An Iterative Substructuring Method for Coupled Fluid-Solid Acoustic Problems 1

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1 An Iterative Substructuring Method or Coupled Fluid-Solid Acoustic Problems 1 Jan Mandel Department o Mathematics and Center or Computational Mathematics, University o Colorado at Denver, Denver, CO , U.S.A., and Department o Aerospace Engineering Science and Center or Aerospace Structures, University o Colorado at Boulder, Boulder, CO , U.S.A November 13, 2 A ast parallel iterative method is proposed or the solution o linear systems arizing rom Finite Element discretization o the time harmonic acoustics o coupled luid-solid systems in luid pressure and solid displacement ormulation. The method generalizes the FETI-H method or the Helmholtz equation to elastic scattering. The luid and the solid domains are decomposed into non-overlapping subdomains. Continuity o the solution enorced by Lagrange multipliers. The system is augmented by duplicating the degrees o reedom on the wet interace. The original degrees o reedom are then eliminated and the resulting system is solved by the GCR method preconditioned by a subspace correction. In each iteration, the method requires the solution o one independent acoustic problem per subdomain, and the solution o a coarse problem with several degrees o reedom per subdomain. Computational results show that the method is scalable with the problem size, requency, and the number o subdomains. The number o iterations was mostly about same as the number o iterations o the FETI-H method or the related Helmholtz problem with Neumann boundary condition instead o elastic scatterer. Convergence is explained rom the spectrum o the iteration operator. Key Words: Lagrange multipliers, domain decomposition, iterative substructuring, elastic scattering, Helmholtz equation, coupled luid-solid acoustics, FETI 1. INTRODUCTION The method o Finite Element Tearing and Interconnecting (FETI) was proposed by Farhat and Roux [11] or solving linearized elasticity problems. The FETI 1 This work has been submitted to Academic Press or possible publication. Copyright may be transerred without notice, ater which this version may no longer be accessible. 1

2 2 JAN MANDEL method consists o decomposing the domain into non-overlapping subdomains. Lagrange multipliers are used to enorce that the values o the degrees o reedom coincide on the interaces between the subdomains interaces. All original degrees o reedom are then eliminated, leaving a dual system or the Lagrange multipliers, which is solved by preconditioned conjugate gradients. For details, urther developments, and theoretical analysis o the FETI method or positive deinite problems, see [1, 4, 5, 9, 1, 13, 14, 16, 17, 18, 2, 21] and reerences therein. Variants o the FETI method or the Helmholtz equation o scattering, were proposed by De La Bourdonnaye et al. [3], Farhat et al. [6, 7, 8], and urther developed by Tezaur et al. [22]. In this paper, we present an extension o the method o [7, 8], called FETI-H, to the case o an elastic scatterer. The paper is organized as ollows. In Section 2, we state the boundary value problem o coupled luid-solid acoustics. Section 3 reviews the variational ormulation and Finite Element discretization. The discrete equations are put in a substructured orm in Section 4, which leads to the iterative method, presented in Section 5. Section 6 contains computational results. An explanation o some o the convergence properties o the method rom the spectrum o the iteration operator is given in Section 7. Section 8 is the conclusion. 2. THE SCATTERING PROBLEM We consider an acoustic scattering problem with an elastic scatterer completely immersed in a luid. Let Ω and Ω e be bounded domains in R n, = 2, 3, Ω e Ω, and let Ω = Ω \ Ω e. Let ν denote the exterior normal. Let Ω be decomposed disjoint subsets, Ω = Γ d Γ n Γ a. The domain Ω is illed with a luid. The acoustic pressure at time t is assumed to be o the orm Re pe iωt, where p is complex amplitude independent o t. The amplitude p is governed by the Helmholtz equation with the boundary conditions p + k 2 p = in Ω, (1) p = p on Γ d, p ν = on Γ n, p ν + ikp = on Γ a, (2) where k = ω/c is the wave number and c is the speed o sound in the luid. The boundary conditions (2) model excitation, sound hard boundary, and outgoing boundary, respectively. The amplitude o the displacement u o the elastic body occupying the domain Ω e satisies the elastodynamic equation τ + ω 2 ρ e u = in Ω e, (3) where τ is the stress tensor and ρ e is the density o the solid. For simplicity, we consider an isotropic homogeneous material with τ = λi( u) + 2µe(u), e ij (u) = 1 2 ( u i x j + u j x i ), (4) where λ and µ are the Lamé coeicients o the solid.

