C. Farhat, P. S. Chen, J. Mandel, A scalable Lagrange multiplier based domain decomposition method for implicit time-dependent problems, International Journal for Numerical Methods in Engineering 38 (1995) 3831--3854.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....component M i involves the solution of a local problem on the substructure# i . Here, P t denotes the transpose of the matrix P . Recalling the definition of P 0 , we see that P t #= P , in general. A full description of the conjugate gradient method applied to Equation (3) can be found in [FCM95, Tos00, TK99]. Here, we only remark that the action of the projection P on a vector can be evaluated at the expense of applying the matrix F and of solving a coarse problem of dimension K. Moreover, the action of P t does not need to be calculated in practice. A suitable choice of the projection P ensures ....

....g is constructed with the local load vectors on the substructures. We then introduce a vector of Lagrange multipliers #, to enforce the constraints, and obtain a saddle point formulation of (5) After eliminating the primal variable u, we obtain the following equation for the dual variable #, see [FCM95, TK99], B # S 1 B t # = B # S 1 g, # # Range(B) 6) We consider a PPCG method for the solution of (6) This corresponds to the choice F = B # S 1 B t , d = B # S 1 g, V = Range(B) in (2) We note that V is the space of jumps of the tangential vectors in # W . We then define the ....

[Article contains additional citation context not shown here]

Charbel Farhat, Po-Shu Chen, and Jan Mandel. A scalable Lagrange multiplier based domain decomposition method for time-dependent problems. Int. J. Numer. Meth. Eng., 38:3831--3853, 1995.

....component M i involves the solution of a local problem on the substructure# i . Here, P t denotes the transpose of the matrix P . Recalling the definition of P 0 , we see that P t #= P , in general. A full description of the conjugate gradient method applied to Equation (3) can be found in [1, 4, 5]. Here, we only remark that the action of the projection 3 P on a vector can be evaluated at the expense of applying the matrix F and of solving a coarse problem of dimension K. Moreover, the action of P t does not need to be calculated in practice. A suitable choice of the projection P ....

....g is constructed with the local load vectors on the substructures. We then introduce a vector of Lagrange multipliers #, to enforce the constraints, and obtain a saddle point formulation of (5) After eliminating the primal variable u, we obtain the following equation for the dual variable #, see [1, 5], B # S 1 B t # = B # S 1 g, # # Range(B) 6) We consider a PPCG method for the solution of (6) This corresponds to the choice F = B # S 1 B t , d = B # S 1 g, V = Range(B) in (2) We note that V is the space of jumps of the tangential vectors in # W . We then define the ....

[Article contains additional citation context not shown here]

Charbel Farhat, Po-Shu Chen, and Jan Mandel. A scalable Lagrange multiplier based domain decomposition method for time-dependent problems. Int. J. Numer. Meth. Eng., 38:3831--3853, 1995.

....of the number of substructures and increases only slowly with the number of unknowns associated to the substructures. In addition, if suitable scaling matrices are introduced, the condition number can also be made independent of possibly large jumps of the coe#cients; see section 3.1. We refer to [14, 16, 26, 10, 38, 34, 25], for Poisson and elasticity problems, to [12] for acoustic scattering problems, to [13, 11, 27, 38] for shell and plates problems, and to [39, 33] for edge element approximations of Maxwell s equations. A number of domain decomposition methods have also been proposed for advection di#usion ....

....i # K II i # 1 f I i . We can easily check that, since our local bilinear forms are positive definite, the local Schur complements S i are always invertible and, consequently, there is no natural coarse space associated to the substructures; we are in a similar case as that considered in [10]. Following [10] we first find u from the first equation in (12) and substitute its value in the second equation. We obtain the system F# = d, 13) where F : B S 1 B t , d : B S 1 f. Following [25, 39] we now define a preconditioner for (13) We introduce the matrices, R : R 1 , R 2 ....

[Article contains additional citation context not shown here]

Charbel Farhat, Po-Shu Chen, and Jan Mandel. A scalable Lagrange multiplier based domain decomposition method for time-dependent problems. Int. J. Numer. Meth. Eng., 38:3831-- 3853, 1995.

