An Iterative Substructuring Method for Coupled Fluid-Solid Acoustic Problems

Transcription

1 An Iterative Substructuring Method or Coupled Fluid-Solid Acoustic Problems 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 2 Revised December 21 Accepted or publication in Journal o Computational Physics as o January 17, 22 A ast parallel iterative method is proposed or the solution o linear equations arizing rom Finite Element discretization o the time harmonic coupled luid-solid systems in luid pressure and solid displacement ormulation. The luid and the solid domains are decomposed into nonoverlapping subdomains. Continuity o the solution is 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 iterations preconditioned by a coarse space correction. In each iteration, the method requires the solution o one independent local acoustic problem per subdomain and the solution o a global 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 method generalizes the FETI-H method or the Helmholtz equation to coupled luid-elastic scattering. The number o iterations is about same as or the FETI- H method or the related Helmholtz problem with Neumann boundary condition instead o an elastic scatterer i enough coarse space unctions are used. Convergence behavior is explained rom the spectrum o the iteration operator and rom numerical near decoupling o the equations in the luid and in the solid regions. Key Words: Lagrange multipliers, domain decomposition, iterative substructuring, elastic scattering, Helmholtz equation, coupled luid-solid acoustics, FETI 1

2 2 JAN MANDEL 1. INTRODUCTION In this paper, a new iterative substructuring method is proposed or the solution o the coupled luid-elastic time harmonic scattering problem. Iterative solution o the coupled problem by alternating between the luid and the elastic problem is known, c., Achil diev and Nazhmidinov [1] or Cummings and Feng [2]. The new method extends the FETI-H domain decomposition method, proposed by De La Bourdonnaye et al. [3] and Farhat et al. [4, 5] or the solution o the Helmholtz equation, to an iterative method or the coupled problem. The new method or the coupled problem is designed so that the interaction between the luid and the solid is resolved at the same time as the solution within the luid and the elastic domains, and the luid part o the iteration operator reduces to the operator in FETI-H in the limit when the scatterer becomes rigid. The method also uses the same basic building blocks in the luid subdomain as FETI-H, making sotware reuse easier. The FETI method was originally proposed by Farhat and Roux [6] or solving linearized elasticity problems. The basic idea o all FETI-type methods is to decompose the domain into non-overlapping subdomains and to use Lagrange multipliers to enorce that the values o the degrees o reedom coincide on the interaces between the subdomains. The original degrees o reedom are then eliminated, leaving a dual system or the Lagrange multipliers, which is solved iteratively. To achieve scalability with the number o subdomains, a coarse problem is required. For elasticity, the coarse problem ormed naturally rom nullspaces o the local subdomain matrices results in a scalable method, c., Farhat et al. [7] and Mandel and Tezaur [8]. The general coarse problem in FETI was introduced by Farhat and Mandel [9, 1]; this is the approach used in FETI-H and in the present method, with plane waves as the generators o the coarse problem. For urther inormation on FETI, see [11, 12, 13, 14, 15] and reerences therein. The modiications o the FETI method or the Helmholtz equation involve replacing the conditions or the continuity o the solution and its normal derivatives on subdomain interaces by their complex linear combination to obtain a radiation condition, inspired by Després [16]. The present modiication o FETI or elastic scattering uses a complex linear combination o continuity conditions or the displacement and the traction, related to a method by Bennethum and Feng [17]. Related variants o the FETI method or the Helmholtz equation o scattering were proposed by De La Bourdonnaye et al. [3], Farhat et al. [18], and Tezaur et al. [19]. For other work on iterative methods or indeinite problems arizing rom scattering, see, e.g., [2, 21, 22, 23, 24]. The distinctive eature o the present method is the treatment o the wet interace. The luid region and the elastic region are divided into nonoverlapping subdomains. The division into subdomains does not need to match across the wet interace. This comes naturally because, due to the selection o the variational orm o the problem, the wet interace conditions are enorced only weakly. The degrees o reedom on the wet interace are duplicated and new equations are added to enorce the equality o the duplicates. The original degrees o reedom in all subdomains are then eliminated. The intersubdomain Lagrange multipliers and the duplicates o the degrees o reedom on the wet interace are retained and orm the reduced problem,

