Non-orthogonal spline wavelets for boundary element analysis

Koro, Kazuhiro; Abe, Kazuhisa

doi:10.1016/s0955-7997(01)00036-4

Cited by 53 publications

(20 citation statements)

References 17 publications

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“…The results shown in Fig. 3 thus correspond to the number of entries of the matrix H. In calculation of BE solution, the approximationsũ andq were defined as the wavelet series with the Haar wavelets 11) . Usingũ andq, the true solutions u and q were expanded in every sub-boundary divided at four corners of the square domain.…”

Section: (1) Problem Descriptionmentioning

confidence: 99%

“…For this type of problems, the coefficient matrix A is assembled through the calculation and truncation of coefficients g ij and h ij . These coefficients were calculated under the discretization of a boundary integral equation with the piecewise constant non-orthogonal spline wavelets 11) . The wavelets used in the present numerical test have the third-order vanishing moments; p = 0 and n = 3.…”

Section: (3) 2-d Mixed Boundary Value Problemsmentioning

confidence: 99%

“…In the earlier stage of the development of wavelet BEM, these schemes were used for matrix compression and its threshold had to be determined empirically, as shown in Ref. 11) . This is because we did not have an appropriate way for determining the threshold.…”

Section: Introductionmentioning

confidence: 99%

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Matrix-Compression Property of Beylkin-Type Truncation Scheme for Wavelet Bem

Koro

Abe

2005

Structural Eng./Earthquake Eng.

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In the present paper, we investigate theoretically and experimentally the number of non-zero matrix entries generated by the wavelet BEM with the Beylkin-type compression algorithm. The Beylkin-type algorithm, which is based on a prescribed level-independent threshold, retains the asymptotic convergence rate of BE solutions, like widely-used level-dependent compression schemes. The coefficient matrix compressed by the Beylkin-type scheme has O(N 1+γ ) (0 < γ < 1, N : degree of freedom (DOF)) non-zero entries; level-dependent schemes enable us to reduce the matrix entries up to O (N (log N ) α ) (α ≥ 1). However, for matrix compression using the Beylkin-type scheme the compression rate is greater than or comparable to that of the Schneider's level-dependent scheme, in the moderate DOF range.

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Section: (1) Problem Descriptionmentioning

confidence: 99%

Section: (3) 2-d Mixed Boundary Value Problemsmentioning

confidence: 99%

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Matrix-Compression Property of Beylkin-Type Truncation Scheme for Wavelet Bem

Koro

Abe

2005

Structural Eng./Earthquake Eng.

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“…These properties make it di cult to apply the BEM to large-scale problems. To overcome this obstacle, the e ective techniques for reducing the computational cost have been developed over the last decade: a wavelet-based BEM (wavelet BEM [1][2][3][4]) and a fast multipole method (FMM [5,6]) are major fast methods for boundary element (BE) analysis.…”

Section: Introductionmentioning

confidence: 99%

“…In many engineering applications, the truncation algorithm proposed by Beylkin et al [12], which is the alternative of the above Dahmen-Schwab matrix compression scheme [4,[13][14][15]. The Beylkin-type truncation scheme enables us to reduce the memory requirements to O(N log N ) using a ÿxed threshold parameter independently of the degree of freedom (DOF).…”

Section: Introductionmentioning

confidence: 99%

A practical determination strategy of optimal threshold parameter for matrix compression in wavelet BEM

Koro

Abe

2003

Numerical Meth Engineering

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SUMMARYA practical strategy is developed to determine the optimal threshold parameter for wavelet-based boundary element (BE) analysis. The optimal parameter is determined so that the amount of storage (and computational work) is minimized without reducing the accuracy of the BE solution. In the present study, the Beylkin-type truncation scheme is used in the matrix assembly. To avoid unnecessary integration concerning the truncated entries of a coe cient matrix, a priori estimation of the matrix entries is introduced and thus the truncated entries are determined twice: before and after matrix assembly. The optimal threshold parameter is set based on the equilibrium of the truncation and discretization errors. These errors are estimated in the residual sense. For Laplace problems the discretization error is, in particular, indicated with the potential's contribution c to the residual norm R used in error estimation for mesh adaptation. Since the normalized residual norm c = u (u: the potential components of BE solution) cannot be computed without main BE analysis, the discretization error is estimated by the approximate expression constructed through subsidiary BE calculation with smaller degree of freedom (DOF). The matrix compression using the proposed optimal threshold parameter enables us to generate a sparse matrix with O(N 1+ ) (06 ¡1) non-zero entries. Although the quasi-optimal memory requirements and complexity are not attained, the compression rate of a few per cent can be achieved for N ∼ 1000.

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Displaced Multiwavelets and Splitting Algorithms

Shumilov

2016

Advanced Engineering Materials and Modeling

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Non-orthogonal spline wavelets for boundary element analysis

Cited by 53 publications

References 17 publications

Matrix-Compression Property of Beylkin-Type Truncation Scheme for Wavelet Bem

Matrix-Compression Property of Beylkin-Type Truncation Scheme for Wavelet Bem

A practical determination strategy of optimal threshold parameter for matrix compression in wavelet BEM

Displaced Multiwavelets and Splitting Algorithms

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