C. Farhat and F. X. Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm. Internat. J. Numer. Meths. Engrg., 32:1205--1227, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the constraint G = e. Two di erent type preconditioners have been proposed in the literature for the FETI methods for positive de nite elliptic problems: the computationally economical lumped preconditioners, cf. 32] and the mathematically optimal Dirichlet preconditioners, cf. 30] and [33]. A condition number bound, C(1 log(H=h) has been proved in [56] for the FETI method with Dirichlet preconditioner, both in two and three dimensions. A scaled Dirichlet preconditioner is proposed in [53] and a condition number bound C(1 log(H=h) is proved. This bound is also ....

Charbel Farhat and Francois-Xavier Roux. A Method of Finite Element Tearing and Interconnecting and its Parallel Solution Algorithm. Int. J. Numer. Meth. Engrg., 32:1205-1227, 1991.

....and various combinations between them and the local preconditioners. In the framework of non overlapping domain decomposition techniques, we refer for instance to BPS (Bramble, Pasciak and Schatz) 3] Vertex Space [7, 17] and to some extent Balancing Neumann Neumann [13, 14, 15] as well as FETI [10, 16], for the presentation of major two level preconditioners. We refer to [6] and [18] for a more exhaustive overview of domain decomposition techniques. CERFACS France and COPPE UFRJ Brazil. This work was partially supported by FAPERJ Brazil under grant 150.177 98. CERFACS, 42 av. Gaspard ....

C. Farhat and F.-X. Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Meth. Engng., 32:1205--1227, 1991.

....depend on the number of subdomains and only weakly depends on the number of mesh points within subdomains. Other non overlapping domain decomposition preconditioners that possess similar optimality properties include the vertex space [5, 14] the balancing NeumannNeumann [9, 10, 11] and the FETI [6, 12] methods. For most of these techniques, this property can also be extended to discontinuous coe#cient problems under the assumption that the jumps in the coe#cients align with the interfaces between subdomains [8, 9, 15, 19] While domain decomposition techniques can be applied to situations where ....

....a strange domain with Neumman boundary conditions everywhere except at the two end points where Dirichlet boundary conditions are enforced. This use of local sti#ness matrices has been found beneficial in other multilevel methods such as the AMGe algebraic multigrid algorithm [2] the FETI method [6], and the balancing NeummanNeumman scheme [10] In all of these algorithms, the local sti#ness matrices provides a local submatrix which is closely connected to the original physical problem (e.g. a substructure) Fig. 3.1. An interface on unstructured meshes. Lemma 4. The operator dependent ....

C. Farhat and F.-X. Roux, A method of finite element tearing and interconnecting and its parallel solution algorithm, Int. J. Numer. Meth. Engng., 32 (1991), pp. 1205--1227.

....or fourth order elasticity problems. The one level FETI method equipped with the Dirichlet preconditioner was shown to be numerically scalable for second order elasticity problems while the two level FETI method was designed to be numerically scalable for fourth order elasticity problems (see [FR94, Far91b, Far91a, FR91, FR92, FM98, Rou95]) The second level coarse grid is an enriched version of the original one level FETI method with coarse grid. The coarse problem is enriched by enforcing transverse displacements to be continuous at the corner points. This coarse problem grows linearly with the number of subdomains. Current ....

Charbel Farhat and Francois-Xavier Roux. A Method of Finite Element Tearing and Interconnecting and its Parallel Solution Algorithm. Int. J. Numer. Meth. Engng., 32:1205--1227, 1991.

....in accuracy, especially in the structure solution. However, this trade off of accuracy for speed may be desirable in a preliminary design. At every time step, the corresponding linearized system of equations is solved via the FETI (Finite Element Tearing and Interconnecting) substructuring method [40,41]. The FETI algorithm is an optimal domain decomposition iterative algorithm which is based on a saddle point variational principle. It incorporates a mechanically sound preconditioner and a natural coarse grid operator that propagate the error globally, accelerate convergence, and ensure ....

C. Farhat and F. X. Roux, A Method of Finite Element Tearing and Interconnecting and its Parallel Solution Algorithm, International Journal for Numerical Methods in Engineering, 32,1205-1227, (1991).

....information to between neighboring substructures. In the absence of a mechanism for exchanging information among all substructures, the condition number of the problem grows with the number of substructures. This problem of lack of numerical scalability has been addressed by both the FETI method [15] and the closely related balancing method [26] through the use of a coarse problem for exchanging information among all the sub domains. These methods have demonstrated numerical scalability for both second order elasticity problems [13] and fourth order plate and shell problems [12] 7] The ....

....error globally during the PCG iteration, thus accelerating convergence. 2. 1 Two Sub domain Case In the simplest case of two sub domains, 1) and (2) Figure 1, the variational form of the three dimensional boundary value problem to be solved can be reduced to the following algebraic system [15]: K (1) u (1) f (1) B (1) T (2.1) K (2) u (2) f (2) B (2) T (2.2) B (1) u (1) B (2) u (2) 2.3) 6 where K (j) u (j) and f (j) j = 1; 2 are the sti ness matrix, the displacement vector, and the prescribed force vector, respectively, associated ....

