Exploiting partial or complete geometrical symmetry in 3D symmetric Galerkin indirect BEM formulations

Bonnet, Marc

doi:10.1002/nme.716

Cited by 19 publications

(20 citation statements)

References 8 publications

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“…Several methods have been proposed in the literature (e.g. [1,4,5]) which allow a reduction of the entire problem to subproblems and following reconstruction of the global solution.…”

Section: Symmetries In the Boundary Element Methodsmentioning

confidence: 99%

“…Section 2 presents the adaptive cross approximation (ACA) algorithm [13] which generates blockwise low rank approximants for the BEM matrices without using an explicit kernel expansion. The exploitation of symmetry is another possibility to reduce computational costs and has been presented in [1,4,5] using linear representation theory for finite groups. The aim is a decomposition of function spaces into orthogonal subspaces of symmetric functions, such that each subproblem is defined on a so called symmetry cell.…”

Section: Introductionmentioning

confidence: 99%

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Application of the adaptive cross approximation technique for the coupled BE-FE solution of symmetric electromagnetic problems

Kurz

Rain

Rjasanow

2003

Computational Mechanics

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Electromagnetic devices can be analysed by the coupled BE-FE method, where the conducting and magnetic parts are discretized by finite elements. In contrast, the surrounding space is described with the help of the boundary element method (BEM). This discretization scheme is well suited especially for problems including moving parts (see [12]). The BEM discretization of the boundary integral operators usually leads to dense matrices without any structure. A naive strategy for the solution of the corresponding linear system would need at least OðN 2 Þ operations and memory, where N ist the number of unknowns. Methods such as fast multipole [6] and panel clustering [9] provide an approximation to the matrix in almost linear complexity. These methods are based on explicitly given kernel approximations by degenerate kernels, i.e. a finite sum of separable functions, which may be seen as a blockwise low-rank approximation of the system matrix. The blockwise approximant permits a fast matrix-vector multiplication, which can be exploited in iterative solvers, and can be stored efficiently. In contrast to the methods mentioned we will generate [2] the low-rank approximant from the matrix itself using only few entries and without using any explicit a priori known degenerate-kernel approximation. Special emphasis is put on the handling of symmetry conditions in connection with ACA. The feasibility of the proposed method is demonstrated by means of a numerical example. IntroductionThe application of the boundary element method for the solution of linear electromagnetic problems has many advantages. Only the boundaries of the considered domains need to be discretized, open boundary problems pose no additional difficulties, and problems including motion can be treated elegantly. The BE-FE coupling procedure is elaborated in [12]. However, application of the BEM leads to dense matrices. The storage requirements and computational costs are of OðN 2 Þ, where N is the number of unknowns. One remedy could be the use of an approximation algorithm with almost linear complexity. Section 2 presents the adaptive cross approximation (ACA) algorithm [13] which generates blockwise low rank approximants for the BEM matrices without using an explicit kernel expansion. The exploitation of symmetry is another possibility to reduce computational costs and has been presented in [1, 4, 5] using linear representation theory for finite groups. The aim is a decomposition of function spaces into orthogonal subspaces of symmetric functions, such that each subproblem is defined on a so called symmetry cell. The global solution can then be reconstructed from these components. Sections 3.1-3.2 give an overview of using symmetry in FE and BE methods. Next, in Sect. 3.3 the major contribution of this paper, a symmetry-exploiting ACA algorithm is discussed, which not only reduces the problem size due to symmetry but also yields additional benefit by compression of BEM matrices and possesses an almost linear complexity w.r.t. the number of unknowns. Numerica...

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“…Several methods have been proposed in the literature (e.g. [1,4,5]) which allow a reduction of the entire problem to subproblems and following reconstruction of the global solution.…”

Section: Symmetries In the Boundary Element Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Application of the adaptive cross approximation technique for the coupled BE-FE solution of symmetric electromagnetic problems

Kurz

Rain

Rjasanow

2003

Computational Mechanics

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“…For example, by rotating the dependence between nodes 1 and 6 (i.e. A 1,6 ) we see that A 3,2 , the dependence between nodes 3 and 2, must be identical.…”

Section: The Triangle Examplementioning

confidence: 99%

“…Bonnet [6] discusses applications where the domain consists of two disconnected components where only one of the components is symmetric. The resulting matrix will then be partially equivariant, and this part may be block-diagonalized.…”

