w h e r e t w o p l a n e s and a quadric f a c e meet. ( i ; j ) i s t h e e d g e of i n t e r s e c t i o n of t h e p l a n e s t h a t p i e r c e s t h e q u a d r i c a t e x t e r i o r i n t e r s e c t i o n p o i n t C.

polynomial vanishes on the plane determined by adjacent edge nodes A and B and ETP C where the edge of intersection of the planes pierces the quadric. (The other triple point of the planes and the quadric is, of course, vertex i.) Further insight is gained by examination of a few simple examples of approximation over p-p. EXAMPLE 7.1. The polycondron shown in Fig. 7.7 is a tetrahedron with one of its faces replaced by a section of an ellipsoid. It is a "4-pol of order five" with quadric surface S2 = (1;2;3)2. Each planar surface is denoted by the three vertices on the plane. The three planes meet at vertex 4 .

226

RATIONAL FINITE ELEMENT BASIS

I /

4

I A \

\ 6

Fig.

7.7.

A p o l y c o n d r o n of o r d e r f i v e .

227

THREE-DIMENSIONAL APPROXIMATION

W7(x,y,z) = k7(1;3;4) (2;3;4)/(A;B;C).

(7-38g)

We verify some of the wedge properties to illustrate the congruence theorems. From (2;3;4) E (3;4) mod (1;3;4), (6;7;C) E (6;C) mod (1;3;4), and (A;B;C) (A;C) mod (1;3;4),we obtain W1 : (3;4)(6;C)/(A;C) mod (1;3;4).

(7.39)

A comparison of Eqs.(7.39) and (3.11) shows that the projection of Wl(x,y,z) on face (1;3;4) is the appropriate two-dimensional wedge. (The vertices in Fig. 3.3 must be relabelled to be consistent with Fig. 7.7.) Congruences on the quadric face require more subtle analysis. The algebraic geometry theorems may be clarified by the ensuing discussion. A polynomial of degree k in x, y, and z is congruent on a quadric to a polynomial in two variables of degree 2k. Suppose, for example, we use the equation of the quadric to solve for z in terms of x and y. Then z = P1(x,y) + t/P2(x,y) Substituting this expression for z in Pk(x,y,z) = 0, we obtain Pk-l(x,y) dP2(x,y) = %(x,y), and squaring both sides of this equation we find that

.

Each wedge in Eq.(7.38) is a quadratic function over a linear function in x, y, and z. On S2, each may be expressed as a quartic function over a quadratic function of two variables, say x and y . 228

RATIONAL FINITE ELEMENT BASIS

For any linear u(x,y,z), there are polynomials P4(x,y) and Q2(x,y) for which

i

mod (1;2;312.

(7.40)

The analysis has already established linearity of the wedges on the planar faces. Hence, P4(x,y) must vanish on each edge of the quadric face. Each of these edges is of order two in x and y. Thus P4 is the zero polynomial, and degree one approximation is attained on the quadric face. We have proved that the wedges form a basis for degree one approximation on all four faces of the polycondron. These faces have a total order of five in x, y, and z . For any linear u, there is a P2(x,y,z) such that

P2 can vanish on the polycondron boundary of order five only if P2 is the zero polynomial. Thus the wedges form a basis for degree one approximation over the polycondron. The projection of the wedges on (1;2;3)2 = S2 are basis functions for degree one approximation on S2 and are.the same for polycondra on either side of this quadric. The projections of wedges on a face depend only on the equation of the face and on the equations of its boundary edges. Continuity of patchwork approximation is thus assured.

229

THREE-DIMENSIONAL APPROXIMATION

EXAMPLE 7 . 2 .

A polycondron of o r d e r

same as i n F i g . 7 . 7 .

Fig.

7.8.

m = 7 is

P o i n t s 1 - 7 , A, B , and C a r e t h e

shown i n F i g . 7.8.

On q u a d r i c s u r f a c e ( 1 ; 2 ; 3 1 2 ,

A polycondron of order s e v e n .

w e have i n a d d i t i o n t o ETP A , B, and C t h e 4 (m - 4 ) = 1 2 t r i p l e p o i n t s on t h e q u a r t i c e x t e r i o r edge. T h i s i s a t o t a l of 1 5 t r i p l e p o i n t s on (1;2;312. Any polynomial of d e g r e e t h r e e t h a t v a n i s h e s on t h e s e 15 p o i n t s i s congruent t o t h e c u b i c a d j o i n t The polycondron i n F i g . 7 . 8 i s generon (1;2;312. a t e d f r o m t h a t i n F i g . 7 . 7 by t r u n c a t i n g pyramid [1,2,3,41 w i t h q u a d r i c ( 8 ; 9 ; 1 0 ) 2 .

The c o n s t r u c t i o n

recipe y i e l d s W1 = k 1 ( 6 ; 7 ; C ) ( 2 ; 3 ; 8 ) ( 8 ; 9 ; 1 0 ) 2/Q3.

(7.42)

W e note t h a t (8;9;1012 contains the 1 2 t r i p l e points

on t h e e x t e r i o r q u a r t i c edge. Q3

Therefore,

(8;9;10)2(A;B;C) mod (1;2;312.

(7.43)

From ( 7 . 4 2 ) and (7.431, W1 5 ( 6 ; 7 ; C ) (2;3;4)/(A;B;C) mod ( 1 ; 2 ; 3 ) 2 ' ( 7 . 4 4 ) [ P l a n e ( 2 ; 3 ; 8 ; 1 0 ) i s t h e same as p l a n e ( 2 ; 3 ; 4 ) . I

230

RATIONAL FINITE ELEMENT BASIS

Comparing (7.44) with (7.381, we see that the projection of W1 on (1;2;3)2 is the same for the polycondra in Figs. 7.7 and 7.8. Construction and analysis of the other wedges is an instructive exercise. EXAMPLE 7.3.

In this last example we consider the polycondron of order six shown in Fig. 7.9. The

A polycondron of o r d e r s i x .

Fig. 7 . 9 .

edges of the spherical surface (1;2;3;412 = P 5 are segments of latitudes and longitudes: P1 = (1;4;10) = z, P2 = (1;2;10) = y, z , P4 = (3;4;10) = x y, P3 = (2;3;9) = (1/2) i x2 y2 - z 2 , where P = 0 and P5 = (1;2;3;412 = 1 on face i. The adjoint for this polycondron is

-

-

-

-

(a- 1)y - z2 + (a- ’(2 - 43) (a- 1)yz.

Q2(x,y,z) = 1

+

x

+

The wedges for vertices 1 and 9 are W1 = kl(x

- y) (1 -

Wg = k g z ( l

-

(2

+

fi)y

m)23/42 (x,Y,z)

+ and

2 ~ ) [ 1+ x

-

x

2

-

y2

231

2

z )/Q2(x,y,z).

21x2

THREE-DIMENSIONAL APPROXIMATION

Derivation of these functions and of the other basis functions is left to the reader as is verification of the various properties. One may also derive basis functions for higher degree approximation over this polycondron. Use of a digital computer could relieve the algebraic tedium of such analysis. The remainder of this chapter is devoted to extensions of these results to general well-set polypoldra. The algebra is in some instances formidable. Deficit points are introduced in various places. The analysis is not easily followed as much of the algebra has been omitted. This material has been included here to indicate the scope of the methodology and to serve as a guide for further study A reader who does not wish to spend a great deal of effort in this area of research is advised to pass over the remainder of this chapter or to skim through it without attempting to grasp the finer points. Sections 7.5-7.7 may be omitted without loss of continuity.

.

7.5 7.5.1

THE ADJOINT OF A WELL-SET POLYPOLDRON Conditions on Qm-4

The adjoint for a p-p of order m is of maximal order m - 4 . Adjoint Qm-4 is not unique when the p-p has any edge of order higher than three. This ambiguity is removed by requiring that Qm-4 contain a selected set of deficit points on boundary Pm of the p-p. As many of these points as possible may be chosen as singular points of Pm of orders greater than two. Four kinds of points that determine the 232

R A T I O N A L F I N I T E ELEMENT BASIS

adjoint will now be defined. (i) Multiple points of Pm. Let ri = m (P,) , Piexcept where pi includes all singular points of Pmf for the vertices, and where neighbors are introduced to account for nonordinary singularities. We demand that m (Qm-4) = > r - 2. By Theorem 7.4, this accountsPifor not more than dl =

1

titjtk +

i<j
1 +c3

-

gi)

-v

(7.45)

it1

degrees of freedom of Q,-4f where the p-p has F faces and V vertices. In Eq.(7.45), gi is the genus defined in Eq. (7.10). (ii) Surface deficit points. Let the subset of the pi that are multiple points of component Pk k tk 1*2*. of boundary Pm be denoted by p j* j = Subset pk is divided into two sets of points. All j points on Pk-Pr (k # r) are in pkr and points not in j the intersection of distinct components of Pm are in k k . The number of conditions imposed in (i) by singular points on Pk that are not in any Pk-Pr is

.. .

1 i2

kk C ., where skk = m kk (Pk). (s. 1 3 3 (Pj 1 J

By Eq.(7.10) and Theorem 7.4, this does not exceed t k ' l C 3 conditions. Let :g be defined by k gk - tk'l

C3

-

kkC (s. ) 3

j

I

2 gk 2

(7.46)

k We choose gk deficit points that are simple points k of Pm on Pk but are not on the face defined by P These points are chosen so that together with the

.

233

THREE-DIMENSIONAL APPROXIMATION

c o n d i t i o n s imposed on Pk i n ( i )there a r e tk-l C c o n d i t i o n s imposed on when w e demand that t h e s e d e f i c i t p o i n t s a l s o l i e on Q,-4. This guarantees t h a t no s u r f a c e of maximal o r d e r t k - 4 c o n t a i n s k d e f i c i t p o i n t s on Pk and has m u l t i p l i c i t y t h e gk kk skk 2 a t each p Summing over t h e F s u r f a c e s , j * j w e o b t a i n as t h e t o t a l number of s u r f a c e d e f i c i t p o i n t s on Pm: F

-

(7.47)

k= 1 (iii) Defkcit p o i n t s on p-p edges. k skr = m k r ( P and d e f i n e g z by 3 (Pj 1

Let

(7.48) J

Then

(7.. 4 9 ) q qfk For each f a c e Pq a d j a c e n t t o f a c e P k l t h e r e i s a p-p edge on t h e space curve Pk-Pq. On t h i s curvel b u t not on t h e p-p edge, w e introduce

d y = t t [ ( t k+ t q ) / 2

-

-

- 21 + 1 g z gq (7.50) k q d e f i c i t p o i n t s . W e demand t h a t Qme4 contain t h e s e p o i n t s . I t i s e a s i l y shown t h a t d];g2 0. Summing over a l l t h e p-p edges, w e have a t o t a l of

234

R A T I O N A L FINITE ELEMENT BASIS

conditions on Qm-4 from deficit points on p-p edges. Summing the conditions in (7.45) and (7.50) , we obtain d? + q:d = t t (m - tk - tq) - 2 k q + t t [(tk + tq)/2 21 + 1 (7.52) k q

= t t [m

k q

-

2

-

-

(tk + tq)/21

-

1.

Referring to Eq.(7.8), we observe that this is the dimension of the space of surfaces of maximal order m - 4 on space curve Pk-Pq. Hence, these conditions determine all elements in the divisor Qm-40 Pk 0 :P q (iv) Deficit points on exterior edges. tk

.

An exterior edge is the space curve that is the intersection of two components of Pm that do not define adjacent faces of the p-p. On exterior edge Pk-Pq we introduce (7.53) + tq)/2 - 21 - (gk q + gq) k k q deficit points, which we require to be on Qm,q. (It should be noted that d y may be negative.) We observe that d? = d? - 1. When we add the contribution on the'exterior edge from (i), we do not have two vertices to subtract. Thus on the exterior edge there is one more condition than in Eq. (7.52) :

d?

= t t [(tk

(7.54) d p = t t [m - 2 - (tk + tq)/21. k q By Eq.(7.8), this is one more than the dimension of the space of surfaces of maximal order m 4 on Pk-Pq. Any Qm,q that satisfies these conditions must contain the exterior edge. The total number of deficit points on exterior edges of the p-p, obtained by summing over all such edges is

dtq

+

-

235

d4

-

-

c

THREE-DIMENSIONAL APPROXIMATION

ltktq (

tk + t q

ext. edges Pk-Pq

2

-

2)

-

g;

-

k

gql

-

(7.55)

A Desargues-type theorem is implicit in Eq.(7.55). There are edges of some p-p on which d y in Eq. (7.53) is negative. When this occurs, we require only dFq + dtq , which is less than dtq, conditions of type (i) to assure containment of exterior edge Pk-Pq in adjoint surface Qm-4. For example, when (t ,t ) = (l,l), (1,2), or (2,1), we have gz = g; = 0 k q and d? = -1. We encountered this situation in Section 7.4. There is one extraneous condition of type (i) on each exterior edge of order less than three. We have not yet proved that all the constraints imposed by these four sets of conditions are independent. This is proved when we demonstrate that a unique Qm-4 is determined. 7.5.2

Existence of a Unique Adjoint

The number of conditions on Qm-4 is not greater than: d = d1 + d2 + d3 + d4. All contributions to dl from singular points of components of Pm are cancelled by reductions in the number of deficit points that contribute to d2 + d3 + d4. A careful accounting yields d =

1 j ,k,q

j
tjtktq

+

F

1

-

k=l tk-lC3 t +t tktq(9 2 236

(7.56)

+

-

2).

RATIONAL FINITE ELEMENT BASIS

Let U be the space of surfaces of maximal order m that satisfy these conditions. By Theorem 7 . 4 , dim

u

m-l C3 - 1 - d .

-

4

(7.57)

inductive argument will be used to prove that the right-hand side of ( 7 . 5 7 ) is zero and that consequently there is at least one Qm-4 in space U. We have already proved that the right-hand side of (7.57) is zero for any well-set polyhedron. Given any well2 .Tn set p-p of order m with boundary Pm = S T1 Tt stl 2 tn for which m-1C3 - 1 - d = 6, we will show that increasing the order of S to s + 1 increases d by It will follow mC3 - m-1 c 3 = m-1 C 2 [see Eq. ( 7 . 4 1 1 . that any p-p with n i1 faces has the same value for m-l C3 - 1 - d as a well-set polyhedron with n + 1 faces. But this value is known to be zero. The change in d when the order of S is increased fnom s to s + 1 is An

..

d(Pm+l)

-

d(Pm) = T ~ . Tnot ~ on

C s-1 3

s

n

-

k=l

S+tk stk (- 2

m = s +

1

tk

k=l and that

-

(C n

s)* =

2)

211.

n

We note that

(m

-

(7.58)

n

k=l

237

(7.59)

THREE-DIMENSIONAL APPROXIMATION

Expanding (7.58), we obtain with the aid of (7.59):

k=l Applying (7.60) to this last equation, we obtain d(Pm+l)

-

-

d(Pm) = s-1C2 + [s (3/2) 1 (m 2 + (m s) /2 = (m2

-

-

3m

+

-

s) (7.61)

2)/2 = m-lC2,

as was to be proved. We have shown that dim U 2 0 and that there is at least one Qm-4 in U that satisfies the enumerated conditions. We now prove that there is only one such 1 element. Let Qm-4 and 4-:Q be in U. Let Ft be the product of the p-p boundary components of surfaces adjacent to Ss and let Fm-t-s be the product of the remaining factors, excluding Ss. Then since all the exterior edges of the p-p are contained in any element of U: (7.62) Q2 =- 0 mod Fm-t-s 0 ss. Q1

-

Moreoverl by construction: 1 Q 0 Ft 0 Ss = Q 2

0

Ft

0

Ss.

