A FETI Method for a Class of Indefinite or Complex Second- or Fourth-Order Problems

Transcription

1 A FETI Method for a Class of Indefinite or Complex Seond- or Fourth-Order Problems Charbel Farhat 1, Jing Li 2, Mihel Lesoinne 1 and Philippe Avery 1 1 University of Colorado at Boulder, Department of Aerospae Engineering Sienes ( 2 Kent State University, Department of Mathematial Sienes Summary. The FETI-DP domain deomposition method is extended to address the iterative solution of a lass of indefinite problems of the form (K σ 2 M)x = b, and a lass of omplex problems of the form (K σ 2 M + iσd)x = b, where K, M, and D are three real symmetri positive semi-definite matries arising from the finite element disretization of either seond-order elastodynami problems or fourth-order plate and shell dynami problems, i is the imaginary omplex number, and σ is a positive real number. 1 Introdution Real linear or linearized systems of equations of the form (K σ 2 M)x = b (1) and omplex linear or linearized systems of equations of the form (K σ 2 M + iσd)x = b (2) are frequent in omputational strutural dynamis. Eq. (1) is enountered, for example, in the finite element (FE) simulation of the fored response of an undamped mehanial system to a periodi exitation. In that ase, K and M are the FE stiffness and mass matries of the onsidered mehanial system, respetively, σ is the irular frequeny of the external periodi exitation, b is its amplitude, (K σ 2 M) is the impedane of the mehanial system, and x is the amplitude of its fored response. Suh problems also arise during the solution by an inverse shifted method of the generalized symmetri eigenvalue problem Kx = ω 2 Mx assoiated with an undamped mehanial system. In that example, K and M have the same meaning as in the previous ase, (ω 2, x) is a desired pair of eigenvalue and eigenvetor representing the square of a natural irular frequeny and the orresponding natural vibration mode of the undamped mehanial system, respetively, and the shift σ 2 is introdued

2 20 Charbel Farhat, Jing Li, Mihel Lesoinne and Philippe Avery to obtain quikly the losest eigenvalues to σ 2. In both examples mentioned here, the matries K and M are symmetri positive semi-definite, and therefore (K σ 2 M) rapidly beomes indefinite when σ is inreased. Eq. (2) is enountered in similar problems when the mehanial system is damped, in whih ase i denotes the pure imaginary number satisfying i 2 = 1 and D denotes the FE damping matrix and is also symmetri positive semi-definite. Domain deomposition based preonditioned onjugate gradient (PCG) methods have emerged as powerful equation solvers in this field on both sequential and parallel omputing platforms. While most suessful domain deomposition methods (DDMs) have been designed for the solution of symmetri positive (semi-) definite systems, some have targeted indefinite problems of the form given in (1) (Cai and Widlund [1992]). The objetive of this paper is to present an alternative DDM that addresses both lasses of indefinite (1) and omplex (2) problems, that is based on the FETI-DP (Farhat, Lesoinne and Pierson [2000], Farhat et al. [2001]) DDM, and that is salable when K, M, and D result from the FE disretization of seond-order elastodynami problems and fourth-order plate and shell dynami problems. 2 The FETI-DP method The dual-primal finite element tearing and interonneting method (FETI- DP) (Farhat, Lesoinne and Pierson [2000], Farhat et al. [2001]) is a thirdgeneration FETI method (for example, see Farhat [1991], Farhat and Roux [1991]) developed for the salable and fast iterative solution of systems of equations arising from the FE disretization of stati, dynami, seond-order, and fourth-order ellipti partial differential equations (PDEs). When equipped with the Dirihlet preonditioner (Farhat, Mandel and Roux [1994]) and applied to fourth-order or two-dimensional seond-order problems, the ondition number κ of its interfae problem grows asymptotially as (Mandel and Tezaur [2001]) κ = O (1 + log m H ), m 2, (3) h where H and h denote the subdomain and mesh sizes, respetively. When equipped with the same Dirihlet preonditioner and an auxiliary oarse problem onstruted by enforing some set of optional onstraints at the subdomain interfaes (Farhat et al. [2001]), the ondition number estimate (3) also holds for seond-order salar ellipti problems (Klawonn, Widlund and Dryja [2002]). The result (3) proves the numerial salability of the FETI methodology with respet to all of the problem size, the subdomain size, and the number of subdomains. More speifially, it suggests that one an expet the FETI-DP method to solve small-sale and large-sale problems in similar iteration ounts. This in turn suggests that when the FETI-DP method is well-implemented on a parallel proessor, it should be apable of solving an n-times larger problem using an n-times larger number of proessors in almost

