Evaluation of the Spectral Finite Element Method With the Theory of Phononic Crystals

TL;DR

This study compares classical and spectral finite element methods in simulating elastodynamic problems, showing spectral methods outperform classical ones at high frequencies in capturing dispersive behavior, with both methods aligning at low frequencies.

Contribution

The paper demonstrates the superior accuracy of spectral finite element methods over classical methods in high-frequency dispersive simulations, using Floquet-Bloch analysis for various materials.

Findings

Spectral methods match analytical dispersion curves at high frequencies.

Classical methods perform well at low frequencies, matching analytical results.

Lumping process in classical FEM affects dispersion predictions, especially at high frequencies.

Abstract

We evaluated the performance of the classical and spectral finite element method in the simulation of elastodynamic problems. We used as a quality measure their ability to capture the actual dispersive behavior of the material. Four different materials are studied: a homogeneous non-dispersive material, a bilayer material, and composite materials consisting of an aluminum matrix and brass inclusions or voids. To obtain the dispersion properties, spatial periodicity is assumed so the analysis is conducted using Floquet-Bloch principles. The effects in the dispersion properties of the lumping process for the mass matrices resulting from the classical finite element method are also investigated, since that is a common practice when the problem is solved with explicit time marching schemes. At high frequencies the predictions with the spectral technique exactly match the analytical dispersion curves, while the classical method does not. This occurs even at the same computational demands. At low frequencies however, the results from both the classical (consistent or mass-lumped) and spectral finite element coincide with the analytically determined curves. Surprisingly, at low frequencies even the results obtained with the artificial diagonal mass matrix from the classical technique exactly match the analytic dispersion curves.