(7.63)

By Theorem 7.3, there is a b for which Q1 - bQ2 = 0 By Eq.(7.62), for this same b, mod Ft 0 S s . Q1 - bQ2 = 0 mod Fm-t-s 0 S,. ible. Hence, there must be a

238

Surface S s is irreducsuch that

Gs-4

R A T I O N A L FINITE ELEMENT BASIS

Q’

-

b~~ = ~s-4~m-t-s Ft mod Ss.

(7.64)

Both Q1 and Q2 vanish on s-1C3 surface nodes on Ss which have been chosen not to lie on any surface of order less than s - 3. These surface points are on neither Fm-t-s nor Ft. Hence, Gs,4 must be the zero polynomial. This argument applies over each surface with the same value for b. Thus Q1 bQ2 = 0 mod Pm. Boundary Pm is a product of simple irreducible components. Therefore, Qm-4 - bQm2,4 must be the zero polynomial. We have proved that the adjoint constructed from the enumerated conditions is unique for the general well-set p-p.

-

7.5.3

Wedge Regularity

Theorems 5.1 and 5.2 generalize to three dimensions. When a given p-p is related to other p-p as in Theorem 5.1 and deficit points are assigned to edges and surfaces independent of the p-p, the theorems provide a basis for establishing regularity over any specific p-p. If possible, deficit points should be chosen on the absolute line in homogeneous coordinates. This should simplify the construction. Regularity has been established for well-set polyhedra. The situation is quite obscure for more varied elements. Placement of deficit points to yield regular wedges has not been investigated. The entire analysis of polypoldra having faces of orders greater than two is so complicated that use of such elements does not appear imminent. Nevertheless, this area may prove fruitful for theorists.

239

THREE-DIMENSIONAL APPROXIMATION

POLYPOLDRA NODES AND ADJACENT FACTORS FOR DEGREE k APPROXIMATION

7.6

7.6.1

Node Placement

The number and location of four types of p-p nodes will now be described: (i) all vertices, (ii) k-1C interior nodes chosen not to all lie on any surface of order less than k 3,

-

-

(iii)s+k-lC3 k-1c 3 - s-1C 3 nodes on each face Pi of order s, chosen so that no surface of maximal 4 contains these nodes together with order s + k C interior nodes and the s-l C exterior the k-1 3 surface points "Section 7.5.1-(ii) I on Pi defined for the adjoint construction,

-

-

-

(iv)k+3c3 k-s+3'3 k-t+3'3 k-s-t+3 c 3 - 2 of adjacent faces Pi and Pj, nodes on edge : . i P S chosen so that no surface of maximal order k contains these nodes and the two vertices on P i.Pj. +

These nodes are used in the construction of adjacent factors for the basis functions. 7.6.2

Adjacent Factors at Interior Nodes

There are k-l C interior nodes. For each of these nodes, the wedge has an adjacent Sactor that vanishes on the unique surface of order k 4 deter1 interior nodes. mined by the remaining k-1C3

-

240

-

RATIONAL FINITE ELEMENT BASIS

Adjacent Factors f o r Face Nodes

7.6.3

-

There a r e s+k-1C3

-

k-lC3

s-l C 3 nodes on f a c e

For e a c h of t h e s e nodes, t h e r e i s a wedge w i t h Pi. a unique a d j a c e n t f a c t o r of d e g r e e s + k - 4 d e t e r mined by :

(1) t h e remaining face nodes, (2) t h e a d j o i n t d e f i c i t p o i n t s on P i of t h e t y p e d e s c r i b e d i n S e c t i o n 7 . 5 . 1 - ( i i ) , and t h e i n t e r i o r nodes.

(3)

Adjacent F a c t o r s f o r Edge Nodes

7.6.4

i ' The a d j a c e n t f a c t o r f o r edge node p on Ps-P: i s

+

t h e polynomial o f d e g r e e s

P:

-

the s+k-lC3 C

t-1 3 ( 2 ) t h e s-1C3

k-1'3

-

k-1'3

t

-

+

k

-

s-1'3

4 determined by: + t+k-lC3

a d j a c e n t f a c e nodes,

+ t-1C 3

s u r f a c e d e f i c i t p o i n t s on

and P i f o r t h e a d j o i n t c m s t r u c t i o n [ S e c t i o n

7.5.1-(ii) I ,

i n t e r i o r nodes,

( 3 ) t h e k-1C3

+

-

t)/2 21 + 1 s i n g u l a r and d e f i i ' c i t p o i n t c o n d i t i o n s on Ps - P i d e s c r i b e d i n S e c t i o n 7 . 5 . 1 - ( i i i ) , and (4)

the s t [ ( s

-

Pi,

exclud i n g t h e t w o v e r t i c e s on P i . P j s t ' where all t h e e l e ments i n t h i s c y c l e are u n i q u e l y determined by t h e edge nodes o t h e r t h a n p and t h e v e r t i c e s on P i - P z . (The wedge i s c o n g r u e n t t o Hk on t h i s edge.) (5) t h e s t k

2 elements of Hk 0 P i 0

241

THREE-DIMENSIONAL APPROXIMATION

The sum of the conditions in (1)-(5) is f = s+k-lC3 + t+k-lC3 + 1 + stk 2.

-

-

k-1 c 3

+

st[(s

+ t)/2 - 21

It can be shown that the above reduces to = k+s+t-lC3

-

(7.65)

By Eq.(7.5), there is at least one surface of maximal order s + t + k 4 that satisfies these conditions. 1 2 Let R and R be any two candidates. Then R1 0 PiPj = R2 0 Pip] and there is a b f o r which i j bR2 = 0 on PsPt. If k 2 3, then R1 bR 2 must R1 be the zero polynomial. If k > 3, there must be a - 2 i j R1 bR = Gk-4PsPt. We note, howGk- 4 for which 1 ever, that RL and RL vanish on the interior nodes at which Pip' # 0 and which were chosen not to lie on any surface of order less than k - 3. Thus Gk-4 is the zero polynomial and a unique adjacent surface is established.

-

-

-

-

-i

7.6.5

Adjacent Factors for Vertex Nodes

Let surfaces of orders sl, s2, and s 3 meet at vertex q. The conditions on adjacent surface Rq are that the following points must be on Rq: 1 2 3 1 points in Ps O Ps2 O ps3- 9 r (1) hl = s 1s2s3 1 21 + 1 (2) h2 = sls21(s1 + s2)/2

-

-

+ +

s3)/2

-

s2s3 [ (s2 + s3)/2

-

s1s3[(s1

+

21

+

1

21 + 1 singular points of components Pl, P 2 , and P 3 , and deficit points on edges P1-P2, P 1 - P3 , and P2.P3.

242

RATIONAL FINITE ELEMENT BASIS

(3)

h3 = k(s1s2

+

s1s3

+

s2s3)

-

3 points in

:H 0 Pi 0 PI, where the nodes on Pimp’ with the exception of vertex q determine surface HZ of maximal order k,

1

k-1C3 face nodes h4 = :=1 si+k-l 3 and surface deficit points [Section 7.5.1- (ii)I , and (4)

(5) h5 = k-1C3 interior nodes.

Summing these five contributions and defining s = s1 + s2 + s3, we find after a great deal of algebra (which the eager reader may duplicate) that h1

+ h2 + h3 + h4 + h5

= s+k-1C3

-

1.

-

There is at least one Rq of maximal order s + k 4 that satisfies these conditions. Uniqueness may be established by the usual procedure. 7.7

ATTAINMENT OF DEGReE k APPROXIMATION

The adjacent factors for the edge and vertex node wedges have been constructed so that for each node q : s : F

-tRqs+ t+k-4

0

pi S

0

pj t = *m-4Hq k

0

Pi

0

Pi.

(7.66)

Hence, there is a b for which FqRq - bqQm-lHq = 0 9 on Pi.Pj, and the approximation to any u(x,y,z) with nodal values u satisfies 9

(7.67)

q

where Hk is of maximal degree k. If u is of maximal degree k, then the approximation is precise on Pi.Pj. This analysis holds for all 243

THREE-DIMENSIONAL APPROXIMATION

of the p-p edges. L e t t h e sum of t h e orders of t h e i f a c e s a d j a c e n t t o Ps be w. Then t h e o p p o s i t e f a c t o r s of a l l wedges a s s o c i a t e d w i t h nodes on f a c e Pi c o n t a i n a common f a c t o r of degree m s w. Thus f o r any u o f maximal d e g r e e k:

-

U(X,Y,Z)

-

1

Uq$(X,Y,Z)

= -

Nm+k-4

Qm- 4

all q

-

-

'm-sGs+k-4 Qm- 4

mod Pi.

(7.68)

a d j a c e n t factors Rq v a n i s h i Hence, on t h e s-1C3 e x t e r i o r p o i n t s i n P oQm-4. Nm+k-4 = 0 on these p o i n t s . A l s o , Nm+k-4 = 0 on t h e C i n t e r i o r nodes and on t h e s+k-l k-1 3 k-1'3 C f a c e nodes. Polynomial Pm-s i n E q . ( 7 . 6 8 ) does s-1 3 n o t v a n i s h a t any o f t h e s e s+k-l C p o i n t s . BY cons t r u c t i o n , no s u r f a c e of maximal o r d e r k + s 4 c o n t a i n s a l l t h e s e p o i n t s (or s a t i s f i e s t h e corresponding number o f c o n d i t i o n s when any of t h e p o i n t s coallesce), so t h a t Gs+k-4 must be t h e z e r o polynomial. W e have shown, a l t h o u g h d e t a i l s have been o m i t ted, t h a t u u Wk v a n i s h e s on t h e p-p boundary 9 9 of o r d e r m when u i s of maximal d e g r e e k . S i n c e t h i s boundary i s a p r o d u c t o f s i m p l e i r r e d u c i b l e components, t h e r e must be a Gkm4 such t h a t For a l l nodes q on Pi,

-

-

-

-

"i+k-4 Qm-4

-

'mGk-4 Qm- 4

S u r f a c e Gk-4 must c o n t a i n t h e i n t e r i o r nodes of t h e T h i s i s n o t possible u n l e s s Gk-4 i s t h e z e r o p-p. polynomial. W e have proved t h a t degree k approximat i o n i s a t t a i n e d o v e r t h e p-p. 244

Chapter 8

A RATIONAL SOLUTION TO AN IRRATIONAL PROBLEM

8.1

IRRATIONAL WEDGES

We have shown how a unique minimal rational basis may be constructed to achieve degree k approximation over any well-set rational polypol. Alternative bases may be found if we permit functions that need not be rational. Consider the element in Fig. 3.1 which has been redrawn in Fig. 8.1 for convenience.

Fig.

8.1.

A

3 - c o n of o r d e r f o u r .

Study of this element motivated the analysis of elements with curved sides, as indicated in Section 3.1. Further examination of this simple 3-con reveals other aspects of wedge construction. We seek a wedge for vertex 1. This wedge must be linear on

245

RATIONALSOLUTION TO AN IRRATIONAL PROBLEM

sides ( 3 ; l ) and (1;2) and must vanish on side (2;3)2. In Section 3.1 we chose

and subsequently discovered how to construct wedges of this type for all well-set rational polypols from geometric properties of the elements. We then generalized to higher degree approximation and to three dimensions. suppose we allow irrational basis functions and choose a wedge numerator of N(x,y) = y l ( 1 - X 2 1/2 1 .

-

( 8 2)

Henceforth, the absolute signs will be omitted, and it is understood that positive square roots are always taken. A "rabbit function" is a function that is pulled out of a hat. When a magician pulls a rabbit out of a hat, the effect astonishes the onlookers. Once the illusion is revealed, the rabbit generation is no more surprising than the generation of rabbits by rabbits. We present the following rabbit function to demonstrate the existence of irrational wedges:

(8.3) We note that and that

w1 w1

= =

( 1(1

y) mod x ,

(8.4a)

mod y.

(8.4b)

XI

Similar wedges may be constnucted for the other three nodes to achieve degree one approximation over this 3-con. The mechanism of this construction will be revealed eventually. One may be puzzled by

246

RATIONAL FINITE ELEMENT BASIS

this path of investigation. Having found so beautiful a structure for rational wedge construction, why do we even consider irrational wedges? The motivation is our desire €or admissibility of a wider class of elements rather than any quest for alternative bases for well-set rational polypols. It will be shown that irrational wedges provide a basis for approximation of any degree over ill-set elements. Just as polynomials are inadequate for all but a few simple elements (of great practical importance), rational functions are inadequate for ill-set elements. Consider the ill-set eLement of Fig. 8.2, in which intersection points A and B are not exterior to the element. The rational function W1(x,y) = kl(l

-

x

2

-

y2)/(A;B)

is not acceptable; the adjoint vanishes along a line

' " q $

2

I- x - Y

-

x=o ' A 8.2.

An i l l - s e t

=o

1,O)

I (OC2) Fi g .

2

Y-2X+2=0

3-con of o r d e r four.

that passes through the element. On the other hand, the irrational rabbit function displayed in Eq.(8.5) is acceptable:

247

RATIONAL SOLUTION TO A N IRRATIONAL PROBLEM

w1 This function vanishes on the curved side. Moreover, W1 -= (1 y) mod x. To show that W1 is linear on side (1;3), we set y = 2 (x - 1) and expand the numerator in Eq. (8.5) , noting that x2 5 - 1 in the region of interest:

-

=

+ 2(1 - x) 1 [(l-x 1 + 1 - XI 2 2 1/2 (1 - x ) + 2(1 - x ) +~ 3(1 - x)(l-x ) (1 - ~ ) [ 3- x + 3 ( 1 -2~) 1/2]

=

3(1

[(l-x 1 =

x)[l

-

(~/3)+ (1-x2) 1/2] 6

Hence,

w1

-

f

(1 - x) [l

-

(x/3) + (1-x2)1/21

1

-

(x/3) + (1-x2 ) 1/2

mod (y

-

2x + 2).

= 1 - x

In Section 8.2, we will construct basis functions of this type for all the nodes of the ill-set polycon in Fig. 8.2 in order to provide a basis for degree one approximation over this element. It will be shown that irrational bases may be constructed to achieve any desired degree approximation. There is some cause for consternation. The title of this work is "A Rational Finite Element Basis". Irrational approximation over ill-set elements could lead us too far afield. We could consider element sides defined by nonalgebraic functions and proceed to transcendental approximation. Where would this end? The reader may be consoled by the following prospectus: (1) we will consider only construction 248

RATIONAL FINITE ELEMENT BASIS

of irrational wedges to provide bases for approximation over ill-set polypols, and ( 2 ) these wedges will be determined by the construction developed for rational wedges. We will demonstrate that irrational wedges are actually members of our family of rational wedges. This paradox is resolved in the next section. 8.2

THE METHOD OF DESCENT

Hadamard introduced the phrase "method of descent" (Hadamard, 1952, p. 4 9 ) to describe a technique for solving differential equations which "consists in noticing that he who can do more can do less: if we can integrate equations with m variables, we can do the same for equations with m 1 variables." We project the solution in m variables on to the space of m 1 variables. It will soon become apparent that the method of descent allows solution of problems not amenable to attack directly in the m - 1 variables. We define z = (1-x21.2'1 Then the element in Fig. 8.1 maps on to a quadric face on the surface over which x2 + z 2 1 = 0. This face may be chosen as a face of a polycondron defined by the four surfaces on which x = O , y = O , z - y = O , a n d x2 + z 2 - 1 = 0 .

