有限元素法(Finite Element Method , FEM)是近代解決土木工程 結構分析問題的方法，若將元素做適當的分割，使用工程計算程式運行可以快速且得到良好的結果。
工程師一般將結構體做切割後分析出數值結果，往往無法得知分析的近似解是否達到收斂或精確，故此論文之目的在於使用者訂定網格後，程式在保持網格數不變、平衡方程式數不變的前提下，利用矩陣濃縮技巧再將網格細分成三種型式:單一網格、網格一分為四元素、網格一分十六元素後，再加以探討對於此網格分割求得之數值解是否達良好品質。
本論文之結論是探討均勻節點排列在對答案精確度是否達到收斂，使用者所分割網格數目以及元素比例是否影響結果，所求得之解來進行品質的比較分析。

FEM is the method to study the elasticity and the structural analysis problems in modern Civil Engineering. For the structural analysis of the large objects, FEM is useful if the adequate number of element for the structural are applied.
Engineers generate the mesh for analysis and get the results, but often neglect the importance about the convergence of results. Therefore, the accuracy and convergence remains unknown. The purpose of the thesis is to help the user find out this point. On the premise that the numbers of element equation remain fixed, the program, with matrix condensation, segments the grid into 3 types: single grid, the grid is divided into four elements, the grid is divided into sixteen elements then to find out whether the segments achieve good quality.
The conclusion of the thesis is to compare and analyze the solution to see if the program can distinguish the accuracy and convergence of the solution converge or not, and also to see whether the number of dividing nodes in FEM and the ratio of the elements will affect the results data or not.

[1]彭國倫, "FORTRAN 95程式設計,ISBN：9789575669591,出版社：碁峰資訊 " 2001/11/28.
[2]O. Ural, "Finite element method: basic concepts and applications,1973,Structural analysis (Engineering); Finite element method; Matrices,101-405-844."
[3]陳俊豪, " "有限元素法導論," 科技圖書股份有限公司.," 1979.
[4]陳正宗, 有限元素分析與工程實例: MSC/NASTRAN 軟體應用: 北門, 1996.
[5]R. W. Clough and J. L. Tocher, "Finite element stiffness matrices for analysis of plates in bending," in Proceedings of conference on matrix methods in structural analysis, 1965.
[6]J. H. Argyris and S. Kelsey, Energy theorems and structural analysis vol. 960: Springer, 1960.
[7] Meek, John L. Matrix structural analysis. New York: McGraw-Hill, 1971.
[8]O. C. Zienkiewicz and P. Morice, The finite element method in engineering science vol. 1977: McGraw-hill London, 1971.
[9]R. J. Guyan, "Reduction of stiffness and mass matrices," AIAA journal, vol. 3, pp. 380-380, 1965.
[10]M. Gockel, MSC/NASTRAN Handbook for Dynamic Analysis: MSC/NASTRAN Version 63: MacNeal-Schwendler Corporation, 1983.
[11]Dhatt, Gouri, Emmanuel Lefrancois, and Gilbert Touzot. Finite element method. John Wiley & Sons, 2012.

[12]Tong, Xingcun Colin. (2011). Advanced Materials for Thermal Management of Electronic Packaging.
[13]潘子瑜, "結構有限元素分析:理論與:AMPS套裝軟體之應用，碩士論文，國立台灣科技大學營建工程學系," p. 7, 2007.
[14]K. Konigsberger, " Analysis 2. Springer-Verlag, Berlin/HeidelbergISBN 3-540-43580-8. (fur Jacobi-Matrizen reeller Funktionen)," 2000.
[15]H. G. Klaus Fritzsche, " From Holomorphic Functions to Complex Manifolds. Springer-Verlag, ISBN 0-387-95395-7. (S. 30-31; fur Jacobi-Matrizen holomorpher Funktionen)."
[16]H. P. Gavin, "STRUCTURAL ELEMENT STIFFNESS MATRICES AND MASS MATRICES,Duke University Department of Civil and Environmental Engineering," 2012.
[17]F. Hildebrand "Introduction to N.A," p. p.387~392.
[18]E. H. O. M. F. W. F. Milton Abramowitz and Irene A. Stegun, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4, 1965.
[19]W. McGuire, Gallagher, R. H., and Ziemian, "R. D. Matrix Structural Analysis, 2nd Ed. New York: John Wiley & Sons, 2000.."
[20]J. Robinson, "Structural Matrix Analysis for the Engineer. New York: John Wiley & Sons, 1966."
[21]C. A. Felippa, "A historical outline of matrix structural analysis: a play in three acts, Computers & Structures 79 (14): 1313–1324, doi:10.1016/S0045-7949(01)00019-6, ISSN 0045-7949."
[22]T. Vejchodsky and P. Šolin, "Static condensation, partial orthogonalization of basis functions, and ILU preconditioning in the hp-FEM," Journal of Computational and Applied Mathematics, vol. 218, pp. 192-200, 8/15/ 2008.
[23]K. A. Atkinson, "An Introduction to Numerical Analysis, 第二版, John Wiley & Sons, New York, 1989年 ISBN 0-471-50023-2."
[24]T. K. a. V. O. I. Gohberg, "Fast Gaussian elimination with partial pivoting for matrices with displacement structure,Primary 15A06; Secondary 15A09, 15A23, 15A57, 65F05 " Math. Comp. 64 (1995), 1557-1576
[25]P. L. Gould and P. K. Basu, "Geometric Stiffness Matrices for the Finite Element Analysis of Rotational Shells," Journal of Structural Mechanics, vol. 5, pp. 87-105, 1977/01/01 1977.
[26]N. Srinil and G. Rega, "The effects of kinematic condensation on internally resonant forced vibrations of shallow horizontal cables," International Journal of Non-Linear Mechanics, vol. 42, pp. 180-195, 1// 2007.
[27]J. J. Connor, Analysis of structural member systems: Ronald Press Co., 1976.
[28]J. M. Gere and S. Timoshenko, Mechanics of materials: PWS-KENT Pub. Co., 1990.
[29]Christian, Joshua M., and Clifford K. Ho. "Finite element modeling of concentrating solar collectors for evaluation of gravity loads, bending, and optical characterization." ASME 2010 4th International Conference on Energy Sustainability. American Society of Mechanical Engineers, 2010.