Efficient Numerical Procedure for Reduction of Computation Time in BIEM for Elastodynamic Analysis of Non-planar Faults

Recent progress in boundary integral equation method (BIEM) have been enabled us to analyze dynamic rupture process on fully non-planar fault geometry, such as steps, bendings, and branchings, in 2D (Kame and Yamashita, 1999) and 3D (Aochi, Fukuyama and Matsu'ura 2000; Fukuyama, Tada and Shibazaki 2002, Eos Trans. AGU 83(47)). In space-time boundary integral formulations, the stress on the fault surface is represented in terms of the slip history on the fault surface only as τ ({x},t) = τ ^0({x},t) + f({x},t) - μ /2c ˙ V({x},t) where μ is the shear modulus, c is the shear wave speed, V is the slip rate, τ ⁰ is the loading stress, and f is a linear function of prior slip-rate over the causality cone. f is expressed as a double convolution integral in space and time. Evaluation of the convolution integrals is the most computationally demanding part of the elastodynamic analysis. In order to reduce computation time, a spectral representation of f in space has been derived for a planar fault and significant reduction has been made by using Fast Fourier Transform algorism for convolution in Fourier domain. It is, however, usually difficult to derive a similar spectral representation for non-planar faults. We thus turn our attention to convolution in time. Here we apply " truncation of convolution integrals" to the time-space BIEM in 2D in-plane problem. This is originally introduced to 2D anti-plane spectral formulation by Lapusta et al.(2000) for a relatively slow process where the effect of wave is relatively small. The principle is very simple: We truncate the convolution integrals in time and replace the stress contribution from the rest of causality cone with the static one due to the accumulated slip, i.e., slip rate summed up inside the rest of the cone. For numerical implementation, we combine dynamic scheme with static scheme. We try to choose truncation time L as short as possible without much loss of the dynamic stress contribution, where we always assume L is smaller than the total time step N in computation. For a large time step N, computation time is expected to decrease from N*N to N*L. The methodology can be readily extended to 3D BIEM without any modifications. Preliminary examples of the application of truncation will be shown. First dynamic rupture with several prescribed speeds is simulated on a planar fault both by the truncated convolution and by the full convolution. The result shows the amount of truncation time allowed in the simulation and the corresponding reduction time in computation. The truncation method will be tested in continuing simulations that require very fine cell size, e.g., rupture with high sub-Rayleigh speeds. For a precise representation of the wave-field around rupture front at that speed, we will make the cell size about to 1/20 of the size that slip-weakening zone would have in very low-speed rupture.