A posteriori error estimation for unilateral contact with matching and non-matching meshes

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1 Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 A posteriori error estimation for unilateral contact wit matcing and non-matcing meses Patrice Coorevits a, *, Patrick Hild b, Jean-Pierre Pelle a a Laboratoire de Mecanique et Tecnologie, ENS de Cacan/CNRS/UPMC 61 avenue du President Wilson, Cacan Cedex, France b Laboratoire de Matematiques, Universite de Savoie, Domaine Scienti que, Le Bourget du Lac Cedex, France Received 18 December 1998 Abstract In tis paper, we consider te unilateral contact problem between elastic bodies. We propose an error estimator based on te concept of error in te constitutive relation in order to evaluate te nite element approximation involving matcing and non-matcing meses on te contact zone. Te determination of te a posteriori error estimate is linked to te building of kinematically-admissible stress elds and statically-admissible stress elds. We ten propose a nite element metod for approximating te unilateral contact problem taking into account matcing and non-matcing meses on te contact zone; ten, we describe te construction of admissible elds. Lastly, we present optimized computations by using bot te error estimates and a convenient mes adaptivity procedure. Ó 2000 Elsevier Science S.A. All rigts reserved. 1. Introduction Te numerical simulation of contact problems is more often carried out by nite element metods. For te user, one important aspect is obviously to evaluate te discretization errors due to te use of tis type of approximation. From te point of view of matematics, a unilateral contact problem corresponds to a variational inequality [1±3]. Te convergence of te associated nite element metods as been studied by numerous autors on te basis of a priori error estimators. In particular, te case of two elastic bodies as been developed in [4] for matcing meses and in [5±8] for non-matcing meses. However, tese a priori error estimations do not allow us to quantify te discretization errors. Tis quanti cation requires te de nition of a posteriori error estimations. For linear problems, various researc e orts ave been performed: estimators based on te residual of te equilibrium equations [9], estimators using te smooting of nite element stresses [10] and estimators based on te concept of error in te constitutive relation [11,12]. For non-linear problems, and especially for te non-linearity of contact, te work available is muc less abundant. We can owever cite Ref. [13] wic, based on a penalty metod, transforms te variational inequality into a variational equality tat allows, witin te classical framework, building an estimator based on te residuals. Neverteless, tis estimator explicitly uses te penalty parameter, wic does represent a major drawback. Te aim of tis paper is to propose, for unilateral contact problems witout friction between elastic bodies under small perturbations, an error measure based on te concept of error in te constitutive relation. Tis error measure can be used for te classical numerical tecniques for local treatment of te non-interpenetration condition, yet is particularly well-adapted to te global treatment of te non-interpenetration condition proposed in [8]. * Corresponding autor /00/$ - see front matter Ó 2000 Elsevier Science S.A. All rigts reserved. PII: S ( 9 9 ) X

2 66 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 Te basis of te metod is presented in Section 2. In particular, te development of an error measure in te constitutive relation relies on a classi cation of te kinematic conditions, equilibrium equations and constitutive relations. In te case of contact, in order to establis tis classi cation, we consider as in [14] te contact zone as a mecanical entity wit its own variables and its own constitutive relations. Te building of te error measure is te purpose of Section 3. Te quality of an approximate admissible solution wic, by de nition, satis es te kinematic conditions and equilibrium equations is evaluated by te manner in wic te constitutive relations are satis ed. A link between te error measure in te constitutive relation and te classical errors in te solution is establised. For tese types of problems, suc an approac can be considered as an extension of Prager±Synge's teorem [15] in elasticity. Te discretization of te problem by nite elements and te application of te proposed error measure are developed in Sections 4 and 5. In particular, te tecnique wic allows building an approximate admissible solution from te nite element solution is detailed. In Section 6 examples of te use of te error estimator as well as examples of adaptive computations are sown. 2. Problem set-up 2.1. General notations We consider te bidimensional unilateral contact problem between two elastic bodies, denoted by X 1 and X 2 (Fig. 1), respectively. We assume tat te boundary ox` of X`, ` ˆ 1; 2 is divided into tree parts: On te rst part, denoted by o 1 X`, we suppose tat te displacement eld is given: U `o 1 X` j ˆ U ` d ; ` ˆ 1; 2: 1 For te sake of simplicity, we suppose in te following tat: U `d ˆ 0; ` ˆ 1; 2: On te second part, denoted by o 2 X`, a surfacic density of forces F `d is given. Te complementary part, denoted by o C X` ˆ ox` o 1 X` [ o 2 X`, is te candidate contact zone. We suppose tat o C X 1 ˆ o C X 2, wic we denote by. Te body X` is submitted to a density of volumic forces f `d. We assume tat te strain tensor e is linearized and we denote K` by te elasticity operator associated wit X`. Te notation n` stands for te unit outward normal on te boundary of X` Formulation of te contact In order to clearly express te error in te constitutive relation, we represent, as in [14], te contact zone ˆ o C X 1 ˆ o C X 2 as a mecanical entity equipped wit its constitutive relation. We coose te orientation of by setting n C ˆ n 1. We ten introduce on te interface te functions W 1 ; W 2 ; R 1 ; R 2 and R C, Fig. 1. Te unilateral contact problem between two elastic bodies.

