On the equivalence of the Trefftz method and the method of fundamental solutions for plate problem

Abstract

In this paper, it is proved that the two approaches for biharmonic equation, known in the literature as the method of fundamental solutions (MFS) and the Trefftz method, are mathematically equivalent in spite of their essentially minor and apparent differences in the formulation. In deriving the equivalence of the Trefftz method and the MFS for plate problem, it is interesting to find that the T-complete set in the Trefftz method for 1-D, 2-D, 3-D Laplace and Helmholtz problems are imbedded in the degenerate kernels of the MFS. The unknown coefficients of each method for plate problems correlate by a mapping matrix after considering the degenerate kernels for the fundamental solutions in the MFS and the T-complete function in the Trefftz method. The mapping matrix is composed of a rotation matrix and a geometric matrix which depends on the source location. Also, the occurring mechanism of the degenerate scale for the plate problems is examined in this paper. 本文主要探討 Trefftz 法與基本解法求解雙諧和方程問題兩者在數學上之等效性。於推導板之靜力問題前，先發現在二維及三維的拉普拉斯方程與漢姆赫茲方程中，Trefftz 的完整解集合不論是在內域問題或外域問題皆可由基本解法中的退化核函數中求得。因此，我們把兩階控制方程的成功案例拓展為四階，並設計一個圓形固端板範例做說明，利用退化核函數展開基本解所得到之係數矩陣與 Trefftz 法中所得到之係數矩陣相互比較後，可產生一映射矩陣。此映射 矩陣與源點的位置分佈有關並可分解為一旋轉矩陣與幾何矩陣。透過此映射矩陣，可看出在數值中所遇到的退化尺度的發生機制。