.... [14] We will also introduce a general family of projections which was first introduced for heterogeneous problems in [17] Such methods have recently been tested in very large scale numerical experiments; see [1] For a more detailed description and extensions beyond scalar elliptic problems, see [12, 13, 25, 27, 33]. Let us point out that there are also other variants of the FETI methods; see, e.g. Park, Justino, and Felippa [26] The relation of one of them to the FETI method developed by Farhat and Roux is discussed in [29] and a convergence analysis of this method can be found in Tezaur s dissertation ....

....spaces V : # # U : ##, Bz# = 0 #z # ker (S) ker (G t ) range (P ) FETI AND NEUMANN NEUMANN METHODS 7 and V # : # U : #, Bz#Q = 0 #z # ker (S) range (P t ) It can be easily shown that V # is isomorphic to the dual space of V . Following Farhat, Chen, and Mandel [12], we call V the space of admissible increments. The original FETI method is a conjugate gradient method in the space V applied to P t F# = P t d, # # # 0 V, 12) with an initial approximation # 0 chosen such that G t # 0 = e. The most basic FETI preconditioner, as introduced in Farhat, ....

Charbel Farhat, Po-Shu Chen, and Jan Mandel. A scalable Lagrange Multiplier Based Domain Decomposition Method for Time-dependent Problems. Int. J. Numer. Meth. Engng., 38(22):3831--3853, 1995.

....problems. This method was introduced by Farhat and Roux [25] a detailed presentation is given in [26] a monograph by the same authors. Originally used to solve second order, self adjoint elliptic equations, it has later been extended to many other problems, e.g. time dependent problems [17], plate bending problems [18, 23, 42] heterogeneous elasticity problems with composite materials [44, 45] acoustic scattering and Helmholtz problems [21, 22, 27, 28] linear elasticity with inexact solvers [31] and Maxwell s equations [43, 50] Another Lagrange multiplier based method, the ....

Charbel Farhat, Po-Shu Chen, and Jan Mandel. A scalable Lagrange multiplier based domain decomposition method for time-dependent problems. Int. J. Numer. Meth. Eng., 38:3831{ 3853, 1995.

....h from now on. 3. A review of the FETI method. In this section, we give a brief review of the original FETI method introduced in Farhat and Roux [7] and the variant with a Dirichlet preconditioner introduced in Farhat, Mandel, and Roux [6] For more detailed descriptions and proofs, we refer to [4, 5, 16, 21] and the references therein. Let the domain Omega ae R 2 be decomposed into N non overlapping subdomains Omega i ; i = 1; N; each of which is the union of elements and such that the finite element nodes on the boundaries of neighboring subdomains match across the interface Gamma : ....

Ch. Farhat, P.S. Chen, and J. Mandel. A scalable Lagrange Multiplier Based Domain Decomposition Method for Time-dependent Problems. Int. J. Numer. Meth. Engrg., 38(22):3831-- 3853, 1995.

....in [29, 30] based on mechanical arguments. A mathematical analysis of these and of some extended FETI algorithms will be given in [22] The family of FETI methods has also been extended to problems that lack natural auxiliary coarse problems, e.g. time dependent problems from elastodynamics, see [9, 12], and acoustic scattering problems, see [11] Furthermore, FETI algorithms have also been developed for plate and shell problems, see, e.g. 12, 10] for algorithmic descriptions and numerical results and [24, 32] for a mathematical analysis. In this paper, we consider a FETI method for the edge ....

....a mathematical analysis. In this paper, we consider a FETI method for the edge element approximation of Problem (3) Here, the local problems are not singular and, as in the case of time dependent problems, there is no natural coarse problem associated with the subdomains. We will proceed as in [9], and propose a set of local functions that will allow us to build a coarse space for the Lagrange multipliers. In addition, following [30, 22] we also propose a family of preconditioners, built from the values of the coecient A in (2) An important feature of our method is that the condition ....

[Article contains additional citation context not shown here]

Charbel Farhat, Po-Shu Chen, and Jan Mandel. A scalable Lagrange multiplier based domain decomposition method for time-dependent problems. Int. J. Numer. Meth. Eng., 38:3831{ 3853, 1995.