-

-

-

(8.6)

We determine a wedge basis for degree one approximation over this polycondron. In each of these wedges that contribute to the approximation on the quadric surface, we replace z by (1-x ) 'I2 to obtain basis functions for the ill-set 3-con in Fig. 8.1. 249

RATIONALSOLUTION TO AN IRRATIONAL PROBLEM

When a polyp01 is related to a three-dimensional element, linear sides in the two-dimensional element may become curved in three dimensions. Side nodes must be introduced on these curved edges of the p-p. In our current example, node 6 is added on side 11;3)2 of the p-p, which corresponds to linear side (1;3) of I

Y

X = 2I tx TOP V I E W FROM Z Fig. 8.3. P r o j e c t i o n of a c u r v e d p o l y c o n d r o n e d g e i n t o a s t r a i g h t s i d e of a p o l y c o n .

the polycon. The wedges projected into the x-y plane are linear in x, y, and by setting z = (1-x ) (1-x2)ll2. The dependence on (1-x ) along the polycon sides is removed by choosing appropriate combinations of wedges when we eliminate node 6. Referring to Fig. 8.3, wedges Wi(x,y) f o r the 3-con are related to wedges W.(x,y,z) of the @-p by: 7

(8.7b) (8.7~)

It will be shown that these wedges are linear in

x and y alone on the 3-con sides. This is true in 250

R A T I O N A L FINITE ELEMENT BASIS

general when p-p wedges are projected on to the x-y plane and extraneous nodes are eliminated. On the curved side of the 3-con in our example, z = y so that linearity in x,y, and z implies linearity in x and y alone. Linearity on the straight sides will be demonstrated by analysis of the actual wedges. The polycondron defined by the surfaces in Eq.(8.6) is shown in Fig. 8.4, in which points are labelled

*f

Point 5 = Fig.

8.4.

(l/d!,l/fi,l/tQj;

7 i s o n (1;2) a t

-.

The p o l y c o n d r o n a s s o c i a t e d w i t h t h e 3-con in Fig. 8.1.

consistently with Fig, 7.7. Quadric face (1;2;3)2 is a section of the surface on which x2 + z2 1 = 0. The adjoint plane for this p-p is

-

(A;B;C) = 1

+ x+

z.

(8.8)

Referring to Eqs.(7.38), we note that adjacent factors are of the planes (6;7;C), (5;7;B), and (5;6;A). Since point 7 is at infinity, plane (6;7;C) must intersect plane x = 0 in a line through B and parallel to (1;2). This is the line (B;C). It follows that the three adjacent factors are:

25 1

RATIONALSOLUTION TO A N IRRATIONAL PROBLEM

and

-

(6;7;C) = (6;B;C) = 1

+

z

(5;7;B) = (5;B;C) = 1

+

z

+

(a-

(5;6;A) = 1 + x

-

-

(2 +fi)x,

(8.9)

(1 + a ) x , (8.10) 1

-

/?!)y.(8.11)

We now obtain from (7.38): W1(x,y,z) =

(2

-

y) 11 +

and W6(X,Y,Z) = 2(1 + fi)x(z

-

2

(2 + Js)xl/(l

+

x

+

2)

,

(8.12)

-

y)/(l + x +

(8.13)

2).

By substituting (8.12) and (8.13) into (8.71, we obtain Wl(x,y) = [ (1-x ) y] [l + (1-x ) 1 1 2 XI/ Rationalizing the denominator, [l + x + (l-~~)~’~]. we obtain W1(x,y) = (l-x2)1/2[(l-x 2) 1/2 yl/(l + x). (8.14)

-

-

-

This is the rabbit function given in Eq.(8.3), which is no longer a rabbit function. The magical hat is replaced by a rational construction. The other wedges are:

w3 (X,Y) = x

-

(1 + l/JZ)y[l

+ x -

+ x),

(1-x ) 1’21/(1

(8.15) W2(x,y) = y

-

(1 + l/JZ)y[l

+ x - (l-x2)q/(l + x), (8.16)

W5(x,y) = (1 + JZ)y[l

+ x - (l-x2)1’21/(l

+ x). (8.17)

It is easily verified that these wedges are linear on the 3-con sides and that they provide a basis for linear functions over the element. We chose this simple element to illustrate the method of descent. Perhaps these irrational wedges have some application,

252

RATIONAL FINITE ELEMENT BASIS

but the rational wedges described in Chapter 3 seem more natural f o r this well-set element. Higher degree approximation over the polycondron projects into higher degree approximation by irrational wedges over the polycon. The theory f o r construction of rational wedges in three dimensions thus provides the means for constructing irrational wedges in two dimensions that achieve any degree approximation. One may speculate about the possibility of projecting four-dimensional rational wedges to obtain three-dimensional irrational wedges for approximation over ill-set polypoldra. We now direct our attention to the 3-con in Fig. 8.2, for which we have not found a rational basis. 8.3

WEDGES FOR AN ILL-SET POLYCON

Wedges for the 3-COn in Fig. 8.2 are obtained from analysis of the ill-set polycondron of Fig. 8.5.

Fig.

8.5.

A n ill-set polycondron.

253

RATIONSL SOLUTION TO AN IRRATIONAL PROBLEM

This polycondron has three planar and one quadric face: (1;2;4) = X, (2;3;4) = z

-

(1;3;4) = 1

-

x + (y/2)

(1;2;3)2 = z2

y,

+

x2

-

(8.18)

1.

Surface (1;2;3) intersects plane (1;2;4) on edge (1;2) and again on line (B;C). The p-p is ill set. Plane (A;B;C), however, does not pass through the quadric face so that the wedges are regular on this face. The adjacent and adjoint factors for these wedges are

-

(13 + 4J3)x/ll + (4 3n)z/ll,

(5;6;A) = 1 (A;B;C) = 1

-

-

(x/3) +

2,

-

(8.19a) (8.19b)

(2 (6;7;C) = (6;B;C) = 1 + z = (5;7;B) = (5;B;C).

= 2(1

+ G ) x ( -z y)/3[1

+ O)X

-

(x/3) +

(8.19~)

18.21) 21.

Substitution of (8.20) and (8.21) into (8.7) yields [(l-x21 1/2 Wl(X,Y) =

3[1

-

y1[1 + (1-x21 112

-

(~/3)+ (1-x2) 1/2]

x]

.

(8.22)

This is the rabbit function in Eq.(8.5) , for which linearity on adjacent sides has already been verified.

254

RATIO NAL FINITE ELEMENT BASIS

The other wedges are obtained in similar fashion. For example,

-

-

-

2fi)/3 + (1-x2)li2 + (1 + 2fi)x/3 (1 + fi)y/31/[l - ( 4 3 ) + (1-x 21 1/2] (8.23)

x[(l

Even for simple elements the wedges are complicated. As the order of the polycon or the number of its sides responsible for irregularity of the rational wedges increases, the construction becomes more complex. 8.4

NONCONVEX QUADRILATERALS

The method of descent may be used to determine irrational wedges for nonconvex quadrilaterals. Such wedges are not unique. They depend on the choice of a three-dimensional element. To illustrate the technique, a quadrilateral that is symmetric about the line on which y x = 0 will be examined. This quadrilateral is shown in Fig. 8.6. The

-

) l . l ( # -x + l

3

C-lpl

Y=O

x=o

F i g . 8.6.

5

1+y-2x=o (O,-l)

A nonconVeX q u a d r i l a t e r a l .

vertices have been labelled to be consistent with

255

RATIONAL SOLUTION TO AN IRRATIONAL PROBLEM

the polycondron in Fig. 7.7 when we relate this quadrilateral to a three-dimensional element. Let 2

= *(x2

+

y2)?

(8.24)

Then sides ( 3 ; 5 ) and (2;s) of the quadrilateral lie in the plane x

+

y

+

2

= 0.

(8.25)

Substituting Ax2 + y2)1’2 for z in (8.25)I we obtain + y = ~ ( x +* y2)1’2 or x2 + y 2 + 2xy = x2 + y2 so that xy = 0. If y = 0 and z = I (x2 + y 2 ) 1’2 I I then x + 1x1 = 0 or x 5 0. If x = 0, then y + I y I = 0 or y 2 0. Thus x + y + I ( x 2 + y2)1’21 = 0 on quadrilateral sides (3;5) and (2;5) but not on the extensions of these sides into the quadrilateral. We determine wedges for this nonconvex quadrilateral from the three-dimensional element shown in Fig. 8.7. This ill-set polycondron is defined by x

Fig. 8.7. An i l l - s e t p o l y c o n d r o n r e l a t e d t o t h e nonconvex q u a d r i l a t e r a l i n F i g . 8 . 6 .

256

RATIONAL FINITE ELEMENT BASIS

t h e f o u r surfaces:

+ y + 2 1 + y - 2x x

and

z

2

+

1

x 2

- x

-

= 0, = 0,

(8.26)

2y = 0 , 2 - y = o .

The polycondron a d j o i n t is ( A i B i C ) = 1 + “1

+

3fl

+ ( 3 a / 2 ) 1 (x + y)

+

(8.27)

(a/2)]z.

The a d j a c e n t factor f o r vertex 1 i s ( 6 ; 7 ; C ) = 1 + 11

-

3[1

-

-

(3fl/2)]

(X

+

y)

(8.28)

(JZ/2)]2.

The a d j a c e n t f a c t o r f o r vertex 2 is

(5;7;B) = y .

(8.29)

W e o b t a i n from E q s . ( 7 . 3 8 )

= (2

-

A d j o i n t Q = (A;B;C)

+ Jz) (1 + x

-

2y) (x

+

y

+

2 ) ?

o f t h e polycondron y i e l d s t h e

denominator D for t h e nonconvex q u a d r i l a t e r a l when 257

RATIONAL SOLUTION TO AN IRRATIONAL PROBLEM

we substitute ( x 2

+

for z in Eq. (8.27) :

y2)1’2

D(x,y) = 1 + 11 + (3a/2)1 (x + y) + 3[1 + (fl/2)1 (x2 + y2l1I2.

(8.35)

The quadrilateral wedges derived from (8.30)-(8.35) with z replaced by (x2 + y2)ll2 are W5(X,Y) = (1 +

X

-

2y) (1 + y

-

2x)/D(x,y),

(8.36)

(In this case, an edge node of a polycondron projects into a vertex node of the two-dimensional element.) 1 w1 (X,Y) = 7 W6 (x,y,(x2 + y2)1’2) 1 + 7 W7(X,Y, (x2 + y2)1/2) =

[l + ( J z / 2 ) ][x + y + (x2 + y2)1/2]. [1 + c 3 n - 4 ) (x + y)/2 + (6 - 3 a ) (x2 + ~~)~/~/21/D(x,y), (8.37)

W2(X,Y) = W2(X,Y, (x2+Y2)1/2) =

(1 + x -2y) [(l

+

+

$){x

2 2 1/2) +J7(x,y,(x +y )

+ (x2 +y2 ) 1/2)

+ qyI/D(x,y)

=

(8.38)

(8.39)

W2(Y,X).

The theory assures us that these wedges are a basis for degree one approximation over the quadrilateral and that they satisfy all the requirements in Section 1.5. To illustrate how these properties are satisfied, we verify linearity of W on ( l i 3 ) . 1 On this side, y = (1 + x)/2 and the numerator in Eq.(8.37) is

258

RATIO NAL FINITE ELEMENT BASIS

3Jzx 3 a 3 2 + (2 - $)x2 + $1 +

=

[a3 +

E

[(Jz -

-

5

$2+

+

a x

;+

2 2 1/2] x ) ( x +y ) mod (1 + x 2y)

(1

+

-

x ) ( x2+y 2) 1/2]

mod (I

+ x

-

(Jz - F1I X + (x2+y 21 1/2] mod (1 + x -

1 + (1 + x) [z

2y)

2y).

The denominator satisfies D E [ 1 + ( 1 + -3Jz)1 + 3x

2

2

= -

17 3 + 7 3 a . 3~ ( +1 -)x 342 2

=

[z 1 + (a- T)X 1 +

Thus N/D

E

+ 3(1 +

mod (1 + x

+

+

-

2y)

(x2+y 2) 1/2] mod (1 + x - 2y)

3(1

%)

(x2+y2) 1/2] mod (1 + x

(1 + x) .mod (1 + x

-

2y)

, as

-

2y).

was to be

shown. 8.5

REMARKS

Ill-set elements and irrational wedges have been described primarily for the theoretical implications. Irrational wedges are not as easily determined as the rational wedges for well-set elements, and their complex form detracts from application. It is, however, interesting to consider the hierarchy of basis functions. Polynomials are adequate for triangles, parallelograms, and a restricted class of 259

RATIONAL SOLUTION TO AN IRRATIONAL PROBLEM

well-set elements. Rational functions suffice for all well-set polypols and polypoldra, but these functions are inadequate for ill-set elements. The irrational basis functions provide the extra freedom needed for ill-set elements. The interrelationships of these basis functions and the close connection between their algebraic form and geometric properties of the elements has been thematic in this development. Recognition of the role of the exterior diagonal of a convex quadrilateral in the construction of a basis for approximation over the quadrilateral was the starting point of this entire study. By means of a sequence of congruence theorems we were able to generalize to elements with curved sides, to higher degree approximation, and to three-dimensional elements. It is indeed satisfying to be able to use the three-dimensional analysis to remove the well-set restriction on two-dimensional elements. In the last section we concluded with analysis of a quadrilateral, and the sophistication required for the construction of wedges for nonconvex quadrilaterals may be contrasted with that required for the convex quadrilateral. The use of restricted nodes as described in Section 1.3 to treat ill-set elements is a practical alternative to irrational basis functions.

260

Chapter 9

FINITE ELEMEN T DISCRETIZATION

9.1

INTRODUCTORY REMARKS

Discretization of a boundary-value problem by the finite element method requires evaluation of various integrals over the elements into which the region of interest is partitioned. Commonly encountered integrands contain products of basis functions and their derivatives (Aziz, 1972; Strang and Fix, 1973; Zienkiewicz, 1971). We have constructed basis functions for a wide class of elements. Analytic integration of expressions encountered in discretization is feasible only for a few simple types of elements such as triangles and rectangles. One must resort to numerical quadrature for the more complicated elements. The effect of quadrature error on the approximate solution to the boundary-value problem is discussed in Section 9.3. In general, a significant characteristic of any patchwork approximation used in finite element computation is the ease with which the integrals can be evaluated. If we restrict our attention to bilinear functionals that yield discrete linear 261

FINITE ELEMENT DISCRETIZATION

systems and for which C o trial functions are adequate, then we are concerned primarily with products of the basis functions and their first derivatives. Some of the quadrature difficulties are discussed in Section 9.2. It will become apparent that higher degree approximation and complicated elements compound the difficulties. Guides for numerical quadrature are provided by the consistency analysis in Section 9.3. The -patch test described in that section is particularly useful. The triangle-averaging method discussed in Section 9 . 4 is an attempt to utilize analytic integrals over triangles for computation of discrete equations for more complicated elements. As used in the past, triangle averaging has limited applicability and poor theoretical justification. A closer scrutiny of concepts related to triangle averaging reveals a powerful alternative that resolves major difficulties encountered in finite element discretization. This "mosaic discretization" procedure, developed in Section 9 . 5 , is crucial to the entire study. The theory in Chapters 1-8 can be brought to bear on real problems in a numerically convenient manner through mosaic discretization. Most of the problems encountered when one departs from simple elements and l o w degree approximation are in the discretization. Numerical solution is often accomplished by standard techniques that do not suffer significantly when more sophisticated elements are used. In Section 9.6, various schemes are compared for discretizing the Laplacian operator over quadrilateral elements by way of illustration.

