Trefftz法求解三維漢姆荷茲方程式

Trefftz Method for Solving 3-D Helmholtz Equation

鐘弘光(Hung-Kuang Chung) ； 簡秋記(Chyou-Chi Chien) ； 黃惠茹(Hui-Ru Hwung)

《中原學報》 27卷3期 (1999/09) Pp. 13-25

https://doi.org/10.6358/JCYU.199909.0013

摘要

本文使用Trefftz法求解三維漢姆荷茲方程式。首先以三維之球座標系統加以推導漢姆荷茲方程式 Trefftz 完全解集，然後再應用比較係數法，邊界積分法，及邊界插值法以求出滿足邊界條件之各項完全解部分和之權重係數。Trefftz比較係數法係利用Trefftz解集之三類線性獨立函數項在邊界上加以比較。Trefftz邊界積分法則將整個邊界當做一個元素乘以權重函數積分，權重函數之項數由解值達收斂而決定，並不是取決於積分點之個數。而Treffiz邊界插值法則是採用邊界上每點之值，將其代入Trefftz解集，以求得各項之權重係數。前兩種方法應用於以平面波與點聲源為入射波之於軟球與硬球散射勢能場。Trefftz邊界插值法則討論單極圓球輻射之勢能場。

關鍵字

Trefftz法 ； 漢姆荷茲方程式 ； 軟球 ； 硬球 ； 入射波 ； 散射 ； 輻射 ； 勢能函數

並列摘要

The Trefftz method was applied to solve for three dimensional Helmholtz equation. The complete Trefftz solution set in spherical coordinate systems was derived, and then the function comparison method, Trefftz boundary integral method and the boundary collocation method were applied on the boundary together with its condition to find the coefficient of each term of the partial sum of Trefftz complete solution set. The function comparison method compares the three types of the linear independent function in the Trefftz solution set on the boundary, while the Trefftz boundary collocation method applied each point on the boundary to Trefftz solution set In the Trefftz boundary integral method, the whole boundary was treated as an element for integration The numbers of unknowns and weighted functions have nothing to do with the number of integration points, but were decided by the absolute convergence of them The scattering fields of a soft sphere and a hard sphere imping by plane wave or wave from a point source were solved by the function comparison method and Trefftz boundary integral method. The method of collocation was applied to find the radiation field of a sphere.

並列關鍵字

Trefftz method ； Helmholtz equation ； soft sphere ； hard sphere ； incident wave ； radiation ； scattering ； potential function