Null-field integral equation approach for structure problems with circular boundaries

Abstract

In this paper, null-field equation approach is developed to deal with the structural problems including multiple circular boundaries, e.g. holes or inclusions. 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 model, of the formulation are achieved. Finally, the general-purpose program in a unified manner is developed for structure problems with circular boundaries including the torsion bar, plate vibration and elasticity problems. 本文使用零場積分方程配合退化核與傅立業級數來處理含多圓洞與夾雜結構問題。為了充分利用圓形的幾何特性，我們將基本解與邊界密度函數分別以退化核與傅立業級數展開。除了直接法外，我們也會使用間接法來求解。由於誤差來自於傅立業級數的項數擷取多寡，故此套方法可被視為是一套半解析法。我們藉由佈點法可建構出一線性代數系統並進而求得未知的傅立業係數。本法將達成五個預期目標: (1) 毋須計算主值問題，(2)指數收歛特性，(3)良態代數系統的建構，(4)邊界層效應的消除與(5)無網格模式。最後，我們將發展一套廣用程式來處理含圓型邊界的扭轉、薄板振動與彈力問題。