本文欲進行三維異類異向性體之邊界元素分析：首先應用了史磋公式，其優點在於經過整理後可以取得簡潔美觀的數學表示式，同時原先以異向性彈性材料推導的數學式，經過合適的矩陣內容調整及維度擴張，即可延伸應用於壓電材料及磁電彈材料，不須重新進行推導；而在數值運算上為了對應材料向量、矩陣維度的不同，透過矩陣調適法，根據材料特性給予適當的矩陣內容並加以擴張矩陣維度，使所欲分析之材料矩陣維度一致，即可利用本師門編撰的邊界元素程式進行運算；而計算方面使用雷冬轉換後的三維邊界元模型及基本解，並利用子區域法將結構中各個子區域的邊界積分方程組及結合條件進行組合，配合前面提到的矩陣調適法，即可用來分析兩不同材料的異向性體作緊密結合的問題。
在實例驗證的部分，展示了異向性彈性材料、壓電材料及磁電彈材料，兩材料間組成一結構體，依據不同材料組合擴張矩陣並適用對應的緊密結合條件，再使用三維邊界元素程式進行分析，同時將計算結果與商用有限元素軟體ANSYS進行比對，驗證程式的正確性及準確性。

This article presents the boundary element analysis for three-dimensional anisotropic heterogeneous solids. At the beginning of the study, Stroh formalism is chosen for its advantages of concise mathematical expressions, and after proper arrangements for its related matrix dimension, these mathematical equations can be further applied to piezoelectric and magneto-electro-elastic materials and the re-derivation of expressions for these materials is not required. To calculate with computer programming corresponding to vectors and matrices in different dimensions, we use the adaptable adjustment technique and give some reasonable contents to the matrices according to material characteristics to expand the matrix dimension. BEM program coded by our teacher can be operated once these matrix dimensions are consistent. To analyze the problems with solids in different material types bonded perfectly together, the modifications for the 3D BEM program combined with the adaptable adjustment technique and the subregion technique are necessary. Several perfectly bonded examples including (1) two different anisotropic materials, (2) two different piezoelectric materials, (3) anisotropic-piezoelectric materials, and (4) anisotropic-piezomagnetic materials are present and comparisons with commercial software ANSYS will also be conducted to verify the 3D BEM program.

[1] T. C.-T. Ting and T. C.-T. Ting, Anisotropic elasticity: theory and applications (no. 45). Oxford University Press on Demand, 1996.
[2] C. Hwu, Anisotropic elastic plates. Springer Science & Business Media, 2010.
[3] C. Hwu and W. J. Yen, "On the anisotropic elastic inclusions in plane elastostatics," 1993.
[4] Y. Liang and C. Hwu, "Electromechanical analysis of defects in piezoelectric materials," Smart Materials and Structures, vol. 5, no. 3, p. 314, 1996.
[5] Y. Chen and C. Hwu, "Green's functions for anisotropic/piezoelectric bimaterials and their applications to boundary element analysis," CMES-Computer Modeling in Engineering and Sciences, vol. 57, no. 1, pp. 31-50, 2010.
[6] E. Pan and W. Chen, Static green's functions in anisotropic media. Cambridge University Press, 2015.
[7] J. Aboudi, "Micromechanical analysis of fully coupled electro-magneto-thermo-elastic multiphase composites," Smart materials and structures, vol. 10, no. 5, p. 867, 2001.
[8] Y. Liu, J. Wan, J.-M. Liu, and C. Nan, "Effect of magnetic bias field on magnetoelectric coupling in magnetoelectric composites," Journal of Applied Physics, vol. 94, no. 8, pp. 5118-5122, 2003.
[9] X. Jiang and E. Pan, "Exact solution for 2D polygonal inclusion problem in anisotropic magnetoelectroelastic full-, half-, and bimaterial-planes," International Journal of Solids and Structures, vol. 41, no. 16-17, pp. 4361-4382, 2004.
[10] Y. E. Pak, "Elliptical inclusion problem in antiplane piezoelectricity: implications for fracture mechanics," International journal of engineering science, vol. 48, no. 2, pp. 209-222, 2010.
[11] M. Siboni and P. P. Castañeda, "A magnetically anisotropic, ellipsoidal inclusion subjected to a non-aligned magnetic field in an elastic medium," Comptes Rendus Mécanique, vol. 340, no. 4-5, pp. 205-218, 2012.
[12] Y. Liu and H. Fan, "Analysis of thin piezoelectric solids by the boundary element method," Computer methods in applied mechanics and engineering, vol. 191, no. 21-22, pp. 2297-2315, 2002.
[13] M. Hsieh and C. Hwu, "Extended Stroh-like formalism for magneto-electro-elastic composite laminates," in International Conference on Computational Mesomechanics Associated with Development and Fabrication of Use-Specific Materials, 2003, pp. 325-332.
[14] L. Hill and T. Farris, "Three-dimensional piezoelectric boundary element method," AIAA journal, vol. 36, no. 1, pp. 102-108, 1998.
