3 7 ? HSlJ

Ato. A9?o

MINIMIZATION OF A NONLINEAR ELASTICITY

FUNCTIONAL USING STEEPEST DESCENT

DISSERTATION

Presented to the Graduate Council of the

University of North Texas in Partial

Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY

By

Terence W. McCabe, B.S., M.A.

Denton, Texas

August, 1988

•to/I

McCabe, Terence W., Minimization of a Nonlinear

Elasticity Functional Using Steepest Descent. Doctor of

Philosophy (Mathematics), August, 1988, 76 pp.,

bibliography, 21 titles.

The method of steepest descent is used to minimize

typical functionals from elasticity. These functionals are

coercive and bounded below but are not defined everywhere on

an appropriate Sobolev space H®(fl) or product of such

spaces. For the case ft = [a,b], suppose w € H2(ft),

u € H2(ft) * H2(ft)HHQ(0) w(a) < w(b), and u'+w' > 0 on [a,b].

The function u + w is called a deformation of the interval

[a,b]. The functional I is defined as follows:

I(u) - fb[[u'(x)+w'(x)]2 + * ldx J a*- [u1 (x)+w* (x) | J

rb

J W(x,u'(x)) dx a

where W is the stored-energy, I(u) is the total

stored-energy and the potential energy is assumed to be

zero. We are interested in minimizing I while preserving

the boundary conditions of w. In the case ft = [a,bj and

s = 2, I is convex; but for bounded regions ft in IR2 with

appropriate smoothness on #ft and s • 3, the corresponding I

is not convex. The choice of s in both cases is made to /

insure I is Frechet differentiable at many points in H2(ft)

and respectively.

The following results are proved. The gradient Vl and

a "modified gradient" Q are derived. It is shown that Q is

locally Lipschitz, Q(u) is a descent direction and Q(u) » 0

if and only if Vl(u) «= 0. The existence of solutions y and

z respectively to the equations y'(t) » -Vl(y(t)) with

y(0) = uQ and z'(t) = -Q(z(t)) with z(0) « uQ, is proved for

t € [0,#) where S > 0 and u£+ w' > 0 on 0. If z exists for

all t in [0,oo), then lim z(t) exists and is a local minimum t-ta>

of I. If ft C R2, if z exists for all t in [0,ao) and if the

range of z is bounded in H|xH®, then there is an increasing

sequence {tj} of numbers such that lim tj = oo , i-too

lim || Q(z(ti)) I] » 0 , and lim zft,) = <J) exists in i-too i-to

where <J> e Under certain conditions, Vl(<J>) = 0.

Finally, this steepest descent process is implemented

in a True Basic code for the ft2 case and graphical output is

displayed.

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS iv

CHAPTER

I. INTRODUCTION 5

II. THE ONE DIMENSIONAL CONVEX PROBLEM 19

III. THE TWO DIMENSIONAL NONCONVEX PROBLEM . . . . 32

IV. NUMERICAL SOLUTIONS OF THE TWO DIMENSIONAL PROBLEM FROM ELASTICITY 52

APPENDIX 68

BIBLIOGRAPHY 74

iii

LIST OF ILLUSTRATIONS

Figure Page

1. Graph of initial pair W and Y 62

2. Graph of approximate solution pair W+Ut and Y+Vt 62

3. Cross sections of initial W, W(I,5) and W(5,J) . 63

4. Cross sections of solutions (W+U|)(I,5) and

(H+Ut)(5rJ) 63

5. Cross sections of initial Y, Y(5,J) and Y(I,5) . 64

6. Cross sections of solutions (Y+Vi)(5,J) and

(Y+Vj)(I,5) 64

7. Initial pair W and Y 65

8. Deformations of W and Y at k = 15 65

9. Graphs of approximate solutions W+U2 and Y+V2 65

10. Cross section W(I,5) 66

11. Cross section of intermediate stage: (W+U)(I,5) 66

12. Cross section of solution: (W+U2)(I»5)

at k • 100 66

iv

CHAPTER I

INTRODUCTION

Solutions to many mathematical problems involve finding

a minimum or critical point of a functional. Such

functionals are often derived as variational principles for

differential equations. Proving the existence of critical

points can be difficult and has required the development of

many sophisticated mathematical methods, such as the

"mountain pass lemma" [14, pp. 7-20] and the theory

developed for convex functionals, earlier by Rosenbloom [16]

and later much more extensively by Brezis and others [5].

However, some problems involve functionals which are

nonlinear, not defined everywhere and not convex. Such

functionals arise in elasticity. In this paper, a typical

functional I from elasticity will be investigated, where

X €<(>) is the total stored-energy for the deformation <j) of the

set 0 C R B , The study will be conf ined to R1 and R2, where

I is convex in R1 and I is not convex in R 2 . A constructive

method using steepest descent will be used with an emphasis

on applicability to computing. The setting will be Sobolev

spaces (see Adams, [1]). Great use will be made of the

Sobolev gradients, an idea used by Neuberger [10, 11, 12,

13], Richardson [15] and Glowinski, Keller and Reinhart [8].

Because many of the obstacles to overcome are present in the

one dimensional case, the key ideas will be developed and

implemented for 0 « [0,1] in this introduction and Chapter

II. The more complicated, nonconvex case will be dealt with

in Chapter III.

Let H2(fi) denote the Sobolev space H2'2(Q) of square

integrable functions on the closed, bounded region Q with L2

second derivatives. When it is clear, H2(fl) will be written

as H2. Similarly, let H§ denote the subspace H2flH^ of H2

which contains all the elements of H2 which satisfy zero

boundary conditions. Denote by C(ft) the space of continuous

functions on 0 and by Cl(Q) the subspace of C(0) with

continuous first derivatives.

For the remainder of this chapter, 0 is assumed to be

[0,1]. Let w be a fixed function in H2(Q) such that

w(0) < w(l) and u be a function in H§(0) such that

w1(x) + u'(x) > 0 for each x € 0. The function u + w is

called a deformation of the interval [0,1]. The functional

1 is defined as follows:

(1.1) I(u) = f T(u'(x)+w> (x) )2 + J 1 dx J0 L Iu' (x) +w1 (x) |J I U1 ( X ) +W 1 ( X ) | •

f M(x,u'(xj)dx •'o

where W is the stored-energy density, I(u) is the total

stored-energy and the potential energy is assumed to be

zero. The goal is to minimize I while preserving the

boundary conditions of w. One of the characteristics of

elasticity problems is that minima are not unique. In the

case of 0 • [0,1] C R1, I is convex (see Ball, [2]).

Another way to show the convexity of I is to calculate the

first and second Frechet derivatives of I on a certain

natural open subset BZ of . A function v f H| is said to

belong to the set BZ if and only if v'(x)+w'(x) > 0 for all

x e 0.

LEMMA 1.1 BZ is an open set.

PROOF: Since HQ is compactly embedded in Cl, there is an

M > 0 such that if v € H$ then ||v|| < M*||vjJ , [1, pp.

C1 HQ

97-98]. Suppose v e and v has property BZ, that is

v'+w' > 0 on ft. Since v is in C1, there is a j > 0 such $

that S = min {v1 (x)+w' (x)}. Let r • W T m ' L e t ** e Ho suc*1

x € 0 that Ijv-qJJ < r. Me know that |!v• — • II < -

H Q Sup C1

M ||v-q|| S M*r • - |. Thus, | v' (x)-q' (x) | < | for

£ £ each x in 0 and so v' (x) ~ 2 < (2> a n d v' (x)+w' (x) - ^ <

s q'CxJ+w'tx). Since v'(x)+w'(x) > 6 then q'(x)+w*(x) > ^ .

Therefore q belongs to BZ and BZ is an open set.

The open set BZ is a natural set with which to begin a

study of I. The functional I is defined for each u in BZ

8

since the second term of the integral is bounded. Of course

there are functions for which I can be defined and which are

not in H§ and so not in BZ. For example, let w(x) « x for

each x € [0,1] and define u by the following: let u(x) » x

for x € [0,l/3)r let u(x) » -2x+l for x 6 {1/a,2/a] and let

u{x) * x-1 for x € (2/3#l]. This function u belongs to Hg

and u+w is said to have cracks or buckles. When following

the steepest descent process using an initial member of BZ,

it is possible that the trajectory leaves the set BZ and

descends to a region of HQ containing functions with such

cracks. From computations, this is not likely in R1 but has

been observed for many initial <|> in the R2 case. This seems

to indicate that some functions $ have boundary conditions

which force a buckling when the method of steepest descent

is used to minimize I.

Returning attention to I and the calculation of its

first and second Frechet derivatives, suppose u is in BZ.

Since BZ is an open subset of HQ and H2 is embedded in C1,

then for each g and h in

(1.2) I'(U)h =• f W + W l h ' - "'I JQ L IU'+W'|2 J

and

(1.3) I"(u)(h,g) - zf^h'g* + h' 'g' 1 Jq L |u1+w'|3J *

9

Jn lu'+W' 3J

Setting g = h,

I«'(u)(h,h) « 2

which is positive if h # 0. Hence, I is convex, although

not strongly convex as defined by Rosenbloom [16* p. 128].

Note also that I is twice continuously differentiable.

In order to calculate the gradient of I, first note

that for each h € H§, ||h||2 » j* |"ha + (h' )2 + (h")2l, the H2

norm for the subspace . Let P2 be the projection of

L2xL2xL2 onto the subspace * 2 consisting of triples of the

form (h,h',b") and h(0) « h(l) « 0. This subspace is

isomorphic to Hjjf. Since u € BZ, sign(u* (x)+w' (x)) • 1 for

each x in 0. Thus,

I'(u)h = C r2(u'+wl)hl + 1 1 J0 L |u»+w'|2J

- f l \ 0»h + r 2 ( u . + w . , + 1 lh' + 0»h" 1 J I L lu'+w'pJ

2 (u1 +w') - * I+w1 I 2 1»

2(U1+W1) - 1

Iu1+w'I2

0

rh '

h'

h" L 2xL 2xL 2

2

h '

h«

h" L2xL2xL2

10

0 h " 2(u1+w1) - 1

•*2 h' lu'+w'I2 •*2 h' o . h". L2xL2xL2

n p. 2(u'+w') - 1

k ' + w f 0 1 1

where H^(q,q*,q") «• q for each triple In the subspace H of

L2XL2XL2. Thus, II P (0, 2(u»+w«)-l/|u'+w«|2 ,0) is the Hg 1 2

gradient of I at u, denoted by (?HgI)u or (?I)u

Since I is twice continously differentiable at u € BZ

and Vl is locally Lipschitz, then the differential equation

( 1 . 4 ) y«(t) = - (?I)(y(t)) and y(0) - u0 € BZ

has a local solution using the usual Picard iteration

technique. Since Vl is a function from Hq to H$, an

examination of the iteration technique used to prove the

existence of y yields that trajectories y stay (locally) in

Hjj. This contrasts with the not uncommon difficulty in

nonlinear semi-group theory in which one starts with a

* "n

generator A and has to prove that the composite (I - A) p

stays in the domain of A. The process generates a nonlinear

semi-group which is not defined everywhere, i.e. defines a

local semi-group.

One may argue that insistence on the use of requires

too strong an hypothesis. Use of Hq would allow for the

buckling observed in elasticity problems in higher

11

dimensions [2, 3]. The functional I is only lower

semi-continuous on Hq, Gateau differentiable in some

directions and not Frechet differentiable anywhere.

