In the direct formulation of boundary integral equation (BIE), thermal loading reveals itself as additional domain integral that will destroy the advantage of boundary modeling feature. The most appealing approach is to analytically transform the domain integral into boundary such that the boundary modeling feature can be restored. Recently, the leading author has presented a direct transformation for two-dimensional anisotropic thermoelasticity, not relying on any domain distortion. However, due to mathematical complexity, such direct transformation has not been achieved for three-dimensional generally anisotropic thermoelasticity. Despite the importance of this topic in BEM, the direct transformation has remained unexplored so far. As the first successful work, this research presents the complete process to make this direct transformation for treating three-dimensional anisotropic thermoelasticity with implementation in an existing code. Additionally, this work also takes into account the presence of constant volume heat sources.

[1] C. V. Camp and G. S. Gipson, Boundary element analysis of nonhomogeneous biharmonic phenomena. Springer Science & Business Media, 2013.
[2] A. Deb and P. Banerjee, "BEM for general anisotropic 2D elasticity using particular integrals," International Journal for Numerical Methods in Biomedical Engineering, vol. 6, no. 2, pp. 111-119, 1990.
[3] D. Nardini and C. Brebbia, "A new approach to free vibration analysis using boundary elements," Applied mathematical modelling, vol. 7, no. 3, pp. 157-162, 1983.
[4] F. Rizzo and D. Shippy, "An advanced boundary integral equation method for three‐dimensional thermoelasticity," International Journal for Numerical Methods in Engineering, vol. 11, no. 11, pp. 1753-1768, 1977.
[5] P. Wen, M. Aliabadi, and D. Rooke, "A new method for transformation of domain integrals to boundary integrals in boundary element method," International Journal for Numerical Methods in Biomedical Engineering, vol. 14, no. 11, pp. 1055-1065, 1998.
[6] V. Sladek and J. Sladek, "Boundary integral equation method in thermoelasticity Part III: Uncoupled thermoelasticity," Applied mathematical modelling, vol. 8, no. 6, pp. 413-418, 1984.
[7] J. Sladek, V. Sladek, and I. Markechova, "Boundary element method analysis of stationary thermoelasticity problems in non‐homogeneous media," International Journal for Numerical Methods in Engineering, vol. 30, no. 3, pp. 505-516, 1990.
[8] Y. Shiah and C. Tan, "Exact boundary integral transformation of the thermoelastic domain integral in BEM for general 2D anisotropic elasticity," Computational Mechanics, vol. 23, no. 1, pp. 87-96, 1999.
[9] Y. Shiah, Y.-C. Chaing, and T. Matsumoto, "Analytical transformation of volume integral for the time-stepping BEM analysis of 2D transient heat conduction in anisotropic media," Engineering Analysis with Boundary Elements, vol. 64, pp. 101-110, 2016.
[10] Y. Shiah and M.-R. Li, "The solution to an elliptic partial differential equation for facilitating exact volume integral transformation in the 3D BEM analysis," Engineering Analysis with Boundary Elements, vol. 54, pp. 13-18, 2015.
[11] Y. Shiah, "Analysis of thermoelastic stress-concentration around oblate cavities in three-dimensional generally anisotropic bodies by the boundary element method," International Journal of Solids and Structures, vol. 81, pp. 350-360, 2016.
[12] Y.-C. Shiah and J.-Y. Chong, "Boundary Element Analysis of Interior Thermoelastic Stresses in Three-Dimensional Generally Anisotropic Bodies," Journal of Mechanics, vol. 32, no. 6, pp. 725-735, 2016.
[13] Y. Shiah et al., "Direct volume-to-surface integral transformation for 2D BEM analysis of anisotropic thermoelasticity," CMES-Computer Modeling in Engineering and Sciences, vol. 102, no. 4, pp. 257-270, 2014.
[14] Y. Shiah, "Analytical transformation of the volume integral in the boundary integral equation for 3D anisotropic elastostatics involving body force," Computer Methods in Applied Mechanics and Engineering, vol. 278, pp. 404-422, 2014.
[15] I. Lifshits and L. Rozentsveyg, "Construction of the Green's tensor for the basic equations of the theory of elasticity in the case of an unbounded elastically anisotropic media environment," ed: JETF, 1947.
[16] R. Wilson and T. Cruse, "Efficient implementation of anisotropic three dimensional boundary‐integral equation stress analysis," International Journal for Numerical Methods in Engineering, vol. 12, no. 9, pp. 1383-1397, 1978.
[17] M. A. Sales and L. Gray, "Evaluation of the anisotropic Green's function and its derivatives," Computers & structures, vol. 69, no. 2, pp. 247-254, 1998.
[18] E. Pan and F. Yuan, "Boundary element analysis of three‐dimensional cracks in anisotropic solids," International Journal for Numerical Methods in Engineering, vol. 48, no. 2, pp. 211-237, 2000.
[19] 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.
[20] A. V. Phan, L. Gray, and T. Kaplan, "On the residue calculus evaluation of the 3‐D anisotropic elastic green's function," International Journal for Numerical Methods in Biomedical Engineering, vol. 20, no. 5, pp. 335-341, 2004.
[21] C.-Y. Wang and M. Denda, "3D BEM for general anisotropic elasticity," International Journal of Solids and Structures, vol. 44, no. 22, pp. 7073-7091, 2007.
[22] Y. Shiah, C. Tan, and V. Lee, "Evaluation of explicit-form fundamental solutions for displacements and stresses in 3D anisotropic elastic solids," CMES: Computer Modeling in Engineering & Sciences, vol. 34, no. 3, pp. 205-226, 2008.
[23] C. Tan, Y. Shiah, and C. Lin, "Stress analysis of 3D generally anisotropic elastic solids using the boundary element method," Computer Modeling in Engineering and Sciences (CMES), vol. 41, no. 3, p. 195, 2009.
[24] T. Ting and V.-G. Lee, "The three-dimensional elastostatic Green's function for general anisotropic linear elastic solids," The Quarterly Journal of Mechanics and Applied Mathematics, vol. 50, no. 3, pp. 407-426, 1997.
[25] V.-G. Lee, "Explicit expression of derivatives of elastic Green’s functions for general anisotropic materials," Mechanics Research Communications, vol. 30, no. 3, pp. 241-249, 2003.
[26] Y. Shiah, C. Tan, and C. Wang, "Efficient computation of the Green's function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis," Engineering Analysis with Boundary Elements, vol. 36, no. 12, pp. 1746-1755, 2012.