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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.
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 126 (19 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...
Comparison Of TwoDimensional Conformal Local Radiation Boundary Conditions
 Electromagnetics
, 1996
"... Numerical solutions for openregion electromagnetic problems based on differential equations require some means of truncating the computational domain. A number of local Radiation Boundary Conditions (RBCs) for general boundary shapes have been proposed during the past decade. Many are generalizatio ..."
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Cited by 8 (4 self)
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Numerical solutions for openregion electromagnetic problems based on differential equations require some means of truncating the computational domain. A number of local Radiation Boundary Conditions (RBCs) for general boundary shapes have been proposed during the past decade. Many are generalizations of the BaylissTurkel RBC for circular truncation boundaries. This paper reviews several twodimensional RBCs for general truncation boundaries. The RBCs are evaluated on the basis of their performance on two separate numerical tests: the annihilation of terms in the Hankel series and the comparison of nearfield and radar cross sections for finite element solutions to scattering problems. These tests suggest that the simpler RBCs can be very competitive with RBCs based on more sophisticated derivations. 1. INTRODUCTION Despite the rapid growth of computer resources over the last few years, the main concern in solving realistic open region electromagnetic problems is to reduce the size of...
Scalable Solutions to IntegralEquation and FiniteElement Simulations
"... Abstract — When developing numerical methods, or applying them to the simulation and design of engineering components, it inevitably becomes necessary to examine the scaling of the method with a problem’s electrical size. The scaling results from the original mathematical development; for example, a ..."
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Abstract — When developing numerical methods, or applying them to the simulation and design of engineering components, it inevitably becomes necessary to examine the scaling of the method with a problem’s electrical size. The scaling results from the original mathematical development; for example, a dense system of equations in the solution of integral equations, as well as the specific numerical implementation. Scaling of the numerical implementation depends upon many factors; for example, direct or iterative methods for solution of the linear system, as well as the computer architecture used in the simulation. In this paper, scalability will be divided into two components—scalability of the numerical algorithm specifically on parallel computer systems and algorithm or sequential scalability. The sequential implementation and scaling is initially presented, with the parallel implementation following. This progression is meant to illustrate the differences in using current parallel platforms and sequential machines and the resulting savings. Time to solution (wallclock time) for differing problem sizes are the key parameters plotted or tabulated. Sequential and parallel scalability of time harmonic surface integral equation forms and the finiteelement solution to the partial differential equations are considered in detail. Index Terms—Finiteelement methods, integral equations.
Calculation of Light Distribution in Optical Devices By a Global Solution of an Inhomogeneous Scalar Wave Equation
"... A numerical scheme is suggested for solving the scalar wave equation for optical devices characterized by a zdependent refractive index. The homogeneous wave equation is converted into a single column inhomogeneous linear system by applying imaginary (absorbing) boundary potentials at the device bo ..."
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Cited by 1 (1 self)
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A numerical scheme is suggested for solving the scalar wave equation for optical devices characterized by a zdependent refractive index. The homogeneous wave equation is converted into a single column inhomogeneous linear system by applying imaginary (absorbing) boundary potentials at the device boundaries. The imposed absorbing boundary conditions enable discrete representation of the device on a compact grid. Our scheme is based on applying an efficient sparse preconditioner to the linear system, which enables its global solution (i.e. simultaneously for all zvalues) by fast iterative methods, such as the Quasi Minimal Residual algorithm. A single solution of the inhomogeneous equation with the imaginary boundary operators allows the calculation of modespecific as well as regionspecific light intensity coupling probabilities for initial mode of interest. Numerical examples illustrate the usefulness of the suggested scheme to the optimization of optical devices.
Wave splitting of Maxwell’s equations with anisotropic heterogeneous constitutive relations. Inverse Probl
 Imag
"... The equations for the electromagnetic field in an anisotropic media are written in a form containing only the transverse field components relative to a half plane boundary. The operator corresponding to this formulation is the electromagnetic system’s matrix. A constructive proof of the existence of ..."
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The equations for the electromagnetic field in an anisotropic media are written in a form containing only the transverse field components relative to a half plane boundary. The operator corresponding to this formulation is the electromagnetic system’s matrix. A constructive proof of the existence of directional wavefield decomposition with respect to the normal of the boundary is presented. In the process of defining the wavefield decomposition (wavesplitting), the resolvent set of the timeLaplace representation of the system’s matrix is analyzed. This set is shown to contain a strip around the imaginary axis. We construct a splitting matrix as a DunfordTaylor type integral over the resolvent of the unbounded operator defined by the electromagnetic system’s matrix. The splitting matrix commutes with the system’s matrix and the decomposition is obtained via a generalized eigenvalueeigenvector procedure. The decomposition is expressed in terms of components of the splitting matrix. The constructive solution to the question on the existence of a decomposition also generates an impedance mapping solution to an algebraic Riccati operator equation. This solution is the electromagnetic generalization in an anisotropic media of a DirichlettoNeumann map. Keywords directional wavefield decomposition, wavesplitting, anisotropy, electromagnetic system’s matrix, generalized eigenvalue problem, algebraic Riccati operator equation, generalized vertical wave number. 1