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256
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 91 (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 85 (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...
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.
Perfectly Matched Layers for Elastodynamics: A New Absorbing Boundary Condition
 J. Comp. Acoust
, 1996
"... The use of perfectly matched layers (PML) has recently been introduced by Berenger as a material absorbing boundary condition (ABC) for electromagnetic waves. In this paper, we will first prove that a fictitious elastodynamic material halfspace exists that will absorb an incident wave for all angle ..."
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Cited by 50 (4 self)
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The use of perfectly matched layers (PML) has recently been introduced by Berenger as a material absorbing boundary condition (ABC) for electromagnetic waves. In this paper, we will first prove that a fictitious elastodynamic material halfspace exists that will absorb an incident wave for all angles and all frequencies. Moreover, the wave is attenuative in the second halfspace. As a consequence, layers of such material could be designed at the edge of a computer simulation region to absorb outgoing waves. Since this is a material ABC, only one set of computer codes is needed to simulate an open region. Hence, it is easy to parallelize such codes on multiprocessor computers. For instance, it is easy to program massively parallel computers on the SIMD (single instruction multiple data) mode for such codes. We will show two and three dimensional computer simulations of the PML for the linearized equations of elastodyanmics. Comparison with Liao's ABC will be given. 1. Introduction Sim...
Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers
, 2010
"... The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variablecoefficient Helmholtz equation including veryhighfrequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Hel ..."
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Cited by 46 (6 self)
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The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variablecoefficient Helmholtz equation including veryhighfrequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The generalized minimal residual method (GMRES) solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the threedimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach. © 2011 Wiley Periodicals, Inc. 1
Perfectly matched layers for hyperbolic systems: General formulation, wellposedness and stability
 SIAM J. Appl. Math
"... Abstract. Since its introduction the Perfectly Matched Layer (PML) has proven to be an accurate and robust method for domain truncation in computational electromagnetics. However, the mathematical analysis of PMLs has been limted to special cases. In particular, the basic question of whether or not ..."
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Cited by 41 (5 self)
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Abstract. Since its introduction the Perfectly Matched Layer (PML) has proven to be an accurate and robust method for domain truncation in computational electromagnetics. However, the mathematical analysis of PMLs has been limted to special cases. In particular, the basic question of whether or not a stable PML exists for arbitrary wave propagation problems remains unanswered. In this work we develop general tools for constructing PMLs for first order hyperbolic systems. We present a model with many parameters which is applicable to all hyperbolic systems, and which we prove is wellposed and perfectly matched. We also introduce an automatic method for analyzing the stability of the model and establishing energy inequalities. We illustrate our techniques with applications to Maxwell’s equations, the linearized Euler equations, as well as arbitrary 2×2 systems in (2 + 1) dimensions. Key words. Perfectly matched layers, stability. AMS subject classifications. 35L45, 35B35 1. Introduction. Many
Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates
 IEEE Microw. Guid. Wave Lett
, 1997
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A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations
, 2008
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Fast Solution Methods in Electromagnetics
, 1997
"... Various methods for efficiently solving electromagnetic problems are presented. Electromagnetic scattering problems can be roughly classified into surface and volume problems, while fast methods are either differential or integral equation based. The resultant systems of linear equations are either ..."
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Cited by 33 (0 self)
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Various methods for efficiently solving electromagnetic problems are presented. Electromagnetic scattering problems can be roughly classified into surface and volume problems, while fast methods are either differential or integral equation based. The resultant systems of linear equations are either solved directly or iteratively. A review of various differential equation solvers, their complexities, and memory requirements is given. The issues of grid dispersion and hybridization with integral equation solvers are discussed. Several fast integral equation solvers for surface and volume scatterers are presented. These solvers have reduced computational complexities and memory requirements. 1. Introduction Computational electromagnetics is a fascinating discipline that has drawn the attention of mathematicians, engineers, physicists, and computer scientists alike. It is a discipline that creates a symbiotic marriage between mathematics, physics, computer science, and various applicatio...