Co"'p. & .I,",M .• i,h Appls. Vol. 8, No. I. pp 35-56, 198~Printed in Great Brittin.

fX117 -494 JI8210 t OOJ5- 21$0 J.00/0Pergamon Pres, Lid.

'i 1(/< (-.. b

INTERIOR PENALTY METHODS FOR FINITEELEMENT APPROXIMATIONS OF THE SIGNORINI

PROBLEM IN ELASTOSTA TICS

1. T, ODEN and S. J. KIM

Texas Institute for Computational Mechanics. Department of Aerospace Engineering and EngineeringMechanics. The University of Texas at Austin. TX 78712. U,S,A.

(Received April198Il

Abstract-The use of interior penalty methods as a basis for developing finite element approximations ofboundary value problems with constraints is explored. Particular attention is given to the Signorini problem ofcontact of an elastic body with a rigid foundation. Error estimates are derived and the results of a numericalexperiment are discussed.

I. INTRODl'CTlOtliThis paper is concerned with the development of interior penalty formulations of a class ofcontact problems in elastostatics and with their use as a basis for the development of new finiteelement methods for the numerical solution of problems of this type. In recent years. the use ofexterior penalty methods has been very popular as a basis for the development of finite elementschemes for the analysis of boundary-value problems with constraints. We mention. in thisregard. the reduced integration methods of Zienkiewicz. Taylor and Tooll]. Malkus andHughes (21 and the studies of their numerical stability and convergence properties by Oden.Kikuchi and Song[3]. These methods arc best \'eiwed as arising from formulations of boundary-value problems as constrained optimization problems. Then many ideas from classical opti-mization theory can be used 10 solve the systems of equations or inequalities characterizing thediscretizcd problems.

For example. consider the prOblem of minimizing a functional F on a closed convex set K of aH ilberl space V. The exterior penalty method involves replacing F by a penalized functional

where f is an arbitrary positive number and P is an exterior penalty functional endowed withthe properties such that

(i) P is weakly lower semiconlinuous(ii) P( r) ~ O.

PII')=Oiffl' E K.P(tl>Oifr$K.

Generally P is designed so that P(v) increases with the distance from r to the constraint set;thus. the more the constraint is violated by a trial vector I' E V. the larger the penalty that mustbe paid. By the addition of the penalty terms f-I P. the original constrained minimizationproblem is reduced to the unconstrained minimization problem of finding Ii, E \" such that

min F,(I') = F,(II,)I'E v

where the minimum is sought throughout the entire space V. Under mild conditions on F. one isw("~lkl~'

guaranteed the existence of a sequence {Ii,} of minimizers such that III ~ Ii in Vas f ....O.where II is a minimizer of the original functional F. A key aspect of stich an approach is that the

35

36 J. T. ODEN and S. J. KI\I

approximate mllllmtZer II, lies exterior to the constraint set K; hence the term "exterior"penalty method.

There are other types of penalty methods employed in optimization theory. In particular. ininterior penalty methods a penalized functional such as

F, = F + EQ. f > 0

is used. but the interior penalty functional Q is designcd so that Ihe minimizers II, of F, lie ininterior to the constraint set K. Thus, the addition of the penalty term Q prevents the violationof the constrain\; the ··trial solutions" II, always remain inside the constraint sct K for anyE > O. Again. under reasonable conditions on F and Q. II, converges to a minimizer II E K asf ~ O. The interior penalty methods are typically used in finite-dimensional problcms with linearinequality constraints. but some generalizations to abstract optimization problems have beenproposed (Fiacco and Jones [4]. Stong [5], Fiacco[6 D. For more dctails on various penaltyapproaches in optimization theory. see Fiacco and McCormick In

The present investigation is concerned with the development of finite element ap-proximations of the Signorini problem of elastostatics: Ihe problem of cquilibrium of a linearlyelastic body in contact with a rigid frictionless foundation. This class of problems. which isdiscussed in grcat detail in the forthcoming book of Kikuchi and Oden (8], is characterized bythc systcm

( EWliu ).j + Ii = 0 ;n n ]IIi =0 on rD

Eijk:IIUllj = ti on rF

(1.1 )

and Ihe incquality constraints

u·n::::s.(rn(u)::::O

(1.(U)(II· n- s) = 0

(TTiIU) = 0

on rc ( 1.2)

wherc the usual notations and conventions of elasticity are used; n is a smooth. open. boundeddomain in RN with boundary an = rD U rF U re. n is a unit cxterior vector normal to an andIIi = lIi(X) (with x = (x,. X2' ...• XN) E fl) are the Cartesian components of the displacementvector u. Ew are the clasticitics of the material. Ii are the components of body forcc per unitvolume, and ti arc thc surface tractions. In (1.2). rc is the contact area. i,e. r c contains thoseportions of an which come in contact with the foundation upon the application of loads. Thefunction "s" is a normalized initial "gap" between the body and thc foundation prior todeformation. Thc notation hcrc is standard and follows that in Kikuchi and Oden [8]. As is \vellknown (see. e.g. Duvaut and Lions [91 or Kikuchi and Oden [80. problcm (1.1) can be fromulatedas the constrained optimization problem.

mllllmlze F on K (1.3)

where F is now the total potential energy and K is the subset of admissiblc displacementssatisfying the constraint (1.2) (in an appropriatc sense). This fact was exploited by Odcn.Kikuchi and Song[3]. who dcveloped and analyzed several numerical schemes for studyingsuch contact problems using an exterior penalty formulation of (\ J), Despitc the fact that (l.3)involves an inequality constraint. therc does not appear to have been any attempts in theliterature to use thc interior penalty ideas as a basis for formulating this problem and solving itnumerically.

