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A 3D Perfectly Matched Medium from Modified Maxwell's Equations with Stretched Coordinates
 Microwave Opt. Tech. Lett
, 1994
"... A modified set of Maxwell's equations is presented that includes complex coordinate stretching along the three cartesian coordinates. The added degrees of freedom in the modified Maxwell's equations allow the specification of absorbing boundaries with zero reflection at all angles of incid ..."
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Cited by 253 (18 self)
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A modified set of Maxwell's equations is presented that includes complex coordinate stretching along the three cartesian coordinates. The added degrees of freedom in the modified Maxwell's equations allow the specification of absorbing boundaries with zero reflection at all angles of incidence and all frequencies. The modified equations are also related to the perfectly matched layer that was presented recently for 2D wave propagation. Absorbing material boundary conditions are of particular interest for finite difference time domain (FDTD) computations on a singleinstruction multipledata (SIMD) massively parallel supercomputer. A 3D FDTD algorithm has been developed on a Connection Machine CM5 based on the modified Maxwell's equations and simulation results are presented to validate the approach. 1. Introduction The finite difference time domain method [1, 2] is widely regarded as one of the most popular computational electromagnetics algorithms. Although FDTD is conceptually v...
On Nonreflecting Boundary Conditions
 J. COMPUT. PHYS
, 1995
"... Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated ..."
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Cited by 220 (3 self)
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Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated condition. Second, the exact DtN boundary condition is derived for elliptic and spheroidal coordinates. Third, approximate local boundary conditions are derived for these coordinates. Fourth, the truncated DtN condition in elliptic and spheroidal coordinates is modified to remove difficulties. Fifth, a sequence of new and more accurate local boundary conditions is derived for polar coordinates in two dimensions. Numerical results are presented to demonstrate the usefulness of these improvements.
Numerical Solution Of Problems On Unbounded Domains. A Review
 A review, Appl. Numer. Math
, 1998
"... While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The issue of setting the ABC's appears most significant in many ..."
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Cited by 129 (18 self)
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While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The issue of setting the ABC's appears most significant in many areas of scientific computing, for example, in problems originating from acoustics, electrodynamics, solid mechanics, and fluid dynamics. In particular, in computational fluid dynamics (where external problems represent a wide class of important formulations) the proper treatment of external boundaries may have a profound impact on the overall quality and performance of numerical algorithms and interpretation of the results. Most of the currently used techniques for setting the ABC's can basically be classified into two groups. The methods from the first group (global ABC's) usually provide high accuracy and robustness of the numerical procedure but often appear to be fairly cumbersome and (computa...
Radiation Boundary Condition for the Numerical Simulation of Waves
 Acta Numerica
, 1999
"... We consider the efficient evaluation of accurate radiation boundary conditions for time domain simulations of wave propagation on unbounded spatial domains. This issue has long been a primary stumbling block for the reliable solution of this important class of problems. In recent years, a number of ..."
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Cited by 90 (3 self)
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We consider the efficient evaluation of accurate radiation boundary conditions for time domain simulations of wave propagation on unbounded spatial domains. This issue has long been a primary stumbling block for the reliable solution of this important class of problems. In recent years, a number of new approaches have been introduced which have radically changed the situation. These include methods for the fast evaluation of the exact nonlocal operators in special geometries, novel sponge layers with reflectionless interfaces, and improved techniques for applying sequences of approximate conditions to higher order. For the primary isotropic, constant coefficient equations of wave theory, these new developments provide an essentially complete solution of the numerical radiation condition problem. In this paper the theory of exact boundary conditions for constant coefficient timedependent problems is developed in detail, with many examples from physical applications. The theory is used to motivate various approximations and to establish error estimates. Complexity estimates are also derived to
The Perfectly Matched Layer in Curvilinear Coordinates
 SIAM J. Sci. Comput
, 1996
"... : In 1994 B'erenger showed how to construct a perfectly matched absorbing layer for the Maxwell system in rectilinear coordinates. This layer absorbs waves of any wavelength and any frequency without reflection and thus can be used to artificially terminate the domain of scattering calculation ..."
