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

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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.

ACADEMIC PRESS, INC.

111 Fifth Avenue. New York, New York 10003

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

This page intentionally left blank

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


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


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


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.


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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+ 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)


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


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


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


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


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~)


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)


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


+ 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


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


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 ) .


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


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


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


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

.


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


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


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

-

-


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


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


‘,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


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.


(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


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


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


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


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


- 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


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


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


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


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, r - 2. By Theorem 7.4, this accountsPifor not more than dl =

1

titjtk +

i<j 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


(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


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


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


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


value of the y-ordinate of vertex 2. For a

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