101.J. Eap, Sci Vol. 20. So. 9. Pl'. 9i7-9llS. 19I1~Prinltd in Great Britain.

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ANALYSIS OF SOME FINITE ELEMENT METHODS FOR ACLASS OF PROBLEMS IN ELASTO-PLASTICITY

1. T. ODENTexas Institute of Computational Mechanics, ASE-EM Dept., The University of Texas at Austin. Austin.

TX 78712. U.S.A.

and

1. R. WHITEMANInstitute of Computational Mathematics. Brunei University. Uxbridge VB8 3PH, England

Abstract-In this work the analysis of certain equilibrium problems in elasto-plasticity by the use of finiteelement techniques is considered. Variational inequalities are established. These characterise the stresseswhich minimise the total complementary energy. subject to the constraint that the stress satisfies theequilibrium and the yield conditions for materials having a continuous convex yield function. An exteriorpenalty formulation of this principle is presented. together with existence, uniqueness and convergencetheorems. Some convergence criteria for certain types of finite element approximations of the penalisedproblem are given. Finally. a finite element method for the analysis of bending of elasto-plastic plates isdescribed. Conditions for the convergence of this method are also proved using the methods of Hlavacek.

I. INTRODUCTIONIN THIS paper. we consider a class of problems in elasto-plasticity that can be characterized asconvex optimization problems of the following type.

Find a stress field U E K such that

,p(U) = inf 1/1('1').TEK

The following notations and conventions are used:

,p( '1') == the total complementary energy functional at the stress 'I'=! aCT. '1')- F(T) (t E K),

a( T. U) == the complementary strain energy produced by stress fields

'I' = (T;j). u=(uij).lsi,jsN.

= !In CijklTkllJ'ij dx.Cilkl == components of the inverse of Hooke's tensor

Cijkl = Cjikl = C,kij• i S i. j, k. I s N

Cijkl E L",(!l) , CijklAklAjj ~ mOAijAj;

for every symmetric matrix Ajj, mo> a.e. in !l.

110, being components of a prescribed displacement field 110

(I.I )

1 av·Til -a ' dx = L(v) 't/v E V.n XIV == space of admissible displacements

= {v = (Vlo V2•... , VN) E (H 1(!l»NIVj = 0 a.e. in fu}.

L(V) == 1 bivi dx +i tiVi ds, v E v, LEV',n ra

UES Vol. lO.So. 9-A

and I( '1') S I a.e. in fl}.

978 J. T. ODEN and J. R. WHITEMAN

I == the yield function for the material.

In the above, n is an open bounded polygonal domain in RN with boundary r = ru u r

Analysis of some finite element methods 979

ProofUnder the stated assumptions, the set K is convex, nonempty and closed in S. The functional

1jJ:K ~ R is clearly strictly convex, coercive, and differentiable on K. Hence, IjJ is coercive andweakly sequentially lower semicontinuous on a non empty weakly sequentially closed set and,therefore, has at least one minimizer in K. Uniqueness follows from the strict convexity of 1jJ.Since

lim! [1jJ(u+ t(1'- u»-IjJ(u)] = a(u, '1'- u)- F('T- u),....0. t

is then non-negative, (1.2) follows. Equations (1.3) follow from (1.2) via Green's formulas andstandard arguments. 0

The minimization problem (1.1) is referred to as the Haar-Karman principle in the theory ofplasticity. Theorem 1.1, which is an immediate consequence of well-known results in convexoptimization theory, was also proved by Hlavacek[l]; a special case of this theorem was notedin a survey paper by Necas[2] and studied in the papers of Mercier[3] and Falk andMercier [4, 5]. Finite element methods for elasto-plasticity problems based on a related principlecovering plastic flow were developed by Mercier[3], Falk and Mercier[4,5], Hlavacek[l]; seealso the related work of Johnson and Mercier[6].