3 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 3 which is solved iteratively. The diagonal blocks or the intersubdomain Lagrange multipliers in the reduced problems are exactly same as in the FETI-H method and its straightorward extension to elasticity. There are no Lagrange multipliers across the wet interace. Instead, the luid and the elastic region communicate by way o the duplicated primal variables on the wet interace only. This primal-dual approach is reminiscent o the Dual-Primal FETI method (FETI-DP) or positive deinite problems [25, 26]. Another important eature o the present method is scaling. While scaling o dierent physical ields is routine, here we beneit rom the act that as the stiness o the scatterer approaches ininity, the luid and elastic parts o the discretized coupled problem become uncoupled in the limit. The coarse space o FETI-H consists o plane waves or the Lagrange multipliers between the luid subdomains. The present method includes also plane waves or the multipliers on interaces between the elastic regions, and or the primal variables on the wet interace. When the same number o coarse unctions are used in each o the two added categories as or FETI-H, the number o iterations is similar to the iteration count or the FETI-H method or the same problem with a rigid scatterer. The present ormulation o the method is or bounded regions only. While the ormulation o the method is dimension independent, numerical results are given only or a prototype implementation in 2D. Perormance in 3D, urther development o the method, theoretical analysis, and parallel implementation will be reported elsewhere. The paper is organized as ollows. Section 2 introduces the notation. In Section 3, we review the FETI-H method as a starting point. In Section 4, a straightorward extension is proposed or scattering in an elastic medium. The new method or the coupled luid-elastic problem is presented in Section 5. Explanations o the convergence properties o the method are given in Section 6. Implementation details are discussed in Section 7. Complexity o the algorithm is addressed in Section 8. Section 9 contains computational results, and Section 1 is the conclusion. 2. NOTATION The space H 1 (Ω) is the Sobolev space o unctions with square integrable irst order generalized derivatives. Functions are denoted by p, u, etc. The corresponding vectors o values o degrees o reedom are p, u. Matrices are denoted by A, B,... The symbol T denotes the transpose. Quantities associated with subdomain Ω s are denoted by superscript s, e.g., p s, p s. Block vectors or matrices where each block consists o subdomain vectors or matrices are denoted as û, Â, and ater scaling, as ũ, Ã. The external normal vector is denoted by ν. The symbol means approximately equal and << means much less than. Finally, is the l 2 norm and o a vector and A is the spectral norm o a matrix. 3. THE FETI-H METHOD FOR THE HELMHOLTZ EQUATION In this section, we review the FETI-H method as a starting point in a orm suitable or our purposes. For more details, see [3, 4, 5]. Consider a time harmonic acoustic scattering problem in a bounded domain Ω R n, n = 2 or 3, illed with luid with speed o sound c and vibrating at

4 4 JAN MANDEL FIG. 1. Model 2D Problem Γ n Γ d Ω e ν Γ a Ω Γ n angular requency ω. The scatterer occupies the domain Ω e Ω e Ω, and, in this section, the scatterer is rigid. The luid domain is Ω = Ω \ Ω e, c., Fig. 1. Let Ω be decomposed into disjoint subsets Ω = Γ d Γ n Γ a. The luid pressure amplitude p is governed by the Helmholtz equation, where k = ω/c, 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) and p ν = n on Ω e. (3) The domain Ω is decomposed into nonoverlapping subdomains Ω = N s=1 Ω s. (4) The normal to Ω s is denoted by νs. It is well-known that under reasonable conditions, the equation (1) is equivalent to the decomposed orm p s + k 2 p s =

5 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 5 in Ω s, s = 1,..., N, with the continuity conditions or p and its normal derivative on the interaces between the subdomains, p s = p t, p s ν s = pt ν t, on Ω s Ω t. (5) Multiplying (1) by a test unction q and integrating by parts using the boundary conditions, one obtains the standard variational orm, p p V, p q + k 2 pq ik pq = np, q V, (6) Ω Γ a Ω e Ω where p is an extension o the boundary data to a unction p H 1 (Ω ), and V = {q H 1 (Ω ) q = on Γ d }. A conorming discretization o (6) is obtained by replacing V by a inite element space V h V. Functions in the inite element space have the representation p = m i=1 p iφ i, where {φ 1,..., φ m } is a basis o V h. The discretized system is ( K + k 2 M ikg )p = r, (7) where K, M, G, r are the stiness matrix, the mass matrix, the boundary stiness matrix, and the right-hand side vector, respectively, given by q p T K q = p q, p T M q = pq, p T G q = pq, (8) Ω Ω Γ a q T r = p q + k 2 p q ik p + nq. Ω Γ a Ω Let the subdomains Ω s consist o union o elements. Deine subdomain matrices by subassembly, p st K s q = p q, p st M s q s = pq, p st G s q s = pq, (9) Γ a Ω s where p s denotes the vector o the entries o p that are associated with Ω s. We will use block vectors and corresponding partitioned matrices, ˆp = p 1. p N, ˆK = K K N Ω s, ˆM = Ω a M M N. (1) Let N = N 1. N N

