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題 名 | 勢能理論在擴散方程式求解之應用 |
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作 者 | 邱金火; 溫志超; | 書刊名 | 中原學報 |
卷 期 | 17 1988.12[民77.12] |
頁 次 | 頁51-70 |
分類號 | 319.4 |
關鍵詞 | 求解; 勢能理論; 擴散方程式; |
語 文 | 中文(Chinese) |
中文摘要 | 本文以疊加原理(The principal of superposition)將擴散方程式分兩部份求解,第一部份以問題的起始條件對整個定義域做積分;第二部份以單層勢能(Single layer potential)及雙層勢能(Double layer potential)理論將其解化成僅含有時間參數且在邊界上的Volterra積分方程式。完成數學模式建立後,再藉數值方法求值。 本文以有限元素法(Finite element method)對第一部份的積分方程求值,此種積分方程為一開顯函數(Explicitfunction) ,以有限元素法中的兩點等參數元數求值甚為容易。第二部份之Volterra積分方程的求值,則以邊界元素法(Boundary element method)處理,因為邊界元素法本身具有節省電子計算機儲存空間的優點,且本研究發現聯立方程式的係數會構成下三角矩陣,在數值求解過程中甚為簡便。以有限元素法及邊界元素法二數值方法的聯合運用,建立一程式架構,並用此程式架構演算出三個例題,所得答案與教學正確解做比較,其結果令人滿意。 沿用本文所建立之程式架構可推廣至定義域中有相同變數的問題,僅需修改部份副程式,對於應用甚方便,且能適用於較小型之電子計算機。 |
英文摘要 | In this study, the solution of the diffusion equation is separated into two parts by using the principal of superposition. The first part of the solution is integrated within the whole domain of problem with the initial condition. The other part of the solution is transformed into the Volterra integral equation, which only involves time variable, by the theory of single layer potential or double layer potential. After the mathematical model has been established, it is solved by the numerical method. The first part of the solution, which is an integral equation and contains explicit function, is easy to solve with the finite element method of linear elements isoparametric transformation. In order to save the space of computer storage, it is convenient to solve the second part of the solution, Volterra integral equation, with the boundary element method. It is found that the coefficient matrix of the simultaneous equations formed from the boundary element method is a lower triangle matrix which simplifies the process of numerical calculation. With the above two numerical methods, a form of computer program is developed. Three examples are compared with their respective exact solution. They show that the results are quite agreeable. It can be easily extended to the problem whose domain contains the same variables as discussed in the study and its solution can be easily solved using the developed program, by very little change to the subroutine. Moreover, the developed computer program is suitable for the microcomputer or pc. |
本系統中英文摘要資訊取自各篇刊載內容。