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Farhat, C., and Roux, F.-X. A Method of Finite Element Tearing and Interconnecting and its Parallel Solution Algorithm. International Journal for Numerical Methods in Engineering 32 (1991), 1205-1227.

....analysis of FETI methods, see [25, 39] FETI algorithms can be employed successfully for advection di#usion problems as well. We are primarily interested in convection dominated problems. FETI methods were first introduced for the solution of conforming approximations of elasticity problems in [15]. In this approach, the original domain# is decomposed into non overlapping subdomains# i , i = 1, N . On each subdomain# i a local sti#ness matrix is obtained from the finite element discretization of local Neumann # Courant Institute of Mathematical Sciences, 251 Mercer Street, New ....

Charbel Farhat and Francois-Xavier Roux. A method of Finite Element Tearing and Interconnecting and its parallel solution algorithm. Int. J. Numer. Meth. Eng., 32:1205--1227, 1991.

....the subregions into which the region of the original problem has been partitioned. However, there are also di#erences and we have seen a need to extend our understanding of the FETI algorithms. The Finite Element Tearing and Interconnecting (FETI) methods were first introduced by Farhat and Roux [16]. An important advance, making the rate of convergence of the iteration less sensitive to the number of unknowns of the local problems, was made by Farhat, Mandel, and Roux a few years later [14] Our own work is based on the pioneering work by Mandel and Tezaur [24] who fully analyzed a variant ....

....h # ## h , by i (x) # 1 i (x) if i (x) #= 0, 0 if i (x) 0. We note that these functions provide a partition of unity: # i # # i (x) i (x) # 1 #x # # h # ## h . 5) 3. A review of the FETI method. In this section, we review the original FETI method of Farhat and Roux [16, 17] and the variant with a Dirichlet preconditioner introduced in Farhat, Mandel, and Roux [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 ....

Charbel Farhat and Francois-Xavier Roux. A Method of Finite Element Tearing and Interconnecting and its Parallel Solution Algorithm. Int. J. Numer. Meth. Engng., 32:1205--1227, 1991.

....have been investigated extensively (see [63] and the references therein) Here we mention only one method which has been well developed and tested on structures problems. 52 4.4. 1 A domain decomposition method The finite element tearing and interconnect method (FETI) developed by Farhat et al.[35, 34] is an iterative substructuring method that tears the monolithic problem into a series of subdomains. These subdomains are interconnected via Lagrange multipliers thus the Schur complement is only solving for the Lagrange multipliers (these become the primary variables in the solve) As the ....

Charbel Farhat and Francois-Xavier Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm. International Journal for Numerical Methods in Engineering, 32:1205--1227, 1991.

....continuity is then enforced by using Lagrange multipliers across the interface defined by the subdomain boundaries. A computationally very efficient member of this class of domain decomposition algorithms is the Finite Element Tearing and Interconnecting (FETI) method introduced by Farhat and Roux [7]. In its original version, a Neumann problem is solved on each subdomain and the method is known to be scalable in the sense that its rate of convergence is independent of the number of subproblems. In a variant of the FETI method introduced in Farhat, Mandel, and Roux [6] an additional Dirichlet ....

....a(u h ; v h ) hF; v h i 8v h 2 W h( Omega Gamma : 3) In what follows, we work exclusively with the discrete problem and we drop the subscript 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; ....

Ch. Farhat and F.-X. Roux. A Method of Finite Element Tearing and Interconnecting and its Parallel Solution Algorithm. Int. J. Numer. Meth. Engng., 32:1205--1227, 1991.

....method with Lagrange multipliers for the solution of linear systems arising from the edge element approximations. Our algorithm belongs to the family of Finite Element Tearing and Interconnecting (FETI) methods which have been rst introduced for the solution of elasticity problems in [14]. In this approach, the original domain is decomposed into nonoverlapping subdomains i , i = 1; N . On each subdomain i a local sti ness matrix is obtained from the nite element discretization of a i ( Analogously, a set of right hand sides is built. The continuity of the ....

Charbel Farhat and Francois-Xavier Roux. A method of Finite Element Tearing and Interconnecting and its parallel solution algorithm. Int. J. Numer. Meth. Eng., 32:1205-1227, 1991.

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C. Farhat and F.-X. Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm. International Journal for Numerical Methods in Engineering, 32:1205--1227, 1991.

....Element Tearing and Interconnection) is the generic name for a suite of DD based iterative solvers with Lagrange multipliers designed with the condition number estimates (1) and (2) in mind. The first and simplest FETI algorithm, known as the one level FETI method, was developed around 1989 [23, 24]. It can be described as a two step Preconditioned Conjugate Gradient (PCG) algorithm where subdomain problems with Dirichlet boundary conditions are solved in the preconditioning step, and related subdomain problems with Neumann boundary conditions are solved in a second step. The one level FETI ....

C. Farhat and F. X. Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm. Internat. J. Numer. Meths. Engrg. 32, 1205-1227 (1991).

