An adapted coarse space for Balancing domain decomposition method to solve nonlinear elastodynamic problems

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1 A adapted coarse space for Balacig domai decompositio method to solve oliear elastodyamic problems Mikaël Barboteu Uiversity of Perpiga, 52 aveue Paul-Alduy, Perpiga, Frace This work is devoted to preset a scalable domai decompositio method to solve oliear elastodyamic problems. Large o liear elastodyamic problems represet a appropriate applicatio field for substructurig methods which are efficiet o parallel computer with the proviso of usig specific precoditioer techiques well adapted to the mechaical modelig. Accordig to this reaso, we develop a adapted Balacig domai decompositio method [Ma93, LeT94] appropriated to solve this kid of systems. By usig the theoretical framework of Schwarz additive decompositio method [LeT94, LMV98] ad by usig argumets developed i [ABLV00], we propose a two level Neuma-Neuma precoditioer based o the costructio of a coarse space of "lower eergy" modes adapted to fiite deformatios problems with dyamic process. I sectio 1, oliear elastodyamic problems ad domai decompositio frameworks are recalled. The sectio 2 is devoted to preset the defiitio of a adapted coarse space by usig Schwarz additive formulatio. The costructio of the two level Neuma-Neuma precoditioer is detailled i sectio 3. I last sectio 4, we test the efficiecy of this updated Balacig domai decompositio method o umerical solutios of a academic o liear dyamic problem. 1 Noliear elastodyamic problems ad domai decompositio frameworks Dyamic deformable body systems i large deformatios are govered by o liear time depedet equatios. A typical o liear elastodyamic problem defied i a referece cofiguratio ca take the followig variatioal form, { Fid u L 2 (]0; T[; U 0 ) such that for each t ]0; T[, Ω ρü(t).v + Ω Π(t) : v Ω f(t).v g(t).v = 0, v U (1) gω 0

2 2 Mikaël Barboteu where ρ deotes the mass desity; Π is the first Piola-Kirchoff tesor; f ad g are the exteral force desities. A dot superscript idicates the time derivative. The set U 0 = {v H 1 (Ω) dim ;v = 0 o 0 Ω} represets the space of kiematically admissible displacemet fields. The problem (1) ca be solved by a eergy coservative time itegratio scheme [Go00] which is appropriate due to his log term time itegratio accuracy ad stability. I the followig, we cosider a collectio of discrete times (t p ) p=1..p which defie a partitio of the time iterval [0; T] = P p=1 [t p; t p+1 ] with t p+1 = t p + t ad t = T P. By usig a secod order time itegratio scheme (adapted midpoit scheme) [Go00], the weak form (1) itegrated betwee the times t p ad t p+1 gives the followig system, { Fid up+1 U 0 such that 1 t Ω ρ( u p+1 u p ).v + Ω Π algo : v Ω f p+ 1.v 2 g gω p+ 1.v = 0, 2 (2) where p+ 1 2 = 1 2 ( p + p+1 ) ad p deotes the approximatio of (t p ). The eergy coservative scheme (2) used i this work, is characterized by the tesor Π algo proposed by Gozalez [Go00]. After a fully discretizatio step (time ad space), we deduce the o liear systems defied by 1 t M( u p+1 u p ) + G algo (u p+1,u p ) q p+ 1 = 0 (3) 2 where M comes from the discretizatio of the iertia term 1 t Ω ρ( u p+1 u p ).v ad G algo is due to the discretizatio of the hyperelastic part Ω Π algo : v ad q p+ 1 comes from the discretizatio of the exteral forces 2 Ω f.v + gω g.v. The o liear system (3) ca be solved by