3 The FETI-DPH method 21 a onstant CPU time. This was demonstrated in pratie for many omplex strutural mehanis problems (for example, see Farhat, Lesoinne and Pierson [2000] and Farhat et al. [2001] and the referenes ited therein). Next, the FETI-DP method is overviewed in the ontext of the generi symmetri positive semi-definite (stati) problem Kx = b, (4) where K has the same meaning as in problems (1,2) and b is an arbitrary vetor, in order to keep this paper as self-ontained as possible. 2.1 Non-overlapping domain deomposition and notation Let Ω denote the omputational support of a seond- or fourth-order problem whose disretization leads to problem (4), {Ω (s) } Ns denote its deomposition into N s subdomains with mathing interfaes Γ (s,q) = Ω (s) Ω (q), and let s=n s Γ = Γ (s,q) denote the global interfae of this deomposition. In the,q>s remainder of this paper, eah interfae Γ (s,q) is referred to as an edge, whether Ω is a two- or three-dimensional domain. Let also K (s) and b (s) denote the ontributions of subdomain Ω (s) to K and b, respetively, and let x (s) denote the vetor of dof assoiated with it. Let N of the N I nodes lying on the global interfae Γ be labeled orner nodes (see Fig. 1), Γ denote the set of these orner nodes, and let Γ = Γ \Γ. If in eah subdomain Ω (s) the unknowns are partitioned into global orner dof designated by the subsript, and remaining dof designated by the subsript r, K (s), x (s) and b (s) an be partitioned as follows K (s) = [ K (s) rr K (s)t r K (s) r K (s) ] [ ], x (s) x (s) r = x (s) [ ] and b (s) b (s) r = b (s). (5) The r-type dof an be further partitioned into interior dof designated by the subsript i, and subdomain interfae boundary dof designated by the subsript b. Hene, x (s) r and b (s) r an be further partitioned as follows [ x (s) r = x (s) i ] T [ x (s) and b (s) b r = b (s) i where the supersript T designates the transpose. Let x denote the global vetor of orner dof, and x (s) to Ω (s). Let also r by and r x(s) r = ±x (s) ] T b (s) b, (6) denote its restrition be the two subdomain Boolean matries defined b and x = x (s), (7)

4 22 Charbel Farhat, Jing Li, Mihel Lesoinne and Philippe Avery C Ω 1 C Ω 2 C Ω 3 C Fig. 1. Example of a definition of orner points. where the ± sign is set by any onvention that implies that Ns r x (s) r represents the jump of the solution x aross the subdomain interfaes. Finally, let b = B (s)t b (s). (8) In Farhat, Lesoinne and Pierson [2000] and Farhat et al. [2001], it was shown that solving problem (4) is equivalent to solving the following domaindeomposed problem K (s) rr x (s) r + K (s) r B (s)t K r (s)t x + B r (s)t x (s) r + λ + B r (s)t Q b µ = b (s) r, s = 1,..., N s (9) B (s)t K (s) B(s) x = b, (10) Q T b r x(s) r = 0, (11) r x (s) r = 0, (12) where λ is an N λ -long vetor of Lagrange multipliers introdued on Γ to enfore the ontinuity (11) of the solution x, and µ is another vetor of Lagrange multipliers introdued to enfore the optional linear onstraints (12). These optional onstraints, a onept first developed in Farhat, Chen, Risler and Roux [1998], are assoiated with a matrix Q b with N Q < N λ olumns defined on Γ. The word optional refers to the fat that Eq. (12) and the vetor of Lagrange multipliers µ are not neessarily needed for formulating the above domain-deomposed problem. Indeed, sine the solution of problem (4) is ontinuous aross the subdomain interfaes, it satisfies Eq. (11) and therefore satisfies Eq. (12) for any matrix Q b. The domain-deomposed problem (9 12) was labeled dual-primal in Farhat, Lesoinne and Pierson [2000] and Farhat et al. [2001] beause it is formulated in terms of two different types of global unknowns: the dual Lagrange multipliers represented by the vetor λ, and the primal orner dof represented by the vetor x. In the remainder of this paper, the j-th olumn of Q b is denoted by q j.

