Improper Integrals Calculations for Fourier Boundary Element Method

摘要

This paper presents a method of regularization for the numerical calculation of improper integrals used in different formulations of Boundary Element Method (BEM). The main attention of the readers we would like to focus on Fourier Formulation of BEM. The singular integrals arise when for discretization the elements of a higher order than zero are used. Very often in the Diffusive Optical Tomography for infant head modeling, triangular or square curvilinear boundary elements of the second order are used [12,14], hence, our interest in the subject of effective and accurate calculation of singular integrals. Even for the classical formulation of BEM such a problem is extremely difficult [1]. Some authors believe that the practical application possesses only flat triangular boundary elements of zero-order, and although there is some truth in this statement, the elements of the second order show a significant advantage [10,12] in Diffusion Optical Tomography (DOT) for example. This issue becomes even more interesting when we deal with the Galerkin BEM formulation offering the possibility of matrix of coefficients symmetrisation, which has fundamental importance for inverse problems. This matter becomes critical when we start to consider the Fourier BEM formulation, introduced by Duddeck [5]. His approach provides the possibility of a solution in the case which has no fundamental solution. The light propagation, which is described by the Boltzmann equation (see Arridge [2]) is such a case. Currently and most commonly, the Boltzmann equation is approximated by the diffusion equation in strongly light scattering media [10]. In the author’s opinion, the problem of numerical integration of improper integrals has not yet been fully exhausted in the classic and Galerkin BEM formulation but the Fourier BEM formulation still expects the proposals of the effective solutions. Such an offer we would like to present in this paper.