6 6 JAN MANDEL be the 1 matrix o the global to local map o the subdomain degrees o reedom, so that ˆp = N p. Then the global matrices can be written as assembly o the subdomain matrices, K = N T ˆK N, M = N T ˆM N, G = N T Ĝ N. (11) It is easy to ind ˆr so that r = N T ˆr. Then the discrete problem (7) is equivalent to ( ˆK + k 2 ˆM ikĝ )ˆp = ˆr, where p = N T ˆp. Now choose a matrix B = [B 1,..., BN ] such that the condition B ˆp = is a discrete version o the irst constraint in (5), that is, the condition that the values o the same degrees o reedom on two dierent subdomains coincide. We assume that B ˆp = i and only i ˆp = N p or some p. Introducing Lagrange multipliers λ or the constraint B ˆp =, we obtain the decomposed problem [ ˆK + k 2 ˆM ikĝ B T B ] [ ˆp λ ] = [ ] ˆr. (12) It is easy to see that (12) is equivalent to (7) in the sense that p is a solution o (7) i and only i ˆp = N p solves (12) with some λ, c., [27]. To obtain an iterative substructuring method, we will eliminate the primary variables ˆp rom the system (12) and solve the resulting reduced system iteratively. However, the matrix ˆK + k 2 ˆM may be close to singular due to near-resonance, except or diagonal blocks that correspond to subdomains with the radiation boundary condition. This motivates replacing the intersubdomain continuity conditions (5) by the complex linear combination p s = p t, σ st ikp s + p s ν s = σ st ikp t p t ν t, on Ω s Ω t, (13) where σ st {, ±1}, σst = σts. This means that the decomposed problem (12) is replaced by the regularized system where [ Â B T B ] [ ˆp λ ] = [ ] ˆr, (14) Â = ˆK + k 2 ˆM ikĝ + ik ˆR, (15) and the regularization matrix ˆR is given by R 1... ˆR = R N, pst R s q s = N t=1 t s σ st Ω s Ωt pq. (16) Clearly, N T R N =, that is, the contributions to the subdomain matrices in the subdomain assembly cancel, so (14) and (12) are equivalent. It is shown in [18] that i or a given s, all σ st or all σst with some σst, then Âs is invertible. For details on strategies or choosing σ st to guarantee this, see [18].

7 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 7 Ater eliminating the primary variables by ˆp = Â 1 (ˆr BT λ ), we obtain the reduced system or the Lagrange multipliers λ, F λ = g, where F = B Â 1 BT, g = B Â 1 ˆr. (17) The reduced system (17) is solved iteratively, with preconditioning by subspace correction, using a coarse space with ew degrees o reedom per subdomain as ollows. Because the same iterative method applies to generalizations to elasticity and to the coupled problem, we drop the subscript and denote the vector o unknowns by x rather than p and b rather than g. Let Q be a matrix with the same number o rows as F. The columns o Q span the coarse space. FETI-H enorces the condition that the residual is orthogonal to the coarse space, Q T (Fx b) =, (18) by adding a correction rom the coarse space in each iteration, which results in the preconditioned system where PFx = Pb, (19) P = I Q(Q T FQ) 1 Q T F. (2) Since it ollows rom (19) that Fx b Range Q, the equations (18) and (19) together imply that Fx = b. The system (19) is solved by Krylov space iterations. The initial approximation is chosen to satisy (18). Since all increments are in the span o residuals o (19) and Q T FP =, all iterates satisy (18). For the Helmholtz equation in Ω, the matrix Q = Q is chosen as Y 1... Q = B Y, Y =....., (21)... Y N where the columns o Y s are inite element interpolants in Ωs o plane waves y k (x) = e ikx d k in several directions d k = 1. In 2D, we choose an even number o directions equidistant on the unit circumerence. Since with each direction, the opposite direction is also included, the column space o Q is same as the column space o the complex conjugate o Q, so it does not matter i the transpose or the conjugate transpose are used in (18). 4. EXTENSION TO ELASTIC SCATTERING Consider the time harmonic elastodynamic equation τ + ω 2 ρ e u = in Ω e, (22)

8 8 JAN MANDEL with natural boundary conditions τ ν = pν on Ω e, (23) Here, u is the displacement, τ is the stress tensor, ρ e is the density o the solid, and p is given boundary data. 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 ), (24) where λ and µ are the Lamé coeicients o the solid. Similarly as in the previous section, Ω e is decomposed into non-overlapping subdomains Ω 1 e,..., Ω Ne e, and, under reasonable conditions [28], equation (22) is equivalent to the decomposed orm τ s +ω 2 ρ e u s = in Ω s e, s = 1,..., N e, with the continuity o the displacement and the traction on intersubdomain interaces, u s = u t, τ s ν s = τ t ν t, on Ω s e Ω t e. (25) We have the standard variational ormulation o (22) or u V e = (H 1 (Ω e )) n, Ω e the discretization ( ) λ( u)( v) + 2µe(u) : e(v) + ω 2 Ω e ( K e + ω 2 M e )u = s, ρ e u v Γ p(ν v) =, v V e, with the matrices and the right-hand side deined by u T ( ) K e v = λ( u)( v) + 2µe(u) : e(v), (26) Ω e u T M e v = ρ e (u v), u T s = Ω e Γ q(ν v), (27) the block orm o the solution and the subdomain matrices u 1 K 1 e... M 1 e... û =., ˆK e =..... ˆM e =..... u Ne... K Ne e... M Ne e, (28) the matrix o the global to local mapping N e, the assembly o subdomain matrices, K e = N T e ˆK e N e, M e = N T e ˆM e N e, (29) the constraint matrix B e such that B e û = i and only i û = N e u or some u, and the decomposed problem [ ˆK e + k 2 ˆM e B T e B e ] [ û λ e ] = [ ] ŝ, (3)