In the present paper, we examine finite-element approximations of (1.1) (or 1.2) for the2-dimensional case in which n is a convex polygon. Following this introduction, we describeseveral alternative variational formulations of problem (1.1) including Lagrange and perturbed-Lagrange methods. We give special consideration to an exterior penalty formulation for whichwe establish an existence theorem and a convergence result. In Section 3, we describe finiteelement methods based on the penalty formulation. These lead to an approximation u~of thesolution of (1.1) which depends upon the mesh parameter h and the penalty parameter f. Inparticular, we consider regular families of finite elements in which the stress is approximated bypiecewise linear polynomials on subtriangles of a triangular element. Interpolation properties forsuch elements have been established by Johnson and Mercier[6]. Using their results and someof the strategy of Hlavacek [I], we are able to establish existance and convergence results forthe penalty-finite-element scheme subject to certain assumptions on the yield function f.

In Section 4, the final section of the paper, we analyse finite element methods forelasto-plastic plates obeying a generalized Hencky law. There the techniques of Hlavacek[l]are used, with minor changes, to establish conditions sufficient to guarantee the convergence ofour method.

2. LAGRANGE MULTIPLIER AND EXTERIOR PENALTY FORMULATIONSWe shall now consider alternative formulations of the elasto-plasticity problems considered

in the previous section. We shall say that q, sljJ in ~(!l) iff q" IjJE L2(!l) and q,(x) $1jJ(x) a.e. inn. We can then describe the positive and negative parts of the function 1(1') - I in ~(!l)according to

(f( '1') - 1)+= sup {O,I( '1') - I)}; (f( 'T) -1)- = (f( '1') - I) - (f( '1') - 1)+.

The constraint I( '1') S I, l' E S, can thus be expressed in the form,

(/( 'T) -1)+ = O. (2.1)

A so-called mixed variational principle for problem (1.1) can be constructed by introducing theLagrangian

L: ExN~R;

L('T, q)= 1jJ('T)- L q(f('T)-l)dxI ( avoE = {'I' E S Jo 'Til a; dx = L(V)

N = {q E L2(!l)lq sO}.'tIv E v}. (2.2)

980

If we introduce the assumption

J. T. ODEN and J. R. WHITEMAN

f: R" -+ R is differentiable in all of Ro

and use the notation

d/(u) '1''' == I'(u) . '1',-'IdTij

(2.3)

then saddle points (u, p) E Ex N of L can be characterized by the system (see Ekeland andTemam(7])

a(u, '1') - 10 pf'(u) . 'T dx = F( '1')

10 (q-p)(f(u)-I)dx~O

't/TE E.

't/ q E N. (2.4)

The rather weak assumptions on I are insufficient to guarantee the existence of a uniquesolution to (2.4) (as far as we know).

We obtain a perturbed Lagrangean principle by introducing the functional

L,: Ex N -+R, E>O, (2.5)

saddle points (u" p,) of which are characterized by

a(u .. 'T) - ( pJ'(U,)' 'T dx = F( 'T)InIn (q - p,)(Ep, + I(u,) - I) dx ~ 0

The approximation p, of the Lagrange multiplier p is then

Ip, = -- (f(u,) - 1)+.

E

't/q E N. (2.6)

(2.7)

The perturbed Lagrangian problem (2.6) is equivalent to a formulation obtained using extcriorpenalty methods. For E > 0, let

Then minimizers u, of 1jJ, satisfy the variational equality

a(u"T)+.!. ( (f(u,)-I).f'(u,)· 'Tdx=F('T)dn 't/'TEE.

(2.8)

(2.9)

Introducing (2.7) into (2.9) gives the first of (2.6). Thus, if u, is a solution of (2,9) and P. iscomputed using (2.7), then (u .. p,) satisfies the system (2.6) characterizing saddle points of thcperturbed Lagrangian L, of (2,5),

With the establishment of (2.9), we have reduced the problem of solving the variationalinequality (1.2) to one of solving a nonlinear variational equality. The solution u, of (2.9) willnot, of course, satisfy the yield condition.

Analysis of some finite element methods 981

Theorem 2.1Let condition (2.3) and the hypotheses of Theorem \.I hold with K ¥ 0. Then(i) 't/ f> 0 there exists at least one solution to the variational problem (2.9) and(ii) there exists a subsequence {u.} of such solutions which converges weakly in E to the stress

u which is the solution of the original variational problem (\.I) (or equivalently (1.2)).If, in addition, the following condition holds: There exists a number a> 0 such that, for any

uEE

allqllos sup lIn qf'(u)' 'TdXITES-{O) 11'1'110

't/ q E N. (2.10)

Moreover, let CT-+ 10 qf'(u)' Tdx be completely continuous for fixed q E N, 'I' E S. Then thereexists a subsequence {p,} of perturbed Lagrange multipliers (computed from the CT. by using(2.7» which converges weakly in N to a solution p of (2.4).

