A domain decomposition convergence for elasticity equations

Transcription

1 A domain decomposition convergence for elasticity equations A. Cakib a, A. Ellabib b, A. Nacaoui c, a Département de Matématiques Appliquées et Informatique FST de Beni-Mellal, Université Sultan My Slimane B.P. 53, Beni-Mellal, Morocco. b Université Cadi Ayyad, Faculté des Sciences et Tecniques, Département de Matématiques et Informatique, Avenue Abdelkrim Elkattabi, B.P 549, Guéliz Marrakec, Maroc. c Laboratoire de Matématiques Jean Leray, UMR 669, Université de Nantes/CNRS/ECN, rue de la Houssinière, BP 98, 443 Nantes, France. Abstract A non-overlapping domain decomposition metod for elasticity equations based on an optimal control formulation is presented. Te existence of a solution is proved and te convergence of a subsequence of te approximate solutions to a solution of te continuous problem is sown. Te implementation based on lagrangian metod is discussed. Finally, numerical results sowing te efficiency of our approac and confirming te convergence result are given. Key words: Convergence, Domain decomposition, Elasticity equations, Optimal control formulation. Introduction Domain decomposition metods is divided into two classes, tose tat use overlapping domain, and tose tat use non-overlapping domains, wic we refer to as substructuring. Various substructuring metods wit non-overlapping can be encountered in literature and fruitful references can be found from [7]. Corresponding autor. addresses: cakib@fstbm.ac.ma (A. Cakib), ellabib@fstg-marrakec.ac.ma (A. Ellabib), nacaoui@mat.univ-nantes.fr (A. Nacaoui). Preprint submitted to Matematics and Computers in Simulation 9 January 8

2 A study of elasticity equations by domain decomposition metod was treated from [4,5,,5]. In [5], te autors ave presented te tecniques for te algebraic approximation of Diriclet to Neumann maps for linear elasticity. Tis tecniques are based on te local condensation of te degree of freedom belonging to a small area-defined inside te sub-domain- on a small patc defined on te interface. In [], te domain decomposition metod wit Lagrange multipliers is introduced by reformulating te preconditioned system of te FETI algoritm as a saddle point problem wit bot primal and dual variables as unknowns. In tis paper, we consider a linear elasticity material wic occupies an open bounded domain Ω R were te boundary is denoted by Γ = Ω. Te linear elasticity problem [] is given, for i =,, by σ ij (u) x j = f i in Ω u i = on Γ. were u = (u, u ) is te displacement vector, f = (f, f ) te volume force vector, σ ij is te stress tensor. Te traction vector t is defined by for i =,, t i = σ ij (u)n j were n is te outward normal unitary vector of te domain Ω along boundary Γ. Te strain tensor ε ij is given by ε ij = ( ui + u ) j x j x i Tese tensors are related by σ ij (u) = G ( ε ij (u) + ν ν ) ε kk (u)δ ij k= () () (3) wit G and ν are te sear modulus and Poisson ratio, respectively, and δ ij is te Kronecker delta tensor. We wis to determine te solution of () by a domain decomposition metod. To tis end and for simplicity we consider ere only te case were Ω is partitioned into two open subdomains Ω () and Ω () suc tat Ω = Ω () Ω (). Te interface between two domains is denoted γ so tat γ = Ω () Ω (). Let Γ = Ω () Γ and Γ = Ω () Γ. Let us denote by f (k) i = f i, for k =,. We consider te problems defined over te subdomains

3 σ ij (u () ) x j = f () i in Ω () u () i = on Γ σ ij (u () )n () j = ψ i on γ (4) σ ij (u () ) x j = f () i in Ω () u () i = on Γ σ ij (u () )n () j = ψ i on γ were n (i) is te outward normal unitary vector of te subdomain Ω (i) along te interface γ, for i =,. (5) In tis work, we are interesting to combine te optimization tecniques and non-overlapping domain decomposition to solve problem (). Tis combination is obtained as a constrained minimization problem for wic te cost functional is te L (γ)-norm of te difference between te dependent variables u (), u () across te common boundaries γ and te constraints are te problems (4) and (5). At tis stage its must be noted tat a similar idea of tis combination was already used for Laplace operator in [6,7], for coupled stokes flows [], for nonlinear sedimentary basin problem [9]. Here, we extend tis idea for te study of elasticity equations. Furtermore, we prove te convergence of approximate optimal solutions to continuous one and we give an algoritm based on gradient conjugate wit variable steeps. Te paper is organized as follows. In section we provide an optimal control formulation equivalent to te model problem (). In section 3, we prove te existence of optimal solution. Te existence of te discrete optimal control problem, obtained by finite element approximation, is given in section 4. Te convergence of approximate solutions to te continuous one is sown in section 5. Section 6 deals wit te description of our optimization algoritm, in section 7, we report some numerical result. Optimal control formulation Define te following convex set : K = {ψ = (ψ, ψ ) (L (γ)) / ψ k L (γ) C, for k =, } were C is a nonnegative given constant. 3

