Dispersion Properties of Explicit Finite Element Methods for Wave Propagation Modelling on Tetrahedral Meshes

Abstract

We analyse the dispersion properties of two types of explicit finite element methods for modelling acoustic and elastic wave propagation on tetrahedral meshes, namely mass-lumped finite element methods and symmetric interior penalty discontinuous Galerkin methods, both combined with a suitable Lax-Wendroff time integration scheme. The dispersion properties are obtained semi-analytically using standard Fourier analysis. Based on the dispersion analysis, we give an indication of which method is the most efficient for a given accuracy, how many elements per wavelength are required for a given accuracy, and how sensitive the accuracy of the method is to poorly shaped elements.

New Higher-Order Mass-Lumped Tetrahedral Elements for Wave Propagation Modelling

“…For the analysis we will follow the same procedure as in [9]. We consider a homogeneous medium with ρ, c = 1 and consider physical plane waves of the form u = eî (κ·x−ωt) , whereî := √ −1 is the imaginary number, κ is the wave vector, and ω is the angular velocity.…”

“…are the eigenvalues of σ(S (κ) ) [9]. The numerical wave propagation speed is given by c For our dispersion analysis, we will consider a congruent, nearly-regular, equifacial mesh, known as the tetragonal disphenoid honeycomb.…”

“…• n vec = n 0 Details on the dispersion analysis and how the quantities listed above are computed can be found in [9].…”

“…We will also compare the mass-lumped methods with the symmetric interior penalty discontinuous Galerkin (SIPDG) method, introduced and analyzed in [Grote et al 2006]. For the penalty term, we use the lower bound derived in [10], since it was shown in [9] that this penalty term results in a significantly more efficient scheme than the penalty terms based on the more commonly used trace inequality of [20]. The quantities for the computational cost are computed in the same way as for the mass-lumped method, but now n 0 denotes the number of basis functions in Ω 0 and n mat is computed as n mat = through a face, including e itself.…”

Maximizing the accuracy of finite element simulation of elastic wave propagation in polycrystals

“…The monograph by Bathe [41] provides an excellent description and evaluation of the basic methodologies of the main schemes in use for time domain simulation. A variety of studies of performance of these and their variants, limiting to relevance to the method used in the present work, can be found in, e.g., [34,[42][43][44][45][46][47][48][49][50]].…”