3 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 3 Let Γ = Ω e be the wet interace. On Γ, the luid pressure and the solid displacement satisy ν u = 1 p ρ ω 2, ν τ ν = p, ν τ ν =, (5) ν where ρ is the luid density. The interace conditions (5) model the continuity o normal displacement, the balance o normal orces, and zero tangential tension, respectively, c., e.g., [23]. 3. VARIATIONAL FORM AND DISCRETIZATION We use the ollowing standard variational orm and discretization by conorming elements, c., e.g., [19]. Deine the spaces V = {q H 1 (Ω ) q = on Γ d }, V e = (H 1 (Ω e )) n, where H 1 is the Sobolev space o generalized unctions with square integrable generalized irst derivatives. Assuming that p on Γ d is extended to a unction in H 1 (Ω ), multiplying equation (1) by a test unction q V, equation (3) by a test unction u V e, and integrating by parts, we obtain the ollowing variational orm o (1) (5): Find p such that p p V, and u V e such that or all q V and all v V e, Ω e p q + k 2 Ω Ω ( λ( u)( v) + 2µe(u) : e(v) ) + ω 2 pq ik pq ω 2 Γ a Γ ρ e u v Ω e ρ (ν u)q =, Γ p(ν v) =. Proceeding as usual in the Finite Element method [2], we replace V and V e with conorming inite element spaces and obtain the algebraic system [ K + k 2 M ikg ρ ω 2 T T K e + ω 2 M e ] [ ] [ ] p r =. (6) u In the coupled system (6), p and u are the vectors o the (values o) degrees o reedom o p and u, i.e., p and u are the inite element interpolations o p and u, respectively. The matrix blocks in (6) are deined by p K q = p q, p M q = pq, Ω Ω p G q = pq, u ( ) K e v = λ( u)( v) + 2µe(u) : e(v), Γ a Ω e u M e v = ρ e (u v), p Tv = p(ν v). Ω e Γ

4 4 JAN MANDEL 4. SUBSTRUCTURED DISCRETE EQUATIONS The luid and solid domains are decomposed into nonoverlapping subdomains that consist o union o elements, Ω = N s=1 Ω s e, Ω e = N e s=1 Ω s e. (7) The vectors o degrees o reedom corresponding to Ω s and Ωs e are denoted by p s and u s, respectively. The subdomain matrices are deined by subassembly, p s K s q = p q, p s M s q s = pq, Ω s p s G s q s = pq, u s K s ev s = λ( u)( v) + 2µe(u) : e(v), Ω s Γa Ω s e u s M s ev = ρ e (u v), p r T rs v s = p(ν v). Ω s Ω s e Ω r Ωs e We will use vectors consisting o all subdomain degrees o reedom, ˆp = p 1. p N, û = and the corresponding partitioned matrices, K 1... ˆK = diag(k s ) = K N u 1. u Ne,, ˆK e = diag(k s e) = The matrices ˆM, Ĝ, and ˆM e deined similarly, and ˆT = (T rs ) rs = T T 1,Ne T N,N e T N,1 K 1 e K Ne e Let N and N e be the matrices with, 1 entries o the global to local maps corresponding to the decompositions o Ω and Ω e, respectively, c., (7), so that Let B = (B 1,..., BN K = N ˆK N, K e = N e ˆK e N e.. ) and B e = (B 1 e,..., B Ne e ) be matrices with entries, ±1 such that the conditions B ˆp = and B e û = express the constraint that the values o the same degrees o reedom on two dierent subdomains coincide, that