4 For te numerical approximation of te problem (), we propose te following optimal control formulation Minimize J(u () (ψ), u () (ψ)) for all ψ K (P O) were J(u () (ψ), u () (ψ)) = ( () u i u () ) i dσ (6) i= γ and u () (ψ), u () (ψ) are respectively te solution of (4) and (5). We ave te following result Proposition Assume tat f and Ω are smoot enoug. Ten te problem () is equivalent to (6). Proof. Let u e be te solution of () and let us denote by u (k) e = u e, for k =,. Assume tat f and Ω are smoot enoug, suc tat ψ i,e = σ ij (u () )n () j in L (γ), for i =,. One can coose te constant C, defining K, suc tat ψ e = (ψ,e, ψ,e ) K ; tis means tat max ( ψ,e L (γ) ; ψ,e L (γ)) C. Tis implies tat (u () e (ψ e ), u () e (ψ e )) is a solution of (6). Conversely, let (u () (ψ ), u () (ψ )) be a solution of (6) for ψ K, ten we ave J(u () (ψ ), u () (ψ )) J(u () (ψ), u () (ψ)) for all ψ K. In particular, we ave J(u () (ψ ), u () (ψ )) J(u () e (ψ e ), u () e (ψ e )) =, tis involves tat u = result. u () in Ω () u () in Ω () is a solution of () and acieves te equivalence is 3 Existence of optimal solution We first give some notations and definitions wic can be useful in te following. We define te spaces, for i =,, H i,d (Ω (i) ) = {υ (H (Ω (i) )) / υ Γi = } 4

5 were (H (Ω (i) )) is te Sobolev space equipped wit te norm,ω (i) defined by ( ( ) υ,ω (i) = υl,ω + υ (i) l,ω (i)), υ l,ω (i) = υ l dx l= Ω (i). ( ) H i,d (Ω (i) ) are equipped wit te following norm υ,ω (i) = υ l,ω. (i) l= For ψ K, we consider te weak formulation of equation (4) and (5) given, for k =,, by Find u (k) (ψ) H k,d ( ) υ = (υ, υ ) H k,d ( ) a (k) (u (k), v) = σ ij (u (k) ) ε ij (υ) = f (k) i υ i dx + ( ) k ψ i υ i dσ. (7) i, i= i= γ We define te space of admissible solutions U ad by : U ad = {(u () (ψ), u () (ψ)) solution of (7) / ψ K }. Te optimal control problem (6) can be rewritten as: (P O) Minimize J((u () (ψ), u () (ψ)) for all (u () (ψ), u () (ψ)) U ad. We define te convergence of te sequence (ψ n ) n = ((ψ,n, ψ,n )) n in K to ψ = (ψ, ψ ) K by ψ n ψ ψ k,n ψ k weakly in L (γ), for k =,. (8) We can ten equip U ad wit te topology defined by te following convergence: let ((u () n, u () n )) n be a sequence of U ad and (u (), u () ) U ad ten: (u () n, u () n ) (u (), u () ) u () k,n u() k weakly in H (Ω () ) u () k,n u() k weakly in H (Ω () ), for k =,. (9) We ave ten te following result. Teorem Te problem (P O) is well posed and admits a solution in U ad. 5