However, the intent here is to establish some theorems, the

hypotheses for which are based largely upon observations of

numerical work. With this foundation, it is hoped to

eventually extend the numerical methods to boundary

conditions associated with buckling.

However, the space H$ will be used in a surprising way

to develop a modified steepest descent. Again starting with

an h € Hjf, u € BZ and using the H1 norm defined by

i

II h ||2 " f (h)2 + {h" )2 , Hq 0

it follows that

11 (u)h « fTafu'+w') - 1 lh' L L lu'+w'PJ

0 "h " 1

|u'+w'I2 # h'

L2xL2

2(u1+w1) - 1

u'+w'•2

where P is the projection of L2xL2 onto the subspace M

consisting of pairs (h,h') for which h(0) » h(l) •» 0, which

is isomorphic to H$. Define Q(u) to be

12

II P (O^fu'+w1 J-l/ju'+w' I2). II I f

Three facts will now be established about Q that will

be useful in the steepest descent scheme.

frgMMA 1.2 If u € BZ then Q(u) € Hg.

PROOF : Suppose u E BZ. Q(u) is the element h of for

which the distance between (0, 2(u'+w'J-l/Ju'+w'|2 ) and

(h,h') is minimum. Let g • 2(u1+w')-1/|u1+w'|2. Since u

and w are both in H2 and u'+w' is in C, g1 is seen to be

2 / til' -f-W1' ) 2fu,,+w") + — — and g* € L2 * Computing Q{u) is thus

J U1+w1 J 3

equivalent to minimizing the functional G(v) => f* (v -J o

0)2+(v*- g)2. Taking the Frechet derivative G'(v)h and

using a standard integration by parts argument, one gets

i

G'(v)h * 2*/Q(v - v" + g')h , where v 6 and h € C0. In

order for v to be the minimum of G, it must be the case that

G'{v)h « 0 for each h € C®. This is equivalent to v

satisfying the equation

(1.5) v - v" + g' = 0 and v(0) = v(1) = 0 .

This is an elliptic second order differential equation for

which it is well known that there is a unique solution.

Since g' € L2, it is known that the solution to this

equation belongs [8, pp. 169-175]. Hence Q(u) is the

solution to (1.5) and Q(u) is in H§.

13

The preservation of the smoothness by Sobolev gradients

is a major factor in the decision to use a steepest descent

method. The advantage of using a Sobolev gradient instead

of an L2 gradient is quite evident computationally and has

been confirmed theoretically by Neuberger [13]. The nature

and form of the projections used in these gradients have

also been investigated by Neuberger ([10, 11, 12]).

The second fact about Q is that for each u € BZ, - Q(u)

is a descent direction in Hjj. In order to show this one

assumes Q is locally Lipschitz (this will be proved in

Chapter II). With this assumption, the differential

equation

(1.6) z'(t) - - Q(z(t)) and z(0) * u0 6 BZ

has a local solution, that is, z is defined on [0,d) for

some d > 0. For t € [0,d), let f(t) » I(z(t)). Taking the

derivative of f,

f'(t) - I'(z(t))z'(t)

- <Q(z(t)),z'(t)>H£

- (Q(z(t))Q(z(t))>H£

- - II «<*<*» ll'j-

Thus, f is a non-increasing function. Unless there is a t

such that Q(z(t)) « 0, f is a decreasing function and as t

14

increases, f • I<z(•>) decreases. The function I provides a

Lyapunov function for solutions z to z1 = - Q(z) (see Brauer

and Nohel, [4 ] ) . Considerable use will be made of Lyapunov

functions throughout this work.

Finally, a close relationship will be established

between (Vl)u and Q(u) for u € BZ.

LEMMA 1.3 If u € BZ, then (Vl)u - 0 if and only if

Q(u) «= 0.

PROOF: Suppose u € BZ and (Vl)u • 0. Let g »

2(u'+w') - 1 . Then, u1+w112

0 - <(?I)u,Q{u))H| » I'(u)O(Q(u))

'o" Qu g # (Qu)' 0. JQu)".

flg'i Qu)« •'o

X'<ra•[,&•]> !.'<>, El-•(&•]> <Q(U),Q(U})HI

This implies that || Q(u) ||2 - 0 and Q(u) - 0 in H|. Thus,

Q(u) « 0 in H§.

15

Similarly, suppose Q(u) = 0. Then,

• • <{<!».<»»

(Vl)u " "o" t(Vl)u]' J 9 J(Vl)u]". 2 0_

<(VX)u,(Vl)u)H2 .

Hence, {Vl)u « 0.

As a result of this lemma, the search for critical

points for I becomes a search for zeroes of Q. Of course a

critical point of a nonconvex functional I is not

necessarily a minimum of I. But in using steepest descent

methods to minimize functionals, critical points that are

not local minimums are usually very unstable numerically.

Critical points computed in numerous examples have shown no

signs of such instability. This is a good indication that a

zero gradient obtained in a stable manner is a local

minimum. In computing for the R2 case in which I is not

convex, there is numerical evidence of local convexity near

stable functions where the gradient is close to zero.

In Chapter II, the existence of a local minimum of I,

namely lim z(t), is proved assuming only that the function z t-tao

exists for each t € [0,oo). In Chapter III, it is assumed

16

that 0 is a closed, bounded region of R2 with a smooth

boundary. A corresponding function Q is defined mid shown

to be locally Lipschitz. Again under the assumption that

the solution z to (1.6) exists for all t € [0,oo), it is

shown that there exists an infinite, unbounded and

increasing sequence of positive numbers {tj J, t such that

lim -f ||Q(as (11) )|l } » 0. Under the condition that the i-*oo 1 HqXHq j

range of z is bounded [10, 11] in it is shown that

there is a limit point 4> of {z(tj)> in Hj}*H$ so that if <J> is

in BZ then Q(4>) * 0.

Chapter IV describes how to implement the steepest

descent for these problems on a computer. A finite

difference scheme is used, employing the Sobolev norms. Key

parts of a True Basic code are discussed, including a

Gauss-Seidel type linear solver for the R2 case. The output

on several test problems is presented, including some

graphical output to illustrate the time-stepping process.

CHAPTER BIBLIOGRAPHY

1. Adams, R., Sobolev Spaces, Academic Press, New York, 1975.

2. Ball, J. M., Constitutive inequalities and existence theorems in nonlinear elastostat ics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium 1 (1977), 187-241.

3. , Singular Minimi zers and their Significance in Blasticity, Directions in Partial Differential Equations, Academic Press, 1987, 1-16.

4. Brauer, F. and Nohel, J., Ordinary Differential Equations, Preliminary Edition, W. A. Benjamin, Inc., New York, 1966.

5. Brezis, H., Operateurs maximaux monotones et semi-groupes nonlineares , Math. Studies 5, North Holland, 1973.

6. Curry, H. B., The method of steepest descent for nonl tnear minimi zation problems, Quart. Appl. Math., 2 (1944), 258-261.

7. Evans, L. C., Quasi convexity and partial regular* ty in the calculus of variations, (to appear).

8. Glowinski, R., Keller, H. and Reinhart, L., Continuatioi Conjugate Gradient Methods for the Least Square Solution of Nonlinear Boundary Value Problems, Rapports de Recherche, Institue National de Recherche en Informatigue et en Automatique, 1982.

9. Miranda, C., Partial Differential Equations of Elliptic Type, 2nd rev. ed., Springer Verlag, 1970.

10. Neuberger, J. W., Gradient Inequalities and Steepest Descent for Systems of Partial Differential Equations, Proceedings of the Conference on Numerical Analysis, Texas Tech University, 1983.

11. , Some Global Steepest Descent iesults

17

18

for Nonlinear Systems, Trends in Theory and Practice of Nonlinear Differential Equations, edited by V. Lakshmikantham, New York, Marcel Dekker, 1984, 413-418.

12. , Steepest Descent for General Systems of Linear Differential Equations in Silbert Space, Proceedings of a Symposium held at Dundee, Scotland March-July 1982, Lecture Notes in Mathematics, Vol. 1032, New York, Springer-Verlag, 1983, 390-406.

13. , Steepest Descent for Systems of Sonlinear Partial Differential Equations, Oak Ridge National Laboratory CSD/TM-161, (1981).

14. Rabinowitz, P. H., Kinimax Methods in Critical Point Theory with Applications to Differential Equations, Regional Conference Series in Mathematics, no. 65 (1984).

15. Richardson, W. B., Sonlinear Boundary Conditions in Sobolev Spaces, Dissertation, North Texas State University, 1984.

16. Rosenbloom, P. C., The method of steepest descent, Proc. Sympos. Appl. Math. VI, Amer. Math. Soc., 1956, 127-176.

CHAPTER II

THE ONE DIMENSIONAL CONVEX PROBLEM

The condition of convexity on functionals has been

widely utilized in minimization problems. Extensive work

has been done on convex elasticity problems by S. S. Antman

[2] and J. M. Ball [3, 4]. Using the strong assumption of

convexity, an extensive theory of semi-groups has been

developed by H. Brezis and others [6]. An early development

by P. C. Rosenbloom [10] proved asymptotic convergence of

trajectories of a semi-group satisfying equations of the

form y'(t) • - ( V j ) y ( t ) where y{0) • u and J satisfies the

strong convexity condition that JM(u)(h,h) > c*|| |{h ' o r eac*1

u and h in the Kilbert space H.

The steepest descent method has been adapted by

J. W. Neuberger for the following least squares problem [7,

8]. Suppose each of H and K is a Hilbert space, F is a

twice continuously differentiate function from H to K and

<j>:H -4 K such that $(u) » (1 /g)|}F(u)jj for each u € H. Under

the assumption that

( 2 . i ) ii<*4» uii„ 2 M - r < u > i i K

for some M > 0 and every u € B Q H, Neuberger proves there

exists a solution y:[0,oo) 4 I to the equation

19

20

y'(t) = -(V((>)y{t) and y(0) » u0, and if Range{y) C B then

lim y(t) exists and is a zero of <j> and of P. This gradient t-ta>

inequality is satisfied by many nonconvex functionals. The

set B is typically a bounded subset of a Sobolev space H and

the above result requires that the trajectory y remain

bounded. The results of this paper are an adaptation of

Neuberger's approach to the variational problem (1.1), but

without the advantage of a gradient inequality. In this

chapter, it is assumed that [0,1] » ft C R1. In this case

the functional I is convex. Although much is known for such

convex functionals, our constructive method may shed some

new light on non affine minimizers for 1 (see Ball [4,

pp. 4-5J). Most of the key difficulties in higher

dimensions are present and may be clearly seen in the one

dimensional case.

Attention will be focused on the "modified" gradient Q.

Recall from chapter I that - Q(u) is a descent direction,

Q(u) = 0 if and only if (Vl)u » 0 and if u € BZ then

Q(u) G H§. An important question arises as to the existence

of a solution z to the differential equation

(2.2) z'(t)"-Q(z(t)) and z(0) » u0 €

where t € [0,oo). In order to prove that there are local

solutions to such equations, it is sufficient to prove that

Q satisfies a Lipschitz condition.

21

THEOREM 2.1 Q is locally Lipschitz on BZ.

PROOF: Recall that I and I* are given by

I(u) • [(u'+w1)2 + 1 1 J 0 L ju'+w'|J

and

riuih. - si3n(u'+w,)-h']

J 0 L |u'+w'|2 J

2 c for each u and h in Ho and signfu'+w') • 1 since u BZ.

