Null-field Approach for Boundary Value Problems with Circular Inclusions

Abstract

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.本文係使用零場積分方程求解含圖形夾雜之邊界值問題。為了充分利用圖形邊界的特性，將基本解以場、源點分離的概念展開為分離(退化)的型式，而邊界物理量則以傅立葉級數展開，搭配自適性觀察座標系統來解析求出邊界積分；因此可以獲得五大好處：矩陣良態模式、避免奇異積分、沒有邊界層效應、指數收斂、不必建構網格。相對於傳統邊界元素法，此半解析法擁有某種程度的優越性。此利用零場積分方程搭配分離核及傅立葉級數之方法，可廣泛地用來求解拉普拉斯、赫姆茲、雙諧和、雙赫姆茲之邊界值問題。最後，為了驗證此方法的有效性，對反平面彈力問題、平面靜電場問題與反平面壓電問題均予以測試。