In thc prescnt study. two new formulations of the general class of contact problemsdescribed by (l.I) and (1.2) are given which employ the concepts of interior penalty and whichdiffer in the form of the interior penalty functional used. The construction and analysis of these

Interior penalty methods for finite element approximations 37

penalty methods is given in Section 2. There wc also prove the existence and uniqueness ofminimizers of the penalized functionals for each f > 0 and show that sequencc of suchminimizer can be constructed which converges to a generalized solution of the contact problcmas E tends to zero. Section 3 is devoted to thc development of a ncw class of finite elementmethods for the analysis of boundary-value problems with inequality constraints of thc type(\.3). Thc intcrior pcnalty mcthods developed in Section 2 arc used as a basis for a finite-clement approximation of problem (1.1) or (1.3).

The convergence of these methods is also investigated. and a priori error estimates arederived. An algorithm for the implementation of thc intcrior pcnalty-finite elcment mcthodsdeveloped in Section 3. is outlined in Section 4. and is applied to a representative cxample. Thenumerical example considered here is the problem of indentation of a rigid punch into an elasticsolid, Solutions to this problem by other method are available and can be used for comparison.It is shown that the methods developed in this study are efficient and produce result~, in goodagreement with those obtained by other methods.

2. INTERIOR PENALTY FORMULATIONS

2.1 Variational ideas for contact problems in elasticityWe shall now establish that the general class of contact problems without friction charac-

tcrizcd by (\.1) and (1.2) can be put into a variational setting that cnablcs us to make use ofresults from the thcory of constrained optimization. We bcgin by introducing thc Hilbert space

(2.1 )

whcrc y(v) = vir and y is the usual tracc operator mapping [H I(m/"" continuously onto[H1/~(afl)]N which is the completion of C(afl) with respect to the norm

IldJlll/2.r = inf (IIIIII.: dJ = y( II)}" EH1tlll

and H I(fl) is the usual Sobolcv space of functions with distributional derivatives in LCW) (sce.e.g. AdamslJ()Il.

We shall assumc throughout that mcas I'D > O. Then V is a Hilbcrt spacc whcn equippedwith the norm.

The potential energy functional of our problem is defined by the functional

where

and

and we assume

II(U. v) = r EijUllkfVi.j dx. U. v E VIn

f(v) = r fil'; dx + r t;t:; ds. v E VJfI JrF

fi E elfl). tj E LC(rFl )

(c n (D = 0; rc is a smooth (e,g. C2) surface,

(2.2)

(2.3)

(2.4)

(2.5)

38 J, T. ODf~ and S. J. K'~l

We shall make the following assumptions concerning the elastic coefficients Eijk/ in (2,1):

(ii) Ejjk/(x) = EkJij(x) = Ejjk/(x)a.e. in n. I :5 i. j. k. 1:5 N

(iii) There exists a constant /110> 0 such that

Ejjkl(x)Ak/A;j ~ /IIoA;jAij

for all tensors A such that A;j = Ajj.

(2,6)

Under these assumptions, one can show that a(· .. ) is V-elliptic and continuous (see Duvaut andLions [9]); i.e,

a(u. ~'):5 Mllull,lIvll, V u. v E V Ia(u. v) ~ 1II11~'III' V~· E V (2.7)

Also. it is easily verified that f is a continuous linear functional on V. i.e, f E V' (the dualspace of V)

The constraints (1.2) enter the variational problem in the form of thc constraint set

K={vE V/y,,(v)-S:50a.e.on rd, (2.8)

Herc YII(V) = 1'" on re. VII = Y( v;)l/j. We remark that the ordcring:5 on (2.7) is wcll-defined andthat the set K is a non-empty. c1oscd. convex subset of V (for more details. see Kikuchi andOdcn[S]).

Finally. we notc thaI. the functional F. rcstrictcd to K has the following properties:

(i) F is strictly convex.(ii) F is Gilleaux diffcrentiable: indccd.

(DF(u). v) = (I(U. \') - f(\'l Vu. \' E K

(iii) F is coercive; i,e., because of (2.71,.

(2.9)

As is well known (see e.g. Ekeland and Temam [II D. these properties are sufficient to guaranteethe existence of a unique solution u to the minimization problem

min F(v) = F(u).eK

(2.10)

and to establish that this minimizer u can be characterized as the solution of the variationalincquality:

u E K: a(u. you) ~f(v-u) V \. E K (2.11)

2.2 Penalty for/llulationIn anticipation of problems connected with the approximation of the constraint set K. Vie

now seek an altcrnative formulation of the variational problem (2.10) (or (2.11)) which make useof interior penalty concepts. As a guide to such formulations. we first establish a basicexistence theorem. Again. we are interested in the problem of minimizing of functional

Interior penalty methods for finite dement appro\imation\

F: K .....R when the following situation prevails:

• V is an arbitrary Hilbert space I• K is a non-cmpty. closed. convex subset of V• F: K .....R is a strictly convex. Gateaux differentiable .

coercive functional defined on the set K.

