Efficient evaluation of integrals in three-dimensional boundary element method using linear shape functions over plane triangular elements

This work provides a preliminary contribution in the context of analytical integrations of strongly and hyper singular kernels in boundary element methods (BEMs) in 3D. It concerns the integral of 1=r 3 over a triangle in R 3 , that plays a fundamental role in BEMs in 3D, especially for the Galerkin implementation. Because the existence of the aforementioned integral depends on the position of the source point, all significant instances of the position of the source point will be considered and detailed. For its interest in the context of BEM, the integral is also considered in the more general sense of finite part of Hadamard.

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“…It can be proved that integral (22) admits of the following closed form expression in the local coordinate system L: …”

Section: Discussionmentioning

confidence: 99%

“…Analytical integrations have been basically performed in 2D (only a few works appeared in the 3D context, see e.g. [21][22][23]), towards different schemes. In the first scheme (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Analytical integrations in 3D BEM: preliminaries

Carini

Salvadori

2002

Computational Mechanics

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“…The numerical implementation requires considerable care [1] because it involves evaluation of singular (weak, strong and hyper) integrals. Some of the notable two-dimensional (while all the devices are 3D by definition, useful insight is often obtained by performing a 2D analysis) and three-dimensional approaches used to evaluate the singular integrals are discussed in [1,2] and [3,4,5,6,7] and the references in these papers. It is wellunderstood that many of the difficulties in the available BEM solvers stem from the assumption of nodal concentration of singularities which leads to various mathematical difficulties and to the infamous numerical boundary layers [8,9,33] when the source is placed very close to the field point ( [2] and references [4][5][6] therein).…”

Section: Introductionmentioning

confidence: 99%

“…Some of the notable two-dimensional (while all the devices are 3D by definition, useful insight is often obtained by performing a 2D analysis) and three-dimensional approaches used to evaluate the singular integrals are discussed in [1,2] and [3,4,5,6,7] and the references in these papers. It is wellunderstood that many of the difficulties in the available BEM solvers stem from the assumption of nodal concentration of singularities which leads to various mathematical difficulties and to the infamous numerical boundary layers [8,9,33] when the source is placed very close to the field point ( [2] and references [4][5][6] therein). While mathematical singularities (that occur when the source and field points coincide) have been shown to be artifacts, several techniques have been used to remove difficulties related to physical or geometrical singularities (that occur when boundaries are degenerate, i.e., geometrically singular, or due to a jump in boundary conditions) such as gaussian quadrature integration, mapping techniques for regularization, bicubic transformation, nonlinear transformation and dual BEM techniques [8].…”

Section: Introductionmentioning

confidence: 99%

A study of three-dimensional edge and corner problems using the neBEM solver

Mukhopadhyay

Majumdar

2009

Engineering Analysis with Boundary Elements

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The previously reported neBEM solver has been used to solve electrostatic problems having three-dimensional edges and corners in the physical domain. Both rectangular and triangular elements have been used to discretize the geometries under study. In order to maintain very high level of precision, a library of C functions yielding exact values of potential and flux influences due to uniform surface distribution of singularities on flat triangular and rectangular elements has been developed and used. Here we present the exact expressions proposed for computing the influence of uniform singularity distributions on triangular elements and illustrate their accuracy. We then consider several problems of electrostatics containing edges and singularities of various orders including plates and cubes, and L-shaped conductors. We have tried to show that using the approach proposed in the earlier paper on neBEM and its present enhanced (through the inclusion of triangular elements) form, it is possible to obtain accurate estimates of integral features such as the capacitance of a given conductor and detailed ones such as the charge density distribution at the edges / corners without taking resort to any new or special formulation. Results obtained using neBEM have been compared extensively with both existing analytical and numerical results. The comparisons illustrate the accuracy, flexibility and robustness of the new approach quite comprehensively.

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“…However, some researchers have been looking for semi-analytical solutions or sophisticated approaches in order to improve the accuracy of BEM models used to solve specific problems [18]. Niu et al [19] proposes a semi-analytical algorithm for 3D elastic problems that require the evaluation of nearly strongly singular and hypersingular integrals on the triangular and quadrilateral elements.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional boundary element method model in the frequency domain for simulating dynamic heat conduction

Prata¹,

Tadeu²,

Simões³

2012

Advanced Computational Methods and Experiments in Heat Transfer XII

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The boundary element method (BEM) is a suitable numerical method used to study heat diffusion by conduction, which only requires the discretization of the boundary of the material discontinuities. However, one of the biggest challenges to this technique is the integration of mathematical singularities. In this work the BEM is used to simulate the heat conduction in the vicinity of 3D inclusions, when heated by dynamic heat point sources. Continuity of temperatures and heat fluxes are prescribed along the boundary interfaces. All boundary interfaces are discretized using constant elements. The proposed algorithm is verified by means of known analytical solutions for cylindrical inclusions.

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Efficient evaluation of integrals in three-dimensional boundary element method using linear shape functions over plane triangular elements

Cited by 11 publications

References 8 publications

Analytical integrations in 3D BEM: preliminaries

Analytical integrations in 3D BEM: preliminaries

A study of three-dimensional edge and corner problems using the neBEM solver

Three-dimensional boundary element method model in the frequency domain for simulating dynamic heat conduction

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