5 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 5 is, B ˆp = ˆp = N p or some p (8) B e û = û = N e u or some u. (9) Multiplying the second equation in (6) by ω 2 ρ to symmetrize the o-diagonal block and introducing Lagrange multipliers λ and λ e or the constraints B p = and B e u =, we get the system o linear equations in block orm, ˆK + k 2 ˆM ikĝ ω 2 ρ ˆT B ω 2 ρ ˆT ω 2 ρ ( ˆK e + ω 2 ˆM e ) B e B B e ˆp û λ λ e = where N ˆr = r. The system (1) is equivalent to (6) in the ollowing sense. ˆr, (1) Lemma 4.1. I (ˆp, û, λ, λ e ) is a solution o (1), then ˆp = N p and û = N e u, where (p, u) solves (6). Conversely, i (p, u) is a solution o (6), then there exist λ and λ e such that (ˆp, û, λ, λ e ) is a solution o (1). Proo. To simpliy notation, write (6) as N ŜNx = N ˆb and (1) as Ŝˆx + B λ = ˆb, Bˆx =. Then (8), (9) become Null B = Range N. Suppose that Ŝˆx + B λ = ˆb and Bˆx =. From the second equation and Null B = Range N, it ollows that there is x such that ˆx = Nx. Multiplying the irst equation by N and using the act that N B = (BN) =, we obtain N ŜNx = N ˆb. Conversely, suppose that N ŜNx = N ˆb. Let ˆx = Nx. Then Bˆx =, and Ŝˆx ˆb Null N = (Range N) = (Null B) = Range B, hence Ŝx ˆb = B λ or some λ. Note that symmetry o the system or the existence o a Lagrangean unctional are not needed. We will want to eventually eliminate the variables ˆp and û. Because the matrices ˆK +k 2 ˆM and ˆK e +ω 2 ˆM e are typically close to singular due to near-resonance, we replace them by regularized matrices  = ˆK + k 2 ˆM + ikĝ + ˆR,  e = ˆK e + ω 2 ˆM e + ˆR e, where the regularization matrices are given by N ˆR = diag(r s ), p s R s q s = ik t=1 t s N e ˆR e = diag(r s e), u s R s ev s = iωρ e σ st t=1 t s Ω s Ωt σ st e Ω s e Ωt e pq, (n u)(n v),

6 6 JAN MANDEL where σ st {, ±1}, σst = σts, and σst e It is shown in [6] that i or a given s, all σ st, then Âs σ st on strategies or choosing σ st simply choose σ st Because = σe ts. or all σ st with some is invertible. The case o solid subdomains is similar. For details = {, ±1}, σ st e to guarantee this, see [6]. In our computations, we = +1 i s > t, σst = 1 i s < t, and similarly or σst the system (1) is equivalent to N R N =, N er e N e =,  ω 2 ρ ˆT B ω 2 ρ ˆT ω 2 ρ  e B e B B e ˆp û λ λ e = Since the value o ˆTû depends on the values o û on Γ only, we have ˆr e. (11) ˆTû = ˆTJ e û Γ, û Γ = J eû, where Ĵ e is the matrix o the operator o embedding a subvector that corresponds to degrees o reedom on Γ into û by adding zero entries. Similarly, ˆT ˆp = ˆT J ˆp Γ, ˆp Γ = J ˆp. Thereore, we obtain the augmented system equivalent to (11),  B ω 2 ρ ˆTJ e ω 2 ρ  e B e ω 2 ρ ˆT J B B e J I J e I ˆp û λ λ e ˆp Γ û Γ = ˆr (12) Because the variables in a coupled system typically have vastly dierent scales, we use symmetric diagonal scaling to get the scaled system à B TJ e à e B e T J B Be J I J e I where the matrices and the vectors scale as p ũ λ λ e p Γ ũ Γ = r, (13) à = D  D, à e = ω 2 ρ D e  e D e, T = ω 2 ρ D ˆTD e, (14) B = E B D, Be = E e B e D e, r = D ˆr, (15) ˆp = D p, û = D e ũ, λ = D λ, λ e = D e λe. (16)

7 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 7 The scaling matrices D, D e, E, and E e, are diagonal. We have chosen scaling matrices with positive diagonal entries such that the absolute values o the diagonal entries o à and Ãe are one and the l 2 norms o the columns o B e and B are one. Other, perhaps better, scaling choices are o course possible. The proposed method consists o eliminating the variables ˆp and û rom the augmented system (13) and solving the resulting reduced system by Generalized Conjugate Residuals, with preconditioning by projection on a coarse space with ew degrees o reedom per subdomain. Computing p and ũ rom the irst two equations in (13) gives p = ũ = à 1 ( r B λ + TJ e ũ Γ ) (17) à 1 e ( B e λ e + T J p Γ ) (18) Substituting p and ũ rom (17), (18) into the rest o the equations in (13), we obtain the reduced system where and F = Fx = b, (19) B à 1 B B à 1 TJ e Be à 1 e B e B e à 1 e T J B I J à 1 TJ e, (2) J e à 1 e B e J e à 1 e T J I J à 1 x = λ λ e p Γ ũ Γ, b = B à 1 r J à 1 r. In equation (2), the irst diagonal block B à 1 B is exactly same as in the FETI-H method or the Helmholtz equation. The second diagonal block B à 1 B is the analogue o FETI-H or the elastodynamic problem. FETI-H is known to converge ast, so one can expect that on the subspace deined by the coarse correction (c., Section 5 below), these two diagonal blocks will be well conditioned. The o-diagonal blocks all contain the inverse o A, resp. A e, which are discretizations o dierential operators. Hence, the o-diagonal blocks are discretizations o compact operators, and the structure o the equation (19) resembles discretization o a Fredholm integral equation o the second kind, which is well conditioned because its eigenvalues cluster about a point dierent rom zero. Such clustering is indeed observed computationally, c., Section 7 below. Evaluating the matrix vector product Fx requires the solution o one independent problem per subdomain, because F λ λ e p Γ ũ Γ = B q B e ṽ J q p Γ J eṽ ũ Γ, where { q = à 1 ( B λ + TJ e ũ Γ ), ṽ = à 1 e ( B e λ e + T J p Γ ).

