Convergence of the domain decomposition finite element–boundary element coupling methods

El-Gebeily, M. A.; Elleithy, Wael; Al-Gahtani, Husain J.

doi:10.1016/s0045-7825(02)00405-x

Cited by 28 publications

(23 citation statements)

References 11 publications

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“…The convergence characteristics of the interface relaxation FEM-BEM coupling methods were studied extensively by Elleithy and co-workers [20,21]. The convergence/optimal convergence of the interface relaxation coupling methods is ensured by properly set relaxation parameters that can be assigned constant values for all iterations.…”

Section: Bem Sub-domainmentioning

confidence: 99%

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An adaptive domain decomposition coupled finite element–boundary element method for solving problems in elasto‐plasticity

Elleithy

Grzhibovskis

2009

Numerical Meth Engineering

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SUMMARYThe purpose of this paper is to present an adaptive finite element-boundary element method (FEM-BEM) coupling method that is valid for both two-and three-dimensional elasto-plastic analyses. The method takes care of the evolution of the elastic and plastic regions. It eliminates the cumbersome of a trial and error process in the identification of the FEM and BEM sub-domains in the standard FEM-BEM coupling approaches. The method estimates the FEM and BEM sub-domains and automatically generates/adapts the FEM and BEM meshes/sub-domains, according to the state of computation. The results for two-and three-dimensional applications in elasto-plasticity show the practicality and the efficiency of the adaptive FEM-BEM coupling method.

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Section: Bem Sub-domainmentioning

confidence: 99%

“…The optimal and applicable ranges of static relaxation parameters may be obtained by experimenting with different values. Alternatively, prior to coupled FEM-BEM calculations, static values of the relaxation parameters may be obtained [20,21]. It requires, however, some sort of intricate matrix manipulations.…”

Section: Bem Sub-domainmentioning

confidence: 99%

An adaptive domain decomposition coupled finite element–boundary element method for solving problems in elasto‐plasticity

Elleithy

Grzhibovskis

2009

Numerical Meth Engineering

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“…It is also possible to use different time steps for every method. The convergence of these algorithms is influenced by the selection of the time step of each method and the choice of the iterative coupling relaxation parameter [39,40,41]. Lastly, Soares Jr. et al [42] have presented a formulation that avoids the iterative procedures but considers the FEM and BEM separately.…”

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confidence: 99%

3D non-linear time domain FEM–BEM approach to soil–structure interaction problems

Romero¹,

Galvín²,

Domínguez³

2013

Engineering Analysis with Boundary Elements

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Dynamic soil-structure interaction is concerned with the study of structures supported on flexible soils and subjected to dynamic actions. Methods combining the finite element method (FEM) and the boundary element method (BEM) are well suited to address dynamic soil-structure interaction problems. Hence, FEM-BEM models have been widely used. However, non-linear contact conditions and non-linear behaviour of the structures have not usually been considered in the analyses. This paper presents a 3D non-linear time domain FEM-BEM numerical model designed to address soil-structure interaction problems. The BEM formulation, based on element subdivision and the constant velocity approach, was improved by using interpolation matrices. The FEM approach was based on implicit Green's functions and non-linear contact was considered at the FEM-BEM interface. Two engineering problems were studied with the proposed methodology: the propagation of waves in an elastic foundation and the dynamic response of a structure to an incident wave field.

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“…This will allow us to prove the convergence of the algorithms and to use the minimum-residual and steepest-descent methods to select optimal parameters. Stationary iterative processes for these algorithms were studied in [12,[18][19][20][21]. The iteration number will be indicated using superscript numerals.…”

mentioning

confidence: 99%

“…Condition (12) ensures convergence of stationary processes. For nonstationary processes, this parameter can be selected using formula (14) or (15).…”

mentioning

confidence: 99%

Domain decomposition method with hybrid approximations applied to solve problems of elasticity

2008

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To solve two-dimensional boundary-value problems of elasticity, two iteration algorithms of the domain decomposition method are proposed: parallel Neumann-Neumann and sequential Dirichlet-Neumann. They are based on the hybrid boundary-finite-element approximations. The algorithms are proved to converge. The optimal parameters are selected using the minimum-residual and steepest-descent methods. Some plane problems of elasticity are solved as examples, and stationary and nonstationary iteration algorithms in these examples are analyzed for efficiency Keywords: domain decomposition method, finite-element method, direct boundary-element method, parallel Neumann-Neumann algorithm, sequential Dirichlet-Neumann algorithm, plane problems of elasticity Introduction. Domain decomposition methods (DDMs) constitute a class of methods that reduce boundary-value problems of mathematical physics on complex domains to problems on simpler subdomains. The subdomains do not overlap, unlike those in the Schwarz method [3,6]. First DDM algorithms were proposed in [5,7] for Poisson's equation and in [11] for equations of elasticity. Studies on the subject are reviewed in [1,6,22]. An ingenious domain decomposition method was proposed in [8] to solve the heat-conduction problem for a body with a thin coating. A number of some other original numerical analytic approaches to study the mechanical behavior of elastic bodies are outlined in [13][14][15][16][17].In constructing parallel and sequential iteration DDM algorithms for boundary-value problems of elasticity, the initial domain is split into a finite number of nonoverlapping subdomains. Boundary conditions (kinematic, static, or mixed) sufficient to solve problems on the subdomains are set at the interfaces between subdomains. After finding solutions on subdomains, we obtain residuals of interfacial displacements or/and stresses. They are used to establish new boundary conditions for the subdomains. The process is continued until achieving prescribed accuracy. This is how sequential DDM algorithms work, with the order of subdomain traversal being set. Parallel and sequential DDM algorithms can be combined.Iteration DDM algorithms should converge. To this end, it is necessary to justify them and to select stable methods to solve problems on subdomains because determining the stresses on the domain boundary from the set displacements is an ill-conditioned mathematical problem of elasticity.An important advantage of the DDM is the possibility of using different mathematical models and different numerical methods to solve problems on subdomains and the possibility of organizing parallel calculations.

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Convergence of the domain decomposition finite element–boundary element coupling methods

Cited by 28 publications

References 11 publications

An adaptive domain decomposition coupled finite element–boundary element method for solving problems in elasto‐plasticity

An adaptive domain decomposition coupled finite element–boundary element method for solving problems in elasto‐plasticity

3D non-linear time domain FEM–BEM approach to soil–structure interaction problems

Domain decomposition method with hybrid approximations applied to solve problems of elasticity

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