9 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 9 where s = N T e ŝ. To avoid nearly singular matrices in the subdomains, the intersubdomain conditions are replaced by their complex linear combination [17] where σ st e where u s = u t, σ st iωρ e u s + τ s ν s = σ st iωρ e u t τ t ν t, on Ω s e Ω t e, {, ±1}, σ st e = σ ts e, and we obtain the regularized system [ Âe B T e B e ] [ û λ e ] = [ ] ˆr, (31)  e = ˆK e + ω 2 ˆM e + iω ˆR e, (32) and R 1 e... ˆR e =.....,... R Ne e N e u st R s ev s = ρ e t=1 t s σ st e Ω s e Ωt e (n u)(n v). Ater elimination o û, the reduced system with the matrix F e λ e = g e, where F e = B e  1 e B T e, (33) can be solved iteratively in the same way as in the previous section, with the matrix o the coarse space generators Y 1 e... Q e = B e Y e, Y e =....., (34)... Ye Ne with the columns o Y s e being interpolants o pressure and shear plane waves in Ω s e, c., e.g., [29]. 5. ITERATIVE SUBSTRUCTURING FOR COUPLED PROBLEM We are now ready to consider the coupled problem. The luid pressure in Ω satisies the Helmholtz equation (1) and the boundary conditions (2). The displacement o the solid in Ω e satisies the elastodynamic equation (22). Let Γ = Ω e be the wet interace. On Γ, the luid pressure and the solid displacement satisy the interace conditions ν u = 1 p ρ ω 2, τ ν = pν, (35) ν where ρ is the luid density. The equations in (35) model the continuity o normal displacement and the balance o orces, respectively, c., e.g., [29].

10 1 JAN MANDEL FIG. 2. Model 2D Problem Decomposed in 5x5 Fluid and 2x2 Solid Subdomains Multiplying the equations (1) and (22) by test unctions and integrating by parts using the boundary and the interace conditions, we obtain the standard variational orm [3] Ω e p p V, u V e, p q + k 2 pq ik pq ω 2 ρ (ν u)q =, Ω Ω Γ a Γ ( ) λ( u)( v) + 2µe(u) : e(v) + ω 2 ρ e u v p(ν v) =, Ω e q V, v V e. Γ The inite element discretization is [ ] [ ] [ ] K + k 2 M ikg ρ ω 2 T p r ρ ω 2 T T ρ ω 2 ( K e + ω 2 =, (36) M e ) u where K and M are as in (8), K e and M e are as in (26), (27), and T is given by q T Tv = q(ν v). Γ The second equation in (36) was multiplied ρ ω 2 to symmetrize the system. The luid and the solid domains are decomposed into nonoverlapping subdomains, c., Fig. 2. Again, we assume that the subdomains that are unions o elements. Introducing Lagrange multipliers λ and λ e or the constraints B p = and

11 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 11 B e u =, we get the system o linear equations in block orm, ˆK + k 2 ˆM ikĝ ω 2 ρ ˆT B T ω 2 ρ ˆT T ω 2 ρ ( ˆK e + ω 2 ˆM e ) B T e B B e ˆp û λ λ e = ˆr, (37) where ˆK, ˆM, Ĝ, ˆK e, and ˆM e are given by (1), (28), and ˆT = T T 1,Ne....., q st ˆT st v t =... T N,N e T N,1 Γ Ω s Ωt e q(ν v). Again, the system (37) is equivalent to (36). Adding the regularization matrices on the interaces between the solid subdomains, we obtain the equivalent regularized system  ω 2 ρ ˆT B T ω 2 ρ ˆT T ω 2 ρ  e B T e B B e ˆp û λ λ e = ˆr, (38) where  and  e are given by (15) and (32), respectively. At this point, we would like to eliminate the primary variables ˆp and û to obtain a system in the Lagrange multipliers λ, λ e only. For the coupled system, however, the elimination does not reduce to independent problems in all subdomains because subdomains across the wet interace are coupled via the matrix ˆT. So, irst we augment the system by duplicating the degrees o reedom on the wet interace as ollows. Since the value o ˆTû depends only on the entries o û that correspond to degrees o reedom on Γ, we have ˆTû = ˆTJ e û Γ, where û Γ = J T e û and J e is the matrix o the operator o embedding a subvector û Γ into û by adding zero entries. Similarly, ˆT T ˆp = ˆT T J ˆp Γ, where ˆp Γ = J T ˆp. Adding p Γ and u Γ as new variables, we obtain the augmented system equivalent to (38),  B T ω 2 ρ ˆTJ e ω 2 ρ  e B T e ω 2 ρ ˆT T J B B e J T I J T e I ˆp û λ λ e ˆp Γ û Γ = ˆr. (39)

