Null-field Approach for Boundary Value Problems with Circular Inclusions

In this study, the boundary value problem with circular inclusions is formulated by using the null-field integral equation. To fully capture circular geometries, separable expressions of fundamental solutions in the polar coordinate for field and source points and Fourier series for boundary densities are introduced to derive the formulation analytically. Intermediate advantages are obtained: (1) well-posed model, (2) singularity free, (3) boundary-layer effect free, (4) exponential convergence and (5) mesh free. The method is basically a numerical approach, and because of its semi-analytical nature, it possesses certain advantages over the conventional boundary element method. The null-field approach employing the degenerate kernel and Fourier expansion can be applied to solve boundary value problems which are governed by the Laplace, Helmholtz, biharmonic and biHelmholtz equations. Problems for the anti-plane elasticity as well as the in-plane electrostatic and anti-plane piezoelectricity study are revisited to demonstrate the validity of our method