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50
2004), Simulations of ground motion in the Los Angeles Basin Based upon the spectralelement method
 Bull. Seismol. Soc. Am
"... Abstract We use the spectralelement method to simulate ground motion generated by two recent and wellrecorded small earthquakes in the Los Angeles basin. Simulations are performed using a new sedimentary basin model that is constrained by hundreds of petroleumindustry well logs and more than 20, ..."
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Cited by 66 (12 self)
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Abstract We use the spectralelement method to simulate ground motion generated by two recent and wellrecorded small earthquakes in the Los Angeles basin. Simulations are performed using a new sedimentary basin model that is constrained by hundreds of petroleumindustry well logs and more than 20,000 km of seismic reflection profiles. The numerical simulations account for 3D variations of seismicwave speeds and density, topography and bathymetry, and attenuation. Simulations
Analysis of a finite PML approximation for the three dimensional timeharmonic maxwell and acoustic scattering problems
 MATH. COMP
, 2006
"... We consider the approximation of the frequency domain threedimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the timeharmonic PML approximation to the acoustic scattering problem. Following work of Lassas and Somersalo in 1998, a transiti ..."
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Cited by 25 (7 self)
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We consider the approximation of the frequency domain threedimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the timeharmonic PML approximation to the acoustic scattering problem. Following work of Lassas and Somersalo in 1998, a transitional layer based on spherical geometry is defined, which results in a constant coefficient problem outside the transition. A truncated (computational) domain is then defined, which covers the transition region. The truncated domain need only have a minimally smooth outer boundary (e.g., Lipschitz continuous). We consider the truncated PML problem which results when a perfectly conducting boundary condition is imposed on the outer boundary of the truncated domain. The existence and uniqueness of solutions to the truncated PML problem will be shown provided that the truncated domain is sufficiently large, e.g., contains a sphere of radius Rt. Wealsoshow exponential (in the parameter Rt) convergence of the truncated PML solution to the solution of the original scattering problem inside the transition layer. Our results are important in that they are the first to show that the truncated PML problem can be posed on a domain with nonsmooth outer boundary. This allows the use of approximation based on polygonal meshes. In addition, even though the transition coefficients depend on spherical geometry, they can be made arbitrarily smooth and hence the resulting problems are amenable to numerical quadrature. Approximation schemes based on our analysis are the focus of future research.
On causality and dynamic stability of perfectly matched layers for FDTD simulations
 IEEE Trans. Microwave Theory Tech
, 1999
"... Abstract—We investigate the spectral properties of the Cartesian, cylindrical, and spherical perfect matched layer (PML) absorbing boundary conditions. In the case of the anisotropicmedium PML formulation, we analyze the analytical properties of the constitutive PML tensors on the complex!plane. In ..."
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Cited by 21 (5 self)
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Abstract—We investigate the spectral properties of the Cartesian, cylindrical, and spherical perfect matched layer (PML) absorbing boundary conditions. In the case of the anisotropicmedium PML formulation, we analyze the analytical properties of the constitutive PML tensors on the complex!plane. In the case of the complexspace PML formulation (complex coordinate stretching formulation), we analyze the analytical properties of field solutions directly. We determine the conditions under which the PML’s satisfy (or do not satisfy) causality requirements in the sense of the realaxis Fourier inversion contour. In the case of the noncausal PML, we point out the implications on the dynamic stability of timedomain equations and finitedifference timedomain (FDTD) simulations. The conclusions have impact both on the design of PML’s for practical FDTD simulations and on the use of PML’s as a physical basis for engineered artificial absorbers on nonplanar (concave or convex) surfaces. Numerical results illustrate the discussion. Index Terms—Absorbing boundary conditions, anisotropic media, dispersive media, FDTD methods, perfectly matched layer. I.
A nonconvolutional, splitfield, perfectly matched layer for wave propagation in isotropic and anisotropic elastic media: Stability analysis
 Bulletin of the Seismological Society of America
, 2008
"... Abstract A nonconvolutional, splitfield, perfectly matched layer, referred to as the multiaxial perfectly matched layer (MPML), is proposed and implemented. The new formulation is obtained by generalizing the classical perfectly matched layer (PML; as originally proposed by Bérenger, [1994]) to a ..."
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Cited by 18 (0 self)
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Abstract A nonconvolutional, splitfield, perfectly matched layer, referred to as the multiaxial perfectly matched layer (MPML), is proposed and implemented. The new formulation is obtained by generalizing the classical perfectly matched layer (PML; as originally proposed by Bérenger, [1994]) to a medium in which damping profiles are specified in more than one direction. Under the hypothesis of small damping and using an eigenvalue sensitivity analysis based on first derivatives, we propose a method to study the stability of the MPML. With this method we demonstrate that the stability of the MPML is related to the ratios of the specified damping profiles. Recognition of this fact leads to a general procedure for constructing robust, stable MPML models for anisotropic media. It is also demonstrated that for any anisotropic medium the classical PML exhibits instabilities related to an eigenvalue with zero real part of multiplicity higher than one. Furthermore, we show that exponential growth due to eigenvalues with positive real part can be present in the classical PML for some orthotropic media. The effectiveness of the proposed MPML and its advantages relative to the classical PML are demonstrated by constructing stable terminations for the aforementioned anisotropic media. The method of stability analysis is developed and demonstrated for twodimensional elastodynamics problems, but its extension to threedimensional configurations is straightforward.
