使用多尺度多方向Trefftz method求解三維Biharmonic equation內域問題之研究

Abstract

本文我們使用多尺度多方向Trefffz Method數值方法求解三維biharmonic方程問題以及Cauchy反算問題。過去對於二維biharmonic 方程問題已經提出許多數值方法求解，然而對於三維問題並沒有一個有效的數值方法去求解。這裡我們利用Trefftz Method求解二維並利用此方法延伸求解三維上的問題，甚至比傳統邊界無網格法更有效且簡單。Cauchy反算問題擁有高度病態的問題，我們提出新的後處理（post-condition）線性系統來克服高度病態問題。然後，在本文後半段分別有二維和三維的算例，這些算例我們均使用Dirichlet邊界條件和Neumann邊界條件，之後利用配點法來求解正算以及Cauchy反算問題，並以Matlab程式語言和Mathematica軟體來進行數值分析模擬。

In this thesis, we develope a multi-scale and multi-directional Trefffz Method numerical method for three-dimensional biharmonic equation Cauchy problem and the inverse problem. In the past, the two-dimensional biharmonic equation has arisen many numerical methods, however, there is still not an efficient numerical method to solve the three-dimensional problem.Here we use Trefftz method for solving the two-dimensional problem and extend this method to solve the problem the three-dimensional.The proposed approach is even moreeffective and simple than the conventient boundary type meshless method. Inverse problem Cauchy problem has a highly morbid, we propose a new post-processing (post-condition) linear system problems to overcome the height of the sick.Then, in the second half of this thesis are respectively two and three dimensional numerical examples, in these examples we use the Dirichlet boundary conditions and Neumann boundary conditions, after which collocation method for solving direct problem and Cauchy inverse problem, and use Matlab programming language and Mathematica software to numerical analysis and simulation.