High-order finite element methods for seismic wave propagation (original) (raw)
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Purely numerical methods based on finite-element approximation of the acoustic or elastic wave equation are becoming increasingly popular for the generation of synthetic seismograms. We present formulas for the grid dispersion and stability criteria for some popular finite-element methods ͑FEM͒ for wave propagation, namely, classical and spectral FEM. We develop an approach based on a generalized eigenvalue formulation to analyze the dispersive behavior of these FEMs for acoustic or elastic wave propagation that overcomes difficulties caused by irregular node spacing within the element and the use of high-order polynomials, as is the case for spectral FEM.Analysis reveals that for spectral FEM of order four or greater, dispersion is less than 0.2% at four to five nodes per wavelength, and dispersion is not angle dependent. New results can be compared with grid-dispersion results of some classical finite-difference methods ͑FDM͒ used for acoustic or elastic wave propagation. Analysis reveals that FDM and classical FEM require a larger sampling ratio than a spectral FEM to obtain results with the same degree of accuracy. The staggered-grid FDM is an efficient scheme, but the dispersion is angle dependent with larger values along the grid axes. On the other hand, spectral FEM of order four or greater is isotropic with small dispersion, making it attractive for simulations with long propagation times.
Seg Technical Program Expanded Abstracts, 2007
Purely numerical methods based on finite-element approximation of the acoustic or elastic wave equation are becoming increasingly popular for the generation of synthetic seismograms. We present formulas for the grid dispersion and stability criteria for some popular finite-element methods ͑FEM͒ for wave propagation, namely, classical and spectral FEM. We develop an approach based on a generalized eigenvalue formulation to analyze the dispersive behavior of these FEMs for acoustic or elastic wave propagation that overcomes difficulties caused by irregular node spacing within the element and the use of high-order polynomials, as is the case for spectral FEM.Analysis reveals that for spectral FEM of order four or greater, dispersion is less than 0.2% at four to five nodes per wavelength, and dispersion is not angle dependent. New results can be compared with grid-dispersion results of some classical finite-difference methods ͑FDM͒ used for acoustic or elastic wave propagation. Analysis reveals that FDM and classical FEM require a larger sampling ratio than a spectral FEM to obtain results with the same degree of accuracy. The staggered-grid FDM is an efficient scheme, but the dispersion is angle dependent with larger values along the grid axes. On the other hand, spectral FEM of order four or greater is isotropic with small dispersion, making it attractive for simulations with long propagation times.
2014
A popular approach is the Spectral Element Method (SEM) (Chaljub et al, 2006) which is high-order accurate (low dispersion error), very flexible to parallelization and computationally attractive due to efficient sum factorization technique and diagonal mass matrix. However SEMs suffer from lack of flexibility in handling complex geometry and multi-physics wave propagation. High-order Discontinuous Galerkin Methods (DGMs), i.e. Dumbser et al (2006), Etienne et al. (2010), Wilcox et al (2010), are recent alternatives that can handle complex geometry, space-and-time adaptativity, and allow efficient multi-physics wave coupling at interfaces. However, DGMs are more memory demanding and less computationally attractive than SEMs, especially when explicit time stepping is used.
Numerical modeling of seismic waves by discontinuous spectral element methods
ESAIM: Proceedings and Surveys, 2018
We present a comprehensive review of Discontinuous Galerkin Spectral Element (DGSE) methods on hybrid hexahedral/tetrahedral grids for the numerical modeling of the ground motion induced by large earthquakes. DGSE methods combine the exibility of discontinuous Galerkin meth-ods to patch together, through a domain decomposition paradigm, Spectral Element blocks where high-order polynomials are used for the space discretization. This approach allows local adaptivity on discretization parameters, thus improving the quality of the solution without affecting the compu-tational costs. The theoretical properties of the semidiscrete formulation are also revised, including well-posedness, stability and error estimates. A discussion on the dissipation, dispersion and stability properties of the fully-discrete (in space and time) formulation is also presented. Here space dis-cretization is obtained based on employing the leap-frog time marching scheme. The capabilities of the present approach ...
2008
Various numerical methods have been proposed and used to investigate wave propagation in realistic earth media. Recently an innovative numerical method, known as the Spectral Element Method (SEM), has been developed and used in connection with wave propagation problems in 3D elastic media (Komatitsch and Tromp, 1999). The SEM combines the flexibility of a finite element method with the accuracy of a spectral method and it is not cumbersome in dealing with non-flat free surface and spatially variable anelastic attenuation. The SEM is a highly accurate numerical method that has its origins in computational fluid dynamics. One uses a weak formulation of the equations of motion, which are solved on a mesh of hexahedral elements that is adapted to the free surface and to the main internal discontinuities of the model. The wavefield on the elements is discretized using high-degree Lagrange interpolants, and integration over an element is accomplished based upon the Gauss-Lobatto-Legendre ...