3 representing two displacement elds W 1 and W 2 (on eac side of te interface), two elds of surfacic densities of forces R 1 and R 2 (stresses transmitted to X 1 and X 2 ) and an ``interior eld'' of a surfacic density of forces R C. Te equilibrium of te interface is represented by: R C ˆ R 1 and R C ˆ R 2 on : Let us de ne te jump in te displacement wic, for te interface, plays a similar role as a strain: W C ˆ W 1 W 2 : For all of vector, set: P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 67 n ˆ T n C and t ˆ n n C ; were te notation T represents transposition. Coulomb's constitutive law, in te frictionless case, can be formulated as follows [1]: W C n 6 0; R C n 6 0; R C n W C n ˆ 0; R C t ˆ 0: 8 Te inequality in (5) expresses te non_interpenetration of te two bodies; eiter contact or separation is allowed. Te inequality in (6) states te sign condition on te normal constraint and (7) represents te complementary condition. Lastly, (8) states te nullity of te tangential component of te stress vector, wic re ects te absence of friction. Let us now introduce te conjugate convex potentials u and u [16]: u V ˆ 0 if V n P 0 and u ˆ 0 if n 6 0 and t ˆ 0; 9 1 oterwise 1 oterwise: ten, we ave: u V u T V P 0 8V ; 8. Te constitutive relation can be written in te tree following equivalent forms: W C 2 ou R C ; R C 2 ou W C ; u W C u R C R CT W C ˆ 0: Problem set-up Te problem of unilateral contact witout friction can be formulated as follows: Find (U 1 ; r 1 ) de ned in X 1,(U 2 ; r 2 ), de ned in X 2 and [(W 1 ; W 2 ), (R 1 ; R 2 ; R C )], de ned on suc tat: U `; W ` ; ` ˆ 1; 2 satisfy te kinematic conditions: U ` ˆ 0ono 1 X` and U ` ˆ W ` on : 11 r`; R`; R C ; ` ˆ 1; 2 satisfy te equilibrium equations: Tr r`e V ` dx f `T d V ` dx F `T d V ` ds R`T V ` ds ˆ 0 8V ` 2 U ` 0 X` X` o 2 X` ˆ U ` defined and enoug regular on X` suc tat U ` ˆ 0ono 1 X` 12 and: R C ˆ R 1 and R C ˆ R 2 on.

4 68 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 U `; W `; r`; R C ; ` ˆ 1; 2 satisfy te constitutive relations: r` ˆ K`e U ` ; u W C u R C R CT W C ˆ 0: Error in te constitutive relation Te concept of error in te constitutive relation is based on te classi cation of te equations in kinematic, equilibrium and constitutive relations Admissible elds De nition. A pair ^s ˆ ^u; ^c, ^u ˆ ^U 1 ; ^U 2 ; ^W 1 ; ^W 2 ), ^c ˆ ^r 1 ; ^r 2 ; ^R 1 ; ^R 2 ; ^R C is said to be admissible if te kinematic relations (11) and te equilibrium equations (12) are satis ed Measure of te error in te constitutive relation For all admissible ^s, let's set: e ^s ˆ " X2 ^r` K`e ^U ` ds# 1=2 2 2 u ^W r;x` u ^R C ^R CT ^W C ; 14 C were krk 2 ˆ RX` Tr r;x` rk 1 ršdx. By de nition, te quantity e ^s is te measure of te error in te constitutive law corresponding to te admissible pair ^s. Property. For ^s ˆ ^u; ^c admissible, we ave: e ^s equal to zero if and only if ^s is te exact solution to te reference problem. Wit te error in te constitutive law, we associate te relative error denoted e and de ned as follows: 2 P 2 ^r` K`e ^U ` 2 2 R 3 C r;x` C u ^W C u ^R C ^R CT ^W C ds e ˆ 6 4 P 2 k^r` K`e ^U ` k r;x` 1=2 : 15 Terefore, e is a global error wic allows evaluating te quality of te admissible pair ^s. Let E be a part of X`. Ten, we de ne te local contribution e E of E to te error (15) as follows: 2 ^r` K`e ^U ` 6 e E ˆ 4 2 r;e 2 R u ^W C u ^R C ^R CT ^W C ds \E P 2 k^r` K`e ^U ` k 2 r;x` =2 : 16 were krk 2 r;e ˆ RE Tr rk 1 ršdx. In practical situations, E is an element of te mes's discretization. Te local contributions enable obtaining information concerning te errors located on te structure. By construction, one as: e 2 ˆ X e 2 E : 17 E

