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53
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...
Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media
, 1998
"... We present and analyze a perfectly matched absorbing layer model for the velocitystress formulation of elastodynamics. This layer has the astonishing property of generating no reflection at the interface between the free medium and the artificial absorbing medium. This allows us to obtain very low ..."
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Cited by 56 (6 self)
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We present and analyze a perfectly matched absorbing layer model for the velocitystress formulation of elastodynamics. This layer has the astonishing property of generating no reflection at the interface between the free medium and the artificial absorbing medium. This allows us to obtain very low spurious reflection even with very thin layers. Several experiments show the efficiency and the generality of the model.
Numerical Generation and Absorption of Fully Nonlinear Periodic Waves
, 1997
"... A new method is proposed for the generation of permanent form periodic waves, in a twodimensional fully nonlinear potential flow model. In this method, a constant volume is maintained in the computational domain ("wave tank") by simultaneously generating a mean current, equal and opposite ..."
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Cited by 56 (37 self)
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A new method is proposed for the generation of permanent form periodic waves, in a twodimensional fully nonlinear potential flow model. In this method, a constant volume is maintained in the computational domain ("wave tank") by simultaneously generating a mean current, equal and opposite to the waves mean mass transport velocity. An absorbing beach is modeled at the far end of the tank, with : (i) an external free surface pressure, proportional to the normal particle velocity, to absorb energy from high frequency incident waves; and (ii) a pistonlike condition, at the tank extremity, to absorb energy from low frequency waves. A new feedback mechanism is proposed to adaptively calibrate the beach absorption coefficient, as a function of time, for the beach to absorb the periodaveraged energy of waves entering the beach. Wave generation and absorption are validated over constant depth, for tanks and beaches of various lengths, and optimal parameter values are identified for which ref...
Rapid Evaluation Of Nonreflecting Boundary Kernels For TimeDomain Wave Propagation
 SIAM J. Numer. Anal
, 2000
"... . We present a systematic approach to the computation of exact nonreflecting boundary conditions for the wave equation. In both two and three dimensions, the critical step in our analysis involves convolution with the inverse Laplace transform of the logarithmic derivative of a Hankel function. The ..."
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Cited by 37 (5 self)
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. We present a systematic approach to the computation of exact nonreflecting boundary conditions for the wave equation. In both two and three dimensions, the critical step in our analysis involves convolution with the inverse Laplace transform of the logarithmic derivative of a Hankel function. The main technical result in this paper is that the logarithmic derivative of the Hankel function H (1) # (z) of real order # can be approximated in the upper half z plane with relative error # by a rational function of degree d # O # log # log 1 # +log 2 #+# 1 log 2 1 # # as ###, # # 0, with slightly more complicated bounds for # = 0. If N is the number of points used in the discretization of a cylindrical (circular) boundary in two dimensions, then, assuming that #<1/N , O(N log N log 1 # ) work is required at each time step. This is comparable to the work required for the Fourier transform on the boundary. In three dimensions, the cost is proportional to N...
Nonreflecting Boundary Conditions for the TimeDependent Wave Equation
 J. Comput. Phys
, 2002
"... this paper, we couple fast nonreflecting boundary conditions, developed in [3] for spherical and cylindrical boundaries and here for planar boundaries, to finitedifference solvers for the wave equation. In Section 2, we describe the exact (nonlocal) formulation, and in Section 3 we develop the fast ..."
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Cited by 27 (3 self)
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this paper, we couple fast nonreflecting boundary conditions, developed in [3] for spherical and cylindrical boundaries and here for planar boundaries, to finitedifference solvers for the wave equation. In Section 2, we describe the exact (nonlocal) formulation, and in Section 3 we develop the fast algorithm for handling the convolution operators that arise. In Section 4, we present simple temporal and spatial discretization schemes, and in Section 5, we present a number of numerical experiments. We compare the performance of our exact scheme, local EngquistMajda conditions [10], and the recently popular PML method [7], which uses an absorbing region to dampen undesired reflections. Our conclusions and directions for future work are discussed in Section 6
Perfectly Matched Absorbing Layers for the Paraxial Equations
, 1996
"... : A new absorbing boundary technique for the paraxial wave equations is proposed and analyzed. Numerical results show the efficiency of the method. Keywords: Paraxial equation, Numerical reflexion coefficient, Migration, Schroedinger equation, Perfectly matched layer, absorbing layers (R'esum ..."