a iterative liearizatio scheme idexed by i which leads to the solutio of liear systems: Ka i,p+1 u i,p+1 = 1 t M( u i,p+1 u p ) G algo (u i,p+1,u p ) + q p+ 1 (4) 2 with Ka i,p+1 = 2 t 2Ma + K i,p+1 ad u i,p+1 = u i+1,p+1 u i,p+1 where Ma = up+1 M represets the mass matrix ad Ka i,p+1 = up+1 G algo the hyperelastic taget matrix. We ca isist o the fact that the matrix Ka i,p+1 of system (4) is o symmetric; the o symmetry comes from the form of tesor Π algo (see [Go00]). The liear systems (4) ca be solved by a domai decompositio method [LeT94] which has to be adapted to the o symmetry but also to the presece of iertia terms. Before givig the adaptatios to o liear dyamic problems (sectios 2 ad 3), we preset briefly ow the pricipal features of the Balacig domai decompositio method [Ma93, LeT94]. We choose to adopt a primal Schur complemet method writte with displacemet variables. The basic idea i ooverlappig domai decompositio methods is to split the domai Ω of study ito N small ooverlappig subdomais Ω

3 A adapted coarse space for Balacig domai decompositio method 3 ad iterfaces Γ ( = 1, N). The Schur complemet method cosist the i reducig the global system to a iterface problem by a block Gaussia elimiatio of iteral degrees of freedom. The iterface problem ca take the followig variatioal form: ū V / < S i,p+1 ū, v >=< f i,p+1, v > v V = tr(v ) Γ, (5) where V is the discrete set defied from the space U 0 ad Γ = N =1 Γ. The matrices S i,p+1 = N =1 R S i,p+1 (R ) t deote the global Schur complemet matrices defied o Γ; (R ) t is the restrictio operator which goes from Γ to Γ. The local Schur complemet matrices S i,p+1 are defied o Γ by S i,p+1 = Ka i,p+1 (B i,p+1) t ( Ka i,p+1) 1 B i,p+1. (6) To do that, we( have cosidered the ) subdomai stiffess matrix formulated Ka by Ka i,p+1 = i,p+1 B i,p+1 (B i,p+1 )t Ka. The blocks Ka i,p+1 ad Ka i,p+1 correspod respectively to the iteral ad iterface degrees of freedom. The i,p+1 matrix B i,p+1 represets the cotributio coectig Γ to Ω. The iterface problem (5) ca be solved by a GMRES method (o symmetric cases) with the multilevel Neuma-Neuma precoditioer [LeT94, ABLV00]. This iterative techique eeds to form the matrix vector products S p ad M 1 r by solvig idepedet auxiliary Dirichlet ad Neuma problems o the local subdomais ad a global coarse problem defied o a space of sigular (rigid body) motios. The adaptatio of the Balacig Method to solve liear systems issued from o liear elastodyamic problems [Bar05] ca be realized by usig the theoritical frameworks of Schwarz additive decompositio method with itroductio of a adapted coarse space. 2 Schwarz additive formulatio: towards a defiitio of a adapted coarse space The two-level Neuma-Neuma precoditioer may be iterpreted as a stadard additive Schwarz algorithm [LeT94]. This method cosists i decomposig the iterface space V ito a coarse ad a fie compoet: V G ad V f. The coarse space V G = N =1 D Z (D is a give partitio of uity defied o the iterface, N =1 D R = Id V ) ca be defied by addig local lower eergy compoets ad the fie space V f is defied by duality as follows: V f = N =1 D V f where V f = { v f V, < SR v,r z >= 0, z Z }. (7) The key poit is the costructio of the local spaces Z of rigid motios. This costructio must, if that is ecessary, regularize local Neuma problems

