Seismic waves scattering in three-dimensional homogeneous media using time-domain boundary element method

It is well established that the seismic ground response of surface topographies may differ from those of free field motion during earthquakes. Complex nature of seismic wave scattering by topographical structures can only be solved accurately and economically using advanced numerical methods under realistic conditions.
Among the numerical methods, the boundary element is a powerful numerical technique for analyses of linear and homogeneous materials for both bounded and unbounded domains. In this paper, the algorithm of seismic wave scattering by homogeneous media using time-domain three-dimensional boundary element method has been presented. Three-dimensional traction elastodynamic kernels for both cases of constant and linear variation of displacement have been presented. The convoluted kernel for constant time variation contains apparent singularities in the wave fronts, while in the linear time-convoluted elastodynamic traction kernel, apparent singularities in the wave's front disappear and a well-behaved kernel is resulted. Behavior of the constant and linear time-convoluted elastodynamic traction kernel have been investigated numerically. Kernel values were calculated at the central point of the three boundary elements considering different time steps. The boundary elements are in different condition of symmetry with respect to the source point. The kernel values in the case of constant time-convoluted elastodynamic traction kernel tend to the elatstostatic fundamental solution as expected, Whereas the kernel values in the case of linear time-convoluted elastodynamic traction kernel is equal to the elastostatic fundamental solution if the time step would be greater than time required to shear waves passed the receiver point on the element. Presented constant and linear time-convoluted elastodynamic traction kernels are casaul and have time-translation property which could be used for optimization of numerical algoritms.
The presented elastodynamic kernels have been used instead of temporal integration for wave scattering analyses of homogeneous medium using BE algorithm. Seismic wave scattering by a semi-spherical canyon subjected to incident compressional and shear waves has been analyzed. The semi-spherical canyon has a radius of 200m, shear wave velocity of 800m/s, Poisson’s ratio of 0.25, and mass density of 2.00gr/cm3. The incident wave of the Ricker type has the predominant frequency of 3.0Hz. Calculated results in time-domain are presented as well as comparison of results with other transformed-domain methods in term of dimensionless frequency, which shows the efficiency of the presented boundary element algorithm for solution of seismic wave scattering by homogeneous media in time domain as well as the accuracy of elastodynamic kernels.