本文應用移動最小二乘法（moving least square method,MLSM）來分析一維邊界值問題及二維彈性力學扭矩問題。本方法透過局部區域內離散點函數值資料，並加上微分方程式及邊界條件三者同時以加權最小二乘法建立滿足邊界值問題之近似函數，再由近似函數與節點值之一致性，即可求出邊界問題在節點上的近似值，依此可求出邊界值問題之近似解。本文最後以一維邊界值問題、二維彈力扭矩在橢圓形、正三角形及矩形作為計算範例，並與真解做比較來驗證及討論本方法之可行性與精度。

In this paper, we present a fully meshless method for solving differential equation governing a certain physical problem. The novelty of this approach is that, using the moving least square technique, we attempt to reduce the weighted sum of the residuals that results from the approximation to the field variable, the governing equation and the boundary conditions. The process lead to an interpolation function which is express in terms of the nodal value of the field variable and the nodal value of the nonhomogeneous terms in the differential equation. According to the requirement of consistency of the interpolation function with its value at nodes, the point collocation technique was employed to determine the unknown nodal values, and so complete the process of determining an approximate solution to given problem. Various example problem include the one-dimensional boundary value problem and the two-dimensional problems of torsion of elastic shaft are solved to demonstrate the accuracy and the rate of convergency of this method.

參考文獻
[1] L. B. Lucy, A numerical approach to the testing of the fission hypothesis,The Astron, J.8, 12, 1013-1024 (1977).
[2] B. Nayroles, G. Touzot and P. Villon, Generalizing the finite element
method diffuse approximation and diffuse elements, Computational
Mechanics, 10, 307-318 (1992).
[3] T. Belytschko, L. Gu and Y. Y. Lu, Fracture and crack growth by
element-free Galerkin methods, Modeling and Simulation in Materials
Science and Engineering,2 , 519-534 (1994).
[4] W. K. Liu, S. Jun and Y. F. Zhang, Reproducing kernel particle methods,Int. J. Numerical methods Engineering, 20, 1081-1106 (1995).
[5] J. S. Chen, C.T. Wu and W.K. Liu, Reproducing kernel particle methods for large deformation analysis of non-linear structures, Computer methods in Applied Mechanics And Engineering, 139, 195-227. (1996).
[6] E. Oñate, S. Idelsohn, O. C. Zienkiewicz, R. L. Taylor and C. Sacco, A stabilized finite point method for analysis of fluid mechanics problems,
Computer Methods in Applied Mechanics & Engineering, 139, 315-346
(1996).
[7] S. Li, W. Hao and W. K. Liu, Numerical simulation of large deformation of thin shell structures using meshfree methods, Computational Mechanics,25, 102-116 (2000).
[8] G. J. Wagner and W. K. Liu, Turbulence simulation and multiple scale subgrid models, Computational Mechanics, 25, 117-136 (2000).
[9] J. S. Chen, H. P. Wang, S. Yoon and Y. You, Some recent improvements in meshfree methods for incompressible finite elasticity boundary value problems with contact, Computational Mechanics, 25, 137-156 (2000)
[10]盛若磐, 元素釋放法積分法則與權函數之改良, 近代工程計算論壇論文集, 國立中央大學土木系, 2000.
[11] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming and P. Krysl, An
Overview and Recent Developments, Computer methods in Applied
Mechanics & Engineering, 139, 3-47(1996).
[12] P. Krysl and T.Belytschko, A Library to Compute The Element Free
Galerkin Shape Functions, Computer methods in Applied Mechanics &
Engineering, 190, 2181-2205(2001).
[13]唐正銓,具對稱系統矩陣之局部最小二乘無網格法在振動分析之應用,碩士論文,國立成功大學,2009
[14]翁啟照,無元素葛勒金法在二維彈力之應用 ,碩士論文,國立成功大學,2008