本文主要以B-spline函數為基底函數解三維不規則形狀彈性力學問題。從文獻中發現，目前尚無任何研究將B-spline應用於三維不規則形狀彈性力學問題。本文利用二元空間分割法與幾何布林運算判別元素落在實體內或邊界上，若是規則形狀元素則可簡單地以六面體形狀直接積分；針對不規則形狀元素處理的方式，如同有限元素法，將不規則幾何映對至主要元素中，再利用高斯積分法進行積分。
首先對於規則實體分別以兩種不同邊界條件測試，使用B-spline有限元素法可用較少的自由度得到與傳統有限元素法相近的位移、應力分析結果。其後對兩個不規則實體分析，由立方實體內部挖圓柱可知，當元素增多時，除了二階分析結果不甚理想之外，其餘高階基底函數收斂效果都比有限元素法佳；在立方實體內部挖圓球分析結果顯示當網格不夠小時，則顯現不出高階收斂快的效果，且當縮小網格時發現部分元素會產生較複雜的形狀，此部分整理的積分型態種類太多仍待克服。
由實例結果，可以確定了使用B-spline解三維不規則形狀問題分析結果比有限元素法更為準確。

We used the B-spline functions as the basis functions to solve the three dimensional linear elastic problems with irregular shapes. From the references, there are no researchers using B-spline functions to analyze three dimensional linear elastic problems with irregular shapes. Binary space partition method and geometry Boolean operation are used to determine which element is within the domain or across the boundary. If the element is within the domain, then it is regular element with a cube shape and can be integrated easily. If the element is across the boundary, then for the element across boundary, we map the irregular geometry to a master element, a regular cube, as the finite element used. The integration of the irregular shape is integrated in the master element using Gauss quadrature.
First, we analyze a regular cube with two different boundary conditions. Using B-spline finite element method can decrease the degree of freedom and obtain the same displacement and stress analysis results compare to the traditional finite element method. A cube with a cylinder hole and a cube with a sphere hole are used as irregular shapes examples in this thesis. In the former case, we obtained better result accuracy from the high-order B-spline basis function than the results from traditional finite element method except for the second-order B-spline basis function case. In the later case, the problems we analyzed are limited to small elements numbers. In the situation, the effect of high-order B-spline basis function is not shown. We still have problems to integrate these irregular domains. It should be overcome in the future.
From the results in these examples, using B-spline finite element method to solve three dimensional problems with irregular shapes has better accuracy than traditional finite element method.

[1]Aggarwal, B., “B-spline finite elements for plane elasticity problems” , Texas A&M university, 2006.
[2]Anand, V.B., Computer graphics and geometric modeling for engineers, John Wiley &Sons,Inc., 1993.
[3]Beissel, S. and Belytschko, T., “Nodal integration of the element–free Galerkin method”, Computer methods in applied mechanics and engineering, Vol. 139, 49-74, 1996.
[4]Clough, R.W., The finite element method in plane stress analysis. Proceedings of American society of civil engineers, 2nd conference on electronic computation, Pittsburgh, PA, 345-378, 1960.
[5]Courant, R., Variational methods for the solution of problems of equilibrium and vibrations. Bulletin of the American mathematical society, Vol. 49, 1-23, 1943.
[6]Fuch, H., Kedem, Z. and Naylor, B., “Predetermining visibility priority in 3-d scenes”, Proceedings of siggraph, Vol.13, 175-181, 1979.
[7]Fuch, H., Kedem, Z. and Naylor, B., “On visible surface generation by a priori tree structures”, Proceedings of siggraph, Vol.14, 124-133, 1980.
[8]Höllig, K., Finite element methods with B-splines, Society for industrial and applied mathematics, Philadelphia, 2003.
[9]Höllig, K., Horner, J. and Pfeil, M., “Parallel finite element methods with weighted linear B-splines”, High performance computing in science and engineering '07: Transactions of the high performance computing center, Stuttgart (HLRS),667,2008.
[10]Hikmet, C., Nazan, C. and Khaled, E., “B-spline interpolation compared with finite difference, finite element and finite volume methods which applied to two-point boundary value problems”, Applied mathematics and computation, Vol. 175, 72 - 79, 2006.
[11]Liang, X., Jian, B. and Ni, G., “The B-spline finite element method in electromagnetic field numerical analysis”, IEEE transactions on magnetics, Vol. 23, 2641 - 2643, 1987.
[12]Liu, G. R., Mesh free methods: moving beyond the finite element method, CRC press, Boca raton, 2003.
[13]Ni, G., Xu, X. and Jian, B., “B-spline finite element method for eddy current field analysis”, IEEE transactions on magnetics, Vol. 26, 723 - 726, 1990.
[14]Reddy, J. N., An introduction to the finite element method, McGraw-Hill, Texas, 2006.
[15]Schoenberg, I. J., “Contributions to the problem of approximation of
equidistant data by analytic functions”, Quart. Appl. Math., Vol. 4, 45-99,112-141 , 1946.
[16]Xiang, J., Chen, X., He, Y. and He, Z., “The construction of plane elastomechanics and mindlin plate elements of B-spline wavelet on the interval”, Finite elements in analysis and design, Vol. 42, 1269 - 1280, 2006.
[17]Zamani, N,G., “A least squares finite element method applied to B-spline”, Journal of the Franklin institute, Vol. 311, (3), 195-208, 1981.
[18]Zamani, N,G., “Least squares finite element approximation of navier’s equation”, Journal of the Franklin Institute, Vol. 311, (5), 311-321, 1981.
[19]陳政德,“B-spline有限元素法於二維平面應力問題收斂性探討”, 國立成功大學機械工程研究所碩士論文, 2008.
[20]趙啟翔,“B-spline有限元素法於二維平面應力問題之研究”, 國立成功大學機械工程研究所碩士論文, 2007.
[21]廖宏哲, “二維B-spline有限元素法不規則邊界形狀處理及於平板上的應用”, 國立成功大學機械工程研究所碩士論文, 2007.