Prool(i) Let P: E -+ R denote the penalty functional

peT) =~ 10 (/(T)-l)i dx" 'I' E E.

We can easily verify that P is convex and G-differentiable on S and, therefore, weaklysequentially lower semicontinuous. Moreover, P is also positive semi-definite on K in that

P('T)~O and P('T) =0 iff TEK.

Since the functional IjJ is coercive on S

and since IjJ is weakly sequentially lower semicontinuous (being strictly convex and differenti-able), the penalty functional 1jJ. is coercive and weakly sequentially lower semicontinuous on S.Thus, from standard results in optimization theory, there exists a minimizer u. of 1jJ, (in fact, aunique minimizer) for each f > 0:

(ii) But if we pick 'I' E K. then P( 'T) = 0 and. hence

't/ 'I' E K.

The coerciveness of 1jJ, implies that there must exist a constant C> 0, independent of f, suchthat Ilu.l~s C for all f > O. This implies that there exists a subsequence, also denoted {u.}, andan element u E S, such that

u•......u (weakly) in E.

But, owing to the weak lower semicontinuity of 1jJ, if 'I' E K.

ljJ(u) slim inf ljJ(u,) slim inC1jJ,(u,),...0 .-0

slim inf 1jJ('T) = 1jJ( 'T),.-0

982

i.e.

1. T. ODEI' and J. R. WHITEMAN

We need only show that u E K. But

't/TE K.

P(u.) s E(IjJ( '1') -1jJ(u.» s C'E

Hence

('I' E K).

P(u) slim inf P(u.) sO.-0

~P(u)=O¢:~uE K.

Therefore, the weak limit u of the subsequence u. is precisely the unique minimizer of IjJ in K.Next, suppose that (2,10) holds, Since each minimizer u. satisfies (2.9), we have

11 pJ'(U.) , T dxlal/p.llos sup n II II

rES-lot 'I' 0

IF( '1') - a(u., '1')/- sup- rES-lOt 1/'1'110

SIIFllo+ MI/u.l/o S C" = constant.

Hence, a subsequence {p.} and an elementary p E L2(!l) exists such that P. ---" p. Since N isweakly sequentially closed, pEN. Moreover,

(P.f'(U.)'TdX-( pf'(U)'TdxIn Joas E -+ 0 since P. ---" p and f'(u) in L2(!l), This completes the proof of the theorem. 0

3. FINITE ELEMENT APPROXIMATIONSWe now examine certain equilibrium finite-element methods for approximating the penalized

problem (2.9), Following Johnson and Mercier [6]. we consider a discretization Tit of !lconsisting of E triangular elements 0 with vertices af, i= 1,2,3, which consist of threesubtriangles OJ obtained by joining the vertices to the centroid. We define subspaces Sit of Sand Vit of V as follows

Sit == {Ult E SIUhlol E (PI{{Oi»3; (ah)jfljobeing continuous across interelement boundariesand sides Oa P for all 0 E Th}

Vit == {Vh E VI(Vh)Jlo E PI(O) 't/ 0 E Tit}.

(3.1)

Here P.(M) is the space of polynomials of degree s I defined on the set M.Johnson and Mercier[6] have shown that for any regular faimily {Tit}, 0< h so, of trian-

gulations, h being the maximum length of all sides of all triangles ir Tit, there exists an elementSit E Sit and a constant C independent of U and h such that

r=1,2 (3.2)

for U E S n (Hr(!l)t, where Iulr.ois the seminorm consisting of derivatives of aij of order r.As our approximation of the set of equilibrium stress fields, we introduce the set

(3.3)

Analysis of some finite element methods 983

To approximate the subset of stresses which satisfy the yield condition, we follow Hlavacek [I]and introduce the function

(3.4)

Then a stress field (Th E Sh will approximately satisfy the yield condition if it belongs to the set

(3.5)

In other words, in our approximation of the yield condition I( '1') S I a.e. in !l, we require thatthe average values of Th on each finite element satisfy the yield condition. Obviously

