楊德良臺灣大學：土木工程學研究所朱哲均Chu, Je-JiumJe-JiumChu2007-11-252018-07-092007-11-252018-07-092007http://ntur.lib.ntu.edu.tw//handle/246246/50269本論文主要在探討基本解法以及數值轉換的結合，去求解非均質材料的勢能和擴散問題。基本解法是屬於邊界類型的無網格方法。對於非均質的勢能或是擴散問題，無法直接使用基本解法去模擬。非均質材料在本篇論文中分為兩種類型，一為功能梯度材料，一是材料內部的熱傳導係數不為定值。功能梯度材料是指構成要素(組成、架構)沿濃度方向由一側向另一側呈現連續梯度變化。熱在功能梯度材料上的擴散問題能藉由指定數學轉換式轉換，在使用基本解法求解。而勢能問題在熱傳導係數不為定值的材料上，能使用柯西荷夫轉換法去轉換，再利用基本解法求解。經由轉換求得非均質材料的勢能以及擴散問題的答案，都能與解析解或者使用有限差分的方法所求得的答案一致，因此，基本解法也許能在非均質問題上做更廣泛的研究與應用。This thesis mainly describes the combination of the method of fundamental solutions (MFS) and numerical transformation to solve potential and diffusion problems in non-homogeneous materials. The MFS is a meshless method which belongs to boundary-type method. For the potential and diffusion problems in non-homogeneous materials, the results can not be simulated by the MFS directly. Non-homogeneous materials can demarcate two types in this thesis, one is functionally graded materials (FGMs); one is the heat conductivity which is not constant inside the material. FGMs is a kind of material which is composed by the materials varying from one side to another in the direction of density continuously. The transient heat diffusion problems in FGMs can be solved by the MFS employing specific the transformation’s formulation. Potential problems in non-homogeneous materials can utilize the Kirchhoff’s transformation to transfer to be linear and the results also can be solved by the MFS. The results of potential and diffusion problems in non-homogeneous materials are simulated after transformation and the results are agreement with using finite difference method or analytical solutions. The MFS is successfully applied to solve potential and diffusion problems.口試委員會審定書i 誌謝ii 中文摘要iii Abstract iv List of Figures vii Symbols ix Chapter 1 Introduction 1.1 Motivations 1 1.2 Objective of the present thesis 4 1.3 Organization of the thesis 4 References 6 Chapter 2 Numerical scheme-The Method of Fundamental Solutions 2.1 Introduction 9 2.2 The theory of MFS 10 2.2.1 Laplace problems 12 2.2.2 Diffusion problems 13 References 16 Chapter 3 The MFS with Parameter Transformation for Functionally Graded Materials Heat Problems 3.1 Introduction 17 3.2 Governing equation 18 3.3 Result and discussions 21 3.4 Conclusions 26 References 34 Chapter 4 The MFS with Kirchhoff’s Transformation for Steady State Nonlinear Material Heat Problems 4.1 Introduction 36 4.2 Numerical scheme 37 4.3 Result and discussions 39 4.4 Conclusions 45 References 58 Chapter 5 Conclusions and Future Works 5.1 Conclusions 59 5.2 Future Works 601598328 bytesapplication/pdfen-US基本解法非均質勢能方程式擴散方程式無網格功能梯度材料熱傳導柯西荷夫轉換法解析解有限差分The method of fundamental solutionsnon-homogeneouspotential equationdiffusion equationmeshlessfunctionally graded materials (FGMs)nonlinear heat conductivityKirchhoff’s transformationfinite difference method (FDM)analytical solutionthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/50269/1/ntu-96-R94521323-1.pdf