本論文應用向量式有限元理論模擬平面固體的幾何非線性行為，並加入頂角與邊之接觸判斷之分析功能，以模擬離散塊體系統之運動行為。
向量式有限元素法是以構件和節點為模擬基礎的物理模式，它將連續體定義成一群質點的組合，利用牛頓運動定律描述質點的運動，因此向量有限元的計算變成一組簡單的向量方程式計算。模擬固體變形運動問題時利用隨動變形座標系統分解剛體位移和變形位移，因此能夠精確的計算多個連續體同時具有大的剛體運動和大的幾何變形行為，也使得固體非線性分析得以簡化，此外，再配合顯式的時間積分公式及對應的向量運動方程式，使向量有限元分析程式不論在計算速度或模擬精度上都有合理的結果。
另外，由於接觸碰撞為工程及物理問題中的重要現象，本論文將
接觸力學分析方法納入向量式有限元之程序中，並透過例題驗證，發
現所開發之數值模擬程序具有極好之精度及能力。

In this study , the Vector Form Intrinsic Finite Element (VFIFE , V-5) method equipped with the contact detection and analysis algorithm is applied to simulate the large deformation behaviors of structures.
The V-5 method models the analyzed domain to be composed by finite particles and Newton''s second law is applied to describe each particle''s motion. Thus, the calculation of the V-5 method becomes solving a set of decoupled vector form equations. In the theory of V-5, a convected reference frame and updated deformation coordinate system are used to separate the rigid body motion and pure deformation of the system. After combining these with explicit time integration scheme,the V-5 method can effectively simulate the dynamic behaviors of multi-bodies system having large deformation.
To study the impact/ contact problems that commonly admitted in engineering and industry applications, the contact detection and contact force calculation algorithms were developed for the solid elements of the V-5 method. Through the numerical analyses of benchmark problems with large rotation, impact, the V-5 method demonstrates its accuracy and efficiency on the analysis of structure with large deformation.

[1] 丁承先、石強、王永康，「向量式有限元的基本理論：(二)平面固體元分析」，國立中央大學及美國普渡大學土木系，近代工程計算論壇（2000，2001，2002）
[2] Ting, E. C., Shih, C., Wang, Y. K., “Fundamentals of a vector form intrinsic finite element: Part I. basic procedure and a plane frame element,” Journal of Mechanics, Vol. 20, No. 2, pp. 113-122, 2004.
[3] Ting, E. C., Shih, C., Wang, Y. K., “Fundamentals of a vector form intrinsic finite element︰PartII .plane solid elements , “Journal of Mechanics”, Vol. 20 ,No. 2, pp. 123-321, 2004.
[4] Shih, C., Wang, Y. K., Ting, E. C., “Fundamentals of a vector form intrinsic finite element: Part III. Convected material frame and examples,” Journal of Mechanics, Vol. 20, No. 2, pp. 133-143, 2004.
[5] Shi, G. H.,“Discontinuous deformation analysis︰a new numerical model for the statics and dynamics of block system ”,PH.D. Dissertarion, University of California ,Berkeley,CA , 1988.
[6] Shi, G. H., Goodman, R. E.,“Two dimensional discontinuous deformation analysis”, International Journal for Numerical and Analytical Methods in Geomechanics, vol. 9, pp. 541-556, 1985.
[7] Koo, C. Y., Chern, J. C.,“Modification of DDA method rigid block problems”,Int. J. RockMech.Min.Sci., Vol. 35 , No. 6, pp. 683-693, 1998.
[8] Lanru Jing ,“Formulation of discontinuous deformation analysis (DDA)— an implicit discrete element model for block systems”, Engineering Geology, Vol. 49, pp. 371-381, 1998.
[9] Chan, S. K. , Tuba, I. S.,“A finite element method for contact problems of solid bodies：part I . theory and validation”,Int. J. mech. Sci., Vol. 13, pp. 615-625, 1971.
[10] Chan, S. K. , Tuba, I. S., “A finite element method for contact problems of solid bodies：part II. application to turbine blade fastenings ”,Int.J.mech.Sci., Vol. 13, pp. 627-639, 1971.
[11] Simo, J. C., Wriggers , P. and Taylor, R. L.,“A perturbed Lagrangian formulation for the finite element solution of contact problems”, Comp. Meth. Appl. Mech. Eng., Vol. 50, pp. 163-180 , 1985.
[12] Hallquist, J. O., Goudreau, G. L. and Benson, D. J., “Sliding interfaces with contact-impact in large-scale Lagrangian computations”, Comp. Methods Appl. Mech. Eng., Vol. 51, pp. 107-137, 1985.
