以基本解法結合數值轉換求解非均質材料上之勢能和擴散導問題

The Method of Fundamental Solutions with Parameter Transformations for Potential and Diffusion in Non-homogeneous Material Problems

朱哲均(Je-Jium Chu)

指導教授 ： 楊德良

國立臺灣大學/工學院/土木工程學研究所/碩士(2007年)

https://doi.org/10.6342/NTU.2007.03364

摘要

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。

關鍵字

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

並列摘要

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.

並列關鍵字

The method of fundamental solutions ； non-homogeneous ； potential equation ； diffusion equation ； meshless ； functionally graded materials (FGMs) ； nonlinear heat conductivity ； Kirchhoff’s transformation ； finite difference method (FDM) ； analytical solution

參考文獻

[1.1] Zheng S. F., Ding K., Denda M., Weng G. J., A Dual Homogenization and Finite-Element Study on the In-Plane Local and Global Behavior of a Nonlinear Coated Fiber Composite, Comput. Methods Appl. Mech. Eng., Vol. 183, pp. 141-155, 2000.

[1.2] Chen H. T. and Lin J. Y., Hybrid Laplace Transform Technique for Non-Linear Transient Thermal Problems, Int. J. Heat Mass Transfer, Vol. 34, pp. 1301-1308, 1991.

[1.3] Serdyuk Y. V., Podoltsev A. D. and Gubanski S. M., Numerical Simulations of Dielectric Properties of Composite Material with Periodic Structure, J. Electrostatics, Vol. 63, pp. 1073-1091, 2005.

[1.4] Sutradhar A. and Paulino G. H., The Simple Boundary Element Method for Transient Heat Conduction in Functionally Graded Materials, Comput. Methods Appl. Mech. Eng., Vol. 193, pp. 4511-4539, 2004.

[1.5] Sutradhar A., Paulino G. H., Gray L. J., Transient Heat Conduction in Homogeneous and Non-homogeneous Materials by the Laplace Transform Galerkin Boundary Element Method, Eng. Anal. Bound. Elem., Vol. 26, pp. 119-132, 2002.