本研究旨在探討三維疊層複合材料於旋轉及自重效應下之層間應力與位移行為，並針對薄層異向性與等向性材料於慣性力作用下之數值模擬方法進行改良。由既有研究可知，薄層複材於材料界面處容易產生應力集中與脫膠現象，且由於膠合層厚度極薄，在數值計算過程中常伴隨近似奇異積分問題，進而影響分析之穩定性與準確性。
為改善上述問題，本研究採用 Power-Law Transformation 技術對近似奇異積分進行處理，以改善傳統數值積分於薄層結構分析中的數值誤差與穩定性問題。透過將邊界積分進行冪次形式之轉換，可有效重新分配積分點於近奇異區域之分布情形，使極薄膠合層條件下仍可採用較粗網格進行穩定且精確之分析。
本研究建立三維邊界元素分析程式，針對不同材料性質、幾何形狀與邊界條件進行多組旋轉與自重效應分析，並與 ANSYS APDL 有限元素分析結果進行比較。結果顯示，本文所提出之方法於位移與應力分析上皆具有良好一致性，且於薄層結構分析中可有效降低元素數量與計算自由度，顯示其於複合材料界面分析問題中具有良好之應用潛力。

This study aims to investigate the interlaminar stresses and displacement behavior of three-dimensional laminated composite materials subjected to rotational and self-weight effects, and to improve the numerical simulation methods for thin-layered anisotropic and isotropic materials under inertial loading conditions. Previous studies have shown that thin laminated structures are prone to stress concentration and delamination near material interfaces. In addition, the extremely small thickness of adhesive layers often leads to nearly singular integrals during numerical computations, which may significantly affect the stability and accuracy of the analysis.
To address these issues, this study adopts a power-law transformation technique to treat nearly singular integrals in boundary element analysis. By reformulating the boundary integrals into a power-based form, the integration points can be redistributed more effectively in near-singular regions, thereby reducing numerical integration difficulties. As a result, stable and accurate analysis can still be achieved using relatively coarse mesh discretization, even for structures containing extremely thin adhesive layers.
In this study, a three-dimensional boundary element analysis program is developed using FORTRAN to investigate rotational and self-weight effects under different material properties, geometric configurations, and boundary conditions. The numerical results are further compared with ANSYS APDL finite element analysis. The results demonstrate that the proposed method achieves good agreement in both displacement and stress analysis while significantly reducing the required number of elements and degrees of freedom for thin-layered structures. These results demonstrate the effectiveness and applicability of the proposed method for interfacial analysis of laminated composite materials.

[1] J.M. Zhang, X.Y. Qin, X. Han, G.Y. Li, A boundary face method for potential problems in three dimensions, Internat J Numer Methods Engrg 2009; 80: 320–337.
[2] H.L. Zhou, Z.R. Niu, C.Z. Cheng, Z.W. Guan, Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems, Comput Struct 2008; 86: 1656-1671.
[3] Y.C. Shiah, C.L. Tan, and Li-Ding Chan, Boundary element analysis of thin anisotropic structures by the self-regularization scheme, CMES-Comp Model Eng 2015; 109-110(1): 15-33.
[4] Y.C. Shiah and M.R. Hematiyan, Interlaminar stresses analysis of three-dimensional composite laminates by the boundary element method, Journal of Mechanics; 34 (6): 829-837, 2018.
[5] Tan, F., Liang, J., Jiao, Y. et al. The BEM based on conformal Duffy-distance transformation for three-dimensional elasticity problems. Sci. China Technol. Sci. 63, 2575–2583 (2020).
[6] Y.M. Zhang, Y. Gu, and J.T. Chen, Boundary Layer Effect in BEM with High Order Geometry Elements Using Transformation, CMES, vol.45, no.3, pp.227-247, 2009
[7] L. Jun, G. Beer, J.L. Meek, Efficient evaluation of integrals of order 1/r, 1/r2 , 1/r3 using Gauss quadrature, Eng. Anal. 2 (1985) 118–123.