6 Proof. For all ψ in K, te result of te existence and unicity of te solution of (7) is ensured by te Lax-Milgram teorem, tis involves tat te problem (P O) is well posed. Te proof of te existence of a solution of (P O) is now reduced to sow tat U ad is compact for te topology defined by (9) and tat J is lower semi-continuous on U ad. In order to sow tat U ad is compact, we consider ((u () n, u () n )) n a sequence of U ad, i.e. u (k) n = u (k) (ψ n ) is te solution of (7) for ψ n K. Since for all n and k =,, we ave ψ k,n L (γ) C, we can extract from (ψ n ) n a subsequence denoted again (ψ n ) n, suc tat ψ k,n converges weakly in L (γ) to ψk and ψ = (ψ, ψ) is in K. Te sequence ((u () n, u () n )) n converges weakly to (u (), u () ) and (u (), u () ) is suc tat (u (), u () ) = (u () (ψ ), u () (ψ )) U ad. Indeed, for all n, u (k) n = u (k) (ψ n ) H k,d ( ) is te solution of a (k) (u (k) n, υ) = i= f (k) i υ i dx + ( ) k i= γ ψ i,n υ i dσ υ H k,d ( )() Taking υ = u (k) n in () and using te inequality ψ i,n L (γ) C, for i =,, and te Korn s inequality, we obtain tat u (k) n, β, were β is a nonnegative constant independent of n. Tus we can extract a subsequence denoted again (u (k) n ) n, suc tat u (k) i,n is weakly convergent to u (k) i, in H ( ), for i =,. Since is smoot enoug, te trace operator from H ( ) to L (Γ k ) is compact, tis implies tat u (k) H k,d ( ). It remains to sow tat u (k) is solution of a (k) (u (k) ), υ) = i= f (k) i υ i dx + ( ) k i= γ ψ i υ i dσ υ H k,d ( )() Tis is obtained by using te weak convergence of u(k) i,n x j to u(k) i, x j in L ( ), for i, j =,, and by passing to te limit in equation (). Consequently (u (), u () ) = (u () (ψ ), u () (ψ )) U ad. Tis acieves te proof of te compactness of U ad for te topology defined by te convergence (9). To sow te continuity of te functional J in U ad, let us consider a sequence ((u () n, u () n )) n U ad wic is convergent to (u (), u () ) U ad. We ave J(u () n, u () n ) J(u (), u () ) = (u () i,n u () i,n) dσ (u () i u () i ) dσ i= γ γ ( ) ( ) (I i,n ) (L i,n ) i= i= 6

7 were I i,n = and L i,n = γ γ ( () u i,n u () i + u () i u () ) i,n dσ ( () u i,n u () i,n + u () i u () i ) dσ. Since u (k) n is uniformly bounded in (H ( )) wit respect to n, we ave tat L i,n is uniformly bounded. Te use of te compactness of te trace operator from H ( ) to L (γ) gives lim I i,n =. Tus lim n n J(u() n, u () n ) J(u (), u () ) =. Tis end te proof. 4 Approximation of te problem In tis section, we use te linear finite element metod for te approximation of (P O). We sow te existence of te solution of te discrete problem and we study te convergence of a subsequence of tese solutions to a solution of te continuous problem. Finally, to confirm te convergence result, we give some numerical results. For te seek of simplicity, we reduce our study, in tis section, to te case were te boundary part γ is assumed to be defined as follows: γ = {(b, x) / x [, a]} () were a > and b are two given constants. In te following, we need additional regularity assumptions on K, namely: K = {ψ = (ψ, ψ ) (C (γ)) / ψ k (b, x) ψ k (b, x ) C x x x, x [, a] and ψ k L (γ) C for k =, } (3) were C and C are nonnegative given constants. Te convergence of a sequence (ψ n ) n = ((ψ,n, ψ,n )) n in K to ψ = (ψ, ψ ) K is defined in tis case by ψ n ψ ψ k,n (b,.) ψ k (b,.) uniformly in [, a], for k =, (4) Remark 3 Note tat te existence result sown in section 3, remains valid in K wit te above convergence. In tis case, te compactness of K is ensured by te use of Ascoli-Arzelà teorem s (see []). 7

8 4. Discretization of te problem Let us consider an uniform partition (a i ) N i= of te interval [, a], suc tat: = a < a <... < a N = a, a i a i = for i =,..., N. We define te discrete space associated to K by K = {ψ = (ψ,, ψ, ) (C(γ)) / ψ k, (b,.) [ai,a i ] P ([a i, a i ]) i =,..., N, ψ k,(b,a i ) ψ k, (b,a i ) a i a i C, i =,..., N and ψ k, L (γ) C + C, for k =, } wit te same constants C and C, as in te definition of K. Let H(Ω) be te finite dimensional space given by H( ) = {υ C( ) / υ K P (K), K T } were T is a regular triangulation of, for k =,. Let H k,d( ) = {υ (H( )) / υ Γk = } be te finite dimensional spaces associated respectively to H k,d ( ). For ψ K, we consider te following discrete problem of (6), for k =, : Find u (k) (ψ ) Hk,D(Ω (k) ) υ Hk,D(Ω k (k) ) a (k) (u(k), υ ) = σ ij (u (k) ) ε ij(υ ) = i, i= f (k) i, υ i, dx + ( ) k i= γ ψ i, υ i, dσ (5) were f (k) i, f (k) i, (k) is an approximation of f i suc tat (k) is uniformly bounded and converges to f i almost every were.(6) Te discrete space of te admissible solutions is given by U ad = {(u () (ψ ), u () (ψ )) solution of (5) / ψ K } 8