8 8 JAN MANDEL 5. ITERATIVE SOLUTION The present method consists o solving the linear system (19) by the Generalized Conjugate Residual (GCR) method [12] preconditioned by a subspace correction. GCR is mathematically equivalent to GMRES and easier to program, but less numerically stable. Because the number o iterations was small, GCR was suicient or our purposes. Let Q be a matrix with the same number o rows as F and linearly independent columns. The columns o Q orm the basis o the coarse space. We wish to enorce the condition that the residual is orthogonal to the coarse space, Q (Fx b) =, (21) by adding a correction rom the coarse space in each iteration. Deine the correction operator C by C(v, b) = v + Qw, Q (F(v + Qw) b) =. We irst make make sure that the initial approximation satisies (21): I v is given, the initial approximation is x () = C(v, b). The system (19) is then solved by the GCR method with let preconditioning by the action o the projection P, given by Pv = C(v, ) = (I Q(Q FQ) 1 Q F)v. Equivalently, the GCR method is applied to the preconditioned system PFx = Pb. (22) In step m, the GCR method computes x (m) by adding to x (m 1) a linear combination o preconditioned residuals P(Fx (k) b), k =,..., m 1, so that the l 2 norm o the residual P(Fx (m) b) is minimal. Because Q FP =, it ollows that the iterates run in a subspace, hence, rom the selection o x (), We choose the matrix Q o the orm Q = Q F(x (m) x () ) =, Q (Fx (m) b) =. D B diag(y s) s D e B e diag(ye) s s D J diag(zs ) s D e J ediag(z s e) s For a luid subdomain Ω s, we choose Ys as the matrix o columns that are discrete representations o plane waves in a small number o equally distributed directions, or discrete representation o the constant unction. For a solid subdomain Ω s e, the.

9 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 9 FIG. 1. Model 2D Problem Γ n Γ d Γ Ω e Γ a Ω Γ n columns o Y s e are discrete representations o plane pressure and shear waves, or o rigid body motions. The matrices Z s and Zs e are chosen in the same way as Y s and Y s e, with possibly dierent selection o the number o directions and selection o constant or rigid body modes. Some o the matrices Y s, Ys e, Z s e, or Y s e may be void. 6. COMPUTATIONAL RESULTS We consider a model 2D problem with a scatterer in the center a waveguide, c., Fig. 1. The luid domain Ω is a square with side 1 m, illed with water with density ρ = 1 kg m 3 and speed o sound c = 15 m s 1. The scatterer is a square in the center o the luid domain, with side.2 m, and consisting o aluminum with dnsity ρ e = 27 kg m 3 and Lamé elasticity coeicients λ = N m 2, µ = N m 2. For numerical stability, the rows o the matrices B e, B, and the columns o Q were orthogonalized. The number o reduced variables reported in the tables is the number o the Lagrange multipliers to enorce the linearly independent constraints. The number o coarse degrees o reedom is the size o the basis o the columns o the matrix Q. The luid domain and the solid domain are discretized by bilinear square elements on a uniorm mesh with meshsize h. Both domains are divided independently into m by n subdomains by dividing their horizontal sides into m intervals o the same length and the vertical sides into n intervals o the same length. The iterations were stopped when the relative residual in the l 2 norm o the reduced and scaled system (19) was less than 1 6. The numerical results or the model problem are summarized in Tables 1 to 3. The lines in the tables are organized in groups. The irst line in the group is the result or the FETI-H method with Neumann condition instead o the elastic scatterer. The ollowing lines are or the elastic scatterer, and they show the number o iterations or increasing the number o coarse space unctions, that is, the columns o Q. In all cases, the constant unction is included in the coarse space or the luid and the rigid body modes are included in the coarse space or the solid, both or