39

(2,12)

A functional Q: V .....R shall be called a barrier (or an interior penalty) flinctional for anobjcctive functional F satisfying (212) if and only if

(i) 0: V .....;~ is convex. proper. and lower semicontinuous(ii) 0((')< +:r; if t' E K and Q(r)= +:1: if v ~ K

(iii) F,(v) = F(r)+ EQ(r) is coercive; F,: V .....R r (2,13)

(iv) for evcry sequencc r, E K converging wcakly toI E K as E ... O. we have lim inf fQ( r,) ~ 0

f- ..n

where K is the intcrior of set K and E is an arbitrary positive number. which is less than a finitcvalue Af.

TUEOREM 2.1Let (2.12) be given and Ict Q be a barrier pcnalty functional (i.e. Q satisfies (2.13 )), Then.

Ii) for every E > O. there exists a unique minimizer Ii, E K of the functional

F,(l')= F(!')+fQ(rI. I E V (2.14)

(ii) if 0 is Gfltcaux differcnti:lbk in dorn (01. then the minimizer Ii, for thc fixed E >0 ischaracterizcd by the cquation.

(DF(II, I. v) + f(DO(Ii,).r) = 0 V t' E V (2.15)

wherc (-. .) denotes duality pairing on V' x V and(iii) thcre exists a subsequence Ii" of solutions of (2.15) such that

Ii,. ~ Ii (weakly) in V

as fk .....0. where II E K is the minimizer of Hi.e. Ii is the <;olution of probkm (2.10)).Proof. (i) It is clear that F, is proper. strictly convex. lower semicontinuous. and coercive for

each f > O.By the generalized Weierstrass minimization theorem (see e.g. Vainberg[ 12]). it followsthat there exist a unique minimizer Ii, for each f > O. Clearly. since F. < + z. Ii.. E K.

(ii) Let Q be Gateaux differentiable with derivative DQ: V .....V'. Since J-~ is convex. lowersemicontinuous. and Gflteaux differentiable. the minimizer satisfies the following variationalinequality:

(DF,(II,).t'-li.)~O VI' E K

Since II, E K. there exists µ > ()such that II, ± µI' E K for I' E V. Thus.

(DF,(II,).V)=O V:;EV

which is precisely (215)(iii) Clearly.

F.(II,)~F,(I') \I I' E V

40 J. T. ODEN and S. J. KIM

so that for some fixed Vo E K. and f < M (positive number).

F,(u,l:5 F(t'o) + fQ(Vo)

< F(ro) + MIO(vo)/

= constant.

This means that F,(u.) is uniformly bounded. independent of f. for f < M. But, since F, iscoercivc. we must also have a constant C. independent of f. such that

This implies that a sequence fk -+0 can be found such that

U't ~ U weakly in V

where U is necessarily in K (because U't E K and K is weakly sequentially closed).Hence. from property (iv).

F(r)= lim inf(F(v)+fkQ(v»~ lim inf F,t(u't)<t-o 't-o

= lim inf(F(u.t)+fkQ(u.t))'t-+O

~F(u)'VvEV.

Thus. thc limit U is a minimizer of F and is in K. 0

2.3 EXIImples of barrier fUllctiollalsTwo examrles of barrier functionals appropriate for contact problems are:

(I) The inverse barrier functional

(2.16)

(2,17)

(2) The logarithmic barrier functional

Q2(V) = 1- Ire In (s - 'Yn(") dt. ,. E K I+cc v$K

We shall now demonstrate thai each of these functionals is well-defined for the space Vof(2.1) and satisfy the requirements (2.13) of barricr functionals for the energy functional F of(2.2).

From the definition of V, v E (H1(fl))N. and by the trace theorem, 'Yn(v) = v- n EH1/2(rcl c L2(f cl since v . n - s <O. a.e. in K. (v . n - srI is bounded in K. Furthermore. themeasure of rc is positive. Thus (using again the simplified notation V· n = 'Yn(v)).

(V'n-srl E e(fel. (2,18)

On the other hand. from the elementary relationship - x < -In x < X-I. 'Vx> 0 and x E R. we have"'n-s<-ln(s-v'n)«s-v'n)-' a.e. in re. But (v-n-s) E U(fd and (v'n-s)-I EC(rd C L2(f d. thus

-In (s- V· II) E L2(rc).

Here we have made use of the following fact:

(2,19)

Interior penalty Illethod~ for finite element approximations 41

if otherwise.

if g(v) E Lltn)

LEMMA 2, I.Let /. gl and g~ be real-valued functions such that gtlx 1< f(x) < g~(x I 'V x E r{. gl.

g~ E e(re), Then f E L~(rc). 0

To study additional properties of these functionals we next call upon a basic lemma due toI3rezis [131 and establish another lemma on convexity which can be easily proved,

LEMMA 2.2.Let g: R'\' --> R be a proper. convex and lower semi-continuous function, Let \' E [utnJ].\' and

let F:(U'(!1))N -->R be defined by

( f g(,,) dxF(v): n

+:x

Then F is proper. convex and lower semicontinuous,

LEM~IA 2.3.Let f; R --> R be convex and decreasing function and g; V --> R be a concave, Then thc

composition f' g is convex. 0

Now. let us establish the following theorem:

THEOREM 2.2.The functionals 0,. Q2: V-->R defined by (2.16) and (2.17) are barrier functionals for the

energy functional F of (2.2).Proof. (i) By using Lemma 2.3. we easily verify the functions

and(/J(")(X): (s(x) - ,,(x)· nIx)) I 1

Q2(\')(X): -In(s(xl- ,,(x) . nIx))(2.20)

are convex in R'" for any choicc of v: that is.

l/j(ITU+(I-lTh')$a(/i(u)+(1 - IT)l/;('')

i = 1.2. u E [0. I]. 'V u. v E K. 'V x f re,

Thus. the functionals 0;: V --> R given hy

arc also convex functionals on K.The functions q\ and q~ of (2.20) are continuous from R2 into R (as can be seen by noting

that each is a composition of a continuous functions fl(x) = x ~I or f2(X) = In x and a continuousfunction p(vl = s - \' . n). Moreover. qi E v(r c). Hence. from Lemma 2.2. 01 and 02 areconvex and lower semi-continuolls on V.