9 We approac te cost functional by te following discrete one : J (u () (ψ ), u () (ψ )) = i= γ ( u () i, (ψ ) u () i, (ψ )) ) dσ, and we state our discrete optimization problem as follows (P O ) (u () inf,u() ) U ad were u (k) J (u (), u() ) = u (k) (ψ ) is solution of (5), for k=,. Note tat te set K can be identified wit te following subset of R N K = { {X} = (X,,..., X,N, X,,..., X,N ) R N / C X l,i X l,i C, i =,..., N, l =, and X l,i C + C, i =,..., N, l =, }. We denote by M () and M γ () te set of nodes lying respectively on and γ. Let m (k) be te number of elements of M (), and define NT (k) = N + m (k), for k =,. Let us now introduce in H( ) te canonical basis (p (k) i ) NT k i= suc tat p () i = p () i = p i, for all i M γ (). For te vector P (k) = [p (k), p (k),..., p (k) ], we define te following matrix [P (k) ] = NT (k) P (k) P (k) Ten u (k) can be written u (k) = [P (k) ] {u (k) T } were {u (k) T } = t [u (k),, u (k),,..., u (k), u (k),nt,, u (k),,..., u (k) ] is te vector of te components (k),nt (k) of u (k) in te basis P (k). Let us denote by D P (k) = p (k) x p (k) y p (k) x p (k) y... p (k) NT (k)... p x (k) NT (k) y and [D P(k) ] = D P (k) D P (k) te gradient of u (k), D u(k) D u (k) = t u(k), can be written in term of [D P (k) ] and {u (k) T } by x, u(k), y, u(k), x, u(k), y = [D P (k) ] {u (k) T } Te tensors ε and σ can be read {ε} = t (ε, ε, ε ) and {σ} = t (σ, σ, σ ) 9

10 {ε} can be written in term of D u (k) {ε} = If we denote by [D] te above matrix, we ave {ε} = [D] D u (k) = [D] [D P (k) ] {u (k) T } u (k), x u (k), y u (k), x u (k), y Using equation (3), we can write {σ} in term of {ε} as follows {σ} = [E] {ε} tus {σ} = [E] [D] [D P (k) ] {u (k) T } were [E] is a 3 3 symmetric matrix. Using te above notations we ave σ ij (u (k) ) ε ij(υ ) = t {υ T } t [D P (k) ] t [D][E] [D] [D P (k) ] dx {u (k) T }, and i, i= f (k) i, υ i, dx = t {υ T } Setting now te matrix A (k) = {B (k) } = t [P (k) ] f (k) G (k) i (X) = ( ) k t [P (k) ] f (k) dx. t [D P (k) ] t [D][E] [D] [D P (k) ] dx, te vectors NT (k) dx and {G (k) (X)} = (G i (X)) i= wit X l,j l= j M γ () γ ( t [P (k) ] [P (k) ] ) ij dσ, (7) it is easy to see tat problem (5) can be rewritten, for k =,, as Find {u (k) T (X)} R NT (k) suc tat A (k) {u (k) T (X)} = {B (k) } + {G (k) (X)} (8) We can identify te set Uad 4NT (k) wit te following subset of R U = {({u () T }, {u () T }) solution of (8) / {X} K }.