....1 =p t k Sp k U k = U k 1 k p k r k = r k 1 k Sp k We remark that the residuals r k are perpendicular to the coarse space, since e RH r k = 0, for every k. In addition, the rst projection can be omitted, since W k = r k for every k, thanks to the choice of the initial vector U 0 . See [14], for a similar algorithm. 8. Numerical results. In this section, we present some numerical results on the performance of the hybrid Neumann Neumann method described in the previous sections, when varying the diameters of the coarse and ne meshes, and the coecients a and B. We only consider ....

Charbel Farhat, Po-Shu Chen, and Jan Mandel. A scalable Lagrange multiplier based domain decomposition method for time-dependent problems. Int. J. Numer. Meth. Eng., 38:3831{ 3853, 1995.

....was introduced by Farhat and Roux [16] a detailed presentation is given in [17] a monograph by the same authors. Originally used to solve second order, self adjoint elliptic equations, it has later been extended to many other problems, e.g. time dependent problems, cf. Farhat, Chen, and Mandel [11], plate bending problems, cf. Farhat et al. [12, 13, 15] and heterogeneous elasticity problems with composite materials, cf. Farhat and Rixen [26, 27] Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012. Email address: stefanic cims.nyu.edu URL: ....

Charbel Farhat, Po-Shu Chen, and Jan Mandel. A scalable Lagrange multiplier based domain decomposition method for time-dependent problems. Int. J. Numer. Meth. Eng., 38:3831-3853, 1995.

....but also a blessing. Farhat, Mandel, and Roux [32] have shown numerically, and proved for the FETI method without preconditioning, that the auxiliary problem plays the role of a coarse problem, namely, it causes the condition number to be bounded independently of the number of subdomains. In [26], Farhat, Chen, and Mandel extended to time dependent problems, which lack the naturally occurring coarse problem. The FETI method is in a sense dual to the Neumann Neumann method with a coarse problem, developed by Mandel under the name Balancing Domain Decomposition [45] BDD) which we ....

....T ; 3.16) where 0 is an initial approximation to the conjugate gradient method chosen so that G T 0 = e, and all search directions are in the space Ker G T ae . 3.2 Generalized FETI There have been several extensions to the original method. Extension to time dependent problems was done in [26]. Generalization to plate bending problems is discussed in [51, 53] and, in detail, in this thesis. For the original method for plate bending problems, the condition number was observed to grow fast with the number of elements per subdomain [32] This is caused by the fact that plate bending is a ....

[Article contains additional citation context not shown here]

C. Farhat, P. S. Chen, and J. Mandel. Scalable Lagrange multiplier based domain decomposition method for time-dependent problems. Int. J. Numer. Meth. Engrg., 38:3831--3853, 1995.

.... by the subdomain corner modes [27, 28] This enrichment transforms the original one level FETI method into a genuine two level algorithm known as the two level FETI method [27, 28] Both the one level and two level FETI methods have been extended to transient dynamics problems as described in [26]. 6 Unfortunately, enriching the coarse problem of the one level FETI method by the subdomain corner modes increases its computational complexity to a point where the overall scalability of FETI is diminished on a very large number of processors, say . 21 1 1 . For this reason, the basic ....

C. Farhat, P. S. Chen and J. Mandel. A scalable Lagrange multiplier based domain decomposition method for implicit time-dependent problems. Internat. J. Numer. Meths. Engrg. 38, 3831-3858 (1995).

....a linear problem with positive definite subdomain matrices in each time step. The coarse space built from null spaces is lost, resulting in deteriorating convergence with growing number of subdomains. Quasi optimal convergence properties were retained by introducing an artificial coarse space [FCM95]. For plate bending problems, the condition number was observed to grow fast with the number of elements per subdomain [FMR94] This was resolved by adding to the coarse space Lagrange multipliers that enforce continuity at the corners [MTF] A related idea has been employed in the Balancing ....

....process on a linear system with the matrices K s from (14) with 0 Deltat 1, and a fixed right hand side. Let k ( Deltat) denote the approximate solution after k iterations of FETI for a given Deltat. Then, for all k, lim Deltat 1 k ( Deltat) k ( 1) For further details, see [FCM95]. FETI for Plates Here, the columns of C are chosen as vectors with a one at the position of the Lagrange multiplier that enforces the continuity of the transversal displacement at a crosspoint, and zeroes elsewhere. A crosspoint is an interface node adjacent to at least three subdomains or to ....