5 The FETI-DPH method Interfae and oarse problems Let K = [ ] K 0, d 0 0 r = and r K(s) 1 rr b = b b (s) r, (K (s) r B(s) )T K rr (s) 1 b (s) r. (13) After some algebrai manipulations aimed at eliminating symbolially x (s) r, s = 1,..., N s, x, and µ, the domain-deomposed problem (9 12) an be transformed into the following symmetri positive semi-definite interfae problem where F Irr = K (s) r = r K rr (s) 1 [ K (s) r K = K (F Irr + F Ir K 1 T r, FIr = T r Q b ], b = (K (s) r (B r (s)t ) T K rr (s) 1 F T I r )λ = d r F Ir K 1 b, (14) [ r K rr (s) 1 K (s) r, b Q T b d r (K (s) r Q b ) T K rr (s) 1 (K (s) ], ) Ns (K (s) r r ) Ns (B r (s)t ) T K rr (s) 1 (T Q b ) T K rr (s) 1 (T The FETI-DP method is a DDM whih solves the original problem (4) by applying a PCG algorithm to the solution of the orresponding dual interfae problem (14). At the n-th PCG iteration, the matrix-vetor produt (F Irr + F Ir K 1 F T I r )λ n inurs the solution of an auxiliary problem of the form K z = F T I r λ n. (16) From the fifth of Eqs. (15), it follows that the size of this auxiliary problem is equal to the sum of the number of orner dof, N dof, and the number of olumns of the matrix Q b, N Q. For N Q = 0 that is, for Q b = 0, the auxiliary problem (16) is a oarse problem, and K is a sparse matrix whose pattern is that of the stiffness matrix obtained when eah subdomain is treated as a superelement whose nodes are its orner nodes. This oarse problem ensures that the FETI-DP method equipped with the Dirihlet preonditioner (see Setion 2.3) is numerially salable for fourth-order plate and shell problems, and two-dimensional seond-order elastiity problems (Farhat et al. [2001], Mandel and Tezaur r Q b ). r Q b ) (15)

6 24 Charbel Farhat, Jing Li, Mihel Lesoinne and Philippe Avery [2001]). However, for Q b = 0, the FETI-DP method equipped with the Dirihlet preonditioner is not numerially salable for three-dimensional seondorder problems. For any hoie of Q b 0, K remains a sparse matrix. If Q b is onstruted edge-wise that is, if eah olumn of Q b is onstruted as the restrition of some operator to a speifi edge of Γ the sparsity pattern of K beomes that of a stiffness matrix obtained by treating eah subdomain as a superelement whose nodes are its orner nodes augmented by virtual mid-side nodes. The number of dof attahed to eah virtual mid-side node is equal to the number of olumns of Q b assoiated with the edge on whih lies this mid-side node. If N Q is kept relatively small, the auxiliary problem (16) remains a relatively small oarse problem. This oarse problem was labeled augmented oarse problem in Farhat, Lesoinne and Pierson [2000] in order to distinguish it from the smaller oarse problem obtained with Q b = 0. Furthermore, eah olumn of Q b is referred to as an augmentation oarse mode. When these augmentation oarse modes are hosen as the translational rigid body modes of eah edge of Γ, the FETI-DP method equipped with the Dirihlet preonditioner beomes numerially salable for three-dimensional seond-order problems (Klawonn, Widlund and Dryja [2002]). 2.3 Loal preonditioning Two loal preonditioners have been developed so far for the FETI-DP method: 1. The Dirihlet preonditioner whih an be written as where F D 1 I rr = S (s) bb = K(s) [ ] 0 0 W (s) r 0 S (s) B r (s)t W (s), bb bb K(s)T ib K (s) 1 ii K (s) ib, (17) the subsripts i and b have the same meaning as in Setion 2.1, and W (s) is a subdomain diagonal saling matrix that aounts for possible subdomain heterogeneities (Rixen and Farhat [1999]). This preonditioner is mathematially optimal in the sense that it leads to the ondition number estimate (3). 2. The lumped preonditioner whih an be written as F L 1 I rr = [ ] 0 0 W (s) r 0 K (s) B r (s)t W (s). (18) bb This preonditioner is not mathematially optimal in the sense defined above; however, it dereases the ost of eah iteration in omparison with the Dirihlet preonditioner often with a modest inrease in the iteration ount.

