A Discontinuous-Galerkin approach to model non-classical nonlinearity observed from lab to global scales

Under dynamic perturbations, it has been observed that materials like sedimentary rocks show complex mechanical behaviors. They include the simultaneous dependence of the elastic moduli and attenuation on strain at the same time scale of the perturbations, as well as the conditioning and recovery of the elastic moduli that may happen at time scales that are much larger. The latter cases were recently referred to as non-classical nonlinearity. Aside from laboratory experiments, comparable observations of the non-classical nonlinearity have been made in the field over the past two decades with the development of long-term continuous monitoring of the velocity field inside the Earth using methods such as ambient noise interferometry. A variety of mathematical models that can potentially quantify the non-classical nonlinearity have already been proposed, e.g., the Damage–Breakage Rheology Model, the Internal Variable Model and the Godunov–Peshkov–Romenski model. However, implementing them in numerical schemes suitable to reproduce nonlinear effects in wave propagation on the local, regional, or global scale is challenging. This can be of interest for constraining a more realistic dynamic rheology for the Earth with the field observations. In this work, wave propagation in different non-classical nonlinear models is implemented in FEniCS using the discontinuous Galerkin (DG) method in 1D. Behaviors of the different models are systematically studied and quantitatively compared against measurements. This work lays the foundation for an extension to the simulation of 2D/3D wave propagation in the Earth on the large-scale DG simulation frameworks, e.g., SeisSol and ExaHyPE.