Section: Other Numerical Algorithms and Issuesmentioning

confidence: 99%

Applications of the Generalized Fourier Transform in Numerical Linear Algebra

Åhlander

Munthe-Kaas

2005

Bit Numer Math

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Abstract.Matrices equivariant under a group of permutation matrices are considered. Such matrices typically arise in numerical applications where the computational domain exhibits geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform. This technique yields substantial computational savings in problems such as solving linear systems, computing eigenvalues and computing analytic matrix functions.The theory for applying the Generalized Fourier Transform is explained, building upon the familiar special (finite commutative) case of circulant matrices being diagonalized with the Discrete Fourier Transform. The classical convolution theorem and diagonalization results are generalized to the non-commutative case of block diagonalizing equivariant matrices.Our presentation stresses the connection between multiplication with an equivariant matrices and the application of a convolution. This approach highlights the role of the underlying mathematical structures such as the group algebra, and it also simplifies the application of fast Generalized Fourier Transforms. The theory is illustrated with a selection of numerical examples.

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“…In the framework of BEMs, where the reduction is particularly important because the discretization matrices are typically full, other approaches can be found in literature, such as the one presented in [3] where a generalization of the Fourier transform for arbitrary finite groups is used, or the one described in [7,8] where the original problem is reduced to a family of smaller ones defined on a reduced geometry, named symmetry cell, with boundary conditions to be assigned on the new boundaries. Both these approaches are based on the group representation theory.…”

Section: Introductionmentioning

confidence: 99%

Restriction matrices for SGBEM applications

Aimi

Diligenti

Freddi

et al. 2003

Computational Mechanics

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In the context of linear elasticity, we consider a symmetric boundary integral formulation associated with a mixed boundary value problem defined on a domain X & R m ; m ¼ 2; 3; with piecewise smooth boundary C. We assume that " X X is mapped onto itself by a finite group G of congruences having at least two distinct elements. The aim of this paper is to present a systematic technique for exploiting geometrical symmetry in the numerical treatment of boundary integral equations with the Symmetric Galerkin Boundary Element Method (SGBEM). This technique will be based upon suitable restriction matrices strictly related to the group G and to the mesh defined on the boundary. Hence, we can decompose the related SGBEM problem into independent subproblems of reduced dimension with respect to the original one. Shape functions for each subproblem can be obtained from classical BEM basis, ordered as a vector, applying restriction matrices suitably constructed starting from group representation theory.Many linear operator equations, in physical and engineering problems, are defined on regions which have various geometrical symmetries, that is, are left unchanged by geometrical transformations such as rotations or mirror reflections and so on. Under appropriate discretizations of such equations, the invariance of the problem domain can give important advantages in the discretization phase, in terms of memory storage and computational cost of classical numerical methods, such as FEM or BEM, without modifying the accuracy of the approximate solution [3, 6, 7, 10]. It is possible to take advantage of the symmetry thanks to group representation theory and a decomposition theorem. From the latter, using the symmetry one can reduce the original problem to a family of smaller ones. A global solution is obtained from superposition of the partial results. A systematic approach of the use of geometrical symmetry, based on the use of group representation theory, has been carried out by Bossavit (9) in the framework of variational formulation of elliptic boundary-value problems.The aim of this paper is to present a systematic technique for exploiting symmetry properties to effect a significant reduction of computational cost in the numerical treatment of boundary integral equations in linear elasticity problems defined on a domain X & R m , m ¼ 2; 3, with the Symmetric Galerkin Boundary Element Method (SGBEM). This technique will be based upon suitable restriction matrices strictly related to the mesh defined on the boundary.Restriction matrices are not new in literature: they have been widely used in the context of parallel multigrid algorithms [14], aggregation/disaggregation techniques [15], domain decomposition reduction methods [16] for the numerical solution of elliptic boundary value problems. In some cases, these matrices were strictly related to the symmetry properties of the domain of the differential problem, but unfortunately the corresponding examples were reduced to the simple geometrical situation of symmetry with respect...

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Exploiting partial or complete geometrical symmetry in 3D symmetric Galerkin indirect BEM formulations

Cited by 19 publications

References 8 publications

Application of the adaptive cross approximation technique for the coupled BE-FE solution of symmetric electromagnetic problems

Application of the adaptive cross approximation technique for the coupled BE-FE solution of symmetric electromagnetic problems

Applications of the Generalized Fourier Transform in Numerical Linear Algebra

Restriction matrices for SGBEM applications

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