5 Wen ˆ;, te de nitions (14), (15) and (16) are already known and correspond to te case of elasticity [12] Relation wit te oter errors P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 69 Proposition. Let U 1 ; U 2 ; r 1 ; r 2 ; W 1 ; W 2 ; R 1 ; R 2 ; R C ) be a solution to te contact problem (11)±(13). For all ^s ˆ ^U 1 ; ^U 2 ; ^r 1 ; ^r 2 ; ^W 1 ; ^W 2 ; ^R 1 ; ^R 2 ; ^R C admissible, one as: e 2 ^s X2 U ` ^U ` 2 u;x` r` ^r` 2 r;x` P 0; 18 were kuk 2 ˆ u;x` RX` Tr e U Ke U ŠdX. Terefore v ux 2 t U ` ^U ` 6 e ^s and 2 u;x` v ux 2 t r` ^r` 6 e ^s : 19 2 r;x` Tis property is an extension of Prager±Synge's teorem in elasticity to te more general unilateral contact case. Proof. For ` ˆ 1; 2, one as: ^r` K`e ^U ` 2 ˆ ^r` r` K`e U ` ^U ` r;x` ˆ ^r` r` 2 r;x` U ` 2 r;x` ^U ` 2 2 u;x` X` Tr ^r` r` e U ` ^U ` i dx: 20 Since r` and ^r` satisfy (12) and U ` and ^U ` satisfy (11), we can write: ^r` K`e ^U ` 2 ˆ ^r` r;x` r` 2 r;x` U ` ^U ` 2 2 ^R` R` u;x` C T W ` ^W ` ds: 21 C Ten X 2 ^r` K`e ^U ` 2 ˆ X2 r;x` ^r` r` 2 X2 U ` ^U ` 2 2 ^R 1 R 1 T W 1 ^W 1 r;x` u;x` i ^R 2 R 2 T W 2 ^W 2 ds: 22 By using (12) and R C t e 2 ^s ˆX2 ^r` ˆ 0, we deduce: r` 2 X2 r;x` U ` ^U ` 2 2 u;x` ^R CT W C ds 2 R C ^W C n n ds: 23 De nition (14) is only valid wen ^s is suc tat u ^W C and u ^R C are nite. Oterwise, te error e ^s is equal to in nity and relation (18) is obviously satis ed. If u ^W C ˆ0 and u ^R C ˆ0, property (18) is establised by observing tat: ^R CT W C ds Tis concludes te proof. R CT ^W C ds ˆ ^R C n W C n ds R C ^W C n n ds P 0: 24

6 70 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 Wen initially non-contact points can come into contact after deformation, we must take into account te initial gap between te bodies. Te unilateral contact conditions (5)±(8) on can ten be written: W C n 6 G n; 25 R C n 6 0; 26 R C n W C n G n ˆ0; 27 R C t ˆ 0; 28 were G n is te non-negative function expressing te distance between te two bodies. It tus becomes possible to de ne te error in te constitutive relation in te case of an initial gap. e ^s ˆ " X2 were G ˆ G n n C. ^r` K`e ^U ` 2 r;x` 1=2; 2 u ^W C G u ^R C ^R CT ^W C G dsi Te continuous and discrete variational formulations for te unilateral contact problem 4.1. Te continuous case: te variational inequality and te mixed formulation Let U 0 ˆ U 1 0 U2 0, were U1 0 and U2 0 ave been introduced in (12). We set, for all U ˆ U 1 ; U 2 and V ˆ V 1 ; V 2 in U 0 a U; V ˆX2 X` Tr e U ` K`e V ` dx: 30 a(.,.) is te bilinear symmetrical form in elasticity. We also set, for all V in U 0 L V ˆX2 X` f `T d V ` dx o 2 X` F `T d V ` ds : 31 Te linear form L : takes into account te external loads f `d and F `d. We ten de ne te convex of admissible displacements denoted U ad comprising te non-interpenetration condition between te bodies n U ad ˆ V ˆ V 1 ; V 2 2U 0 ; V 1T n 1 V 2T n 2 6 0on o: 32 Te variational formulation associated wit problem (11)±(13) is ten [1,4,3]: ndu suc tat: U 2 U ad ; a U; V U P L V U 8V 2 U ad : 33 Tis problem is well-posed and admits a unique solution in te case were o 1 X 1 and o 1 X 2 are positive. If not, su cient conditions tat state te existence and uniqueness for well-oriented loads [4] are also available. Next, we introduce te mixed variational formulation of te unilateral contact problem wic consists of nding U; k 2U 0 N tat satis es:

7 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 71 a U; V k V 1T n 1 V 2T n ds 2 ˆ L V 8V 2 U 0 ; C C l k U 1T n 1 U 2T n ds 2 P 0 8l 2 N ; 34 were N is te convex cone of negative functions de ned on in some dual sense (for a detailed study, see [4]). Wen o 1 X 1 and o 1 X 2 are positive, it is clear tat problem (34) as a unique solution U; k were U is te solution to (33) and k ˆ R C n (7) Finite element discretization for matcing and non-matcing meses We now consider te general case of non-matcing meses on te contact zone. A detailed study of contact problems wit non-matcing meses on te contact zone can be found in [8]. In te present case, bot polygonally-saped bodies X 1 and X 2 are discretized independently, tereby leading to nodes wic do not coincide on te contact zone. We will denote te nite element translation of te space U` 0 by U` 0; and write U 0; ˆ U 1 0; U2 0;. Te 0;, bot of wose components are continuous on X` and polynomial of degree one on eac functions V ` 2 U` triangle, satisfy te embedding conditions on o 1 X`. Let us denote te discretization parameter associated wit X` by ` and set ˆ ( 1 ; 2 ) (see Fig. 2). Next, we introduce te approximation convex cone, denoted by U ad; and de ned as follows: U ad; ˆ V ˆ V 1 ; V 2 2U 0;; V 1T n 1 V 2T n v 2 ds P 0 8v 2 N 1 ; 35 were N 1() is te closed convex cone of non-positive continuous functions, piecewise linear on te mes of X 1 on. Te discretized mixed variational formulation ten becomes: nd U 2 U 0; and k 2 N 1 ) suc tat: a U ; V k C C l k U 1T n1 V 1T n 1 V 2T n2 ds ˆ L V 8V 2 U 0; ; U 2T n2 ds P 0 8l 2 N 1 : It is simple to sow tat problem (36) admits a unique solution, denoted by (U, k ). It is also simply sown tat U belongs to te convex de ned in (35) and tat te pair (U, k ) tends towards (U, R C n ) if te discretization parameter ˆ ( 1 ; 2 ) tends towards 0. Tese convergence results and te a priori error estimates can be found in [5,8]. 36 Fig. 2. Non-matcing meses.

8 72 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 Remark. Te mixed problem (36) is equivalent to finding te saddle point on U 0; N 1 ) associated wit te following Lagragian: L V ; l ˆ1 2 a V ; V V 1T n1 V 2T ds n2 L V : 37 l In te general case of nonmatcing meses, we coose te mes of X 1 wen defining te Lagrangian multipliers in N 1. In te case of matcing meses, tere is obviously no coice in defining te set of te multipliers Te matrix formulation In order to provide te matrix formulation of te previous mixed problem, we old xed. Tus, we consider a discretization comprising N` nodes belonging to X`, ` ˆ 1; 2; let N ˆ N 1 N 2. For te sake of simplicity, we assume te absence of embedding conditions. Let m be te number of nodes on belonging to te mes of X 1, and let numbers 1 troug m correspond to tese nodes. We denote te number of nodes on belonging to te mes of X 2 by n, and let te numbers between N 1 1 and N 1 1 n correspond to tese nodes. Let w k ; 1 6 k 6 m and u k ; 1 6 k 6 n be te scalar basis functions on te mes of X 1 and X 2, respectively, on. Te matrix formulation of Eq. (36) ten becomes: 0! M 1 1 K 1 0 q 1 0 B C 0 K 2 q C 2;1 A K ˆ F 1 38 F 2 or, wit obvious notations: 0 Kq AK ˆ F in R N ; 39 were q 1 is te vector of components U 1 i ; 1 6 i 6 N 1; q 2 te vector of components U 2 i ; N i 6 N; K te vector of components k i ; 1 6 i 6 m; K 1 (resp. K 2 ) te N 1 -by-n 1 (resp. N 2 -by-n 2 ) sti ness matrix corresponding to X 1 (resp. X 2 ); M 1 te m-by-m matrix of coef cients m 1 j;k ˆ R w k w j ds; 1 6 j; k 6 m; C 2;1 te n-by-m rectangular matrix of coef cients c j;k ˆ R w k u j ds; N j 6 N 1 n 1; 1 6 k 6 m, and F 1 and F 2 te vectors representing te external loads. Te inequality in (36) yields te remaining conditions. Denoting te vector corresponding to te normal displacements of te nodes of X` on by qǹ, we deduce te matrix formulation corresponding to (36) as follows: Find q 2 R N and K 2 R m satisfying: Kq AK ˆ F in R N ; M 1 q 1 N t C 2;1 q 2 N 6 0 in Rm ; K 6 0 in R m ; M 1 q 1 N t C 2;1 q 2 N T K ˆ 0 in R: 40 In te current case, te integral condition in (35) can be written: M 1 q 1 N t C 2;1 q 2 N 6 0. Let us note tat M1 represents te symmetrical mass matrix on te mes of X 1 on. We denote te coupling matrix between te two meses on te contact zone as C 2;1. Remark. 1. Wen te nodes of bot bodies fit togeter on te contact zone, M 1 ˆ C 2;1 and te previous condition becomes M 1 q 1 N q2 N 6 0; tis does not represent te node-on-node condition wic is q1 N q2 N 6 0. Te condition M 1 q 1 N q2 N 6 0 allows some sligt interpenetration of te bodies. 2. Te multiplier K expressing te contact pressure on te contact zone satisfies te non-positiveness condition on.