[8] Zhongrong Niu, Zongjun Hu, Changzheng Cheng, Huanlin Zhou, A novel semi-analytical algorithm of nearly singular integrals on higher order elements in two dimensional BEM, Engineering Analysis with Boundary Elements, Volume 61, December 2015, Pages 42-51.
[9] Zhilin Han, Wei Pan, Changzheng Cheng, Zongjun Hu, Zhongrong Niu, A semi-analytical treatment for nearly singular integrals arising in the isogeometric boundary element method-based solutions of 3Dpotential problems, August 2022, Computer Methods in Applied Mechanics and Engineering 398(39–41):115179
[10] Y.C. Shiah, Yue-Fang Hsiao, Mohammad-Rahim Hematiyan, Efficient Modeling of Heat Conduction in Multiply Bonded Composites Covered with Thermal Barrier Coating, Engineering Analysis with Boundary Element Method, Volume 164, July 2024, 105763.
[11] Y.C. Shiah, BEM Simulation of the Interlaminar Stresses of 3D Thin Composites Subjected to Inertial Loads, Engineering Analysis with Boundary Elements, Volume 183, 2026, 106595.
[12] C.L. Tan, Y.C. Shiah, C.Y. Wang, Boundary Element Elastic Stress Analysis of 3D Generally Anisotropic Solids Using Fundamental Solutions Based on Fourier Series, International Journal of Solids and Structures, Vol.50, pp.2701-2711, 2013
[13] S. Wu, On the evaluation of nearly singular kernel integrals in boundary element analysis, Numer. method Eng. 11 (1995) 331–337.
[14] P.R. Johnston, Application of sigmoidal transformation to weakly singular and nearly-singular boundary element integrals, Internat. J. Numer. Methods Engrg. 45 (1999) 1333–1348.
[15] Y.C. Shiah, Nguyen Anh Tuan, M.R. Hematiyan, Direct transformation of the volume integral in the boundary integral equation for treating three-dimensional steady-state anisotropic thermoelasticity involving volume heat source, International Journal of Solids and Structures 2018, 143: 287-297.
[16] Y.C. Shiah, C.L. Tan, and C.Y. Wang, Efficient Computation of the Green’s Function and its Derivatives for Three-Dimensional Anisotropic Elasticity in BEM Analysis, Engineering Analysis with Boundary Elements 2012, 36: 1746-1755
[17] T. Kant, K. Swaminathan, Estimation of transverse/interlaminar stresses in laminated composites—a selective review and survey of current developments, Composite Structures, 49, pp. 65–75, (2000).
[18] A.H. Puppo, H.A. Evensen, Interlaminar shear in laminated composites under generalized plane stress, Journal of Composite Materials, 4, pp. 204–220, (1970).
[19] N.J. Pagano, On the calculation of interlaminar normal stress in composite laminate, Journal of Composite Materials, 8, pp. 65–81, (1974).
[20] P.W. Hsu, C.T. Herakovich, Edge effects in angle-ply composite laminates, Journal of Composite Materials, 11, pp. 422–428, (1977).
[21] S. Tang, A. Levy, A boundary layer theory. Part II: extension of laminated finite strip, Journal of Composite Materials, 9, pp. 42–52, (1975).
[22] R.B. Pipes, N.J. Pagano, Interlaminar stresses in composite laminates an approximate elasticity solution, ASME Journal of Applied Mechanics, 41, pp. 668–672, (1974).
[23] N.J. Pagano, Stress fields in composite laminates, International Journal of Solids and Structures, 14, pp. 385–400, (1978).
[24] S.S. Wang, I. Choi, Boundary-layer effects in composite laminates. Part I: free-edge stress singularities, ASME Journal of Applied Mechanics, 49, pp. 541–548, (1982).
[25] Y.C. Shiah, Kuo-Wei Hsu, Two-dimensional analysis of interlaminar stresses in thin anisotropic composites subjected to inertial loads by regularized boundary integral equation, Composites Part B: Engineering, 159, pp. 105–113, (2019).