Farhat C., Chen P. S., and Mandel J. (1995) Scalable Lagrange multiplier based domain decomposition method for time-dependent problems. Int. J. Numer. Meth. Engrg. 38: 3831--3853.

....subdomains interfaces. All original degrees of freedom are then eliminated, leaving a dual system for the Lagrange multipliers, which is solved by preconditioned conjugate gradients. For details, further developments, and theoretical analysis of the FETI method for positive de nite problems, see [1, 4, 5, 9, 10, 13, 14, 16, 17, 18, 20, 21] and references therein. Variants of the FETI method for the Helmholtz equation of scattering, were proposed by De La Bourdonnaye et al. 3] Farhat et al. 6, 7, 8] and further developed by Tezaur et al. 22] In this paper, we present an extension of the method of [7, 8] called FETI H, to the ....

C. Farhat, P. S. Chen, and J. Mandel, Scalable Lagrange multiplier based domain decomposition method for time-dependent problems, Int. J. Numer. Meth. Engrg., 38 (1995), pp. 3831-3853.

....between the subdomains and results in a method which, for elasticity problems, has a condition number that grows only polylogarithmically with the number of elements per subdomain, and is bounded independently of the number of subdomains. Extension to time dependent problems was done in [9]. However for plate bending problems, Center for Computational Mathematics, University of Colorado at Denver, Denver, CO 802173364, USA, and Department of Aerospace Engineering Sciences, University of Colorado at Boulder, Boulder, CO 80309 0429, USA. Email: jmandel colorado.edu y Center for ....

....i ) Gamma rw s (x i ) 2.19) where oe rs = 1 or oe rs = Gamma1. Here, rw r (x i ) means the values of 1 and 2 degrees of freedom at node x i . In particular, the entries of B are Gamma1; 0; 1, and they are constant along an edge between two subdomains. Remark 2.3. In an earlier paper [9], we have studied the case of time dependent problems where the subdomain stiffness matrices K s are perturbed by the addition of a multiple of the subdomain mass matrix, thus making the new local matrix positive 6 Fig. 2.1. Definition of B t t t t t t t tt t t t t t t t t r r r r r r r r r r ....

[Article contains additional citation context not shown here]

C. Farhat, P. S. Chen, and J. Mandel, Scalable Lagrange multiplier based domain decomposition method for time-dependent problems, Int. J. Numer. Meth. Engrg., 38 (1995), pp. 3831--3853.

No context found.

C. Farhat, P. S. Chen, and J. Mandel. A scalable Lagrange multiplier based domain decomposition method for implicit time-dependent problems. Internat. J. Numer. Meths. Engrg., 38:3831--3854, 1995.

....a linear problem with positive definite subdomain matrices in each time step. The coarse space built from null spaces is lost, resulting in deteriorating convergence with growing number of subdomains. Quasi optimal convergence properties were retained by introducing an artificial coarse space [FCM95]. For plate bending problems, the condition number was observed to grow fast with the number of elements per subdomain [FMR94] This was resolved by adding to the coarse space Lagrange multipliers that enforce continuity at the corners [MTF] A related idea has been employed in the Balancing ....

....process on a linear system with the matrices K s from (1.14) with 0 Deltat 1, and a fixed right hand side. Let k ( Deltat) denote the approximate solution after k iterations of FETI for a given Deltat. Then, for all k, lim Deltat 1 k ( Deltat) k ( 1) For further details, see [FCM95]. 1.5.3 FETI for Plates Here, the columns of C are chosen as vectors with a one at the position of the Lagrange multiplier that enforces the continuity of the transversal displacement at a crosspoint, and zeroes elsewhere. A crosspoint is an interface node adjacent to at least three subdomains or ....

Farhat C., Chen P. S., and Mandel J. (1995) Scalable Lagrange multiplier based domain decomposition method for time-dependent problems. Int. J. Numer. Meth. Engrg. 38: 3831--3853.