7 The FETI-DPH method 25 3 The FETI-DPH method In the ontext of Eq. (1), K (s) rr beomes K (s) rr σ 2 M (s) rr. Hene, the extension of the FETI-DP method to problems of the form given in (1) or (2) requires addressing the following issues: 1. K (s) rr σ 2 M (s) rr is indefinite and therefore the dual interfae problem (14) is indefinite. 2. Independently of whih interfae points are hosen as orner points, K (s) rr σ 2 M (s) rr is in theory singular when σ 2 oinides with an eigenvalue of the penil (K (s) rr,m (s) rr ). 3. How to onstrut augmentation oarse modes and extended Dirihlet and lumped preonditioners that address the speifis of problems (1,2). For problems of the form given in (2), only the third issue is relevant. The first issue an be addressed by solving the dual interfae problem (14) by a preonditioned generalized minimum residual (PGMRES) algorithm rather than a PCG algorithm. The seond and third issues were addressed in Farhat, Maedo and Lesoinne [2000] in the ontext of the basi FETI method and aousti sattering appliations that is, for the exterior Helmholtz salar problem where σ 2 = k 2 and k denotes the wave number. More speifially, a regularization proedure was developed in that referene to prevent all subdomain problems from being singular for any value of the wave number k, without destroying the sparsity of the loal matries K (s) rr k 2 M (s) rr and without affeting the solution of the original problem (1). Furthermore, for the salar Helmholtz equation, the oarse modes were hosen in Farhat, Maedo and Lesoinne [2000] as plane waves of the form e ikθt j X b, j = 1, 2,, where θ j denotes a diretion of wave propagation and X b the oordinates of a node on Γ. The resulting DDM was named the FETI-H method (H for Helmholtz). Unfortunately, the regularization proedure haraterizing the FETI-H method transforms eah real subdomain problem assoiated with Eq. (1) into a omplex subdomain problem. For aousti sattering appliations, this is not an issue beause the Sommerfeld radiation ondition auses the original problem to be in the omplex domain. However, for real-valued problems suh as those represented by Eq. (1), the regularization proedure of the FETI-H method is unjustifiable from omputational resoure and performane viewpoints. In pratie, experiene reveals that K (s) rr σ 2 M (s) rr is non-singular as long as K (s) rr is non-singular. This observation is exploited here to design an extension of the FETI-DP method for indefinite problems of the form given in (1) and omplex problems of the form given in (2) by: 1. Replaing the PCG solver by the PGMRES solver. 2. Adapting the Dirihlet and lumped preonditioners to exploit an interesting harateristi of problems (1,2).

8 26 Charbel Farhat, Jing Li, Mihel Lesoinne and Philippe Avery 3. Construting a new augmentation oarse spae that is effetive for seondorder elastodynami problems as well as fourth-order plate and shell dynami problems. The extension of FETI-DP outlined above is named here the FETI-DPH method. 3.1 Adapted Dirihlet and lumped preonditioners Consider the subdomain (impedane) matrix Z (s) rr = K(s) rr σ2 M (s) rr. (19) For the appliations outlined in the introdution, M (s) rr is a mass matrix; hene, in three dimensions and at the element level, this matrix is proportional to h 3. On the other hand, for the same appliations, K (s) rr is a stiffness matrix; in three dimensions and at the element level, it is proportional to h for seond-order elastiity problems, and to 1/h 2 for fourth-order plate and shell problems. It follows that for a suffiiently fine mesh, Z (s) rr is dominated by K (s) rr. These observations, the optimality of the Dirihlet preonditioner and the omputational effiieny of the lumped preonditioner established for the solution of problem (4) suggest preonditioning the loal matries Z (s) rr by Dirihlet and lumped onstruts that are based on K (s) rr (see Setion 2.3) and not Z (s) rr. When Rayleigh damping is used, D (s) rr = KK (s) rr + MM (s) rr, (20) where K and M are two real onstants, and the same reasoning an be invoked to advoate preonditioning the loal matries Z (s) rr = K (s) rr σ 2 M (s) rr + iσd (s) rr (21) by Dirihlet and lumped onstruts that are based on (1+iσ K )K (s) rr and not Z (s) rr. Finally, it is pointed out that the ad-ho reasoning outlined above an be mathematially justified, at least in the ontext of the salar Helmholtz equation (for example, see Klawonn [1995] and the referenes ited therein). 3.2 Wavy augmented oarse problem Let r denote the residual assoiated with the iterative solution of the dual interfae problem (14). From Eqs. (9 12) and Eq. (14), it follows that r = d r F Ir K 1 b (F Irr + F Ir K 1 F T I r )λ = r x (s) r, (22)