The Finite Integration Technique as a General Tool to Compute Acoustic, Electromagnetic, Elastodynamic, and Coupled Wave Fields
"... This review paper presents a unified treatment of the numerical timedomain modeling of acoustic, electromagnetic, and elastodynamic waves and of the combination thereof: this means coupled wave fields, like piezoelectric (elastoelectric) and electromagneticultrasonic wave fields. Since the famous ..."
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Cited by 16 (3 self)
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This review paper presents a unified treatment of the numerical timedomain modeling of acoustic, electromagnetic, and elastodynamic waves and of the combination thereof: this means coupled wave fields, like piezoelectric (elastoelectric) and electromagneticultrasonic wave fields. Since the famous paper by Yee [1966], the timedomain solution of Maxwell’s equations has become very popular, and this initiated the FiniteDifference TimeDomain (FDTD) Method [Taflove and Hagness, 2000; fdtd.org, 2001]. Ten years later, Madariaga [1976] independently developed a similar method in elastodynamics. Both methods start from the governing equations in differential form, using standard secondorder finitedifference stencils in space and time. In electromagnetics, Weiland [1977] introduced a different approach, which starts from the full set of Maxwell’s equations in integral form. Today, this method is commonly called the Finite Integration Technique (FIT). In the 1990s, based on Weiland’s ideas, Fellinger [1991] adapted the FIT to the elastodynamic case. Today, a toolbox of several modeling codes – called AFIT, EMFIT, EFIT, PFIT, and EMUSFIT, which stand for acoustic (A), electromagnetic (EM), elastic (E), piezoelectric (P), and electromagneticultrasonic (EMUS) Finite Integration Technique (FIT)
A variational formulation of a stabilized unsplit convolutional perfectly matched layer for the isotropic or anisotropic seismic wave equation
 CMES
, 2008
"... In the context of the numerical simulation of seismic wave propagation, the perfectly matched layer (PML) absorbing boundary condition has proven to be efficient to absorb surface waves as well as body waves with non grazing incidence. But unfortunately the classical discrete PML generates spurious ..."
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Cited by 16 (7 self)
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In the context of the numerical simulation of seismic wave propagation, the perfectly matched layer (PML) absorbing boundary condition has proven to be efficient to absorb surface waves as well as body waves with non grazing incidence. But unfortunately the classical discrete PML generates spurious modes traveling and growing along the absorbing layers in the case of waves impinging the boundary at grazing incidence. This is significant in the case of thin mesh slices, or in the case of sources located close to the absorbing boundaries or receivers located at large offset. In previous work we derived an unsplit convolutional PML (CPML) for staggeredgrid finitedifference integration schemes to improve the efficiency of the PML at grazing incidence for seismic wave propagation. In this article we derive a variational formulation of this CPML method for the seismic wave equation and validate it using the spectralelement method based on a hybrid first/secondorder time integration scheme. Using the Newmark time marching scheme, we underline the fact that a velocitystress formulation in the PML and a secondorder displacement formulation in the inner computational domain match perfectly at the entrance of the absorbing layer. The main difference between our unsplit CPML and the split GFPML formulation of Festa and Vilotte (2005) lies in the fact that memory storage of CPML is reduced by 35 % in 2D and 44 % in 3D. Furthermore the CPML can be stabilized by correcting the damping profiles in the PML layer in the anisotropic case. We show benchmarks for 2D heterogeneous thin slices in the presence of a free surface and in anisotropic cases that are intrinsically unstable if no stabilization of the PML is used.
Accelerating a threedimensional finitedifference wave propagation code using GPU graphics cards
, 2010
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Parsimonious finitevolume frequencydomain method for 2D PSVwave modelling
"... A new numerical technique for solving 2D elastodynamic equations based on a finitevolume frequencydomain approach is proposed. This method has been developed as a tool to perform twodimensional (2D) elastic frequencydomain fullwaveform inversion. In this context, the system of linear equations ..."
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Cited by 7 (7 self)
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A new numerical technique for solving 2D elastodynamic equations based on a finitevolume frequencydomain approach is proposed. This method has been developed as a tool to perform twodimensional (2D) elastic frequencydomain fullwaveform inversion. In this context, the system of linear equations that results from the discretisation of the elastodynamic equations is solved with a direct solver, allowing efficient multiplesource simulations at the partial expense of the memory requirement. The discretisation of the finitevolume approach is through triangles. Only fluxes with the required quantities are shared between the cells, relaxing the meshing conditions, as compared to finiteelement methods. The free surface is described along the edges of the triangles, which can have different slopes. By applying a parsimonious strategy, the stress components are eliminated from the discrete equations and only the velocities are left as unknowns in the triangles. Together with the local support of the P0 finitevolume stencil, the parsimonious approach allows the minimising of core memory requirements for the simulation. Efficient perfectly matched layer absorbing conditions have been designed for damping the waves around the grid. The numerical dispersion of this FV formulation is similar to that of O(∆x 2) staggeredgrid finitedifference formulations when considering