262

RATIONAL FINITE ELEMENT BASIS

Rational wedges are not the only basis functions that satisfy the conditions in Section 1.5 over wellset elements. A valuable alternative for finite element computation is discussed in Section 9.7. Mosaic discretization with this alternative basis is particularly suited for finite element computation. The discussion in this chapter is confined to planar elements. Generalization to higher dimension is straightforward. In chapters 2-8 we have treated the general problem of continuous patchwork representation, and the analysis pertains to a broad area of research. Analysis in this chapter is more specifically concerned with finite element application. 9.2

SOME SIMPLE QUADRATURE FORMULAS

When the values of a function are known only on a discrete set of points, there is no alternative to numerical quadrature for approximating the integral of the function over an element. Even when the function is known it may be necessary to approximate its integral. Either a complicated function or an irregular element boundary can cause difficulties. To obtain a quadrature formula, one may approximate the actual function by some simpler function €or which analytic integration is possible. A patchwork approximation over a collection of elements need not be continuous across element boundaries when used only for integration. If the approximation is of degree k over element Em with maximum chord length h, and if U is the approximation to u over Em, then

263

FINITE ELEMENT DISCRETIZATION

For a fixed domain, the number of elements increases as h-2 when h + 0 so that

Salient features of quadrature analysis are illustrated by study of the membrane eigenvalue problem. Let R be a region with smooth enough boundary to yield a properly posed problem. Let u = 0 on this boundary. We seek the smallest nonzero eigenvalue X and corresponding eigenfunction u(x,y) in C2 (R) such that

V 2 u(x,y) + Au(x,y)

= 0

in R.

(9.2)

Let V be the space of functions that vanish on the boundary of R and are continuous with piecewise continuous first derivatives in R. It is known that I

X

= min

V€V

F(v)E

// lVvI2 dx dy R //

(9.3)

v2 dx dy

and that the minimum is achieved only when v = uI the function that satisfies Eq.(9.2). If we cover R with a set of nonoverlapping elements and define v in each element in terms of nodal values viI then discrete equations are 264

RATIONAL FINITE ELEMENT BASIS

obtained from the stationarity conditions

- aF - avi

... .

i = 1,2,

0;

(9.4)

Solution of these equations yields the approximation v* to u that is best over the approximation space in the sense that F(v*) is closest to X when v = v*. The approximation space is a subspace of V of dimension equal to the number of nodes. In general, v* # u. Convergence of v* to u as the number of nodes is increased and the maximal chord length is decreased depends on the degree of approximation within the elements (Section 1.4). Within each element v =

1

ViWi(X,Y).

(9.5)

i

Thus the contributions to the integrals in ( 9 . 4 ) from element Em are

IllVx

viWi(x,y)

l2

dx dy.

Hence, we must

i Em approximate for all i,j,m:

These integrals are often approximated, even for simple elements where closed form analytic integrals are known. For example, let K be the area of a 265

FINITE ELEMENT DISCRETIZATION

triangular element with vertices i,j, and k. degree one quadrature over the triangle,

JJ WiWj dx dy

=

{

K/6 for i = j K/12 for i # j.

For

(9.7)

Let n be a neighbor of vertex i and let Kin be the sum of the areas of the triangles that share side (iin). Eqs.(9.4) define a discrete eigenvalue problem, and the coefficient of A in the ith equation ~ ~ + vi)/12. ~ ( This v term ~ is replaced in is many computer programs by

1

(vi/6)

1 n

Kin

.

(9.8)

When integrating with respect to only one variable, we may use Gaussian quadrature to achieve high accuracy with relatively few nodes. A suitable choice of nodes can lead to increased accuracy in two-variable integration as well. Degree one accuracy is achieved over a triangle by

where K is the triangle area and ui is the value of u at vertex i. The same accuracy is achieved by multiplying the value of u at the triangle centroid by the triangle area. Let (9.10

be the integral of wedge function Wi over element Em. Let the Wi be a basis for degree k approximation over n. the element; i = 1,2,

...,

266

RATIONAL FINITE ELEMENT BASIS

suppose u is approximated by n U(x,y) = UiWi(X,Y) i=l

1

(9.11)

We define wedge quadrature by n v Iw(u) = uiwi. i=l

(9.12)

This is the exact integral of U over the element, and the quadrature error is O(h k + 3 ) . We note that IW(W.W.) =

{

wi when

i = j

(9.13)

0 when i # j . c [p and q are exponents in Eq. (9.13) .I 1 3

Therefore, IW(UP) =

n

1

(9.14)

UfWi.

i=l

The approximation in Eq.(9,8) is obtained by wedge quadrature with degree one wedges. Although wedge quadrature has application, more efficient schemes are usually preferable. For general application to problems with degree one approximation within elements, we now describe a method that yields degree two quadrature formulas that have reasonably low errors for cubics. We connect the element vertices with straight lines to form a polygon and choose interior point P with coordinates

xp

-

= (l/n)

n

n

xj

.

and

yp = (l/n)

j=1

j=1

261

yj.

(9.15)

FIN ITE ELEMENT D ISCRETI ZATlON

We then partition the polygon into triangles by connecting its vertices to P. Integrals over the polycon may be estimated by summing quadrature . formulas over the triangles and by then adding or subtracting (for convex or concave polycon sides) estimated integrals over the segments bounded by polycon boundaries and polygon sides. For example, the integral of f over the element shown in Fig. 9.1

Fig. 9 . 1 .

E l e m e n t p a r t i t i o n for q u a d r a t u r e .

is approximated by I ( f ) = I(f;polygon)

-

+ I(f;segment

I(f;segment [2,3,6]).

[4,5,71)

(9.16)

We seek quadrabure formulas for triangles and segments. There are three points, say r, s, and t, in each triangle or segment such that reasonable accuracy is achieved by the formula I ( f ; segment

or triangle) = ( K / 3 ) (fr + fs + ft),

(9.17) where K is the area of the segment or the triangle. For any triangle, there is a family of formulas like (9.17) for which degree two quadrature is achieved. The ellipse in Fig. 9.2 has the triangle centroid as its center and contains the feet of the

268

RATIO NAL FINITE ELEMENT BASIS

I Fig. 9.2.

& 3

Degree t w o t r i a n g l e quadrature.

triangle medians. I f we choose any point on the ellipse as one of the quadrature points, then two other points on the ellipse are uniquely determined f o r use in a quadrature formula like (9.17). The side midpoints (feet of the medians) are one set. A particular choice that has been widely used is the median trisection points labeled r , s, and t in Fig. 9.2. Quadrature formula (9.17) with these median trisection points is well suited for the integrals that occur in finite element studies. The coordinates of the median trisection points are 2 1 xr = 3 x1 + g (x2 + x3), 1 (xl + x3), x = 2 x2 + (9.18) S

Xt =

6

2 1 5 x3 + g (xl + xz)

#

and similarly for the y values. Segment quadrature formulas are not as easily obtained. I t is convenient to introduce the iso,parametric segment to.obtain wedge integrals. This is the segment in which the true arc is replaced by a parabolic arc through the segment vertices and the 269

FINITE ELEMENT DISCRETIZATION

point of intersection of the perpendicular bisector of the chord with the arc. The bisector is the axis of the isoparametric parabola(Fig. 9.3). We have for

Fig.

9.3.

T h e isoparametric s e g m e n t .

this segment:

We evaluate this integral for U(x,y) equal to each of 1, x, y, x2, y2, and xy to obtain: area K of the segment = 4bc/3, centroid for segment = (0,-3b/5),

(9.20a) (9.20b)

(9.20d) We seek the coordinates of points r, s 8 and t for Eq.(9.17). By symmetry, we must have xr = -xt’ yr - yt, and xs = 0. For (9.17) to yield ( 9 . 2 0 ~ ) ~ we require that xr = - A m c, and for (9.17) to yield (9.20b) we must have (1/3) (yr + ys + y ) = t (2/3)yr + (1/3)yS = -(3b/5), or

270

RATIONAL FINITE ELEMENT BASIS

Y S = -[2yr

+ (9b/5)].

To o b t a i n (9.20d) from ( 9 . 1 7 ) ,

2(yr)

2

+ (ySl2

(9.21)

we require t h a t (9.22)

= 9b2/7.

There are two d i s t i n c t s o l u t i o n s t o ( 9 . 2 1 ) One o f t h e s e ( F i g . 9 . 4 ) i s y , = -0.9696b y r = -0.4152b. and y,

and ( 9 . 2 2 ) .

and

The o t h e r ( F i g . 9 . 5 ) i s yr = -0.7848b

= -0.2304b.

Fig. 9.4. Integration points; solution A .

Fig. 9.5. Integration points; solution B.

As b/c increases, t h e segment a p p r o a c h e s a t r i a n g l e and t h e p o i n t s i n F i g . 9.4 are c l o s e t o t h e m i d p o i n t s of t h e sides.

Although t h e r e s u l t i n g

q u a d r a t u r e f o r m u l a i s exact f o r a l l p o l y n o m i a l s t h r o u g h d e g r e e t w o , t h e median t r i s e c t i o n p o i n t s i n F i g . 9.2 have been found to be s u p e r i o r i n t h i s

limit.

The p o i n t s i n F i g . 9.5 move closer to t h e s e

median t r i s e c t i o n p o i n t s a s b / c i n c r e a s e s , and w e t h e r e f o r e c h o o s e t h e s e p o i n t s for segment q u a d r a t u r e :

r = (-0.5477c8-0.7848b) s = (0,-0.2304b) t = (0.5477~,-0.7848b)

(9.23)

W e may u s e t h e t r u e segment area i n E q . ( 9 . 1 7 )

r a t h e r t h a n t h e i s o p a r a m e t r i c area i f t h e t r u e area can be d e t e r m i n e d w i t h o u t undue d i f f i c u l t y .

27 1

FINITE ELEMENT DISCRETIZATION

Analytic integration is not difficult f o r isoparametric wedge functions (that is, rational wedges €or isoparametric segments). We consider, for example, the element in Fig. 9.6. The rational

F i g . 9.6.

Degree t w o rational approximation over an isoparametric segment.

wedges for this segment are the polynomials 2 2 2 2 W1 = [b x(c + x) + 2c y(b + y ) + 2bcxy]/2b c , (9.24a) (9.24b) (9.24~) (9.24d) W5(XIY) = W4(-x,y) I 2 2 w6 = -I(Y/b) + (X /c 1 1 -

(9.24e (9.24f

The integrals of these wedges over the segment are W1

= -K/14,

W4 =

12K/35,

W2 = -K/141

w5

The segment area is K

= 12K/35, =

4bc/3.

272

3 = 2K/35, w6 = 2K/5.

W

(9.25)

R A T I O N A L F I N I T E ELEMENT BASIS

9.3

CONSISTENT QUADRATURE AND THE PATCH TEST

Functional analysis has been applied with great sophistication to the study of convergence of finite element discrete solutions as element dimensions shrink to zero (Aziz, 1972). A model problem that has received much attention is the diffusion equation: -V.pVu

+

qu = f (inhomogeneous), = Xu (eigenvalue),

(9.26)

over region R, excluding its boundary on which appropriate conditions are specified on u. Values for p and q are given, subject to the constraint that For the inhomogeneous (or p > 0 and q _L 0. source) problem, f is given. It has been established that standard finite element techniques with degree one basis functions yield an approximation uh to the true solution u that satisfies

Hey; h is the maximum chord length of the elements Various matrix coefand K is a generic constant ficients must be evaluated for the computation of uh. Crucial parameters in the discretization include (9.28) m' where i and j vary over all node indices of element and where the BY are the basis functions for the Em element. Eq.(9.26) is reduced to the discrete form = ABhyh fh (inhomogeneous problem) or Ahgh = (eigenvalue problem). The underbars denote vectors with a component corresponding to each node.

%sh

273

FINITE ELEMENT DISCRETIZATION

The convergence bound in (9.27) is valid when integrals such as (9.28) are computed with no error to yield the discrete equations. Roundoff error may be neglected in the convergence analysis for most . problems that occur in practice. Even in the absence of roundoff error, however, precise determination of Ah, Bh, and gh is often not feasible. Numerical quadrature is a major source of error in the discretization. A discretization procedure is said to be "consistent" (Herbold, et al, 1969; Strang and Fix, 1973; Aziz, 1972) when the computed approximation uh differs from truth by the same order of h as the ideal approximation (with no error in discretization). Thus degree one consistent approximation for (9.26) yields such that 11 Gh 2 Kh2 The value of K may differ from that of Eq.(9.27). Henceforth, we suppress the superscript m relating byj to element m. We are concerned here with consistent approximation of bij by bij. This has been studied by many investigators, and at least three authors examine this in the work edited by Aziz (1972; Aziz and Babuska on pp. 285-303, Fix on pp. 525-556, and Strang on pp. 689-709). A stability criterion described by Aziz and Babuska that appears as 9.2.6 on p. 291 of Aziz (1972) reduces in this situation to 2 (9.29) Kh a(h), lbij - b.. 11 I -

1.11

.

A

I\

where a(h) -t 0 as h + 0. Although Strang and Fix presume polynomial bases for much of their analysis, experience suggests more general significance of the "patch test" (e.g., Irons and Razzaque on pp.557-587 214

RATIONAL FINITE ELEMENT BASIS

of the book edited by Aziz). for any linear function L:

This test is passed if

1

(9.30) all nodes j of the element for each node i. This is equivalent to the following three conditions for each i: (9.31a)

1 j

and

c i

x.b I ij = c x j g i j , j

(9.31b)

yjbij = cyjGij.

(9.31~)

i

We seek gij to satisfy both (9.29) and (9.31). It is not clear that these are either necessary or sufficient conditions for consistent approximation, but both analysis and experience indicate that these are reasonable criteria, satisfaction of which should yield acceptable discrete equations. Moreover, it will be shown that these conditions can be attained without difficulty. It is interesting to compare some of the consequences of conditions (1.16) on the basis functions and (9.31) on the Gij. We first consider the triangle. Vertices 1, 2 , and 3 are not collinear. Hence, Eqs.(l.l6) determine unique triangle basis functions [see also Eqs. (3.13)-(3.16)]:

275

FINITE ELEMENT DlSCRETlZATlON

(9.32)

Thus the bij in (9.28) are uniquely defined by (1.16) since the Bi are determined by (9.32) as the standard triangle wedges. Even though the integrals in (9.28) are easily evaluated for triangles, it is instructive to show how the bij may be determined directly from (9.31). Let i = 1 in (9.31a) to obtain 3

C

1 3

= I/VWl

-V

Wj dx dy

j=1

j=1 =I/VWl.V1

(9.33)

dx dy = 0.

From (9.31b),

We choose the coordinate axis (Fig. 9.7) to simplify evaluation of this integral. The y-axis is along

Fig.

9.7.

T r i a n g l e c o o r d i n a t e s f o r i = 1.

276

RATIONAL FINITE ELEMENT BASIS

side (2;3) and the positive x-axis contains vertex 1. Let sij be the length of side (i;j) of the triangle. Then 0

J2

dx dy =

I

-

dy [W,(x,y)

Wl(0,y)l on (1;2)

y2 +f3dy 0

[Wl(x,y)

- Wl(O,y)l

+ (y3/2)

= -(y2/2)

= ~

on (1;3)

~ ~ / 2 5

Since x2 = x3 = 0, we have

23 2

so that

The values of W1 on both (1;2) and (1;3) are x/xl

so that in (9.31c),

(9.35) By substituting (9.34) into (9.33) and then solving (9.33) and (9.35) simultaneously, we obtain b12 = -y3/2x1 = -(cot e 3 ) / 2 ,

(9.36a)

b13 = -Y2/2X1 = -(cot e2)/2.

