Systematic development of upwind numerical fluxes for the space discontinuous Galerkin method applied to elastic wave propagation in anisotropic and heterogeneous media with physical interfaces

Abstract

This research work presents, within a unified and wave oriented variational framework, systematic development of upwind numerical fluxes for the space discontinuous Galerkin methods to model elastic wave propagation in multidimensional anisotropic media with discontinuous material properties. Both first-order velocity-stress and velocity-strain wave formulations are considered. The proposed approach allows the derivation of upwind numerical fluxes in a well structured and hierarchical way according to the degree of inhomogeneity across a physical interface. The numerical fluxes that are exact solutions of a relevant Riemann problem defined at a physical interface are obtained. The developed explicit and intrinsic tensorial expressions of upwind numerical fluxes in multidimensional case allow a better understanding and analysis of the physical meaning of involved terms. As numerical applications, an example with a physical interface separating two materials, one anisotropic and the other isotropic, and an example of polycrystalline material that presents a particular case with a larger number of physical interfaces, are considered. The proposed numerical fluxes are numerically investigated and validated.

Multi-parameter optimization of attenuation data for characterizing grain size distributions and application to bimodal microstructures

“…The numerical model is based on the following system of conservation laws, composed of the balance equation of linear momentum with no source term and geometrical conservation laws: In system (32), the unknowns are respectively σ and v, the Cauchy stress tensor and the velocity vector. The solid domain presented in figure 9 is then discretized as a Cartesian grid of space step ∆x = 12.0µm, in such a way that the semi-discrete form of the above governing equations can be written by means of the discontinuous Galerkin (DG) first-order approximation [29,30,40]. Note that such a finite element size leads to good convergence properties for continuous Galerkin finite element schemes [41,42], which have the same (second-order) accuracy as the considered DG approach.…”

“…At last, an explicit two-step second-order Runge-Kutta time-discretization is used to derive the discrete system. The reader interested in more details about formulation, implementation and stability of DG methods for anisotropic and piecewise homogeneous media should refer to references [30,40,43].…”