(ii) Property (ii) of (2.13) is obviously satisfied by definition of the OJ,(iii) It is ohvious that F + fOI is coercive: indeed. since QI!\') ~ O. then

and. by hypothesis. F(\')--> +::c as 11,,111--> + 'X..

To verify that 02 satisfies (iii). note that since In (s - l'n) < S - Vn•

42

Thus, for any v E V.

1. T. ODEN and S, 1. K'M

where CI. C2 and c) are positive constants. Hence.

(iv) Let us examine property (iv) of (2.13). Let VEt E K converge weakly to v E K asfk ~O, Then.

(a) Inverse barrier function

lim inf fkQ,(V •• )~O since Q,(v»O, 'rj\' E K•• -01)

(b) Logarithmic barrier functionSince -In x > - x, x> 0, we have

and

fr,(s-v •• ' n)ds = lie (5 -v •• ' n)d/l

~ cllv'lll, + 115110.I. 1'( ~ CJ

where (.'1 is a positive constant. Hcre we uscd the fact that cvery sequence \'" weaklyconvergent in V must be bounded in V. Thcreforc.

~ lim inf (- fkC,)•• ....0

= 0, oNow let us examine the Gateaux differentiability of functionals O. and O~.To do this. we

will make use of the following Generalized Leibnitl rule:

LHI~IA2.4,If g: n x (0. µ)~R (µ E R. µ > 0) satisfies the following conditions,(i) Caratheodory conditions

l.-.g(x.1) E el(!l) a,c. x E fl

x~g(x. 1) E LI(!l) 'rj I E (0. µ)

(ij) there exists II E L '(fA) such that

Iilg(x. 1) I ~ IIlx) a,e. in fl. I E (0, µ)(it

Then.

:1 L g(x. t) dx = f. ilK(> 1) .n til d.\, o

Interior penalty methods for finite element appro\imalion, 43

THEORnl 2.3.Functionals Q, and Q, are Gitteaux differentiable,Proof. In our cases. £{I(V)= (.I' - V· n)-' and q,l\'l = -In (.I' - v' n). First. it is clear that the

functions t-+ql(u+h'). q:lu+J\') E CI(fcland that \'-+£{I((U+tVl. l/,(u+tv) E LI(fe). 'VI E(0. µ) since ql and q2 are in e(f (') and the functions X-I and In x. x > 0 are continuous. Second. wenote that

lilq,(U + Iv'l = 1 ,. n ~I E LI(re)ilt 2( u ' n + J\ . n - s)

and

for all u + h' E K because v . n E e(f d and (u' n - ~rlE L"(fcl for all u E K.Therefore.

(by Lemma 2.4)

where [,.,1 denotes duality pairing on W' x W with W = HI12(fcl. Similarly.

i' . J \'. nlim":,,\()~(u+'\l')= .tltA-+ll (III I'e(s -11'11

2.4 Calculation of contac( pressuresOnc of thc advantagcs of penalty formulations such as ollrs is thaI the unconstraincd

problem contains only one dcpendent variable u.; unlike Lagrange multiplier techniques. it isnot necessary to introduce a multiplier corresponding to the constraint. Nevertheless. suchmultiplicrs may have an important physical significance and one would hope that it is possibleto evaluate them in penalty formulations,

In the contact problem considered here. the multiplier corresponding to the contactconstraint is precisely the contact pressure normal to the contact surface. To ascertain how itmight be computed. wc rcwrite (2.15) in the form.

a(u,.\')+€(ilQ(u,l.y,,(v)]=f(vl 'Vv E V

whereas the Lagrange multiplier formulation of the problem is of the form

(2.21 )

a(lI. v) - [po y"(\n = f(,·)

Iq-p.y,,(II)-sJ~O

where N is the nonempty. closed convex set

'V \. E V I'VqEN (2.22)

N =lq E W'lq~O} (2.23)

In (2.23) the ordering" $ ,. is understood to define the positive cone in Hi' conjugate to thecone C in W whereon" $" signifies that if u. l' E H I(m and y( u) $ y( v). then y(u) $ y( t)

almost everywhere on aO.

CAMWA Vol. 8, So, 1-0

44 1. T, ODEN and S, j, K'M

A comparison of (2.21) with (2.22) suggests that a possible approximation p, of the contactpressure p might be given by

wherc

a(u,. v) - [p,. Yn(~')] = f(v) 'r/v E V (2.24)

p, =-f(Yn(U,)-sr2 for 0= 011p,=f(Yn(U,)-s)-1 forQ=02

(2.25)

Such approximations of contact pressures are always computable: since a unique solution u,of (2.21) exists for every f > O. we need only compute the normal trace Yn(u)( = U . n foru E C(n») and then use (2.26) to obtain p,.

