A Neumann--Neumann Domain Decomposition Algorithm for Solving Plate and Shell Problems

Tallec, Patrick Le; Mandel, Jan; Vidrascu, Marina

doi:10.1137/s0036142995291019

Cited by 78 publications

(65 citation statements)

References 25 publications

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“…Most of the early theoretical and numerical work has been carried out for scalar symmetric positive definite second order problems; see for example [6], [13]- [15], [23]. Then, the method was extended to other problems, like the advection-diffusion equations [1,7], plate and shell problems [27] or the Stokes equations [26,22].…”

Section: Introductionmentioning

confidence: 99%

Deriving a new domain decomposition method for the Stokes equations using the Smith factorization

Dolean

Nataf

Rapin³

2008

Math. Comp.

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Abstract. In this paper the Smith factorization is used systematically to derive a new domain decomposition method for the Stokes problem. In two dimensions the key idea is the transformation of the Stokes problem into a scalar bi-harmonic problem. We show, how a proposed domain decomposition method for the bi-harmonic problem leads to a domain decomposition method for the Stokes equations which inherits the convergence behavior of the scalar problem. Thus, it is sufficient to study the convergence of the scalar algorithm. The same procedure can also be applied to the three-dimensional Stokes problem.As transmission conditions for the resulting domain decomposition method of the Stokes problem we obtain natural boundary conditions. Therefore it can be implemented easily.A Fourier analysis and some numerical experiments show very fast convergence of the proposed algorithm. Our algorithm shows a more robust behavior than Neumann-Neumann or FETI type methods.

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Section: Introductionmentioning

confidence: 99%

Deriving a new domain decomposition method for the Stokes equations using the Smith factorization

Dolean

Nataf

Rapin³

2008

Math. Comp.

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“…Many numerical verifications have confirmed that property. For fourth-order problems (plates and shells), displacement continuity at the "corners" of the substructures must be enforced in the macrospace in order to maintain scalability [39,40]. Since the coarse problem at the corner is sufficient to ensure the invertibility of the Steklov operators, strategies such as the FETI-DP [26] and the BDDC [41,42] enable one to remove the rigid-body modes from the macrospace.…”

Section: Comparison With Other Multiscale Strategiesmentioning

confidence: 99%

On a mixed and multiscale domain decomposition method

Ladevèze

Néron

Gosselet

2007

Computer Methods in Applied Mechanics and Engineering

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This paper presents a reexamination of a multiscale computational strategy with homogenization in space and time for the resolution of highly heterogeneous structural problems, focusing on its suitability for parallel computing. Spatially, this strategy can be viewed as a mixed, multilevel domain decomposition method (or, more accurately, as a "structure decomposition" method). Regarding time, a "parallel" property is also described. We also draw bridges between this and other current approaches.Key words: domain decomposition, multiscale, computational mechanics IntroductionIn the structural mechanics field, one can observe a surge of interest in the multiscale analysis of structures with complex microstructural geometry and/or complex behavior. When accurate solutions are required, calculations must be performed on a finely discretized model of the structure (defined on what is called the "micro" level) consistent with short lengths of variation, both in space and in time. A major application is the analysis of composite structures described on the microscale or on the mesoscale [1]; there are also other applications, such as in [2,3]. These types of situations lead to problems with very large numbers of degrees of freedom and computation costs which are prohibitive if one uses classical finite element codes. One of the main objectives * Corresponding author. E-mail: pierre.ladeveze@lmt.ens-cachan.fr Preprint of the last few decades has been to develop efficient and robust computational strategies suitable for these types of problems. One of these strategies uses the theory of the homogenization of periodic media [4][5][6]. Other developments and the associated computational approaches can be found in [7][8][9][10][11][12][13][14]. Besides periodicity, these strategies rely on the fundamental assumption that the ratio between the two scales is small. Thus, the boundary zones require specific treatment because in these zones the microstructure cannot be homogenized. Here, we follow a recently introduced multiscale computational strategy for nonlinear evolution problems. This strategy involves an automatic homogenization technique in space as well as in time [15,16] which is an extension of previous works limited to space alone [17,18]. This strategy, developed in a general framework, makes no a priori assumption regarding the form of the solution and, therefore, does not suffer from the limitations of standard homogenization techniques; moreover, it relies on an iterative algorithm. Until now, this strategy has been developed in the framework of small displacements of (visco)plastic structures under possible contact with or without friction. This paper is a reconsideration of this computational strategy in order to assess its adaptability to parallel computing. Therefore, we examine its three main characteristics. The first characteristic is that it can be viewed as a mixed, multilevel domain decomposition method or, more precisely as will explained further on, as a "structure decomposition" method. Ind...

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“…In an implementation of the BDD preconditioner (3), (9) , it is advantageous to use an initial approximation as u B0 = R 0 S −1 0 R T 0 g, then the Step 1 and Step 2 in every iteration can be omitted since after using the residual of the initial approximation in the Step 1, the unknown vector λ becomes the zero vector. As a result, the BDD preconditioner Eq.…”

Section: Construction Of the Bdd Preconditionermentioning

confidence: 99%

“…This method is introduced by Mandel (3) by adding a coarse problem to an earlier method of De Roeck and Le Tallec (4) known as the NeumannNeumann method and then is extended for problems with large jumps in coefficients (5) . Mandel's BDD algorithm is now exclusively employed for the solution of huge structural problems (6) , Stokes problems (7) , semiconductor simulation (8) and plate and shell problems (9) . While this method shows good performances to solve large scale elastic problems (10) , it still suffers from the high computation costs and the large required memory to solve the 3-dimensional heat conductive problems (11) .…”

Section: Introductionmentioning

confidence: 99%

A Scalable Balancing Domain Decomposition Based Preconditioner for Large Scale Heat Transfer Problems

Mukaddes

Ogino

Kitamura

et al. 2006

JSME international journal. Ser. B, Fluids and thermal engineer

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An efficient and scalable Balancing Domain Decomposition (BDD) type preconditioner for large scale linear systems arising from 3-dimensional heat transfer problems is presented. The new method improves parallel scalability of BDD by employing an incomplete balancing technique to approximate a coarse space problem and a diagonal scaling to precondition the local fine space problems instead of the Neumann-Neumann preconditioner. It may increase the number of iterations but reduces the computation costs of the precondition process for each iteration. Consequently, total computation time and required memory are expected to be reduced. The convergence estimates may also be independent of the number of subdomains. We have implemented this algorithm on the parallel processors and have succeeded in solving some ill-conditioned large scale heat transfer problems.

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A Neumann--Neumann Domain Decomposition Algorithm for Solving Plate and Shell Problems

Cited by 78 publications

References 25 publications

Deriving a new domain decomposition method for the Stokes equations using the Smith factorization

Deriving a new domain decomposition method for the Stokes equations using the Smith factorization

On a mixed and multiscale domain decomposition method

A Scalable Balancing Domain Decomposition Based Preconditioner for Large Scale Heat Transfer Problems

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