Recursive evaluation of time convolution integrals in the spectral boundary integral method for mode III dynamic fracture problems

The shear crack, propagating spontaneously on a frictional interface, is a useful idealization of a natural earthquake. However, the corresponding boundary value problems are quite challenging in terms of required memory and processor power. While the huge computation amount is reduced by the spectral boundary integral method, the computation effort is still huge. In this paper, a recursive method for the evaluation of convolution integrals was tested in the spectral formulation of the boundary integral method applied to 2D anti-plane crack propagation problems. It is shown that analysis of a 2D anti-plane crack propagation problem involving Nt time steps, based on the recursive evaluation of convolution integrals, requires O(αNt) computational resources for each Fourier mode (as opposed to O(Nt2) for a classical algorithm), where α is a constant depending on the implementation of the method with typical values much less than Nt. Therefore, this recursive scheme renders feasible investigation of long deformational processes involving large surfaces and long periods of time, while preserving accuracy. The computation methodology implemented here can be extended easily to 3D cases where it can be employed for the simulation of complex spontaneously fault rupture problems which carry a high computational cost.