The question of the convergence of p, to the contact pressure pEW' satisfying (2.22)pivots of the Babuska-Brezzi condition for this unilateral constraint (see. for example.Babuska[14] and Brezzi(15)). For the present problem. this condition assumes the form

There exists a constant a > 0 such that

allqllw':S sup [q. 'Yn(v)]'E \' Ilvll, for all q E W'. (2,26)

This condition is equivalent to the requirement that the trace operator 'Yn: V -+ W be surjectiveand that therefore. its transpose Y~ be bounded below.

THEOREM 2.4,Let (2.26) hold. Then there exists a sequence of positive numbers f -+ 0 such that p, defined

by (2.25) converges weakly in the set NeW' to the contact pressure p. in (2.22), where N isdefined in (2.23).

Proof. If p,~p in W' and u.-"u in V. then it is obvious from (2.24) that (u. p) is a solutionof (2.22). Thus. we nced to show that p, is uniformly bounded in f. But this follows (2.24):

Sincc (2.26) holds.

[P •. YII(~')] = f(v) - a(u,. v)

:S (1Ifll* + mllu,III)IIvll,:S Cllvlll 'r/ v E V. (2.27)

Thus. a subsequence exists which converges weakly to pEW': but pEN. since N is closedand convex in W'.

DStronger condilions are needed to guarantee that p, converges strongly to p in W' and to

estimate rates of convergence in terms of f

By substracting (2.24) from (2.22). we have

a(u-u,.v)-IP,-P.Yn(v)]=O 'r/v E V

Thus. from (2.26) and (2.7)1.

lip - p,IIw':S ~ sup Ip - p,. Yn(v)]a, E V /Ivlll

:S .~ IIu - U.lll

(2.28)

(2.29)

It follows that p, will converge strongly to p in W' whenever u, converges strongly to u in V.

Interior penalty methods for finite element approximations

Note also that. by virtue of (2.28) and (2.7h.

45

mllu- u,r ~ Ip, - p. y.(u, -u)]

= Ip. - p. y.(uJ - 5] + [p - p" y"(u) - 5]

~ [p, - p. y" (U,) - 5] (2.30)

(since [p - p,. 'Y,,(u) - s] ~ 0 by virtue of (2.22h.).Consider the case of the inverse barrier functional 01 and let us assume that the exact

contact pressure p and its perturbation P. = - f(y,(U,)- 5r2 are such that

(2.31)

Then. for the inverse barrier functional 01' we have

[p, - p. 'Y,,(u,) - s] = [p, - p. - Y(f)( - p,tl/2]

= - Y(f)[p, - p, (- pf'/2]

- Ylf)[p, - p. ( - p, )-1/2 _ ( _ P )-112]

~ - Y(f)[p, - p. (- ptl/2]

the last stcp having used the fact that - Y(f)[p, - p. (- p,rl/2 - ( - p r1/2] ~ O. Hence. in thiscase.

(2.32)

Combining this result with (2.29) and (2.30) yields

(2.33)

Similarly. for the logarithmic barrier functional Q~.if

then a repctition of the above manipulation yields

Ilu-u,I"~~I(-prlll~am

In summary. we have proved the following result:

(2.34)

(2.35)

THEOREM 2.5.Let (u. p) E V x W' and (u,. Pi) E V x W' be solutions of problems (2.2]) and (2,24).

respectively. with p, given by (2,23). In addition. let (2.7) hold. Then, for all f > O.

(2.36)

Moreover. if the inversc barrier functional is used and (2.31) holds. then

(2.37)

46 J. T, OllEN and S. J. KIM

whcrcas if the logarithmic barrier functional o~is used and (2,34) holds.

(2,38)

oWe remark that assumptions (2,3 I) and (2,34) can be considcrably weakened. Obviously.

(2.31) automatically holds for p, and it need hold for p only on some space H UM'c). rr being ameasurable subset of rc on which PIx) < O. Similar comments apply to (2,34),

Wc also remark at the dramatic increase in the rate of convergence of u, to u in the case oflogarithmic barrier functionals as opposed to the inverse functional. the ratc bcing O(E) comparedwith O( f 1/2).

1 FINITE EI.EMENT APPROXIMATIONS

3.1 COllstruction of fillite elemellt approximatiollsWc construct finite element approximations of the penalizcd variational problem (2.21) in

the usual fashion; the domain n is partitioned into a collection of E finite c1emcnts nc overwhich the displaccmcnt field u is approximatcd by polynomials of degree k. By regularrefinements of the mesh. we can. in this way. produce a family {VdO"IlO:;I of finite-dimensionalsubspaccs of thc space V. whcre h, the mcsh parameters. is the largest diameter of an elementin the mcsh. Typically.

\lh ={l.1l E (Co(O))"'lv"llle=V/ E ~dfl,,)

e = I. 2, .. , . E} (11)

where gJdn,.) is the space of polynomials of dcgrec :5 k on n, .. We aSSllme thaI these spnces arcendowed with standard interpolation propcrties: i.e. if v E IH'(DnN n V. then therc exists,.Il E Vil sllch that

II,' - \,/'11. :5.CI1,,",·II,. .. \' = O. I )

µ = min Ik+ 1- s. r-s).

In this manner. we approximatc the variational problem (2.21) by sceking u/' E Vil C V stichthat

h.l, 1. Vii . n(t) d - f ,Ila(u,.' )+f ( h "'",, t - (, )rc S -u, . n(3,3)

whcrc a = 2 if O. is used and a = 1 if Q: is uscd.Most of the criteria sufficient to guarantee the existcnce of a unique solution u, of the

"continuous" problem (2.21) carryover to the discrcte problem (3.3): the functional F, of (2.14)is clearly strictly convex. GiHeaux-differentiable. and coercive on each \lh, Thus. for each f >()and each h > 0, there exists a unique u/' E V" satisfying (3.3).

