Wave propagation in a fractional viscoelastic Andrade medium: Diffusive approximation and numerical modeling

• The prototypical fractional viscoelastic Andrade model with power-law attenuation is considered.
• The featured fractional derivative is recast using a diffusive representation to reduce memory cost.
• An accurate quadrature scheme is analyzed to compute a stable diffusive approximation.
• The model is implemented in an efficient numerical scheme to perform transient wave simulations.
• Numerical results and a semi-analytical solution demonstrate the performances of the approach.

This study focuses on the numerical modeling of wave propagation in fractionally-dissipative media. These viscoelastic models are such that the attenuation is frequency-dependent and follows a power law with non-integer exponent within certain frequency regimes. As a prototypical example, the Andrade model is chosen for its simplicity and its satisfactory fits of experimental flow laws in rocks and metals. The corresponding constitutive equation features a fractional derivative in time, a non-local-in-time term that can be expressed as a convolution product whose direct implementation bears substantial memory cost. To circumvent this limitation, a diffusive representation approach is deployed, replacing the convolution product by an integral of a function satisfying a local time-domain ordinary differential equation. An associated quadrature formula yields a local-in-time system of partial differential equations, which is then proven to be well-posed. The properties of the resulting model are also compared to those of the Andrade model. The quadrature scheme associated with the diffusive approximation, and constructed either from a classical polynomial approach or from a constrained optimization method, is investigated. Finally, the benefits of using the latter approach are highlighted as it allows to minimize the discrepancy with the original model. Wave propagation simulations in homogeneous domains are performed within a split formulation framework that yields an optimal stability condition and which features a joint fourth-order time-marching scheme coupled with an exact integration step. A set of numerical experiments is presented to assess the overall approach. Therefore, in this study, the diffusive approximation is demonstrated to provide an efficient framework for the theoretical and numerical investigations of the wave propagation problem associated with the fractional viscoelastic medium considered.