Applications of hypersingular equations to free-surface seepage problems

Abstract

Free surface boundary is not only unknown in prior but also the boundary conditions are overspecified. In this paper, the Laplace problem with overspecified boundary conditions on the free surface is solved by using the hypersingular equation. In the conventional boundary element method, only singular integral equation is used. The boundary of free surface can be determined by trial and error after initial guess and iterations. By introducing the hypersingular equation, the convergence rate of free surface can be accelerated. Finally, numerical examples were demonstrated and compared with others to show the validity of the present method. 自由水面屬於一未定邊界的問題，本文以超奇異積分方程求解此 Laplace 問題。傳統的邊界元素法在分析此問題時，僅使用奇異積分方程式，需經過多次的試誤疊代才能得到收斂的自由水面。因此我們將自由水面視為一過定條件的邊界，且自由水面之邊界為未知；引入超奇異積分方程式來求解自由水面，可加速收斂速度。在本文中我們將提出測試例，並與前人所做的結果做一比較，以驗證此法之優越性。