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標題: | 應用區域分割及基本解法於偏微分方程叢集計算之研究 Applications of Domain Decomposition Method and the Method of Fundamental Solutions for Some Partial Differential Equations with Clustering |
作者: | Dar-Der Hsieh 謝達德 |
指導教授: | 楊德良 |
關鍵字: | 區域分割,基本解法,平行計算,尖角計算域,拉普拉斯方程式,史托克斯方程式,擴散方程式,對流-擴散方程式, domain decomposition,the method of fundamental solutions,parallel computing,cusp domain,Laplace equation,Stokes’ equation,diffusion equation,advection-diffusion equation, |
出版年 : | 2006 |
學位: | 碩士 |
摘要: | 本論文主要在探討區域分割法以及無網格基本解法的結合,並應用於求解拉普拉斯方程式、史托克斯方程式、擴散方程式以及對流-擴散方程式的成果。基本解法是一種無網格且具高效率計算的數值方法,但此種方法,在計算點數越多的情況下,會導致基本解所構成的線性系統矩陣越趨於病態。此問題在基本解法的發展上,造成對於計算規模的限制。因此,本論文提出此建立在區域分割理論上的基本解法,來解決這個病態矩陣的問題。應用此種方法,本文針對尖角計算域、穩態史托克斯流場、以及跟時間等相關的問題,做一系列的探討。值得注意的是,對於尖角計算域問題的計算,本篇論文是先應用保角轉換,使尖角計算域轉換為半無窮計算域,再將此區域分割並用基本解法來求解。由於區域分割法具有應用在平行計算的潛力,因此,本文也將此方法應用於平行計算上之效率討論。 This thesis mainly describes the combination of the Domain Decomposition Method (DDM) and the Method of Fundamental Solutions (MFS) as a meshless numerical method (DDM-MFS) to solve problems governed by various Partial Differential Equations (PDEs), including Laplace equation, Stokes’ equations, diffusion equation and advection-diffusion equation. The MFS is an efficient meshless method, which its application can be extensively found in the literature. However, the resultant matrix in the MFS will approach to ill-conditioned as computational points increasing, which creates restrictions for development of this method. The DDM-MFS is thus developed in order to solve this problem. Applied the DDM-MFS, problems including the cusp domain problems, the steady Stokes’ flow problems and the time-dependent problems are successfully addressed. It is to be noted that there is a new approach to solve the cusp problems proposed in this thesis, which is to transform the cusp domains in to semi-infinite domains by conformal mapping techniques. Since the DDM is developed as a preparation for parallel computing, the efficiency of parallel computing applied the DDM-MFS is also tested for the problems mentioned above in the thesis. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/31726 |
全文授權: | 有償授權 |
顯示於系所單位: | 土木工程學系 |
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