Multi-level hierarchical preconditioners for boundary element systems

Barra, Luis Paulo da Silva; Coutinho, Álvaro L. G. A.; Telles, J. C. F.; Mansur, Webe João

doi:10.1016/0955-7997(93)90004-5

Cited by 17 publications

(12 citation statements)

References 12 publications

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“…In sparse systems, matvec products may consume more than 90% of the total CPU time spent by the solver, as in the case of the PCG with the main diagonal of the system taken for preconditioner. Even when more complex preconditioning techniques are employed, such as ILU factorization [2], polynomial [3], multi-level [4] or domain decomposition [5] techniques, the task of performing matvec products consumes a significant amount of the total CPU time. Therefore, the optimization of this kind of operation is of extreme importance.…”

Section: Introductionmentioning

confidence: 99%

Comparison between element, edge and compressed storage schemes for iterative solutions in finite element analyses

Ribeiro¹,

Coutinho²

2005

Int. J. Numer. Meth. Engng

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SUMMARYThis paper is concerned with optimization techniques for the iterative solution of sparse linear systems arising from finite element discretization of partial differential equations. Three different data structures are used to store the coefficient matrices: the usual element-based data structure, the compressed storage row format and the edge-based approach. A comparison between these storage schemes is performed, quantifying for most common linear elements the number of floating points operations, indirect addressing and memory requirements necessary to perform matrix-vector products. The overall performance of the preconditioned conjugate gradient method is measured for different situations involving 2D and 3D diffusion and elasticity problems, highlighting the pros and cons of each storage scheme.

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Section: Introductionmentioning

confidence: 99%

Comparison between element, edge and compressed storage schemes for iterative solutions in finite element analyses

Ribeiro¹,

Coutinho²

2005

Int. J. Numer. Meth. Engng

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“…The reasons for that is the exceptionally good convergence properties of their preconditioned versions. Important results for non-hermitian BE systems are reported by , Mansur, Araújo and Malaghini (1992), Barra et al (1993), Prasad et al (1994), and Valente and Pina (2001). In Saad and van der Vorst (2000), a complete review on iterative solvers is provided; more than 200 references are cited.…”

Section: General Aspectsmentioning

confidence: 99%

New developments on BE/BE multi-zone algorithms based on krylov solvers: applications to 3D frequency-dependent problems

Araújo¹,

Mansur²,

Dors³

et al. 2004

J. Braz. Soc. Mech. Sci. & Eng.

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In this paper, new developments concerning the use of BE/BE coupling algorithms for solving 3D time-harmonic problems are reported. The algorithms are derived by considering different iterative solvers. Their chief idea is to work with the global sparse matrix of the coupled system, however without considering the many zero blocks associated with the non-coupled nodes of different subregions. The use of iterative solvers makes it possible to store and manipulate only the block matrices with non-zero coefficients. Preconditioned iterative solvers based on the Lanczos, bi-conjugate gradient, and GMRES (generalized minimal residual) methods are used to derive the BE/BE coupling algorithms. A description of the formulation of these solvers, which are completely general and can be applied to any non-singular, non-hermitian systems of equations, is provided. The performance of the algorithms is verified by solving some foundation-soil interaction problems. Important parameters for estimating the efficiency of the algorithms as required CPU times, matrix sparsity, and accuracy of the obtained responses are presented in the results of the paper

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“…Some issues in the analysis of larger three-dimensional BEM systems through preconditioned GMRES were evidenced by Leung and Walker (1997), who also proposed a strategy to overcome some limitations and extend the applicability of the algorithm to systems with some thousands of unknowns. Barra et al (1993) proposed a strategy for the construction of preconditioners for GMRES solved BEM problems, while Wang et al (2005) investigated a class of preconditioners for fast multipole BEMs.…”

Section: Introductionmentioning

confidence: 99%

A fast 3D dual boundary element method based on hierarchical matrices

Benedetti

Aliabadi

Davi

2008

International Journal of Solids and Structures

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In this paper a fast solver for three-dimensional BEM and DBEM is developed. The technique is based on the use of hierarchical matrices for the representation of the collocation matrix and uses a preconditioned GMRES for the solution of the algebraic system of equations. The preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. Special algorithms are developed to deal with crack problems within the context of DBEM. The structure of DBEM matrices has been efficiently exploited and it has been demonstrated that, since the cracks form only small parts of the whole structure, the use of hierarchical matrices can be particularly advantageous. Test examples presented show that, with the proposed technique, substantial increase in number of elements over the crack surfaces leads only to moderate increases in memory storage and solution time.

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Multi-level hierarchical preconditioners for boundary element systems

Cited by 17 publications

References 12 publications

Comparison between element, edge and compressed storage schemes for iterative solutions in finite element analyses

Comparison between element, edge and compressed storage schemes for iterative solutions in finite element analyses

New developments on BE/BE multi-zone algorithms based on krylov solvers: applications to 3D frequency-dependent problems

A fast 3D dual boundary element method based on hierarchical matrices

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