12 12 JAN MANDEL Because the variables in a coupled system typically have vastly dierent scales, we use symmetric diagonal scaling to get the scaled system à BT TJ e à e BT e T T J B Be J T I J T e I where the matrices and the vectors scale as p ũ λ λ e p Γ ũ Γ = r, (4) à = D  D, à e = ω 2 ρ D e  e D e, T = ω 2 ρ D ˆTD e, (41) B = E B D, Be = E e B e D e, r = D ˆr, (42) ˆp = D p, û = D e ũ, λ = D λ, λ e = D e λe. (43) The scaling matrices D, D e, E, and E e are diagonal with positive diagonal entries. We irst choose D and D e so that the absolute values o the diagonal entries o à and Ãe are one. Then the matrices E and E e are deined by the property that the l 2 norms o the columns o B e and B are one. It is easy to see that the scaling results in a dimensionless system. In addition, the scaling reveals numerical near decoupling o the luid and the elastic problem, c., Section 6 below. We now eliminate the variables ˆp and û rom the augmented system (4) and solve the resulting reduced system iteratively. Computing p and ũ rom the irst two equations in (4) gives p = ũ = à 1 ( r B T λ + TJ e ũ Γ ), (44) à 1 e ( B T e λ e + T T J p Γ ). (45) Substituting p and ũ rom (44), (45) into the rest o the equations in (4), we obtain the reduced system Fx = b, (46) where and F = B à 1 J T à 1 B T B à 1 TJ e B e à 1 e B T e B e à 1 e T T J B T I J T à 1 TJ e, (47) J e à 1 e B T e J e à 1 e T T J I x = λ λ e p Γ ũ Γ, b = B à 1 r J T à 1 r.

13 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 13 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 T q p Γ J T e ṽ ũ Γ, where { q = à 1 ( B T λ + TJ e ũ Γ ), ṽ = à 1 e ( B T λ e e + T T J p Γ ). The present method consists o solving the linear system (46) by GMRES. For the coupled problem, the matrix Q o coarse space generators is Q = D B Y D e B e Y e D J T Z D e J T e Z e, (48) where Y and Y e are given by (21) and (34), respectively, as matrices o columns that are discrete representation o plane waves in the subdomains. The matrices Z and Z e are o the same orm as Y and Y e, respectively. Because the multiplication by J T and JT e selects only values o degrees o reedoms on the wet interace, the only blocks o Z and Z e that contribute to Q are blocks or subdomains adjacent to the wet interace. 6. EXPLANATION OF CONVERGENCE PROPERTIES 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 [31], 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 the eigenvalues o PF. It is well-known that r(m) decreases ast with m i the eigenvalues are clustered around a point other than zero. Such clustering has been observed in FETI-H [3]. We argue that because o the orm o the operator F, the same clustering can be expected or the coupled problem. Aside rom the diagonal scaling, the irst diagonal block o F in (47) is same as in the FETI-H method or the Helmholtz equation, c. (17) and second diagonal block is same as in the extension o FETI-H or the elastodynamic problem, c., (33). FETI-H is known to converge ast, so one can expect that on the subspace deined by the coarse correction, these two diagonal blocks will be well conditioned. The o-diagonal blocks all contain the inverse o à or Ãe, which are discretizations o dierential operators. Hence, the o-diagonal blocks resemble discretizations o compact operators, and the structure o equation (46) 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.

14 14 JAN MANDEL For a very rigid scatterer, the problem becomes numerically decoupled in the limit. Indeed, i the Lamé coeicient µ +, the scaling matrix D e, hence rom (41), T, and one has F B à 1 J T à 1 B T B e à 1 e B T e B T, as µ +. (49) I J e à 1 e B T e I We show that the decoupling indicated by (49) takes place or a wide range o parameters important in practice. Assume that the mesh is quasiuniorm with characteristic spacing h. Then T h n 1. Further assuming that kh << 1, which should be the case or approximation reasons, it is easy to see that D (h n 2 ) 1/2. Similarly, i ω 2 h 2 ρ e << µ, we have D e (ω 2 ρ µh n 2 ) 1/2, and it ollows that T = ω 2 ρ D ˆTD e ω 2 ρ D ˆT D e ω 2 ρ (h n 2 ) 1/2 h n 1 (ω 2 ρ µh n 2 ) 1/2 = ωhρ 1/2 µ 1/2 = khc ρ 1/2 µ 1/2. Consequently, the system will become numerically decoupled i khc ρ 1/2 µ 1/2 << 1. (5) In the computations reported in Section 9, the condition (5) is satisied, because khc ρ 1/2 µ 1/2 was o the order o at most 1 3. Figures 3 and 4 show one representative case o the spectrum o the iteration operator PF rom (19) or Neumann boundary condition on the scatterer, and or the elastic scatterer, respectively. The problem setting was as in Section 9, with h = 1/4, k = 1, 4 4 luid subdomains and, or Figure 4, also 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 the origin and some clustered around 1. The eect o adding the elastic scatterer shows in ew extra eigenvalues close to the cluster and a minor change in other eigenvalues. 7. IMPLEMENTATION CHOICES Some implementation details o the algorithms described in Sections 3-5 are slightly dierent than in FETI-H or were let open and will be speciied in this section. The dierences are essentially only dierent choices in implementation. They are not related to the treatment o the coupled problem and do not change the essence o either method, though they o course inluence numerical results. We have generated the matrices B and B e by creating one equation with coeicients, 1, 1 or each pair o subdomains and a degree o reedom they share, in the same way as in FETI, e.g. [27]. To avoid redundant constraints, we have

15 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS FIG. 3. Spectrum o the FETI-H iteration operator Complete spectrum Im Re 14 FIG. 4. Spectrum o the iteration operator or coupled problem Complete spectrum Im Re