Let u € BZ. Since Hg is embedded in C1 and u'+w1 > 0 on

[0,1], there is a neighborhood B(u,rt) and positive numbers

and M2 such that for each v € B(u,rt)f JJl/tv'+w' )|j < Mt

and JJv'+w'Jj < M 2 . Let v € B(u,rt).

Let f » 2(u'+w1) - 1 and |u'+w'12

2(v'+wf) - * and note that f1 and g1 are in L2

|v'+w'|2

Let F - Q(u) ® II ? 0

x proof of Lemma 1.2, F and G satisfy the differential

and G » Q(v) = I! P [°j . From the 1 1 u

equations -FM+F+fl « 0 and -G"+G+g' • 0, respectively.

Using the equivalent norm of Hjj, ^ = /*(hw)2, it follows %

that

||Q(u)-Q(v)||^

22

n p "o" - n p 0 i i JP. .9.,

( 2 . 2 )

« f 1 (F,,-G" )2 = f 1 {F+f ' -G~g1 )2 •'O •'O

- f1<F-G)2 + 2(P-G)(f1-g') + (f'-g')2

•'o

< 2 /" * (F-G)2 + 2 /*1 (f'-g')2 »0 *0

The two terms of the right side of (2.2) will be

considered separately. Since is embedded in L2 and Hjj is

embedded in , there are positive constants Kt and K2 such

that for each h t€ and h a€ , jjh4 j|2 < Kt -||h| J|'

2 and L2

||h2|j2 < K2.||h2||

2 . For the first term from the right side "h§

of (2.2), it follows that

(2.3) 1 (F-G) 2 - HP-Gil^ < Kj •||F-G|(~

K, P "o" - f

f Sm

i

"o" "o'

.f ss.

HA

2

L2XL2

2

L2xL2

< K,-l

- K,/'(f-g)2

- K,J j2(u'-v'

< 2K, r ' m . - v i ' L r i»i-w'i-fiv'*w,irl

Jq \ [(u'+W1 ) a (V1 +w' ) 2J J

(u'-v1) (u'+v'+^w1 )"|2

(u'+w ')2(V

if1 +2w' )"|

•+W)'i J

23

l f u'-V •*0

< 2Kj f (u'-v')2C22+(2M1M24)2]

< 2Ki'Hu-vll2 [22 + (2M1M24)2]

H0

< 2Ki K2-||u-v|j2 [ 4+4M2M| ] < K-||u-vj|2

" "H§ " Hq

where K - 2KtK2[4+4M?M|].

Consider now the second term f (f'-g1)2 from the right Jo

side of (2.2). Differentiating f and g, gives that

f - 2u"+2w,,+2(u"+W") € L2 and a' - 2v"+2w,,+2*V ,+W") € L2. ]u'+w'|3 |v'+w'|»

Thus,

(2.4) rNf-g') 2

n2{u"-v'M+2r(u,,+w"Hv,+w,)8"(v,,+w,,>(u,+w,,3n2

L |u'+w'|3 Jv'+w113 J J < s f i i u•.-v>M2+r<

^,Hw,,>(v,•,^w,

JQ [ Iu'+w * J s1v1+w

) 3 _ { V . ( U . + W ' )8-|2

(2.5)

5 8'llu"vll^

+ 8-m|2 j*1 [(VL"+VI") (v'+w1 )-(v"+w")(u'+w1 )]2.

Let S » u'-v1, 9 • u'+w' and a • v'+w'. It is clear that

||ff|| < M and lUII < m . Since H(| is embedded in C1, Sup 2 Sup 2

there is a K3 > 0 such that ||tf|| < K3*||j|j * K3«J|u-vJ| Sup H|

24

< k 3 T j . Let L » K3*r1. Focusing on the integral of the

second term of the right side of (2.5), the function

pointwise satisfies the following:

t(u"+w"){v'+w1)-{v"+w")(u1+w') ]2

»|u" (v'+w1) 3+ w" (v1 +w') ® — v'^u'+w1 )3- w'^u'+w1)8!2

< j |-V"(T3+ u" ( f f - S ) 3 l + jw"!®3-^3] |J

< pu"-vH | |#|8 + |u" | I ( - 3 < r 2 6 + 3 f f $ 2 - $ 3 ) |

+ Jw" j | | (<r2+ff<r+*2 ) |J

< |ju"-v"|M§ + |u"||^|(3M^+3M2L+L2) + |wM|J #|3M|J 2

< 3J Ju"—v"|2M| + |u"|2|£|2(3Ml+3M2L+L2)2

+ |w" |a|tf|29M|J

Integrating, it is clear that

J[(u"+w")(v'+w1)-(v"+w")(u'+w1)]2

o

< (u"-V,,)2+ 3(3M1+3M2L+L2)2 (U")2j

<£ 3M|.[|u-v||2 + 3(3M|+3M2L+L2 )2||u||2 *K§-jju-vlj2

Hi UH§

FsM| + 3{3Ml4-3M2L+L2)2||u||2 K§

1

+ 9M24||»||^]||U-V||^

25

M-IML n -

2

L 0

where M is the constant from the previous step. Using this

inequality in (2.4), and then using (2.3) and (2.4), it

follows that

2 f 1 (F-G)2 + 2 f *(f•-g')2 •'O •'O

< 2-K.||u-v||^ + 2.[a-||u-v||^ + 8-Mj2-M-||u-v||^]

• (2»K + 16 + 16«M|2«M ) • j|u-v||2 . Ho

Thus,

IJQ{tl) —Q(V)|j2 < {2*K + 16 + 16'M|2«M )*||u-V||2 H0

for each v € B(u,rj). Therefore, Q is locally Lipschitz on

BZ and the theorem is proved.

Using this result and a standard existence argument

(see Brauer and Nohel, [5, pp. 364-417]), for each u0 € BZ

there is an e > 0 and a function z:[0,e) -> H2 such that

z'(t) « -Q(z (t)) and z(0) =» u0 . This is of course a local

result. But in numerical experiments for this steepest

descent problem, the global existence of solutions z to such

differential equations has always been observed. This

phenomenon can hold even for problems which have no

solution. For example, the least squares problem where

F:H^([0,1]) -+ L2([0,1]) such that F(u) « u'-l-u2 and

26

<|>(u) « { ^ / 2 ) | J ' F ( u ) ||2 produces a globally defined trajectory

z, but F has no zero for an Interval 0 of length more than

it . The global existence of trajectories shall be part of

the hypothesis of the following theorem.

THEOREM 2.2 Suppose z:[0,oo) -» H$ such that for each

t € [0,oo) z'(t) • -Q(z (t)) and z(0) = u0 . Then, 11m z(t) t-ta)

exists and is a member u of and lim ||Q(z(t))|| = 0. t-to Hq

PROOF: Let f: [0,co) -4 R such that for each t € £0,co),

f(t) • I(z(t)). Using the differentiability of I,

(2.6) f»(t) - l>(z(t))0z'(t) - <Q(z(t)),z'(t)>Hi

« <Q(z{ t)) ,-Q(z(t) )>H^ - - ||Q(z( t) )||2

and

(2.7) f"(t) = I"(z(t))[Q(z(t)),Q(z(t))]

- 2f 1[l+ - rl EQ(z{ t))1 J2

JQ L |Q(z(t))*+w'|8J

> 2^1[Q(z(t))']2 - 2 • Jj'Q (z {t}) ||

- 2•(- f'<t)).

Therefore,

(2.8) f"(t) > -2•f*(t)

for each t € [0,oo). Dividing both sides of (2.8) by the

27

negative quantity f1(t) and integrating, it is clear that f

satisfies the inequalities:

(2.9) f"(t) K -f'(t) < -21 7TJt) ~ 2 a n - f ' ( 0 ) - ®

and hence,

(2.10) 0 < -f'(t) < -f1(0)e"2t.

The following is an argument that lim z(t) exists in t-400

H|. Suppose s and t are in [0,co) and s < t. Let k and n be

the non-negative integers such k: 5 8 < an<* n~l < * 1 n.

Thus,

||z(t)-i(«)||Hj - || /a z' jfH 1

. r i»k K J

< if* f /1+1-f'(0)e_2i 1 / 2

i-k L 1 J

<ff l [f-(0)]i/2[-e-2i-2 ->• e-2i] i -k 2

l/2r - ~il/2 |,-1

i rnk < [ - n o , ] 1 ' ' ^ ] 1 ' 8 . ft- e"1

Since the series X e-i converges, then the trajectory z{t) i >o

28

is Cauchy in as t approaches oo. Therefore, the lim z(t) t-to

exists and is in . Call this limit u. Observe that

lim f'(t) and is zero because of (2.10). Since t-ta)

f ( t) a -||Q(z( t) )|j2 for each t € [0,oo}# then

lim ||Q(z(t} )||2 « 0. t-*oo

Even though the functional I is coercive in the Hg norm

but not in the H§ norm, it has been observed that for many

trajectories z the Range(z) is bounded. The following

theorem uses the hypothesis that z remains bounded in HQ.

THEOREM 2.3 Suppose z satisfies the hypothesis of Theorem

2.2 and u = lim z(t) € . Then, t-taj

(i) if Range(z) is bounded in H§, then u € , and

<ii) if u € B2I then Vl(u) = 0 and u is a local minimum of I.

PROOF; (i) Suppose the Range(z) is bounded in H§. Let

{tj> 00 be an increasing sequence in [0,oo) such that i»I

lim (t|> -» GO. There is a subsequence <z(Sj)} of (ztt;)} i-to

which converges weakly in H§ to a point v € HQ. Thus,

{z(S|)> converges weakly to v in . But, u is the strong

limit of z(t) in and so is the weak limit of {zfsj)}.

Since weak limits are unique, v = u and u G H§.

(ii) Suppose u € BZ. Since Hjf is compactly embedded

in C1 [1], there is a subsequence of {z(si)) which converges

29

strongly to ari element vi In C1. Strong convergence in C1

implies strong convergence in , so that vt = u. Suppose

{uk} is the convergent subsequence in C1. Since u € BZ and

<uk} converges strongly to u in C1, there is an M4 > 0 and

a positive integer n such that if k > n then

JujJ (x)+w* (x) | ~ an(* Ju1 (x)+w' (x) | - * o r ®acl1

x € [0,1]. Let H2 be the maximum of <||uk+wJjci}

a n d | j u + w | | c 1 '

Let h € such that ||h|jH = 1» ^or each k > n.

|<Q(uk)-Q(u),h>Hi|

< I f ^ - u ' J h ' + [("'-Vk'Xu'+Uk'+Zw'rtb, |J0 1 (UjJ+W1 )'2 (u« +w' )2 J

< 2* jj'Nu^-u'Jh' |

+ I f 1 j"[u'~uk ' 1 |u'+uk '+2w' |*[ jh,j Ko 1 (u£+W* ) * (u'+W« ) '* J

< 2-|K"UIIHJ IMIHJ + I-2Mj • |h* I -Ml

S 2-|K"UIIHJ IMIHJ + .WJ-IK IIH} IMIHJ =• (2+2H, ) - ||u-ut [|H, .

Since ||u-ut||H, < ||u'-ui||0, , then {| <Q(uk )-Q(u) ,h)H, | J

converges to 0. Hence, (Q(uk)} converges weakly to Q(u) in

H$. Since j||Q(uk -» 0 and ||Q(u)|| < lim inf {||Q(uk)||H J ,

then ||Q(u>)|H£ = 0 (see [9, p. 60]). Therefore, Q(u) • 0.