(9.36b)

277

FINITE ELEMENT DlSCERflZATlON

We have shown that just as Eq.(1.16) determines unique triangle basis functions, Eq.(9.31) determines unique values of bij. This uniqueness is a property peculiar to triangles, and this is another beneficial, property of triangles for finite element computation. A quadrilateral has four nodes. There are only three equations in (1.16) so that a unique basis is no longer determined. In fact, we shall presently describe a useful alternative to the rational wedges for discretization. Despite lack of uniqueness of the wedges, it will be shown that each of the three expressions in (9.31) is unique. For i = 1, we choose the coordinate axes as shown in'Fig. 9 . 8 .

I

Fig. 9.8.

f+X

Q u a d r i l a t e r a l c o o r d i n a t e s f o r i = 1.

We recall that W1 is linear on sides (1;2) and (1;4) and vanishes on sides (2;3) and (4;3). The same analysis that has just been applied to the triangle yields for the quadrilateral: 4 4 4 blj = 0, xjblj = s24/2, yjblj = 0. j=1 j=1 j=1 Therefore, the following three conditions on the four $ are obtained from (9.31):

-

-

1

11

278

RATIONAL FINITE ELEMENT BASIS

(9.37c) The coordinate axes in Fig. 9.8 are referred to vertex 1. If we designate the coordinates referred to vertex i by (xi,y!), then patch conditions (9.31) for i = 1,2,3,4 are:. 4

1 iij

=

0,

(9.38a)

j=1

f

in xjbij = (si+l i+3 )/2, (9.38b)

j=l 4

c

j=1 9.4

yjbij = 0. iA

(9.38~)

TRIANGLE AVERAGING h

In some finite element computations, the bij are determined by the method of triangle averaginq (Zienkiewicz and Cheung, 1967). A quadrilateral may be partitioned into two triangles with either diagonal as their comon side. Each of these partitions leads to different contributions to the discrete equations, and the average of the two is used in the triangle-average method. Fair results have been obtained for some problems with this method, but we are concerned with this approach primarily as a springboard to better discretization techniques. 279

FINITE ELEMENT DISCRETIZATION

Let (i,j,k) denote the interior angle between lines (i;j) and (j;k), where i, j, and k are vertices of the quadrilateral. Triangle averaging yields

(9.39)

The patch test is passed for each triangle partition and Eqs. (9.37) are linear in the Gij. Hence, the averages in (9.39) must also satisfy the patch test. This may be verified directly, but we omit the somewhat tedious algebra. The trapezoid in Fig. 9.9 is convenient for illustrating various discretization techniques.

=I

s34

Fig. 9 . 9 .

An i l l u s t r a t i v e t r a p e z o i d .

For this trapezoid, Eqs.(9.39) give

280

RATIONAL FINITE ELEMENT BASIS

Analytic values for this trapezoid are found in Section 9.6 for the rational wedges. Referring to Eq.(9.61), we note that b44 = 0.3856 for rational basis functions and that this differs appreciably from the triangle-average value of b44 = 0 . 5 O f i . It is difficult to assess the effect on convergence of the triangle-average approximation. A less heuristic procedure which involves more arithmetic is to use the piecewise linear basis functions obtained by averaging the basis functions associated with each vertex for the two triangulations. Although this provides a basis for continuous degree one approximation, the basis functions are creased along the quadrilateral diagonals. For the illustrative trapezoid, the basis function associated with vertex 4 is linear over each of the four triangles determined by the sides and the diagonals (triangles A # B, C I and D in Fig. 9.9) and has values: B4(x4,y4) = 1, B4(x1,y1) = B4(x2,y2) = B4(x3,y3) = 0, and B4(x7,y,) = 1/6. (Point 7 is the point of intersection of the diagonals.) We denote the Gij obtained with these piecewise linear basis functions by bij. Values for the b i j may be determined by expanding in terms of triangle bases and summing contributions from triangles A-D. For example, let W1, W4, and W7 be the wedges for triangle A. Then in triangle A, B4(x,y) = W4 + W7/6 and the contribution to bi4 from integration over A is bi4(A) = b44 + b77/36 + b47/3, where the bij are the triangle values given in Eqs. (9.34) and (9.36) : A

1 1 + cot 3 21T + -(cot 1 bi4(A) = z[COt 36 6 - -13 cot KIT ] = 7

@.

28 1

IT 6

+

cot gIT )

FINITE ELEMENT DlSCR ETlZATlON

Summing the contributions from the four triangles, we obtain bi4 = 5a/12 = 0.416715. (9.41) The remaining b' are readily computed from (9.38): ij bi2 = -J3/3, bi3 = -J5/6, bi4 = J5/12, (9.42) bi3 = 20/3, bi3 = -J5/6, bi4 = -J5/3. We observe that this latter procedure gives values that are much closer to the rational wedge integrals ;computed in Section 9.6) than the triangle-average values. This suggests that averaging the basis functions may be superior to averaging the values determined by the two triangulations. 9.5

MOSAIC DISCRETIZATION

A careful scrutiny of the second triangle averaging method introduced in Section 9.4 suggests generalizations. The integration in this approach is precise for a piecewise linear basis, although this basis is not as smooth as the rational wedges. Averaging of triangle basis functions is a trivial example of a general technique that we will call mosaic discretization. Let D be a well-set algebraic element over which degree k approximation is achieved by the rational n. Let F be the basis functions Wi with i = 1,2, bilinear functional associated with a particular boundary-value problem. We seek the iij that approximate bij = F(Wi,W.). (9.43)

...,

3

Suppose bij is easily evaluated for any triangular element within prescribed tolerances for each i,j. 282

RATIO NAL FINITE ELEMENT BASIS

Rectangles or other appropriate simple elements could be used instead of or in addition to triangles, but we shall confine this study to triangles. We triangulate element D as for a conventional finite element computation over region D. Let the triangle m nodes be s = 1,2,. ..,t and let Ts(x,y) be the triangle basis function associated with node s of triangle m. The triangle bases are chosen to yield degree k' approximation over each triangle in terms of its nodal values, where k' - k. Let

The sum is over all triangles that contain both s and s'. Let matrix C be defined as the matrix of order t with elements cssl. Numerical solution of the boundary-value problem could be attempted with the triangle grid replacing the single element D. It is assumed, however, that the triangles are smaller than needed for the desired accuracy and that algebraic element D is o f a more appropriateasize. The triangulation is introduced only for evaluation of the gij. Let Wis denote Wi(xs,ys), the value of the rational wedge associated with node i of D at node s of the triangle network. We define T

wi =

mil,Wi2I .. ., Wit) ,

(9.45)

and consider the following approximation to bij: bij = EiCW.. T (9.46) -J It will be shown that this is the precise value of F(Mi,Mj), where the Mi (i = i,2, ,n) are a basis A

...

283

F INITE ELEMENT D ISCR ET IZATlON

for degree k approximation over D, similar to the basis constructed in Section 9.4 by the second triangle averaging method. As the maximum chord length of the triangles decreases, the bi approach the bij. Mosaic basis functions Mi are defined by Eqs. ( 9 . 4 7 ) and ( 9 . 4 8 ) : Mi(Xs'Ys) - wis (9.47) for i = 1,2,...,n and s = 1,2,.. .,t, and (9.48)

s (m) for (x,y) in triangle m with nodes s(m). We observe that Mi is the patchwork degree k' approximation to Wi over D, where k' is the degree of approximation (not less than k) attained by the.:T It will now be shown that the Mi are a basis for degree k approximation over D. For any polynomial Pk of maximal degree k, n

n

(9.49)

for all (x,y) in triangle m. Since the Wi are a basis for degree k approximation over D:

for (x,y) in D. n

In particular, for each s

284

RATIONAL FINITE ELEMENT BASIS

Summing over i on the right-hand side of (9.49), we obtain

for all (x,y) in triangle m. Since the Tt are a basis f o r at least degree k approximation over this triangle, the right-hand side of (9.52) is equal to Pk(x,y) and we have proved that n

we now define

I

for (x,y) not in m. m m F(Mi,Mj)

1 0

Then F(Mi,Mj) = all m in D

s,s'

m (9.55)

285

FINITE ELEMENT DISCRETIZATION

Referring to Eq. (9.46) , we observe that Gi = F(Mi,M.) as asserted. By Eq.(9.53), the Mi are a 7 basis for degree k approximation over D. There may be preferable approximations for specific elements, but mosaic discretization has many commendable features. It is applicable to any algebraic element for any degree of approximation. Any program designed for simple elements can be extended to treat more complicated elements with relative ease. One may use the existing program to compute the C matrix. When solving an inhomogeneous problem, one uses the triangle grid to determine components fS =

1

G(f(x,y) ,T;(X,Y)

1

(9.56)

m

of the vector (9.57)

where G is a known functional. The contribution from integration over D to the inhomogeneous term in the equation associated with node i is T f for i = 1,2, fi = W. n. (9.58) -1 -

...,

A quadrature correction factor may be applied

near curved boundaries. Let nodes s1 and s 2 of a triangulated well-set algebraic element lie on a curved boundary (Fig. 9-10). We introduce vertex s3 between s1 and s 2 on the boundary curve in order to facilitate quadrature over the element. Vertex s3 is at the intersection of the perpendicular bisector of line (sl;s2) with the curved boundary. The curve is then approximated by the isoparametric arc through

286

RATIONAL FINITE ELEMENT BASIS

Fig.

9.10.

A n i s o p a r a m e t r i c s e g m e n t of a

t r i a n g u l a t e d e l emen t

.

the three boundary nodes. In Section 3.6, we observed that the degree one basis functions for any two-sided polycon are identical to those of the triangle with the same nodes. The area of an isoparametric segment is 4/3 times the area of the triangle [Eq.(9.20a)]. The gradient of each degree one basis function is constant over the segment. Hence, when one is concerned with discretization of the Laplacian, bs.s

(

isoparametric segment

dx dy

=I

= (4/3)bses (triangle). 1 1

For any functional, we may estimate the segment contribution as 4 / 3 times the contribution from the inscribed triangle to obtain =SS'

(segment) = (4/3)css, (triangle). (9.59)

If the boundary arc is part of a concave boundary of D, then the segment is exterior to D and the value in (9.59) must be subtracted from cSS,. Even though rational basis function Wi of D can be singular in such an escribed segment, Mi is well defined here.

287

F INlTE ELEMENT D ISCR ET IZATION

For any specific functional, one may develop quadrature formulas for isoparametric segments. The polynomial basis functions and parabolic arc make the associated inbegrals tractable for a broad range of functionals. This analysis generalizes to three dimensions. Rectangular parallelepipeds are suitable elements for integration. Tetrahedra may also be used. It might be feasible to devise special elements for use along boundaries, but in the absence of such elements one may use a sufficiently fine partition to model boundaries within prescribed tolerances. 9.6

A DISCRETE LAPLACIAN FOR QUADRILATERALS

The convex quadrilateral is a simple straightsided two-dimensional element for which precise integration is often impractical. This element has great theoretical importance and is of considerable practical interest. We examine in this section discretization of the Laplacian over networks that include convex quadrilateral elements. Closed form evaluation of bij

=))

VWi.VW. dx dy 3

(9.60)

can be accomplished over any convex quadrilateral through use of projective coordinates. Application of these formulas involves considerable algebra and alternative evaluations seem preferable. For degree one approximation over the trapezoid in Fig. 9 . 9 , we have W4(x,y) = [(Val - (x/y)l [y - (fi/2)1, VW4 = [(fi/2y) - 112 + [(l/fl) - (J5x/2y2)Ii, and

288

RATIONAL FINITE ELEMENT BASIS

We perform the integration in (9.60) to obtain b44

Y/fi

-

[VW412 dx = 5fi In 2. 9 (9.61)

This integration is not so easily performed f o r the general quadrilateral. Having f.oundb 4 4 , we may use the patch conditions [Eq.(9.38)1 to determine the remaining bij. To compute the bi4 for i = 1,2, and 3 from b44 we use the coordinates in Fig. 9.11.

Fig. 9 . 1 1 .

C o o r d i n a t e s for d e t e r m i n i n g b

i4'

From (9.38a) : b14

+

+ b34 -

b24

-

From ( 9 . 3 8 ~ )-nbl4 ~ bZ4

-

2.

(9.62)

(fi/2)b24 = 0 , or

+

2b14 = 0.

From (9.38b), -bZ4/2 + b44

b24 = 2[b44

-543 ln 9

-

(9.63)

J5/2 and

- 09' In

(JS/2)l =

2).

(9.64)

By substihuting (9.64) into (9.631, we obtain b14 = ( , 6 / 2 ) (1 -

% In

289

2).

(9.65)

FINITE ELEMENT DISCRETIZATION

Substitution of (9.64) and (9.65) into (9.62) yields

(9.66)

To compute the bi3 for i = 1,2, and 3, we use the coordinates in Fig. 9.12. From (9.38a), we obtain

Fig.

C o o r d i n a t e s for d e t e r m i n i n g b

9.12.

b13 + b23

+

10 In 2 b33 = -b43 = fi(T

- z). 1

i3'

(9.67)

From (9.38b): -b13

+ (b33/2)

From (9.38c), (nb13/2) and

(9.68)

= a/2.

+ (J5b23/2)

-

(t'%43/2)

-

10 In 2 1 = -1. b13 i-b23 = b43 2 Substituting (9.69) into (9.67) , we obtain

b33 =

J5(-20 In 2

Then (9.64) yields b13 = -D(1

-

1).

- - In

2).

= 0,

(9.69)

(9.70) (9.71)

Substituting this value for b13 into (9.69), we obtain 3 In 2). (9.72) b23 - b43 b13 = f i ( 2

-

-

290

R A T I O N A L F I N I T E ELEMENT BASIS

The remaining coefficients may be determined in the same manner. It is hoped that this example clarifies use of the patch conditions in computation of the discrete equations. An alternative to analytic integration that is no more difficult for the general quadrilateral than for this illustrative trapezoid is evaluation of 644 by trimedian quadrature. We may partition the quadrilateral along diagonal (1;3) and apply (9.17) with the coordinates in (9.18) for each of the two triangles. For the trapezoid in Fig. 9.9, this yields b44 = 0.3800, which compares quite favorably with the triangle-average approximations. The remaining b may be determined from (9.38). By ij determining the other Gij from bd4 in this manner, we assure satisfaction of the patch conditions independent of the quadrature error in evaluating b44. Trimedian quadrature has quadratic accuracy so that a(h) in (9.29) is equal to h and stability is assured as well. The trapezoid in Fig. 9.9 and the functional associated with the Laplacian provide a convenient illustration of mosaic discretization. We seek an approximation for bij of degree one rational basis functions. If we use degree one triangle basis functions over the triangulation in Fig. 9.9, we obtain with the aid of Eqs. (9.34) and (9.36) for equation ordering 1, 2, 3, 4 , 7 the matrix given in Eq.(9.73). We note that f o r the rational wedge = ~0,0,0,1,~1/6~1 and , that G44 W4(x,y): therefore depends on the elements c44, c45, c54, and cS5. A

A

A

A

3

291

FIN ITE ELEMENT DISCRETIZATION

(9.73) 1 3

0

1 -7

1

-1

-2

-2

-1

\

From Eq.(9.46),

-1

6 ,

we obtain

in agreement with Eq. (9.41). The triangulation in Fig. 9.13 yields identical coefficients. If on the other hand we partition the quadrilateral into 12 equilateral triangles as shown in Fig. 9.14, we obtain h

b44 = (85/216)n = 0.3940.

(9.75)

The difference between this value and the rational wedge integral is about one quarter of the difference between the result of the coarse triangulation and the rational wedge integral.

Fig. 9.13.

Three t r i a n g l e s . F i g . 9 . 1 4 . 292

Twelve t r i a n g l e s .