The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion
Recently, there has been an increased interest in applying the discontinuous Galerkin method (DGM) to wave propagation. In this work, we investigate the applicability of the interior penalty DGM to elastic wave propagation by analysing it's grid dispersion properties, with particular attention to the effect that different basis functions have on the numerical dispersion. We consider different types of basis functions that naturally yield a diagonal mass matrix. This is relevant to seismology because a diagonal mass matrix is tantamount to an explicit and efficient time marching scheme. We find that the Legendre basis functions that are traditionally used in the DGM introduce numerical dispersion and anisotropy. Furthermore, we find that using Lagrange basis functions along with the Gauss nodes has attractive advantages for numerical wave propagation.
Geophysical Journal International, 2008
Recently, there has been an increased interest in applying the discontinuous Galerkin method (DGM) to wave propagation. In this work, we investigate the applicability of the interior penalty DGM to elastic wave propagation by analysing it's grid dispersion properties, with particular attention to the effect that different basis functions have on the numerical dispersion. We consider different types of basis functions that naturally yield a diagonal mass matrix. This is relevant to seismology because a diagonal mass matrix is tantamount to an explicit and efficient time marching scheme. We find that the Legendre basis functions that are traditionally used in the DGM introduce numerical dispersion and anisotropy. Furthermore, we find that using Lagrange basis functions along with the Gauss nodes has attractive advantages for numerical wave propagation.
Geophysical Journal International, 2000
Numerical modelling of seismic motion in sedimentary basins often has to account for P-wave to S-wave speed ratios as large as five and even larger, mainly in sediments below groundwater level. Therefore, we analyse seven schemes for their behaviour with a varying P-wave to S-wave speed ratio. Four finite-difference (FD) schemes include (1) displacement conventional-grid, (2) displacement-stress partly-staggered-grid, (3) displacement-stress staggered-grid and (4) velocity-stress staggered-grid schemes. Three displacement finite-element schemes differ in integration: (1) Lobatto four-point, (2) Gauss four-point and Gauss one-point. To compare schemes at the most fundamental level, and identify basic aspects responsible for their behaviours with the varying speed ratio, we analyse 2-D second-order schemes assuming an elastic homogeneous isotropic medium and a uniform grid. We compare structures of the schemes and applied FD approximations. We define (full) local errors in amplitude and polarization in one time step, and normalize them for a unit time. We present results of extensive numerical calculations for wide ranges of values of the speed ratio and a spatial sampling ratio, and the entire range of directions of propagation with respect to the spatial grid.
Journal of Computational Physics
The mass-lumped method avoids the cost of inverting the mass matrix and simultaneously maintains spatial accuracy by adopting additional interior integration points, known as cubature points. To date, such points are only known analytically in tensor domains, such as quadrilateral or hexahedral elements. Thus, the diagonal-mass-matrix spectral element method (SEM) in non-tensor domains always relies on numerically computed interpolation points or quadrature points. However, only the cubature points for degrees 1 to 6 are known, which is the reason that we have developed a p-norm-based optimization algorithm to obtain higher-order cubature points. In this way, we obtain and tabulate new cubature points with all positive integration weights for degrees 7 to 9. The dispersion analysis illustrates that the dispersion relation determined from the new optimized cubature points is comparable to that of the mass and stiffness matrices obtained by exact integration. Simultaneously, the Lebesgue constant for the new optimized cubature points indicates its surprisingly good interpolation properties. As a result, such points provide both good interpolation properties and integration accuracy. The Courant-Friedrichs-Lewy (CFL) numbers are tabulated for the conventional Fekete-based triangular spectral element (TSEM), the TSEM with exact integration, and the optimized cubature-based TSEM (OTSEM). A complementary study demonstrates the spectral convergence of the OTSEM. A numerical example conducted on a half-space model demonstrates that the OTSEM improves the accuracy by approximately one order of magnitude compared to the conventional Fekete-based TSEM. In particular, the accuracy of the 7th-order OTSEM is even higher than that of the 14th-order Fekete-based TSEM. Furthermore, the OTSEM produces a result that can compete in accuracy with the quadrilateral SEM (QSEM). The high accuracy of the OTSEM is also tested with a non-flat topography model. In terms of computational efficiency, the OTSEM is more efficient than the Fekete-based TSEM, although it is slightly costlier than the QSEM when a comparable numerical accuracy is required.