....by the National Science Foundation under grants ASC 9217394 and ASC 9121431. This paper has been submitted for journal publication elsewhere. subdomains. The method was further extended to time dependent problems, which lack the naturally occurring coarse problem, by Farhat, Chen, and Mandel [9]. In this paper, we show that the condition number of the preconditioned FETI method is bounded independently of the number of subdomains and polylogarithmically in terms of subdomain size, as is the case for other optimal non overlapping domain decomposition methods [3, 6, 8, 16, 17] We refer to ....

C. Farhat, P. S. Chen, and J. Mandel, Scalable Lagrange multiplier based domain decomposition method for time-dependent problems, Tech. Report CUCAS -94-21, Center for Aerospace Structures, University of Colorado at Boulder, November 1994.

....1. In [20] it was shown that such a strategy is cost effective for substructure problems because reorthogonalization is applied only to the interface Lagrange multiplier unknowns. The FETI method has also been extended to problems with multiple right hand sides [9, 12] transient dynamic analysis [10, 13,14], fourth order plate and shell problems [8, 17, 30] non linear structural analysis [36, 41] and problems with multipoint constraints [15] The specific extension of FETI to plate and shell problems has initiated the development of what is known today as the twolevel FETI method (see [8, 17,30] ....

C. Farhat, P.S. Chen, and J. Mandel. A scalable Lagrange multiplier based domain decomposition method for time-dependent problems. Internat. J. Numer. Meths. Engrg., 38(22):3831--3853, November 1995.

....between the subdomains and results in a method which, for elasticity problems, has a condition number that grows only polylogarithmically with the number of elements per subdomain, and is bounded independently of the number of subdomains. Extension to time dependent problems was done in [9]. However for plate bending problems, the condition number was Center for Computational Mathematics, University of Colorado at Denver, Denver, CO 802173364, USA, and Department of Aerospace Engineering Sciences, University of Colorado at Boulder, Boulder, CO 80309 0429, USA. Email: ....

....(x i ) Gamma rw s (x i ) 2.19) where oe rs = 1 or Gamma1. Here, rw r (x i ) means the values of the 1 and 2 degrees of freedom at node x i . In particular, the entries of B are Gamma1; 0; 1, and they are constant along any edge between two subdomains. Remark 2.3. In an earlier paper [9], we have studied time dependent problems where the subdomain stiffness matrices K s are perturbed by adding a multiple of the subdomain mass matrix, thus making the local matrices positive definite. Consequently, all matrices Z s are void and the natural coarse problem is lost in time dependent ....

[Article contains additional citation context not shown here]

C. Farhat, P. S. Chen, and J. Mandel, Scalable Lagrange multiplier based domain decomposition method for time-dependent problems, Int. J. Numer. Meth. Engrg., 38 (1995), pp. 3831--3853.

....that the auxiliary problem plays the role of a coarse problem, namely, it causes the condition number to be bounded independently of the number of subdomains. The method was further extended to time dependent problems, which lack the naturally occurring coarse problem, by Farhat, Chen, and Mandel [8]. In this paper, we show that the condition number of the preconditioned FETI method is bounded independently of the number of subdomains and polylogarithmically in terms of subdomain size, as is the case for other optimal non overlapping domain decomposition methods [3, 5, 7, 15, 16] We refer to ....

C. Farhat, P. S. Chen, and J. Mandel, Scalable Lagrange multiplier based domain decomposition method for time-dependent problems, Int. J. Numer. Meth. Engrg., (1995). To appear.

....This approach is also suitable for solving problems arising from mixed formulations. For the application of domain decomposition techniques to mixed finite element methods, see [43] 34] 25] and the references therein. The dual approach has been introduced in [39] and further studied in [69] [36] and [55] For domain decomposition methods on nonconforming finite element discretizations, see [4] 2] 53] 57] 19] and their references. 3. Fully Black box Overlapping Schwarz Method In this chapter, we propose a practical solver based on the framework of overlapping Schwarz methods. ....

C. Farhat, P. S. Chen, and J. Mandel, Scalable Lagrange multiplier based domain decomposition method for time-dependent problems, Int. J. Numer. Meth. Engrg., (1995). To appear.

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C. Farhat, P. S. Chen, J. Mandel, A scalable Lagrange multiplier based domain decomposition method for implicit time-dependent problems, International Journal for Numerical Methods in Engineering 38 (1995) 3831--3854.

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