10 1 JAN MANDEL TABLE 1 Decreasing mesh size and wavelength, 1 solid subdomain Problem description Coarse directions Number o Subdomains Multipliers Wet inter. degrees o reedom h k Fluid Solid Fl. Sol. Fl. Sol. Orig. Red. Coarse Iter. 1/4 1 4x4 rigid /4 1 4x4 1x /4 1 4x4 1x /8 2 4x4 rigid /8 2 4x4 1x /8 2 4x4 1x /16 4 4x4 rigid /16 4 4x4 1x /16 4 4x4 1x /16 4 4x4 rigid /16 4 4x4 1x /16 4 4x4 1x TABLE 2 Increasing number o solid subdomains Problem description Coarse directions Number o Subdomains Multipliers Wet inter. degrees o reedom h k Fluid Solid Fl. Sol. Fl. Sol. Orig. Red. Coarse Iter. 1/25 2 4x4 rigid /25 2 4x4 1x /25 2 4x4 1x /25 2 4x4 rigid /25 2 4x4 2x /25 2 4x4 2x /25 2 4x4 rigid /25 2 4x4 3x /25 2 4x4 3x /25 2 4x4 rigid /25 2 4x4 1x /25 2 4x4 1x /25 2 4x4 rigid /25 2 4x4 2x /25 2 4x4 2x /25 2 4x4 rigid /25 2 4x4 3x /25 2 4x4 3x

11 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 11 TABLE 3 Increasing number o luid and solid subdomains Problem description Coarse directions Number o Subdomains Multipliers Wet inter. degrees o reedom h k Fluid Solid Fl. Sol. Fl. Sol. Orig. Red. Coarse Iter. 1/ x3 rigid / x3 1x / x3 1x / x5 rigid / x5 3x / x5 3x / x3 rigid / x3 1x / x3 1x / x5 rigid / x5 3x / x5 3x the multipliers and or the wet interace. The prototype implementation was done in MATLAB, hence we do not report timings. Table 1 shows that the method is scalable when the requency is increased and the mesh is correspondingly reined. Since there is only one solid subdomain, the eect o increasing the coarse space or the solid and or the wet interace was minor. The results in Table 2 show that the number o iterations does not increase when the scattered is decomposed into increasing number o subdomains. Finally, Table 3 shows the number o iterations does not grow when a ixed problem is decomposed into a growing number o both luid and solid subdomains. 7. SPECTRUM OF THE ITERATION OPERATOR Convergence o the method can be explained by an analysis o the spectrum o the iteration operator. From well-known properties o Krylov space methods [12], it ollows that in m iterations, the residual is reduced by the actor o at least Cr(m), where r(m) = min deg(p)=m p()=1 max p(λ). λ σ(pf) λ Here, the minimum is taken over complex polynomials and the maximum over eigenvalues o PF. Figures 2 and 3 show one representative case o the spectrum o the iteration operator PF rom (22) or Neumann boundary condition on the scatterer, and or the elastic scatterer, respectively. The problem setting was h = 1/4, k = 1, 4 4 luid subdomains and, or Figure 3, 2 2 solid subdomains. The coarse space had 4 directions or luid waves, both or the multipliers and or the wet interace. The igures show that in both cases, the spectrum is clustered around a point other than the origin. For the coupled problem, there are ew more eigenvalues near

12 12 JAN MANDEL 14 FIG. 2. Complete spectrum Spectrum o the FETI-H iteration operator 1.5 Spectrum detail Im 6 Im Re Re 14 FIG. 3. Spectrum o the iteration operator or coupled problem Complete spectrum Spectrum detail Im 6 Im Re Re the origin and some clustered around 1. Because the eigenvalues are clustered around a point dierent rom zero, a polynomial with a low value o r(m) exists, and convergence is ast. The eect o adding the elastic scatterer shows in ew extra eigenvalues close to the cluster and a minor change in other eigenvalues. The change in the spectrum can be heuristically explained as ollows, c., also the remarks at the end o Section 5. The last two by two diagonal block (with I on the diagonal) in (2) is a discretization o a Fredholm integral equation o the second kind, with eigenvalues clustered around 1. The inluence o the other o-diagonal blocks is relatively weak. The FETI-H method already has eigenvalues around 1, so the overall eect o adding the elastic scatterer on the spectrum o the iteration operator is small. 8. CONCLUSION We have presented a new ast substructuring method or coupled luid-solid acoustic problems. The computational results indicate that the method is scalable with respect to mesh size, requency, and the number o subdomains. The growth o the number o iterations can be controlled by increasing the size o the coarse space. Numerical calculation o the spectrum o the iteration operator suggests that the ast convergence is due to clustering o the spectrum. In most cases, the resulting number o iterations is same or slightly larger than or the FETI-H method

