This paper proposes a novel algorithm to accelerate the process of modal analysis in 3D elastodynamic problems in BEM (boundary element method) with high accuracy. Because of low efficiency and high cost, conventional BEM is rarely used for solving 3D elastodynamics problems in engineering problems. With applying the DRBEM (dual reciprocity boundary element method) to form new integral equations of 3D elastodynamics problems to reduce time complexity by using reciprocity method twice, we introduce modified FMM (fast multipole method) to simplify the computation process and improve the efficiency from O(n2) to O(n) in matrix multiplication. The main features in this method are: (1) Position Location (PL) algorithm is used to eliminate one layer of nested loops in conventional FMM, and which achieve a good performance in efficiency; (2) time dimension integrations in the element of matrices are canceled for high efficiency; (3) instead of the interaction between points, we apply point to element interaction method for saving plenty of the CPU cost in modified FMM; (4) it does not need to compute complex dynamic fundamental solutions which are necessary in conventional BEM. In this algorithm, the corresponding eigenvalue problem is solved by Hessenberg matrix and QR reduction algorithm iteratively. We have tested our method in numerical examples during last section, and have observed significant optimal results in efficiency and accuracy.
Nonsmooth modal analysis via the boundary element method for one-dimensional bar systems