11 Ten te discrete cost functional reads : J (u (), u() ) = J({u() T }, {u () T }) = () [R] ({u T } {u () T }), ({u () T } {u () T }) NT (k) were.,. is te inner product in R and were R = (r i j ) i,j NT (k) [R] = R R is given by r i j = p i p j dσ if i, j M γ () r i j = γ oterwise. te matrix [R] is defined by Te matrix form of te optimization problem reads: (P M) inf J({u () T }) U T }, {u () T }) ({u () T },{u() s/c A (k) {u (k) T (X)} = {B (k) } + {G (k) (X)} for k =, 4. Existence of te solution of te discrete problem (9) It is easy to see tat (P O ) is equivalent to (P M), tus we sow tat (P M) as a solution in U. Teorem 4 Te problem (P M) admits a solution on U, for all >. Proof. Let us consider a minimizing sequence (({u () T } n, {u () T } n )) n of J in U, suc tat lim n J({u() T } n, {u () T } n )) = inf J(w (), w () ). (w (),w () ) U We ave tat for all n and k =,, {u (k) T } n = {u (k) T }(X n ) is te solution of A (k) {u (k) T (X n )} = {B (k) }+{G (k) (X n )}. Using te fact tat K is bounded and closed (compact) in R N, we can extract from ({X} n ) n a subsequence denoted again ({X} n ) n wic converges in R N to {X } K. From te definition of {G (k) } in equation (7), we can sow tat te sequence ({G (k) (X n )}) n

12 converges to {G (k) (X )} in R NT (k). Let {u (k) } be te solution of A (k) {v} = {B (k) }+{G (k) (X )}, we sow tat te sequence ({u (k) T } n ) n converges to {u (k) } in R NT (k), for k =,. Indeed, we ave tat A (k) {u (k) T } n, {u (k) T } n {u (k) } = {B (k) }, {u (k) } n {u (k) } T + {G (k) (X n )}, {u (k) T } n {u (k) } () and A (k) {u (k) }, {u (k) T } n {u (k) } = {B (k) }, {u (k) } n {u (k) } T + {G (k) (X )}, {u (k) T } n {u (k) }. () Subtracting equation () from (), and using te fact tat te matrix A (k) is symmetric and positive definite, we obtain tat tere exists a nonnegative constant α suc tat α {u (k) T } n {u (k) } {G (k) (X NT (k) n )} {G (k) (X )} NT (k) {u (k) T } n {u (k) } NT (k) () te result is obtained by passing to te limit in (). Te main result of tis teorem follows from te fact tat J({u () T } n, {u () T } n ) converges to J({u () }, {u () }), wic is obtained by passing to te limit in te following equation J({u () T } n, {u () T } n )) J({u () }, {u () }) = [R] ({u () T } n {u () T } n ), ({u () T } n {u () T } n ) [R] ({u () } {u () }), ({u () } {u () }) = [R] ({u () T } n {u () T } n ), ({u () T } n {u () T } n ) [R] ({u () } {u () }), ({u () T } n {u () T } n ) + [R] ({u () } {u () }), ({u () T } n {u () T } n ) [R] ({u () } {u () }), ({u () } {u () }) = [R] (({u () T } n {u () T } n ) ({u () } {u () })), ({u () T } n {u () T } n ) + [R] ({u () } {u () }), (({u () T } n {u () T } n ) ({u () } {u () })).

13 5 Convergence result In tis section, we are interested in sowing te existence of a subsequence of te solutions of te discrete problems wic converges to a solution of te continuous one. For tis we introduce te following definitions: Let (ψ ) be a sequence suc tat ψ K for all, we define te convergence of (ψ ) to ψ K as by ψ ψ ψ i, (b,.) ψ i (b,.) uniformly in [, a] for i =,.(3) For a sequence ((u () te sequence ((u () (u (), u(), u(), u() )) suc tat (u (), u() ) U ad, te convergence of )) to (u (), u () ) U ad, as, is defined by ) (u(), u () ) u () i, u() i weakly in H (Ω () ) Our convergence result is based on te following lemma. u () i, u() i weakly in H (Ω () ) for i =,. Lemma 5 (i) For any (u (), u () ) U ad, suc tat u (k) = u (k) (ψ) for ψ K, tere exists a sequence ((u (), u() )) suc tat u (k) = u (k) (ψ ) for ψ K and (u (), u() ) (u(), u () ). (ii) Let ((u (), u() )) be a sequence of Uad suc tat u (k) = u (k) (ψ ) for ψ K. Ten tere exists a subsequence of ((u (), u() )) denoted again by ((u (), u() ψ K and (u () (iii) If ((u () )) and an element (u (), u () ) U ad suc tat u (k) = u (k) (ψ) for, u() ) (u(), u () )., u() )) is a sequence suc tat (u (), u() ) U ad, and (u (), u () ), u() ) (u(), u () )., u() ) J(u(), u () ) as. U ad suc tat (u () Ten J ((u () (4) Proof. In order to sow (i), let (u (), u () ) U ad suc tat suc tat u (k) = u (k) (ψ) for ψ K. For > and k =,, we construct te sequence (ψ ) = (ψ,, ψ, ) as follows: ψ k, C(γ) suc tat ψ k, (b,.) [ai,a i ] P for i =,..., N, (i+ ) ψ k, (b, a i ) = ψ k (b, τ) dτ for i =,..., N, (i ) 3