16 16 JAN MANDEL orthogonalized the rows o the matrices using the QR algorithm. The resulting matrices are still sparse. FETI-H does not orthogonalize the constraint matrices and either leaves redundant constraints in or takes care not to create them. The columns o the matrix o coarse space generators Q rom (48) are usually linearly dependent. One reason is aliasing o modes rom two subdomains on their interace. Another reason is that the basis o plane waves is ill-conditioned. To obtain a stable numerical algorithm, we have irst orthogonalized the columns o Y s and Ys e and then the matrix Q, dropping linearly dependent columns in each step. The resulting matrices are still sparse. Since the columns o Q are now linearly independent, there is a chance that the coarse system matrix Q T FQ, whose decomposition is needed in the coarse coarse correction (2), may be nonsingular. Though this is not guaranteed, the coarse matrices in numerical experiments were always nonsingular. In the case o a singular matrix, one can use a pseudoinverse. FETI-H does not orthogonalize the coarse space generators and deals with the resulting singular coarse matrix by masking zero pivots. We have used GMRES provided with MATLAB as the iterative solver, while FETI-H uses GCR. The two methods are mathematically equivalent, minimizing the l 2 norm o the residual over the Krylov space. GMRES is more complicated and more numerically stable. For more inormation on Krylov space solvers, see [31]. 8. COMPLEXITY To interpret computational results, one needs to estimate the eect o the number o subdomains, subdomain size, and coarse space size on the on the computational complexity o the method. We omit the eect o the orthogonalization o the constraint matrices and coarse space generators on their nonzero structure, and assume that the asyptotic complexity o the algoritm used or the LU decomposition o a block sparse matrix o order nm, where m is the order o dense blocks, is proportional to n α m 3, and the asymptotic complexity o the multiplication o a vector by the inverse o the LU actors (i.e., the solution) is n β m 2. For example, or topologically two-dimensional problems and band or proile methods with natural ordering, α = 2 and β = 1.5; or ully sparse methods, the complexity is less [32]. The complexity the present method is dominated by the LU decomposition o the subdomain matrices, i.e., the diagonal blocks o Ãe and Ã, and the creation and the LU decomposition o the coarse matrix Q T FQ. Denote by N the total number o degrees o reedom, by N s the number o subdomains, by N i the total number o degrees o reedom on subdomain interaces, and by N c the number o coarse space basis vectors per subdomain. Assume that all subdomains are about the same size, and there are N t interaces between the subdomains. The number o degrees o reedom per subdomain is N/N s on the average. The total cost o the LU decomposition o the subdomain matrices is about N s (N/N s ) α and the cost o one solution is N s (N/N s ) β. We now estimate the cost o creating the irst diagonal block o the coarse matrix, C = (D B Y ) T ( B Ã 1 B T )D B Y. This matrix has ull blocks which arize by the interaction o N c coarse space vectors rom two subdomains with a common neighbor, hence it is a topologically n dimensional block matrix o order asymptotically N s N c, with average block size N c. Computing

17 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 17 ( B Ã 1 B T )D B Y requires N c N t solutions o subdomain problems at (N/N s ) β each. Multiplication o the result by (D B Y ) T adds about Nc 2 N s (N i /N s ). The LU decomposition o C costs Ns α Nc 3 and the solution o a system with C costs Ns β Nc 2. Hence, we get asymptotic estimate o the dominant cost o the setup phase T setup N s (N/N s ) α + N c N t (N/N s ) β + N α s N 3 c + N 2 c N i, (51) and the dominant cost per iteration, T iteration N s (N/N s ) β + N β s N 2 c. (52) The estimates or the luid and elastic subdomains should be done separately ollowing (51), (52) and added together. The complexity o the terms added by the coupling is easily seen to be o lower order. It should be noted that the complexity estimates (51), (52) are not speciic to the present method and apply to any nonoverlapping domain decomposition method with a coarse space applied in a similar manner as here, such as the FETI or the Balancing Neumann-Neumann [33] method or positive deinite problems. 9. COMPUTATIONAL RESULTS We consider a model 2D problem with a scatterer in the center o 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 unless speciied otherwise in some problems, and consisting o aluminum with density ρ e = 27 kg m 3 and Lamé elasticity coeicients λ = N m 2, µ = N m 2. 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, c., Fig. 2. 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 the multipliers and or the wet interace. The stopping criterion or GMRES iterations was Fx b < 1 6. The stopping criterion or the iterations does not involve the original variables in the coupled problem (36), which we write here as as Kz = d. Since the variables have dierent scales, and, in addition, the right-hand side in the second block in (36) is zero, the usual relative residual Kz d / d is meaningless. Instead, we have evaluated the component-wise scaled residual Res = max i d i j K ijz j j K. (53) ij z j The numerical results or the model problem are summarized in Tables 1 to 4. 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