RATIONAL FINITE ELEMENT BASIS

Instead of refining the triangulation, we may use higher degree approximation over the coarse triangulation. The bij for degree two basis functions over the equilateral triangle in Fig. 9.15 are

(Values not displayed above may be obtained from symmetry considerations.)

F i g . ,9.15.

A degree two t r i a n g l e .

We consider degree two approximation over the triangulation of Fig. 9.13 with nodes chosen as in Fig. 9.16. The vector associated with node 4 in

1

2

3

4

10 I I

12

5

Fig. 9.16. Nodes f o r d e g r e e two approximation over a t r a p e z o i d p a r t i t i o n e d i n t o t h r e e t r i a n g l e s .

Fig. 9.13 is

Matrix C for the 12 nodes in Fig. 9.16 is:

293

FINITE ELEMENT DISCRETIZATION

‘ I -72 3

1

-_2

0

0

0

0

z

c = -a 3

,

g1

0

0

-7

2

0

0

-_4 --4

2

1

0

2

0

-6 - -3

1

0

0

0

0

4 - -

0

o - -

‘5

3

6

-3

-_ -3 0

--4 3

4 -_ 3

0

0

--

0

0

0

1 6

0

1 -

0

o

0

0

0

1 3o

$

o

o

3

3

-

--3

0

--3 --3

3

3

3 4 3

0’

0

0

0

3 .

0

0

0

o

o

-1

o - - 23

o

o

-_4 -_4

0

0

--4

--4

0

0

-32

2

1

0

0

-- --43 0 --4 --4 3 3 0

3

1

0 - 23 0

8

4 -7

4

8

-$

0 - 43

4

0 - 3 0

0

0

0 - - - 3 3 4 4 0 0 - -3 - - 3 4 1 0 0 3

0

o - -

-- --23

3

3

3

1

6

3

2

3 1 -2 6

3

(9.78) h

In order to evaluate b44 we must compute the first T eight components of 3 C :

ZiC = ( f l / 3 ) ( W 4 , 8 / 9 , 1 / 3 , 4 / 9 , - 1 / 1 2 , - 1 / 9 , -1/9, ...). We then compute T &44 = %C% = (125/324)fi = 0 . 3 8 5 8 a .

1/9,

This value is quite close to the rational wedge integral in Eq. ( 9 . 6 1 ) : b44 = ( 5 / 9 ) (In 2 ) fi = 0.385la. A four node isoparametric basis and fourpoint Gaussian quadrature (see Strang and Fix, 1 9 7 3 , See. 4 . 3 €or details) yields G44 = 0 . 3 8 4 6 0 . 294

2

RATIONAL FINITE ELEMENT BASIS

The values for &44/fi for the trapezoid computed by various schemes are displayed in the following table : Analytic rational wedge integral Triangle-averaging of elements Triangle-averaging of bases 4-node isoparametric element with 4-point Gaussian quadrature Mosaic discretization 1. Linear triangle bases a. Three triangles b. Twelve triangles 2 . Quadratic triangle bases Three triangles Trimedian quadrature

0.3851 0.5000 0.4167 0.3846

0.4167 0.3935 0.3858 0.3800

Although it is difficult to estimate the effect of quadrature error on the overall discretization error, one would expect that the methods for which the tabulated value is close to 0.385 introduce less deviation from the solutiorl in the absence of error in quadrature. Numerical studies are recommended for each class of problems. For general application of mosaic discretization, the above table suggests that one should use degree two mosaic approximation of degree one rational basis functions. In the next section, further insight is given by analysis of "optimum" bases and associated discrete equations.

295

F INlTE ELEMENT DlSCR ETlZATlON

9.7

HARMONIOUS DISCRETIZATION

Basis functions are not uniquely defined by the properties enumerated in Section 1.5. This raises the question of whether or not there is an optimum basis for a given degree approximation over an element. We must, of course, have some measure of optimality. There i s justification for the "least stiffness matrix" criterion posed by Strang and Fix (1973, pp. 99-100). Finite element equations are commonly derived from a variational principle. Computed nodal values minimize a functional over a prescribed approximation space. For example, when solving Laplace's equation over region R subject to the constraint that the solution have prescribed variation on boundary a R of R, we minimize over space T the functional

{V

E

T

I

dx dy, (9.79) F[vl =JJlVvI' 1 v E C (R); v(x,y) = u(x,y) on aR),

P 1 where C (R) is the space of continuous functions P over R (including aR) with piecewise continuous first derivatives. The finite element approximation subspace S of T depends on the basis functions B and is restricted to functions w for which F[w;B] =

xT K(B) w_,

(9.80)

where K(B) is the "stiffness matrix" derived with basis functions B and where the components of w - are the values of w(x,y) in space S at the element nodes. Let Kr be the contribution to K from integration over element Er in R and let contain the values

xr

296

R A T I O N A L F I N I T E E L E M E N T BASIS

of w at the nodes of E from Er is Fr[w;BI = yr of yr be wi, as obtained finite element equations w(x,y;B) =

1

Then the contribution to F Kr K ~ .Let the components by solution of discrete for basis B. Then

WiBi(x,Y)r

(X,Y)

E

Ere

(9.81)

i&Er The value of Fr is least when V 2w = 0 over Er. This is achieved by the basis H for which V 2 Hi = 0 in Er. Thus Hi is the unique Green's function over Er associated with the boundary conditions on the wedge normalized to unity at node i. It will now be shown that these harmonic wedges achieve degree one approximation. By virtue of the boundary conditions on the Hi (Fig. 9-17], for any linear u(x,y):

on boundary aEr of Er.

Fig. 9 . 1 7 .

Moreover, V 2 g = 0 in Er.

Boundary-value

problem for H

i'

The harmonic function g vanishes on aEr and so must be identically zero in Er. Thus the H. are a basis 1 €or degree one approximation over Er. Harmonic wedges cannot provide a basis for higher degree 297

FINITE ELEMENT DISCRETIZATION

approximation since polynomials of degree greater than one are in general not harmonic. For given nodal values, F[w;HJ = min F[w;B]. B

(9.83)

Let wiB) have as its components nodal values obtained by minimizing F[w;B] From (9.83) we obtain

.

F[w(H) ;HI = min F[w;H] - F[w(B) ;B].

(9.84)

W

A fundamental property of the variational method is that the harmonic basis functions minimize functional F. The Hi are related to the Peano kernels that appear in the general theory of L-splines (Varga, 1971). Even though harmonic wedges do not form a degree k basis when k > 1, the minimum principle guarantees that the harmonic wedge integral is less than that obtained with degree k interpolation of the true nodal values within each element. In general, harmonic bases yield the least possible F for a given element structure and degree of approximation on element boundaries. Moreover, the maximum principle for harmonic functions assures us that the maximum error in the approximation over Er must be on 3Er. This need not be true for any other choice of basis functions. For any homogeneous elliptic problem, the "optimum" basis functions satisfy the partial differential equation that is the Lagrangian of the functional. We call such basis functions "harmonic". In general, harmonic functions are not known explicitly for elements that occur in finite element 298

FINITE ELEMENT DISCRETIZATION

computations. Any scheme in which an attempt is made to approximate discrete equations determined from harmonic basis functions will be called harmonious discretization. The method of mosaic discretization would assume new significance if we could devise some means for evaluating the harmonic basis functions at all the interior nodes of a partitioned element. (The values are known on the element boundary.) It turns out that e v e n t h o u g h t h e a n a l y t i c f o r m of t h e r a t i o n a l w e d g e s i s more e a s i l y d e t e r m i n e d than t h a t of t h e harmonic w e d g e s ,

m o s a i c d i s c r e t i z a t i o n is more e a s i l y i m p l e m e n t e d w i t h the harmonic wedges.

Let C be the coefficient (or stiffness) matrix for the partitioned element as defined in Section 9 . 5 . Let basis function Bi have components on aEr as the elements of K~ and components at nodes interior We express matrix C in to Er as elements of block 2x2 form and obtain from ( 9 . 4 6 ) :

xi.

This is minimized when -1

(9.86) Ei * We, therefore, use ( 9 . 8 6 ) to approximate values of the harmonic basis functions at the interior nodes. This elimination of interior nodes in terms of boundary values is known in the finite element literature as "static condensation". This method of discretization has been used to eliminate interior degrees of freedom of "macroelements" -1 = -Cz2 v.

299

'21

FINITE ELEMENT DISCRETIZATION

(Strang and Fix, 1973, Sec. 1.9). Theoretical ramifications described now suggest broader application of this technique. Having determined by (9.861, we may compute all the bij: T v.] T [w.

xi

"bj

-1 -1

(9.87)

=

Let Er have n nodes before partitioning and define -2 W = [ w-1 w

.

*

En]*

(9.88)

By substituting (9.86) into (9.87), we obtain the fundamental equation for mosaic harmonious discretization:

Kr = [bijl = WT (Cll

-

- c12c22C'

21) W. (9.89)

The word "harmonious" is particularly appropriate. The basis functions are tuned to the problem at hand. We now show that the harmonic basis functions are appropriate even when an inhomogeneous term is introduced. Although we restrict this discussion to Poisson's equation, the results apply to a much more general class of elliptic boundary-value problems. We seek a finite element approximation to the function u E C2 (R) that satisfies V 2u + f = 0 in R for given f in R and u on boundary aR of R. The commonly used functional Flv] = 11 (IVv( - 2fv)dx dy R is stationary at v = u. Finite element computation requires evaluation for each nodal value vr of each element Er of

300

RATIONAL FINITE ELEMENT BASIS

a

-

JJ(lvv12

t 2fv) dx dy =

b11 r vr j

- :f ,

j=1

(9.90) Er where bTj and fT are defined by Eq. (9.90) . Over element Er' let v equal the sum of a homogeneous component vH and a particular component vp. The basic finite element approximation is restriction of variation of v(x,y) on element boundaries. Once the element boundary variation is set, appropriate nodes are selected and a unique harmonic basis function may be associated with each node. The homogeneous component of v in Er is

t

The particular solution solves the boundary-value problem V 2vp + f = 0 over Err vP

E

C 2 (R) and vp = 0 on aEr.

(9.92)

The quadratic component of the integrand in (9.90) is lvv12 = IvvH/2+ Ivvp12

+ 2VVH'VVP.

(9.93)

The second term on the right-hand side of (9.93) does not depend on the vi. That the integral of the last term in (9.93) over Er is zero is seen from

Jj

vVH. VvP

r

dx dy =

[v'(vpvVH) Er

-

VpV

2 vH] dx dy

.

(9.94) 301

FIN ITE ELEMENT D ISCR ETlZAT 1 0N

We note that V 2vH

=

0 in Er and that

V - (vPVvH) dx dy =

Er since vp vanishes on aEr.

J

VPVVH.%

= 0

aEr It follows that

bTj =JJVHi.VHj

dx dy.

(9.95)

Er The linear component of the integrand in (9.90) is -2f(x,y)[vp(x,y) + vH(x,y)]. The particular solution vp does not depend on the vi so that : f

=JJf(x,Y)Hi(x,Y)

dx dy.

(9.96)

Even though F[v] depends on the particular solutions within the elements, the discrete equations from which nodal values vi are determined do not depend on vP' The static condensation procedure outlined in the discussion of Eqs. (9.85)- (9.89) yields Kr of Eq. (9.89) for any f (x,y) (Hoppe, 1973). Let I , and gv be the inhomogeneous terms defined by Eq.(9.58) for the boundary and interior nodes, respectively, of the equations before condensation. Let the inhomogeneous vector of the condensed equations be = ( f:, f;, ..., f : ) . Static condensation yields -1 f = --w f + c 12 c22 (9.97) ZvA s the partition of E is refined, the elements of r matrix Kr approach the bTj in (9.95) and the f approach the : f in (9.96). elements of -

zT

302

The discrete system is solved for the nodal values. The approximation to u within Er is r ri V i ~ (x,y) + v,r(x,y). If we store or recompute the matrices used for the static condensation over *r we can determine values for v(x,y) at the interior nodes. Alternatively, we can use the rational wedges Wi to approximate v(x,y) by r viWi (x,y) The error in this approximation is of the same order of magnitude as the error in the approximation of u by v. Although static condensation eliminates need for rational basis functions in obtaining the discrete equations, the rational bases may be quite useful as interpolation functions. It will be shown in Chapter 10 that the rational wedges play a crucial role in applications where interpolation is essential. We illugtrate harmonious discretization with degree one approximation over the quadrilaterals analyzed in Section 9.6. Similar equations are derived by Felippa and Clough (1970). A convenient partitioning of a quadrilateral into four triangles is achieved by choosing the "isoparametric origin" as the interior triangle vertex (Fig. 9.18). This

1

i=l

Fig.

.

9.18.

A

partitioned quadrilateral.

is point 9 with coordinates equal to the averages of

303

FINITE ELEMENT DISCRETIZATION

the quadrilateral vertex coordinates. We recall that degree one basis functions are approximated by piecewise degree k' mosaic functions. We refer to the case where k' = 1 as "degree one mosaic approximation" and to the case where k' = 2 as "degree two mosaic approximation". For a square element, we compute with little effort for degree one mosaic approximation : 3 bll = T'

b12 = b13 = b14

= - -1

4'

(9.98)

The harmonic basis functions for the square are the conventional bilinear functions so that precise values are obtained with degree two mosaic bases: - 2 1 1 (9.99) bll - 3, b12 = b14 - - 7;' b13 = - 3' Although the coefficients in (9.98) satisfy the patch test, the values in (9.99) should yield less discretization error for a given finite element representation of a region by a collection of squares. The rational and the harmonic mosaic approximation with k' = 2 are the same for the square. We shall see that a radical departure of coefficients for these two choices of basis functions can occur for some quadrilaterals. The analysis in Section 9.6 and the discrepancy between coefficients in (9.98) and (9.99) motivated use of degree two mosaic approximation for further studies. Harmonious discretization with the fourtriangle partitioning shown in Fig. 9.19 yields coefficients that are quite close to the true harmonic coefficients. Node 5 is the isoparametric center. Nodes 6-9 are midpoints of the segments connecting the vertices to node 5. 304

RATIONAL FINITE ELEMENT BASIS

Fig. 9 -19.

D e g r e e t w o mosaic approximation

A computer program was written to compute coefficients based on Eq.(9.89). For the trapezoid analyzed in Section 9 . 6 , a value of b44 = 0.3847J3 was computed. We note that the analytic value for the rational wedges is close to this "optimum" value. Degree one harmonious mosaic approximation yields a value of bq4 = 0.410. The rational wedges are poor when used with nearly ill-set elements. As the interior angle of a quadrilateral approaches 180° at vertex i, the analytic rational wedge value for bii grows without bound. The harmonic basis function integral is bounded even though the gradient of the harmonic wedge at i becomes infinite. A study was made of the coefficients computed by various schemes for the quadrilateral in Fig. 9.20 as a function of the h

h

F i g . 9.20.

.A

nearly ill-set q u a d r i l a t e r a l .