9 The FETI-DPH method 27 whih shows that the residual r represents the jump of the iterate solution aross the subdomain interfaes. From Eq. (12), Eq. (15), Eq. (14) and Eq. (11), it follows that at eah iteration of the PGMRES algorithm applied to the solution of problem (14), FETI-DPH fores the jump of the solution aross the subdomain interfaes to be orthogonal to the subspae represented by the matrix Q b. This feature is a strategy for designing an auxiliary oarse problem whih, when Q b is well hosen, aelerates the onvergene of a DDM (Farhat, Chen, Risler and Roux [1998]). In this work, the searh for a suitable matrix Q b is driven by the following reasoning. Suppose that the spae of traes on Γ of the desired solution of problem (1) is spanned by a set of orthogonal vetors {v je } N λ j=1, where the subsript E indiates that v je is non-zero only on edge E Γ. Then, the residual r defined in Eq. (22) an be written as N λ r = α j v je, (23) j=1 where {α j } N λ j=1 is a set of real oeffiients. If eah augmentation oarse mode is hosen as q j = v je, j = 1,, N Q, (24) Eq. (12) simplifies to α j = 0, j = 1,, N Q. (25) In that ase, Eq. (25) implies that at eah iteration of the PGMRES algorithm, the first N Q omponents of the residual r in the basis {v je } N λ j=1 are zero. If a few vetors {v je } NQ j=1, N Q << N λ, that dominate the expansion (23) an be found, then hoosing these vetors as oarse augmentation modes an be expeted to aelerate the onvergene of the iterative solution of the dual interfae problem (14). Hene, it remains to exhibit suh a set of orthogonal vetors v je and onstrut a omputationally effiient matrix Q b. A seond-order elastodynami problem is governed by Navier s displaement equations of motion µ u + (Λ + µ) ( u) + b = ρ 2 u t 2, (26) where u R 3 denotes the displaement (vetor) field of the elastodynami system, Λ and µ its Lamé moduli, b R 3 its body fores, ρ its density, and t denotes time. If a harmoni motion is assumed, that is, if u(x, t) = v(x)e iωt, (27) where X R 3 denotes the spatial variables, and ω denotes a irular frequeny, the homogeneous form of Eq. (26) beomes

