The application of a two -dimensional upwind leapfrog scheme to linear elastodynamics.

Abstract

Two-dimensional linear isotropic elastodynamic wave propagation problems were investigated numerically using an upwind leapfrog scheme. In addition, this simulation involved system separation of the body waves. There is substantial improvement in the simulation fidelity, along with less computational costs when compared with a non-separation approach. The upwind leapfrog scheme, which is based on the idea of bi-characteristics, has the desirable attribute for the simulation of waves, in that it has no dissipation and excellent phase speed accuracy. Using this scheme elastodynamic wave propagation problems are solved on a single mesh ('single mesh scheme') to capture the P-wave and the S-waves. One difficulty with this initial simulation, is that the S-waves are poorly resolved, showing advancing phase error. As long as the computation is carried out on a single mesh, since there are two waves with distinct speeds, full advantage of the upwind leapfrog scheme in the elastodynamic system cannot be exploited. In order to evade the difficulty of two wave speeds, the system is separated by a Helmholtz decomposition. By this decomposition the two types of waves are completely separated into individual systems, where each problem is reduced to an acoustic type problem. By applying the upwind leapfrog scheme to each individual system ('double mesh scheme'), each wave is resolved with highly accurate solutions with comparable error for both waves. The only significant issue of this system separation, is to re-coupling the waves on the boundary. With a careful consideration of stationary characteristics and the use of a low pass filter for stabilization, the boundary re-coupling was accomplished. Reflection coefficients for free surface and rigid boundary conditions are presented. As opposed to a 'single mesh scheme,' the upwind leapfrog scheme on double mesh with proper CFL number, shows higher resolution of the propagated waves, as well as significant reduction of computational costs. Finally some application problems are presented using the 'double mesh scheme.' Wave propagation and diffraction by a rigid inclusion, as well as the reflection and transmission in two connected semi-infinite media, are simulated and various waves associated with the interface are captured.