Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method.

*(English)*Zbl 1003.65133Summary: The accurate numerical solution of near singular boundary integrals was an issue of major concern in most of the boundary element analysis next to the singular boundary integrals. The problem was solved in this paper by a kind of nonlinear transformation, namely, the distance transformation for the accurate evaluation of near singular boundary integrals with various kernels for both the two- and three-dimensional problems incorporated with the distance functions defined in the local intrinsic coordinate systems. It is considered that two effects la the role in the transformation. They are the damping out of the near singularity and the rational redistribution of integration points.

The actual numerical computation can be performed by standard Gaussian quadrature formulae and can be easily included in the existing computer code, along with its insensitivity to the kind of the boundary elements. Numerical results of potential problem were presented, showing the effectiveness and the generality of the algorithm, which makes it possible, for the first time, to observe the behavior of various boundary integral values with numerical means, when the source point is moving across the boundary with fine steps.

The actual numerical computation can be performed by standard Gaussian quadrature formulae and can be easily included in the existing computer code, along with its insensitivity to the kind of the boundary elements. Numerical results of potential problem were presented, showing the effectiveness and the generality of the algorithm, which makes it possible, for the first time, to observe the behavior of various boundary integral values with numerical means, when the source point is moving across the boundary with fine steps.

##### MSC:

65N38 | Boundary element methods for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

31A10 | Integral representations, integral operators, integral equations methods in two dimensions |

##### Keywords:

Laplace equation; boundary element method; order of near singularity; boundary layer effect; numerical results; near singular boundary integrals; Gaussian quadrature formulae
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\textit{H. Ma} and \textit{N. Kamiya}, Eng. Anal. Bound. Elem. 26, No. 4, 329--339 (2002; Zbl 1003.65133)

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