14 ψ k, (b, ) = It is easy to see tat ψ k (b, τ) dτ and ψ k, (b, a) = a a ψ k (b, τ) dτ. ψ k, (b, a i ) ψ k, (b, a i ) C for i =,..., N (5) wic leads, wit some elementary calculations to te following estimate ψ k, ψ k L (γ) C. (6) We deduce from tis tat ψ k, L (γ) C + C. (7) Ten ψ K and ψ converges to ψ. Let ((u (), u() )) be in Uad suc tat u (k) = u (k) (ψ ), tis means tat u (k) Hk,D(Ω (k) ) is te solution of a(u (k), υ ) = i= f (k) i, υ i, dx + ( ) k ψ i, υ i, dσ υ Hk,D(Ω k (k) ). (8) i= γ Using equations (7) and (8), we can sow tat (u (k) ) is uniformly bounded in (H ( )) and tus we can extract a subsequence denoted again (u (k) ), suc tat u (k) (k) i, is weakly convergent to V i in H ( ), for i =,. From te compactness of te trace operator from H ( ) to L (Γ k ) we ave tat V (k) = (V (k), V (k) ) H k,d ( ). To conclude tat V (k) = u (k), it suffices to sow tat V (k) is solution of te equation: a(v (k), υ) = i= f (k) i υ i dx + ( ) k i= γ ψ i υ i dσ υ H k,d ( ). (9) Let υ in H k,d ( ), and denote by Φ linear interpolant of υ, we ave: = Π υ H k,d( ) te piecewise a(u (k), Φ ) = i= f (k) i, Φ i, dx + ( ) k i= γ ψ i, Φ i, dσ (3) 4

15 By passing to te limit in equation (3) as, we obtain tat V (k) is a solution of equation (9). Indeed, we ave i, ( σij (u (k) ) ε ij(φ ) σ ij (V (k) ) ε ij (υ) ) = I + I were I = i, ( σij (u (k) ) σ ij(v (k) ) ) ε ij (υ) and I = i, σ ij (u (k) ) (ε ij(φ ) ε ij (υ)) From te weak convergence in H ( ) of u (k) (k) i, to V i, for i =,, we ave tat I converges to as. By virtue of te convergence result of Φ = Π υ to υ in (H ( )), as (see [3]) and since u (k) is uniformly bounded in (H ( )), we get tat I converges to. In similar fasion using te convergence (6) and (3), we can sow tat lim f (k) i, Φ i, dx f (k) i υ i dx i= Ω (k) = lim (f (k) i, f (k) i ) υ i dx + (Φ i, υ i ) f (k) i, dx = i= lim i= γ i= γ = lim ψ i, Φ i, dxσ ψ i υ i dσ (f (k) i, f (k) i ) υ i dσ + γ γ (Φ i, υ i ) f (k) i, dσ = Tis acieve te proof of assertion (i). To sow (ii), Let ((u (), u() )) be a sequence of Uad suc tat u (k) = u (k) (ψ ), for ψ K. We ave tat for all and i =,, ψ i, T, were T is te space defined by T = {χ C(γ) / χ(b, x) χ(b, x ) C x x x, x [, a] and χ L (γ) C + C}. 5