305

FINITE ELEMENT DISCRETIZATION

value of the y-ordinate of vertex 2. For a << 1, the true rational wedge coefficient varies as N/a with N approximately equal to 0.18. Degree two mosaic approximation of the rational wedge W1 with the partitioning of Fig. 9.19 gave in the limit as a -+ 0 a value of b 11 = 6.75. For a = 0, degree two mosaic harmonious discretization with this same partitioning gave Gll = 1.62526. This is quite close to the true harmonic integral. Isoparametric coordinates and four-point Gaussian quadrature gave for a = 0 a value of Gll = 2.1667. The upper triangular part of the symmetric matrix Kr = is displayed below for three techniques:

kijl

Rational mosaic discretization (four triangles, k'=2) 6.7500 -2.8750 -1.0 -2.8750 0 1.1875 1.6875 0 1.0 1.6875 Isoparametric coordinates (4-point Gaussian quad.) 2.1667 -0.5833 -1.0 -0.5833 0.5417 0 0.0417 1.0 0 0.5417 Harmonious mosaic discretization (4 triangles, k'=2) 1.6253 -0.3126 -1.0 -0.3126 0 -0.0937 0.4063 1.0

0

0.4063 Harmonious discretization has significant theoretical ramifications. Static condensation may be viewed as a means for obtaining an accurate

306

R A T I O N A L FINITE ELEMENT BASIS

evaluation of harmonic coefficients. Mosaic approximation of the harmonic basis functions often yields nearly optimal discrete equations with few interior nodes in each element. More crucial to the analysis than degree approximation within the elements (once variation on element boundaries is limited by the placement of boundary nodes) is the accuracy with which the harmonic basis functions are represented. Further insight is gained by analysis of Laplace's equation with triangle elements. The 3-node linear basis functions for a triangle are harmonic and hence yield optimum equations. For this simple case we have already seen that there is only one possible basis and that the patch conditions define unique coefficients (Section 9.3). If we allow quadratic variation on each triangle side, we require six nodes. The analysis parallels that of the quadrilateral. One might suppose that the patch conditions for degree two approximation over a triangle are 6

-

h

b. . ) = 0 =I

(9.100)

j=l for 0 5 s + r 5 - 2 and i = 1, 2, 6. If degree two approximation were our objective, then this would be appropriate. Just as for degree one approximation over a triangle, the degree two coefficients would be uniquely determined by the patch conditions. Harmonic basis functions do not yield degree two approximation. Patch conditions need only be imposed for the harmonic components of polynomials of maximal degree two. Instead of

...,

307

FINITE ELEMENT DISCRETIZATION

(9.100), the harmonic patch conditions are

c 6

pj(bij

-

j=1

h

b. . ) = 0 13

(9.101)

2 for p = 1, xj, yj, x.Y., and (x2 - y.) and for j 3 3 j 3 i = 1, 2, 6. There are only five conditions for each i in (9.101). Just as for the degree one quadrilateral, there is one fewer condition than there are coefficients. In general, any one coefficient together with the patch conditions and symmetry of the bij suffices f o r determination of all the coefficients. (There are special cases where the patch conditions are reducible and specification of any of a certain subset of the coefficients does not determine the other coefficients. This will be clarified by example.) A computer program was written to compute the triangle coefficients €or 6-node quadratic and €or 6-node mosaic harmonic basis functions. Degree two mosaic approximation was used with the partitioning shown in Fig. 9.21, in which the triangle centroid

...,

h

2

Fig. 9.21.

A partitioned triangle.

RATIONAL FINITE ELEMENT BASIS

is the interior vertex and all side nodes are at midpoints. A standard degree two approximation was used over each of the three sub-triangles. Values at nodes 7-10 were eliminated by static condensation [Eq.(9.89)1. A more refined partitioning for the static condensation did not yield significantly different coefficients. The piecewise quadratic approximations of the harmonic basis functions are quite accurate. Striking differences were observed between the conventional 6-node coefficients and those derived by mosaic harmonic discretization. This is true even for the equilateral triangle. It is possible that one may realize a significant improvement in computational efficiency through use of harmonic basis functions. This is an important area for further research. Coefficients for a few triangles will now be displayed. The upper triangular part of Kr = [b..] is given for each case. h

17 EXAMPLE 9.1.

An equilateral triangle (Fig. 9.22)

Fig. 9 -22.

An e q u i l a t e r a l t r i a n g l e .

(1) Quadratic basis. 0.5773

0.0962 0.5773

0.0962 0.0962 0.5773

-0.3849 -0.3849 0 2.3094

0 -0.3849 -0.3849 -0.7698 2.3094

-0.3899 0 -0.3899 -0.7698 -0.7698 2.3094

309

FINITE ELEMENT DISCRETIZATION

(2) Mosaic harmonic basis

(three sub-triangles).

0.4491 -0.0321 -0.0321 -0.2566 0.1283 -0.2566 0.4491 -0.0321 -0.2566 -0.2566 0.1283 0.4491 0.1283 -0.2566 -0.2566 2.1811 -0.8981 -0.8981 2.1811 -0.8981 2.1811

.

( 3 ) Mosaic harmonic basis (nine sub-triangles) To assess the accuracy of the three-triangle

partitioning, the equilateral triangle was partitioned into nine triangles as shown in Fig. 9.23 with degree two mosaic approximation.

Fig. 9.23.

A nine

triangle partition.

For this partitioning, the coefficients were: 0.4326 -0.0485 -0.0485’-0.2402 0.1448 -0.2402 0.4326 -0.0485 -0.2402 -0.2402 0.1448 0.4326 0.1448 -0.2402 -0.2402 2.1648 -0.9146 -0.9146 2.1648 -0.9146 2.1648 In the following table, percentage differences from the nine-triangle results are indicated: degree two 3 triangles 9 triangles 0.5773 (33%) 0.4490 (3.8%) 0.4326 bll 2.3094 (6.7%) 2.1811 (0.76%) 2.1648 644 A

310

RATIONAL FINITE ELEMENT BASIS

EXAMPLE 9.2.

A 45O

-

Fig. 9 . 2 4 .

45O

-

90° triangle(Fig. 9.24).

A 45°-450-900

triangle.

(1) Quadratic basis.

1.0000

0.1667 0.5000

0,1667 -0.6667 0 -0.6667 0 -0.6667 0 0 0.5000 0 0 -0.6667 0 2.6667 -1.3333 2.6667 -1.3333 2.6667

(2) Mosaic harmonic basis. 0 0.6444 -0.0111 -0.0111 -0.3111 0.4111 -0.0889 -0.4889 0 0 0.4111 0.1778 2.3111 -1.3333 2.6667

-0.3111 0.1778 -0.4889 -0.3556 -1.3333 2.3111

For this right triangle, the quadratic basis function associated with node 5 is harmonic so that the fifth columns of the above matrices are identical. The patch conditions are reducible and a value of brs with neither r nor s equal to five is required for unique determination of the remaining coefficients from the patch conditions. A

311

FINITE ELEMENT OlSCRETlZATlON

EXAMPLE 9.3.

Fig.

An

obtuse triangle (Fig. 9.25).

9.25.

An obtuse . t r i a n g l e .

(1) Quadratic basis. 10.101 1.7000 1.6667 -6.8000 0 -6.6667 0 2.6000 -0.8333 -6.8000 3.3333 2.5000 0 3.3333 -6.6667 20.267 -13.333 6.6667 2.0.267 -13.600 20.267 (2) Mosaic harmonic basis. 4.1602 0.1709 0.1964 -0.8014 -2.9405 -0.7857 2.2064 -1.2118 -5.2558 2.5764 1.5139 2.1361 1.4848 2.6055 -5.2110 14.2087 -10.364 0.7275 18.811 -10.689 14.444. These limited studies suggest numerous paths €or investigation, but further analysis of harmonious discretization would lead us too far afield. For application of harmonious discretization to a general algebraic element, one could partition the element into 6-node isoparametric triangles with a common interior vertex. A conventional finite element program could be used to obtain coefficients for the partitioned element. Then Eq.(9.89) would yield the contribution of the element to the discrete equations. A more refined partitioning could be used 312

RATIONAL FINITE ELEMENT BASIS

for a better representation of curved boundary arcs of the element by the isoparametric arcs. Harmonious discretization provides a practicable means for determining nearly optimal coefficients for any algebraic reticulation. When less timeconsuming schemes are preferred, harmonious discretization provides a standard for comparison in preliminary analysis. In any event, the rational wedge functions provide a regular basis for interpolation within an element of values and derivatives in terms of computed nodal parameters regardless of the basis used in the discretization.

313

Chapter I 0

TWO-LEVEL COMPUTATION

10.1 RECAPITULATION We have developed an extensive theory for

construction of basis functions over algebraic elements. The fundamental theorem of Noether and associated algebraic geometry concepts were invaluable in the analysis. It was demonstrated that by appropriate choice of nodes any prescribed degree basis can be constructed for any well-set algebraic element. Practical guidelines were presented for numerical quadrature over elements. The analysis is of interest €or its mathematical content and f o r its resolution of questions of existence and uniqueness of rational basis functions for continuous patchwork approximation over algebraically reticulated regions. The practical utility of this development has yet to be demonstrated. When performing a finite element computation, one chooses elements appropriate for the configuration being studied. For most two-variable problems of concern, triangles and rectangles (and less often parallelograms) are used. In some situations 314

RATIONAL FINITE ELEMENT BASIS

isoparametric coordinates are introduced to model curved boundaries and interior interfaces. In three dimensions, isoparametric parallelepipeds allow considerable modeling flexibility. There seems to be no great need for the more general algebraic elements and associated rational basis functions. Before the discovery of isoparametric coordinates there seemed to be no great need for more general elements than triangles and parallelograms. Now isoparametric elements are considered indispensable for solving a wide class of problems. Computation sophistication rises to meet the limits of the available methodology. It is not unreasonable to expect that eventually problems will arise for which the wider class of algebraic elements will be essential. For example, element geometry may be dictated by physical considerations that preclude three- or four-sided two-dimensional elements. The theory awaits such application. There are computational techniques of current concern for which the more versatile algebraic elements provide a needed flexibility. In this chapter we describe two such techniques. 10.2

SYNTHESIS

The term "synthesis" has been used in the nuclear reactor physics literature to denote a class of computations in which the solution to a problem in n variables is approximated by linear combinations of solutions to auxiliary problems in n - 1 variables. The coefficients of combination are a function of the nth variable. A common example is the solution of the neutron group diffusion equa315

TWO- LEV EL COMPUTAT 10N

tions in three space dimensions (Kaplan, 1965). Let Hm(x,y) for m = l121...,M be a collection of precomputed two-dimensional basis functions. These functions are often obtained by solving two-dimensional problems characteristic of different elevations (constant value for z) of a reactor. Let Wi(z) for i = 1 1 2 1 . . . r 1 be the one-variable hat functions [see Eq.(1.5)]. The synthesis approximation space is I

M

The free parameters U (sometimes called combining im or mixing coefficients) are determined by the usual Ritz-Galerkin procedure to yield a "best" approximation in space S to the solution to the threedimensional problem. By applying interior boundary conditions on the Uiml we can select subsets of the Hm for use in each of several axial zones, thereby reducing the number of unknowns at each elevation zi. This type of synthesis computation is extremely useful and provides a flexible model for analysis of complex configurations in a far more efficient manner than would be possible with detailed threedimensional discretization. An additional degree of flexibility is introduced by multichannel synthesis, where a coarse finite element network is superimposed on the (x,y)-plane as shown in Fig. 10.1. Let Ws (x,y) be the wedge basis function associated with 7 node j over element s . The trial function within the x,y region of element s is

316

R A T I O N A L FINITE ELEMENT BASIS

I

M

J

Suppose Hm is obtained by solving (for each m) a two-dimensional problem having N nodes in the (x,y)plane. Three-dimensional discretization without synthesis would require I N nodes. The singlechannel synthesis approximation of Eq.(lO.l) has I M free parameters. The multichannel mockup of Eq. (10.2) has I M J free parameters. In reactor computations it is not uncommon for N to exceed 1 0 , 0 0 0 and for M to be equal to (about) 5 . The full freedom of (10.2) is not used in practice.

Fig. 10.1.

S u p e r i m p o s e d coarse structure.

Some of the synthesis computation advantages are lost when M J becomes large. The multichannel structure is introduced to permit a more realistic representation of gross tilts in (x,y) variation. This is accomplished even if we restrict groups of points to have the same combining coefficients. Within selected elements, we restrict U imj to be 317

TWO-LEVEL COMPUTATION

the same for all nodes j and each im.

6

In Fig. 10.2,

6

6

5

Fig. 10.2.

4

R e s t r i c t e d multichannel nodes.

there are four elements over which this restriction is imposed. The values for j are indicated in the figure. Thus the 26 nodes in the plane have only six degrees of freedom. We note that element 1 is ill set but that this poses no problems. In this element we have I

We have eliminated the difficulty accompanying illset elements by the method of restricted variation described in Section 1.3. The superimposed coarseelement trial function is equal to unity over the ill-set element. The fine-element structure over which Hm(x,y) was computed in advance is convenient for use as an integration grid when applying the Ritz-Galerkin method to obtain the discrete three-dimensional For equations with the trial function of Eq.(10.2). this discretization we must evaluate Ws(x,y) at each 3 node of the integration grid. This is easily done 318

H A 1 I U N A L FINITE ELEMENT B A S I S

with the rational wedge functions. On the other hand, isoparametric basis functions would lead to difficulties. These functions cannot be readily evaluated in terms of x and y. The rational basis functions are ideally suited for multichannel synthesis application. They permit a wide class of algebraic elements. They may be evaluated easily for integration over the fine structure. The rational wedges provide precisely the flexibility needed to bridge the gap between detailed three-dimensional computations and the single-channel synthesis of Eq.(lO.l). In the next section, we consider another application which is closely related to multichannel synthesis but which arises in entirely different circumstances. 10.3

COARSE MESH REBALANCING

new technique for accelerating convergence of linear iterative procedures was introduced by the author (Wachspress, 1966, Chap. 9). One interrupts the linear iteration periodically with a variational acceleration computation in which the last iterate appears as a base function multiplied by a patchwork coarse mesh correction function. Nakamura (1971) subsequently analyzed use of a coarse mesh finite element representation for the multiplicative correction. He and Froehlich (1967) provided more extensive theoretical foundations for the method and performed many numerical experiments. They called this class of computations "coarse mesh rebalancing" The two-level structure is analogous to that of multichannel synthesis, but the motivation is quite A

.

3 19

TWO-LEVEL COMPUTATION

different. Let u - be the solution to a discrete fine mesh problem and let u ( x , y ) be the continuous patchwork approximation with nodal values equal to the components of g. Let &L be the approximation to y obtained after t iterations. Let the coarse finite element basis functions be W . (x,y). Then the trial 3 function for the coarse mesh rebalancing is

c J

Ut(X,Y)

=

j=1

U . W . (X,Y)U,(X,Y).

7 3

(10.4)

There are J free parameters in the rebalancing computation, and the coarse representation is chosen so that these.parameters may be determined by an efficient direct method such as block Gaussian elimination (Wachspress, 1966, p. 26). When the "best" values for the U are substituted into (10.41, j the nodal values of the resulting Ut(x,y) replace the components of gt and the linear iteration is resumed. Periodic coarse mesh rebalancing has been demonstrated to be an effective means for accelerating convergence for a wide assortment of problems. The rational basis functions permit use of much more diverse coarse mesh representations than was hitherto possible. Physical boundaries and interfaces may be represented by algebraic curves. Thus availability of general algebraic elements broadens the scope and enhances the effectiveness of coarse mesh rebalancing.

320

RATIONAL FINITE ELEMENT BASIS

10.4

CONCLUDING REMARKS

This monograph has dealt primarily with the theory of construction of rational basis functions for continuous patchwork appro#imation over any algebraically reticulated region. In this last chapter we have indicated possible applications, some of which have already been implemented. Even in this applications-oriented chapter the analysis has not been supported by extensive numerical studies. Such studies are not essential in a work devoted to the laying of theoretical foundations. This is a research monograph in which further areas f o r study have been indicated. Refinements, modifications, and new concepts will undoubtedly be introduced as this research is pursued. It is hoped that this work has awakened a new appreciation of the interdependence of diverse pure and applied mathematics disciplines as tools in our endeavors to model natural phenomena.

