Theoretically supported scalable BETI method for variational inequalities

Bouchala, Jiří; Dostál, Zdeněk; Sadowská, Marie

doi:10.1007/s00607-008-0257-3

Cited by 19 publications

(8 citation statements)

References 31 publications

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“…After proving optimality results for the semimonotonic augmented Lagrangians for bound and equality constraints (SMALBE) algorithm [33], they have recently presented a theoretically supported scalable algorithm for scalar variational inequalities [34]. Similar results were obtained by Dostál et al [35,36] for FETI-DP and by Bouchala et al [37] for a boundary element variant of FETI. The aim of this paper is to show optimality results for multibody contact problems of elasticity using our TFETI (Total FETI) variant [38] of the FETI method which enforces the prescribed displacements by Lagrange multipliers.…”

Section: Introductionsupporting

confidence: 59%

Scalable TFETI algorithm for the solution of multibody contact problems of elasticity

Dostál

Kozubek

Vondrák

et al. 2009

Numerical Meth Engineering

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SUMMARYA Total FETI (TFETI)-based domain decomposition algorithm with preconditioning by a natural coarse grid of the rigid body motions is adapted to the solution of multibody contact problems of elasticity in 2D and 3D and proved to be scalable. The algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition parameter and the discretization parameter is bounded. The analysis is based on the classical results by Farhat, Mandel, and Roux on scalability of FETI with a natural coarse grid for linear problems and on our development of optimal quadratic programming algorithms for bound and equality constrained problems. The algorithm preserves parallel scalability of the classical FETI method. Both theoretical results and numerical experiments indicate a high efficiency of our algorithm. In addition, its performance is illustrated on a real-world problem of analysis of the ball bearing.

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Section: Introductionsupporting

confidence: 59%

Scalable TFETI algorithm for the solution of multibody contact problems of elasticity

Dostál

Kozubek

Vondrák

et al. 2009

Numerical Meth Engineering

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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: 99%

“…In similar way to Eq. (41), the derivative of the displacement fundamental solution may be expressed aš U P J,q (x) = 1 4πr 2Ũ P Jq (ê) (49) where the modulation function is U P Jq (ê) = −ê q H P J + C r K Ms π M qs P K M Jêr + M qr P K M Jês , (50) that only depends on the orientation of x (ê) but not on its modulus r . The M i j P K M N components have the following integral representation in terms of the parameter p M i j P K M N (ê)…”

Section: Appendix 2: Fundamental Solutionsmentioning

confidence: 99%

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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“…The experiments presented here were carried out by the codes developed originally for the research in the preconditioning of variational inequalities by M. Domorádová [9] and for developing BETI based scalable algorithms for variational inequalities by M. Sadowská [5]. To illustrate the effect of the steplength in the expansion step, we give here only two examples, a 2D inner obstacle problem discretized by the finite element method and a 3D contact problem of elasticity discretized by the boundary element method in combination with the BETI domain decomposition method.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…The second problem arises from the application of the TBETI (total boundary element tearing and interconnecting) domain decomposition method [5] to the solution of a 3D contact problem of elasticity. The TBETI method proved to be an efficient scalable algorithm for the solution of variational inequalities.…”

Section: Numerical Experimentsmentioning

confidence: 99%