Free and smooth boundaries in 2-D finite-difference schemes for transient elastic waves

A method is proposed for accurately describing arbitrary-shaped free boundaries in single-grid finite-difference schemes for elastodynamics, in a time-domain velocity-stress framework. The basic idea is as follows: fictitious values of the solution are built in vacuum, and injected into the numerical integration scheme near boundaries. The most original feature of this method is the way in which these fictitious values are calculated. They are based on boundary conditions and compatibility conditions satisfied by the successive spatial derivatives of the solution, up to a given order that depends on the spatial accuracy of the integration scheme adopted. Since the work is mostly done during the preprocessing step, the extra computational cost is negligible. Stress-free conditions can be designed at any arbitrary order without any numerical instability, as numerically checked. Using 10 grid nodes per minimal S-wavelength with a propagation distance of 50 wavelengths yields highly accurate results. With 5 grid nodes per minimal S-wavelength, the solution is less accurate but still acceptable. A subcell resolution of the boundary inside the Cartesian meshing is obtained, and the spurious diffractions induced by staircase descriptions of boundaries are avoided. Contrary to what occurs with the vacuum method, the quality of the numerical solution obtained with this method is almost independent of the angle between the free boundary and the Cartesian meshing.Comment: accepted and to be published in Geophys. J. In

An axisymmetric time-domain spectral-element method for full-wave simulations: Application to ocean acoustics

The numerical simulation of acoustic waves in complex 3D media is a key topic in many branches of science, from exploration geophysics to non-destructive testing and medical imaging. With the drastic increase in computing capabilities this field has dramatically grown in the last twenty years. However many 3D computations, especially at high frequency and/or long range, are still far beyond current reach and force researchers to resort to approximations, for example by working in 2D (plane strain) or by using a paraxial approximation. This article presents and validates a numerical technique based on an axisymmetric formulation of a spectral finite-element method in the time domain for heterogeneous fluid-solid media. Taking advantage of axisymmetry enables the study of relevant 3D configurations at a very moderate computational cost. The axisymmetric spectral-element formulation is first introduced, and validation tests are then performed. A typical application of interest in ocean acoustics showing upslope propagation above a dipping viscoelastic ocean bottom is then presented. The method correctly models backscattered waves and explains the transmission losses discrepancies pointed out in Jensen et al. (2007). Finally, a realistic application to a double seamount problem is considered.Comment: Added a reference, and fixed a typo (cylindrical versus spherical

A 2.5D BEM-FEM using a spectral approach to study scattered waves in fluid–solid interaction problems

42nd International Conference on Boundary Elements and other Mesh Reduction Methods, BEM/MRM 2019; ITeCons-University of Coimbra, Coimbra; Portugal; 2 July 2019 through 4 July 2019. - Publicado en WIT Transactions on Engineering Sciences, Volume 126, 2019, Pages 111-123This work presents a two-and-a-half dimensional (2.5D) spectral formulation based on the finite element method (FEM) and the boundary element method (BEM) to study wave propagation in acoustic and elastic waveguides. The analysis involved superposing two dimensional (2D) problems with different longitudinal wavenumbers. A spectral finite element (SFEM) is proposed to represent waveguides in solids with arbitrary cross-section. Moreover, the BEM is extended to its spectral formulation (SBEM) to study unbounded fluid media and acoustic enclosures. Both approaches use Lagrange polynomials as element shape functions at the Legendre–Gauss–Lobatto (LGL) points. The fluid and solid subdomains are coupled by applying the appropriate boundary conditions at the limiting interface. The proposed method is verified by means of a benchmark problem regarding the scattering of waves by an elastic inclusion. The convergence and the computational effort are evaluated for different h-p strategies. Numerical results show good agreement with the reference solution. Finally, the proposed method is used to study the pressure field generated by an array of elastic fluid-filled scatterers immersed in an acoustic mediumMinisterio de Economía y Competitividad BIA2016-75042-C2-1-

