本文主旨係利用邊界元素法來處理三維彈性力學中 ， 無限區域 和有限區域的問題 。
對於無限區域的問題 ， 邊界元素法比有限元素法更能夠得到精確的解 。 但邊界積分方程式中 ， 當場點和源點重合時 ， 會造成奇異性積分的問題 。 本文提出的方法是在實體邊界的外部 ， 假想有一輔助面源 ， 此假想的源具有實際的物理意義 ， 並且在輔助邊界上以八節點的曲面等參元素做建立網格 。 藉由此法 ， 可避免源點和場點的重合，不會造成奇異性的積分 ， 再加上邊界元素法能將維度降一階 ， 減少了龐大的運算時間 。
由數值結果與解析解的比較 ， 本文的方法證實 ， 應用在彈性力學問題上是一個有效 、 可靠的方法 。

The aim of this research is to deal with three dimensional elasticity problem using the BoundaryElement Method (BEM). The problems contain infinite and finite domain problems. For infinite domain it can obtain more accurate solution than Finite Element Method.
BEM can reduces the dimension of problem by one it can be frugal computation time. In this research, we suppose an auxilliary surface source exterior to the actual boundary. The auxilliary surface is discretized by element mesh with eight-noded isoparametric elements. This method cancels the singular integral problem because the field point do not coincide with the source point.
The numerical results are very accurate compared with the analytical solution. It is proved that this method is an efficient and reliable numerical method in handling the elasticity problem.

中文摘要．．．．．．．．．．．．．．．．．．．．．．．．．．．．I
英文摘要．．．．．．．．．．．．．．．．．．．．．．．．．．．．II
誌謝．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．III
目錄．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．IV
圖目錄．．．．．．．．．．．．．．．．．．．．．．．．．．．．．VI
表目錄．．．．．．．．．．．．．．．．．．．．．．．．．．．．．VII
符號說明．．．．．．．．．．．．．．．．．．．．．．．．．．．．VIII
第一章 緒論 ．．．．．．．．．．．．．．．．．．．．．．．．．．．1
1.1 研究動機．．．．．．．．．．．．．．．．．．．．．．．．．．1
1.2 文獻回顧．．．．．．．．．．．．．．．．．．．．．．．．．．2
1.3 本文架構．．．．．．．．．．．．．．．．．．．．．．．．．．4
第二章 邊界元素公式 ．．．．．．．．．．．．．．．．．．．．．．．5
2.1 統御方程式．．．．．．．．．．．．．．．．．．．．．．．．．5
2.2 基本解．．．．．．．．．．．．．．．．．．．．．．．．．．．8
2.3 計算場點位移的邊界積分式．．．．．．．．．．．．．．．．．．10
2.4 計算場點應力的邊界積分式．．．．．．．．．．．．．．．．．．13
第三章 公式推導．．．．．．．．．．．．．．．．．．．．．．．．．15
3.1 元素與形狀函數．．．．．．．．．．．．．．．．．．．．．．．15
3.2 公式之離散化．．．．．．．．．．．．．．．．．．．．．．．．17
3.3 分析原理．．．．．．．．．．．．．．．．．．．．．．．．．．22
第四章 實例測試與結果．．．．．．．．．．．．．．．．．．．．．．27
4.1 外部問題．．．．．．．．．．．．．．．．．．．．．．．．．．27
4.2 內部問題．．．．．．．．．．．．．．．．．．．．．．．．．．32
4.3 輔助面與實體邊界的距離之影響．．．．．．．．．．．．．．．．41
4.4 收斂性．．．．．．．．．．．．．．．．．．．．．．．．．．．41
第五章 結論．．．．．．．．．．．．．．．．．．．．．．．．．．．45
參考文獻．．．．．．．．．．．．．．．．．．．．．．．．．．．．47

1. Rizzo, F.J. (1967) ``An Integral Equation Approach to Boundary Value Problems of Classical Elastotatics'', Quarterly of Applied Mathematics, Vol. 25, pp. 83-95.
2. Cruse, T.A. (1969) ``Numerical Solutions in Three Dimensional Elastostatics'', International Journal Solids Structures, Vol. 5, pp. 1259-1274.
3. Cruse, T.A. (1974) ``An Improved Boundary-Integral Equation Method
for Three Dimensional Elastic Strees Analysis'', Computers and Structures, Vol. 4, pp. 741-754.
4. Qliveria, E.R.A. (1968) ``Plane Stress Analysis by a General Integral Method'', Proceedings ASCE, Journal of Engineering Mechanics Division, Vol. 94 No. EM1
pp. 79-85.
5. Heise, U. (1978) ``Numerical Properties of Integral Equations in which the Given Boundary Values and the Sought Solutions Are Defined on Different Curves'', Computers and Structures, Vol. 8, pp. 199-205.
6. Butterfield, R. and Banerjee, P.K. (1971) ``The Elastic Analysis of Compressible Piles and Pile Groups'', Geotechnique, Vol. 21, pp. 43-60.
7. Tomlin, G.R. and Butterfield, R. (1974) ``Elastic Analysis of Zoned Orthotropic Continus'', Proceedings ASCE, Journal of Engineering Mechanics Division, Vol. 94 No. EM1 pp. 511-529.
8. Banerjee, P.K. (1976) ``Integral Equation Methods for Analysis of Piece-Wise Non Homogeneous Three-Dimensional Elastic Solids of Arbitrary Shape'', International Journal of Mechanical Sciences, Vol. 18, pp. 293-303.
9. Gary, B. and Enayat M. (1984) ``A Comparison of the Boundary Element and Superposition Methods'', Computers and Structures, Vol. 19, pp. 697-705.
10. Koopmann, G.H., Song, L. and Fahnline, J.B. (1989) ``A Method for Computing Acoustic Fields Based on the Principle of the Wave Superposition'', The Journal of Acoustical Society of America, Vol. 86, pp. 2433-2438.
11. Jeans, R. and Mathews, I.C. (1992) ``The Wave Superposition Method
as a Robust Technique for Computing Acoustic Fields'', The Journal of Acoustical Society of America, Vol. 92, No. 2, pp. 1156-1166.
12. Brebbia, C.A., and Dominguez, J. (1992) Boundary Element an Introductory Course, Second Edition, Computational Mechanics Publications, Southampton Boston.
13. Adel, S.S. (1974) Elasticity Theory and Applications, Pergamon Press, New York.
14. Reismann, H. and Pawlik, P.S. (1980) Elasticity Theory and Applications, Wiley, New York.
15. Brebbia, C.A , Telles, J.C.F. and Wrobel, L.C. (1984) Boundary Element Techniques : Theory and Applications in Engineering, Berlin