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42
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...
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
1998] The spectral element method: an efficient tool to simulate the seismic response of 2d and 3d geological structures
 Bulletin of Seismological Society of America
"... Abstract We present the spectral element method to simulate lasticwave propagation in realistic geological structures involving complicated freesurface topography and material interfaces for two and threedimensional geometries. The spectral element method introduced here is a highorder variat ..."
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Cited by 87 (7 self)
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Abstract We present the spectral element method to simulate lasticwave propagation in realistic geological structures involving complicated freesurface topography and material interfaces for two and threedimensional geometries. The spectral element method introduced here is a highorder variational method for the spatial approximation of elasticwave equations. The mass matrix is diagonal by construction in this method, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energymomentum conserving scheme that can be put into a classical explicitimplicit predictor/multicorrector format. Longterm energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The associated Courant condition behaves as At c < 0 (n ~ lind N2), with nel the number of elements, na the spatial dimension, and N the polynomial order. In practice, a spatial sampling of approximately 5 points per wavelength is found to be very accurate when working with a polynomial degree of N = 8. The accuracy of
Highorder nonreflecting boundary scheme for timedependent waves
 Journal of Computational Physics
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Highorder nonreflecting boundary conditions for dispersive waves
, 2003
"... Problems of linear timedependent dispersive waves in an unbounded domain are considered. The infinite domain is truncated via an artificial boundary B, and a highorder nonreflecting boundary condition (NRBC) is imposed on B. Then the problem is solved by a finite difference (FD) scheme in the fin ..."
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Cited by 24 (5 self)
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Problems of linear timedependent dispersive waves in an unbounded domain are considered. The infinite domain is truncated via an artificial boundary B, and a highorder nonreflecting boundary condition (NRBC) is imposed on B. Then the problem is solved by a finite difference (FD) scheme in the finite domain bounded by B. The sequence of NRBCs proposed by Higdon is used. However, in contrast to the original loworder implementation of the Higdon conditions, a new scheme is devised which allows the easy use of a Higdontype NRBC of any desired order. In addition, a procedure for the automatic choice of the parameters appearing in the NRBC is proposed. The performance of the scheme is demonstrated via numerical examples.
Improving the performance of perfectly matched layers by means of hpadaptivity. Numerical Methods for Partial Differential Equations
"... We improve the performance of the Perfectly Matched Layer by using an automatic hpadaptive discretization. By means of hpadaptivity, we obtain a sequence of discrete solutions that converges exponentially to the continuum solution. Asymptotically, we thus recover the property of the PML of having ..."
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Cited by 15 (6 self)
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We improve the performance of the Perfectly Matched Layer by using an automatic hpadaptive discretization. By means of hpadaptivity, we obtain a sequence of discrete solutions that converges exponentially to the continuum solution. Asymptotically, we thus recover the property of the PML of having a zero reflection coefficient for all angles of incidence and all frequencies on the continuum level. This allows us to minimize the reflections from the discrete PML to an arbitrary level of accuracy while retaining optimal computational efficiency. Since our hpadaptive scheme is automatic, no interaction with the user is required. This renders tedious parameter tuning of the PML obsolete. We demonstrate the improvement of the PML performance by hpadaptivity through numerical results for acoustic, elastodynamic and electromagnetic wavepropagation problems in the frequency domain and in different systems of coordinates.
Perfectly matched layers for transient elastodynamics of unbounded domains
 Int. J. Numer. Meth. Engng
"... One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave e ..."
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Cited by 14 (0 self)
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One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all nontangential anglesofincidence and of all nonzero frequencies. In a recent work [Computer Methods in Applied Mechanics and Engineering 2003; 192:1337
1375], the authors presented, inter alia, timeharmonic governing equations of PMLs for antiplane and for planestrain motion of (visco)elastic media. This paper presents (a) corresponding timedomain, displacementbased governing equations of these PMLs and (b) displacementbased nite element implementations of these equations, suitable for direct transient analysis. The nite element implementation of the antiplane PML is found to be symmetric, whereas that of the planestrain PML is not. Numerical results are presented for the antiplane motion of a semiinnite layer on a rigid base, and for the classical soil
structure interaction problems of a rigid stripfooting on (i) a halfplane, (ii) a layer on a halfplane, and (iii) a layer on a rigid base. These results demonstrate the high accuracy achievable by PML models even with small bounded domains. Copyright 2004 John Wiley & Sons, Ltd.
Recent Developments in Finite Element Methods for Structural Acoustics
, 1996
"... This paper reviews recent progress in finite element analysis that renders computation a practical tool for solving problems of structural acoustics ..."
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Cited by 14 (3 self)
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This paper reviews recent progress in finite element analysis that renders computation a practical tool for solving problems of structural acoustics
A mathematical framework of the bridging scale method
 International J. Numer. Methods Engrg
"... In this paper, we present a mathematical framework of the bridging scale method (BSM), recently proposed by Liu et al. Under certain conditions, it had been designed for accurately and efficiently simulating complex dynamics with different spatial scales. From a clear and consistent derivation, we i ..."
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Cited by 12 (2 self)
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In this paper, we present a mathematical framework of the bridging scale method (BSM), recently proposed by Liu et al. Under certain conditions, it had been designed for accurately and efficiently simulating complex dynamics with different spatial scales. From a clear and consistent derivation, we identify two error sources in this method. First, we use a linear finite element interpolation, and derive the coarse grid equations directly from Newton’s second law. Numerical error in this length scale exists mainly due to inadequate approximation for the effects of the fine scale fluctuations. An modified linear element (MLE) scheme is developed to improve the accuracy. Secondly, we derive an exact multiscale interfacial condition to treat the interfaces between the molecular dynamics region �D and the complementary domain �C, using a time history kernel technique. The interfacial condition proposed in the original BSM may be regarded as a leading order approximation to the exact one (with respect to the coarsening ratio). This approximation is responsible for minor reflections across the interfaces, with a dependency on the choice of �D. We further illustrate the framework and analysis with linear and nonlinear lattices in onedimensional space. Copyright � 2005 John Wiley & Sons, Ltd. KEY WORDS: multiscale computation; bridging scale method; coarse–fine decomposition; molecular