The question of thc existence of approximate contact pressures is more delicate. Cor-responding to each choice of Vil and barrier functional O. there is a space Wh of approximatepressures. For example. if Vh is spanned be piecewise quadratics and Q is the logarithmicbarrier functions, qh E Wh will mean that qi.1 is piecewise quadratic. If p/ is the ap-proximation of p, given by

PI" = - f~O(U.") (Q = 01 or 0;)

then p,h will be a stable approximation in H~, if a discrete Babuska-Brezzi condition of the typein (2.26) holds for the subspace \lh and Who

Interior penalty methods for finile element approximations 47

To demonstrate the role of such a discrete condition. consider a class of approximations forwhich the following condition holds:

(i) The family {Wit} of approximations of the contact pressure is such that

~Vlt C HI V II > 0 and U ~Vlt = WIt ..·c,

(ii) There exists a constant lXlt > 0 such that(l41

where

Let us introduce one more set Kit which is a finite elemenl approximation of K such that

Now thc following theorem can be establishcd,

THEOREM ll.Lel Nil be a finite elemcnt approximation of N of (2.23). i.e.

Then the discrete problem

(}.5)

has a unique solution u." for all f > O. Moreover. if condition (ii) of 0.4) holds. then (u/,. P.").where p,1t = - E8Q(u,lt) converges weakly to a pair (u/'. ph) E K" X ~VI' and (uh. pit) is thesolution of

a(ult. \,It)_ [pit. YII(,It)] = f(\'I) V,h E \/1,)~ [qlt_plt'Y/I(U/')-S]~O Vqlt E N'I' (3.6)

Proof. The fact that a solution lI,h exists to (3.11 for each f > 0 was noted earlier. Also. (2,9)and (2.13) provide the uniform bound ness of IIu."ll, in f in KII' On the other hand, from (ii) of(3.4). we also have the uniform boundness of IIIp/'11I in f in Nh. Therefore. there exist subsequencesof {/lh} and {ph} which converge to ult and pl. as f-+O. i.e.

a(II~l'vlt)-[P~l.Ynh·It)]=fh,h) V,II E \/It

~ alul'. ,h)_[pll. y,,(v.l')] = f(vlt) V v'It E V", as fk-+O.

Now. we will show that pi, and 1111 satisfy thc inequality in 0.6). First we considcr:(i) The logarithmic barrier functional. Then

And. for arbitrary qh E NIl'

48

But

J, T. ODEN and S. 1. KIM

and

(3.7)

Passing to the limit as f tends to zero gives

(ij) The inverse barrier functional. In this case.

Similarly.

[p,h. "Yn(u,h)- s] = r (- p,hh/(E)(-p,hr'12dtJrc=Y(f) r (-p/)'12dtJrc$ C1Y(f)

because IlIp,hlll is uniformly bounded in f. Thus. again taking the limit as f tends to zero yields

By subtracting (3.6) from (3.5). we have

Thus. from (2.7)1 and (ii) of (3.4)

o

(3.8)

(3.9)

By arguments similar to those used in analyzing the continuous problem. we can establishthe following theorem on the rate of convergence in f.

THEOREM 3.2.Assume that solution Ph of (3.6) and p,h = - €<5Q(u,h) are such that

for the inverse barrier functional or (3.IQ)

Interior penally mel hods for finite element approximations

for the logarithmic barrier functional and (3.6) holds. Then we have

for the inverse barrier functional and

49

(3.11)

(3.12)

for the logarithmic barrier functional. 0There is still a deficiency in these estimates that can be overcome under certain additional

hypothesis; the estimates (3, II) and (3.12) involve ph on the r.h.s. rather than p and. hence,may depend on h. This is avoided in the case in which

for (3.1 I) or

for (3.12). Then. for instance,

and

Thus. whenever (3.11). (3.12) and (3.13) hold.

(3.13)

1

O(hU\!(E)+\!(f))

lIuh- u/II, =

O(hUf + f)

for the inverse barrierfunctional

for the logarithmic barrierfunctional.

3.2 A priori error boundsThe error between the penalized finite element solution (U,h. p,h) and the true solutions (u. p)

can be divided into two parts:(a) The errors with the fixed mesh size h

(b) The errors of the finite approximation obtained as f -+ 0

Clearly.

(3.14)

and

(3.15)

50 J. T. ODES and S. 1. KIM

But we have already estimated the first parts of these errors in Theorem 3.2. Now. the finiteelement approximation errors can be estimated in the following manner.

THEOREM 3.3.Assume that (2.22). (2.31). (2.34). (ii) of (3.4) and (3.10) hold. Then the following estimates

hold:

IIIuh - ull, S C,Ulu - vhlll + Illp - qhl/l, +-I1I'>',,(u - vh)1I1l" ah

+ C2[lllp - q"IIlIII'Y,,(u - ~,h)11I+ [q - ph + q" - p. 'Y,,(u)- s nil!

cIlip" - pili slllq" - pili + a: {llIq" - plll* + Ilu" - ull.}

where 111·11/ .. = 11·1111'"Proof. B'y subtracting (2.22), from (3.6)). we have

and let v" = u" - u. Then. since

the proof is completed in the following steps:

(3.16)

(i) [ph - p. 'Y,,(u" - v")] = [p" - p. 'Y,,(u- v")] + [p" _1/". 'Y,,(I/" -II)]

