The time-harmonic solution of the linear elastic wave equation is needed for a variety of applications. The typical procedure for solving the time-harmonic elastic wave equation leads to difficulties solving large-scale indefinite linear systems. To avoid these difficulties, we consider the original time dependent equation with a method based on an exact controllability formulation. The main idea of this approach is to find initial conditions such that after one time-period, the solution and its time derivative coincide with the initial conditions. The wave equation is discretized in the space domain with spectral elements. The degrees of freedom associated with the basis functions are situated at the Gauss-Lobatto quadrature points of the elements, and the Gauss-Lobatto quadrature rule is used so that the mass matrix becomes diagonal. This method is combined with the second-order central finite difference or the fourth-order Runge-Kutta time discretization. As a consequence of these choices, only matrix-vector products are needed in time dependent simulation. This makes the controllability method computationally efficient. © 2008 Elsevier Inc. All rights reserved.
Mönkölä, S., Heikkola, E., Pennanen, A., & Rossi, T. (2008). Time-harmonic elasticity with controllability and higher-order discretization methods. Journal of Computational Physics, 227(11), 5513–5534. https://doi.org/10.1016/j.jcp.2008.01.054