(3.6)

One possible approximation to (1.1) is to seek u" E K" such that

(3.7)

This finite element method was studied by Hlavacek[I], who showed that if the actual solutionu E (H1(!l)t, then Ilu - uhllo ~ 0 as h ~ 0 for any regular family of triangulations. Weshall consider instead of (3.7) a penalty approximation based on (2.8) and (2.9). Let

(3.8)

and

(3.9)

Our exterior penalty approximation of problem (l.l) consists of seeking ul, E S" such that

(3.10)

Theorem 3.1Let there exist a stress 'T E K n (H1(!l»4. In addition, let (3.2) hold and let the conditions of

Theorem 2.1 hold. Then there exists a unique solution ul, E E" of problem (3.10) for everyE>O.

ProofHlavacek [1] has shown that the existence of a stress 'I' E K n (H 1(!l»4 implies that K" is

nonempty. The result then follows immediately from arguments in the proof of Theorem2.1. 0

We next set up an auxiliary problem:

Find tTl. E E" such that for each E > O},p.(Uh)S,p.(T,,) 'rJ'T"EE".

Theorem 3.2Let the assumptions of Theorem 3.1 hold. Then:(i) There exists a unique solution tTl. to (3.11) V E > 0,

(ji) If u. E En (H1(!l»4 and tTl. E E" are the unique solutions to the problems

(3.11)

,p,(U.)S,p.(T)

(3.12)

984 J. T. aDEN and J. R. WHITEMAN

and, if we have a regular family {T,,} of triangulations. h E (0, hoJ, then

I/u. - 0.;,1/0-+ 0 as h -+ O.

Prool(i) The functional

is coercive strictly convex, and differentiable on Eh' which is closed and convex. Hence, (i)follows,

(ii) To prove this assertion, we call upon a lemma due to Cea[8]:Lemma 3.1. Let u and Uh be solutions of the minimization problems

~(U) = minimum over K.(Uh) = minimum over Kh'

where is a quadratic functional on a real Hilbert space H with a positive-definite seconddifferential. K C H a closed convex set and Kh C H a closed convex subset for any h,0< h ~ ho. In addition, let the following assumptions hold:

't/ hE (0, lIoJ. 3 Vh E Kh such that 111/- vhll-o as 11---+0. (H.1)"Vh E Kh' Vh -~ u* E H weakly for h-O"~ 1/* E K. (H.2)

Then

Thus, we need to verify hypotheses H.I and H.2 for problems (3.12). Towards this end, werecord a result of Hlavecek [I].

Lemma 3.2. Under the conditions of Theorem 3.1. hypotheses H,I and H.2 of Lemma 3.1hold for = ,p., K = E and Kh = Eh'

ProofCondition H.I holds trivially since, by assumption, u. E En (HI(!l» and, by (3.2),

Ilu. - Shl!s Chlu."-n. To verify H.2, suppose '1'" E Eh' 'Th -- 'T weakly. But 't/ v E V 3{Vh} suchthat v" - v strongly in V. Hence av"J aXI ---+ av;/ aXj strongly in S. Thus

i aVhl i aVjT"ij -a dx---+ Tjj -a dx = L(v) as h -+ O.n XI n XITherefore, 'T E E. 0

Completion 01 Prool 01 Theorem 3.2. Theorem 3.2 now follows immediately from Lemmas3.1 and 3.2 (with U = U,. Uh = U;" = 1/1., etc.). 0

We also have the following result since Eh is nonempty and closed:

Theorem 3.3Under the conditions of Theorem 3.2,11... ;, - u"l!o- 0 as € -+0 where u" is the solution of

(3.7). 0We have established conditions sufficient to guarantee that Uh -+ U, U;, -+ Uh, as €, h -+ O.

The determination of € as a function of h which will produce a stable scheme depends stronglyon the form of f and will not be taken up in this paper. Since

(3.13)

Analysis of some finite element methods 985

we would expect Uh---+U as e,h---+O for stable schemes provided Iluh-Uhllo---+O. Weshall conclude this phase of our study by establishing a condition sufficient to guarantee thislatter property; a detailed analysis of the stability of these schemes shall be postponed until alater investigation.