10 28 Charbel Farhat, Jing Li, Mihel Lesoinne and Philippe Avery µ v + (Λ + µ) ( v) + ρω 2 v = 0. (28) The free-spae solutions of the above vetor equation are v = a p sin(k p θ X), v = a p os(k p θ X), (29) v = a s1 sin(k s θ X), v = a s1 os(k s θ X), (30) v = a s2 sin(k s θ X), v = a s2 os(k s θ X), (31) where θ R 3 is an arbitrary vetor of unit length ( θ 2 = 1), a p R 3 is a vetor that is parallel to θ, (a s1, a s2 ) R 3 R 3 are two independent vetors in the plane orthogonal to θ, k p = ρω 2 Λ + 2µ, and k s = ρω 2 µ. (32) The free-spae solutions (29) are known as the elasti pressure or longitudinal waves, and the free-spae solutions (30) and (31) are known as the elasti shear or transverse waves. Consider next the following fourth-order PDE assoiated with a given elasti body 2 u m D ω2 u = 0, where m = ρτ, D = Eτ 3 12(1 ν 2 ), (33) E denotes the Young modulus of the elasti body, ν its Poisson ratio, τ its thikness, and all other variables have the same meaning as before. The reader an hek that the free-spae solutions (29,30,31) with k p = k s = 4 m D ω2 (34) are also free-spae solutions of Eq. (33). The PDE (33) an model the harmoni transverse motion of a plate. In that ase, u is a salar representing the transverse displaement field. However, for the purpose of onstruting an augmented oarse problem for the FETI-DPH method, and only for this purpose, it is assumed here that when u R 3, Eq. (33) models the harmoni motion of a shell in all three dimensions. Hene, a general solution of either Eq. (28) or Eq. (33) an be written as ( v = {a pj 1j sin(k p θ j X) + 2j os(k p θ j X) )} + + j=1 ( {a s1j 3j sin(k s θ j X) + 4j os(k s θ j X) )} j=1 j=1 { a s2j ( 5j sin(k s θ j X) + 6j os(k s θ j X) )}, (35)

11 The FETI-DPH method 29 where θ j R 3 is an arbitrary vetor of unit length defining the diretion of propagation of an elasti pressure or shear wave, 1j, 2j, 3j, 4j, 5j, and 6j are real oeffiients, and k p and k s are given by Eq. (32) for a seondorder elastodynami problem and by Eq. (34) for a fourth-order plate or shell dynami problem. From Eq. (35) and Eq. (24), it follows that the desired matrix Q b is omposed of bloks of six olumns. The olumns of eah blok are assoiated with one diretion of propagation θ j and one edge E of the mesh partition, and an be written as 3(m 1) + 1 3(m 1) + 1 q bl 3(m 1) + 2 = a pj sin(k p θ j X m ), q bl+1 3(m 1) + 2 = a pj os(k p θ j X m ), 3(m 1) + 3 3(m 1) + 3 (36) 3(m 1) + 1 3(m 1) + 1 q bl+4 3(m 1) + 2 = a s2j sin(k s θ j X m ), q bl+5 3(m 1) + 2 = a s2j os(k s θ j X m ), 3(m 1) + 3 3(m 1) + 3 l = 6(j 1) + 1, m = 1,, N I N, where q b [3(m 1) + 1] designates the entry of q b assoiated with the dof in the x-diretion attahed to the m-th node on an edge E Γ, q b [3(m 1) + 2] designates the entry assoiated with the dof along the y-diretion, q b [3(m 1) + 3] designates the entry assoiated with the dof along the z- diretion, and X m R 3 denotes the oordinates of this m-th node. Hene, if N E denotes the number of edges of the mesh partition, and N θ the number of onsidered diretions of wave propagation, the total number of augmentation oarse modes is given in general by N Q = 6N E N θ. To these modes an be added the edge-based translational rigid body modes as these are free-spae solutions of Eq. (28) when ω = 0. In this paper, the number of diretions is limited by Nθ max = 13, and the diretions θ j are generated as follows. A generi ube is disretized into points. A diretion θ j is defined by onneting the enter point to any of the other 26 points lying on a fae of the ube. Sine eah diretion θ j is used to define both a osine and a sine mode, only one diretion θ j is retained for eah pair of opposite diretions, whih results in a maximum of 13 diretions. 4 Performane studies and preliminary onlusions Here, the FETI-DPH method is applied to the solution of various problems of the form given in (1) or (2) and assoiated with: (a) the disretization by quadrati tetrahedral elements (10 nodes per element) of a wheel arrier fixed at a few of its nodes, and (b) the disretization by linear triangular shell elements of an alloy wheel lamped at a few enter points (Fig. 2). When the struture is assumed to be damped, Eq. (20) is used to onstrut D and K

