Boundary particle method for Cauchy inhomogeneous potential problems

Abstract

This article makes the first attempt to apply the boundary particle method (BPM) to the solution of Cauchy inhomogeneous potential problems. Unlike the other boundary discretization meshless methods, the BPM does not require any inner nodes to evaluate the particular solution, since the method uses the recursive composite multiple reciprocity technique to reduce an inhomogeneous problem to a series of higher order homogeneous problems. Thanks to its truly boundary-only meshless merit, the BPM is particularly attractive to solve inverse problems. In this study, the inner source term, i.e., right-hand side of Poisson equation, is of polynomial, exponential and trigonometric functions, or a combination of these functions, which frequently appear in inverse engineering problems. This article investigates numerical convergence and stability of the BPM in conjunction with the truncated singular value decomposition regularization technique, and presents sensitivity analysis with respect to the ratio parameter of accessible boundary length, as well as varied physical domains, e.g., simply connected domain with smooth or piecewise smooth boundary, amoeba-like irregular shape, annulus and L-shaped domains and also compares the BPM results with those by the dual reciprocity-boundary element method solutions, the dual reciprocity boundary knot method (DR-BKM) solutions and the analytical solutions. Our numerical experiments demonstrate that the proposed BPM is highly accurate, computationally efficient and numerically stable for Cauchy inhomogeneous potential problems.