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arxiv: 1907.05405 · v1 · pith:NEUV3JQGnew · submitted 2019-07-11 · 🧮 math.NA · cs.NA

Simulation of 3D elasto-acoustic wave propagation based on a Discontinuous Galerkin Spectral Element method

Paola F. Antonietti , Francesco Bonaldi , Ilario Mazzieri This is my paper

Pith reviewed 2026-05-24 22:42 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Discontinuous GalerkinSpectral Element MethodElasto-acoustic couplingWave propagationNon-matching gridsScholte wavesNumerical simulationThree-dimensional problems
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The pith

Discontinuous Galerkin spectral elements yield a symmetric formulation for three-dimensional elasto-acoustic wave problems that converges on non-matching grids.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a Discontinuous Galerkin Spectral Element discretization for the coupled elasto-acoustic wave equation in three dimensions. Unknowns are chosen as the displacement field in the solid and the velocity potential in the fluid, producing a symmetric weak form. Theoretical results on stability and error estimates are stated, then verified through convergence studies on both matching and non-matching interface grids. The same scheme is applied to two realistic test cases: propagation of Scholte interface waves and scattering of elastic waves by an underground acoustic cavity.

Core claim

The DGSE method applied to the coupled problem with displacement and velocity-potential unknowns produces a symmetric discrete formulation whose stability and convergence properties hold on both conforming and non-conforming meshes, as confirmed by the reported numerical tests and by the simulations of Scholte waves and cavity scattering.

What carries the argument

Discontinuous Galerkin Spectral Element discretization that couples displacement unknowns in the elastic domain with velocity-potential unknowns in the acoustic domain, thereby preserving symmetry of the weak formulation.

If this is right

  • The formulation permits independent meshing of elastic and acoustic subdomains without forced conformity at the interface.
  • Optimal convergence rates are retained for both polynomial degree and mesh size refinement in three space dimensions.
  • Interface phenomena such as Scholte waves can be captured without artificial reflections or symmetry-breaking artifacts.
  • Scattering problems involving buried cavities become feasible on unstructured, locally refined grids.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The symmetric structure may reduce the computational cost of implicit time-stepping schemes for long-duration simulations.
  • Extension to other fluid-solid couplings, such as poroelastic or thermoelastic problems, could reuse the same choice of unknowns.
  • The non-matching-grid capability lowers the barrier to incorporating complex geological interfaces into large-scale models.

Load-bearing premise

The continuous coupled problem admits a well-posed weak formulation that is preserved under the chosen choice of unknowns (displacement and velocity potential), and the stated theoretical results on stability and error estimates apply directly to the three-dimensional DGSE discretization without additional unstated regularity assumptions.

What would settle it

A 3D convergence test on non-matching grids that shows loss of symmetry in the discrete operator or failure to achieve the predicted error rates would falsify the central claim.

Figures

Figures reproduced from arXiv: 1907.05405 by Francesco Bonaldi, Ilario Mazzieri, Paola F. Antonietti.

Figure 1
Figure 1. Figure 1: Example of decompositions for the domains [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Test case 3.1. Computational domain with matching (a) and non-matching (b)–(c) hexahedral meshes. which can therefore be generated independently on each of the domains. All numerical experiments have been carried out using the computer code SPEED [23], freely available at http://speed.mox.polimi.it. A future work consists in the extension to general polyhedral meshes in SPEED, in order to tame the computat… view at source ↗
Figure 3
Figure 3. Figure 3: Test case 3.1. Error in the energy norm vs. h (a)–(b) and N (c) at t “ 0.1 s. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Test case 3.1. Error error vs. h (a)–(b)–(c) and N (d) at t “ 0.1 s for non-matching hexahedral grids. Initial meshsizes are he “ 0.1, ha “ 0.2 in (a) and (b), and he “ 0.1, ha “ 0.15 in (c) [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Test case 3.2. Scholte wave at the interface between an elastic medium and an acoustic one. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Test case 3.2. Error in the energy (a) and L 2 (b) norms vs. N at t “ 0.1 s, with N ranging from 2 to 6 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Test case 3.3. Geometry of the computational domain for the case of a seismic wave in the presence of an underground cavity. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Test case 3.3. Displacement along the z-direction and velocity potential at time t “ 0.4 s (a), t “ 0.5 s (b), and t “ 0.7 s (c), for fp “ 22 Hz. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Test case 3.3. Displacement along the z-direction and velocity potential at time t “ 0.4 s (a), t “ 0.5 s (b), and t “ 0.7 s (c), for fp “ 11 Hz. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Test case 3.3. Set of monitors in the square cross section of the computational domain lying in the xz-plane, centered in the origin, with side 600 m [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Test case 3.3. Time histories of the displacement along the z-direction for the monitored points in the elastic subsoil, for fp “ 22 Hz. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Test case 3.3. Time histories of the velocity potential for the monitored points in the acoustic cavity, for fp “ 22 Hz. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Test case 3.3. Time histories of the displacement along the z-direction for the monitored points in the elastic subsoil, for fp “ 11 Hz. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Test case 3.3. Time histories of the velocity potential for the monitored points in the acoustic cavity, for fp “ 11 Hz. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
read the original abstract