12 30 Charbel Farhat, Jing Li, Mihel Lesoinne and Philippe Avery and M are determined by requiring that the ritial damping ratio of the first 10 modes of the struture be equal in a least-squares sense to a speified value, ξ. In all problems, the shift is set to σ 2 = ω 2 = 4π 2 f 2, where ω 2 is the square of a (possibly natural) irular frequeny of the struture and f is the orresponding frequeny in Hz. To help the reader appreiate the magnitude of a hosen shift value, the natural frequenies of both strutures are haraterized in Table 1. In order to investigate the performane, potential, and various salability properties of the FETI-DPH method, various values of σ 2 are onsidered, three meshes with different resolutions are employed for the wheel arrier seond-order problem (504,375 dof, 1,317,123 dof, and 2,091,495 dof), and one mesh with 936,102 dof is employed for the alloy wheel fourthorder shell problem. In all ases, the right-sides of problems (1,2) are generated by a distributed load, the omputations are performed on a Silion Graphis Origin 3800 system with 40 R MHz proessors, and onvergene is delared when the relative residual satisfies RE n = (K σ2 M + iσd) x n b 2 b (37) Fig. 2. FE disretizations of a wheel arrier (left) and an alloy wheel (right). First, attention is direted to the wheel arrier undamped problem, and for eah generated mesh, N s is hosen to keep the subdomain problem size onstant. Two frequenies, 500 KHz and 2 MHz, are onsidered: the latter value of the shift σ 2 arises, for example, when exiting the struture by its 200 th natural frequeny, or shifting around it during the solution of an eigenvalue problem. The number of wave diretions is set to N θ = 2, and the three translational rigid body modes are inluded in the onstrution of the augmentation matrix Q b. The performane results of the FETI-DPH solver obtained on N p = 12 proessors are reported in Table 2 where N itr reords the iteration ount. For eah onsidered frequeny, the iteration ount assoiated with the hosen number of subdomains and hosen preonditioner is almost independent of the mesh size, whih highlights the numerial salability of the FETI-DPH method with respet to both the subdomain problem

13 The FETI-DPH method 31 Table 1. Eigenvalue/Frequeny partial spetrum of the penil (K,M). Wheel Carrier (2 nd -order) Alloy Wheel (4 th -order) Mode Number Eigenvalue (ω 2 ) Frequeny Eigenvalue (ω 2 ) Frequeny 1 2.6e e+04 Hz 7.6e e+02 Hz e e+06 Hz 1.0e e+03 Hz e e+06 Hz 3.0e e+03 Hz e e+06 Hz 5.7e e+04 Hz e e+06 Hz 9.5e e+04 Hz e e+06 Hz e e+06 Hz size and the total problem size. For this seond-order problem, the lumped and Dirihlet preonditioners deliver similar CPU performanes; hene, the lumped preonditioner is preferable sine it requires less memory. Table 2. Performane of the FETI-DPH solver: wheel arrier, undamped, 2 nd -order problem; fixed subdomain problem size; N θ = 2 (+ the three translational rigid body modes); N p = 12. Frequeny Shift (σ 2 ) Mesh size N s N itr CPU N itr CPU Lumped Lumped Dirihlet Dirihlet 504,375 dof s s Hz 9.8e+12 1,317,123 dof s s. 2,091,495 dof s s. 504,375 dof s s Hz 1.6e+14 1,317,123 dof s s. 2,091,495 dof s s. To illustrate the performane of the FETI-DPH solver for problems of the form given in (2), the wheel arrier is next assumed to have a Rayleigh damping. The mesh with N dof = 1, 317, 123 is onsidered, the number of subdomains is set to N s = 600, the shift is set to σ 2 = 10 5 Hz, the number of wave diretions is set to N θ = 2, the three translational rigid body modes are inluded in the onstrution of the augmentation matrix Q b, and the number of proessors is set to N p = 16. For these parameters, the performane results of FETI-DPH equipped with the lumped preonditioner are reported in Table 3 for the undamped ase (ξ = 0), and for realisti damping senarios (ξ = 1%, ξ = 2%, and ξ = 5%). These results suggest that the intrinsi performane of FETI-DPH improves with the amount of damping. For the undamped ase, FETI-DPH operates in the real domain. This explains why in that ase, eah iteration is 2.7 times faster than in the damped ase where FETI-DPH operates in the omplex plane.

