本文採用移動Trefftz近似法式結合了無元素法之移動近似及邊界積分法中Trefftz法滿足微分方程求解基底。在本文利用H-R變分原理求得平衡方程式及基底函數配合邊界條件計算求得變數。
文中分析懸臂梁受剪力、簡支梁受均部荷重、無限板中央裂縫受拉力及無限板中央開孔受拉力等二維彈性力學問題，並且透過不同的佈點方式及不同的基底階數分析配合解析解分析其結果精度及誤差收斂性，驗證此法的可靠性及可行性。
分析算例後，在利用均勻佈點方式，隨著節點數越多，可有良好的精確性及收斂性，若有應力集中現象，會隨著佈點間距越小，越明顯。均勻佈點與隨機佈點兩者經度相差不大，驗證了本文所使用的方法受到佈點方式之影響不大，並且從誤差表中可發現位移及應力之精確度相近，與一般數值方法中位移精確度往往大於應力精度相比較，本文之數值方法在應力分析上更為精準。
總結以上幾點結論，本文之移動Trefftz 近似法，在分析二維彈性力學之問題上皆能得到良好的結果精度，為穩定及具有實用性之數值模擬方法。

In this thesis, we use the moving Trefftz method to analyze the two dimensional elasticity problems. The method is combined with the meshless method and the Trefftz method in the boundary integral equation to satisfy the differential equation. In this paper, the modified H-R variational principle is used to obtain the integral equation relative to the boundary conditions. Using the bases that satisfy the equilibrium equation and by the moving approximation techniques, the numerical solution can be obtained.
Using this method, we simulate a cantilever beam loaded by the shear force, a simply support beam loaded by the uniform load, an infinite plate with a crack in the middle loaded by the tensile force and an infinite plate with a hole loaded by the tensile force, and we use different point distribution and different order of base function. By analyzing the numerical results with the exact solution and comparing the accuracy and error convergence of the results, we can verify the reliability and feasibility of this method.

[1]L.B. Lucy, A numerical approach to the testing of the fissionhypothesis, The Astron, Journal, (1977). 82: 1013-1024.
[2]P. Lancaster & K. Salkauskas, Surfaces generated by moving least squares methods. Mathematics of computation, (1981). 37(155), 141-158.
[3]B. Nayroles, G.Touzot. , & P. Villon., Generalizing the finite element method: diffuse approximation and diffuse elements. Computational mechanics, (1992). 10(5), 307-318.
[4]T. Belytschko., L. Gu. and Y. Y. Lu, Fracture and crack growt by element-free Galerkin methods, Modeling and Simulation in Materials Science and Engineering, (1994). 2, 519-534.
[5]W. K. S. Jun Liu and Y. F. Zhang, Reproducing kernel particle methods, International journal for Numerical methods Engineering, (1995). 20, 1081-1106.
[6]Chen, J.-S., Pan., C., Wu, C.-T., & Liu, W. K., Reproducing kernel particle methods for large deformation analysis of non-linear structures. Computer methods in applied mechanics and engineering, (1996). 139(1), 195-227.
[7]T. Belytschko., Y. Krongauz., D. Organ., M. Fleming., & P. Krysl. , Meshless methods: an overview and recent developments. Computer methods in applied mechanics and engineering, (1996). 139(1), 3-47.
[8]Y. Krongauz. & T. Belytschko., A Petrov-Galerkin diffuse element method (PG DEM) and its comparison to EFG. Computational mechanics, (1997). 19(4), 327-333.
[9]T. Zhu, J,D, Zhang, S.N. Atluri, A Meshless Local Boundary Integral Equation (LBIE) Method for solving Nonlinear Problems, Computational Mechanics, (1998). 22, 174-186.
[10]S.N. Atluri, T. Zhu, A New Meshless Local Petrov-Galerkin(MLPG) Approach in Computational Mechanics, Computational Mechanics, (1998). 22, 117-127.
[11]盛若磐, 元素釋放法積分法則與權函數之改良，近代工程計算論壇文集，國立中央大學土木系(2000).
[12]Long. S. &S. Atluri., A meshless local Petrov-Galerkin method for solving the bending problem of a thin plate. Computer Modeling in Engineering and Sciences, (2002). 3(1), 53-64.
[13]Y.M. Wang, S.M. Chen, C.P. Wu, A meshless collocation method based on the differential reproducing kernel interpolation, Computational Mechanics, (2010). 45, 585-606.
[14]楊佳蓉.齊次基底移動最小二乘法在二維彈性力學上之應用.國立成功大學土木工程研究所碩士論文, (2011).
[15]吳建勳.受束制之移動最小二乘法在二維彈性力學問題上之應用.國立成功大學土木工程研究所碩士論文, (2013).
[16]陳昱任.基於狀態變數與Hermite型置點法之移動最小二乘法在二維彈性力學之應用.國立成功大學土木工程研究所碩士論文, (2014).
[17]王重凱.移動最小功法在二維彈性力學問題分析之應用.國立成功大學土木工程研究所碩士論文, (2017).
[18]E. Trefftz., Ein Gegensttick zum ritzschen Verfahren. Proc. 2ndZnt. Cong. Appl. Me&, Zurich, (1926), 131-37.
[19]W. G. Jin , Y. K. Cheung, & Zienkiewicz, O. C. Application of the Trefftz method in plane elasticity problems. Int. J. numer. Meth. Engng, 1990, 30, 1147-61..
[20]E. Kita, N. Kamiya, Trefftz method: An overview. Advances in Engineering Software, 24 (1995), 3-12.
[21]Qing-Hua Qin, Trefftz finite element method and its applications. Appl. Mech. Rev. 58(5), (2005) , 316–337.
[22]Vladimír Kompiš and Mário Štiavnický Trefftz functions in FEM, BEM and meshless methods. Computer Assisted Mechanics and Engineering Sciences, 13, (2006), 417–426.
[23]S. P. Timoshenko, J. N. Googier ,Theory of Elasticity, 3rd Edn, McGraw-Hill, New York, (1970).
[24]I.N. Sneddon, M. Lowengrub Crack Problems in the Classical Theory of Elasticity John Wiley and Sons, New York (1969)
[25]M. H. Sadd. Elasticity : theory, applications, and numerics. Amsterdam; Boston : Elsevier Butterworth-Heinemann, c, (2005).