321

References Aziz, A. K. (1972). "The Mathematical Foundations of the Finite Element Method." Academic Press, New York. Bocher, M. (1907). "Introduction to Higher Algebra." MacMillan, New York. Carr, G. S . (1970). "Formulas and Theorems in Mathematics." Chelsea, Bronx, New York. Ciarlet, P. G. and Raviart, P. A. (1972a). Interpolation theory over curved elements, with applications to finite element methods,"Computer Methods in Applied Mechanics and Engineering," pp. 217-249. North-Holland Publ., Amsterdam. Ciarlet, P. G. and Raviart, P. A. (1972b). General Lagrange and Hermite interpolation in Rn with applications to finite element methods, A r c h i v e f o r Rational Mechanics and Analysis 111 -222.

46,

pp.

Courant, R. and Hilbert, D. (1953). "Methods of Mathematical Physics," Vol. I. Interscience, New York. Coxeter, H. S . M. (1961). Geometry." Wiley, New York.

"Introduction to

Ergatoudis, J. (1966). Quadrilateral elements in plane analysis. Masters thesis, University of Wales, Swansea. Faulkner, T. E. (1960). Oliver L Boyd, Edinburgh.'

"Projective Geometry."

Felippa, C. A . and Clough, R. W. (1970). The finite element method of solid mechanics, "Numerical Solution of Field Problems in Continuum Physics," Vol. 11, SIAM-AMS Proc., pp. 210-252. Providence, Rhode Island.

322

RATIONAL FINITE ELEMENT BASIS

Froehlich, R. (1967). A theoretical foundation for coarse mesh variational techniques, P r o c . I n t e r n .

C o n f . R e s . Reactor U t i l D. F . 1, p . 2 1 9 .

.

Fulton, W. (1969) Benjamin, New York.

. and

Reactor Math., Mexico,

"Algebraic Curves. 'I

Hadamard, J. (1952). "Lectures on Cauchy's Problem in Linear Partial Differential Equations." Dover, New York. Herbold, R. J., Schultz, M. H., and Varga, R. S. (1969). Quadrature schemes for the numerical solution of boundary value problems by variational techniques, A e q u . M a t h . J, p p . 9 6 - 1 1 9 . Hodge, W. V. D. and Pedoe, D. (1968). "Methods of Algebraic Geometry," Vols. 1 and 2. Cambridge Univ. Press, London and New York. Hoppe, V. (1973). Finite elements harmonic interpolation functions, "The of Finite Elements with Applications," J. R. Whiteman, pp. 131-142. Academic London.

with Mathematics edited by Press,

Irons, B. M. (1966). Numerical integration applied to finite element methods, C o n f . on u s e of

D i g i t a l C o m p u t e r s i n S t r u c t u r a l E n g . U n i v . of Newcastle.

Jordan, W. B. (1970). Plane isoparametric structural element, KAPL Memo M-7112, UC-32, Math and C o m p u t e r s TID-4500, 5 4 t h e d .

Kaplan, S. (1965). Synthesis methods in reactor analysis, "Advances in Nucl. Sci. and Eng.," Vol.111. Academic Press, New York. Macaulay, F. S. (1916). "Algebraic Theory of Modular Systems," Cambridge tracts in Math & Math Physics #19. McLeod, R. and Mitchell, A. R. (1972). The construction of basis functions for curved elements in the finite element method, J. I n s t . Math A p p l .

10,

pp.

382-393.

323

REFERENCES

McLeod, R. and Mitchell, A. R. (1975). The use of parabolic arcs in matching curved boundaries in the finite element method, J . I n s t . M a t h . A p p l . (in publication). Mitchell, A. R., Phillips, G., and Wachspress, E. L. (1971). Forbidden elements in the finite element method, J. I n s t . M a t h . A p p l . g , pp. 260-269. Muir, T. (1960). "Theory of Determinants," Dover, New York. four vols

.

Nakamura, S. (1971). Coarse mesh acceleration of iterative solution of neutron diffusion equations, Nucl. S c i . and E n g .

43,

pp. 1 1 6 - 1 2 0 .

Strang, G. J. and Fix, G. (1973). "An Analysis of the Finite Element Method." Prentice Hall, Englewood Cliffs, New Jersey. Synge, J. L. (1957). "The Hypercircle in Mathematical Physics." Cambridge, London and New York. van der Waerden, B. L. (1950). "Modern Algebra," Vol. 2 (Engl. trans.). Ungar, New York. van der Waerden, B. L. (1939). "Algebraische Geometrie." Springer Publ., New York. Varga, R. S . (1971). "Functional Analysis and Approximation Theory in Numerical Analysis." SIAM Publ., Philadelphia, Pa. Verdina, J. (1971). "Projective Geometry and Point Transformations." Allyn & Bacon, Rockleigh, New Jersey. Wachspress, E. L. (1966). "Iterative Solution of Elliptic Systems. Prentice Hall , Englewood Cliffs, New Jersey. Wachspress, E. L. (1971). A rational basis for function approximation, Proc. C o n f . on A p p l . N u m e r i c a l Anal., Dundee. "Springer Verlag Lecture Notes in Math." Vol. 228, pp. 223-252.

324

R A T I O N A L F I N I T E ELEMENT BASIS

Wachspress, E. L. (1973). A rational basis for function approximation: Part 11, curved sides, J. I n s t . Math. A p p l .

11,pp.

83-104.

Wachspress, E. L. (1974). Algebraic geometry foundations for finite element computation, C o n f . N u m e r i c a l Sol . D i ff. E q s . , D u n d e e , "Springer Verlag Lecture Notes in Math." Vol. 363, pp. 177-188. Wait, R. (1971). A finite element for three dimensional function approximation, Proc. C o n € . on A p p l . N u m e r i c a l A n a l . , D u n d e e . "Springer Verlag Lecture Notes in Math." Vol. 228, pp. 348-352. Walker, R. (1962). Dover, New York.

"Algebraic Curves."

Zienkiewicz, 0. C. (1971). "The Finite Element Method in Engineering Science" (2nd ed.). McGraw Hill, New York. Zienkiewicz, 0 . C. and Cheung, Y. K. (1967). "Finite Element Methods in Structural Mechanics." McGraw Hill, New York.

325

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INDEX

A Absolute line, 91 Adjacent factors, 126-127, 167 polyen, 126-127 polyp~l,163-167 polypoldron, 240-243 polynomial, 5455,126 side, I 1 Adjoint, 135-136 polycon, 135 p0lyp01, 163-167 polypoldron, 208,232-239 special, 135 Adjunct point, 168,198 Affine plane, 67,89 Algebraic curve component, 7,92 plane, 7 space, 206 element, 9 geometry, 88-124 reticulation, 205 transform, 113 Approximation space, 1, 18-20 degree of, 19,243 Areal coordinates, 3, 37,41 Adz, A. K.,261,213-274 and Babuika, I., 274 B Barycentric coordinates, 41

Basis degree of, 20 harmonic, 297-298 irrational, 245-259 minimal, 22 mosaic, 284 rational, 24 regular, 23 Bezout, 64,98,210 theorem, 98,210 Bigradient matrix, 95 Birational correspondence, 118 equivalence, 118 transformation, 118 Bocher, M.,8,94 C

Can, C.S.,125 Ciarlet, P. G., and Raviart, P. A., 31 Circles, intersection, 74 Coarse mesh rebalancing, 319-321 Complete quadrangle, 41, 70 Component, curve, 92 Congruence elementary, 70-75 notation, 8 theorems, 101-108,124 Continuity, 15-19 order of, 19 Convergence, 18-19, 273-275 Coordinates affine, 90

327

INDEX

areal, 3, 37, 41 homogeneous, 41,90-92 homogenizing, 90 isoparametric, 2 4 2 8 projective, 40-49,90-92 Coxeter, H. S. M.,35,40-42,64,110,209. Cramer, 109,136 Cubic-independent, 110 Curve algebraic, 7 component, 7,92 simple, 29,92 multiplicity, 92 order of, 7 relatively prime, 92 space, 206 cusp, 120 Cycle intersection, 122 virtual, 144 Cyclic points, 98 D Data fitting, 177-178,194-197 Deficit points, 163,233-236 Degrees of freedom conic, 63,71 bubic, 7475 curve, 111 line, 63 linear form, 15 polynomial, 109,209 quadratic, 63, 71 Desargues theorem, 21 3-214 generalizations of, 214,216,221,236 Descent, method of, 249-253 Dimension of curves, 110-111, 131 polynomial space, 112, 208-209 projective space, 112 space curves, 209 surfaces, 209 Discretization, 261-313 harmonious, 296-31 3 mosaic, 282-295 Divisor, 108, 122-123 Duality, principle of, 43 w e

E

3 20

boundary, 206 exterior, 207 node, 208 Edge, W.. 45 EIP (exterior intersection points), 10 deficiency, 59-60,67,68,129 multiple, 67, 133 nonordinary, 115, I34 ordinary, 108, 131 sets, 128 Ergatoudis, J., 25,87 Euler’s equation, 206 Exterior diagonal of polygon, 50 quadrilateral, 35-36 Exterior intersection points (see HP)

F

Faulkner, T.E., 213 Felippa, C. A., and Clough, R.W.,303 Fix, G., 274 and Strang-see Strang Forbidden elements, 25, 27 Froehlich, R, 319 Fultcin, W., 92-94,98, 110, 124-125 Fundamental points, 113 G Geometric properties, 9 Genus of plane curve, 118-119, 122 surface, 210-21 1 H Hadamard, J., 249 Harmonic basis, 296-313 discretization, 296-31 3 Hat function, 3 Herbold, R J., Schultz, M. H., and Varga, R. S., 274 Hexahedron, 30,211, 215-219 Higher degree approximation polycons, 179-196 polypols, 197-204 polypoldra, 243-244 Hodge, W. V. D., and Pedoe, D., 94,96,125 Homogeneity, variable of, 95 Homogeneous coordinates, 90-92 EIoppe, V., 302 Hybrid node, 16

INDEX

I Ill-set polycon, 9 polypol, 29 polypoldron, 206-207 vertices, 141 Intermediate approximation, 194-197 Irons, B. M., 25,28, 87 and Razzaque, A., 274 Intersection, 7,89-100 conics, 73-74 cycle, 98, 101 degree, 99 double point, 93 exterior, - see EIP multiple, 7 , 8 9 nonordinary, 115, 134 number, 93-98 order, 99 ordinary, 108 plane curves, 7,92-100 proper, 93 quadrics, 209-210 set, 7 simple, 7 surfaces, 209 transverse, 7, 94 triple point, 55,93 Irrational wedges, 245-259 Irreducible conic, 64 curve, 7 polynomial, 7 surface, 210 Irregular lines, 1 13 bparametric coordinates, 2428 degree of approximation, 180-181, 185-186 hexahedron, 21 2 origin, 303-304 parabola, 25,28 segment, 270 tetrahedron, 212 J Jordan, W.B.,25 K &plan, S., 316

L

Laplacian, discrete, 288-313 Linearity of wedges on curves, 15

329

on element sides, 15 Linear subvariety, 112 system of curves, 1 12 Lune, 85

M

Macaulay, F. S., 94, 125 MacMillan, D. B., 28,185 Macroelement, 299 McIxod, R., 87 and Mitchell, A.R., 87 Median trisection, 269 Mitchell, A. R, 25,31,87 Mosaic approximation, 282-295 basis for, 284 degree of,304 Muir, T., 94,109,125 N Nakamura, S., 319 Neighborhoods, 120 Nodes edge, 218 face, 2 18 hybrid, 16 interior, 11, 190-193,220-221 polypoldra, 240 restricted, 17,260,318 side, 11,16 vertex, 11,206 Noether, M., conditions of, 101-102,108 theorem of, 101-102 0 Opposite factor, 126-127 polynomial, 5455,126 side, 11 Order of CUNB, 7 element, 10 vertex, 207 P Parallelepiped, 30,212 Patch test, 262, 273-279,289-291 Patchwork approximation, 1-32 Pentagon, 50,69-70 Points adjunct, 168,198 associated, 109.112 cyclic, 98 deficit, 163 double, 93

INDEX

fundamental, 1 13 independent, 1 11 cubic, 110 isolated, 148 multiple, 92 nonordinary, 115 ordinary, 108 neighboring, 120 nonsingular, 92 simple, 7,92 singular, 7, 92 triple, 55, 207 Polycon, 9, 52-62 order of, 10, 16 two-sided, 83-86 Polycondron, 223-232 Polycube, 162, 172-175 Polygon, 49 Polyhedron, 220-223 Polynomial adjacent, 54-55, 208 denominator, 54,208 equivalent, 8 irreducible, 7 normalization, 8,12 opposite, 5455,208 real coefficients of, 82-83 wedges, 24,212 Polypol, 29, 162-176 order of, 29 rational, 122, 166 Polypoldron (P-p), 232-244 simple, 206 Polyquadron, 223 Projection on surface, 216 Projective coordinates, 40-49 invariant, 119 plane, 67,91 Pyramid, 1-4

Quadric independence, 209 Quadrilateral, 32-51 diagonal points, 41 exterior diagonal, 35-36 nonw nvex , 255-260 wedges for, 36-40 R Rabbit function, 246 Rational , curve, 119 polYpo1, 122 wedge, 24,33 Raviart, P. A. - see Ciarlet Rebalancing, coarse mesh, 319-320 Rectangle basis, 5 Regularity of wedges polycon, 147-162 polygon, 147,154 polypol, 166,171 polypoldron, 239 theorems, 150, 153 Residual set, 11 1 Resultant, 95-96 Reticulation, 205 regular algebraic, 205 Ritz-Galerkin, 1-2, 18-19,316,318 S

Segment, quadrature, 269-272, 287 isoparametric, 270 Sides adjacent, 11 opposite, 11 Singularities, resolution of, 112-124 Singular points, 89 Specialization, 96 Spline, 2 Static condensation, 299;302 Stationarity, 265 Stiffness matrix, 296 Strang, G. J., 19,274 and Fix, C., 20,28, 30,83,261,274,294,296,300 Sylvester’s dialytic method, 94-95 Synge, J. L., 30 Synthesis, 31 5-319 multichannel, 316-319

Q

Quadrangle, complete, 4, 70 Quadratic transformation, 113-118, 139-141 Quadrature, consistent, 273 segment, 269-272 triangle-average, 279-282, 289-291 trimedian, 269, 291 wedge, 267

T

Tacnode, I21 Tetrahedron, 30,212 truncated, 211-215

330

INDEX

Transform, algebraic, 113 Triangle averaging, 279-282 diagonal, 42 of reference, 41,91 Trimedian quadrature, 269,291 Triple point, 55, 207 exterior (ETP), 21 3 Two-level computation, 314-321 Two-sided elements, 83-86 Unit point, 42,91

Wait, R., 30, 21 5 Walker,R.,9,90, 111,112-125, 135 Wedge, 1-5 degree of, 179-199 harmonic, 296-313 integrals, 47-49 irrational, 245-259 mosaic, 284 properties of, for degree one approximation, 23 quadrature, 267 rational, 29, 33 regularity, 147-171, 239 Well set polycon, 9,61 polygon, 9 polyhedron, 206-207 pol~poL29 polypoldron, 206-207 Z Zienkiewicz, 0.C., 30, 261 and Cheung, Y . IC, 25,28 30,179,279

U

V van der Waerden, B. L., 121, 125 Varga, R. S., 298 Variable of homogeneity, 95 Variational acceleration, 319-320 Variety, 209 Verdina, J., 125 Vertex, 9,206 Virtual cycle, 144 W Wachspress, E. L, 49,199,319-320

A 5 8 6

c 7 0 8 E 9 F O G 1

H Z

1 3 1 4

331

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