On Multiple Modes of Propagation of High-Order Finite Element Methods for the Acoustic Wave Equation
Earlier analyses of numerical dispersion of high-order finite element methods (HO-FEM) for acoustic and elastic wave propagation pointed out the presence of multiple modes of propagation. The number of modes increases with the polynomial degree of the finite element space, and since they were regarded as numerical artifacts, the use of HO-FEM was discouraged on wave propagation problems. Later on, alternative techniques showed that numerical dispersion decreases with the polynomial degree, and were supported by the success of spectral element methods on seismic wave propagation. This work concerns the interpretation of multiple propagation modes, which are solutions of an eigenvalue problem arising from the HO-FEM discretization of the wave equation as approximations to an eigenvalue problem associated with the continuous wave equation. By considering a continuous version of the standard periodic plane wave whose amplitude depends on the element grid, there are multiple combinations of the amplitude coefficients that yield exact solutions to the acoustic wave equation. Hence, modes regarded as non-physical can be associated with feasible propagation modes. Under this point of view, one can separately analyze each propagation mode or focus on the acoustical (constant amplitude) mode.
The author is supported by CNPq under grant 306083/2014-0 and is a collaborator of INCT-GP.
- 4.T. Belytschko, R. Mullen, On dispersive properties of finite element solutions, in Modern Problems in Elastic Wave Propagation, ed. by J. Miklowitz, J. Achenbach (Wiley, New York, NY, 1978), pp. 67–82Google Scholar
- 5.L. Brillouin, Wave Propagation in Periodic Structures. Electric Filters and Crystal Lattices, 2nd edn. (Dover publications, New York, NY, 1953)Google Scholar
- 7.G. Cohen, P. Joly, N. Tordjman, Eléments finis d’ordre élevé avec condensation de masse pour l’équation des ondes en dimension 1. Rapport de recherche RR-2323 (1994)Google Scholar
- 14.K. Marfurt, Appendix - analysis of higher order finite-element methods, in Numerical Modeling of Seismic Wave Propagation, ed. by K. Kelly, K. Marfurt. Geophysics Reprint Series, vol. 13 (Society of Exploration Geophysicists, Tulsa, OK, 1990), pp. 516–520Google Scholar