Approximation of Optimal Interface Boundary Conditions for Two-Lagrange Multiplier FETI Method

Transcription

1 Aroxmaton of Otmal Interface Boundary Condtons for Two-Lagrange Multler FETI Method F.-X. Roux, F. Magoulès, L. Seres, Y. Boubendr ONERA, 29 av. de la Dvson Leclerc, BP72, Châtllon, France, Unv. Henr Poncaré, BP239, Vandoeuvre-les-Nancy, France, Summary. Interface boundary condtons are the key ngredent to desgn effcent doman decomoston methods. However, convergence cannot be obtaned for any method n a number of teratons less than the number of subdomans mnus one n the case of a one-way slttng. Ths otmal convergence can be obtaned wth generalzed Robn tye boundary condtons assocated wth an oerator equal to the Schur comlement of the outer doman. Snce the Schur comlement s too exensve to comute exactly, a new aroach based on the comutaton of the exact Schur comlement for a small atch around each nterface node s resented for the two-lagrange multler FETI method. 1 Introducton Interface boundary condtons are the key ngredent to desgn effcent doman decomoston methods, see Chevaler and Nataf 1998], Benamou and Desrés 1997], Gander et al. 2002]. However, convergence cannot be obtaned for any method n a number of teratons less than the number of subdomans mnus one n the case of a one-way slttng. For the two-lagrange multler FETI method, ths otmal convergence can be obtaned wth generalzed Robn tye boundary condtons assocated wth an oerator equal to the Schur comlement of the outer doman, see Roux et al. 2002]. In ractce ths otmal condton cannot be mlemented snce the Schur comlement s too exensve to comute exactly. Furthermore, the Schur comlement s a dense matrx on each nterface and even f t were comuted, usng t would create a very large ncrease of the bandwdth of the local subroblem matrx. Hence the ssue s how to buld a sarse aroxmaton of the Schur comlement that s not exensve to comute and that leads to good convergence roertes of the two-lagrange multler FETI teratve method. Dfferent aroaches based on aroxmate factorzaton or nverse comutaton of the subroblem matrx have been tested, see Roux et al. 2002]. Here,

2 284 F.-X. Roux, F. Magoulès, L. Seres, Y. Boubendr a new aroach based on the comutaton of the exact Schur comlement for a small atch around each nterface node aears to be a very effcent method for desgnng aroxmatons of the comlete Schur comlement. Furthermore ths aroach can be easly mlemented wthout any other nformaton than the local matrx n each subdoman. 2 Revew of the Two-Lagrange Multler FETI Method 2.1 Introducton of Two-Lagrange Multler on the Interface Consder a slttng of the doman Ω as n Fgure 1 and note by subscrts and the degrees of freedom located nsde subdoman Ω (s), s = 1, 2, and on the nterface Γ. Then, the contrbuton of subdoman Ω (s), s = 1, 2 to the matrx and the rght-hand sde of a fnte element dscretzaton of a lnear artal dfferental equaton on Ω can be wrtten as follows: K (s) = K (s) K (s) K (s) K (s) ] ], b (s) b (s) = b (s) where K (1) and K (2) reresent the nteracton matrces between the nodes on the nterface obtaned by ntegraton on Ω (1) and on Ω (2). The global roblem s a block system obtaned by assemblng local contrbuton of each subdoman: 0 0 K (2) K (2) K (2) K x (1) x (2) x = b (1) b (2) b. (1) The block K s the sum of the two blocks K (1) and K (2). In the same way, b = b (1) + b (2) s obtaned by local ntegraton n each subdoman and sum on the nterface. Fg. 1. Non-overlang doman slttng. The two-lagrange multler FETI method, see Farhat et al. 2000], s an teratve based doman decomoston method whch conssts to determne the soluton of the followng couled roblem:

3 Aroxmaton of Otmal Interface Boundary Condtons 285 ] ] ] K (2) K (2) + A (1) ] K (2) K (2) + A (2) x (1) x (1) x (2) x (2) λ (1) + λ (2) (A (1) + A (2) )x (1) = 0 λ (1) + λ (2) (A (1) + A (2) )x (2) = 0 b (1) = b (1) + λ (1) ] ] b (2) = b (2) + λ (2) where the free matrces A (1) and A (2) are to be determned for the best erformance of the algorthm. It s clear that ths couled roblem s equvalent to the global roblem (1), see Roux et al. 2002]. The elmnaton of x (s) n favor of x (s) n the two frst equatons and substtuton n the two last equatons leads to the followng lnear system uon the varable λ := (λ (1), λ (2) ) T : Fλ = d (2) wth F and d the matrx and rght hand sde defned as: ] I I (A F := (1) + A (2) )S (2) + A (2) ] 1 I (A (1) + A (2) )S (1) + A (1) ] 1 I ] (A d := (1) + A (2) )S (2) + A (2) ] 1 c (2) (A (1) + A (2) )S (1) + A (1) ] 1 c (1) The teratve soluton of ths system s usually erforms wth a Krylov method. 2.2 Otmal Interface Boundary Condtons It s shown n Roux et al. 2002] that the best choce for the free matrces A (s), s = 1, 2 corresonds to the comlete outer Schur comlement,.e. the dscretzaton of the otmal contnuous boundary condtons assocated to the Steklov-Poncaré oerator, see Ghanem 1997], Collno et al. 2000] and Boubendr 2002]. An extenson of ths result n the case of a one way slttng can be obtaned n the dscrete case, see Roux et al. 2002], and n the contnuous case, see Nataf et al. 1994]. Theorem 1. In a case of a two-doman slttng, the Jacob teratve algorthm for the two-lagrange multler FETI method wth augmented term equal to the comlete outer Schur comlement converges n one teraton at most. Theorem 2. In a case of a one way slttng, the Jacob teratve algorthm for the two-lagrange multler FETI method wth augmented term equal to the comlete outer Schur comlement converges n a number of teraton equal to the number of subdoman mnus one.

4 286 F.-X. Roux, F. Magoulès, L. Seres, Y. Boubendr 3 Aroxmaton of Otmal Interface Boundary Condtons In the revous secton, we have recalled that the best choce for the augmented matrx n the case of a one way slttng doman decomoston s the comlete outer Schur comlement matrx. Ths choce can not be done n ractce snce the comutatonal cost of the comlete outer Schur comlement matrx s too exensve. 3.1 Neghbor Schur Comlement From a hyscal ont of vew, the comlete outer Schur comlement matrx reresent the nteractons of all the degree of freedom of the subdomans condensed on the nterfaces. The restrcton of the nteractons only wth the neghborng subdomans, leads to aroxmate the comlete outer Schur comlement wth the neghbor Schur comlement. The comutatonal cost and the exchange of data are thus reduced to the neghborng subdomans only. Unfortunately, ths aroach stll leads to an exensve comutatonal cost. Hence the ssue s how to buld a sarse aroxmaton of the Schur comlement that s not exensve to comute and that gves good convergence for the two-lagrange multler FETI method. 3.2 Lumed Aroxmaton We have shown n Roux et al. 2002] that an aroxmaton of the neghbor Schur comlement matrx K (s) bb K (s) b K(s) ] 1 K (s) b wth ts frst term,.e. wth the matrx K (s) bb gves good results. Such an aroxmaton, resents the advantage to be very easy to mlement snce ths matrx s comuted by the neghborng subdoman durng the assembly rocedure and the ntegraton of the contrbuton of the nterface nodes. Only an exchange wth the neghborng subdoman s requred for ths regularzaton rocedure. 3.3 Sarse Aroxmaton based on Overlang Layers In ths secton we resent a new aroach for the aroxmaton of the neghborng Schur comlement wth a sarse matrx, whch leads to a better aroxmaton than the lumed aroxmaton, as shown n the numercal results. The goal s to obtaned a sectral densty of the aroxmated matrx close to the sectral densty of the neghbor Schur comlement matrx. We frst defne the followng subsets of ndexes: V Ω (2) = {ndexes of nodes nsde the subdoman Ω (2) } V Γ = {ndexes of nodes on the nterface Γ } V l = {ndexes of the nodes j such that the mnmum connectvty dstance between and j s lower or equal than l, l N} V l Γ, = V Γ V l

