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40
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 219 (4 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.
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...
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 59 (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.
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...
Complex Coordinate Stretching as a Generalized Absorbing Boundary Condition
 Microwave and Optical Technology Letters
, 1997
"... By complex coordinate stretching and a change of variables, it is shown simply that PML is reflectionless for all frequencies and all angles. Also, Maxwell's equations for PML media reduce to ordinary Maxwell's equations with complex coordinate systems. Many closed form solutions for Maxwe ..."
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Cited by 27 (1 self)
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By complex coordinate stretching and a change of variables, it is shown simply that PML is reflectionless for all frequencies and all angles. Also, Maxwell's equations for PML media reduce to ordinary Maxwell's equations with complex coordinate systems. Many closed form solutions for Maxwell's equations map to corresponding closed form solutions in complex coordinate systems. Numerical simulations with the closed form solutions show that metallic boxes lined with PML media are highly absorptive. These closed form solutions lend a better understanding to the absorptive properties of PML media. For instance, they explain why a PML medium is absorptive when a dielectric or metallic interface extends to the edge to a simulation region where PML media reside. More importantly, the complex coordinate stretching method can be generalized to nonCartesian coordinate systems, providing absorbing boundary conditions in these coordinate systems. 1. Introduction The perfectly matched layer (PML) ...
PMLFDTD in cylindrical and spherical grids
 IEEE Microwave and Guided Wave Letters
, 1997
"... ..."
Finitedifference timedomain simulation of ground penetrating radar on dispersive, inhomogeneous, and conductive soils
 IEEE Trans. Geosci. Remote Sensing
, 1998
"... Abstract — A threedimensional (3D) timedomain numerical scheme for simulation of ground penetrating radar (GPR) on dispersive and inhomogeneous soils with conductive loss is described. The finitedifference timedomain (FDTD) method is used to discretize the partial differential equations for tim ..."
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Cited by 24 (4 self)
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Abstract — A threedimensional (3D) timedomain numerical scheme for simulation of ground penetrating radar (GPR) on dispersive and inhomogeneous soils with conductive loss is described. The finitedifference timedomain (FDTD) method is used to discretize the partial differential equations for time stepping of the electromagnetic fields. The soil dispersion is modeled by multiterm Lorentz and/or Debye models and incorporated into the FDTD scheme by using the piecewiselinear recursive convolution (PLRC) technique. The dispersive soil parameters are obtained by fitting the model to reported experimental data. The perfectly matched layer (PML) is extended to match dispersive media and used as an absorbing boundary condition to simulate an open space. Examples are given to verify the numerical solution and demonstrate its applications. The 3D PMLPLRCFDTD formulation facilitates the parallelization of the code. A version of the code is written for a 32processor system, and an almost linear speedup is observed. Index Terms—Absorbing boundary conditions, dispersive media, electromagnetic underground propagation, finitedifference timedomain (FDTD) methods. I.
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.
Analytical Derivation of a Conformal Perfectly Matched Absorber for Electromagnetic Waves
 Microwave Opt. Technol. Lett
, 1997
"... We present an analytical derivation of a 3D conformal perfectly matched layer (PML's) for mesh termination in general orthogonal curvilinear coordinates. The derivation is based on the analytic continuation to complex space of the normal coordinate to the mesh termination. The resultant fields ..."
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Cited by 18 (6 self)
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We present an analytical derivation of a 3D conformal perfectly matched layer (PML's) for mesh termination in general orthogonal curvilinear coordinates. The derivation is based on the analytic continuation to complex space of the normal coordinate to the mesh termination. The resultant fields in the complex space do not obey the Maxwell's equations. However, it is demonstrated that through simple field transformations, a new set of fields can be introduced so that they obey Maxwell's equations for an anisotropic medium with a properly chosen constitutive parameters depending on the local radii of curvature. The formulation presented here recovers, as particular cases, the previously proposed Cartesian, cylindrical, and spherical PML's. A previously employed anisotropic (quasi) PML for conformal terminations is shown to be the large radius of curvature approximation of the anisotropic conformal PML derived herein. Key words  Absorbing boundary condition, absorbing media, electromagn...
Perfectly Matched Layers in the Discretized Space: An Analysis and Optimization
 Electromagn
"... The perfectly matched layer (PML) has recently been introduced by Berenger as a material absorbing boundary condition (ABC) for electromagnetic waves. Recently, it has been pointed out that this absorbing boundary condition is the same as coordinate stretching in the complex space. In this paper, th ..."
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Cited by 18 (5 self)
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The perfectly matched layer (PML) has recently been introduced by Berenger as a material absorbing boundary condition (ABC) for electromagnetic waves. Recently, it has been pointed out that this absorbing boundary condition is the same as coordinate stretching in the complex space. In this paper, the corresponding coordinate stretching is analyzed in the discretized space of Maxwell's equations as described by the Yee algorithm. The corresponding dispersion relationship is derived for a PML medium and then the problem of reflection from a single interface is solved. A perfectly matched interface is shown not to exist in the discretized space, even though it exists in the continuum space. Numerical simulations both using finite difference method and finite element method confirm that such discretization error exists. A numerical scheme using the finite element method is then developed to optimize the PML with respect to its parameters. Examples are given to demonstrate the performance o...