Analysis of circular torsion bar with circular holes using null-field approach

Abstract

The degenerate kernels and Fourier series expansions are adopted in the null-field integral equation to solve torsion problems of a circular bar with circular holes. The main gain of using degenerate kernels is free of calculating the principal values. An adaptive observer system is addressed to fully employ the property of degenerate kernels in the polar coordinate. After moving the null-field point to the boundary and matching the boundary conditions, a linear algebraic system is obtained without boundary discretization. The unknown coefficients in the algebraic system can be easily determined. The present method is treated as a "semi-analytical" solution since error only attributes to the truncation of Fourier series. Finally, several examples are given to demonstrate the validity of the proposed method. 本文使用零場積分方程搭配分離核函數與傅立葉級數求解含圓形邊界之扭轉問題。藉由分離核函數的表示式，可解析計算所有的邊界積分而免於計算主值的困擾。文中採用自適性觀察座標系統的想法，來充分掌握分離核函數的特性。透過零場積分方程將零場點推向邊界且均勻佈點，滿足邊界條件後可以得到一線性代數方程式，其中的未知傅立葉係數均可輕易地求得。由於誤差僅來自於擷取有限項的傅立葉級數，故本方法可視為“半解析法”。最後，為了驗證此方法的可行性與正確性，提出含圓孔洞的受扭桿問題予以測試。