Transient elastic wave analysis of 3-D large-scale cavities by fast multipole BEM using implicit Runge-Kutta convolution quadrature

Existing boundary element methods (BEMs) have been proven to be efficient numerical techniques particularly in modeling wave propagation problems but still remain limitations in solving large-scale problems. The main disadvantages may be caused by several problems, including high cost and large amount of computer memory for the computation. A time-domain BEM in which a collocation method is used for the time discretization also shows numerical instability for small CFL (Courant-Friedrichs-Lewy) number. In this work, we propose a new three-dimensional (3-D) approach for solving transient elastic wave propagation problems of large-scale spherical and spheroidal cavities by the convolution quadrature BEM accelerated by fast multipole method (FMM) using an implicit Runge-Kutta scheme (in abbreviation, we name it as IRK-based CQ-FMBEM). Two special techniques pertaining to the FMM including the scaling of modified spherical Bessel functions and the truncation method are used to enhance the efficiency and stability of our proposed method. This approach is found to be particularly suitable for 3-D large-scale problems as its high accuracy, stability of time-marching process, and computational efficiency. These properties are subsequently illustrated through numerical examples dealing with large-scale wave problems of 3-D single spherical and spheroidal cavities and multiple equally and irregularly spherical cavities. The accuracy of the present formulation is verified by comparing the obtained results with available reference solutions in the literature. Some numerical aspects of time increments, sizes and topological shapes of cavities, and their influences on the deformations and the waveforms are investigated. Also, detailed investigation of the computational efficiency of our proposed method, such as CPU time, required memory, etc. is presented. All the implementation tasks are carried out using the Tokyo Tech. supercomputer which is called TSUBAME 2.5 [42].