9 Let V ˆ V 1; V 2 2U 0; and l 2 N 1(). Denoting te vector of components V 1 i ; 1 6 i 6 N 1 by ~q 1, te vector of components V 2 i ; N i 6 N by ~q 2 and te vector of components l i ; 1 6 i 6 m by ~K and in applying te Remark from Section 4.2, it becomes straigtforward tat te matrix formulation in (40) can also be written as: max ~K min ~q 2 ~qt K~q! T ~q T F AT~q K; ~ were ~q T ˆ ~q1 ~q 2 : 41 Using te property Kq AK ˆ F (see (39)) implies tat (41) becomes a minimization problem of a quadratic function wit convex constraints: 1 K 2 ~ T A T K 1 A K ~ K ~ T A T K 1 F 1 2 F T K 1 F : 42 min ~K 6 0 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 73 Te problem in (42) is a classical minimization problem and is solved by using an iterative Frank and Wolfe algoritm. Wit te values of K now available, we can obtain q by simply computing K 1 F AK. 5. Construction of admissible elds Te aim of tis section is to describe te construction of te admissible elds by applying te properties of te nite element solution [12] Building of te kinematically-admissible displacement eld If bot domains are discretized wit matcing or non-matcing meses on te contact zone, te only di culty lies in building te displacement elds ^U 1 and ^U 2 wic satisfy te non-interpenetration condition in te case were te integral conditions in (35) ave been adopted. Te eld U ˆ (U 1; U 2 ) given by te algoritm does not satisfy te non-interpenetration condition (see Remark 1, Section 4.3). We tus set: ^U 1T n 1 ˆ ^W 1T n 1 ˆ U 1T n1 E 2 max U 1T E 1 E n1 U 2T ; n2 0 ; 2 ^U 2T n 2 ˆ ^W 2T n 2 ˆ U 2T n2 E 43 1 max U 1T E 1 E n1 U 2T ; n2 0 ; 2 were E 1 (resp. E 2 ) denotes te Young's modulus of X 1 (resp. X 2 ). Wit respect to te tangential displacements, we can write: ^U 1 t ˆ ^W 1 t ˆ U 1 t and ^U 2 t ˆ ^W 2 t ˆ U 2 t : 44 For te nodes i wic are not located on, we set: ^U 1 i ˆU 1 i and ^U 2 i ˆU 2 i 45 and ^U 1 (resp. ^U 2 ) is built on X 1 (resp. X 2 ) in a piecewise linear fasion on eac element by using te previous values Building of te statically-admissible stress eld Te algoritm used to solve problem (42) yields a normal contact pressure K wic is te vector introduced in (39) corresponding to te nodal values of k (36). We recall tat k is continuous, non-positive and piecewise linear on te mes of X 1 on. We set ^R 1 ˆ k n 1 ; ^R 2 ˆ k n 2 and ^R C ˆ ^R 1 so tat te equilibrium of te interface (2) is satis ed. One ten obtain: ^R C n ˆ ^R CT n C ˆ k 6 0 and ^R C t ˆ ^R C ^R C n nc ˆ 0; conditions (6) and (8) are tus ful lled. Te rst equation in (36) becomes:

10 74 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 Tr e U 1 K1 e V 1 X1 dx ˆ f 1T X 1 X2 dx ˆ f 2T X 2 Tr e U 2 K2 e V 2 d V 1 d V 2 dx F 1T d V 1 o 2 X 1 dx F 2T d V 2 o 2 X 2 ds ds ^R CT V 1 1 ds 8V 2 U 1 0; ; 46 ^R CT V 2 2 ds 8V 2 U 2 0; : Denoting r 1 ˆ K1 e U 1, it becomes obvious tat U 1; r1 is te nite element solution to a linear elasticity problem on X 1 witout unilateral contact conditions and wit given loads fd 1, F d 1 and ^R C. Supposing tat te volume loads fd 1 are constant (on eac triangle) and tat te surface loads F d 1 are piecewise linear on te mes of o 2 X 1, te construction of te statically-admissible stress eld ^r 1 on X 1 is ten classical [12]. Denoting r 2 ˆ K2 e U 2, it is also clear tat U 2; r2 is te nite element solution to a linear elasticity problem on X 2 wit given loads fd 2, F d 2 and ^R C. In te case of matcing meses on te contact zone, te construction of te statically-admissible stress eld ^r 2 on X 2 is also classical. Wen considering te construction of te statically-admissible stress eld ^r 2 in te general case of nonmatcing meses, te situation is more complicated tan for te construction of ^r 1. As a matter of fact, ^R C n is piecewise linear on on te mes of X 1 but, in te general case, not on te mes of X 2. Remark. 1. If te mes of X 2 on is strictly finer tan te mes of X 1 on, ten te construction is also classical. 2. If te mes of X 1 is strictly finer tan te mes of X 2 on, it is possible to provide a symmetrical definition of te closed convex cone N 1 () of te Lagrangian multipliers in (35) and ten to apply multipliers, wic are non-positive and piecewise linear on te mes of X 2 on. If we wis to reconstruct te admissible eld ^r 2 in te most general case of non-matcing meses, ten we must take into account tat ^R C n is only continuous and piecewise linear on eac edge of E2, a boundary element of X 2 (see Fig. 3). Te teoretical construction of ^r 2 by using te classical metod ten becomes possible by dividing te triangles into subtriangles. In practical cases, tis is only feasible wen te number of subtriangles is small and te triangles are not very at (Fig. 4 represents te kinds of subtriangles tat lead to di culties) Practical constructions We ave observed in te previous sections tat te teoretical construction of an admissible ^s ˆ ^U 1 ; ^U 2 ; ^r 1 ; ^r 2 ; ^W 1 ; ^W 2 ; ^R 1 ; ^R 2 ; ^R C ) satisfying u ^R C ˆ0andu ^W 2 ^W 1 ˆ0 is always possible by applying te appropriate properties of te nite element solution (36). In order to ``simplify'' te numerical implementation, we now propose two reasonable alternatives wic will be tested and ten compared to te analytical construction. Te latter is considered rst. Fig. 3. Te given contact pressure on te triangle E 2.

11 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 75 Fig. 4. Flat subtriangles Analytical construction of a strictly-admissible stress eld Tis building process is classically carried out in two steps: during te rst step, densities of forces ^F are constructed on te edge of eac element; tese densities are in equilibrium wit te body forces f `d, and during te second step, te strictly statically-admissible eld ^r` is constructed element-by-element, using te densities as boundary conditions. Tis construction is easy to accomplis wen tere are matcing meses on te contact zone or if one mes is a submes of te oter. Oterwise, te teoretical construction is practically possible wen te subdivision of te triangles does not lead to overly- at triangles. Te oter case (leading to practical di culties) appears wen some nodes of X 1 and X 2 are very close in comparison wit te discretization parameters 1 and 2 ; in tis case, very at triangles cannot be avoided. Next, we propose two alternatives for treating te latter case Numerical construction by use of iger-degree polynomials Te cosen tecnique consists of searcing te stress eld ^r` on te element E of X` in te following form [17]: ^r` je ˆ K`e V E ; 47 were V E is a displacement eld de ned on E, a polynomial of degree p k wit p being te degree of te element used to carry out te nite element analysis and k being a strictly positive integer. More precisely, V E is te solution to te following nite element problem on a single element E: nd V E,a polynomial of degree p k on E, suc tat: 8V displacement field of degree p k on E; Tr K`e V E e V de ˆ f ` d V de ^F joe V ds: E E oe 48 Tis problem can be solved, apart from one displacement eld of a solid, on account of te equilibrium of te densities ^F wit te body forces f `d. On eac element E of te mes, a small linear system must be solved: K p k U ˆ F p k ; 49 were K p k denotes te classical rigidity matrix constructed wit te interpolation polynomials u p k i of degree p k and F p k denotes te vector of generalized forces. Tus, we obtain an approximate eld wic is no longer strictly admissible, yet wic leads to e cient error estimators as sown in [17].

