- 學位論文
數值全域轉換法於二維彈性力學之應用
Applications of Numerical Global Domain Transformation Method to 2D Elasticity Problems
摘要
本研究旨在應用Tsay等人(Tsay and Hsu, 1997; Wang and Tsay, 2005; Tsay, Wang and Huang, 2006; 黃永德，1998)發展出來的「數值全域轉換法」(Numerical Global Domain Transformation Method, NGDTM)，在任意二維幾何區域求解彈性力學的問題(Airy stress function)，並發展出一套通用的數值求解策略。 「數值全域轉換法」為一個將整體解題空間由物理發生的平面轉換至另一個平面求解的方法。利用複變映射原理(Complex mapping)以及邊界元素積分法(Boundary integral element method)求解Laplace方程式，將一個任意二維幾何形狀轉換至一矩形區域，使得有限差分法(Finite difference method)得以在此矩形區域方便地運用。 Airy stress function為求解彈性力學的控制方程式之ㄧ(Timoshenko and Goodier, 1986)，由於定義的關係，traction邊界條件只能提供二階的偏導數值。過去在矩形空間應用FDM求解Airy stress function時，必須透過沿著幾何邊界對traction邊界條件積分獲得一階及零階偏導數的數值邊界條件(Timoshenko and Goodier, 1986)。然而當幾何邊界複雜時，這個積分的動作會變得相當地困難，並且程式不容易撰寫成一般化，因此未被廣泛地應用在數值方法。 本研究透過Tsay等人發展出來的區域數值轉換理論，將任意的幾何邊界條件及traction邊界條件轉換至矩形區域，然後在該矩形區域上將traction邊界條件透過數值積分轉換成運用有限差分法需要之數值邊界條件。控制方程式亦透過此區域數值轉換理論由不規則區域轉換至矩形區域。運用轉換過的控制方程式和邊界條件在矩形區域進行有限差分法之計算。進行控制方程式和邊界條件的轉換時，將透過「柯西-里曼方程式」(Cauchy-Riemann conditions)化簡方程式，並且轉換後的Jacobian， 、 不為常數。在矩形區域進行離散時，本研究採取在區域內(不含邊界)設置以中央差分離散之控制方程式，邊界點上的自由度每個點設置兩條邊界條件，四個角落分別設置三條邊界條件的策略，使得未知數的數目等於方程式的數目，獲得唯一且收歛的解。以一兩端承受彎矩的栱型樑為例，轉換至矩形區域後，在網格點為 ，網格大小為 的情況下，取中間的剖面的數值結果與解析解相較，勢能(Potential)的誤差在 之內，應力 的誤差在 之內，應力 的誤差在 之內，應力 最大誤差為0.004(解析解為0)。 本研究推導出控制方程式、邊界條件及應力式正確的轉換關係，並提出一套通用的數值求解策略，奠定Tsay等人發展出來的「數值全域轉換法」在彈性力學領域應用的基礎。同時經由本研究建立之求解模式，有限差分法可以很方便地運用在二維彈性力學的求解上，提昇了運用有限差分法求解二維彈性力學時適用的廣度。
並列摘要
The objective of this research is to develop a general numerical scheme to solve elasticity problems in an arbitrary two dimensional domain by “Numerical global domain transformation method (NGDTM),” a domain transformation theorem developed by Tsay et al. (Tsay and Hsu, 1997; Wang and Tsay, 2005; Tsay, Wang and Huang, 2006; 黃永德，1998). NGDTM transforms the problem solving domain from an arbitrary two dimensional domain, the physical domain, to a rectangular domain by applying complex mapping theorem and solving Laplace equations by boundary integral element method (BIEM). When the domain is transformed to a rectangle domain, it becomes very convenient to solve the problems with finite difference method (FDM). Airy stress function is one of the governing equations in elasticity (Timoshenko and Goodier, 1986). Based on the Airy stress function, the traction boundary conditions can only provide the second order derivative of the primary variable. It is thus necessary to integrate the traction boundary condition along the geometry to obtain the numerical boundary condition when one tries to solve the Airy stress function in a rectangular domain (Timoshenko and Goodier, 1986). Such integration becomes relatively difficult when the geometry is complex. Partly due to such shortcoming, FDM is not popular in solving elasticity problems. In this work, an arbitrary two dimensional domain and traction boundary conditions were transformed to a rectangular domain through the domain transformation theorem. Traction boundary conditions in the rectangular domain were integrated to obtain the numerical boundary condition. The governing equation was also transformed form the arbitrary domain to the rectangular one by the domain transformation theorem. After applying the transformed governing equation and numerical boundary conditions, the finite difference method calculation was performed in a rectangular domain. During the transformation derivative, Cauchy-Riemann conditions were applied in which the Jacobians and , were not constants. The governing equation and numerical boundary conditions were both expressed in the first-order central difference. The governing equation was discreted to a 13 point formulation and applied on the internal meshes excluding the boundary. Each node on the boundary has two boundary conditions. Each corner has three boundary conditions. This scheme makes the number of unknowns equal to the number of equations and result in an unique and convergent solution. An arc with a couple of moment applied was used to verify the proposed scheme. The rectangle domain was discreted in nodes and the mesh size was. . In comparison with the analytical solution on the middle of the arc, the error of the potential was less than , the error of the normal stress was less than , the error of the normal stress was less than , and the maximum error of the shear stress was 0.004 (the analytical solution is zero). A NGDTM for the Airy stress function has been developed in this work. A transformed governing equation, boundary conditions, stress formulation and a general numerical scheme have been derived. Through the machinery of NGDTM developed herein, the FDM can be used conveniently to solve two-dimensional elasticity problems.