30

By lemma 1.3, this implies ( V l ) u = 0. Since 1 is convex,

then u is a local minimum for I on HQ.

CHAPTER BIBLIOGRAPHY

1. Adams, R., Sobolev Spaces, Academic Press, New York, 1975.

2. Antman, S. S., Monotonicity and invertibility conditions id one-dimensional nonlinear elasticity, Nonlinear Elasticity, ed. R. W. Dickey, Academic Press, New York,1973.

3. Ball, J. M., Constitutive inequalities and existence theorems in nonlinear elastostatics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium 1 (1977), 187-241.

4 . , Singular Minimizers and their Significance in Elasticity, Directions in Partial Differential Equations, Academic Press, 1987, 1-16.

5. Brauer, F. and Nohel, J., Ordinary differential Equat ions, Preliminary Edition, W. A. Benjamin, Inc., New York, 1966.

6. Brezis, H., Operateurs m x i m n u x mono tones et semi-QJdupes noftlin&iTt,S, Math. Studies 5, North Holland, 1973.

7. Neuberger, J. W., Gradient Inequalities and Steepest Descent for Systems of Partial Differential E q u a t i o n s , Proceedings of the Conference on Numerical Analysis, Texas Tech University, 1983.

8. , Some Global Steepest Descent iesults for Honlinear Systems, Trends in Theory and Practice of Nonlinear Differential Equations, edited by V. Lakshmikantham, New York, Marcel Dekker, 1984, 413-418.

9. Riesz, F. and Sz.-Nagy, B., Functional Analysis, Ungar, New York, 1955

10. Rosenbloom, P. C., The method of steepest descent, Proc. Sympos. Appl. Math. VI, Amer. Math. Sac., 1956, 127-176.

31

CHAPTER III

THE TWO DIMENSIONAL NONCONVEX PROBLEM

This elasticity problem in two dimensions has a

nonconvex functional I. However the main difficulties

encountered in two dimensions are of the same nature as

those in Chapter II for the one dimensional case. Let G be

a closed and bounded region in IR2 with a smooth boundary.

Let Hg(ft) » H3(fl)flH£(G). Suppose (w,y) € H3(fl)xH3(fl),

(u,v) € Hg(fl)xHjj(fl) and « u v(x) «= ff(x) • det D[j}ty]

- det["u*+w* u2 +w 21( x ) f o r ' Lvi +Yi v2+y2j

each x € (I. E'or each appropriate {u,v), define I(u,v) by

!u,v) - J (u I(U,V) - I (U|+w t)2+(U 2+W 2)

2+(V 1+Y 1)2+<v 2+y 2)

2+ 1

Q |ffU,vl

811 f W(Dd>)

where (j) = (u,v). Note that I is not defined for all

<u,v) € HjjxH|j.

An affine deformation <|> is one of the form

<|><x} • A0(x) + x0 where A0€ M+xn and x € Rn (see Ball, [4,

page 4]). W as defined above is H 1 , p quasiconvex for

1 < p < m. The condition of quasiconvexity, combined with

certain growth conditions on W, implies I is sequentially

32

33

weakly lower semicontinuous (swlsc). This Implies by

standard arguments that I has minimizers (Ball [3, p. 190]).

Ball has pointed out that, although non-affine minimizers

supposedly abound, little is known about such minimizers or

their smoothness [4, p. 5J. L. G. Evans [5, p. 3] has

proven partial regularity for Junctionals which satisfy a

uniformly strictly quasiconvexity condition, that is,

(3.1) jfw(A)+4 |D<J)|dY < jfw(A+D<j»dY

for some i >0, every smooth, bounded open O C R 2 , each 2X2

matrix A, and every <|> € Cl(Q#R2) with <|> » 0 on #Q. If

\ - 0, this is the definition of guasiconvexity. As Ball

points out, this does not allow for the singular behavior of

W(A) as det(A) -+ 0+ [4, p. 5]. In this paper the approach

is constructive with hypotheses suggested by observations of

computations. Conditions will be given that imply existence

of <|> in c^xc1 for which (Vl)«f» » 0. Many such functions

observed in computing are non-affine. Initial boundary

conditions which are planar do give rise, as expected, to

affine solutions.

Let BZ be the subset of H{xh{ such that (u,v) € BZ if

and only if #u v(x) > 0 for each x in ft.

LEMMA 3.1 BZ is an open subset of H(jxHj|.

PROOF: Using the fact that H§xe^ is compactly embedded in

34

the proof is similar to that of Lemma 1.1.

The choice of H|XH| is made so as to insure that X is

Frechet differentiate on BZ. The strategy for

investigating I will be the same as in Chapter II. For each

(u,v) and (g,h) in BZ, differentiating I yields

(3.2)

I'(u,v)(g,h)

' J -2 I (Uj +Wj Jgi + tUa+WaJga+tVt+yt Jh^va+ya )h2

0

J-sign(g) \« I2 I u,vl ft

t («i+wt )h2+(v2+y2)gt

(v1+y1 )g2-(u2+w2>hi 1

(Uj+Wt )' r - (v 2 +Y 2r ffl (u 2+w 2 ) + 1 + ( V i + y t ) 92 (v t+Yt ) +(u 2 +w 2 ) 9 hi

.(V2+Y2), CI - (Ui+W t )J ^2, (Uj+Wj )' (u2+w2) (V 1+y 1) ,(v2+y2)J

-(v2+y2)1

+{u2+w2) - (U i+Wj) )

where 9> is the projection from L2XL2XL2XL2XL2XL2 onto the

subspace consisting of elements of the form

(9rh,g< ,g2 ,h< ,h*} and n :R6-* IR such that i»l

II (xt ,x2 ,x3 ,x4 ,x5 ,x6 ) a (xj,x2>. Note that the first two l > i

components in the left side of the above inner product

expression have been deleted because they are (0,0) and the

two equivalent norms of H^xH^ have the same value in this

case. Let Q be the function from BZ to HqXH^ defined by

35

Q(U,V) = n v i * i

( ((Ui+W!)] 12 (U2+W2) 1 ItVi+Y!)

-(v2+y2

+ 1 +<v1+y1) T

&2 +(u2+w2) -(Ui+Wt )j J

(3.3) H 9> 1 # 1

o (JT 91 I l 34>J

''i^1 ;• yi (F,G) where is the projection of L2xL2xL2 onto the subspace

consisting of elements of the form (g>0i*g2)' ^ = ^iX9>i and

ir1 is the function on R3 such that 1t (Xi ,x2 ,x3) » x t. Since

Q is a projection, finding Q(u,v) is equivalent to

minimizing the functional y:BZ -f R defined by

?(P»q) = J*(P-0)2 + ( P i - a 4 > ) 2 + (p2-b<t>)2

a

+ {q-0)2 + {q1-c<j>)2 + {q2-d<j>)

2.

A variational argument gives that a minimum of f is a pair

(F,G) satisfying the pair of differential equations

(3.4) AF * 2A(u+w) + 2 det D

F P " [v+y] + F

with F « 0 on d(i, and

AG = 2A(v+w) + 2 ^ det + G (3.5)

with G = 0 on $0. By the theory of second order elliptic

partial differential equations (see Miranda [6]), for each

g 6 H1 there exists a unique solution y to the equation

36

Ay « y + g and y = 0 on $0

such that y € H$. Since (u,v) € BZ, then for each equation

(3.4) and (3.5) the corresponding g is in HQ. Thus, there

exists a unique pair of solutions (FrQ) to (3.4) and (3.5)

such that Q(u,v) • (F,G) € HgxH§. Hence, Q is a function

from BZ to HjjxHjf •

Following the same strategy as in Chapter II, the next

step is to show that Q is locally Lipschitz on BZ. Several

technical lemmas are required to begin this effort.

LEMMA 3.2 Suppose (u,v) = <j) € BZ. Suppose At , A2, A3 and

A4 are functions defined on HffxH|f by the following:

U> *!«)» - q = 2. 3 or 4

(ii) A2 (4>) - [det D(<(>) ]p = <tp , p - 1 or 2, ap -

(iii) A,(+) - det D[v*J

(iv) A4«j» » [A3«J»|p , p - 1 or 2 .

Then, each of At, A2, A3 and A4 is locally Lipschitz as a

function from H$XH§ to C(FL), H1(FT), H 2(0) and L2(FI)

respectively.

PROOF: Since H C R2, H1(0) and Hjj(Q) are algebras (see

Adams, [1]) and H$ is embedded in C1. By elementary

calculations, one has that each of Aj, A2, A3 and A4 is

37

/

Prechet differentiable. Thus, they are each locally

Lipschitz.

In showing the desired properties of Q, the norm of

HjjxHg must be considered carefully. As is well known, an

equivalent norm for H§(ft) is given by

where A is the Laplacian operator (see Agmon, Douglis and

Nirenberg, [2]}. A typical calculation shows that for Hjj,

the usual norm is equivalent to one that depends only on the

non-cross partial derivatives Ujjj and u222 * Thus, it is

easy to see that a third equivalent norm for H® is given by

INI2 38 / (Au)2 + ( [Au]j )2 + ([Au]2 )2 ,

with the first term being necessary. Use of this equivalent

norm will be made in showing that Q is locally Lipschitz.

Another technical but elementary lemma is needed to

show the consequences of products of such functions as At,

A2 » A3 and A4.

LEMMA 3.3 Suppose (U,v) - 4> € BZ, 0 > 0, ? :BZ -4 H2,

*8:BZ -4 H1, «:BZ -» H1 and each of r, * and at is Lipschitz on

the open ball B[<J>,y?] C BZ. Then, there is a N > 0 such that

for each ff € B[<J), ],

3 8

/ [ y { (O<§ U ) W ( < r ) - s r (<|>)<S < < j > ) * ( < j > ) ] 2 < N - | | < J > - f f | | 2

Q H g x H g

P R O O F : L e t M > 0 b e a L i p s c h i t z c o n s t a n t f o r e a c h o f ^

a n d w o n B [ ( | > , ^ ] . S i n c e t h e r a n g e s o f ? , "3 a n d # a r e

c o n t a i n e d i n H ^ f i ) a n d H ^ f Q ) i s e m b e d d e d i n L 4 ( f i ) , t h e n

t h e r e i s a M t > 0 s u c h t h a t | | < > H ' ) | | 2 < M t f o r e a c h 9 i n

B [ < f > , ) ? ] a n d A = 9 , <s o r » . S i n c e r a n g e o f y C H 2 ( f t ) a n d

H 2 ( f l ) i s e m b e d d e d i n C ( 0 ) , t h e n ? ( B [ < } ) , ; # ] ) i s u n i f o r m i l y

b o u n d e d o n U b y a c o n s t a n t M 2 • H e n c e , f o r e a c h 9 fc B [ < j > , ^ ]

f [* (*)* (*)»(*) -» ($)* ($)* ($) 1*

< 3 / [ y U ) - * « l » ] 2 [ < 9 U ) * U ) ] 2

• ' a

+ 3i\

0

+ | r < $ ) | 2 [ * ( 0 * ( $ > - * ( $ > * ( < • ) ] *

0

< 3 | | y ( f f ) - y ( < j > ) | | 2 E * U ) ] 4 ) *

+ 3 M 2 [ * f ( f f ) - « « l » ] 4 J T

+ 3 M 2 t < 9 ( i r ) - ^ ( < J » 3 4 ^

< 3 M | | 4 > - d | 2 ( M 1 ) 2 ( M 1 ) 2

H ^ x H g

+ 3 M 2 ( M t ) 2 | | » ( ) - W (4>) | | 2

H $ * H ©

39

+ 3M 2 (Mt > 2 ||«S {<T }-<« «|>) ||2

H^XH I

+ 3M 2(M t )2M||^-4>||2

H||XH§

» [3M(Mj ) 4 + 6M2 (Ma ) 2M3 H -4>1|2

Thus, N « 3M(M|)4 + 6M2(Mj[)2M and the lemma is proved.