5 Aroxmaton of Otmal Interface Boundary Condtons 287 The sarse aroxmaton nvestgated here consst to defne a sarse augmented matrx obtaned through an extracton of some coeffcents and local condensaton along the nterface. The comlete algorthm to comute the augmented matrx n the case of a two doman slttng n subdoman Ω (1) can be defne as: Algorthm 1. sarse aroxmaton] 1. constructon of the structure of the nterface matrx A 1 R dmvγ dmvγ. 2. constructon of the sarse structure of the subdoman matrx K (2) R dmv Ω (2) dmv Ω (2). 3. assembly of the matrx K (2). 4. for all n V Γ do 4.1. extracton of the coeffcents K mn, (m, n) V l V l, and constructon of the sarse matrx A 2 R dmv l dmv l wth these coeffcents comutaton of the dense matrx A 3 R dmv Γ, 1 dmv Γ, 1 by condensaton of the matrx A 2 on the degree of freedom VΓ, extracton of the coeffcents of the lne assocated wth the node from the matrx A 3 and nserton nsde the matrx A 1 at the lne assocated wth the node. 5. constructon of the symmetrc matrx A 4 = (AT 1 +A1) regularzaton of the matrx wth the matrx A 4. where l denotes the number of layers consdered. Smlar calculaton erformed n the subdoman Ω (2) gves the augmented matrx A (2) to add to the subdoman matrx K (2). As an examle the regular mesh wth Q 1 -fnte elements resented Fgure 2 leads to the subsets of Fg. 2. Numberng of the nodes n subdoman Ω (2). ndexes V7 1 = {1, 2, 7, 8, 13, 14}, V7 2 = {1, 2, 3, 7, 8, 9, 13, 14, 15, 19, 20, 21} and VΓ,7 1 = {1, 7, 13}. These subsets corresond to the overlang layers reresented Fgure 3. 4 Numercal Results 4.1 The Model Problem In ths secton, a two dmensonal beam of length L 1 and hgh L 2 submtted to flexon s analyzed. The Posson rato and the Young modulus are resectvely ν = 0.3 and E = N/m 2. Homogeneous Drchlet boundary condtons

6 288 F.-X. Roux, F. Magoulès, L. Seres, Y. Boubendr Fg. 3. On the left one nterface node, on the mddle one nterface node wth one layer, and on the rght one nterface node wth two layers. are mosed on the left and homogeneous Neumann boundary condtons are set on the to and on the bottom. Loadng, model as non homogeneous Neumann boundary condton are mosed on the rght of the structure. The beam s meshed wth trangular elements and dscretzed wth P 1 fnte elements. The doman s then slt nto two or ten subdomans n a one way slttng and the condensed nterface roblem s solved teratvely wth the orthodr Krylov method. The stong crteron s set to r n 2 < 10 6 r 0 2, where r n and r 0 are the nth and ntal global resduals. 4.2 Sectral Analyss Fgure 4 reresent the sectral densty of the egenvalues of the matrx of the condensed nterface roblem (2) for dfferent augmented matrces. An augmented matrx equal to the neghbor Schur comlement wll leads to egenvalues equal to one whch corresond to a sectral densty equal to a Drac functon. 80 Lumed arox. of Schur cmt 180 Sarse arox. of Schur cmt number of egenvalues number of egenvalues egenvalue egenvalue Fg. 4. Sectral densty of the condensed nterface roblem wth an augmented matrx ssue from the lumed aroxmaton (left) vs from the sarse aroxmaton (rght) (L 1 = 10, L 2 = 1, h = 1/160). Case of two subdomans. We can see on Fgures 4 that a sarse aroxmaton erformed wth a number of layers equal to four leads to a sectral densty close to a Drac functon. Ooste, a lumed aroxmaton leads to sectrum much more dfferent. Ths result can be exlan by the fact that the sarse aroxmaton s based on local condensaton.e. on local Steklov-Poncaré oerators whch s not the case of the lumed aroxmaton.

