SOLUTION OF BIHARMONIC PROBLEMS WITH CIRCULAR BOUNDARIES USING NULL-FIELD INTEGRAL EQUATIONS

Abstract

The null-field integral equation method in conjunction with Fourier series and degenerate kernels are proposed to solve the biharmonic equations with circular boundaries. The degenerate kernels in the BIEM are expanded by using the separation of field point and source point. The improper boundary integrals are novelly avoided since the appropriate interior and exterior expansion of degenerate kernels are used. The unknown boundary densities are expressed in terms of Fourier series and the unknown Fourier coefficients can be easily determined by using the collocation method. Finally, the numerical solutions for problems of plate and Stokes flow are compared with the data of finite element solution (ABAQUS) and previous results to demonstrate the validity of the present method. 本文採用零場積分方程法結合傅立葉級數與退化核來求解含圓形邊界雙諧和方程問題。其中退化核即為分離核，係將基本解中場、源點分離導得級數形式，藉由使用退化核的內外域表示式可避免主值積分的計算。未知的邊界密度函數以傅立葉級數展開，並配合選點法，未知的傅立葉係數即可輕易求得。針對板與史托克斯流問題，本文所得之數值結果將與有限元素法套裝軟體 (ABAQUS) 結果和前人研究做一比較，以驗證本法的可行性。