16 According to te Ascoli-Arzelà teorem s, we can extract a subsequence noted again (ψ ), suc tat ψ i, converges in T to ψ i T, for i =,. Furtermore, by passing to te limit in equation (7), we ave tat ψ = (ψ, ψ ) K. Using te same tecniques as in te proof of (i), we sow tat for k =,, u (k) i, converges weakly in H ( ) to u (k) i = u (k) i (ψ) and tat u (k) is solution of (5). Te proof of te assertion (iii) uses mainly te same tecnique as in te proof of continuity of J in Teorem. Tis ends te proof of te lemma. We can now prove our main result of convergence stated in te following teorem Teorem 6 Let ((u (),, u(), )) be a sequence suc tat (u (),, u(), ) is solution of (P O ) and and (u (),, u(), ) U ad. Ten, tere exists a subsequence denoted again ((u (),, u(), )) and an element (u (), u () ) U ad suc tat (u (),, u(), ) (u(), u () ) furtermore (u (), u () ) is solution of (P O). Proof. Let (u (), u () ) be an element of U ad, from te assertion (i) of Lemma, tere exists a sequence ((u (), u() )) suc tat (u (), u() ) U ad and (u (), u() ) (u(), u () ) According to te assertion (iii), we ave tat J (u (), u() ) J(u(), u () ) as Now, Let ((u (),, u(), )) be a sequence suc tat is solution of (P O ) and (u (),, u(), ) U ad. From te assertion (ii) of Lemma, tere exists a subsequence denoted again ((u (),, u(), )) and an element (u (), u () ) U ad suc tat (u (),, u(), ) (u(), u () ) According to te assertion (iii), we ave tat J (u (),, u(), ) J(u(), u () ) as 6

17 owever, we ave tat J(u (),, u(), ) J(u(), u() ) for all (3) Te main result is ten obtained by passing to te limit in equation (3), as. 6 Optimization algoritm We use te Lagrange multiplier rule to derive an optimality system of equations from wic solutions of te optimization problem (PO) may be determined. Let u (i), λ (i) H i,d (Ω (i) ), for i =,, and ψ (L (γ)) we define te Lagrangian L(u (), u (), ψ, λ (), λ () ) = J(ψ, u (), u () ) Ω () + f () (x)λ () (x) dx + ψ(x)λ () (x) dx Ω () γ + Ω () f () (x)λ () (x) dx ψ(x)λ () (x) dx γ Ω () σ ij (u () (x))ε ij (λ () (x)) dx σ ij (u () (x))ε ij (λ () (x)) dx Setting to zero te first variations wit respect to te multipliers λ ans λ yields te constraints (7). Setting to zero te first variations wit respect to u () and u () yield te adjoint equations a () (v, λ () ) = (u () u (), v) γ v H,D (Ω () ) (3) and a () (v, λ () ) = (u () u (), v) γ v H,D (Ω () ) (33) respectively. Ten te adjoint equations is given by 7

18 σ ij (λ () ) = in Ω () x j λ () = on Γ (34) σ ij (λ () )n j = u () u () on γ σ ij (λ () ) = in Ω () x j λ () = on Γ (35) σ ij (λ () )n j = (u () u () ) on γ Let J (ψ) = J(ψ, u (), u () ) were, for given ψ, u (i) : ψ (L (γ)) H i,d (Ω (i) ) for i =, are defined as te solution of (4) and (5) respectively. Ten, te minimization problem is equivalent to te problem of determining ψ (L (γ)) suc tat J (ψ) is minimized. Now, te first derivative of J is defined troug its action on variations ψ by dj dψ, ψ = (u () u (), ũ () ũ () ) γ ψ (L (γ)) (36) were ũ () H,D (Ω () ) and ũ () H,D (Ω () ) are te solution of and a () (ũ (), v) = ( ψ, v) γ v H,D (Ω () ) (37) a () (ũ (), v) = ( ψ, v) γ v H,D (Ω () ) (38) respectively. Set v = λ () in (37), v = λ () in (38), v = ũ () in (3) and v = ũ () in (33). Combning te results yields tat dj dψ = λ() λ () on γ. (39) we now present our domain decomposition algoritm Algoritm k = and ψ, is given For k =,... 8

19 Solve σ ij (u (),k ) x j = f () i in Ω () u (),k = on Γ σ ij (u (),k )n j = ψ i,k on γ (4) σ ij (u (),k ) x j = f () i in Ω () u (),k = on Γ σ ij (u (),k )n j = ψ i,k on γ (4) σ ij (λ (),k ) x j = in Ω () Solve λ () = on Γ (4) σ ij (λ (),k )n j = u (),k u(),k on γ Compute J(ψ,k ) = λ (),k (ψ,k) λ (),k (ψ,k) Update γ k = J(ψ,k) J(ψ,k ) d,k = J(ψ,k ) + γ k d,k Solve Compute σ ij (D (),k ) x j = in Ω () D (),k = on Γ σ ij (D (),k )n j = d i,k on γ ρ k = (u(),k u(),k, D(),k D(),k ) D (),k D(),k ψ,k+ = ψ,k ρ k d,k End For (44) σ ij (λ (),k ) x j = in Ω () λ (),k = on Γ (43) σ ij (λ (),k )n j = (u (),k u(),k ) on γ σ ij (D (),k ) x j = in Ω () D (),k = on Γ (45) σ ij (D (),k )n j = d i,k on γ 7 Numerical results In order to illustrate te performance of te numerical metod described above, we solve te linear elasticity problem (), in two-dimensional domain Ω = (, ) (, ), wit u = u an on Γ and f =. We assume tat te boundary is split into two parts Γ = [,.5] {} [,.5] {} {} [, ] and Γ = [.5, ] {} {} [, ] [.5, ] {}. For tese data, te analytical 9