7 4.3 Asymtotc Analyss Aroxmaton of Otmal Interface Boundary Condtons 289 The asymtotc analyss of the roosed methods uon dfferent arameters s now analyzed. The analyss uon the doman sze reorted Fgure 5 show the resectve deendence of the methods. The asymtotc behavor of the ro Neghbor Schur cmt Sarse arox. of Schur cmt Lumed arox. of Schur cmt number of teratons Fg. 5. Asymtotc behavor for dfferent augmented matrces and dfferent subdoman sze. (L 1 = 64, L 2 = 1, h = 1/20). osed methods uon the mesh sze s resented Fgure 6. On the left cture, four layers are consdered for the sarse aroxmaton. A lnear deendence uon the mesh sze can be notced for all the methods. On the rght cture, the number of layers of the sarse aroxmaton ncrease roortonally wth the mesh sze. A lnear deendence stll occurs for the sarse aroxmaton but the sloe of the curve s lower than wth a constant number of layers equal to four. The asymtotc results obtaned wth ths last aroxmaton H 10^3 Neghbor Schur cmt Sarse arox. of Schur cmt Lumed arox. of Schur cmt 10^3 Neghbor Schur cmt Sarse arox. of Schur cmt Lumed arox. of Schur cmt number of teratons 10^2 number of teratons 10^2 10^1 10^-3 10^-2 10^-1 10^0 10^1 10^-3 10^-2 10^-1 10^0 h h Fg. 6. Asymtotc behavor for dfferent augmented matrces and dfferent mesh sze on the left for a constant number of layers, and on the rght for a number of layers ncreasng roortonally wth the mesh sze. (L 1 = 10, L 2 = 1). Case of ten subdomans. are stll less effcent than those obtaned wth a contnuous aroach, see Gander et al. 2002], but the mlementaton of the revous method doesn t deends on a ror knowledge of the roblem to be solved (coeffcents of the

8 290 F.-X. Roux, F. Magoulès, L. Seres, Y. Boubendr artal dfferental equaton, mesh sze,... ) and thus hels ts use as a black box routne! 5 Conclusons In ths aer the rncle of the two-lagrange multler FETI method wth otmal nterface boundary condtons has been reman. A new method for the aroxmaton of these otmal condtons has been ntroduced. Ths new method s based on the comutaton of the exact Schur comlement for a small atch around each nterface node. Ths method aears to be a very effcent method for desgnng aroxmatons of the comlete Schur comlement that gve robust teratve algorthms for solvng many dfferent knds of roblems. References J.-D. Benamou and B. Desrés. A doman decomoston method for the Helmholtz equaton and related otmal control roblems. J. of Com. Physcs, 136:68 82, Y. Boubendr. Technques de décomoston de domane et méthode d équatons ntégrales. PhD thess, INSA de Toulouse, Jun P. Chevaler and F. Nataf. Symmetrzed method wth otmzed second-order condtons for the Helmholtz equaton. Contem. Math., 218: , F. Collno, S. Ghanem, and P. Joly. Doman decomoston method for harmonc wave roagaton: a general resentaton. Comuter methods n aled mechancs and engneerng, 184: , C. Farhat, A. Macedo, M. Lesonne, F.-X. Roux, F. Magoulès, and A. de La Bourdonnaye. Two-level doman decomoston methods wth lagrange multlers for the fast teratve soluton of acoustc scatterng roblems. Comut. Methods n Al. Mech. and Engrg., 184(2): , M. Gander, F. Magoulès, and F. Nataf. Otmzed schwarz method wthout overla for the Helmholtz equaton. SIAM J. Sc. Comut., 24(1):38 60, S. Ghanem. A doman decomoston method for Helmholtz scatterng roblems. In P. E. Bjørstad, M. Esedal, and D. Keyes, edtors, Nnth Int. Conf. on Doman Decomoston Methods, ages ddm.org, F. Nataf, F. Roger, and E. de Sturler. Otmal nterface condtons for doman decomoston methods. Techncal Reort 301, CMAP (Ecole Polytechnque), F.-X. Roux, F. Magoulès, S. Salmon, and L. Seres. Otmzaton of nterface oerator based on algebrac aroach. In Doman Decomoston Methods n Sc. Engrg., ages , 2002.