Applications of Numerical Global Domain Transformation Method to 2D Elasticity Problems
Date Issued
2007
Date
2007
Author(s)
Lo, Wei-Lin
DOI
zh-TW
Abstract
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.
Subjects
數值全域轉換法
區域轉換
邊界符合保角網格
保角網格
彈性力學
有限差分法
數值計算
Numerical global domain transformation method
NGDTM
Boundary-fitted conformal grid
Conformal grid
Airy stress
Biharmonic
Finite difference method
FDM
Numerical solution
Type
thesis
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