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Cited by 27 (4 self)
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: A new absorbing boundary technique for the paraxial wave equations is proposed and analyzed. Numerical results show the efficiency of the method. Keywords: Paraxial equation, Numerical reflexion coefficient, Migration, Schroedinger equation, Perfectly matched layer, absorbing layers (R'esum'e : tsvp) This work has been performed as part of the IFPINRIA / Consortium Project. This paper has been submited to the Journal of Computational Physics Unit de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Tlphone : (33 1) 39 63 55 11  Tlcopie : (33 1) 39 63 53 30 Couches absorbantes parfaitement adapt'ees pour les 'equations paraxiales R'esum'e : Une nouvelle technique de conditions absorbantes pour les 'equations paraxiales est pr'esent'ee et analys'ee. L'id'ee est d'interpr'eter puis de g'en'eraliser le mod`ele de couches propos'e par J.P. B'erenger pour l"electromagn'etisme aux 'equations de type Schroedinger. Quelques r'esult...
Generation of Waves in Boussinesq Models using a Source Function Method
, 1998
"... A method for generating waves in Boussinesqtype wave models is described. The method employs a source term added to the governing equations, either in the form of a mass source in the continuity equation or an applied pressure forcing in the momentum equations. Assuming linearity, we derive a tr ..."
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Cited by 25 (6 self)
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A method for generating waves in Boussinesqtype wave models is described. The method employs a source term added to the governing equations, either in the form of a mass source in the continuity equation or an applied pressure forcing in the momentum equations. Assuming linearity, we derive a transfer function which relates source amplitude to surface wave characteristics. We then test the model for generation of desired incident waves, including regular and random waves, for both one and two dimensions. We also compare some model results with analytical solution and available experiment data. Keywords: wave generation, source function, boundary condition, Boussinesq model. 1 Introduction The problem of generating and absorbing waves at the boundary of models based on Boussinesqtype equations is essentially an unsolved one, due to the fact that the exact structure of the wellposed initial boundary value problem is unknown for most forms of the model equations. Though it is...
2004 A twolayer approach to wave modelling
 Proc. R. Soc. Lond. A
"... A set of model equations for waterwave propagation is derived by piecewise integration of the primitive equations of motion through two arbitrary layers. Within each layer, an independent velocity profile is derived. With two separate velocity profiles, matched at the interface of the two layers, ..."
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Cited by 20 (2 self)
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A set of model equations for waterwave propagation is derived by piecewise integration of the primitive equations of motion through two arbitrary layers. Within each layer, an independent velocity profile is derived. With two separate velocity profiles, matched at the interface of the two layers, the resulting set of equations has three free parameters, allowing for an optimization with known analytical properties of water waves. The optimized model equations show good linear wave characteristics up to kh ≈ 6, while the secondorder nonlinear behaviour is captured to kh ≈ 6 as well. A numerical algorithm for solving the model equations is developed and tested against one and twohorizontaldimension cases. Agreement with laboratory data is excellent.
Fully Nonlinear Potential Flow Models Used For Long Wave Runup Prediction
"... A review of Boundary Integral Equation methods used for long wave runup prediction is presented in this chapter. In Section 1, a brief literature review is given of methods used for modeling long wave propagation and of generic methods and models used for modeling highly nonlinear waves. In Section ..."
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Cited by 19 (9 self)
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A review of Boundary Integral Equation methods used for long wave runup prediction is presented in this chapter. In Section 1, a brief literature review is given of methods used for modeling long wave propagation and of generic methods and models used for modeling highly nonlinear waves. In Section 2, fully nonlinear potential flow equations are given for the Boundary Element Model developed by the author, including boundary conditions for both wave generation and absorption in the model. In Section 3, details are given for the generation of waves in the model using various methods (wavemakers, free surface potential, internal sources). In Section 4, the numerical implementation of the author's model based on a higherorder Boundary Element Method is briefly presented. In Section 5, many applications of the model are given for the computation of wave propagation, shoaling, breaking or runup on slopes, and interaction with submerged and emerged structures. The last application presente...