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73
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...
Fictitious Domain Methods For The Numerical Solution Of ThreeDimensional Acoustic Scattering Problems
 J. Comput. Phys
, 1999
"... . Efficient iterative methods for the numerical solution of threedimensional acoustic scattering problems are considered. The underlying exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a nonreflecting boundary condition on the artificial boundary. ..."
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Cited by 29 (17 self)
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. Efficient iterative methods for the numerical solution of threedimensional acoustic scattering problems are considered. The underlying exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a nonreflecting boundary condition on the artificial boundary. The finite element discretization of the approximate boundary value problem is performed using locally fitted meshes, and algebraic fictitious domain methods with separable preconditioners are applied to the solution of the arising mesh equations. These methods are based on imbedding the original domain into a larger one with a simple geometry (for example, a sphere or a parallelepiped). The iterative solution method is realized in a lowdimensional subspace, and partial solution methods are applied to the linear systems with the preconditioner. Results of numerical experiments demonstrate the efficiency and accuracy of the approach. Key words. Acoustic scattering, nonreflecting boundary ...
Preconditioning Helmholtz linear systems
, 2009
"... Linear systems which originate from the simulation of wave propagation phenomena can be very difficult to solve by iterative methods. These systems are typically complex valued and they tend to be highly indefinite, which renders the standard ILUbased preconditioners ineffective. This paper present ..."
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Cited by 22 (1 self)
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Linear systems which originate from the simulation of wave propagation phenomena can be very difficult to solve by iterative methods. These systems are typically complex valued and they tend to be highly indefinite, which renders the standard ILUbased preconditioners ineffective. This paper presents a study of ways to enhance standard preconditioners by altering the diagonal by imaginary shifts. Prior work indicates that modifying the diagonal entries during the incomplete factorization process, by adding to it purely imaginary values can improve the quality of the preconditioner in a substantial way. Here we propose simple algebraic heuristics to perform the shifting and test these techniques with the ARMS and ILUT preconditioners. Comparisons are made with applications stemming from the diffraction of an acoustic wave incident on a bounded obstacle (governed by the Helmholtz Wave Equation).
Iterative Solution Of The Helmholtz Equation By A SecondOrder Method
 SIAM J. Matrix Anal. Appl
, 1996
"... . The numerical solution of the Helmholtz equation subject to nonlocal radiation boundary conditions is studied. The specific problem is the propagation of hydroacoustic waves in a twodimensional curvilinear duct. The problem is discretized with a secondorder accurate finitedifference method, resu ..."
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Cited by 21 (6 self)
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. The numerical solution of the Helmholtz equation subject to nonlocal radiation boundary conditions is studied. The specific problem is the propagation of hydroacoustic waves in a twodimensional curvilinear duct. The problem is discretized with a secondorder accurate finitedifference method, resulting in a linear system of equations. To solve the system of equations, a preconditioned Krylov subspace method is employed. The preconditioner is based on fast transforms, and yields a direct fast Helmholtz solver for rectangular domains. Numerical experiments for curved ducts demonstrate that the rate of convergence is high. Compared with band Gaussian elimination the preconditioned iterative method shows a significant gain in both storage requirement and arithmetic complexity. This research was supported by the U. S. National Science Foundationunder grant ASC8958544 and by the Swedish National Board for Industrial and Technical Development (NUTEK). y Department of Scientific Computi...
A Perfectly Matched Layer for the Helmholtz Equation in a Semiinfinite Strip
"... The Perfectly Matched Layer (PML) has become a widespread technique for preventing reflections from far field boundaries for wave propagation problems in both the time dependent and frequency domains. We develop a discretization to solve the Helmholtz equation in an infinite two dimensional strip. ..."
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Cited by 20 (3 self)
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The Perfectly Matched Layer (PML) has become a widespread technique for preventing reflections from far field boundaries for wave propagation problems in both the time dependent and frequency domains. We develop a discretization to solve the Helmholtz equation in an infinite two dimensional strip. We solve the interior equation using highorder finite differences schemes. The combined HelmholtzPML problem is then analyzed for the parameters that give the best performance. We show that the use of local highorder methods in the physical domain coupled with a specific second order approximation in the PML yields global highorder accuracy in the physical domain. We discuss the impact of the parameters on the effectiveness of the PML. Numerical results are presented to support the analysis.