[15] E. Pan, B. Yang, G. Cai, and F. Yuan, "Stress analyses around holes in composite laminates using boundary element method," Engineering Analysis with Boundary Elements, vol. 25, no. 1, pp. 31-40, 2001.
[16] F. Tonon, E. Pan, and B. Amadei, "Green’s functions and boundary element method formulation for 3D anisotropic media," Computers & Structures, vol. 79, no. 5, pp. 469-482, 2001.
[17] S. Kurz, O. Rain, and S. Rjasanow, "The adaptive cross-approximation technique for the 3D boundary-element method," IEEE transactions on Magnetics, vol. 38, no. 2, pp. 421-424, 2002.
[18] K. Liew and J. Liang, "Modeling of 3D transversely piezoelectric and elastic bimaterials using the boundary element method," Computational mechanics, vol. 29, no. 2, pp. 151-162, 2002.
[19] H. Ding, W. Chen, and A. Jiang, "Green's functions and boundary element method for transversely isotropic piezoelectric materials," Engineering analysis with boundary elements, vol. 28, no. 8, pp. 975-987, 2004.
[20] M.-H. Zhao, P.-Z. Fang, and Y.-P. Shen, "Boundary integral–differential equations and boundary element method for interfacial cracks in three-dimensional piezoelectric media," Engineering analysis with boundary elements, vol. 28, no. 7, pp. 753-762, 2004.
[21] X. Chen and Y. Liu, "An advanced 3D boundary element method for characterizations of composite materials," Engineering analysis with boundary elements, vol. 29, no. 6, pp. 513-523, 2005.
[22] M. Denda and C.-Y. Wang, "3D BEM for the general piezoelectric solids," Computer methods in applied mechanics and engineering, vol. 198, no. 37-40, pp. 2950-2963, 2009.
[23] P. Fedeliński, G. Górski, T. Czyż, G. Dziatkiewicz, and J. Ptaszny, "Analysis of effective properties of materials by using the boundary element method," Archives of Mechanics, vol. 66, no. 1, pp. 19-35, 2014.
[24] I. Pasternak, R. Pasternak, V. Pasternak, and H. Sulym, "Boundary element analysis of 3D cracks in anisotropic thermomagnetoelectroelastic solids," Engineering Analysis with Boundary Elements, vol. 74, pp. 70-78, 2017.
[25] C.-L. Hsu, C. Hwu, and Y. Shiah, "Three-dimensional boundary element analysis for anisotropic elastic solids and its extension to piezoelectric and magnetoelectroelastic solids," Engineering Analysis with Boundary Elements, vol. 98, pp. 265-280, 2019.
[26] K.-C. Wu, "Generalization of the Stroh formalism to 3-dimensional anisotropic elasticity," Journal of elasticity, vol. 51, no. 3, pp. 213-225, 1998.
[27] E. Pan and F. Tonon, "Three-dimensional Green’s functions in anisotropic piezoelectric solids," International Journal of Solids and Structures, vol. 37, no. 6, pp. 943-958, 2000.
[28] E. Pan, "Three-dimensional Green's functions in anisotropic magneto-electro-elastic bimaterials," Zeitschrift für angewandte Mathematik und Physik ZAMP, vol. 53, no. 5, pp. 815-838, 2002.
[29] F. C. Buroni and M. Denda, "Radon–stroh formalism for 3d theory of anisotropic elasticity," Advances in Boundary Element Techniques, pp. 295-300, 2014.
[30] L. Xie, C. Zhang, C. Hwu, J. Sladek, and V. Sladek, "On two accurate methods for computing 3D Green׳ s function and its first and second derivatives in piezoelectricity," Engineering Analysis with Boundary Elements, vol. 61, pp. 183-193, 2015.
[31] L. Xie, C. Hwu, and C. Zhang, "Advanced methods for calculating Green's function and its derivatives for three-dimensional anisotropic elastic solids," International Journal of Solids and Structures, vol. 80, pp. 261-273, 2016.
[32] L. Xie, C. Zhang, C. Hwu, and E. Pan, "On novel explicit expressions of Green’s function and its derivatives for magnetoelectroelastic materials," European Journal of Mechanics-A/Solids, vol. 60, pp. 134-144, 2016.
[33] C. Hwu and C. Liao, "A special boundary element for the problems of multi-holes, cracks and inclusions," Computers & structures, vol. 51, no. 1, pp. 23-31, 1994.
[34] C. A. Brebbia, J. C. F. Telles, and L. C. Wrobel, Boundary element techniques: theory and applications in engineering. Springer Science & Business Media, 2012.
[35] N. N. Rogacheva, The theory of piezoelectric shells and plates. CRC press, 2020.
[36] A. Soh, Liu, and JX, "On the constitutive equations of magnetoelectroelastic solids," Journal of Intelligent Material Systems and Structures, vol. 16, no. 7-8, pp. 597-602, 2005.