A similar result can be proved in which rsBZ -* H2,

<«:BZ -» H2 and »:BZ -* L4 and each locally Lipschitz. These

results are essential for the following theorem.

THBQRBM 3t4 Q is locally Lipschitz on BZ.

PROOF: Let (u,v) « (J) € BZ. In analyzing Q» functions such

as kt , A2, A3 and A4 from lemma 3.2 and similar ones in

which u+w and v+y are interchanged, must be considered.

Choose 0 > 0 and N > 0 such that B[(4>),j?] C BZ and this ball

of radius $ and the constant N satisfies the hypotheses and

conclusions for each of the ten applications of lemma 3.3

that will used in the next paragraph. Let

(r,s) = ff € B[ (<J>),/? ]. Let Q(u,v) ® (F,G) and

Q(r,s) » (H,K) . Let ff„ „ • <r and ff_ * $. Thus, u>v r,s

2 i w . p n i

2

[aj LKJ|

40

• r-«n; 3 + II°-kii^

Considering the first term from above, it follows that

(3.6) I|*-hH2 " / C(AF)-(AH)]2

Hjj ''ft

+ / [ (AF) j - (AH) j ]2 + / [(AF)2-(AH)2]2,

Ju

Considering the first term on the right side of (3.6) and

using (3.4) for each of F and H,

/ [(AF)-(AH)]2

= jTl^Mu+wl+jJiyr d.t D[v'y]+F

- ^(r+wl-yfpr det D [s+y] ~H] 2

(3.7) < 3j£«<A»-Ar)» + 3^^-jip aet D ^ ]

" fijT let D[J y]]2 + 3jf(F-H)*.

If for each (p,q) € r(p,q) - ift 1 ia, *(p,q)(x) « 1 p-q|

and *(p,q) = det DJ p,q , then lemma 3.3 applies and the [q+y J

right side of (3.7) is less than or equal to

12 • ||u~r||2^ + 12jf «J»*(<j» - r (0*(/)j2+ 3 • ||F-H||

- 12-|^'"HSxhS + + 3-||P"H|1^

2

L2

41

(3.8) < 12(1+N) • IH*~'llggxjjg+ 3 - | M I ^

The second term of the right side of (3.8) will be dealt

with as it appears again in the second term of ||f-h||2 . " l!Hg

Next, considering the second term of |Jf-H||2 from

(3.6), it follows that

jfC(AF)j-(AH)|]2 ft

I

2 - 2A (r+w) i - [jj-pde tD [s+y] ] t ~

H l ]

(3.9) i 4jT[A(u+w)i-A(r+w)i ]2+ 4 f (Fl-Hi)2

II II

+ 4[jffr•" 1 • a®tD[vly] * | f p " V d e t D U y ] ] 2

+ " - l V [ d e t D U y ] ] J 2 '

Focusing on the third term of the right side of (3.9), if

At(r,s) = * 14 , A2 (r,s) » £det D[g+y]J • *i» a n d

A3 (r ,s) = det D|g+y] f o r each (r,s) € B[<[>, 3, then these

three functions satisfy the hypotheses of lemmas 3.2 and 3.3

with /? and N as the constants from lemma 3.3. Thus, the

third term of the right side of (3.9) is less than or equal

to 4*N'||<b- || " "HJxh$

42

Similarly, for the fourth term of the right side of

(3.9), if Aj(r,s) « i a 2 ia , A2(r,s)(x) - 1, and I r, s I

A4(r,s) * j det D[s+yJJ1 ^o r ®ach (r,s) € then these

three functions satisfy the hypotheses of lemmas 3.2 and 3.3

with and N as the constants from lemma 3.3. Thus, the

third term of the right side of (3.9) is less than or equal

to 4" N • ||<J) <r ||2 •

The first term of the right side of (3.9) satisfies

4/* [A(u+w) t -A(r+w) 1 ]2 - 4/*[A(u)1-A(r)1 ]

2

11 HH§xHg

This leaves only the second term of the right side of (3.9)

to be analyzed. Recall from (3.3) that Q(u,v) is defined as

Thus,

Q (u, v) » II 9" 111

I *0 »4>

ld<M

» fjr I 1 b* „b4>. "•"I])

4 / (Fj-H,)2 < 4[IF—H|j2

" V

- (F,G)

SB n '0 0 " a<l> - n a<r

K J 1 H'

SS 4 I V

L2XL2XL2

43

(3.10)

< 4 • 1 "o " "o

b4, L<i>J

a<r Lvl

L2XL2XL2

= 4j^2(u1+w1 )-Jr(v2+y2 )-2(r,+w1 )+$r(s2+y2 )J 2

+ j^2{u2+w2)+|T(v1+y1 )J

< sajf^-rj >2 + (u2-p2)2 + [^r(v2+y2) - *^2-(s2+y2)j

+ [jr(vi+Yi) " fr(si+Yi>]

< 3 2 . | | u - r | | » + 3Z.||vr-s||2 kl Q tlQ

+ 32jr[jT(V2+y2) - ^t(s2+Y2 )j 2

+ 32 [i?<vi+yi) - i?(si+Yi )]2•

Focusing on the third term of the right side of (3.10), if

A^r.s) * |fl 1 »y , A2(r,s)(x) = 1 and A8(r,s) » (s2+y2> for I r,sl

each (r,s) € B[<|>,4], then these three functions satisfy the

hypotheses of lemmas 3.2 and 3.3 with fi and N as the

constants from lemma 3.3. Thus, the third term of the right

side of (3.10) satisfies

3 2Jfl[i T { V 2 + Y 2 ) " i*(S2+Y2)]2

< 32J^[Ai <4>)A2 (4>)A3 (<J>) - At (<r)A2(<r)A3(<r)J2

< 32 'N* {kb-ffjl2

HjjxH I

44

Similarly, the fourth term of the right side of (3.10)

satisfies the following inequality

32jdD7<V,+Y'1 - ^ < s ' + Y i ' ] 2 S 8-N-||<t>-<r||^^.

Combining these inequalities involving the right side

of (3.10), it has been shown that

4f(Ft-Hj)2

* 32-|lu-rll',+ 32-IMI*,+

1*0 H0 W0xW§

- (32»N,+ 64 *N) •||<M||2

H§xH§

where N| > 0 is an embedding constant for H$XH|| in H^XH§.

These calculations and inequalities hold also for the second

term of the right side of (3.8).

The third term of the right side of (3.6) can be

bounded in the same way as the second term. Combining all

of these inequalities, it follows that there exists an

M_ > 0 such that for each # € B[<b,/] F

In an analogous way, there exists an MQ > 0 such that

for each r € B[<j), ]

45

IMI;I * VIMI H3

Combining these two inequalities, it follows that

CM; H ; x H ? <

Therefore, Q is locally Lipschitz on BZ and the theorem is

proved.

Using a standard existence argument, for each (u0,v0)

in BZ there exists an c > 0 and a function z:[0,e) BZ such

that z*(t) • -Q(z{t)} and z(0) » (u0,v0). In many numerical

experiments using steepest descent and an initial pair

(w,y), evidence of the global existence of z has been

observed. This has been the case even for (w,y) with

nonplanar boundary conditions. A more complete discussion

of these results will be given in Chapter IV. The global

existence of trajectories shall be part of the hypothesis of

the following theorem.

THEOREM 3.4 Suppose {u0#v0) = <j>0 € BZ and z:[0,oo)->BZ such

that for each t € [0,oo) z' (t) *» -Q(z(t)) and z(0) « <J)0.

Then,

(i) There exists an increasing, unbounded sequence

{11 >.00 C [0,co) such that lim ||Q(z(t{ )) || • 0,

1 - 1 i-to H Q XHQ

(ii) If Eange(z) is bounded in then there is a

46

sequence {t.-} 00 satisfying part (i) such that 1 i • 1

{z(tj )}.""^converges strongly in C^XCQ to a point <{> in

and

(iii) If (J> from part (ii) is in BZ, then Vl{<|>) « 0.

PROOF: Suppose the hypothesis of the theorem is true. If

there is a t0 € [0,oo) such that ]|Q(z( t0)) II • 0, then

for each t > t0, z'(t) = 0 and z(t) « z(t0). In this case

the theorem in clearly true. Therefore, suppose that

||Q{z(t0))|| > 0 for each t € [0,a>). H£XH£

Part (i): Let f:[0,oo)-»IR such that f{t) » I(z{t)).

Differentiating, it follows that

f'(t) « I1(z(t))z«(t) » <Q(z(t)),z»{t))H^XH^

« o X i i o

Thus, f is a decreasing function. Note also that f is

bounded below by 0 and — f • {t) > 0 for each t € [0,oo) .

Assume - f is bounded away from zero by some 8 > 0. Then

for each t € [0,oo)# f'{t) < -S and f(t) < -£t+f(0). But

then there must be an s € [ 0 , j such that f(s) = 0 . The

fact that f(t) > 0 for each t implies f(t) » 0 for each

t > s. Thus, f' (s+1) = 0. But f' (s+1) • -l|Q(z(s+l) ) II2

< 0. This is a contradiction and so - f' is not bounded

47

away from 0. Since - f'(t) > 0 for each t € [0,co), an

CO m

increasing unbounded sequence c a n be constructed

such that lim - f'ftj) » lim ||Q(z{ (ti )) II2 exists and is

i-to i-ta H^XHJ

0, which is the conclusion of part (i).

Part (ii): Suppose the Range(z) is bounded in

There exists a bounded sequence in H$xH$ satisfying part

(i). By a well known property of Hilbert Spaces [8, p. 89],

there exists a subsequence which converges weakly to an

element <|> in HjjxHjj. Since this sequence is also bounded in

and H$xHjj is compactly embedded in C^xc^, there exists

a subsequence {<j>j} of which converges strongly in CqXC$

to an element $ of C$xc^. Since «|>i) converges weakly in

H^XH| to <|>, it converges weakly to <f> in H^XH^. But, since

{<J>i > converges strongly to $ in C^XCq, it converges weakly

to $ in CQXCQ and hence it converges weakly to $ in H^xH^.

Thus, {(j } converges weakly to both q> and <p in HlxHl,

implying that $ = <J>. Therefore, {{j } converges strongly in

C^xc^ to an element (|> € which is the conclusion of

part (ii).

Part (iii): Suppose «j)i } • {z(tj)} = {(u1 ,v' )} is the

sequence generated in part (ii) and <|> » (u,v) € BZ. The

strategy is to show that Q{<|>) is the weak limit of the

sequence {Q(z(t-)> « {(Mu^v*)} in H^xHq. Let e > 0.

