A wavelet-based method for simulation of two-dimensional elastic wave propagation

Abstract

S U M M A R YA wavelet-based method is introduced for the modelling of elastic wave propagation in 2-D media. The spatial derivative operators in the elastic wave equations are treated through wavelet transforms in a physical domain. The resulting second-order differential equations for time evolution are then solved via a system of first-order differential equations using a displacement-velocity formulation. With the combined aid of a semi-group representation and spatial differentiation using wavelets, a uniform numerical accuracy of spatial differentiation can be maintained across the domain. Absorbing boundary conditions are considered implicitly by including attenuation terms in the governing equations and the traction-free boundary condition at a free surface is implemented by introducing equivalent forces in the semi-group scheme. The method is illustrated by application to SH and P-SV waves for several models and some numerical results are compared with analytical solutions. The wavelet-based method achieves a good numerical simulation and shows an applicability for an elastic-wave study.

On a Wavelet-Based Method for the Numerical Simulation of Wave Propagation

“…In [34] we have demonstrated the way in this scheme can be used when the physical parameters are varying with depth or distance. Here we focus on the applicability of a wavelet-based method in media with surface topography or complex internal heterogeneity.…”

“…This enables the discrete time solution (5) to be used for wave propagation. When various boundary conditions and nonlinear effects are considered at every time step in a domain, the consideration via a set of forcing terms can increase the efficiency of computation and can reduce the numerical instability (e.g., [10,34]). …”

A 2D wavelet-based spectral finite element method for elastic wave propagation

“…Hence, the time interval Dt ¼ 3:0 ms causes the sampling density L in the wavelet domain apparently to be sufficiently high to accurately capture the response characteristics generated by the loading specified in Figure 4. Note that, for a given time window t of interest, the connection between L and Dt is explicitly set through Equation (18). For generating a FEM reference solution (constructed with 9-noded quadrilateral elements) with a high spatial accuracy, the total number of nodes is taken considerably larger than for the other three solutions, namely 58,081 nodes (corresponding to 120 Â 120 ¼ 14,400 elements), while the discrete time step is taken significantly smaller, i.e.…”

“…Daubechies wavelets have further been applied for representing the spatial derivative operators in one-dimensional [17] and two-dimensional [18,19] elastic wave propagation formulations, leading to an accurate description of the spatial response. In addition, Mitra and Gopalakrishnan have used Daubechies wavelets for a spectral analysis of wave propagation phenomena, where 2D problems were solved by transforming both the time parameter and one of the spatial coordinates with DWT, and subsequently solving the ordinary differential equation in terms of the remaining spatial coordinate with FEM [20,21].…”

Wavelet Based Simulation of Elastic Wave Propagation

“…The wavelet-based projection methods were successfully used for simulation of wave propagation problems in infinite and semi-infinite medias [12,[47][48][49][50][51][52]. Another important usage is wave propagation in structural engineering elements; e.g.…”