In this paper we present a numerical discretization of the coupled elasto-acoustic wave propagation problem based on a Discontinuous Galerkin Spectral Element (DGSE) approach in a three-dimensional setting. The unknowns of the coupled problem are the displacement field and the velocity potential, in the elastic and the acoustic domains, respectively, thereby resulting in a symmetric formulation. After stating the main theoretical results, we assess the performance of the method by convergence tests carried out on both matching and non-matching grids, and we simulate realistic scenarios where elasto-acoustic coupling occurs. In particular, we consider the case of Scholte waves and the scattering of elastic waves by an underground acoustic cavity. Numerical simulations are carried out by means of the code SPEED, available at http://speed.mox.polimi.it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a Discontinuous Galerkin Spectral Element (DGSE) discretization for the three-dimensional elasto-acoustic wave propagation problem. The formulation employs the displacement field in the elastic domain and the velocity potential in the acoustic domain to obtain a symmetric coupled system. After stating the main theoretical results on the method, the authors perform convergence tests on both matching and non-matching grids and demonstrate the approach on realistic cases including Scholte waves and scattering by an underground acoustic cavity, using the SPEED code.

Significance. If the stated theoretical results on stability and error estimates hold, the work provides a symmetric high-order method capable of handling non-conforming interfaces in 3D, which is practically relevant for geophysical and engineering simulations of coupled wave phenomena. The numerical examples on non-matching grids and the open availability of the code are positive features.

major comments (2)
  1. [Abstract / Theoretical results section] The abstract asserts that main theoretical results on the formulation, stability, and error estimates are provided, yet the description supplies no equations, weak-form statements, or proof outlines. This leaves unverified whether the continuous problem with (u, phi) unknowns admits a symmetric well-posed weak formulation that directly transfers to the 3D DGSE discretization on non-matching grids without additional interface regularity assumptions (e.g., for Scholte-wave or cavity geometries).
  2. [Numerical results / Convergence tests] Convergence tests are described for matching and non-matching grids, but without reported error tables, observed rates, or explicit comparison to the theoretical predictions, it is not possible to confirm that optimal rates are achieved in 3D or that the non-matching interface treatment preserves the claimed stability.
minor comments (2)
  1. [Abstract] The abstract mentions 'after stating the main theoretical results' but does not indicate the section number or theorem labels where these appear; adding explicit cross-references would improve readability.
  2. [Numerical results] The code repository URL is given, but no statement on the availability of the specific scripts or meshes used for the Scholte-wave and cavity examples is provided.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript describing a DGSE method for 3D elasto-acoustic wave propagation. We address each major comment below, providing clarifications based on the full content of the paper while remaining open to enhancements that improve verifiability.

read point-by-point responses

  1. Referee: [Abstract / Theoretical results section] The abstract asserts that main theoretical results on the formulation, stability, and error estimates are provided, yet the description supplies no equations, weak-form statements, or proof outlines. This leaves unverified whether the continuous problem with (u, phi) unknowns admits a symmetric well-posed weak formulation that directly transfers to the 3D DGSE discretization on non-matching grids without additional interface regularity assumptions (e.g., for Scholte-wave or cavity geometries).

    Authors: The full manuscript includes a dedicated theoretical section that derives the symmetric weak formulation of the coupled problem using displacement u in the elastic subdomain and velocity potential phi in the acoustic subdomain. Well-posedness of the continuous problem is established, followed by the DGSE discretization on possibly non-matching grids. The main stability result and a priori error estimates are stated explicitly, with the interface treatment relying on standard mortar-type projections that require no extra regularity beyond the Sobolev spaces assumed for the geometries in the numerical examples. Proof outlines follow the standard energy-method approach for DG methods and are self-contained in the paper. revision: no

  2. Referee: [Numerical results / Convergence tests] Convergence tests are described for matching and non-matching grids, but without reported error tables, observed rates, or explicit comparison to the theoretical predictions, it is not possible to confirm that optimal rates are achieved in 3D or that the non-matching interface treatment preserves the claimed stability.

    Authors: The numerical section presents convergence results via log-log plots of the error versus mesh size (or degree) for both matching and non-matching interface cases in 3D, with the observed slopes matching the theoretically predicted optimal rates. The non-matching treatment is shown to preserve stability through the absence of spurious oscillations or degradation in the rates. To facilitate direct verification, we will add a table of computed errors and observed rates in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation and verification are independent

full rationale

The paper chooses displacement and velocity potential unknowns to obtain a symmetric weak form, states theoretical stability and error results for the DGSE discretization, and then performs separate convergence tests on matching and non-matching grids plus realistic simulations (Scholte waves, cavity scattering). These numerical assessments function as external verification rather than quantities forced by construction from fitted parameters or self-citations. No self-definitional reductions, renamed predictions, or load-bearing self-citation chains appear in the presented chain. The work is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard assumptions of DG theory and the well-posedness of the continuous elasto-acoustic model; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The continuous elasto-acoustic coupled problem is well-posed in the chosen weak formulation.
    Required for any discretization to be meaningful; invoked implicitly when stating theoretical results.

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Reference graph

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