A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element

Qin, Xianyun; Zhang, Jianming; Xie, Guizhong; Zhou, Fenglin; Li, Guanyao

doi:10.1016/j.cam.2011.03.012

Cited by 74 publications

(46 citation statements)

References 17 publications

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“…Example 4 The computation of the nearly singular integral on a curved surface element is considered in this example [16]. The chosen surface element, named as spherical surface element [24], is represented in parametric form with the usual spherical polar system (θ, ϕ).…”

Section: Variable Transformationmentioning

confidence: 99%

“…The chosen surface element, named as spherical surface element [24], is represented in parametric form with the usual spherical polar system (θ, ϕ). And the element's geometric parameters are given as follows:θ ∈ [0, π/4], ϕ ∈ [π/4, π/2], the sphere radius r = 0.1, and with center (0, 0, 0).…”

Section: Variable Transformationmentioning

confidence: 99%

“…transformation [4], sigmoidal transformation [5,6], sinh transformation [6][7][8][9][10][11][12], rational transformation [13], distance transformation [14][15][16][17], exponential transformation [18][19][20][21][22][23] and combinations of the polar coordinates approach and variable transformations [6]. In this work, we focus on the sinh transformation, based on the sinh function, proposed by Johnston and his collaborators [7].…”

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confidence: 99%

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Use of the sinh transformation for evaluating 2D nearly singular integrals in 3D BEM

Zhang

Gong

et al. 2015

Acta Mech

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This paper presents a new strategy for the numerical evaluation of 2D nearly singular integrals that arise in the solution of 3D BEM using eight-node second-order quadrilateral surface elements. The strategy is an extension of the previous sinh transformation techniques, which is used to evaluate the 1D or 2D nearly singular integrals on simple geometry elements, such as linear or planar elements. The novel feature of the proposed method is that a new distance formula is introduced here, and based on this an ingenious combination of the distance formula and the sinh transformation is developed, and hence, the rapid variation of the distance formula on the integration interval can been smoothed out. Several numerical examples involving boundary layer effect and thin-body problems in 3D potential theory are given to verify the accuracy and efficiency of the presented method.

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Section: Variable Transformationmentioning

confidence: 99%

Section: Variable Transformationmentioning

confidence: 99%

mentioning

confidence: 99%

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Use of the sinh transformation for evaluating 2D nearly singular integrals in 3D BEM

Zhang

Gong

et al. 2015

Acta Mech

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“…In the conventional subdivision method, the integration element is divided directly into sub-triangles by simply connecting the singular point with each vertex of element as shown in Fig. 5, thus the shape of sub-triangles will poor when the singular point is located near the edge or in the edge especially for slender elements, which will result in the numerical results become less accurate [6,7]. However, with the proposed adaptively element subdivision technique, integration element is broken up into triangular and quadrilateral sub-elements through a sphere of decreasing radius, consequently, shape of sub-element is good which are beneficial conditions for using   The conventional subdivisions of quadrilateral element.…”

Section: Numerical Examplesmentioning

confidence: 99%

An adaptive element subdivision technique for evaluating 3D weakly singular boundary integrals

Zhang¹,

Lu²,

Zhong³

et al. 2014

WIT Transactions on Modelling and Simulation

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A general adaptive element subdivision technique is presented for the numerical evaluation of weakly singular integrals, which often appear in three-dimensional boundary element analysis equations. In this algorithm, the weakly singular boundary element is broken up into a few sub-elements through a sphere of decreasing radius. The sub-elements involving the singular point are evaluated numerically after using a coordinate transformation to remove the singularities, while other quadrilateral sub-elements and triangular sub-elements are evaluated numerically by the standard Gaussian quadrature and Hammer quadrature respectively. The number of sub-elements and their size are determined adaptively according to the position of the singular point. Numerical examples are presented for both planar and curved surface element. The results demonstrate our method can provide higher accuracy and efficiency than the conventional method.

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“…Jabłoński 9 solved 3D Laplace and Poisson equations by proposing the analytical evaluation of the surface integrals appearing in BEMs. Qin et al 10 implemented changes to the conventional distance transformation technique to evaluate nearly singular integrands on 3D boundary elements, including planar and curved surface elements and very irregular elements of slender shape.…”

Section: Introductionmentioning

confidence: 99%

Closed Form Integration of Singular and Hypersingular Integrals in 3D BEM Formulations for Heat Conduction

Tadeu

Prata

Simões

2012

Mathematical Problems in Engineering

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The evaluation of the singular and hypersingular integrals that appear in three-dimensional boundary element formulations for heat diffusion, in the frequency domain, is presented in analytical form. This improves computational efficiency and accuracy. Numerical integrations using existing techniques based on standard Gaussian integration schemes that incorporate an enormous amount of sampling points are used to verify the solutions of singular integrals. For the hypersingular integrals the comparison is evaluated by making use of an analytical solution that is valid for circular domains, combined with a standard Gaussian integration scheme for the remaining boundary element domain. Closed form solutions for cylindrical inclusions with null temperatures and null heat fluxes prescribed on the boundary are then derived and used to validate the three-dimensional boundary element formulations.

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A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element

Cited by 74 publications

References 17 publications

Use of the sinh transformation for evaluating 2D nearly singular integrals in 3D BEM

Use of the sinh transformation for evaluating 2D nearly singular integrals in 3D BEM

An adaptive element subdivision technique for evaluating 3D weakly singular boundary integrals

Closed Form Integration of Singular and Hypersingular Integrals in 3D BEM Formulations for Heat Conduction

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