Approximate formulation of the hypersingular boundary integral equation in potential theory
The basis of hypersingular boundary integral equation (HBIE) development is singular and hypersingular field solutions of kernel functions. Instead of taking traditional local or global regularization strategies, the current paper proposes an approximate formulation of the HBIE in potential theory, designed for numerical solution when collocating at irregular places such as corners and edges. This is because any CPV boundary integral can be approximated by the mean value of two corresponding nearly singular boundary integrals at boundary points where the continuity requirements are met. The nearly singular boundary integrals can be evaluated accurately with the previously developed distance transformation techniques. In consequence, both the analytical and numerical efforts can be reduced associated with the difficult problems of the singularity removal and especially the free term representation, thus the resulting algorithm can be greatly simplified. The 2D and the 3D examples are presented, showing that the algorithm works very well. With the algorithm, both the primary and secondary field variables can be computed accurately at the places near boundaries, on smooth boundaries or on irregular boundaries such as corners and edges.
approximate formulation hypersingular boundary integral equation distance transformation bem continuity condition