A fast dual boundary element method for 3D anisotropic crack problems

Abstract

In the present paper a fast solver for dual boundary element analysis of 3D anisotropic crack problems is formulated, implemented and tested. The fast solver is based on the use of hierarchical matrices for the representation of the collocation matrix. The admissible low rank blocks are computed by adaptive cross approximation (ACA). The performance of ACA against the accuracy of the adopted computational scheme for the evaluation of the anisotropic kernels is investigated, focusing on the balance between the kernel representation accuracy and the accuracy required for ACA. The system solution is computed by a preconditioned GMRES and the preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. The effectiveness of the proposed technique for anisotropic crack problems has been numerically demonstrated, highlighting the accuracy as well as the significant reduction in memory storage and analysis time. In particular, it has been numerically shown that the computational cost grows almost linearly with the number of degrees of freedom, obtaining up to solution speedups of order 10 for systems of order 104. Moreover, the sensitivity of the performance of the numerical scheme to materials with different degrees of anisotropy has been assessed.