Generalized Galerkin approximations of elastic waves with absorbing boundary conditions

Quarteroni, Alfio; Tagliani, Aldo; Zampieri, E.

doi:10.1016/s0045-7825(98)00022-x

Cited by 57 publications

(30 citation statements)

References 14 publications

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“…Fewer works have studied spectral elements for wave propagation problems. We recall here the paper on explicit Fourier-Legendre or full-Legendre domain decomposition methods by Faccioli et al [11], where the focus is on the numerical validation of the method on significant tests in geophysics rather than on the theoretical analysis; the works on monodomain spectral methods for acoustic waves by Maggio and Quarteroni [24] and for elastic waves by Zampieri and Tagliani [33] and Quarteroni et al [26]; the paper by Casadei et al [8] studying a mortar coupling between spectral and finite elements for elastodynamic problems on complex geometries; the work by Hesthaven and Warburton [18] proposing a very efficient unstructured spectral element discretization based on a discontinuous Galerkin formulation of Maxwell equations and other wave problems. We refer to [16] for a review of previous work on spectral methods for hyperbolic problems.…”

Section: Introductionmentioning

confidence: 99%

An explicit second order spectral element method for acoustic waves

Zampieri

Pavarino

2006

Adv Comput Math

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The acoustic wave equation is here discretized by conforming spectral elements in space and by the second order leap-frog method in time. For simplicity, homogeneous boundary conditions are considered. A stability analysis of the resulting method is presented, providing an upper bound for the allowed time step that is proportional to the size of the elements and inversely proportional to the square of their polynomial degree. A convergence analysis is also presented, showing that the convergence error decreases when the time step or the size of the elements decrease or when the polynomial degree increases. Several numerical results illustrating these results are presented.

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Section: Introductionmentioning

confidence: 99%

An explicit second order spectral element method for acoustic waves

Zampieri

Pavarino

2006

Adv Comput Math

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“…In an isotropic medium, this may be accomplished based on a paraxial equation (Clayton and Engquist, 1977;Quarteroni et al, 1998):…”

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confidence: 99%

Finite-Frequency Kernels Based on Adjoint Methods

Liu

Tromp

2006

Bulletin of the Seismological Society of America

270

205

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Complications due to artificial absorbing boundaries for regional simulations as well as finite sources are accommodated. We construct the 3D finite-frequency "banana-doughnut" kernel K m by simultaneously computing the so-called "adjoint" wave field forward in time and reconstructing the regular wave field backward in time. The adjoint wave field is produced by using timereversed signals at the receivers as fictitious, simultaneous sources, while the regular wave field is reconstructed on the fly by propagating the last frame of the wave field, saved by a previous forward simulation, backward in time. The approach is based on the spectral-element method, and only two simulations are needed to produce the 3D finite-frequency sensitivity kernels. The method is applied to 1D and 3D regional models. Various 3D shear-and compressional-wave sensitivity kernels are presented for different regional body-and surface-wave arrivals in the seismograms. These kernels illustrate the sensitivity of the observations to the structural parameters and form the basis of fully 3D tomographic inversions.

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“…The fluid domain is bounded by the boundary Γ f = Γ 0f Γ ef Γ i , and Γ s = Γ 0s Γ es Γ i constitutes the boundary for the structure domain Ω s . On the boundaries Γ 0f and Γ 0s we use the Dirichlet boundary conditions, whereas on the artificial boundaries Γ ef and Γ es we impose the conventional first order absorbing boundary conditions [29,30]. On the interface Γ i between fluid and structure domains, we assume the continuity of normal components of displacements and traction forces.…”

Section: Coupled Elastic-acoustic Wave Equationsmentioning

confidence: 99%

“…The stress tensor is expressed as σ( T , the identity matrix I, and the Lamé parameters µ and λ. The symmetric positive definite matrix B is defined as [29,30] …”

Section: Coupled Elastic-acoustic Wave Equationsmentioning

confidence: 99%

An optimization-based approach for solving a time-harmonic multiphysical wave problem with higher-order schemes

Mönkölä¹

2013

Journal of Computational Physics

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An optimization-based approach for solving a time-harmonic multiphysical wave problem with higher-order schemes Mönkölä, Sanna Mönkölä, S. (2013). An optimization-based approach for solving a time-harmonic multiphysical wave problem with higher-order schemes. Journal of Computational Physics , 242 (1 June), 439-459. doi:10.1016/j.jcp.2013.02 AbstractThis study considers developing numerical solution techniques for the computer simulations of time-harmonic fluid-structure interaction between acoustic and elastic waves. The focus is on the efficiency of an iterative solution method based on a controllability approach and spectral elements. We concentrate on the model, in which the acoustic waves in the fluid domain are modeled by using the velocity potential and the elastic waves in the structure domain are modeled by using displacement.Traditionally, the complex-valued time-harmonic equations are used for solving the time-harmonic problems. Instead of that, we focus on finding periodic solutions without solving the time-harmonic problems directly. The time-dependent equations can be simulated with respect to time until a time-harmonic solution is reached, but the approach suffers from poor convergence. To overcome this challenge, we follow the approach first suggested and developed for the acoustic wave equations by Bristeau, Glowinski, and Périaux. Thus, we accelerate the convergence rate by employing a controllability method. The problem is formulated as a least-squares optimization problem, which is solved with the conjugate gradient (CG) algorithm. Computation of the gradient of the functional is done directly for the discretized problem. A graph-based multigrid method is used for preconditioning the CG algorithm.

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Generalized Galerkin approximations of elastic waves with absorbing boundary conditions

Cited by 57 publications

References 14 publications

An explicit second order spectral element method for acoustic waves

An explicit second order spectral element method for acoustic waves

Finite-Frequency Kernels Based on Adjoint Methods

An optimization-based approach for solving a time-harmonic multiphysical wave problem with higher-order schemes

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