On the block wavelet transform applied to the boundary element method

Bucher, Henrique F.; Wrobel, L.C.; Mansur, Webe João; Magluta, Carlos

doi:10.1016/j.enganabound.2003.10.002

Cited by 12 publications

(2 citation statements)

References 17 publications

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“…Applying wavelet compression to BEM for Laplace equation was developed by Bucher, et al (2004Bucher, et al ( , 2003 and Bucher and Wrobel (2002). In the earlier work of authors, development of method to elasticity problems was presented (Ebrahimnejad, 2007;Ebrahimnejad and Attarnejad, 2009).…”

Section: Introductionmentioning

confidence: 99%

A novel way of using fast wavelet transforms to solve dense linear systems arising from boundary element methods

Ebrahimnejad

Attarnejad

2009

Engineering Computations

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Purpose -The purpose of this paper is to introduce a novel approach to solving linear systems arising from applying a Boundary Element Method (BEM) to elasticity problems. Design/methodology/approach -The key idea is based on using wavelet transforms as a tool to change dense and fully populated matrices of BEM systems into sparse matrices. Wavelets are then used again to produce an algorithm to solve the resultant sparse linear systems. The wavelet transformation part of the method can be added as a black box to existing BEM codes. Findings -Numerical results focusing on the sensitivity of the solution for various physical variables to the thresholding parameters, and savings in computer time and memory are presented. The results show that the proposed method is efficient for large problems. Research limitations/implications -Application of the proposed method is restricted to problems with number of DOF equal to an integer power of 2. Originality/value -The novel algorithm to solve transformed algebraic linear equations uses NSform of the modified matrix, taking the advantage of the hierarchical nature of Multi-Resolution Analysis (MRA) to decompose a parent system into descendant systems with reduced size. These smaller systems are then solved iteratively using generalized minimal residual method.

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Section: Introductionmentioning

confidence: 99%

A novel way of using fast wavelet transforms to solve dense linear systems arising from boundary element methods

Ebrahimnejad

Attarnejad

2009

Engineering Computations

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“…One can observe there are intersections between the performance curves of the LU solver and the GMRES(50) solver with ε = 10 −6 or 10 −16 around N = 1023 or N = 2499, which demonstrate that the investigation on the asymptotic performance behaviour of these solvers is indeed necessary for acquiring a more complete knowledge on their performance. Since the trends of variation of these performance curves are rather regular in the current coordinates, we can extrapolate them to predict the solution time of the corresponding solvers for problems with more degrees of freedom (see also Bucher et al 2002). We estimate that, for a problem with N = 20 000 degrees of freedom, the Gauss solver requires about 22.9 days to get the solution, and the LU solver needs about 3.36 h for the same thing, while the GMRES(50) solver can bring out its solution in about 50 min for ε = 10 −16 , about 11 min for ε = 10 −6 , or less than 2 min for ε = 10 −2 .…”

Section: Efficiency and Accuracymentioning

confidence: 99%

The GMRES method applied to the BEM extrapolation of solar force-free magnetic fields

Yan²,

Devel³

et al. 2007

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Context. Since the 1990's, Yan and colleagues have formulated a kind of boundary integral formulation for the linear or non-linear solar force-free magnetic fields with finite energy in semi-infinite space, and developed a computational procedure by virtue of the boundary element method (BEM) to extrapolate the magnetic fields above the photosphere. Aims. In this paper, the generalized minimal residual method (GMRES) is introduced into the BEM extrapolation of the solar forcefree magnetic fields, in order to efficiently solve the associated BEM system of linear equations, which was previously solved by the Gauss elimination method with full pivoting. Methods. Being a modern iterative method for non-symmetric linear systems, the GMRES method reduces the computational cost for the BEM system from O(N 3 ) to O(N 2 ), where N is the number of unknowns in the linear system. Intensive numerical experiments are conducted on a well-known analytical model of the force-free magnetic field to reveal the convergence behaviour of the GMRES method subjected to the BEM systems. The impacts of the relevant parameters on the convergence speed are investigated in detail. Results. Taking the Krylov dimension to be 50 and the relative residual bound to be 10 −6 (or 10 −2 ), the GMRES method is at least 1000 (or 9000) times faster than the full pivoting Gauss elimination method used in the original BEM extrapolation code, when N is greater than 12 321, according to the CPU timing information measured on a common desktop computer (CPU speed 2.8 GHz; RAM 1 GB) for the model problem.

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A novel approach to solve linear systems arising from BEM using fast wavelet transforms

Ebrahimnejad

Attarnejad

2009

Math. Meth. Appl. Sci.

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A novel approach to solve linear systems arising from BEM using fast wavelet transforms Latif Ebrahimnejad * † and Reza Attarnejad Communicated by V. Volpert This paper uses Daubechies orthogonal wavelets to change dense and fully populated matrices of boundary element method (BEM) systems into sparse and semi-banded matrices. Then a novel algorithm based on hierarchical nature of multiresolution analysis is introduced to solving resultant sparse linear systems. This algorithm decomposes NS-form of transformed parent matrix into descendant systems with reduced sizes and solves them iteratively using GMRES algorithm. Both parts, changing dense matrices to sparse systems and the novel solver, can be added as a black box to the existing BEM codes. Transforming matrices into wavelet space needs less time than saved by solving sparse large systems. Numerical results with a precise study on sensitivity of solution for physical variables to the thresholding parameter, and savings in computer time and memory are presented. Also, the suitable value for thresholding parameter is recommended for elasticity problems. The results indicate that the proposed method is efficient for large problems. Copyright © 2009 John Wiley & Sons, Ltd.Keywords: boundary element method; wavelet transform; MRA; GMRES solver; wavelet solver IntroductionBoundary element method (BEM) can be considered as the eldest mesh reduction method. BEM is an efficient boundary type method, where only the boundary of domain is required to be discretized. Then the dimension of problem is reduced by one in comparison with FE and FD [1,2]. Although total degrees of freedom for BE is much less than FE and FD, because of arising dense and fully populated matrices instead of sparse and banded matrices in FE and FD, the efficiency of BEM for large problems is decreased considerably.Wavelets were first used by Beylkin, Coifman and Rokhlin (BCR) [3] to change the dense and fully populated matrices of BEM. Then wavelet compression was applied to BEM for Laplace equation by . In the earlier works, authors have developed the method to elasticity problems [7, 8]. Permutation technique introduced by Chen [9] changes finger-like matrices of transformed space to semi-banded matrices.Applying wavelet compression to BEM can be implemented as a black box in existing BEM codes. While standard BEM uses Gaussian elimination with O(N 3 ) operations to solve algebraic linear equations, where N is total number of equations, iterative solvers such as Lanczos, CG and GMRES need O(N 2 ) operations per iteration [10--13].Wavelet solver can reduce complexity order to O((N Log N) ), which is impressively motivating. In this paper Daubechies orthogonal wavelets are used to transform original system matrix to a semi-banded sparse one, which then is solved by using wavelet solver. This solver solves linear algebraic systems using NS-form of transformed matrix, hierarchical nature of multiresolution analysis (MRA) and GMRES algorithm.In what follows, required equations used in BEM for elastic...

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On the block wavelet transform applied to the boundary element method

Cited by 12 publications

References 17 publications

A novel way of using fast wavelet transforms to solve dense linear systems arising from boundary element methods

A novel way of using fast wavelet transforms to solve dense linear systems arising from boundary element methods

The GMRES method applied to the BEM extrapolation of solar force-free magnetic fields

A novel approach to solve linear systems arising from BEM using fast wavelet transforms

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