Volume integration in the hypersingular boundary integral equation

The evaluation of volume integrals that arise in conjunction with a hypersingular boundary integral formulation is considered. In a recent work for the standard (singular) boundary integral equation, the volume term was decomposed into an easily computed boundary integral, plus a remainder volume integral with a modified source function. The key feature of this modified function is that it is everywhere zero on the boundary. In this work it is shown that the same basic approach is successful for the hypersingular equation, despite the stronger singularity in the domain integral. Specifically, the volume term can be directly evaluated without a body-fitted volume mesh, by means of a regular grid of cells that cover the domain. Cells that intersect the boundary are treated by continuously extending the integrand to be zero outside the domain. The method and error results for test problems are presented in terms of the three-dimensional Poisson problem, but the techniques are expected to be generally applicable.     

Galerkin formulation and singularity subtraction for spectral solutions of boundary integral equations

A new spectral Galerkin formulation is presented for the solution of boundary integral equations. The formulation is carried out with an exact singularity subtraction procedure based on analytical integrations, which provides a fast and precise way to evaluate the coefficient matrices. The new Galerkin formulation is based on the exact geometry of the problem boundaries and leads to a non-element method that is completely free of mesh generation. The numerical behaviour of the method is very similar to the collocation method; for Dirichlet problems, however, it leads to a symmetric coefficient matrix and therefore requires half the solution time of the collocation method. © 1998 John Wiley & Sons, Ltd.     

体积源边界点法及其在声辐射计算中的应用

利用作者提出的体积源边界点法，对结构振动声辐射的计算问题进行了研究。给出了该法在全频率范围内存在唯一解的数学证明；通过诸多不同边界曲面和不同边界量分布的声辐射算例，从计算精度、计算稳定性、对振动体任意表面几何形状的适应性以及克服解的非唯一性等方面，对该方法的有效性进行了检验。     

A general regularization of the hypersingular integrals in the symmetric Galerkin boundary element method

The symmetric Galerkin boundary element method is used to solve boundary value problems by keeping the symmetric nature of the matrix obtained after discretization. The matrix elements are obtained from a double integral involving the double derivative of Green's operator, which is highly singular. The paper presents a regularization of the hypersingular integrals which depend only on the properties of Green's tensor. The method is presented in the case of Laplace's operator, with an example of application. The case of elasticity is finally addressed theoretically, showing an easy extension to any case of anisotropy. Copyright © 2009 John Wiley & Sons, Ltd.     

On solvability of a boundary integral equation of the first kind for Dirichlet boundary value problems in plane elasticity

The solution of a Dirichlet boundary value problem of plane isotropic elasticity by the boundary integral equation (BIE) of the first kind obtained from the Somigliana identity is considered. The logarithmic function appearing in the integral kernel leads to the possibility of this operator being non-invertible, the solution of the BIE either being non-unique or not existing. Such a situation occurs if the size of the boundary coincides with the so-called critical (or degenerate) scale for a certain form of the fundamental solution used. Techniques for the evaluation of these critical scales and for the removal of the non-uniqueness appearing in the problems with critical scales solved by the BIE of the first kind are proposed and analysed, and some recommendations for BEM code programmers based on the analysis presented are given.