Let

and denote

and

f(e, h) = sup a~'A ra(e, h, Th)ThEEdO) IITh/lO

Since

for any Uh, 'Th E Sh' we also have

a(uh.'Th) II IIsup II II ~ m (fh 0 't/ Uh E Sh.Th E Sh-{O} Th 0

We then have:

(3.15)

(3.16)

Theorem 3.4Let the assumptions of Theorem 3,2 and (3.16) hold and let u1, and u1, be the solutions of

(3.11) and (3. 10), respectively. Then Ilu1,- u1,llo---+ 0 as e.h ---+ 0 if

lim reE, h) = O.•.h-+O

ProofThe stresses u1, and u1, satisfy the equalities

(f) -I ~ (IG(~ (' 1) aIG(q,A(uh» ~ ( ) F( )a UA, Th + E "'" 'l'h Uh)- + a 'l'hij 1'h = 'TIlaE~ ~u

a(u;,. Th) + E-1 ~ I. (f(u;,) - I)+('(u;,) . 'Thdx = F( TA)aETh a

Subtracting the first of these from the second yields

Thus

(3.17)

~ ra(E. h. Th)aETh

II'Th/lo= nE. II)

from which the assertion follows. 0

986 J. T. ODEN and 1. R. WHITEMAN

ISi,js2},

4. A FINITE ELEMENT METHOD FOR ELASTO-PLASTIC PLATES

An analysis similar to that recorded earlier can be developed for the study of finiteelement approximations of the problem of equilibrium of thin elasto-plastic plates obeying aHencky-type law, To describe one such method, we introduce the definitions

M = {M = (Mij) E L2(!l)IMil = Mji,

IIMII~==In MilMjldx,

E== {M E MIL Mil a~::xidx = In bv dxI == convex, continuous yield function defined on the moment

tensor M(f: R4_ R),

P == {M E MI/(M) s I a.e. in !l}

(4.1)

Uo E H5(fl,), - auo = 0 on r.Uo - an

The equilibrium problem for these elasto-plastic plates consists of seeking the momentM E K = E n P such that

where

IjJ(M) s inf IjJ(Q)QEK

IjJ(Q) = the complementary energy

== ( C1lklQIIQkl dx - ( QilKOil dx,In In

(4.2)

K = E n P and Cilkl E L",(!l) has the usual symmetry anddefiniteness.

Problem (4.2) is equivalent to

ME K: L Cjjk/Mkl(Qjl-Mij)dx~ In KOII(Qjj-Mij)dx 't/QE K. (4.3)In order to construct a finite-element method for problem (4.2), we introduce a triangulation ffhof composite triangles of precisely the same type as introduced in Section 3. We then define

M1lnl are continuous across

interelement boundaries

alai' Oaj. ;,j = I.2, 't/ G E Yh},

Vh = {Vh E cl(m n H5(ll)lVhla E Ps(G) , 't/ G E Yh}, (4.4)

Eh = {Mh E -«hi!In Mhii:;~~Idx = L dx = !o bVh dx 't/ Vh E Vh},Ph = {Mh E -«hl/(me~G fa Mh dX) sl 't/ G E Yh}.

Our finite element approximation of (4.3) consists of seeking Mh E Eh n Ph such that

(4.5)

or, equivalently

Analysis of some finite elements methods 987

Note that owing to properties (4.4), we are guaranteed the existence of a moment Qh E.M.hsuch that

r = I.2, (4.7)

for any M E .M. n (H1(O)tOur existence and convergence results on method (4.6) are summed up in the following

theorem:

Theorem 4.1Let conventions (4.1) hold. (i) Then, if K = E n P# 0. there exists a unique solution to

problem (4.2) (or, equivalently, (4.3».If, in addition, there exists a Q E E n P n (H 1(!l»4, and if the solution M of (4.3) E

(HI(fl)4. then (ii) there exists a unique solution Mh E Kh = Eh n PI, to the discrete problem(4.6) and (iii)

for a regular family of triangulations {~h}'

ProolConditions (i) and (ii) follow easily from arguments given in the proofs of Theorems 1.1, 3.1

and Lemma 3.2. To prove (iii) we make use of Lemma 3.1. Hypothesis H.I holds trivially byvirtue of (4.7). To verify H.2, we choose Qh E Eh' Qh-----"Q* E ,(( follows fromthe weak closed ness of E (see the proof of Lemma 3.2).