[26] Y.C. Shiah, Shang-Yu Ye, 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, 73, pp. 170–180, (2016).
[27] Y.C. Shiah, Wen-Sheng Hwang, and Guan-Chyun Shiah, “BEM Stress Analysis for Thin Multilayered Composites Subjected to Inertial Loads,” Journal of Composite Materials, Vol. 43, No. 4, pp. 349-366, (2009).
[28] Y.C. Shiah, Analytical Transformation of the Volume Integral for the BEM Treating 3D Anisotropic Elastostatics Involving Body Force, Computer Methods in Applied Mechanics and Engineering, pp. 404-422, (2014).
[29] Scuderi Letizia, On the computation of nearly singular integrals in 3D BEM collocation, International Journal for Numerical Methods in Engineering. 74, pp. 1733–1770, (2007).
https://doi.org/10.1002/nme.2229.
[30] H. Ma, N. Kamiya, A general algorithm for the numerical evaluation of nearly singular boundary integrals of various orders for two- and three-dimensional elasticity, Computational Mechanics, 29, pp. 277–288, (2002). https://doi.org/10.1007/s00466-002-0340-0.
[31] H. Ma, N. Kamiya, Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method, Engineering Analysis with Boundary Elements, 26, pp. 329–339, (2002). https://doi.org/10.1016/S0955-7997(02)00004-8.
[32] Xianyun Qin, Jianming Zhang, Guizhong Xie, Fenglin Zhou, Guanyao Li, A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element, Journal of Computational and Applied Mathematics,235,pp.4174–4186,(2011). https://doi.org/10.1016/j.cam.2011.03.012.
[33] J.C.F. Telles, A self-adaptive co-ordinates transformation for efficient numerical evaluation of general boundary element integrals, International Journal for Numerical Methods in Engineering, 24, pp. 959–973, (1987). https://doi.org/10.1002/nme.1620240509.
[34] X.L. Chen, Y.J. Liu, An advanced 3D boundary element method for characterizations of composite materials, Engineering Analysis with BoundaryElements,29,pp.513–523,(2005). https://doi.org/10.1016/j.enganabound.2004.12.013.
[35] Y.C. Shiah and C. L. Tan, Higher-Order Green’s Function Derivatives and BEM Evaluation of Stresses at Interior Points in a 3D Generally Anisotropic Solid, CMES – Computer Modeling in Engineering & Sciences, Vol. 78, pp. 95-108, (2011).
[36] Hayami, Ken. (2005). Variable Transformations for Nearly Singular Integrals in the Boundary Element Method. Publications of the Research Institute for Mathematical Sciences. 41. 10.2977/prims/1145474596.
[37] Ye, Wenjing. (2008). A new transformation technique for evaluating nearly singular integrals. Computational Mechanics. 42. 457-466. 10.1007/s00466-008-0262-6.
[38] ohnston, Barbara & Johnston, Peter & Elliott, David. (2007). A sinh transformation for evaluating two-dimensional nearly singular boundary element integrals. International Journal for Numerical Methods in Engineering. 69. 10.1002/nme.1816.
[39] Sladek, Vladimir & Sladek, Jan & Tanaka, M.. (2000). Optimal transformation of the integration variables in computation of singular integrals in BEM. International Journal for Numerical Methods in Engineering. 47. 1263 - 1283. 10.1002/(SICI)1097-0207(20000310)47:7<1263::AID-NME811>3.0.CO;2-I.
[40] Zhang, Yaoming & Gu, Yan & Zheng, Bin. (2010). A general nonlinear transformation for the evaluation of nearly singular integrals.
[41] Lv, Jiahe & Miao, Yu & Zhu, Hong-ping. (2013). General distance transformation for the numerical evaluation of nearly singular integrals in BEM. Computer Modeling in Engineering and Sciences. 91. 101-117.
[42] Xie, Guizhong & Zhang, Jianming. (2014). Application of exponential transformation coupled with adaptive subdivision technique for nearly singular boundary integrals in elasticity problems. WIT Transactions on Modelling and Simulation. 56. 10.2495/BEM360321.