A unified variational framework for the space discontinuous Galerkin method for elastic wave propagation in anisotropic and piecewise homogeneous media

Abstract

We present a unified multidimensional variational framework for the space discontinuous Galerkin method for elastic wave propagation in anisotropic and piecewise homogeneous media. Based on an elastic wave oriented formulation and using a tensorial formalism, the proposed framework allows a better understanding of the physical meaning of the terms involved in the discontinuous Galerkin method. The unified variational framework is written for first-order velocity-stress wave equations. An uncoupled upwind numerical flux and two coupled upwind numerical fluxes using respectively the Voigt and the Reuss averages of elastic moduli are defined. Two numerical fluxes that are exact solutions of the Riemann problem on physical interfaces are also developed and analyzed in the 1D case. The implemented solvers are then applied to different elastic media, especially to polycrystalline materials that present a particular case of piecewise homogeneous media. The use of the three upwind numerical fluxes, which only solve approximately the Riemann problem at element interfaces, is investigated.

Some comparisons and analyses of time or space discontinuous Galerkin methods applied to elastic wave propagation in anisotropic and heterogeneous media

“…In the present paper, the velocity-stress space dG formulation is considered in the general case of anisotropic elastic media with physical interfaces and its stability is analyzed. The result allows explaining the following phenomena previously observed in [17]: when the degree of discontinuity at physical interfaces is high, the use of approximate numerical fluxes leads to instability problems, which cannot be resolved by simply reducing the time step used by the time discretization scheme.…”

“…Otherwise, we recall that space-time meshes generally used for the time dG method are composed of a classical finite element mesh in space and one linear element in time in each space-time slab and so an implicit solver is actually obtained [4][5][6][7][8]17].…”

“…The governing equations of the same model problem of elastic wave propagation defined in the preceding are firstly given within the framework of the velocity-stress space dG method. The unified and wave oriented variational framework proposed in [17] is recalled. Then the exact upwind numerical fluxes developed in [17,18] are given before the presentation of the main result of the present work: the stability of the space dG method in the general case of anisotropic heterogeneous media with physical interfaces of discontinuous material properties.…”

“…It has been widely used for computational fluid dynamics, and its application to solid elastodynamics is more recent. We refer to [9][10][11][12][13][14][15][16][17][18] for a non-exhaustive review of related research works. To apply the space dG method, the second-order elastic wave equations are transformed to a first-order hyperbolic system, both velocity-stress [10,11] and velocity-strain [15] formulations can be used.…”

“…Recently, in the most general case of multidimensional anisotropic elastic media with discontinuous material properties, a unified and wave oriented variational framework has been proposed by the author [17], which allows a systematic development of the exact upwind numerical fluxes for both velocity-stress and velocity-strain formulations [18]. Owing to the compact and intrinsic forms expressed in terms of tensors, the derivation of upwind numerical fluxes can be done in a well structured way, which allows better understanding of the physical meaning of the developed upwind numerical fluxes.…”