DirichlettoNeumann Boundary Conditions for Multiple Scattering Problems
, 2004
"... A DirichlettoNeumann (DtN) condition is derived for the numerical solution of timeharmonic multiple scattering problems, where the scatterer consists of several disjoint components. It is obtained by combining contributions from multiple purely outgoing wave fields. The DtN condition yields an ex ..."
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Cited by 16 (0 self)
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A DirichlettoNeumann (DtN) condition is derived for the numerical solution of timeharmonic multiple scattering problems, where the scatterer consists of several disjoint components. It is obtained by combining contributions from multiple purely outgoing wave fields. The DtN condition yields an exact nonreflecting boundary condition for the situation, where the computational domain and its exterior artificial boundary consist of several disjoint components. Because each subscatterer can be enclosed by a separate artificial boundary, the computational effort is greatly reduced and becomes independent of the relative distances between the different subdomains. The DtN condition naturally fits into a variational formulation of the boundary value problem for use with the finite element method. Moreover, it immediately yields as a byproduct an exact formula for the farfield pattern of the scattered field. Numerical examples show that the DtN condition for multiple scattering is as accurate as the wellknown DtN condition for single scattering problems [6,7], while being more efficient due to the reduced size of the computational domain.
Efficient Iterative Solution of the ThreeDimensional Helmholtz Equation
, 1998
"... We examine preconditioners for the discrete indefinite Helmholtz equation on a threedimensional boxshaped domain with Sommerfeldlike boundary conditions. The preconditioners are of two types. The first is derived by discretization of a related continuous operator that differs from the original on ..."
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Cited by 9 (2 self)
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We examine preconditioners for the discrete indefinite Helmholtz equation on a threedimensional boxshaped domain with Sommerfeldlike boundary conditions. The preconditioners are of two types. The first is derived by discretization of a related continuous operator that differs from the original only in its boundary conditions. The second is derived by a block Toeplitz approximation to the descretized problem. The resulting preconditioning matrices allow the use of fast transform methods and differ from the discrete Helmholtz operator by an operator of low rank. We present experimental results demonstrating that when these methods are combined with Krylov subspace iteration, convergence rates depend only mildly on both the wave number and discretization mesh size. In addition, the methods display high efficiencies in an implementation on an IBM SP2 parallel computer.
Coupling of a nonoverlapping domain decomposition method for a nodal finite element method with a boundary element method
, 2002
"... Nonoverlapping domain decomposition techniques are used both to solve the finite element equations and to couple them with a boundary element method. A suitable approach dealing with finite element nodes common to more than two subdomains, the socalled crosspoints, endows the method with the foll ..."
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Cited by 9 (2 self)
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Nonoverlapping domain decomposition techniques are used both to solve the finite element equations and to couple them with a boundary element method. A suitable approach dealing with finite element nodes common to more than two subdomains, the socalled crosspoints, endows the method with the following advantages. It yields a robust and efficient procedure to solve the equations resulting from the discretization process. Only small size finite element linear systems and a dense linear system related to a simple boundary integral equation are solved at each iteration and each of them can be solved in a stable way. We also show how to choose the parameter definining the augmented local matrices in order to improve the convergence. Several numerical simulations in 2D and 3D validating the treatment of the crosspoints and illustrating the strategy to accelerate the iterative procedure are presented.
Spherical perfectly matched absorber for finitevolume 3D domain truncation
 IEEE Trans. Microwave Theory Techniques
, 2007
"... Abstract – Different implementations of planar perfectly matched absorbers are studied under the unified framework of the FiniteVolume TimeDomain (FVTD) method. This comparative analysis allows to discuss the similarities existing between the theoretical models and explores the differences in the ..."
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Cited by 9 (4 self)
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Abstract – Different implementations of planar perfectly matched absorbers are studied under the unified framework of the FiniteVolume TimeDomain (FVTD) method. This comparative analysis allows to discuss the similarities existing between the theoretical models and explores the differences in their practical implementation and numerical performance in the framework of the FVTD method. Numerical experiments for performance analysis of the different PML models are conducted in terms of discretization and angle of incidence using waveguide models. The results are compared to theoretically expected values and to the firstorder Silver Müller absorbing boundary condition. I.