Spectral-Element and Adjoint Methods in Seismology

We provide an introduction to the use of the spectral-element method (SEM) in seismology. Following a brief review of the basic equations that govern seismic wave propagation, we discuss in some detail how these equations may be solved numerically based upon the SEM to address the forward problem in seismology. Examples of synthetic seismograms calculated based upon the SEM are compared to data recorded by the Global Seismographic Network. Finally, we discuss the challenge of using the remaining differences between the data and the synthetic seismograms to constrain better Earth models and source descriptions. This leads naturally to adjoint methods, which provide a practical approach to this formidable computational challenge and enables seismologists to tackle the inverse problem

Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes

Accurate and efficient numerical methods to simulate dynamic earthquake rupture and wave propagation in complex media and complex fault geometries are needed to address fundamental questions in earthquake dynamics, to integrate seismic and geodetic data into emerging approaches for dynamic source inversion, and to generate realistic physics-based earthquake scenarios for hazard assessment. Modeling of spontaneous earthquake rupture and seismic wave propagation by a high-order discontinuous Galerkin (DG) method combined with an arbitrarily high-order derivatives (ADER) time integration method was introduced in two dimensions by de la Puente et al. (2009). The ADER-DG method enables high accuracy in space and time and discretization by unstructured meshes. Here we extend this method to three-dimensional dynamic rupture problems. The high geometrical flexibility provided by the usage of tetrahedral elements and the lack of spurious mesh reflections in the ADER-DG method allows the refinement of the mesh close to the fault to model the rupture dynamics adequately while concentrating computational resources only where needed. Moreover, ADER-DG does not generate spurious high-frequency perturbations on the fault and hence does not require artificial Kelvin-Voigt damping. We verify our three-dimensional implementation by comparing results of the SCEC TPV3 test problem with two well-established numerical methods, finite differences, and spectral boundary integral. Furthermore, a convergence study is presented to demonstrate the systematic consistency of the method. To illustrate the capabilities of the high-order accurate ADER-DG scheme on unstructured meshes, we simulate an earthquake scenario, inspired by the 1992 Landers earthquake, that includes curved faults, fault branches, and surface topography

A Simple Numerical Absorbing Layer Method in Elastodynamics

The numerical analysis of elastic wave propagation in unbounded media may be difficult to handle due to spurious waves reflected at the model artificial boundaries. Several sophisticated techniques such as nonreflecting boundary conditions, infinite elements or absorbing layers (e.g. Perfectly Matched Layers) lead to an important reduction of such spurious reflections. In this Note, a simple and efficient absorbing layer method is proposed in the framework of the Finite Element Method. This method considers Rayleigh/Caughey damping in the absorbing layer and its principle is presented first. The efficiency of the method is then shown through 1D Finite Element simulations considering homogeneous and heterogeneous damping in the absorbing layer. 2D models are considered afterwards to assess the efficiency of the absorbing layer method for various wave types (surface waves, body waves) and incidences (normal to grazing). The method is shown to be efficient for different types of elastic waves and may thus be used for various elastodynamic problems in unbounded domains

Spectral element modeling of three dimensional wave propagation in a self-gravitating Earth with an arbitrarily stratified outer core

This paper deals with the spectral element modeling of seismic wave propagation at the global scale. Two aspects relevant to low-frequency studies are particularly emphasized. First, the method is generalized beyond the Cowling approximation in order to fully account for the effects of self-gravitation. In particular, the perturbation of the gravity field outside the Earth is handled by a projection of the spectral element solution onto the basis of spherical harmonics. Second, we propose a new formulation inside the fluid which allows to account for an arbitrary density stratification. It is based upon a decomposition of the displacement into two scalar potentials, and results in a fully explicit fluid-solid coupling strategy. The implementation of the method is carefully detailed and its accuracy is demonstrated through a series of benchmark tests.Comment: Sent to Geophysical Journal International on July 29, 200