A Semi-Analytical Approach for Boundary Value Problems with Circular Boundaries

Abstract

In this paper, a semi-analytical approach is developed to deal with problems including multiple circular boundaries. The boundary integral approach is utilized in conjunction with degenerate kernel and Fourier series. To fully utilize the circular geometry, the fundamental solutions and the boundary densities are expanded by using degenerate kernels and Fourier series, respectively. Both direct and indirect formulations are proposed. This approach is a semi-analytical approach, since the error stems from the truncation of Fourier series in the implementation. The unknown Fourier coefficients are easily determined by solving a linear algebraic system after using the collocation method and matching the boundary conditions. Five goals: (1) free of calculating principal value, (2) exponential convergence, (3) well-posed algebraic system, (4) elimination of boundary-layer effect and (5) meshless, of the formulation are achieved. The proposed approach is extended to deal with the problems containing multiple circular inclusions. Finally, the general-purpose program in a unified manner is developed for BVPs with circular boundaries. Several examples including the torsion bar, water wave and plate vibration problems are given to demonstrate the validity of the present approach.