Efficiency of the spectral element method with very high polynomial degree to solve the elastic wave equation

Lyu, Chao; Capdeville, Yann; Zhao, Liang

doi:10.1190/geo2019-0087.1

Cited by 17 publications

(61 citation statements)

References 32 publications

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“…Therefore, α(P ) is calculated based on the square root of the number of DoF (n DoF ) per number of elements (n e ) in the mesh, n DoF /n e . When this metric is applied to SEM quadrilateral elements, it gives results that match the values reported in Lyu et al (2020).…”

Section: Waveform Adapted Triangular Meshessupporting

confidence: 59%

spyro: a Firedrake-based wave propagation and full waveform inversion finite element solver

Roberts¹,

Olender²,

Franceschini³

et al. 2021

Preprint

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Abstract. In this article, we introduce spyro, a software stack to solve acoustic wave propagation in heterogeneous domains and perform full waveform inversion (FWI) employing the finite element framework from Firedrake, a high-level Python package for the automated solution of partial differential equations using the finite element method. The capability of the software is demonstrated by using a continuous Galerkin approach to perform FWI for seismic velocity model building, considering realistic geophysics examples. A time-domain FWI approach is detailed that uses meshes composed of variably sized triangular elements to discretize the domain. To resolve both the forward and adjoint-state equations, and to calculate a mesh-independent gradient associated with the FWI process, a fully-explicit, variable higher-order (up to degree k = 5 in 2D and k = 3 in 3D) mass lumping method is used. We show that, by adapting the triangular elements to the expected peak source frequency and properties of the wavefield (e.g., local P-wavespeed) and by leveraging higher-order basis functions, the number of degrees-of-freedom necessary to discretize the domain can be reduced. Results from wave simulations and FWIs in both 2D and 3D highlight our developments and demonstrate the benefits and challenges with using triangular meshes adapted to the material properties.

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Section: Waveform Adapted Triangular Meshessupporting

confidence: 59%

spyro: a Firedrake-based wave propagation and full waveform inversion finite element solver

Roberts¹,

Olender²,

Franceschini³

et al. 2021

Preprint

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show abstract

“…This means that the SEM can flexibly reduce the spatial-dispersion error by tuning the number of elements and the internal polynomial degree N in each tensorial direction for each element. Lyu et al (2020) validates the high efficiency of using the very high degree SEM (N = ∼ 24) both for the required memory and the computation time, which makes SEM with very high degree attractive and competitive for solving the wave equation.…”

Section: Introductionsupporting

confidence: 58%

“…The errors of numerical wave simulation come from the spatial-and time-dispersion errors, and these two kinds of dispersion errors have been proven not relevant (Stork, 2013;Koene et al, 2018;Lyu et al, 2020). That means they accumulate independently during the numerical simulation.…”

Section: Introductionmentioning

confidence: 99%

“…For a model with sufficiently accurate spatial meshing, a converged waveform without spatial and time-dispersion errors can be obtained by using an extremely small ∆t but with low efficiency. While with the high computational efficiency, the time-dispersion error of using a relatively large time step ∆t ≤ ∆t CFL would also contaminate the numerical waveforms for the finite difference (FD) method (Gao et al, 2016) and as well as SEM especially with N = 4 (Lyu et al, 2020) and even worse for long duration simulations owing to its accumulation over time. Stork (2013) demonstrate that the time-dispersion error can be removed by time-varying filters and interpolation after the wave modelling.…”

Section: Introductionmentioning

confidence: 99%

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Removing the Courant-Friedrichs-Lewy stability criterion of the explicit time-domain very high degree spectral-element method with eigenvalue perturbation

et al. 2021

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The explicit time-domain spectral-element method (SEM) for synthesizing seismograms hasgained tremendous credibility within the seismological community at all scales. Althoughthe recent introduction of non-periodic homogenization has addressed the spatial meshing difficulty of the mechanical discontinuities, the Courant-Friedrichs-Lewy (CFL) stability criterionstrictly constrains the maximum time step, which still puts a great burden on the numericalsimulation. In the explicit time-domain SEM, the source of instability of using a time stepbeyond the stability criterion is that some unstable eigenvalues of the updated matrix are largerthan what can be accurately simulated. We succeed in removing the CFL stability condition inthe explicit time-domain SEM by combining the forward time dispersion-transform method,the eigenvalue perturbation, and the inverse time dispersion-transform method. Our theoretical analyses and numerical experiments both in the homogeneous, moderate and strong heterogeneous models, show that this combination can precisely simulate waveforms with timesteps dozens of the CFL limit even towards the Nyquist limit especially for the efficient veryhigh degree SEM, which abundantly saves the iteration times without suffering from the time-dispersion error. It demonstrates a potential application prospect in some situations such as thefull waveform inversion which requires multiple numerical simulations for the same model.

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“…Nevertheless, if one needs to model the nearsource field, especially for complex-fault systems, homogenized sources present a clear interest and make this possible based on a mesh that can ignore the fault complexity. It can also useful if one wishes to take advantage of large very high degree SEM elements, for which the probability to have a receiver in the element of the source is significant (Lyu et al 2020). For strong-form numerical solvers such as FD, this work is interesting even for the far-field: it gives a rigorous solution to obtain a distributed force map for the source, including in the case of a complex heterogeneous medium around the source.…”

Section: Discussionmentioning

confidence: 99%

Homogenization of seismic point and extended sources

Capdeville

2021

Geophysical Journal International

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SUMMARY Seismic sources are mostly modelled as point sources: moment tensors associated with the gradient of a Dirac distribution. Such sources contain an infinite range of scales and induce a discontinuity in the displacement wavefield. This makes the near-source wavefield expensive to model and the event location complex to invert, in particular for large events for which many point sources are required. In this work, we propose to apply the non-periodic two-scale homogenization method to the wave equation source term for both force and couple-sources. We show it is possible to replace the Dirac point source with a smooth source term, valid in a given seismic signal frequency band. The discontinuous wavefield near-source wavefield can be recovered using a corrector that needs to be added to the solution obtained solving the wave equation with the smooth source term. We show that, compared to classical applications of the two-scale homogenization method to heterogeneous media, the source term homogenization has some interesting particularities: for couple-sources, the leading term of the homogenization asymptotic expansion is dependent on the fine spatial scale, depending on the source type, only one or two first terms of the expansion are non-zero and there is no periodic case equivalent (the source term cannot be made spatially periodic). For heterogeneous media, two options are developed. In the first one, only the source is homogenized while the medium itself remains the same, including its discontinuities. In the second one, both the source and the medium are homogenized successively: first the medium and then the source. We present a set of tests in 1-D and 2-D, showing accurate results both in the far-source and near-source wavefields, before discussing the interest of this work in the forward and inverse problem contexts.

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Efficiency of the spectral element method with very high polynomial degree to solve the elastic wave equation

Cited by 17 publications

References 32 publications

spyro: a Firedrake-based wave propagation and full waveform inversion finite element solver

spyro: a Firedrake-based wave propagation and full waveform inversion finite element solver

Removing the Courant-Friedrichs-Lewy stability criterion of the explicit time-domain very high degree spectral-element method with eigenvalue perturbation

Homogenization of seismic point and extended sources

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