....differential equations. It is based on using direct solvers in subdomains and enforcing continuity at subdomain interfaces by Lagrange multipliers. The dual problem for the Lagrange multipliers is solved by a preconditioned conjugate gradient (PCG) algorithm. The FETI method was developed in [Far91, FR91, FR92], and discussed in detail in the monograph [FR94] Unlike other related domain decomposition methods using Lagrange multipliers as unknowns [GW88, Rou90] the FETI method uses the null spaces of the subdomain stiffness matrices (rigid body modes) to construct a small coarse problem that is ....

....simply by its leading term K bb . This is equivalent to lumping each subdomain stiffness on its interface boundary. The resulting preconditioner is given by QD = QB b K bb B T b (12) 4 Special Instances of FETI FETI for Solid Mechanics (Second Order Elasticity) The original FETI algorithm [Far91, FR91, FR92] is obtained by omitting the condition (6) Then, Q becomes the identity, and an initial approximation 0 is only SUBSTRUCTURING BY LAGRANGE MULTIPLIERS 25 required to satisfy G T 0 = e. It was proved in [MT96] that for the Laplace equation, P1 conforming elements, and the Dirichlet ....

Farhat C. and Roux F.-X. (1991) A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Meth. Engng. 32: 1205--1227.

....via Lagrange multipliers applied at the subdomain interfaces. This gluing phase generates a smaller size symmetric dual problem where the unknowns are the Lagrange multipliers, and which is best solved by a preconditioned conjugate gradient (PCG) algorithm. The FETI method was developed in [8, 13, 14], and discussed in detail in the monograph [15] In contrast with other related domain decomposition methods using Lagrange multipliers as unknowns [17, 27] the FETI method distinguishes itself with the treatment of the null spaces of the subdomain stiffness matrices (rigid body modes) associated ....

....with (2.5) show that satisfies the system of equations P (F Gamma d) 0 G T = e; 2.7) where G = BZ; F = BK B T ; d = BK f; P = I Gamma G(G T G) Gamma1 G T ; e = Z T f: 2.8) See Lemma 3.1 for justification that G T G is nonsingular. The original FETI method [13] is the method of preconditioned conjugate gradients for the equation, equivalent to (2.7) PF = P d; 2.9) where the initial approximation 0 is chosen so that G T 0 = e, and all search directions are in Ker G T . We will show later (Section 3.1) that the solution of (2.7) is unique up to ....

C. Farhat and F.-X. Roux, A method of finite element tearing and interconnecting and its parallel solution algorithm, Int. J. Numer. Meth. Engng., 32 (1991), pp. 1205--1227.

....severe ill conditioning that renders the method ineffective in practice. It is still unclear whether a current implementation of PUM for acoustic waves [47] can circumvent this difficulty. Motivated by PUM, RFB, the FETI method for nonconforming domain decomposition with Lagrange multipliers [24, 25, 29, 31], and recent work on discontinuous Galerkin methods (DGM) for second order equations [3, 8, 9, 52] we propose herein a discretization method in which the standard finite element polynomial field within each element is enriched by free space solutions of the governing homogeneous, ....

.... of structural systems modeled by different types of elements [53] to the investigation of contact problems [55] to the synthesis of independently discretized subdomains and modeled substructures [1, 10, 26, 27] and to the 2 design of fast, domain decomposition based, iterative solvers [24, 30, 31]. The remainder of this paper is organized as follows. The Discontinuous Enrichment Method (DEM) is presented in Sec. 2 as a general approach for improving finite element computation. Implementational issues related to static condensation of the enrichment field, approximation of Lagrange ....

C. Farhat and F.-X. Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm. Internat. J. Numer. Methods Engrg., 32(6):1205--1227, 1991.

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C. Farhat and F. X. Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm. Internat. J. Numer. Meths. Engrg., 32:1205--1227, 1991.

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Farhat, C., Roux, F.-X.: A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Meth. Engng., 32, 1205--1227 (1991)

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C Farhat and F-X Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm. International Journal for Numerical Methods in Engineering, 32:1205--1227, 1991. 21

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C. Farhat, F.-X. Roux, A method of finite element tearing and interconnecting and its parallel solution algorithm, International Journal for Numerical Methods in Engineering 32 (1991) 1205--1227. 29

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C. Farhat, F.-X. Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm. International Journal for Numerical Methods in Engineering, 32:1205--1227, 1991.

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C. Farhat and F.-X. Roux. A Method of Finite Element Tearing and Interconnecting and its Parallel Solution Algorithm. Int. J. Numer. Meth. Engrg., 32:1205--1227, 1991.

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C. Farhat and F.-X. Roux, A method of finite element tearing and interconnecting and its parallel solution algorithm, International Journal for Numerical Methods in Engineering, 32 (1991), pp. 1205--1227.

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C. Farhat and F.-X. Roux. A Method of Finite Element Tearing and Interconnecting and its Parallel Solution Algorithm. Int. J. Numer. Meth. Engrg., 32:1205--1227, 1991.

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Farhat, C. and Roux, F. X., "A Method of Finite Element Tearing and Interconnecting and its Parallel Solution," Internat. J. Numer. Maths. Engrg., Vol. 32, 1991, pp. 1205-1227.

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