Optimal blended spectral-element operators for acoustic wave modeling

Abstract

Spectral-element methods, based on high-order polynomials, are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media. However, to retain computational efficiency, very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided. In the present work, we devise an approach for reducing such errors by considering modified discrete wave operators. We analyze consistent and lumped operators together with blended operators (weighted averages of consistent and lumped operators). Furthermore, using the operator-blending approach and a novel dispersion analysis method, we develop optimal spectral-element operators that have increased numerical accuracy, without resorting to very high order operators. The new operators are faster and computationally more efficient than consistent operators. Our approach is based on the tensor product decomposition of the element matrices into 1D factors. We apply standard lumping to the factor associated with the 1D mass matrix. A simplified numerical dispersion analysis of arbitrary order and spatial dimension provides a practical criterion for weighting consistent and lumped matrices. The approach is general and is valid for solving both the time-dependent and the stationary (Helmholtz) wave equations.

Finite Element Modeling of Linear Elastodynamics Problems With Explicit Time-Integration Methods and Linear Elements With Reduced Dispersion. Comparative Study of Different Finite Element Techniques Used for Elastodynamics.

“…(5) is used for the analysis of the numerical dispersion of the finite element formulations with the averaged mass matrix and modified integration rule techniques. In order to decrease the numerical dispersion of finite element results, we consider the following two possibilities for the calculation of the mass and stiffness matrices: the mass matrix M M M is calculated as a weighted average of the consistent M M M cons and lumped D D D mass matrices with the weighting factor γ (similar to that used in [26,27,30])…”

“…Due to the space discretization, the exact solution to Eq. (1) contains the numerical dispersion error; e.g., see [7,9,11,26,27,28,30,31,32,14,12] and others. The space discretization error can be decreased by the use of mesh refinement.…”

“…(1) as a weighted average of the consistent and lumped mass matrices M M M ; see [26,27,28,30] and others. For the 1-D case and linear finite elements, this approach reduces the error in the wave velocity for harmonic waves from the second order to the fourth order of accuracy.…”

Optimal Reduction of Numerical Dispersion for Wave Propagation Problems. Application to Isogeometric Elements.

“…(2) contains the numerical dispersion error. Usually the analysis of the numerical dispersion error and its improvement for many space-discretization techniques such as the finite elements, spectral elements, isogeometric elements and others starts with the analysis and modifications of the elemental mass and stiffness matrices; see [10,1,2,7,8,15,16,17,18,22,11,14,20,21]. For example, one simple and effective finite-element technique for acoustic and elastic wave propagation problems is based on the calculation of the mass matrix M M M in Eq.…”

“…For example, one simple and effective finite-element technique for acoustic and elastic wave propagation problems is based on the calculation of the mass matrix M M M in Eq. (2) as a weighted average of the consistent and lumped mass matrices; see [15,16,17,18] and others. For the 1-D case and the linear finite elements, this approach reduces the error in the wave velocity for harmonic waves from the second order to the fourth order of accuracy.…”