[13] Benson, D. J., Hallquist, J. O., “A single surface contact algorithm for the post-buckling analysis of shell structures”, Comput. Appl. Mech. Eng., Vol. 78, pp. 141-163, 1990.
[14] Belytschko, T., Yeh, I. S., “The splitting pinball method for contact-impact problems”, Comput. Appl. Mech. Eng.,Vol. 105, pp. 375-393, 1993.
[15] Zhong, Z. H., Nilsson, L., “A contact searching algorithm for general contact problems”, Computers & Structures, Vol.33, No.1,pp.197-209,1989.
[16] Zhong, Z. H., Nilsson, L., “A contact searching algorithm for general 3-d contact-impact problems”,Computers & Structures, Vol. 34, No. 2, pp. 327-335, 1989.
[17] Zhong, Z. H., Nilsson, L., “Automatic contact searching algorithm for dynamic finite element analysis”Computers & Srrucrures , Vol. 52, No. 2, pp. 187-197, 1994.
[18] Belytschko, T., Lin, J. I., “A three-dimensional impact-penetration algorithm with erosion”,Computers & Structures, Vol. 25, No. 1, pp. 95-104 , 1987.
[19] Belytschko, T., Near, M. O., “Contact-impact by the pinball algorithm with penalty and Lagrangian methods”, Int. J.Num. Meth. Eng., Vol. 31, pp. 547-572, 1991.
[20] Taylor, R. L., Papadopoulos, P., “On a finite element method for dynamic contact/impact problems”, Int. J. Num. Meth. Eng., Vol. 36, pp. 2123-2140, 1993.
[21] Carpenter, N. J., Taylor, R. L., Katona, M. G., “Lagrange constraints for transient finite element surface contact”,Int. J. Num. Meth. Eng., Vol. 32, pp. 103-128, 1991.
[22] Laursen, T. A., Chawla, V., “Design of energy conserving algorithms for frictionless dynamic contact problems”, Int. J. Num. Meth. Eng., Vol. 40, pp. 863-886, 1997.
[23] Wriggers, P., Van, T. V., Stein, E., “Finite element formulation of Large deformation impact-contact problems with friction”, Computers & Structures, Vol. 37, No. 3, pp. 319-331, 1990.
[24] Yang, T.Y., Lianis, G., “Large displacement analysis of Viscoelastic beams and frames by the finite-element method”, J. Appl. Mech., vol. 74, pp. 635-640 , 1974.
[25] Bathe, K.J.,“Finite element formulations for large deformation dynamic analysis”, Int. J. Numer. Meth.Engng., Vol. 9, pp. 353-386, 1975.
[26] Pai, P.F., Palazotto, A.N., “Large-deformation analysis of flexible”,Int. J. Solids Structures, Vol. 33, No. 9,pp. 1335-1353 , 1995.
[27] Ren, W. X., Tan, X., Zheng, Z., “Nonlinear analysis of plane Frames using rigid body-spring discrete element method”, Comput. Struct., Vol. 71, pp. 105-119 , 1999.
[28] Pai, P. F., Palazotto, A. N., “Large deformation analysis of flexible beams”, Int. J. Solids Structures ,Vol. 3, No. 9, pp. 1335-1353 , 1996.
[29] Rice, D. L., Ting, E. C., “Large displacement transient analysis of flexible structures”, Int. J. Numer. Meth. Engrg. , Vol. 36, pp. 1541-1562 , 1993.
[30] Pai P. F., Anderson, T. J., Wheater, E. A., “Large deformation tests and total Lagrangian finite element analysis of flexible beams”, Int. J. Solid Structures, Vol. 37, pp. 2951-2980 , 2000.
[31] 莊清鏘，「二維可變形多體系統的動、靜態分析」，博士論文，國立中央大學土木工程研究所，中壢，1999。
[32] Cheng, Y. M. , “Advancements and Improvement in Discontinuous deformation Analysis”, Computers and Geotechnics, Vol. 22, No. 2, pp. 153-163, 1998.
[33] Antonio Munjiza ,The combined finite-discrete element method , John Wiley & Sons Ltd ,England, 2004.
[34] S. Mohammadi , Discontinuum mechanics using finite and discrete elements , Wit Press ,UK, 2003.
[35] 賴建豪，「向量式有限元素法於平面構架幾何非線性之應用」，碩士論文，私立中原大學 土木工程研究所，中壢，2003。
[36] 吳思穎，「向量式剛架有限元於二維結構之大變位與接觸行為分析」，碩士論文，國立中央大學 土木工程研究所，中壢，2005。