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# Analysis of 3D Anisotropic Solids Using Fundamental Solutions Based on Fourier Series and the Adaptive Cross Approximation Method

R. Q. Rodríguez1,2, C. L. Tan2, P. Sollero1, E. L. Albuquerque3
Faculty of Mechanical Engineering, University of Campinas, Brazil.
Department of Mechanical and Aerospace Engineering, Carleton University, Canada.
Faculty of Technology, University of Brasilia, Brazil.

Computer Modeling in Engineering & Sciences 2014, 102(5), 359-372. https://doi.org/10.3970/cmes.2014.102.359

### Abstract

The efficient evaluation of the fundamental solution for 3D general anisotropic elasticity is a subject of great interest in the BEM community due to its mathematical complexity. Recently, Tan, Shiah, andWang (2013) have represented the algebraically explicit form of it developed by Ting and Lee (Ting and Lee, 1997; Lee, 2003) by a computational efficient double Fourier series. The Fourier coefficients are numerically evaluated only once for a specific material and are independent of the number of field points in the BEM analysis. This work deals with the application of hierarchical matrices and low rank approximations, applying the Adaptive Cross Approximation (ACA) to treat 3D general anisotropic solids in BEM using this Green’s function based on Fourier series. The use of hierarchical format is aimed at reducing the storage requirements of the system matrices and the computational effort in the BEM analysis of large systems. Numerical examples are presented to show the successful implementation of using ACA and the formulation based on Fourier series for BEM analysis of 3D anisotropic solids.

### Keywords

Adaptive Cross Approximation, Boundary Element Method, Hierarchical Matrices, Anisotropic Fundamental Solution, Fourier Series.

Rodríguez, R. Q., Tan, C. L., Sollero, P., Albuquerque, E. L. (2014). Analysis of 3D Anisotropic Solids Using Fundamental Solutions Based on Fourier Series and the Adaptive Cross Approximation Method. CMES-Computer Modeling in Engineering & Sciences, 102(5), 359–372.

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