+ [q" - p. 'Y,,(u" -u)1

(ii) [p" - q". 'Y,,(u" - u)] s - [ph - I{". 'Y,,(u)- 51

s [q - p" + q" - p. 'Y" ( u) - s I(iii) [ph - p. 'Y,,(u- vh

)) sllip" - pllllli'Y,,(u - vh 1111(iv) II/ph - pill slllp" - qhl/l + IlIq" - pili(v) a"lIlph -I/hlll s sup [ph - q: 'Y,,(v")j

t·h E Vh IIv IIIa(uh -u v") + Ip - q" 'V (v")]= sup • I • III

t,h E Vh Ilv 'IIIs mdlu" -ulll + m211lp -1/"111*,

By combining (il. (ii). (iii). (iv) and (v) and using the Young's inequality. we get (3,16). 0Finally. we obtain the total error estimates by simple applications of (3.14) and (3.15) and the

results of Theorem 3.2 and Theorem 3.3, The further lise of estimates of the type derived hereto determine asymptotic rate of convergence must await more detailed information on theinterpolation properties of the spaces Who

4. NUMERICAL EXPERIMENTS

4.1 All algorithmWe shall now describe an algorithm for implementing the finile element methods described

in the previous section. Toward this end. let us consider a two-dimensional finite-element modelof V with N -degrees of freedom. Then typical test functions v" E V" and the finite-elementapproximation u," can be expressed in the form

(4, ))

Interior penalty methods for finite element approximations 5\

where (Vh)i' (U");. i= 1. 2 are the Cartesian components of the vectors vh and u,", 1/>" are theusual conforming finite-element basis functions. and the repeated nodal indices ll' arc summedfrom I to N. Notc that if xfl = (Xlfl• xl) denotes node (3 in the finite-element mesh. then

Q;a(x{3) ~ 8/" so that, (.\'~)(xa) = v;" Ill' - I. _., ... N. 1 - 1.2.

(4.2)

Wc shall use the representations (4, I) and the finite-elcment formulation 13,)) to obtain asystem of cquations for the nodal values lit (which depend upon f). For definiteness. Ict usassume that the logarithmic barricr functional is employed. Our first step is to introduce anumerical quadrature rule 1(') to integrate approximately the penalty tcrm:

Here 1(') is a quadrature rule of the type.

E .\f

l(/) = L !,(j): !,(/l= L ll'/f(t/l· I E CO(fcl,=1 j=1

(4.3)

(4.4)

where E is the number of elements modeling rc. 11'/ are quadrature weights. and tf arequadrature points in element e. Typically. wc employ the trapezoidal rule or Simpson's rule forthe examples described below.

With these conventions. our discrete model assumes the form

(4.5)

Hence. introducing (4.1) into (4.5) yields the nonlinear systcm of equations.

where 1/ arc arbitrary and

K~/l = a(4)aij. I/>flij)

Qj(lla)=~ W nj<tm).pB(tm)fl' m~1 m £(tm)-lltni(tm)4>/I<tm)

Ii = 1(4)flij); i. j = I.2: a. (3 = 1. 2 ... , . N.

(4.6)

(4.7)

Here i; are orthonormal basis vectors in R~. Wm are (global) quadrature weights. and tm are(global) quadrature points. N, being the total number of quadrature points on r/. By a simplerenumbering of the unknowns 11;". we can rewrite the nonlinear system (4.6) in tcrms of 2Nunknown nodal displaccment components. "l' "> .... Il~N:

(4.8)

There p. l/ = 1.2 .. , , .2N. Kpq is the usual stiffness matrix. Qp(u) and f,. arc the result ofreordering unknowns in the penalty term Qflj(Ili') and the load term f/ and u ={Ill. "2 •...• 1l2Nl

T is the vector of unknown nodal displacements.Our algorithm for solving this system is given in the following stcps:(I) Choose a trial solution UI which satisfies the constraint conditions.\' - U . n :5 () at nodal

points on rei,.(2) If R denotes the rcsidual.

p = 1. 2 ..... N

52 J. T. ODEN and S, 1. KIM

then we use standard Newton-Raphson iteration as described by the recurrance formula.

(4.9)

(3) Next. we check to sec if the solutions 1I~+1 satisfy the constraint condition at each contaclnodal point. If solutions violate the constraint. the process is reinitiatcd with a scaled-downinitial starting vcctor u;,

(4) Steps 2 and 3 are repeated until the relative errOrs

and (4.10)

are less than a preassigned tolerance frJ and frC' respectively, wherc tlu,q = u~" - u,q. and LN isthe set of nodes on rch. The case of the inverse barrier functional can be handled in preciselythe samc manner.

4.2 A Numerical exampleTo test the effectiveness of our methods. we now consider a represcntative numcrical

example,As an cxample. we consider the problem of indentation of a rectangular body by a rigid

cylindrical punch. The diameter of punch is 8 units and the dimension of clastic body is 4 unitsby 16 units. as shown in Fig. I. The problem is taken 10 be one of planc strain and we chooseYoung's modulus E = 1000 and Poisson ratio I' = OJ.

Taking advantagc of the symmetry of the problem. thc left half of the rcctangular elasticbody is discrctizcd by 18 nine-node quadrilateral isoparamelric clcments. as indicatcd in Fig. 2.

R = 8

Elastic Haterial

Young's modulus

Poisson's ratio

E =:1000

\)=10.3

16

Fig, I. Rigid punch problem.

