Scalable Total BETI based algorithm for 3D coercive contact problems of linear elastostatics

Bouchala, Jiří; Dostál, Zdeněk; Sadowská, Marie

doi:10.1007/s00607-009-0044-9

Cited by 13 publications

(3 citation statements)

References 40 publications

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“…b Boundary element mesh details around the portential contact zone solved using the same iterative techniques. To the best of the authors' knowledge, in spite of the BETI algorithm has been successfully extended to contact problems using SGBEM formulation [50,51], its application using a non-symmetrical boundary element formulation has not been completed. The extension of BETI technique to non-symmetrical boundary element formulations has only been only considered in domain decomposition elastic problems by González et al [52], but its extension to frictional contact problems is not straightforward.…”

Section: ψ (Z) ≤ ε Being ψ (Z) = H T (Z)h(z)/2)mentioning

confidence: 96%

3D BEM for orthotropic frictional contact of piezoelectric bodies

2015

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A numerical methodology to model the threedimensional frictional contact interaction of piezoelectric materials in presence of electric fields is presented in this work. The boundary element method (BEM) is used in order to compute the electro-elastic influence coefficients. The proposed BEM formulation employs an explicit approach for the evaluation of the corresponding fundamental solutions, which are valid for general anisotropic behaviour meanwhile mathematical degeneracies in the context of the Stroh formalism are allowed. The contact methodology is based on an augmented Lagrangian formulation and uses an iterative Uzawa scheme of resolution. An orthotropic frictional law is implemented in this work so anisotropy is present both in the bulk and in the surface. The methodology is validated by comparison with benchmark analytical solutions. Some additional examples are presented and discussed in detail, revealing the importance of considering orthotropic frictional contact conditions in the electro-elastic analysis of this kind of problems.

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Section: ψ (Z) ≤ ε Being ψ (Z) = H T (Z)h(z)/2)mentioning

confidence: 96%

3D BEM for orthotropic frictional contact of piezoelectric bodies

2015

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“…These algorithms have elements of the conjugate gradient method and modifications of active set methods (and possibly of augmented Lagrangian methods as well) and possess the property that the number of iterations needed to find the approximate solution is uniformly bounded provided that the spectrum of the Hessian matrix of the cost function is confined to a given positive interval. Numerical results on asymptotically linear complexity of these algorithms can be found, for instance, in [13] (for two-dimensional scalar variational inequality) and [3,14] (for three-dimensional elasticity). Convergence rates of these nonlinear algorithms, applied in conjunction with the FETI methods, have been bounded in terms of C(H/h) or C(H/h) 2 (see, for instance, [15,12] and the references therein), where C is a constant independent of H and h.…”

Section: Introductionmentioning

confidence: 99%

Two Domain Decomposition Methods for Auxiliary Linear Problems of a Multibody Elliptic Variational Inequality

Lee¹

2013

SIAM J. Sci. Comput.

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Elliptic variational inequalities with multiple bodies in two dimensions are considered. It is assumed that an active set method is used to handle the nonlinearity of the inequality constraint, which results in auxiliary linear problems. For solving such linear problems we study two domain decomposition methods called the finite element tearing and interconnecting (FETI-FETI) and hybrid methods in this paper. Bodies are decomposed into several subdomains in both methods. The FETI-FETI method combines the one-level FETI and the dual-primal FETI (FETI-DP) methods. We present a proof that this combined method has a condition number that depends linearly on the number of subdomains across each body and polylogarithmically on the number of elements across each subdomain. Our numerical results, and those of others, suggest that this is the best possible bound. The hybrid method combines the one-level FETI and the balanced domain decomposition by constraints (BDDC) methods; we prove that the condition number of this method has two polylogarithmic factors depending on the number of elements across each subdomain and across each body. We present numerical results confirming this theoretical finding.Key words. domain decomposition, variational inequalities, one-level finite element tearing and interconnecting, dual-primal finite element tearing and interconnecting, balanced domain decomposition by constraints 1. Introduction. We consider domain decomposition methods for elliptic variational inequalities with multiple bodies. It is assumed that an active set method, such as the primal-dual active set strategy in [24], is used to reduce the variational inequality into a sequence of linear problems; we focus on solving such auxiliary linear problems in a scalable manner and examine two domain decomposition methods for this purpose. Both methods combine existing domain decomposition methods, and each body is decomposed into subdomains that in turn are unions of finite elements in both methods.The first method, called the finite element tearing and interconnecting (FETI-FETI) method in this paper, is a combination of the original FETI (one-level FETI) and the dual-primal FETI (FETI-DP) methods. We prove that the condition number of the FETI-FETI method depends linearly on the number of subdomains across each body and polylogarithmically on the number of elements across each subdomain under some restrictions. Currently we have this result only in two dimensions (2D); 1 see section 3.1 for details. Such results were previously shown numerically in [29] and [30] without a proof.The second method, called the hybrid method in this paper, is a combination of the FETI-DP and the balanced domain decomposition by constraints (BDDC)

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“…The BETI algorithm has been successfully extended to the case of infinite subdomains and contact problems . But implementation of SGBEM, principal ingredient of BETI, is not straightforward, and if it is replaced by a non‐symmetrical BEM formulation to approximate the Steklov–Poincaré operators of the floating substructures, the obtained flexibility equations become non‐symmetric, and different solution strategies should be followed.…”

Section: Introductionmentioning

confidence: 99%

The nsBETI method: an extension of the FETI method to non‐symmetrical BEM‐FEM coupled problems

González

Rodríguez-Tembleque

Park

et al. 2012

Numerical Meth Engineering

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SUMMARYThe finite element tearing and interconnecting (FETI) method is recognized as an effective domain decomposition tool to achieve scalability in the solution of partitioned second-order elasticity problems. In the boundary element tearing and interconnecting (BETI) method, a direct extension of the FETI algorithm to the BEM, the symmetric Galerkin BEM formulation, is used to obtain symmetric system matrices, making possible to apply the same FETI conjugate gradient solver. In this work, we propose a new BETI variant labeled nsBETI that allows to couple substructures modeled with the FEM and/or non-symmetrical BEM formulations. The method connects non-matching BEM and FEM subdomains using localized Lagrange multipliers and solves the associated non-symmetrical flexibility equations with a Bi-CGstab iterative algorithm. Scalability issues of nsBETI in BEM-BEM and combined BEM-FEM coupled problems are also investigated.

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Scalable Total BETI based algorithm for 3D coercive contact problems of linear elastostatics

Cited by 13 publications

References 40 publications

3D BEM for orthotropic frictional contact of piezoelectric bodies

3D BEM for orthotropic frictional contact of piezoelectric bodies

Two Domain Decomposition Methods for Auxiliary Linear Problems of a Multibody Elliptic Variational Inequality

The nsBETI method: an extension of the FETI method to non‐symmetrical BEM‐FEM coupled problems

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