Efficient multigrid solution of boundary element models of cracks and faults

Abstract

AbstractWe consider the multigrid solution of a discretized singular integral equation whose solution represents the displacement discontinuity distribution across a pressurized crack or the slip on a fault subjected to a prescribed shear stress. The multigrid technique reduces the operation count for a crack model having N degrees of freedom from O(N3) operations for standard stationary iterative methods to O(N2) operations. In the numerical simulations performed the multigrid approach proves to be extremely efficient even for small values of N. We use Fourier analysis to determine the spectral properties of the coarse grid correction process and the effect of a number of different interpolation operators on the multigrid algorithm. We show that the multigrid technique can be combined with the process of lumping remote influences to yield an algorithm that involves O(N) operations. The performance of multigrid iteration in a nonlinear environment is explored by considering seams filled with nonlinear material. For this purpose a segmented multigrid algorithm is developed that allows for different seam constitutive relations to be used along different line segments. The O(N2) operation count characteristic of linear multigrid iteration is shown to persist in this nonlinear environment and the segmented multigrid approach is shown to provide significant computational savings for values of N well within the range for typical problems that occur in practice.