Condition number of matrices derived from two classes of integral equations

Christiansen, Søren; Meister, E.

doi:10.1002/mma.1670030126

Cited by 46 publications

(20 citation statements)

References 39 publications

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“…(36) Figure 2 shows the graph of the condition number of A as a function of radius R. We perform calculations for N = 8 (red), N = 12 (blue), N = 16 (black), and N = 20 (green), cf [2]. We observe that for increasing N the condition number also increases.…”

Section: Dirichlet Conditionsmentioning

confidence: 98%

“…It is a well-known fact that the condition number of the system matrix is at least order N , where N is the number of boundary elements, [1]. In [2] the Dirichlet BVP for the Laplace equation is studied, where two domains are taken into account: a circle and an ellipse. In both cases analytical expressions for the condition number are derived.…”

Section: Introductionmentioning

confidence: 99%

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The condition number of the BEM-matrix arising from Laplace's equation

Dijkstra

Mattheij

2007

EJBE

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We investigate the condition number of the matrices that appear in the boundary element method. In particular we consider the Laplace equation with mixed boundary conditions. For Dirichlet boundary conditions, the condition number of the system matrix increases linearly with the number of boundary elements. We extend the research and search for a relation between the condition number and the number of elements in the case of mixed boundary conditions. In the case of a circular domain, we derive an estimate for the condition number of the system matrix. This matrix consists of two blocks, each block originating from a well-conditioned matrix. We show that the block matrix is also wellconditioned.

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Section: Dirichlet Conditionsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

The condition number of the BEM-matrix arising from Laplace's equation

Dijkstra

Mattheij

2007

EJBE

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“…has a non-trivial solution for 1 (s), where B d is the boundary of degenerate scale using the fundamental solution U(s, x) = ln(r) [13,[15][16][17][32][33][34]. For determining the degenerate scale systematically from one trial of a normal scale, we provide a flowchart as shown in Figure 2(a) and the numerical results as shown in Table I.…”

Section: Dual Boundary Integral Formulation and Dual Bem For Torsion mentioning

confidence: 99%

“…Chen et al [4] 235 methods to eliminate degenerate scale, scaling method and restriction method were discussed by Christiansen [33]. He also investigated the condition number of the influence matrix of the fictitious BEM and null-field approach [34]. Here, three regularization techniques will be employed to avoid the zero singular value.…”

Section: Introductionmentioning

confidence: 99%

Degenerate scale problem when solving Laplace's equation by BEM and its treatment

Chen

Lin

Chen

2004

Numerical Meth Engineering

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SUMMARYIn this paper, Laplace problems are solved by using the dual boundary element method (BEM). It is found that a degenerate scale problem occurs if the conventional BEM is used. In this case, the influence matrix is rank deficient and numerical results become unstable. Both the circular and elliptical bars are studied analytically in the continuous system. In the discrete system, the Fredholm alternative theorem in conjunction with the SVD (Singular Value Decomposition) updating documents is employed to sort out the spurious mode which causes the numerical instability. Three regularization techniques, method of adding a rigid body mode, hypersingular formulation and CHEEF (Combined Helmholtz Exterior integral Equation Formulation) concept, are employed to deal with the rankdeficiency problem. The addition of a rigid body term, c, in the fundamental solution is proved to shift the original degenerate scale to a new degenerate scale by a factor e −c . The torsion rigidities are obtained and compared with analytical solutions. Numerical examples including elliptical, square and triangular bars were demonstrated to show the failure of conventional BEM in case of the degenerate scale. After employing the three regularization techniques, the accuracy of the proposed approaches is achieved.

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“…13,14 Here, rectilinear elements are used and a constant approximation of the functions at each element. It can be shown that the condition number depends on the radius R of the circle and the number of boundary elements N according to…”

Section: Large Condition Numbersmentioning

confidence: 99%

Condition Numbers and Local Errors in the Boundary Element Method

Dijkstra¹,

Kakuba²,

Mattheij³

2010

Computational and Experimental Methods in Structures

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In this chapter we investigate local errors and condition numbers in the BEM. The results of these investigations are important in guiding adaptive meshing strategies and solvability of linear systems in BEM. We show that the local error for the BEM with constant or linear elements decreases quadratically with the boundary element mesh size. We also investigate better ways of treating boundary conditions to reduce the local errors. The results of our numerical experiments confirm the theory. The values of the condition numbers of the matrices that appear in the BEM depend on the shape and size of the domain on which a problem is defined. For certain critical domains these condition numbers can even become infinitely large. We show that this holds for several classes of boundary value problems and propose a number of strategies to guarantee moderate condition numbers.

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Condition number of matrices derived from two classes of integral equations

Cited by 46 publications

References 39 publications

The condition number of the BEM-matrix arising from Laplace's equation

The condition number of the BEM-matrix arising from Laplace's equation

Degenerate scale problem when solving Laplace's equation by BEM and its treatment

Condition Numbers and Local Errors in the Boundary Element Method

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