Bi-directional sinh transformations based on the generalized Duffy space for nearly singular integrals

2020 ◽  
Vol 380 ◽  
pp. 112981 ◽  
Author(s):  
Guizhong Xie ◽  
Fenglin Zhou ◽  
Yudong Zhong ◽  
Hongrui Geng ◽  
Changjun Wu
Keyword(s):  
Singular Integrals ◽  
Nearly Singular Integrals

scholarly journals Efficient Computation of Highly Oscillatory Fourier Transforms with Nearly Singular Amplitudes over Rectangle Domains

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1930
Author(s):  
Zhen Yang ◽  
Junjie Ma
Keyword(s):  
Adaptive Mesh Refinement ◽  
Numerical Experiments ◽  
Singular Integrals ◽  
Fourier Transforms ◽  
Adaptive Mesh ◽  
Oscillatory Integrals ◽  
Efficient Computation ◽  
New Approach ◽  
Highly Oscillatory ◽  
Nearly Singular Integrals

In this paper, we consider fast and high-order algorithms for calculation of highly oscillatory and nearly singular integrals. Based on operators with regard to Chebyshev polynomials, we propose a class of spectral efficient Levin quadrature for oscillatory integrals over rectangle domains, and give detailed convergence analysis. Furthermore, with the help of adaptive mesh refinement, we are able to develop an efficient algorithm to compute highly oscillatory and nearly singular integrals. In contrast to existing methods, approximations derived from the new approach do not suffer from high oscillatory and singularity. Finally, several numerical experiments are included to illustrate the performance of given quadrature rules.

scholarly journals A Generation of Special Triangular Boundary Element Shape Functions for 3D Crack Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Guizhong Xie ◽  
Fenglin Zhou
Keyword(s):  
Boundary Element ◽  
Crack Front ◽  
Singular Integrals ◽  
Triangular Element ◽  
Crack Problems ◽  
Nearly Singular Integrals ◽  
Triangular Elements ◽  
Displacement Extrapolation Method ◽  
Element Shape ◽  
Displacement Extrapolation

This paper focuses on tackling the two drawbacks of the dual boundary element method (DBEM) when solving crack problems with a discontinuous triangular element: low accuracy of the calculation of integrals with singularity and crack front element must be utilized to model the square-root property of displacement. In order to calculate the integrals with higher order singularity, the triangular elements are segmented into several subregions which consist of subtriangles and subpolygons. The singular integrals in those subtriangles are handled by the singularity subtraction technique in the integration space and can be regularized and accurately calculated. For the nearly singular integrals in those subpolygons, the element subdivision technique is employed to improve the calculation accuracy. In addition, considering the location of the crack front in the element, special crack front elements are constructed based on a 6-node discontinuous triangular element, in which the displacement extrapolation method is introduced to obtain the stress intensity factors (SIFs) without consideration of orthogonalization of the crack front mesh. Several numerical results are investigated to fully verify the validation of the presented approach.

Computation of nearly singular integrals in 3D BEM

2014 ◽  
Vol 48 ◽  
pp. 32-42 ◽  
Author(s):  
Yaoming Zhang ◽  
Xiaochao Li ◽  
Vladimir Sladek ◽  
Jan Sladek ◽  
Xiaowei Gao
Keyword(s):  
Singular Integrals ◽  
Nearly Singular Integrals
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