12 76 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65± Approximation of te construction by applying te speci c properties of te nite element solution in (36) Tis tecnique consists of building ^s ˆ ( ^U 1 ; ^U 2 ; ^r 1 ; ^r 2 ; ^W 1 ; ^W 2 ; ^R 1 ; ^R 2 ; ^R C ) tat satis es (11), u ^R C ˆ0, u ^W 2 ^W 1 ˆ0, (12) wit a weak interpretation of condition (2) wic will be speci ed later. Te construction of ^U 1 ; ^U 2 ; ^W 1 ; ^W 2 represents te teoretical construction proposed in (43)±(45). Next, we set ^R 1 ˆ k n 1 (wit k corresponding to te solution of (42)), ^R C ˆ ^R 1 and ^R 2 ˆ k 2 n2, were k 2 is te continuous, piecewise linear function on te mes of X 2 on satisfying: k k 2 v 2 ds ˆ 0 for all v 2 wic is continuous and piecewise linear on te mes of X2 on. Te rst equation of (36) ten becomes: Tr e U 1 K1 e V 1 X1 dx ˆ f 1T d V 1 dx F 1T d V 1 ds X 1 o 2 X 1 C C Tr e U 2 K2 e V 2 X2 dx ˆ f 2T d V 2 dx F 2T d V 2 ds X 2 o 2 X 2 ^R CT V 1 1 ds 8V 2 U 1 0; ; ^R 2T V 2 2 ds 8V 2 U 2 0; : Te last equality results from te construction of ^R 2, wose components are piecewise linear on te mes of X 2 on. Te construction of ^r 1 and ^r 2 is ten classical. Tis tecnique consists of approximating te teoretical construction wit elds ^R 1 ; ^R 2 and ^R C, wic satisfy, albeit weakly (see (50)), Eq. (2). 6. Numerical studies 6.1. Example In te rst example, we consider te problem of two elastic bodies initially in contact. Te upper body is submitted to a uniform load (see Fig. 5). In tis problem, we ave adopted symmetry conditions in order to avoid a greater number of singularities. Te material caracteristics are: E ˆ 200 GPa, t ˆ 0.3. Due to te lack of an analytical solution for suc a problem, we use a reference solution, denoted U ref, wic is a nite element solution associated wit a very re ned mes. Ten, te exact error e ex can be de ned as follows: e ex ˆ " P # 2 ku ref U k 2 1=2 u;x` P : 2 ku ref U k 2 u;x` 52 Fig. 5. Problem set-up.

13 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 77 Fig. 6. Initial matcing mes and map of contributions e E to te error. Fig. 7. Convergence of te error estimators as a function of te number of elements on te contact zone wit matcing meses. By subcutting te initial mes (see Fig. 6), te trend in te errors can be studied. In Fig. 7, te convergence rates of te exact error e ex and te computed error e de ned in (15) are compared as a function of te number of elements on te contact zone. We compare te two errors e estimated wit strictly-admissible stress elds (see Section 5.3.1) and e p 3 wit numerical constructions (see Section 5.3.2). We can observe tat te convergence rate of te exact error is te same as te convergence rate of te errors in te constitutive relation in te case of matcing meses. Moreover, te lower bound and te upper bound of te e ectivity index c ˆ e=e ex are 1.28 and 1.40, respectively (see Fig. 8), wereas for numerical constructions, te bounds of c p 3 ˆ e p 3 =e ex are 1.26 and 1.36, respectively. In Fig. 10, te convergence rates are compared in te case of non-matcing meses (see Fig. 9). Te mes of X 2 is a submes of X 1 on te contact zone tat allows us to build strictly-admissible elds. Moreover, te lower and te upper bound of te e ectivity index c are 1.17 and 1.47, respectively (see Fig. 11), wereas te bounds of c p 3 are 1.16 and 1.42, respectively Mes adaptivity Algoritm Te goal of a mes adaptation procedure is to guarantee to te nite element user a certain level of precision by minimizing te computation costs. We will use te -generation, wic is te most

14 78 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 Fig. 8. E ectivity of te error estimators as a function of te number of elements on te contact zone wit matcing meses. Fig. 9. Initial non-matcing mes and map of contributions e E to te error. Fig. 10. Convergence of te error estimators as a function of te number of elements of X 1 on te contact zone wit non-matcing meses.

15 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 79 Fig. 11. E ectivity of te error estimators as a function of te number of elements of X 1 on te contact zone wit non-matcing meses. frequently-employed procedure: te size and topology of te elements are canged wile te type of nite element functions on te di erent meses remains te same. A mes T is said to be optimal [11] for an error measure e if: e ˆ e 0 ; te accuracy prescribed by te user; N is minimized element number of mes T : 53 In order to solve te problem in (53), we adopt te following tecnique [18]: computation on a coarse mes T; computation of bot te global relative error e and te local contributions e E ; determination of te optimized mes T and second computation on te new mes T Examples For te rst example, we once again take te problem of two elastic bodies initially in contact (see Fig. 5). Te problem is optimized for a desired error of 5%. Te initial mes comprises 640 tree-node triangles and tis yields a computed error e of 7.64% (see Fig. 12). Te de ection is sown in Fig. 13. Te optimized mes comprises 514 triangles for a computed error of 4.42% (see Fig. 14). For te second example, we take te case of a cork (see Fig. 15). Te initial mes comprises 112 triangles and tis yields an error of 32.39% (see Fig. 16). Te desired error is set at 10%. Te map of contributions e E to te error is sown in Fig. 17. Fig. 12. Initial mes: node elements, 370 nodes, e ˆ 7.64%, e ex ˆ 6:12%, e 0 ˆ 5%.