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Cited by 88 (5 self)
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: In 1994 B'erenger showed how to construct a perfectly matched absorbing layer for the Maxwell system in rectilinear coordinates. This layer absorbs waves of any wavelength and any frequency without reflection and thus can be used to artificially terminate the domain of scattering calculations. In this paper we show how to derive and implement the B'erenger layer in curvilinear coordinates (in two space dimensions). We prove that an infinite layer of this type can be used to solve time harmonic scattering problems. We also show that the truncated B'erenger problem has a solution except at a discrete set of exceptional frequencies (which might be empty). Finally numerical results show that the curvilinear layer can produce accurate solutions in the time and frequency domain. Keywords: Perfectly Matched Layer, computational electromagnetics, Absorbing layers (R'esum'e : tsvp) Research funded in part by a grant from AFOSR, USA. This paper has been submited to SIAM Scientific Computin...
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
The Finite Volume, Finite Element, and Finite Difference Methods as Numerical Methods for Physical Field Problems
 Journal of Computational Physics
, 2000
"... The present work describes an alternative to the classical partial differential equationsbased approach to the discretization of physical field problems. This alternative is based on a preliminary reformulation of the mathematical model in a partially discrete form, which preserves as much as possi ..."
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Cited by 65 (2 self)
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The present work describes an alternative to the classical partial differential equationsbased approach to the discretization of physical field problems. This alternative is based on a preliminary reformulation of the mathematical model in a partially discrete form, which preserves as much as possible the physical and geometrical content of the original problem, and is made possible by the existence and properties of a common mathematical structure of physical field theories. The goal is to maintain the focus, both in the modeling and in the discretizati on step, on the physics of the problem, thinking in terms of numerical methods for physical field problems, and not for a particular mathematical form (for example, a partial differential equation) into which the original physical problem happens to be translated.
On a class of preconditioners for solving the Helmholtz equation
, 2004
"... In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Precondi ..."
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Cited by 61 (11 self)
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In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year’s Report, St. Hugh’s College, Oxford, 2001] proposed a preconditioner where an extra term is added to the Laplace operator. This term is similar to the zeroth order term in the Helmholtz equation but with reversed sign. In this paper, both approaches are further generalized to a new class of preconditioners, the socalled “shifted Laplace ” preconditioners of the form ∆φ − αk2φ with α ∈ C. Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with
On Absorbing Boundary Conditions for Linearized Euler Equations by a Perfectly Matched Layer
 J. Comput. Phys
, 1995
"... waves. In the present paper, a perfectly matched layer is proposed for absorbing outgoing twodimensional waves in a uniform mean flow, governed by linearized Euler equations. It is well known that the linearized Euler equations support acoustic waves, which travel with the speed of sound relative ..."
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Cited by 58 (1 self)
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waves. In the present paper, a perfectly matched layer is proposed for absorbing outgoing twodimensional waves in a uniform mean flow, governed by linearized Euler equations. It is well known that the linearized Euler equations support acoustic waves, which travel with the speed of sound relative to the mean flow, and vorticity and entropy waves, which travel with the mean flow. The PML equations to be used at a region adjacent to the artificial boundary for absorbing these linear waves are defined. Plane wave solutions to the PML equations are developed and wave propagation and absorption properties are given. It is shown that the theoretical reflection coefficients at an interface between the Euler and PML domains are zero, independent of the angle of incidence and frequency of the waves. As such, the present study points out a possible alternative approach for absorbing outgoing waves of the Euler equations with little or no reflection in computation. Numerical examples that demonstrate the validity of the proposed PML equations are also presented.
Stability of Perfectly Matched Layers, Group Velocities and Anisotropic Waves
, 2003
"... Perfectly matched layers (PML) are a recent technique for simulating the absorption of waves in open domains. They have been introduced for electromagnetic waves and extended, since then, to other models of wave propagation including waves in elastic anisotropic media. In this last case, some numeri ..."
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Cited by 57 (11 self)
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Perfectly matched layers (PML) are a recent technique for simulating the absorption of waves in open domains. They have been introduced for electromagnetic waves and extended, since then, to other models of wave propagation including waves in elastic anisotropic media. In this last case, some numerical experiments have shown that the PMLs are not always stable. In this paper, we investigate this question from a theoretical point of view. In the first part, we derive a necessary condition for the stability of the PML model for a general hzperbolic system. This condition can be interpreted in terms of geometrical properties of the slowness diagrams and used for expflfi[HJ) instabilities observed with elastic waves but also with otherperJz]z]zJ models(anisotropJ Maxwell s equations, linearized Euler equations) . In the second part, we spA]zflJ) our analysis to orthotropV elastic waves and obtain sepnJVfifl a necessary stability condition and a sufficient stability condition that can be expressed in terms of inequalities on the elasticity coefficients of the model.