Analytical study and numerical experiments for radiation and scattering problems using the chief method

Abstract

In this paper the well-known failure in numerical computations of the exterior Helmholtz integral equation at certain characteristic frequencies is investigated analytically and numerically. The combined Helmholtz integral equation formulation (CHIEF) method is a very popular technique that can overcome the non-unique problem, but this method breaks down if the interior points are not properly chosen. The CHIEF method in conjunction with the singular value decomposition (SVD) technique is an easy and efficient method to ensure a unique solution for the exterior problem. Based on the circulant properties and degenerate kernels, an analytical scheme in discrete system of a cylinder is achieved. The optimum numbers and proper positions for the collocation points in the interior domain are analytically studied and suggested in the numerical scheme. Numerical experiments are designed to verify the analytical results. One analytical example for a circular cylinder and one numerical example for a square rod are demonstrated to show the validity of the CHIEF method.