20 solution is given by u an (x, y) = ν G σ xy, u an (x, y) = 4G σ (( ν)(x ) + νy ) (46) t an (x, y) = σ yn, t an (x, y) = (47) wit σ =.5, G = 3.35 and ν =.34. Tis example consists to split te domain Ω into two rectangular subdomains Ω () = (.,.5) (, ) and Ω () = (.5, ) (, ) wit interface γ = {.5} [, ]. In tis section we investigate te convergence of te proposed metod by te evaluation at every iteration te accuracy errors denoted for i, j =, by G (i) k (u j) = u (i) j,k u(i)an j L (γ), G (i) k (t j) = t (i) j,k t(i)an j L (γ). (48) Te following stopping criterion is considered J(ψ,k ) < η J(ψ, ) (49) were η is a small prescribed positive quantity. For all numerical experiments, we take η =. Te mes of discretization is taken as = /4. Te initial guess ψ i, on γ as been cosen as ψ i, =. Wen starting wit tis initial guess, wic is not too close to te exact traction, a sequence of displacements { (u (),k ) } and { k () (u,k ) } of approximation functions for u k γ is obtained and tis sequences converge to te exact solution. We observe from Figure (a), (b) tat te 3 - Grad J Cout J Accuracy errors G_k^{()} G_k^{()}(u_) G_k^{()}(u_) G_k^{()}(t_) G_k^{()}(t_) - -9 (a) Iterations k (b) Iterations k (c) Iterations k Fig.. Computed norm of gradient (a), cost functional (b) and te accuracy errors (c) given by (48) as a function of te number of iterations k. norm of gradient and te cost decrease as a function of number of iterations. Figure (c) and Figure (a) sows te evaluation of accuracy errors as function of number of iterations. Te discrepancy u (),opt u ()an L (γ) between te

21 Iterations k 5 3 Accuracy errors G_k^{()} G_k^{()}(u_) G_k^{()}(u_) G_k^{()}(t_) G_k^{()}(t_) u_^{()} 4 3 Iteration Iteration 5 Iteration Optimal Analytical.5 u_^{()} - Iteration - Iteration 5 Iteration Optimal Analytical (a) (b) y (c) y Fig.. Te accuracy errors (a) given by (48) as a function of te number of iterations k, results of u () (b) and u () (c) on interface γ optimal x -displacement and te exact one is equal to and te discrepancy t (),opt t ()an L (γ) between te optimal x -traction and te exact one is equal to.9 4. Figure (c) and Figure (a) sows te evaluation of accuracy errors as function of number of iterations. Figure - 4 proves te well convergence of te proposed optimal control algoritm. 7 u_^{()} (a) t_^{()} -5 - Iteration Iteration 5 Iteration Optimal Analytical y u_^{()} (b) Iteration Iteration 5 Iteration Optimal Analytical y t_^{()} 5 (c) Initial Iteration 5 Iteration Optimal Analytical Fig. 3. Results of u () (a), u () (b) and t () (c) on interface γ Initial Iteration 5 Iteration Optimal Analytical t_^{()} 5 Initial Iteration 5 Iteration Optimal Analytical t_^{()} -5 - y Initial Iteration 5 Iteration Optimal Analytical (a) y (b) y (c) y Fig. 4. Results of t () (a), t () (b) and t () (c) on interface γ 8 Conclusion In tis paper, te Problem of linear elasticity equations is formulated into an optimal control problem. Te linear finite element is used for te approximation of tis problem. Te convergence of te solutions of discrete problems to a solution of te continuous one is proved. Te numerical results obtained were found to be good in agreement wit te exact solution.

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