It remains only to show that if Qh E Ph and Qh ~ Q*, then Q* E P. Toward this end, weemploy the ideas of Hlavavcek [I]: Let 4>h:Ai-d{ be defined by

Let Qh E Ph. Qh - Q* E Ai. Then for any L E .Ii,

I(L, 4>h(Qh)-Q*)1 == lIn Lil(4)hij(Qh)- Qa)dxlsl(L, 4>h(Qh) - - 4>h(Q*»)1+ I(L, 4>h(Q - Q*»)I

= L f Lil dx f (Qh - Q*Mmes G)-I dx + I(L, 4>h(Q) - Q*)IGETh G G

Thus

But Qh E Ph ¢:::} 4>h(Qh) E P from the definitions of P. Since P is weakly sequentially closed(being closed and convex)

Hence, if Qh E Eh n Ph. Qh - Q* in, we must have Q* E E n P. Hypothesis H.2 of Lemma3.1 is thus verified and the proof is complete. 0

988 1. T. ODEN and 1. R. WHITEMAN

REFERENCES[1) r. HLAVAcEK, A finite element analysis for elasto-plastic bodies. To be published.[21 J. NECAS, ZAMM 60. no (\980).[31 B. MERCIER, Sur la Theorie e/ /' Analyse Numirique de Problimes de Plas/id/e. Thesis, Universite Paris VI (\977).[4] R. FALK and B. MERCIER. Es/imalion d'meur en elasloplas/ieile. 282A, 645 (1976)[5J R. FALK and B. MERCIER. R.A.IR.o. Anal. Numer. II, 135 (1977).[61 C. JOHNSON and B. MERCIER. Numer. Malh. 30. 103 (1978).[71 r. EKELAND and T. TEMAM, Convex Analysis and Varia/ional Problems. North-Holland. Amslerdam (1976).[8] 1. CEA, Oplimisalion. Theorie el Algorililmes. Dunod, Paris (197\).

(Reeeh'ed I October 198\)

page1titlesANALYSIS OF SOME FINITE ELEMENT METHODS FOR A IN THIS paper. we consider a class of problems in elasto-plasticity that can be characterized as Find a stress field U E K such that The following notations and conventions are used: ,p( '1') == the total complementary energy functional at the stress 'I' =! aCT. '1')- F(T) (t E K), a( T. U) == the complementary strain energy produced by stress fields Cilkl == components of the inverse of Hooke's tensor for every symmetric matrix Ajj, mo> a.e. in !l. 110, being components of a prescribed displacement field 110 (I.I ) 1 av· Til -a ' dx = L(v) 't/v E V. V == space of admissible displacements = {v = (Vlo V2 •... , VN) E (H 1(!l»NIVj = 0 a.e. in fu}. L(V) == 1 bivi dx + i tiVi ds, v E v, LEV', n ra and I( '1') S I a.e. in fl}.

page2titles1 avo .. a( 'T, u) = a(u, 'T), a( '1', u) $ ml/Tllol/ul/o, and a( 'T, '1') ~ milTI15 'tiT, u E S. au .. -a'l + bi = 0 in n, I {au; aUj} .

page3titles,....0. t I ( avo E = {'I' E S Jo 'Til a; dx = L(V) 'tIv E v}.

page4page5page6titlesP(u.) s E(IjJ( '1') -1jJ(u.» s C' E ('I' E K). P(u) slim inf P(u.) sO ~P(u)=O¢:~uE K. 11 pJ'(U.) , T dxl al/p.llo s sup n II II IF( '1') - a(u., '1')/ (P.f'(U.)'TdX-( pf'(U)'Tdx In Jo Sit == {Ult E SIUhlol E (PI{{Oi»3; (ah)jfljo Vit == {Vh E VI(Vh)Jlo E PI(O) 't/ 0 E Tit}. r=1,2 (3.2)

page7titlesTheorem 3.1 Proof Theorem 3.2

page8titlesI/u. - 0.;,1/0-+ 0 as h -+ O.

page9titleslim reE, h) = O . II'Th/lo = nE. II)

page10titlesISi,js2}, - auo = 0 on r. In In

page11titlesr = I. 2, Prool

page12