18 18 JAN MANDEL TABLE 1 Decreasing h, k 3 h 2 constant, constant number o elements per subdomain Problem description Coarse directions Number o Subdomains λ Wet degrees o reedom h k Fl. Sol. Fl. Sol. Fl. Sol. Orig. Red. Crs. Iter. Res. 1/4 1 2x2 rigid e-7 1/4 1 2x2 1x e-7 1/4 1 2x2 1x e-7 1/4 1 2x2 1x e-7 1/4 1 2x2 1x e-7 1/4 1 2x2 rigid e-7 1/4 1 2x2 1x e-7 1/4 1 2x2 1x e-7 1/4 1 2x2 1x e-7 1/4 1 2x2 1x e-7 1/8 16 4x4 rigid e-7 1/8 16 4x4 2x e-7 1/8 16 4x4 2x e-7 1/8 16 4x4 2x e-7 1/8 16 4x4 2x e-6 1/8 16 4x4 rigid e-7 1/8 16 4x4 2x e-7 1/8 16 4x4 2x e-7 1/8 16 4x4 2x e-7 1/8 16 4x4 2x e-6 1/ x8 rigid e-6 1/ x8 4x e-7 1/ x8 4x e-6 1/ x8 4x e-6 1/ x8 4x e-7 1/ x8 rigid e-7 1/ x8 4x e-6 1/ x8 4x e-6 1/ x8 4x e-6 1/ x8 4x e-6

19 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 19 TABLE 2 Decreasing h, k 3 h 2 constant, same subdomains Problem description Coarse directions Number o Subdomains λ Wet degrees o reedom h k Fl. Sol. Fl. Sol. Fl. Sol. Orig. Red. Crs. Iter. Res. 1/4 1 4x4 rigid e-7 1/4 1 4x4 2x e-7 1/4 1 4x4 2x e-6 1/4 1 4x4 2x e-6 1/4 1 4x4 2x e-7 1/4 1 4x4 rigid e-7 1/4 1 4x4 2x e-7 1/4 1 4x4 2x e-7 1/4 1 4x4 2x e-7 1/4 1 4x4 2x e-6 1/8 16 4x4 rigid e-7 1/8 16 4x4 2x e-7 1/8 16 4x4 2x e-7 1/8 16 4x4 2x e-7 1/8 16 4x4 2x e-6 1/8 16 4x4 rigid e-7 1/8 16 4x4 2x e-7 1/8 16 4x4 2x e-7 1/8 16 4x4 2x e-7 1/8 16 4x4 2x e-6 1/ x4 rigid e-6 1/ x4 2x e-6 1/ x4 2x e-7 1/ x4 2x e-6 1/ x4 2x e-6 1/ x4 rigid e-7 1/ x4 2x e-7 1/ x4 2x e-6 1/ x4 2x e-6 1/ x4 2x e-6 1/32 4 4x4 rigid e-6 1/32 4 4x4 2x e-7 1/32 4 4x4 2x e-7 1/32 4 4x4 2x e-7 1/32 4 4x4 2x e-7 1/32 4 4x4 rigid e-6 1/32 4 4x4 2x e-7 1/32 4 4x4 2x e-7 1/32 4 4x4 2x e-7 1/32 4 4x4 2x e-7

20 2 JAN MANDEL TABLE 3 Increasing the number o luid and elastic subdomains, obstacle size.5 Problem description Coarse directions Number o Subdomains λ Wet degrees o reedom h k Fl. Sol. Fl. Sol. Fl. Sol. Orig. Red. Crs. Iter. Res. 1/2 8 2x2 rigid e-6 1/2 8 2x2 2x e-7 1/2 8 2x2 2x e-7 1/2 8 2x2 2x e-7 1/2 8 2x2 2x e-7 1/2 8 2x2 rigid e-7 1/2 8 2x2 2x e-7 1/2 8 2x2 2x e-7 1/2 8 2x2 2x e-7 1/2 8 2x2 2x e-7 1/2 8 4x4 rigid e-7 1/2 8 4x4 4x no convergence 1/2 8 4x4 4x no convergence 1/2 8 4x4 4x no convergence 1/2 8 4x4 4x e-7 1/2 8 4x4 rigid e-7 1/2 8 4x4 4x e-6 1/2 8 4x4 4x e-7 1/2 8 4x4 4x e-7 1/2 8 4x4 4x e-6

21 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 21 TABLE 4 Increasing the number o elastic subdomains, obstacle size.9 Problem description Coarse directions Number o Subdomains λ Wet degrees o reedom h k Fl. Sol. Fl. Sol. Fl. Sol. Orig. Red. Crs. Iter. Res. 1/2 32 2x2 rigid e-6 1/2 32 2x2 2x e-7 1/2 32 2x2 2x e-7 1/2 32 2x2 2x e-7 1/2 32 2x2 2x e-7 1/2 32 2x2 rigid e-7 1/2 32 2x2 2x e-7 1/2 32 2x2 2x e-7 1/2 32 2x2 2x e-6 1/2 32 2x2 2x e-6 1/2 32 2x2 rigid e-6 1/2 32 2x2 4x e-7 1/2 32 2x2 4x e-7 1/2 32 2x2 4x e-7 1/2 32 2x2 4x e-7 1/2 32 2x2 rigid e-7 1/2 32 2x2 4x e-7 1/2 32 2x2 4x e-7 1/2 32 2x2 4x e-7 1/2 32 2x2 4x e-7

