Pseudo-Spectral simulation of 1D nonlinear propagation in heterogeneous elastic media

Abstract

In this work, a 1D Pseudo-Spectral Time Domain (PSTD) algorithm has been developed for solving elastic wave equation in nonlinear heterogeneous solids using FFTs for calculation of the spatial differential operator on staggered grid. The solver uses a staggered fourth order Adams–Bashforth method, by which stress and particle velocity are updated at alternating half time steps, to integrate forward in time. To circumvent wraparound inherent to FFT-based pseudo-spectral simulation, Convolution Perfectly Matched Layer (CPML) boundary condition has been used to eliminate implementation problems linked in classical PML to the introduction in nonlinear elasticity of a time dependent bulk modulus. Different kinds of nonlinear elastic models (quadratic and cubic nonlinearity, Nazarov hysteretic nonlinearity, bi-modular nonlinearity, PM-Space nonlinearity) have been implemented. The present study will focus on the comparison of nonlinear signature (harmonics generation, shock, frequency shift and attenuation) of these different kinds of nonlinearity for rod resonance, shock wave generation. These results are expected to be useful in helping to determine the predominant nonlinear mechanism in a specific experiment.