13 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 13 or the same problem with the Neumann boundary condition instead o an elastic scatterer. Perormance study o the method or realistic problems, 3D computations, parallel implementation, and theoretical analysis will be reported elsewhere. ACKNOWLEDGMENT This research was supported in part by the Oice o Naval Research under grant N , and the National Science oundation under grants ECS and DMS The author would like to thank Charbel Farhat, Rabia Djellouli, and other coworkers at the Center or Aerospace Structures at the University o Colorado at Boulder or useul discussions, stimulating remarks, and their patience during several presentations o early versions o this work. The prototype MATLAB code used parts o a code originally written by Mirela Popa and the author [15]. REFERENCES 1. S. C. Brenner, An additive Schwarz preconditioner or the FETI method. Submitted. 2. P. G. Ciarlet, Basic error estimates or elliptic problems, in Handbook o Numerical Analysis, P. Ciarlet and J. L. Lions, eds., vol. II, North-Holland, Amsterdam, 1989, pp A. De La Bourdonnaye, C. Farhat, A. Macedo, F. Magoulès, and F.-X. Roux, A nonoverlapping domain decomposition method or the exterior Helmholtz problem, Contemporary Mathematics, 218 (1998), pp Proceedings o the Tenth Conerence on Domain Decomposition, J. Mandel, C. Farhat, and X.-C. Cai, eds. 4. C. Farhat, P. S. Chen, and J. Mandel, Scalable Lagrange multiplier based domain decomposition method or time-dependent problems, Int. J. Numer. Meth. Engrg., 38 (1995), pp C. Farhat, P.-S. Chen, J. Mandel, and F.-X. Roux, The two-level FETI method - Part II: Extension to shell problems, parallel implementation and perormance results, Comp. Meth. Appl. Mech. Engrg., 155 (1998), pp C. Farhat, A. Macedo, and M. Lesoinne, A two-level domain decomposition method or the iterative solution o high requency exterior Helmholtz problem, Numerische Mathematik, 85 (2), pp C. Farhat, A. Macedo, M. Lesoinne, F.-X. Roux, F. Magoulès, and A. De La Bourdonnaye, Two-level domain decomposition methods with Lagrange multipliers or the ast solution o acoustic scattering problems, Comput. Methds. Appl. Mech. Engrg., 184 (2), pp C. Farhat, A. Macedo, and R. Tezaur, FETI-H: a scalable domain decomposition method or high requency exterior Helmholtz problems, in Eleventh International Conerence on Domain Decomposition Method, C.-H. Lai, P. Bjørstad, M. Cross, and O. Widlund, eds., DDM.ORG, 1999, pp C. Farhat and J. Mandel, Scalable substructuring by Lagrange multipliers in theory and practice, in Proceedings o the 9th International Conerence on Domain Decomposition, Bergen, Norway, June 1996, DDM.ORG, 1998, pp , The two-level FETI method or static and dynamic plate problems - Part I: An optimal iterative solver or biharmonic systems, Comp. Meth. Appl. Mech. Engrg., 155 (1998), pp C. Farhat and F.-X. Roux, An unconventional domain decomposition method or an eicient parallel solution o large-scale inite element systems, SIAM J. Sci. Stat. Comput., 13 (1992), pp A. Greenbaum, Iterative Methods or Solving Linear Systems, SIAM, Philadelphia, A. Klawonn and O. B. Widlund, A domain decomposition method with Lagrange multipliers and inexact solvers or linear elasticity, SIAM J. Sci. Comput., 22 (2), pp , FETI and Neumann-Neumann iterative substructuring methods: Connections and new results, Comm. Pure Appl. Math., 54 (21), pp J. Mandel and M. Popa, A multigrid method or elastic scattering, UCD/CCM Report 19, Center or Computational Mathematics, University o Colorado at Denver, September

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