22 22 JAN MANDEL o iterations or increasing the number o coarse space unctions, that is, the columns o Q. The headings in the tables are: 1. 1/h - the mesh spacing 2. k - the wave number 3. Subdomains Fl. - the number o luid subdomains 4. Subdomain Sol. - the number o elastic subdomains, or rigid i only the Helmholtz equation in the luid is solved 5. Coarse directions λ Fl. - the number o directions per luid subdomain o plane waves or multipliers in the coarse problem 6. Coarse directions λ Sol. - the number o directions per elastic subdomain o plane pressure and shear waves and plane shear waves or multipliers in the coarse problem 7. Coarse directions Wet Fl. - the number o directions per luid subdomain o plane waves or the wet interace in the coarse problem 8. Coarse directions Wet Sol. - the number o directions per elastic subdomain o plane pressure and shear waves or the wet interace in the coarse problem 9. Number o degrees o reedom Orig. - the size o the algebraic problem solved 1. Number o degrees o reedom Red. - the number o degrees o reedom in the reduced problem, i.e., the order o the matrix F 11. Number o degrees o reedom Crs. - the size o the coarse problem ater orthogonalization o the coarse space generators 12. Iter. - the number o iterations 13. Res. - the component-wise scaled residual o the coupled problem rom (53) The scaled residual (53) was o the same order o magnitude as the tolerance in the stopping criterion or smaller. The results in Tables 1 and 2 show scalability when the mesh is reined and the requency increased while keeping k 3 h 2 constant, which is needed to avoid pollution by the phase error and to keep the error decreasing with h or the Helmholtz equation, c., Ihlenburg and Babuška [34]. We irst keep the number o elements per subdomain constant and then we keep the number o subdomains constant. Tables 3 and 4 show convergence or increasing the number o elastic subdomains. Because we want to isolate the eect o the elastic domain, the obstacle is made larger. The luid domain is smaller (in the test problems in Table 4, the luid domain consists o just ew layers o elements), so the number o iterations or the rigid case is much lower. Clearly, increasing the size o the coarse space decreases the number o iterations. The prototype implementation was done in MATLAB, hence we do not report timings. However, we should note that the most time-consuming operation was the generation o the data and assembling the subdomain matrices, ollowed by the setup o the iterative method, and the iterations themselves were only a small raction o the overall time (when the method converged). 1. CONCLUSION We have presented a new iterative substructuring method or coupled luid-solid scattering problems. The computational results indicate that the method is scalable

23 ITERATIVE SUBSTRUCTURING FOR COUPLED ACOUSTICS 23 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 or the same problem with the Neumann boundary condition instead o an elastic scatterer. 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, and to anonymous reerees or their helpul comments. The prototype MATLAB code used parts o a code originally written by Mirela Popa and the author [35]. REFERENCES 1. A. I. Achil diev and Kh. S. Nazhmidinov. An iterative method or solving a diraction problem in an unbounded medium. Dokl. Akad. Nauk Tadzhik. SSR, 31(7): , In Russian. 2. Peter Cummings and Xiaobing Feng. Domain decomposition methods or a system o coupled acoustic and elastic Helmholtz equations. In Eleventh International Conerence on Domain Decomposition Methods (London, 1998), pages (electronic). DDM.org, Augsburg, 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, 218:42 66, Proceedings o the Tenth Conerence on Domain Decomposition, J. Mandel, C. Farhat, and X.-C. Cai, eds. 4. 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(2-4): , C. Farhat, A. Macedo, and R. Tezaur. FETI-H: a scalable domain decomposition method or high requency exterior Helmholtz problems. In Choi-Hong Lai, Petter Bjørstad, Mark Cross, and Olo Widlund, editors, Eleventh International Conerence on Domain Decomposition Method, pages DDM.ORG, Charbel Farhat and Francois-Xavier Roux. An unconventional domain decomposition method or an eicient parallel solution o large-scale inite element systems. SIAM J. Sci. Stat. Comput., 13: , Charbel Farhat, Jan Mandel, and Francois-Xavier Roux. Optimal convergence properties o the FETI domain decomposition method. Comput. Methods Appl. Mech. Engrg., 115: , Jan Mandel and Radek Tezaur. Convergence o a substructuring method with Lagrange multipliers. Numerische Mathematik, 73: , Charbel Farhat and Jan Mandel. 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: , Charbel Farhat and Jan Mandel. Scalable substructuring by Lagrange multipliers in theory and practice. In Proceedings o the 9th International Conerence on Domain Decomposition, Bergen, Norway, June 1996, pages 2 3. DDM.ORG, Axel Klawonn and Olo B. Widlund. A domain decomposition method with Lagrange multipliers and inexact solvers or linear elasticity. SIAM J. Sci. Comput., 22(4): , October Axel Klawonn and Olo B. Widlund. FETI and Neumann-Neumann iterative substructuring methods: Connections and new results. Comm. Pure Appl. Math., 54:57 9, January 21.

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