4

Interior penalty mcthods for finite element approximations

RIGID PUNCH R-8

~

• • • • • •

• • • • • •

• • • • • •

Fig. 2. Rigid punch problem 3 x ~ 9·node elemenls (4 x 8),

53

We set the penalty parameter f = 1O-~. 10-3, 10-4 and 10-5 for the logarithmic barrier functionaland f = 10-3• 10-4• 10-7 and 10-8 for the inverse barrier functional and utilize the algorithmoutlined in the previous section. The computed results are compared with an analysis of thesame problcm by an exterior penalty method used by Song [16] and Kikuchi and Oden [8]. Aprcscribcd centerline deflection of l) = 0.6 was chosen,

In our computations. iterations were stopped when the relative errors eRI and eR~ in (4,10)betwccn r - Ith and rth iteration rcachcd 10-5.

From Tablc I and Fig. 3. we can see that the distribution of contaet pressure computedusing either barricr functional is almost the same and the computed deformed configuration isin excellent agreement with that obtained using an exterior penalty algorithm. Notice that fora choice of f = 10-5 in thc case of the logarithmic barrier functional and f = Ilr8 in thc case ofthc invcrse barricr functional. we obtained slightly different solutions. This is due to the factthat for the logarithmic penalty more iterations are required to reach the oplimum solution sincethe presence of the penally term fQ~ scverely distorts the form of the objcctive functional F

Table I. Contact pressure distribution (9 node·elcments)

~;ode J !:ode 81 :;.,de 11110de 221

Error Limit

-I10-2 1204.8 10-5193.9 1:6.9 111.2

I10- 3 203. :. 193.5 1:6.9 111.2 10-5

In'.rlor 110

-4203.4 193.5 116.9 111.2 10-5

(Logarithmic) -5221.0 255.2 116.0 131..6 I 10-610

2 itt:'rations

10-51208.81195.71119.5 I 112.5 I 10-7

9 iterations-

10- 3 212,4 195,8 180,1 113,9 10-5

10-4 204.8 193,5 117,3 111.2 10-5

Interior110-7 203.4 10-5193.5 176,9 110,7

( Inverse) -8 10-510 1216,0 207.9 178.6 118.82 iteratiollH

10-81 208,4 1195.8 I 179,1 I 1I2.5 I 10-6

8 iterations~Exterior 110-31209.3 1196,2 I 179.1 I 112.5 I 10-3

54 J. T. ODEN and S, 1. KIM

0:

41~VlVll.:.I:J::ll.

I ~0

u<I ~0

z... 8I

! •

0.0' I 8, .i "D.5r I ~

LO:- ~~

I

~a~<

I ~

o I ~100 l-

I

200 ,-

Fig. J(al. Contact pressure distribution at contact node .

• : In~e'rse border funr:t;on~l

0.0 ;

0' ~

1.0 '

IIII

io .

100

i200 ~

II

II)

I~0Zt-,u

I <~Z0

~

u~<

a1 t-

<X

'"I 0"-~

I ILl

'";:)II)III

J 4l

'"'"t-U, <......,Z• 0U

Fig, J(b), Contact pressure distribution at contact node.

I

IIO.ot

0.5 I,l.°rI

J'00 ~

200 '-

I

Interior penalty methods for finite element approximations

til, '", 0

• t • I ~. . .... ~~ 5u.....<

I z

IioI ~

I '"Ck::Jtiltil

1 '"Ck:"-...$u<

1...$

$ z0u

Fig. 3(c), Contact pressure distribution at contact node,

This effect is cured by assigning smaller relative error limits and. concomitantly. allowing mOreitcrations. As shown in Table I. contact pressures in the cascs of f = I()~ and 10-8 wilh 1{ or 9iterations are in good agrcemcnt with the results of the exterior penalty method.

One computational ditliculty thaI remains with our interior penalty method is the absencc ofa systematic method for choosing good slarting values for the itcrativc process. If we pick thetrial points too close to the boundary re. many iterations may be required for convergcnce.whereas an initial displacement field choscn too far from this boundary may be outside theradius of convergcnce of Ihe scheme and the penalty method may fail. In the calculationsdescribed here. the appropriate trial points are chosen by checking the relativc magnitude ofpcnalty term to objectivc function; roughly speaking. if this term is too large. wc readjust thestarting vector to reduce the relative size of the penalty functional.

Acknowledgemellt- This work was complete during a project supported by the Air Forcc Office of Scientific Researchunder Contract F-49620·78-C-0083.

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approximations of contact problems in incompressible linear elalticity. TICOM Report 80-2. The University of Texasat Austin (1980).

4. A. V, Fiacco and A, P. Jones. Generalized penalt~; methods in topological spaces. SIAM 1. Appl, Marlt 17(5). (Sept.1969),

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56 1. T. ODEN and S. J. KIM

10. R. A. Adams. Sobolev Spaces, Academic Press. New York (1975)II. I. Ekeland and R. Teman. Variational Problems in Convex Analysis. North Holland. Amsterdam (1977).12. M. M. Vainberg. Variational Method of Monotont Operators in the TheaI')' of Nonlinear Equations, Halsted Press.

New York (1973).13. H. Brezis. Problemes uniiattraltX. 1. Math, Purl's Appl. 51. t-168 (1972).14, I. Babuska, The finite element method with Lagrange multipliers. NUI/ler. Math .. 20. (1973).15, F. Brezzi. "On the Existence. Uniqueness and Approximation of Saddle· Point Problems Arising from Lagrange

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