An FC-based spectral solver for elastodynamic problems in general three-dimensional domains

Abstract

This paper presents a spectral numerical algorithm for the solution of elastodynamics problems in general three-dimensional domains. Based on a recently introduced "Fourier continuation" (FC) methodology for accurate Fourier expansion of non-periodic functions, the proposed approach possesses a number of appealing properties: it yields results that are essentially free of dispersion errors, it entails mild CFL constraints, it runs at a cost that scales linearly with the discretization sizes, and it lends itself easily to efficient parallelization in distributed-memory computing clusters. The proposed algorithm is demonstrated in this paper by means of a number of applications to problems of isotropic elastodynamics that arise in the fields of materials science and seismology. These examples suggest that the new approach can yield solutions within a prescribed error tolerance by means of significantly smaller discretizations and shorter computing times than those required by other methods.

High-Performance Computational Electromagnetics in Frequency-Domain and Time-Domain

“…Significant progress occurred over the life of this contract in the area of FC (Fourier Continuation) methods for Partial Differential Equations in the time-domain [1][2][3][4][5][6][7]. Our efforts in these areas have resulted in explicit and implicit solvers for high-and low-frequency problems, for linear and nonlinear equations, and including media such as fluids, solids and vacuum-and combinations thereof.…”

“…In the contribution [2], in turn, methods for FC solution of problems containing variable coefficients were introduced; in particular it was found that certain numerical boundary layers need to be adequately represented in order to ensure accurate solution. Reference [3] introduced an approach that allows for treatment of traction boundary conditions in wave propagation problems in solids [3,4] (Navier's elastic wave equation) 1 . The thesis [5] uses the Fourier Continuation method in multiple ways: to solve equations, to propagate to distant regions without meshing, etc.…”

Higher-order implicit-explicit multi-domain compressible Navier-Stokes solvers

“…[11]). More recently, the FC methodology was combined with an overset grid approach for the solution of the compressible Navier-Stokes equations in two dimensions [12] and the elasticity equations in three dimensions [13]. A key development in those contributions is the extension of the FC method [14] to overlapping "sub-patch" block-decompositions of larger meshes.…”

“…A key development in those contributions is the extension of the FC method [14] to overlapping "sub-patch" block-decompositions of larger meshes. Although the contributions [12,13] have successfully used the overset method in the context of explicit solvers, the goal of extending the framework to implicit and multi-domain implicit-explicit solvers had not been realized until now.…”

“…However, achieving high-order spatial accuracy with FD, in practice, is not without challenges. As is well known, a reduction of the dispersion error inherent in FD methods requires an increasing number of points per wavelength for a fixed level of accuracy or use of higher-order methods which typically entail higher costs and restrictive CFL constraints [12,13,20]. In addition, the use of high-order biased stencils near and at domain boundaries can lead to numerical instability.…”