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 摘要本論文主要以基本解法配合Trefftz方法與間接邊界積分方程法求解含圓形及球形邊界拉普拉斯方程之邊界值問題。在基本解法中，我們專注在最佳佈點位置的探討，以及基本解法中解的表示式所含自由常數項扮演的角色這兩個議題。在二維偏心圓例子中，利用映像法的觀點找出佈點的最佳位置。然而，在三維偏心球的例子中，兩焦點仍是佈點的重要位置。除此之外，Trefftz基底與基本解法集中源的關係在本文中也一併討論。我們發現基本解法中的集中源可以利用分離核展開型式找到對應的全部內外域Trefftz基底。反之，利用基本解法中的集中源可以疊加出單一Trefftz基底的解。基於這些發現，可以藉由間接邊界積分方程法與基本解法來建立兩種方法虛擬強度間的關係。關於橢圓領域的格林函數問題可以利用間接邊界積分方程法推導出解析解，此解析解比Lebedev的解更為通用，並與傳統邊界元素法的結果做比較，得到吻合的結果。
 AbstractIn this thesis, we solve the boundary value problems (BVPs) without sources for the 2D and 3D Laplace problems containing circular and spherical boundaries by using the method of fundamental solutions (MFS) in conjunction with the Trefftz method and the indirect boundary integral equation method (BIEM). In the MFS, we focus on the two main issues. Not only optimal location of the source distribution in the MFS but also the two versions of the MFS by adding a free constant term are discussed. In the eccentric annulus, the optimal location of the source distribution in the MFS can be found by using the image concept. Nevertheless, the location of the source at two foci for the problem of eccentric sphere plays an important role. Besides, the relationship between the Trefftz base and the singularity in the MFS is constructed by using the indirect BIEM and degenerate kernel. It is found that one source of the MFS contains all interior and exterior Trefftz sets through a degenerate kernel. On the contrary, one single Trefftz base can be superimposed by putting some sources in the MFS. Based on this finding, the relationship between the fictitious boundary densities of the indirect BIEM and the singularity strength in the MFS can be constructed due to the fact that the MFS is a lumped version of an indirect BIEM. Regarding the Green’s function of elliptic domain, the analytical solution is derived by using the indirect BIEM and is compared with the numerical solution of the BEM. It is found that our solution is more general than Lebedev’s case. Agreement is made after comparing with the BEM solutions.
 ContentsContents…………………………………………………………ITable captions………………………………………………IIIFigure captions………………………………………………IVNotations………………………………………………………VIII摘要…………………………………………………………………XAbstract…………………………………………………………XIChapter 1 Introduction………………………………………11.1 Motivation of the research and literature review………………………………………………………………11.2 Organization of the thesis…………………………3Chapter 2 Solutions of two-dimensional Laplace problems containing circular or elliptic boundaries using the Trefftz method, the MFS and the indirect BIEM……………………………………………………………………7Summary………………………………………………………………72.1 Introduction…………………………………………………72.2 Problem statements…………………………………………92.3 The method of fundamental solutions (MFS)……………………………………………………………………………………92.4 Trefftz method…………………………………………102.5 Indirect boundary integral equation method (Indirect BIEM)………………………………………………………………112.6 Relationship between the Trefftz bases and the singularity in the MFS………………………………………122.7 Illustrative examples and discussions…………122.8 Conclusions…………………………………………………19Chapter 3 Solutions of three-dimensional Laplace problems containing spherical boundaries using the Trefftz method, the MFS and the indirect BIEM………………………………………………………………………39Summary………………………………………………………………393.1 Introduction……………………………………………………393.2 Problem statements……………………………………………413.3 The method of fundamental solutions (MFS)………………………………………………………………………………413.4 Indirect boundary integral equation method (Indirect BIEM)………………………………………………………………423.5 Relationship between the Trefftz bases and the singularity in the MFS………………………………………433.6 Illustrative examples and discussions………443.7 Conclusions…………………………………………………47Chapter 4 Conclusions and further research…………674.1 Conclusions……………………………………………………674.2 Further research……………………………………………68References……………………………………………………………70