For each g € , ||g|| = Su^ j |gi (x) J , Jg2 (x) Jj. Define

48

F:IR5-*|? by F(Xj ,x2 ,x3 ,x4 ,x5 ) = (XJXJ-X2X4) 2x5. Since

{(u^v1)) converges to (u,v) in C Q * C Q , there is a compact

interval S containing the ranges of u2+w2, Vj+yj,

*f»

v2+Y2- u"+wt, u®+w2, v"+yt and V2+y2 for each m 6 Z . Thus,

F is uniformly continuous on S5. Since F is uniformly

continuous on S8 and € > 0, there exists a > 0 such that

if |jum-u|| < #1 and ||v"-vj| < S i t then for each x 6 0

(3.11) |F(ut+wt Ua+WafVt+Yi,v2+y2,a)(x)

-F(u?+w1,u^+w2,v?+yi,vf+y2,b)(x)| <

where (a,b) is one of the pairs (v2+y2,v2+y2),

(vf+yj jVi+yi), {u2+w2 ,u2+w2) or (uf+Wj ,VLt +wt) . Note that

(3.11) is the same as

(3.12) |U2a)<x)-(Uffl)2b)<x)| < l l H

for each appropriate pair (a,b). Let $ be the minimum of 6x

and e .

Since <{> € BZ, {z(ti)) converges to <J> strongly in CqXC^ C

and ^ > 0, there exist positive numbers fi and n such that if

jm *4" m € Z and m > n, then

I-<»I - H S : ; : 3 3 H >'•

K " " i • (—< [ 3 ~ , • > —

49

l | [ - ) I L , s • • I I S - I I L , HqXHQ

Let (g,h) € H^xHq such that

m > n,

C^XCq

= 1. For each

Hi*H &

/ (uf+Wj f

<2 (uf+w2)

\ (vf+Yt) \ l(v^+y2); (ffm) m » 2

'{Ui+wj r (U2+W2) (Vt+Yi ) {V2+Y2)J

H^xh^

-(v?+y2)' +(v?+Yi) + (uf+w2) -(uf+Wi )J

- ( v ^ y ^ +(V1+Y1) +{u2+w2) l-(u1+w1 )J

HoxHQ

H$xh£

|2Jq(U"~Ui)9i+(u2-u2)g2+(vi-vi)hi+(v§-v2)h2

+ ^ [ l ^ F ( V 2 " Y 2 ) + ^ ( V2 +Y2)]si

+ ^[uST2- ( V ? _ Y i ) " 7*<Vi+Yi>]ff2

~~ Y2 ) - J7(v2+Y2)]hl

+ * + F^ Vl + yl )]h2 I

« 2 | < P : v ] •[?]),

t jj-|i!i2!i^}jji2aa»i||Bt|

50

+ J R I , 2 < N ^ : > + W ' ' I I H

+ + L H I I + I H S I

< » - F 1 + I ( J [ J I A ' I 2 + R + ^ I H I R + fa\h*\*]

Thus {Q(z(tm))} converges weakly to Q(<j>) « Q(u,v). Using

the fact that ||Q(U,V)|| < 11m inf {||Q(z(tm) )|| }, [8, HoxH^ H^XH£

p. 60], and lim J|Q(z(tm))It • 0, It follows that nHoo HqXHQ

| |Q(U,V)|| » 0. Thus, Q(u,v) » 0. By a result analogous

H£xh£

to lemma 1.3, this implies (Vl)(u,v) = 0. Thus, (u,v) = (j>

is a critical point of I.

CHAPTER BIBLIOGRAPHY

1. Adams, R., Sobolev Spaces, Academic Press, New York, 1975.

2. Agmon, S., Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math., 12 (1959), 623-727.

3. Ball, J. M., Constitutive inequalities and existence theorems in nonlinear elastostatics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium 1 (1977), 187-241.

Singular Unimizers and their Signif icance in Elasticity, Directions in Partial Differential Equations, Academic Press, 1987, 1-16.

5. Evans, L. C. , fjuas iconvexi ty and partial regularity in the calculus of variations, (to appear).

6. Miranda, C., Partial differential Equations of Elliptic type, 2nd rev. ed., Springer Verlag, 1970.

7. Rabinowitz, P. H., Mini max Methods in Critical Point Theory with Applications to Differential Equations, Regional Conference Series in Mathematics, no. 65 (1984).

8. Riesz, F. and Sz.-Nagy, B., Functional Analysis , Ungar, New York, 1955

51

CHAPTER IV

NUMERICAL SOLUTIONS TO THE TWO DIMENSIONAL

PROBLEM FROM ELASTICITY

Motivation for most of the results in the first three

chapters comes from numerical experiments using a finite

dimensional version of the problem. In some respects, the

theorems of chapters II and III are an indication of the

reliability of the numerical method used. In this chapter,

an explanation of this method will be given, as well as

results of computer runs using several initial pair (w,y).

A self contained True Basic program for the two dimensional

setting is given in appendix A.

Let n be a positive integer such that n > 4, 0 be

[0,ljx[0,l] C R2 and $ - £. Let G «• |(i,j>: i,;j€{0,1,••,n}j

and G' » i, j€{l,2, • • ,n-l}|. Let H^n+l) denote the

2

space of functions from G to R<n+1> and L2(n~l) denote the

space of functions from G' to R<n~l) . If u € H^n+l), u{i,j) = u. . will represent the value of the corresponding

1, j

function u at the point {g-, } € 0, where i, j€{0,l, • • ,n>.

In order to define a norm for K1(n+1), it is necessary

to define two difference operators. Let D1 :H* (n+l)-+L2 (n-1)

such that for each u € H1(n+1)

52

53

(4.1) Dlu(i,j) » n f > i + 1'^ 2] U(i-l>j)

Let DaiH1 (n+l)-»L2 (n~l > such that for each u € H1(n+1)

(4.2) Dlu(i, j) = U{1'^+1)2S

The norm for Hl(n+1) shall be the Sobolev norm defined In

the following way: if u € H1(n+1) a H* then

(4.3) jjujl2 » S 2 \ t) [u( i, j) ]2 + £ [Dlu(i,j)]2

H1 "-j »J -o i ' j • i

+ [D2u(i,j)]2J.

The norm for L2(n-1) is the Euclidean norm giveen by

||u|!2 - S2 nS1[u(i,j)32. L2 (n-1) i,j.i

It is easy to see that each of L2(n-1) and H1 Is a finite

dimensional Kilbert space with the appropriate inner

product. A special subspace of H1 is H^(n+1) = j u € H1:

u(i,j) = 0 if i or j belongs to (o,n}|. Denote H$(n+1) by

Ho •

The problem to be solved is to minimize a functional

I: HQXHQ -• E. Suppose (wfy) € HlxB*. For each

(u,v) € H^xh^, define I by

n - l

(4.4) I(u,v) - S 2 ^ [tDllu+w)(i,j)]2

i # j « 1

54

+ [D2(u+w)(i,j)J2 + [Dl<v+y)<i,j>]2

,2 . 1 + [D2(v+y)( i, j ) ] + • f • I J #j H

where a (u v) - d t[Dl(u+w)(i,j) D2(u+w)(i,j)l . . wnere ar1:Jtu,vj

aet|Dl(v+y)(i,j) D2(v+y) (i, J)J " T n e o n I y

problem with the differentiability of I at a point (u,v) in

this finite dimensional setting is if ar^(u,v) • 0 for any

(i,j) € G', which would also imply that I is not defined at

(u,v). Let BZ be the subset of HqXHq such that (u,v) 6 BZ

if and only if ®^(u,v) > 0 for each (i,j) € G* . Again, BZ

is an open subset of H^xh^. It is easy to see that I is

Frechet differentiable on BZ. Performing this calculation,

one gets

I'(u,v)(g,h)

(4.5)

'Dlg^ 'DlU+Wj • - X i j D2v+yjj D29ij D2U4-Wj j + X ij Bfvfyj j

Dlhj, 9 2 Dlv+y4 j + X ij D2u+Wjj

D2hj j D2v+yj j - X H Dlu+Wj j

(4.6) *2 X <[SaI*j]• [S*;]),ir3

+ <&£i j ] •&!]>, i»j =i

IR2

1

|R2

where for each (i,j) €6', = jj—j2 a n d Ai j ' Bi j » cij

and Pij are the obvious values from the right side of (4.5)

depending on (u,v). Let Auv - {A^}, Buv » {B^},

Cuv = {C£ j } and P u v = <E*£ j ), noting that each of these is a

55

member of L2(n-1).

Let B be the linear transformation from H1<n+1) to the

space K « L2(nfl)xL2(n-l)xL2(n-1) such that B { u ) is the

triple (u,Dlu,D2u). Let P be the orthogonal projection of K

onto its subspace K0 consisting of elements of the form

(g,Dlg,D2g). Note that K0 is isomorphic to HQ . Then (4.6)

becomes

(4.7)

I'(ufv)(g,h)

+ <

+ I Bh,P

0 " \ uv ) uv

U U

0 "

p Cuv i F

uv

Thus, the gradient

' " 0 " ' 0

t p Auv , * / Cuv

1 LV

l Fuv ^(H^XH^)1]!?]

Denote this gradient by (?HI)(u,v).

The strategy for the numerical scheme is to

investigate the differential equation

56

(4.8) z'(t) - -(VHI)(z(t)) and z(0) - (u0,v0)

for (u0,v0) € BZ and z: [0,ooHH^xH$ . Following the notation

used by Neuberger in [5], let E = 3r0fc/>i where sr0 is the

H

orthogonal projection of H1 onto . E is non-negative and

invertible. The following theorem gives an explicit

formula for (VHI)(u,v).

THEOREM 4.1 Suppose (u,v) € H^xH$. Then P « DE~li0Dt .

The proof involves showing that P satisfies the necessary

and sufficient conditions for an orthogonal projection,

that is, P 2 = P, /'(u,Dlu,D2u) = (u,Dlu,D2u) for

(u,Dlu,D2u) 6 K0, P * • P and the R(/) C K0. A careful

examination of P reveals that

(4.9)

and

(4.10)

* P i

uv B

JT P 1

UV

UV

UV B

P. uv

* E~XJ (yBt

uv

UV F_ uv

An equivalent norm for the subspace HqXH^ is given by

the following: for each (u,v) € HqXH* o~no

(4.11) n - 1

||(u,v)||2 - 62 E [Dlu( i, j)J 2 + [D2u( i, j) ]2

HqxH^ i t j •> i

57

+ 62 "S^Dlvd, j) ]2+ [D2v(i,j) ]2 . i ' ;j * 1

Consequently, I is coercive and as well as bounded below on

BZ. Although I is not defined on all of HqXH^, in

particular the boundary of BZ, steepest descent avoids

these places because I is lower semi-continuous [2] and

blows up near boundary points of BZ. Prom the theory of

ordinary differential equations arguments, there exists a

positive number S and a function z : [ 0 , £ s u c h that z

satisfies (4.8). Using these ideas and an argument similar

to that of lemma 3 from [4, p. 191], the following result

can be proved.

THEOREM 4.2 Suppose (u0,v0) G BZ. There exists a solution

z to (4.8) such that z: [O,oo)-*BZ.

A key to this argument is that I(z) is a decreasing

function for every such z. This means that I has some

properties of a Lyapunov function.

The problem of looking for the asymptotic limit of z

is reformulated numerically as constructing the sequence

(4.12) Zn + 1 - Zn - hn-(VHI)(Zn) n » 0,1,2,3,•••

or equivalently, using Zn • (UB«Vn) € H$xH$ and equalities

(4.9) and (4.10),

58

(4.13) Un + 1 » U„ - hn-[Fn]

and

(4.14) Vn+1 - Vn - hn.[Q„]

where for each n

F - E-'Ioi>«<0,A^ ,B ) n u n v n u n n

and

G,, - ,F ) U I I

v n n v n

and {hj} Is an sequence of positive time steps which must

carefully selected.