14 32 Charbel Farhat, Jing Li, Mihel Lesoinne and Philippe Avery Table 3. Performane of the FETI-DPH solver: wheel arrier, damped, 2 nd -order problem; N dof = 1, 317, 123; N s = 600; σ 2 = 10 5 Hz; lumped preonditioner; N θ = 2 (+ the three translational rigid body modes); N p = 16. ξ K M N itr CPU 0% s. 1% 3.42e s. 2% 6.85e s. 5% 1.71e s. Next, attention is direted to the undamped alloy wheel problem to investigate the performane for a fourth-order shell problem of the FETI-DPH solver equipped with the Dirihlet preonditioner. Two different frequenies, 5 KHz and 20 KHz, are onsidered: the upper value of the shift σ 2 arises, for example, when exiting the onsidered alloy wheel by a frequeny that is higher than its 400 th natural frequeny, or shifting around that frequeny during the solution of an eigenvalue problem. The number of subdomains is varied between N s = 100 and N s = 400 and the number of proessors is fixed to N p = 8. Table 4, where N oarse denotes the total size of the augmented oarse problem, ontrasts for eah value of N s the performane of FETI-DP (with PGMRES as a solver) and the best performane of FETI-DPH obtained by varying N θ. The reported performane results suggest that the FETI-DPH solver is numerially salable for dynami shell problems of the form given in (1). They also highlight the superiority of FETI-DPH over FETI-DP whih fails to onverge in a reasonable iteration ount for large values of the shift σ 2. Table 4. FETI-DPH vs. FETI-DP: alloy wheel, undamped, 4 th -order problem; N dof = 936, 102; Dirihlet preonditioner; N p = 8. Frequeny Shift (σ 2 ) N s N θ N oarse N itr CPU , s , s , s Hz 9.8e , s , s , s ,258 > , s ,372 > Hz 1.6e , s ,129 > , s.

15 The FETI-DPH method 33 Aknowledgement. This researh was supported by the Sandia National Laboratories under Contrat No The first author also aknowledges the support by the Sandia National Laboratories under Contrat No , and the seond author also aknowledges partial support by the National Siene Foundation under Grant No. DMS Any opinions, findings, and onlusions or reommendations expressed in this material are those of the authors and do not neessarily reflet the views of the Sandia National Laboratories or the National Siene Foundation. Referenes X. C. Cai and O. Widlund. Domain deomposition algorithms for indefinite ellipti problems. SIAM J. Si. Statist. Comput., 13: , C. Farhat, M. Lesoinne, and K. Pierson. A salable dual-primal domain deomposition method. Numer. Lin. Alg. Appl., 7: , C. Farhat, M. Lesoinne, P. LeTalle, K. Pierson, and D. Rixen. FETI-DP: a dual-primal unified FETI method - part I: a faster alternative to the two-level FETI method. Internat. J. Numer. Meths. Engrg., 50: , C. Farhat. A Lagrange multiplier based divide and onquer finite element algorithm. J. Comput. Sys. Engrg., 2: , C. Farhat and F. X. Roux. A method of finite element tearing and interonneting and its parallel solution algorithm, Internat. J. Numer. Meths. Engrg., 32: , C. Farhat, J. Mandel and F. X. Roux. Optimal onvergene properties of the FETI domain deomposition method. Comput. Meths. Appl. Meh. Engrg., 115: , J. Mandel and R. Tezaur, On the onvergene of a dual-primal substruturing method. Numer. Math., 88: , A. Klawonn, O. B. Widlund and M. Dryja. Dual-primal FETI methods for three-dimensional ellipti problems with heterogeneous oeffiients. SIAM J. Numer. Anal., 40: , C. Farhat, P. S. Chen, F. Risler and F. X. Roux. A unified framework for aelerating the onvergene of iterative substruturing methods with Lagrange multipliers. Internat. J. Numer. Meths. Engrg., 42: , D. Rixen and C. Farhat. A simple and effiient extension of a lass of substruture based preonditioners to heterogeneous strutural mehanis problems. Internat. J. Numer. Meths. Engrg., 44: , C. Farhat, A. Maedo and M. Lesoinne. A two-level domain deomposition method for the iterative solution of high frequeny exterior Helmholtz problems. Numer.Math., 85: , A. Klawonn. Preonditioners for Indefinite Problems. Ph. D. Thesis, Westfalishe, Wilhelms-Universitat, Munster, 1995.