4 4 Mikaël Barboteu but more especially set the costats or the lower eergy modes (like rigid body motios) to zero i the solutio of local Neuma problems. For more details o the presetatio of the Schwarz additive method, we ca refer to [LeT94, LMV98] for symmetric cases ad [ABLV00] for o symmetric cases. With fiite deformatios ad dyamic problems, some lower eergy modes caot be detected i the factorizatio step of the local taget matrices of Neuma problems [Bar05]. These drawbacks come from the fiite deformatios modelig but also from the regularizig cotributio of the mass matrix. Moreover, we also eed to improve the cotiuity betwee subdomais by takig ito accout specific modes (like corers modes...). So we have to itroduce a specific costructio of these lower eergy modes. I the followig, we preset i detail the costructio of the coarse space V G for o liear dyamic problems. The orthogoality relatio used i (7) characterizig the space V f permits to obtai iformatios i order to defie the local coarse space Z. Ideed the expasio of this orthogoality relatio by usig S i,p+1 = N =1 R S i,p+1 (R ) t ivolves oly terms issued from the subdomais eighbourig to the th ad oe term from the th itself, eigh() l=1 < v, (R ) t R l (S l i,p+1 )t (R l ) t R z > } {{ } (8i) + < v, (S i,p+1 )t z > }{{} (8ii) = 0 (8) The relatio (8) ca be verified by imposig the terms (8i) ad (8ii) to zero. Let us see ow what the use of these relatios imply: - use of the term (8ii): this term may be elimiated by imposig that the kerel of (S i,p+1 )t is icluded i the local coarse space Z, (Ker(S i,p+1 )t Z ). Such a choice leads to the same simplificatio tha those obtaied with the kerel of S for the more commo symmetric case [LeT94]. For o symmetric cases, this choice leads to the itroductio of Dual Rigid Modes (DRM) defied through the kerel of (S i,p+1 )t (see [ABLV00] for more precisios). I a pratical poit of view ad accordig to the form S i,p+1 give i (6), the dual rigid modes (oted by vgα ) defied o Ω ca be calculated through the local matrices (Ka i,p+1 )t by the solutio of this followig Neuma systems: v Gα V / (Ka i,p+1 )t v Gα = 0, α = 1, bdrm. (9) where bdrm represets the total umber of dual rigid modes of subdomais Ω. Oe ca easily prove that the modes v Gα Ker(Ka i,p+1 )t are coectig to the elemets z Ker(S i,p+1 )t by the relatio z = v Gα (where v Gα represets the cotributio of v Gα o Γ ). - cotributio of the term (8i): a simple maer to cacel the term (8i) is to fix all the terms of the sum to zero; the elemets z of Z could the be caracterized by the solutio of (R ) t R l (S l i,p+1 )t (R l ) t R z = 0. That makes it possible to esure the cotiuity of the coarse space elemets through the iterface Γ of Ω by cosiderig the cotributios relatig to corers, edges

5 A adapted coarse space for Balacig domai decompositio method 5 ad faces of the eighbourig subdomais Ω l. Ideed, the elemets z ca be foud respectively by these followig relatios : z l Z l / (S l i,p+1 )t z l = 0 l = 1, eigh() (10) z Z / (R l ) t R z = z l l = 1, eigh() (11) The use of the (8ii) ad (10) permits to calculte the dual rigid modes of the subdomais Ω ad his eighbours Ω l ; furthermore the relatio (11) represets the cotiuity costrait of dual modes through the iterface (corers, edges ad faces) betwee Ω ad Ω l. This last poit makes it possible to coect this approach with the Balacig Domai Decompositio Method with Costraits [MD03]; the eforcemet of these kid of costrait leads to expesive computatioal cost. A iexpesive way, ispired by [LMV98] ad [MD03] would be to impose oly the cotiuity o the corers of subdomais Ω. That ca be doe by the computatio of the bdcm Dual Corer Modes (DCM) of subdomais Ω by eforcig a same arbitrary Dirichlet boudary value o the corer iterface degrees of freedom for all cocered subdomais Ω ad Ω l (where bdcm is the total umber of DCM). I coclusio, the coarse space Z ca be geerated by cosiderig the bdrm dual rigid modes defied by solutios of the systems (12) ad particulary by the bdcm dual corer modes give by the systems (13) (see the ext sectio 3 for more details o the computatio of these modes). 3 Adaptatio of the 2-level Neuma-Neuma precoditioer Accordig to the defiitio of the coarse space itroduced i sectio 2, we propose a adapted costructio of the two level Neuma-Neuma precoditioer based o the followig steps: 1. Prelimiary step : We idetify the local iteral degrees of freedom {Prα; α = 1, bdrm } which cacel all bdrm rigid motios of subdomai Ω. This detectio ca be realized durig the factorizatio of the stiffess matrix (K e ) comig from the liear elastostatic system associated to the o liear elastodyamic problem (3). Notice that we ca also make arbitrary this detectio. 2. For each Newto iteratio i of each time step p a) We costruct the local regularized matrices Ka i,p+1 by usig the degrees of freedom {Prα} detected i step (1). These matrices ca be writte by usig the cotributios accordig to iteral ad iterface degrees of freedom; the oly the iteral cotributio Ka i,p+1 of the matrix Ka i,p+1 is regularized by usig the matrix Q α : Ka i,p+1 = Ka i,p+1 + Q α where ( Q α ) jk = { BV if j = k = Prα 0 else

6 6 Mikaël Barboteu where BV is a arbitrary big value (like for example). This regularizatio is ot ecessary for dyamic problems due to the cotributio of the iertia terms Ma which esures the matrices Ka t 2 i,p+1 2 to be o sigular. O the other had, we eed to costruct the regularized matrices Ka i,p+1 i order to impose the boudary coditios o the corers degrees of freedom oted by {Pc γ; γ = 1, bdcm }: Ka i,p+1 = Ka i,p+1 + Q γ where ( Q γ ) jk = { BV if j = k = Pc γ 0 else b) We compute the dual rigid modes {vgα ; α = 1, bdrm } by solvig the (regularized) local Neuma problems set o the space V of subdomai displacemets fuctios, ( Ka i,p+1 B i,p+1 (B i,p+1 )t Ka i,p+1 ) ) ( v Gα = v Gα ( e α0 ) ; α = 1, bdrm (12) where the j th compoet ( e α ) j of the vector e α is equal to the arbitrary big value BV if j = Prα ad to the value zero if ot. c) We compute the dual corer modes {vgγ ; γ = 1, bdcm } by solvig local Neuma problems i which the cotiuity of modes o corers ca be realized by eforcig the same arbitrary Dirichlet boudary value (1 for example) o the corers iterface degrees of freedom {Pc γ; γ = 1, bdcm } for all cocered subdomais Ω : ( ) Ka i,p+1 B ) i,p+1 ( v Gα )t Ka v i,p+1 Gα (B i,p+1 = ( 0 ē γ ) ; γ = 1, bdcm (13) where the j th compoet (ē γ) j of the vector ē γ is equal to the arbitrary big value BV if j = Pc γ ad to the value zero if ot. 3. We defie the local coarse space by: Z = vect ( { v Gα; α = 1, bdrm }, { v Gγ; γ = 1, bdcm } ). With this costructio of lower eergy modes, the two-level Neuma-Neuma precoditioer is classically defied for each time step p ad each Newto iteratio i by M 1 i,p+1 = P G + N (I P G )D i,p+1( S i,p+1) 1 (D i,p+1) t (I P G ) t, (14) =1 where ( S i,p+1 ) 1 is the regularized Schur iverse matrix ad P G deotes the orthogoal S-projectio of V o V G.