16 80 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 Fig. 13. De ection. Fig. 14. Optimized mes: node elements, 303 nodes, e ˆ 4:42%, e ex ˆ 4:08%. Fig. 15. Problem set-up.

17 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 81 Fig. 16. Initial mes: node elements, 87 nodes, e ˆ 32:39%, e ex ˆ 22:3%, e 0 ˆ 10%. Fig. 17. Map of contributions e E to te error. Fig. 18. Optimized mes wit matcing mes: node elements, 411 nodes, e ˆ 11:78%, e ex ˆ 9:27%.

18 82 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 Fig. 19. De ection. Fig. 20. Map of Von Mises stresses. Fig. 21. Optimized mes wit non-matcing mes: node elements, 414 nodes, e ex ˆ 8:96%.

19 P. Coorevits et al. / Comput. Metods Appl. Mec. Engrg. 186 (2000) 65±83 83 By using a tecnique of mes automation [19], in tree steps we obtain an optimized mes wit 696 elements, a computed error of 11.78% and an exact error of 9.27% (see Fig. 18) in te case of matcing meses. Te de ection and te map of Von Mises stresses are sown in Figs. 19 and 20. In two steps, in te case of non-matcing meses, we obtain an optimized mes wit 709 elements and an exact error of 8.96% (see Fig. 21). Te computed error is 11.22% using te metod presented in Section and 11.42% wit te metod in Section Tese various examples sow tat optimized meses for te level of accuracy prescribed by te user, as well as e cient e ectivity indexes, can be obtained. References [1] G. Duvaut, J.-L. Lions, Les inequations en mecanique et en pysique, Dunod, [2] G. Ficera, in: S. Flugge (Ed.), Encyclopedia of Pysics, vol. VIa/2, Springer, Berlin, 1972, pp. 391±424. [3] N. Kikuci, J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Metods, SIAM, Piladelpia, [4] J. Haslinger, I. Hlavacek, J. Necas, Numerical metods for unilateral problems in solid mecanics, in: P.G. Ciarlet, J.L. Lions (Eds.), Handbook of Numerical Analysis, vol. IV, Part 2, Nort Holland, [5] F. Ben Belgacem, P. Hild, P. Laborde, Te mortar nite element metod for contact problems, Mat. Comput. Modelling 28 (1998) 263±271. [6] F. Ben Belgacem, P. Hild, P. Laborde, Approximation of te unilateral contact problem by te mortar nite element metod, C.R. Acad. Sci. Paris, Serie I 324 (1997) 123±127. [7] F. Ben Belgacem, P. Hild, P. Laborde, Extension of te mortar nite element metod to a variational inequality modelling unilateral contact, Mat. Modelling Met. in Appl. Sci., vol. 9, [8] P. Hild, Problemes de contact unilateral et maillages elements nis incompatibles, Tese de l'universite Paul Sabatier, Toulouse 3, [9] I. Babuska, W.C. Reinboldt, Error estimates for adaptive nite element computation, Siam. J. Num. Anal. 15 (1978) 736±754. [10] O.C. ienkiewicz, J.. u, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Num. Met. Engrg. 24 (1987) 337±357. [11] P. Ladeveze, G. Co gnal, J.P. Pelle, Accuracy of elastoplastic and dynamic analysis, Accuracy estimates and adaptative re nements, in: Babuska, Gago, Oliveira, ienkiewicz (Eds.), Finite Element Computations, Wiley, New York, 1986, pp. 181±203. [12] P. Ladeveze, J.P. Pelle, P. Rougeot, Error estimates and mes optimization for nite element computation, Engrg. Comp. 8 (1991) 69±80. [13] P. Wriggers, O. Scerf, C. Carstensen, Adaptive tecniques for te contact of elastic bodies in: Huges, Onate, ienkiewicz (Eds.), Recent Developments in Finite Element Analysis, CIMNE, 1994, pp. 78±86. [14] P. Ladeveze, Nonlinear computational structural mecanics, New Approaces and Non-incremental Metods of Calculation, Springer, Berlin, [15] W. Prager, J.L. Synge, Approximations in elasticity based on te concept of function space, Quart. Appl. Mat. 5 (1947) 261±269. [16] I. Ekeland, R. Temam, Convex Anal. Variational Problems, Amsterdam, Nort-Holland, [17] P. Coorevits, J.P. Dumeau, J.P. Pelle, Control of analysis wit isoparametric elements in 2D and in 3D, submitted. [18] P. Coorevits, P. Ladeveze, J.P. Pelle, Mes optimization for problems wit steep gradients, Engrg. Comp. 11 (1994) 129±144. [19] P. Coorevits, P. Ladeveze, J.P. Pelle, An automatic procedure for nite element analysis in 2D elasticity, Comp. Met. Appl. Mec. Engrg. 121 (1995) 91±120.