本文應用移動Trefftz近似法模擬一階剪切變形理論假設之圓柱殼受力後位移場及合應力場變化情形，方法的特點結合Trefftz法的概念，選擇滿足微分方程式之局部近似基底，以簡化泛函積分式，及配合移動近似法之優點找出局部節點位移關係，建立聯立方程組求得各節點位移變量及合應力。
本文之數值算例為分析封閉及開放圓柱殼於不同邊界條件，外力為三角函數之載重形式及多項式解析解當成邊界條件，比較均勻及隨機佈點下，不同基底階數及點數的誤差及收斂情況。在基底階數選擇上，以六階基底為最有效率階數，有一定精度且有較佳收斂。而點數方面，在高階基底中，點數增多效益高於增加基底階數。

In this thesis, the moving Trefftz method is applied to simulate the displacement field and the result stress field of the cylindrical shell under the assumption of the first-order shear deformation theory. The characteristics of the method combine with the concept of the Trefftz method to select the local basis function that satisfies the differential equation to simplify the functional integral equation. And the advantage of the moving approximation method is used to find the displacement relationship of the local nodes, and the simultaneous equations are established to obtain the displacement variables and result stresses of each node.
The numerical example is open and close cylindrical shells under different boundary conditions, the load form of the trigonometric function and the analytical solution of the polynomial as boundary conditions. We compare the error and convergence with different the order of basis function and the number of nodes. In another case, to prove solution reproducing condition, we choose the high order basis function as exact solution, and approximate it with lower basis. Compared with different order of basis function, the sixth order is the most effective, because of higher accuracy and better convergence. In terms of node number, increasing the number is more effective than increasing the order of basis functions.

[1] Lucy, L. B. (1977). “A numerical approach to the testing of the fission hypothesis.” Astronomical journal, Vol. 82, pp. 1013-1024.
[2] Nayroles, B., Touzot, G. & Villon, P. (1992). “Generalizing the finite element method: diffuse approximation and diffuse elements.” Computational mechanics, Vol. 10, Issue. 5, pp. 307-318.
[3] Belytschko, T., Gu, L. & Lu, Y. Y. (1994). “Fracture and crack growth by element free Galerkin methods.” Modelling and Simulation in Materials Science and Engineering, Vol. 2, No. 3A, p. 519.
[4] Liu, W. K., Jun S., & Zhang, Y. F. (1995). “Reproducing kernel particle methods.” International journal for numerical methods in fluids, Vol. 20, Issue. 8‐9, pp. 1081-1106.
[5] Chen, J. S., Pan, C. T. & Liu, W. K. (1996). “Reproducing kernel particle methods for large deformation analysis of non-linear structures.” Computer methods in applied mechanics and engineering, Vol. 139, Issue. 1-4, pp.195-227.
[6] Onate, E., Idelsohn, S., Zienkiewicz, O. C., Taylor, R. L., & Sacco, C. (1996). “A stabilized finite point method for analysis of fluid mechanics problems.” Computer Methods in Applied Mechanics and Engineering, Vol. 139, Issue. 1-4, pp.315-346.
[7] Krongauz, Y., & Belytschko, T. (1997). “A Petrov-Galerkin diffuse element method (PG DEM) and its comparison to EFG.” Computational Mechanics, Vol. 19, Issue. 4, pp.327-333.
[8] Zhu, T., Zhang, J. D., & Atluri, S. N. (1998). “A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach.” Computational mechanics, Vol. 21, Issue. 3, pp.223-235.
[9] Atluri, S. N., & Zhu, T. L. (2000). “The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics.” Computational Mechanics, Vol.25, Issue.2-3, pp.169-179.
[10] Zhang, X., Liu, X. H., Song, K. Z., & Lu, M. W. (2001). “Least‐squares collocation meshless method.” International Journal for Numerical Methods in Engineering, Vol. 51, Issue. 9, pp.1089-1100.
[11] Long, S., & Atluri, S. N. (2002). “A meshless local Petrov-Galerkin method for solving the bending problem of a thin plate.” Computer Modeling in Engineering and Sciences, Vol. 3, Issue. 1, pp.53-64.
[12] Wang, D., & Chen, J. S. (2004). “Locking-free stabilized conforming nodal integration for meshfree Mindlin–Reissner plate formulation.” Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issue. 12-14, pp.1065-1083.
[13] Wang, Y. M., Chen, S. M., & Wu, C. P. (2010). “A meshless collocation method based on the differential reproducing kernel interpolation.” Computational Mechanics, Vol. 45, Issue 6, pp.585-606.
[14] 林均威，受束制之移動最小二乘法在古典板上之應用，國立成功大學土木工程研究所碩士論文。2013。
[15] 吳梓瑋，應用移動最小二乘法於圓柱體薄殼大變形分析，國立成功大學土木工程研究所碩士論文。2015。
[16] 張芳瑜，移動最小功法在圓柱殼分析之應用，國立成功大學土木工程研究所碩士論文。2018。
[17] Trefftz, E. (1926). “Ein gegenstuck zum Ritzschen verfahren.” Proc. 2nd Int. Cong. Appl. Mech. Zürich. pp.131-137.
[18] Jirousek, J., & Leon, N. (1977). “A powerful finite element for plate bending.” Computer Methods in Applied Mechanics and Engineering, Vol. 12, Issue. 1, pp.77-96.
[19] Cheung, Y. K., Jin, W. G., & Zienkiewicz, O. C. (1989). “Direct solution procedure for solution of harmonic problems using complete, non‐singular, Trefftz functions.” Communications in applied numerical methods, Vol. 5, Issue. 3, pp.159-169.
[20] Jin, W. G., Cheung, Y. K., & Zienkiewicz, O. C. (1990). “Application of the Trefftz method in plane elasticity problems.” International Journal for Numerical Methods in Engineering, Vol. 30, Issue. 6, pp.1147-1161.
[21] Kita, E., & Kamiya, N. (1995). “Trefftz method: an overview.” Advances in Engineering Software, Vol. 24, Issue. 1-3, pp.3-12.
[22] Washizu, K. (2002). “Variational methods in applied mechanics.” 2nd edition. John Wiley & Sons, New Jersey.