Two-Dimensional Fast Multipole Indirect Boundary Element Method-Based Solution to P-Wave Scattering by a Mountain with Large-Scale Random Cracks in an Elastic Half-Space

Abstract

In this study, the fast multipole indirect boundary element method (FM-IBEM) was adopted for solving the scattering problem of incident P-waves using a two-dimensional (2D) mountain terrain with large-scale random cracks. By adopting the FM-IBEM, this study resolves the bottleneck of the conventional boundary element method (BEM) for computing complex multi-degree-of-freedom models, and for the first time quantitatively analyzes the seismic wave scattering problem of large-scale narrow mountain cracks. After verifying the calculation accuracy, the horizontal and vertical displacement amplitudes of the mountain surface and its adjacent surface were accurately calculated. The results suggest the following: (1) Cracks significantly amplify the displacement amplitude of the mountain surface. The amplification mostly depends on the number of cracks and on the frequency of the incident P-wave. Compared with homogeneous mountains (no cracks), under the most unfavorable circumstances, the horizontal displacement is amplified more than 9.5-fold, while the vertical displacement is amplified more than 6.5-fold. (2) When the wave field passes through the crack area, a large amount of energy is trapped. Moreover, this energy is highly concentrated, which amplifies the local ground motion more than 7-fold. (3) The ground motion generated by the P-wave scattering by a mountain with large-scale random cracks exhibits a significant spatial difference. (4) The resonance of multi-order modes can emerge owing to large-scale random cracks. Compared with homogeneous mountains, the ground motion caused by the P-wave scattering by mountains with large-scale random cracks is much more severe in many cases. This phenomenon must be considered in practical seismic engineering applications.