7 A adapted coarse space for Balacig domai decompositio method 7 4 A oliear dyamic problem: the catilever beam I this sectio, we illustrate umerically the previous adaptatios i the case of the solutio of a 2-dimesioal o liear elastodyamic problem. The applicatio relates to the dyamic evolutio of a catilever beam i plae displacemets. To do that, we cosider a elastic beam clamped o oe of hyperelastic beam g subdomais corers domai decompositio iterfaces Fig. 1. Deformed sequece of a catilever beam. Fig. 2. Substructuratio of the beam i 40 subdomais. its tips ad a exteral time idepedet loadig g o the opposite tip. The compressible material respose cosidered is govered by a Ogde costitutive law. The mesh ad its deformed cofiguratios durig the time are preseted i figure 1. From this umerical experimet, we aalyse the scalability of the iterface solver (GMRES) with some versios of the two-level Neuma-Neuma precoditioers. The cosidered precoditioers are : - the o symmetric Neuma-Neuma precoditioer give i [ABLV00] (curve ) without o liear dyamic adaptatios, - the o symmetric Neuma-Neuma precoditioer give i [Bar05] (curve ) preseted i sectio 3 but without dual corer modes (step (c)), - the improved o symmetric Neuma-Neuma precoditioer itroduced i sectio 3 (curve ) with all the features (steps (a), (b) ad (c)). The figure 3 gives the evolutio of average umber of GMRES iteratios (per Newto iteratios) for a beam decomposed i 2, 5, 10, 20, 40, 80 ad 160 subdomais (see figure 2 for a decompositio i 40 subdomais). We observe that the umber of iteratios obtaied with the 2-level Neuma-Neuma precoditioer without adaptatios (curve ) grows up with respect to the umber of subdomais. So the iterface solver with this precoditioer loses its classical scalability. We ca remark that the precoditioer (curve ) give i [Bar05] (without corers modes) permits to strogly decrease the solver iteratios but the depedece with respect to the umber of subdomais is already preset. O the other had, the improved Neuma-Neuma precoditioer

8 8 Mikaël Barboteu iteratios of iterface solver catilever beam umber of subdomais N.-N. precoditioer without adaptatios N.-N. precoditioer give i [Bar05] (without corer modes) improved N.-N. precoditioer (preseted i sectio 3) umber of subdomais N.-N. precoditioers Fig. 3. Numerical scalability with Neuma-Neuma (N.-N.) precoditioers. (curve ) leads to recover the umerical scalability properties i.e. the idepedece of the solver iteratios with respect to the umber of subdomais. Furthermore, the performaces of this precoditioer are pratically the same as the oes obtaied to solve liear elastostatic problems. Ideed, the average umber of iteratios is equal to 7 for a decompositio i 160 subdomais (see table i figure 3) ad if we cosider the associated liear elastostatic problem the umber of iteratios is equal to 6. I order to validate these performaces, we must test this precoditioer o other less academic simulatios. Refereces [ABLV00] Alart, P., Barboteu, M., Le Tallec, P., Vidrascu, M.: Additive Schwarz method for osymmetric problems: applicatio to frictioal multicotact problems. Domai decompositio methods i sciece ad egieerig (Lyo, 2000), Theory Eg. Appl. Comput. Methods, Iterat. Ceter Numer. Methods Eg. (CIMNE), Barceloa, 3-13, (2002) [Bar05] Barboteu, M.: Costructio du précoditioeur Neuma-Neuma de décompositio de domaie de iveau 2 pour les problèmes élastodyamiques e grades déformatios. C. R. Acad. Sci., 340 (I), (2005) [Go00] Gozalez, O.: Exact eergy ad mometum coservig algorithms for geeral models i o liear elasticity. Comput. Meth. Appl. Mech. Egrg., 190, (2000) [LeT94] Le Tallec, P.: Domai decompositio methods i computatioal mechaics. Comput. Mech. Adv., 1, (1994) [LMV98] Le Tallec, P., Madel, J., Vidrascu, M.: A Neuma-Neuma domai decompositio algorithm for solvig plate ad shell problems SIAM J. Numer. Aal., 35, (1998). [Ma93] Madel, J.: Balacig domai decompositio. Comm. Numer. Methods Egrg., 9, (1993) [MD03] Madel, J., Dohrma, C. R.: Covergece of a Balacig Domai Decompositio by Costraits ad Eergy Miimizatio, Numer. Li. Alg. Appl., 10, (2003)