The program in Appendix A is divided into six

sections. Section 1 defines (W,Y), the initial (U0,V0) and

the desired accuracy for several processes in the code.

Section 2 contains the Call statement for the subrountines

which compute Fn = E" ifo9t (°»A

n v rBf| ) and Gn «• u n v n "n v n

E"l5r0fc {0,Cn v ,F„ v ). The description of Section 6 will

u n n n n

give more details on each subroutine. Section 3 contains

the step defining the new pair (Un+i»Vn+1) using (4.13) and

(4.14). It also provides an opportunity to monitor the

choice of the timestep for the first three steps in the

main process. This section calls a subroutine to check the

convexity of I at (Un,Vn), using the quantity

59

)(Zn)]. Section 4 implements the

graphic subroutines which help monitor the process.

Section 5 prints the approximate solution whenever the error

tolerances have been achieved. This section contains the

END statement.

Section 6 contains 4 subroutines, three of which are

used in computing Fn and G0. The subroutine "ROUGH_GRAD"

computes (0,A ,B ) and (0,C ,F_. .. ). The °n vn un va un vn ua vn

subroutine ,,TRANS_D" computes Bt(0,A.. v ,BtI „ ) and Uu vn un v«

St{0,Cn fFn ). The subroutine "INVJDTD" computes an un vn un vn

approximation to E" 1jr0i't (0,A. „ ,B. „ ) and

un vn un vn

E" (0,CTT u ,Fn v ). This is done using the Qauss-Siedel un "n °n vn

iteration technique (see Dahlquist C7, pp. 1016-1026]). In

this technique, E is decomposed into the sum of three

matrices, a strictly lower triangular matrix L, a diagonal

matrix M and the strictly upper triangular matrix Lfc.

Letting ? • ^ ( O ^ , „ ,Bri v ), the iterative scheme is un vn un vn

generally given by

(4.15) M(Xk+1) - M(Xk) - 7 - E(Xk)

and

(4.16) x k + 1 = Xk + M~1[^ - E(Xk)]

with an updating occuring after the calculation of each

component of Xk and (Xk) converges to x where E(^) =» 9. The

60

other component of (V^I)(Un+1,Vn+1) is calculated In the

same way.

Table I contains the graphic output of two runs of the

program in Appendix A. Each pair of pictures depicts z(tk)

for some k and the value of the norm of the gradient at that

point. The first pair of initial functions (W,Y) is a

discretization of a pair (w,y) that satisfies the hypotheses

in Chapter III and are given by w{S,T) » S2+S-T and

y(S,T) .» S+T+T2. Using the initial (U0,V0) " (0,0),

Figure 1 on page 62 is the initial pair (W,Y). Letting

lim z(t) « (Uj,Vj}, Figure 2 is the approximation of the

pair (W+U^Y+Vj). Figures 3 and 4 on page 63 are the middle

cross sections of functions W and W+Uj, respectively.

Figures 5 and 6 on page 64 are the middle cross sections of

the functions Y and Y+V$, respectively. Notice that the

graphs of W+Uj and Y+Vj indicate that the critical point is

not affine. The square of the norm of the gradient of this

approximate solution is 7.212 x 10"5.

The second pair of initial functions considered is a

discretization of a pair (w,y) that does not satisfy the

hypotheses of Chapter III. The trajectory z in this case

can be observed to crack and the lim z(t) = (U2,V2) belongs

to but not HfjxHjj. The formulas for this second pair

(w,y) are w(S,T) • S-S3 and y(S,T) • T. Again using the

initial pair (U0,V0) « (0,0), Figure 7 on page 65 is the

61

initial pair (W,Y). Figure 8 is the deformation of (W,Y)

after 15 time steps (k « 15). Figure 9 is the approximation

of the pair (W+U2fY+V2), for which k » 100. Figures 10, 11,

12 on page 66 are the corresponding middle cross sections of

the functions W, its deformation at step k = 15 and W+Ut,

respectively. The square of the norm of the gradient of

this approximate solution is 5.8:13 x 10~5.

62

Fig. 1—Graph of initial pair W and Y

Fig. 2—Graph of approximate solution pair W+Ut and Y+Vt

63

Pig. 3—Cross sections of initial W, W(I,5) and W(5,J)

Fig. 4—Cross sections of solutions (W+Uj)(I,5)

and (W+U,)(5,J)

64

Fig. 5—Cross sections of initial Y, Y(5,J) and Y(If5)

Fig. 6—Cross sections of solutions (Y+V!)(5,J)

and (Y+Vj)(I,5)

65

Fig. 7—Initial pair W and Y

Fig. 8—Deformations of W and Y at k = 15

Fig. 9—Graphs of approximate solutions W+U2 and Y+V2

66

Fig. 10—Cross section W(I,5)

Pig. 11—Cross section of intermediate stage: (W+U)(I,5)

Fig.12—Cross section of solution: (W+U2)(I,5) at k = 100

CHAPTER BIBLIOGRAPHY

1. Adams, R., Sobolev Spaces, Academic Press, New York, 1975.

2. Ball, J. M., Constitutive inequalities and existence theorems in nonlinear elastostaties, Nonlinear Analysis and Mechanics: Herlot-Watt Symposium 1 (1977), 187-241.

3. Glowinski, R., Keller, H. and Reinhart, L., Continuation Conjugate Gradient Methods for the least Square Solution of Nonlinear Boundary Value Problems, Rapports de Recherche, Institue National de Recherche en Informatique et en Automatique, 1982.

4. Neuberger, J. W., Steepest descent and differential equations, J. Math. Soc. Japan, Vol. 37, No. 2, 1985

5» » Steepest Descent for General Systems of Linear Differential Squat ions in iilbert Space, Proceedings of a Symposium held at Dundee, Scotland March-July 1982, Lecture Notes in Mathematics, Vol. 1032, New York, Springer-Verlag, 1983, 390-406.

6. , Steepest Descent for Systems of Honlinear Partial Differential Equations, Oak Ridge National Laboratory CSD/TM-161, (1981).

7. Young, D. M. and Gregory, R. T., A Survey of Numerical M a t h e m a t i c s , Yol. II, Addison-Wesley Publ. Co., 1973.

67

APPENDIX

APPENDIX

! PROGRAM NAME: 2D ELST ! ELASTICITY - PHANTOM GRID,ITERATIVE LINEAR SOLVER, GRAPHICS ! J(U,V)=* INTEGRAL [ ||D(U,V)||~2 - 1/(DET{U+W,V+Y)) J

SECTION 1

LIBRARY "3DLIB" PRINT "WHAT SIZE GRID? N=10" LET N=10 PRINT "WHAT ACCURACY FOR SOLUTION? El=.OOOOOOOl" LET El=.00000001 PRINT "WHAT ACCURACY FOR GRADIENT? E2=.00001" LET E2=.00001 PRINT "WHAT MESH IN T-DIRECTION"; INPUT H LET K=1 DIM U(0 TO 10,0 TO 10),V{0 TO 10,0 TO 10),W(0 TO 10,0 TO 10) DIM Y(0 TO 10,0 TO 10) DIM X1(0 TO 10,0 TO 10),DT_DU(9,9),RG1{10,10),RG2(10,10) DIM RG3(10,10),RG4(10,10),X2(0 TO 10,0 TO 10) LET S=N/2 FOR I«1 TO N-l

FOR J=1 TO N-l LET U(I,J)-0 LET V(I,J)=0

NEXT J NEXT I FOR I»0 TO N

FOR J=0 TO N LET W(I,J)«(I/N)"2+(I/N)-(J/N) LET Y(I,J)»(I/N)++(J/N)+(J/N)"2 ! LET W(I,J)«={I/N)-{I/N)~3 ! LET Y(I,J)°(J/N)

NEXT J NEXT I LET ERR1=-1 LET ERR2—1

SECTION 2

DO UNTIL MAX{ABS(ERR1),ABS{ERR2))<E1 CALL ROUGH_GRAD(N,S,U(,),V(,),W{,)rY(,),

RG1(,),RG2{,),RG3{,),RG4(,))

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! COMPUTE FIRST COMPONENT OF ROUGH GRADIENT CALL TRANS_D(N,S,RG1(,),RG2(f),DT_DU{,)) ! COMPUTE FIRST COMPONENT OF SMOOTH GRADIENT LET ERROR®1 MAT X1»0 ! RESET X1=0 CALL INV_DTD(N,S,DT_DU(,),X1(,),E2) LET ERR1—1 LET GRAD1=0

! COMPUTE SECOND COMPONENT OF ROUGH GRADIENT CALL TRANS_D(N,S,RG3(,),RG4(,),DT_DU(,)) ! COMPUTE SECOND COMPONENT OF SMOOTH GRADIENT LET ERROR** 1 MAT X2=0 ! RESET X2=0 CALL INV_DTD(N,S,DT_DU(,),X2(,),E2) LET ERR2=-1 LET GRAD2-0

SECTION 3

! SELECT TIMESTEP IN RELATIONSHIP TO SIZE OF GRADIENT SELECT CASE K CASE 1,2 ! CHECK SIZE OF GRADIENT

PRINT "K-";K,"GRADIENT OF FIRST VECTOR - START" MAT PRINT XI PRINT "K«";K,"GRADIENT OF FIRST VECTOR - FINISH" PRINT "K»";K,"GRADIENT OF SECOND VECTOR - START" MAT PRINT X2 PRINT "K=";K,"GRADIENT OF SECOND VECTOR - FINISH" ! CHECK CONVEXITY OF I AT (U(K),V(K)) CALL CONVEX(XI{,),X2(,)fU(,),W(,),V(,),Y(,),N,S) PRINT "WHAT INITIAL TIME STEP"; INPUT H3 FOR 1=1 TO N-l

FOR J=1 TO N-l LET U(I,J)»U(I,J)-H3*X1(I,J) LET T»X1(I,J) LET ERR1=MAX(ERR1,ABS(T)) LET V(I,J)=V(I,J)-H3*X2(I,J) LET T=X2(I,J) LET ERR2-MAX{ERR2,ABS(T))

NEXT J NEXT I

CASE 3 PRINT "WHAT TIME STEP AT STEP 3"; INPUT H

CASE ELSE CALL CONVEX(XI{,),X2(,),U(,),W(,),V(,),¥(,) ,N,S) FOR 1=1 TO N-l

FOR J=1 TO N-l LET T=X1(I,J)

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LET ERRl=MAX(ERR1,ABS(T)) LET T«X2(I, J) LET ERR2=MAX(ERR2,ABS(T))

NEXT J NEXT I LET TIMESTEP=MIN(1/ERR2,1/ERR1) LET H1-MIN(TIMESTEP».001,H) FOR 1=1 TO N-l

FOR J«1 TO N-l LET U (I, J)«U(I,J)-HI*X1(I,J) LET V(I,J)»V(I,J)-HI*X2(I, J)

NEXT J NEXT I

END SELECT

SECTION 4

! PLOT GRAPHS OF U{K)+W AND V(K)+Y SET MODE "HIRES" CALL PARAWINDOW (0, 1, 0, 1, 0, .5, WORK$) SETCAMERA (1, -2, 2) FOR 1-0 TO 10

FOR J=0 TO 10 CALL PL0T0N3 (1/10, J/10, W (I,J)+U(I,J), WORK$)

NEXT J PLOT

NEXT I

CALL PARAWINDOW (0, 1, 0, 1, -.5, 1, WORK$) FOR I»0 TO 10

FOR J=0 TO 10 CALL PL0T0N3 (1/10, J/10, Y(I,J)+V(I,J), WORK$)

