無網格法分析彈性靜力問題之研究

本文主旨在使用無網格局部彼得洛夫－葛勒金(簡稱MLPG)法分析彈性靜力問題。MLPG法係基於局部對稱之弱式(簡稱LSWF)以及變動最小平方近似法(簡稱MLS)來形成形狀函數，因此於物理量的近似與能量的積分，甚至必要邊界條件的引入，完全無須求助有限元素之建立，是一個真實地無網格法。而其積分式可以在規則的形狀領域(一般，在二維問題中為圓形，在三維問題中為球形)及其邊界中容易地求得，且吾人利用懲罰法引入必要邊界條件。本文的數值算例顯示，MLPG法在處理近似不可壓縮材料時不會有體積鎖死之現象，而對於位移及能量範數(norms)可達到極高的收斂率，因為由MLPG法得到之原始解是經由MLS近似法處理，已經足夠平穩以至於不需要後處理的過程來計算應力與應變。關鍵詞：無網格局部彼得洛夫－葛勒金法、局部對稱之弱式、變動最小平方近似法。

關鍵字

變動最小平方近似法；葛勒金法；無網格法

並列摘要

In this study, the Meshless Local Petrov-Galerkin (MLPG) method for solving problems in elasto-statics is developed and numerically implemented. The present MLPG approach is based on a local symmetric weak form and shape functions from the moving least squares approximation. This approach is a truly meshless method, as it does not need a “finite element mesh”, either for purposes of interpolation for the solution variables, or for the integration of the energy. All integrals in the formulation can be easily evaluated over regularly shaped domains (in general, circles in two-dimensional problems, or spheres in three -dimensional problems) and their boundaries. The essential boundary conditions in the present formulation are imposed by a penalty method. The numerical examples show that the present MLPG approach does not exhibit any volumetric locking for nearly incompressible materials, and high rates of convergence with mesh refinement for the displacement and energy norms are achievable. No post-processing procedure is required to compute the strain and stress, since the original solution from the present method using the moving least squares approximation is already smooth enough. Key Words: Meshless Local Petrov-Galerkin method, local symmetric weak form, moving least squares approximation.

並列關鍵字

moving least squares ； Galerkin ； Meshless

參考文獻

1. Atluri S.N., Zhu T.L., “A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics,” Comput. Mech., Vol. 22, pp.117-127 (1998).

2. Atluri S.N., Zhu T.L., “A New Meshless Local Petrov-Galerkin (MLPG) Approach for Solving Nonlinear Problems in Computer Modeling and Simulation,” Comput. Modeling Simul. Eng., Vol. 3, (1998).

3. Atluri S.N., Kim H.G., Cho J.Y., “A Critical Assessment of the Turly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) Methods,” Comput. Mech., Vol. 24, pp.348-372 (1999).

4. Atluri S.N., Zhu T.L., “The Meshless Local Petrov-Galerkin (MLPG) Approach For Solving Problems in Elasto-Statics,” Comput. Mech., Vol. 25, pp.169-179 (2000).

5. Belytschko T., Lu Y.Y., Gu L., “Element-Free Galerkin Methods,” Int. J. Num. Meth. Eng., Vol. 37, pp.229-256 (1994).

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無網格法分析彈性靜力問題之研究

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