本研究針對二維半無窮異向性彈性體於自重作用下之結構行為，建立一套以邊界元素法為基礎之數值分析方法。考量實際工程中地表形狀不規則及地下孔洞之存在，本研究採用有限域模型近似半無窮體，以提升建模彈性並降低邊界條件影響。
在數值處理方面，當外部邊界為模擬無窮域而設置於遠場時，邊界元素尺寸將顯著增大，原以內插函數進行之線積分方法，易因元素尺度過大而產生數值誤差，進而影響節點之應力與位移計算精度。為改善此問題，本研究提出一種混和數值策略：對外部邊界採用高斯積分進行計算，以降低大尺度元素所引致之內插誤差；對內部孔洞邊界採用傅立葉級數轉換方法，並搭配形狀函數進行離散化，以維持局部應力場之解析能力。此外，透過高斯散度定理將體內力項之體積分轉換為邊界積分，使自重效應得以納入分析架構。透過上述方法之建立，可兼顧遠場邊界之數值穩定性與近場應力分析之精度。

This study investigates the structural behavior of a two-dimensional semi-infinite anisotropic elastic body subjected to self-weight and develops a numerical analysis approach based on the Boundary Element Method (BEM). Considering that real engineering conditions often involve irregular surface geometries as well as underground cavities or excavations, a finite-domain model is adopted to approximate the semi-infinite medium. This approach enhances modeling flexibility and reduces the influence of artificial boundary conditions.
　In the numerical implementation, when the external boundary is extended to the far field to simulate an infinite domain, the size of boundary elements becomes significantly large. Conventional line integration methods based on interpolation functions tend to introduce numerical errors under such conditions, thereby affecting the accuracy of stress and displacement calculations at nodal points. To address this issue, a hybrid numerical strategy is proposed. For the external boundary, area integration combined with Gaussian quadrature is employed to reduce interpolation errors associated with large-scale elements. For the internal cavity boundary, a Fourier series transformation is adopted and coupled with shape function discretization to preserve the accuracy of local stress field representation.
　Furthermore, the divergence theorem is utilized to transform the body force term into an equivalent boundary integral form, allowing the self-weight effect to be effectively incorporated into the BEM framework. The proposed approach achieves a balance between numerical stability in the far-field boundary and accuracy in near-field stress analysis.

[1] Brebbia, C. A., Telles, J. C. F., and Wrobel, L. C., Boundary Element Techniques: Theory and Applications in Engineering, Springer-Verlag, Berlin, 1984.
[2] Cruse, T. A., “Boundary Element Analysis in Computational Fracture Mechanics. Kluwer Academic Publisher, Dordrecht, Netherlands, " 1968.
[3] Lekhnitskii, S. G., "Anisotropic plates," 1968.
[4] Zhang, J., Tan, C. L., and Afagh, F. F., “A general exact transformation of body-force volume integral in BEM for 2D anisotropic elasticity, " Computational mechanics, vol. 19, no. 2, pp. 1-10, 1996.
[5] Pan, E., Chen, C. S., & Amadei, B., A BEM formulation for anisotropic half-plane problems, Engineering Analysis with Boundary Elements, 20(3), 185-195, 1997.
[6] Suo, Z., “Singularities, Interfaces and Cracks in Dissimilar Anisotropic Media,” Proceedings of the Royal Society of London A, Vol. 427, No. 1873, pp. 331–358, 1990.
[7] Hwu, C., Anisotropic Elastic Plates, Springer, Springer Science & Business Media, 2010.
[8] Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff International Publishing, Leiden, 1977.
[9] Shiah, Y. C. and Ye, S. Y., “New treatment of the self-weight and the inertial effects of rotation for the BEM formulation of 2D anisotropic solids," Engineering Analysis with Boundary Elements, vol. 73, pp. 170-180, 2016.
[10] 夏育群、陳春來，「邊界元素法之入門介紹」，高立圖書有限公司，中華民國93年12月15日。
[11] Pan, E., & Amadei, B., Stress concentration at irregular surfaces of anisotropic half-spaces, Acta Mechanica, 113(1-4), 119-135, 1995.
[12] Ting, T. C. T., Anisotropic Elasticity: Theory and Applications, Oxford University Press, New York, 1996.
[13] Rizzo, F. J., “An Integral Equation Approach to Boundary Value Problems of Classical Elastostatics,” Quarterly of Applied Mathematics, Vol. 25, No. 1, pp. 83–95, 1967.
[14] Muskhelishvili, N. I., Some basic problems of the mathematical theory of elasticity. Noordhoff Groningen, 1953.
[15]Lekhnitskii, S. G., Theory of elasticity of an anisotropic body, Holden Day, San Francisco, 1963.
[16]Stroh, A. N., “Dislocations and Cracks in Anisotropic Elasticity,” Philosophical Magazine, Vol. 3, pp. 625–646, 1958.
[17] Ting, T.C.T. Image singularities of Green's functions for anisotropic elastic half-spaces and bimaterials. Q. J. Mech. Appl. Math., 1992, 45, 119-139.
[18] Ting, T.C.T. and Barnett, D.M. Image force on line dislocations in anisotropic elastic half-spaces with a fixed boundary. Int. J. Solids Structures, 1993, 30, 313-323.
[19] Wei, L. and Ting, T.C.T. Explicit expressions of the BarnettLothe tensors for anisotropic materials. J. Elasticity, 1994, 36.67-83.
[20] Eshelby, J.D., Read, W.T. and Shockley, W. Anisotropic elasticity with applications to dislocations theory. Acta Metal., 1953, 1, 251-259.
[21] Stroh, A.N. Dislocations and cracks in anisotropic elasticity. Philos. Mag.. 1958, 7, 625-646.
[22] Hwu, C., Anisotropic elasticity with Matlab, Springer International Publishing, 2021