Improvement of hierarchical matrices for 3D elastodynamic problems with a complex wavenumber

Abstract

It is well known in the literature that standard hierarchical matrix (${\mathscr{H}}$-matrix)-based methods, although very efficient for asymptotically smooth kernels, are not optimal for oscillatory kernels. In a previous paper, we have shown that the method should nevertheless be used in the mechanical engineering community due to its still important data compression rate and its straightforward implementation compared to ${\mathscr{H}}^{2}$-matrix, or directional, approaches. Since in practice, not all materials are purely elastic, it is important to be able to consider visco-elastic cases. In this context, we study the effect of the introduction of a complex wavenumber on the accuracy and efficiency of ${\mathscr{H}}$-matrix-based fast methods for solving dense linear systems arising from the discretization of the elastodynamic (and Helmholtz) Green’s tensors. Interestingly, such configurations are also encountered in the context of the solution of transient purely elastic problems with the convolution quadrature method. Relying on the theory proposed in Börm et al. (IMA Journal of Numerical Analysis 12, 2020) for ${\mathscr{H}}^{2}$-matrices for Helmholtz problems, we study the influence of the introduction of damping on the data compression rate of standard ${\mathscr{H}}$-matrices. We propose an improvement of ${\mathscr{H}}$-matrix-based fast methods for this kind of configuration and illustrate how the introduction of a complex wavenumber can, as expected, improve further the efficiency of such methods. This work is complementary to the recent report (Börm et al., IMA Journal of Numerical Analysis 12, 2020). Here, in addition to addressing another physical problem, we consider standard ${\mathscr{H}}$-matrices, derive a simple criterion to introduce additional compression and we perform extensive numerical experiments.