NEXT J PLOT

NEXT I

! PRINT OUT U(K)+W AND V(K)+Y IN NUMERICAL TABLE IF INT<K/5)«K/5 THEN

PRINT "0(1,J) IS THE FOLLOWING AT STEP =";K FOR J»N TO 0 STEP -1

FOR 1=0 TO N STEP 1 PRINT USING "###.####":U(I,J)+W<I,J);

NEXT I PRINT

NEXT J PRINT " " PRINT PRINT "V(I,J) IS THE FOLLOWING AT STEP =";K FOR J=N TO O STEP -1

FOR 1=0 TO N STEP 1 PRINT USING "###.####":V(I,J)+Y(I,J);

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#

NEXT I PRINT

NEXT J PRINT " "

END IF PRINT "STEP = ";K;"MAX GRAD1";ERR1;"MAX GRAD2";ERR2 LET K=K+1

LOOP

i SECTION 5

! PRINT THE SOLUTION PAIR (U+W,V+Y) PRINT PRINT "U(I,J) IS THE FOLLOWING" FOR J=»N TO 0 STEP -1

FOR I»0 TO N STEP 1 PRINT USING "###.####":U(I,J)+W(I,J);

NEXT I PRINT

NEXT J PRINT " " PRINT PRINT "V(I,J) IS THE FOLLOWING" FOR J-N TO 0 STEP -1

FOR 1=0 TO N STEP 1 PRINT USING "###.####":V(I,J)+Y(I,J);

NEXT I PRINT

NEXT J PRINT " PRINT END

I SECTION 6

SUB ROUGH_GRAD (N, S, U(,), V(,), W(,), Y{,), RG1(,), RG2<,),

RG3(,), RG4(,)) FOR 1-1 TO N

FOR J»1 TO N ! A=U1+W1 C=V1+Y1 B=U2+W2 D=V2+Y2 LET A»S*(U(IfJ-1)+W(I,J-l)-0(1-1,J-1)-W(I-1,J-l)) LET A-A+S*(U(I,J)+W(l,j>-U(l-I,jy-W(l-I,j)) LET B™S*(-U(I,J-l)-W(I,J-l)-U(1-1,J-l)-W{1-1,J-l)) LET B«B+S*(U(I,J)+W(I,J)+U(I-1,J)+W(I-1,J)) LET C"S*(V(I,J-l)+Y(I,J-l)-V(1-1,J-l)—Y(1-1,J—1)) LET C^C+S*(V(I,J)+Y(I,J)-V(1-1,J)-Y(1-1,J)) LET D"S*(—V(I,J-l)-Y(I,J-l)-V{1-1,J-l)-Y(1-1,J-l)) LET D»D+S*(V(IfJ)+Y(I,J)+V(I-1,J)+Y(I-1,J)) LET DET3«(A*D-B»C)"2

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LET RG1(I,J)»2*(A-{D/DET3)) LET RG2(X,J)=2 *(B-(C/DET3)) LET RG3(I,J)*2 *(C-(B/DET3)) LET RG4(I,J)=2*(D-(A/DET3))

NEXT J NEXT I

END SUB

SUB CONVEX (XI( , ) , X2(,), U{,)f W(,), V( ,} , Y(,), N, S) LET DET2-1E+6 LET SUM1=0 LET SUM2=0 LET SUM3=0 LET SUM4-0 FOR 1=1 TO N

FOR J=1 TO N ! A-U1+W1 C=V1+Y1 B=U2+W2 D*=V2+Y2 I D1X1=FIRST PARTIAL OF GRAD OF XI ,ETC. LET A^S*(U(I,J-l)+W(I,J-l)-U{1-1»J-l)-W{1-1,J-l)) LET A=A+S*(U{I,J)+W{I,J)-U{I-1,J)-W{1-1,J)) LET B=S*(-U(I,J-l)-W(I,J-l)-U(1-1, J-l)-W(1-1, J-l)) LET B™B+S*(U(I,J)+W{I,J)+U{1-1,J)+W(1-1,J)) LET C=S*(V{I,J-l)+Y(I,J-l)-V(1-1,J-l)-Y(X-l, J-l)) LET C=C+S*(V(I,J)+Y(I,J)-V(1-1,J)-Y(X-l, J)) LET D*S*(-V(I,J-l)-Y(I,J-l)-V(1-1,J-l)—Y(1-1,J-l)) LET D»D+S*(V(I,J)+Y(I,J)+V(1-1,J)+Y(1-1,J)) LET DET1=(A*D-B*C) ! PRINT I ; J ; ; DET1, LET DIXIES*{XI(I,J-l)-XI(I—1,J-l)+X1(I,J)-XI{1-1,J)) LET D2X1»S*(X1(I,J)-X1(I,J-1)+X1(I-1,J)-X1(I-1,J-1>) LET D1X2-S*(X2(I,J-l)-X2(1-1,J-l)+X2(I,J)-X2(1-1,J)) LET D2X2a»S* (X2 (I, J) -X2{I, J-l)+X2(1-1, J) -X2 (1-1, J-l > ) LET DETGRAD«D1X1*D2X2-D2X1*D1X2 ! PRINT M DETGRAD( " ; I; J; " ) * " ;DETGRAD LET

SUM1=SUM1+(2/DET1"3)*(D1X1*D-D2X1*C+A*D2X2-B*D1X2)~ 2 LET SUM2»SUH2+{2/DET1"2)*DETGRAD LET SUM3=SUM3+(A"2+B"2+C *2+D * 2)+1/DET1~ 2 LET SUM4«SUM4+(D1X1)~2+(D2X1)~2+(D1X2)*2+(D2X2)*2 LET DET2-MIN(DET1,DET2)

NEXT J NEXT I PRINT PRINT "MIN DET =";DET2 PRINT "SUM1 • ";SUM1; PRINT "SUM2 =";SUM2 PRINT "NORM OF GRAD";SUM4 PRINT "I«'(U{K),V(K)>(GRAD,GRAD) - ";2*SUM4+SUM1-SUM2 PRINT "I(U(K), V ( K)) -";SUM3

END SUB

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SUB TRANSJD (N, S, A(,), B(,), DT_DU(,)) ! D TRANSPOSE WITH ZERO BOUNDARY CONDITION ! THIS IS THE ROUGH GRADIENT FOR I«1 TO N-l FOR J=1 TO N-l LET T=S*(A(I,J)+A(I,J+1)-A(I+1,J)-A(I+1,J+l)) LET

DT_DU(I,j)=T+S*(B(I,J)-B(I,J+1)+B(I+1,J)-B(I+1,J+l)) NEXT J

NEXT I END SUB

SUB INV_DTD (N, S, DT_DU{,), X( , ) , E2)

! (DTD) • L + M + N ! M(X(K)-X(K-1)) « Y - (DTD)(X(K-l)) : ITERATION TO SOLVE

LINEAR SYSTEM

! A « X(K) - INV(M)[ X(K-l) + Y - DTD(X(K-1)] ! - INV(M)[ Y - (L+N)(X(K-1))] ! X(I,J) = THE GRAD OF U(I,J) OR V(I,J) LET L=1 LET E=10 DO UNTIL E<E2 LET ERROR-O IF INT(L/2)<L/2 THEN FOR 1=1 TO N-l FOR J=1 TO N-l LET A«DT_DU(I,J)+S*S*2*(X(I-1,J-1)+X(I-1,J+l)) LET A»(A+S*S*2*(X(I+1,J-l)+X(1+1,J+l)))/(8*S*S) LET B«A-X(I,J) LET X(I,J)-A LET ERROR^NAX(ERROR,ABS(B))

NEXT J NEXT I LET E=ERROR

ELSE FOR I=N-1 TO 1 STEP -1 FOR J-N-l TO 1 STEP -1 LET A=DT_DU(I,J)+S*S*2*(X(I-1,J-l)+X(1-1,J+l)) LET A»(A+S*S*2*(X(I+1,J-l)+X{1+1,J+l)))/(8*S*S) LET B=A-X(I,J) LET X(I,J)»A LET ERROR'MAX(ERROR,ABS(B))

NEXT J NEXT I LET E'ERROR

END IF LET L-L+l

LOOP END SUB

BIBLIOGRAPHY

Books

Adams, R., Sobolev Spaces, Academic Press, New York, 1975.

Brauer, P. and Nohel, J., Ordinary d i f f e r e n t i a l Equations, Preliminary Edition, W. A. Benjamin, Inc., New York, 1966.

Brezis, H. , (fperateurs maximaux monotones et seai-groupes aonl in tarts, Math. Studies 5, North Holland, 1973.

Miranda, C., Partial Differential Equations of Elliptic Type, 2nd rev. ed., Springer Verlag, 1970.

Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, Regional Conference Series in Mathematics, no. 65 (1984).

Riesz, F. and Sz.-Nagy, B., Functional Analysis , Ungar, New York, 1955

Young, D. M. and Gregory, R. T., A Survey of Jfuaerical Mathematics, Vol. I I , Addison-Wesley Publ. Co., 1973.

Articles

Agmon, S., Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions of e l l i p t i c partial d i f f e r e n t i a l equations satisfying general boundary conditions, I, Comm. Pure Appl. Math., 12 (1959), 623-727.

Antman, S . S . , Mono t on i c i t y and invertibility conditions in one-diaensional nonlinear elasticity, Nonlinear Elasticity, ed. R. W. Dickey, Academic Press, New York, 1973.

Ball, J. M., Constitutive inequalities and existence theorems in nonlinear elastostatics , Nonlinear Analysis and Mechanics: Heriot-Watt Symposium 1 (1977), 187-241.

Singular Minimizers and their Significance in El asticity, Directions in Partial Differential Equations, Academic Press, 1987, 1-16.

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Curry, H. B., The method of steepest descent for nonlinear minimization p r o b l e m s . Quart. Appl. Math., 2 (1944), 258-261.

Evans, L. C. , fjuasiconvexity and partial regularity in the calculus of variations, (to appear).

Neuberger, J. W., Gradient Inequalities and Steepest Descent for Systems of Partial Differential Equations, Proceedings of the Conference on Numerical Analysis, Texas Tech University, 1983.

, Some Global Steepest Descent Mesults for Jfonl inear Systems, Trends in Theory and Practice of Nonlinear Differential Equations, edited by V. Lakshmikantham, New York, Marcel Dekker, 1984, 413-418.

Steepest Descent for General Systems of Linear Differential Squat ions in Milbert Space, Proceedings of a Symposium held at Dundee, Scotland March-July 1982, Lecture Notes in Mathematics, Vol. 1032, New York, Springer-Verlag, 1983, 390-406.

Steepest Descent for Systems of Jfonlinear Partial Differential Equations, Oak Ridge National Laboratory CSD/TM-161, (1981)

Steepest descent and differential equations, J. Math. Soc. Japan, Vol. 37, No. 2, 1985

Richardson, W. B., Jfonl inear Boundary Conditions in Sobole\ Spaces, Dissertation, North Texas State University, 1984.

Rosenbloom, P. C., Ike method of steepest descent, Proc. Sympos. Appl. Math. VI, Amer. Math. Soc., 1956, 127-176.

Reports

Glowinski, R., Keller, H. and Reinhart, L., Continuation Conjugate Gradient Methods for the Least Square Solution of Jfonl inear Boundary Value Problems, Rapports de Recherche, Institue National de Recherche en Informatique et en Automatique, 1982.