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15
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.
Building SpaceTime Meshes over Arbitrary Spatial Domains
 Proc. 11th Int. Meshing Roundtable
, 2002
"... We present an algorithm to construct meshes suitable for spacetime discontinuous Galerkin finiteelement methods. Our method generalizes and improves the 'Tent Pitcher' algorithm of /lngSr and Sheffer. Given an arbitrary simplicially meshed domain X of any dimension and a time interval [0 ..."
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Cited by 26 (5 self)
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We present an algorithm to construct meshes suitable for spacetime discontinuous Galerkin finiteelement methods. Our method generalizes and improves the 'Tent Pitcher' algorithm of /lngSr and Sheffer. Given an arbitrary simplicially meshed domain X of any dimension and a time interval [0, T], our algorithm builds a simplicial mesh of the spacetime domain X x [0, T], in constant time per element. Our algorithm avoids the limitations of previous methods by carefully adapting the durations of spacetime elements to the local quality and feature size of the underlying space mesh.
Spacetime Meshing with Adaptive Refinement and Coarsening
 SCG'04
, 2004
"... We propose a new algorithm for constructing finiteelement meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain# and a target time value T , our method constructs a tetrahedral mesh of the spacetime domain [0, T] i ..."
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Cited by 20 (10 self)
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We propose a new algorithm for constructing finiteelement meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain# and a target time value T , our method constructs a tetrahedral mesh of the spacetime domain [0, T] in constant running time per tetrahedron in IR using an advancing front method. Elements are added to the evolving mesh in small patches by moving a vertex of the front forward in time. Spacetime discontinuous Galerkin methods allow the numerical solution within each patch to be computed as soon as the patch is created. Our algorithm employs new mechanisms for adaptively coarsening and refining the front in response to a posteriori error estimates returned by the numerical code. A change in the front induces a corresponding refinement or coarsening of future elements in the spacetime mesh. Our algorithm adapts the duration of each element to the local quality, feature size, and degree of refinement of the underlying space mesh. We directly exploit the ability of discontinuous Galerkin methods to accommodate discontinuities in the solution fields across element boundaries.
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 MultiField SpaceTime Finite Element Method for Structural Acoustics. Symposium on Acoustics of Submerged Structures and Transduction Systems
 ASME 15th Biennial Conference on Mechanical Vibration and Noise
, 1995
"... A spacetime nite element method for solution of the exterior structural acoustics problem involving the interaction of vibrating elastic structures submerged in an innite acoustic fluid is formulated. In particular, new timediscontinuous Galerkin and Galerkin LeastSquares (GLS) variational formu ..."
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Cited by 13 (8 self)
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A spacetime nite element method for solution of the exterior structural acoustics problem involving the interaction of vibrating elastic structures submerged in an innite acoustic fluid is formulated. In particular, new timediscontinuous Galerkin and Galerkin LeastSquares (GLS) variational formulations for coupled structural acoustics in unbounded domains are developed and analyzed for stability and convergence. The formulation employs a nite computational fluid domain surrounding the structure and incorporates timedependent nonreflecting boundary conditions on the fluid truncation boundary. Energy estimates are obtained which allow us to prove the unconditional stability of the method for the coupled fluidstructure problem with absorbing boundaries. The methods developed are especially useful for the application of adaptive solution strategies for transient acoustics in which unstructured spacetime meshes are used to track waves propagating along spacetime characteristics. An important feature of the spacetime formulation is the incorporation of temporal jump operators which allow for nite element interpolations that are discontinuous in time. For additional stability, leastsquares operators based on local residuals of the structural acoustics equations including the nonreflecting boundary conditions are incorporated. The energy decay estimates and highorder accuracy predicted by our a priori error estimates are demonstrated numerically in a simple canonical example.
Wave propagation near a fluidsolid interface: A spectralelement approach
, 2000
"... We introduce a spectralelement method for modeling wave propagation inmediawith both fluid (acoustic) and solid (elastic) regions, as for instance in offshore seismic experiments. The problem is formulated in terms of displacement in elastic regions and a velocity potential in acoustic regions. M ..."
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Cited by 12 (6 self)
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We introduce a spectralelement method for modeling wave propagation inmediawith both fluid (acoustic) and solid (elastic) regions, as for instance in offshore seismic experiments. The problem is formulated in terms of displacement in elastic regions and a velocity potential in acoustic regions. Matching between domains is implemented based upon an interface integral in the framework of an explicit predictionmulticorrection staggered time scheme. The formulation results in a mass matrix that is diagonal by construction. The scheme exhibits high accuracy for a 2D test case with known analytical solution. The method is robust in the case of strong topography at thefluidsolid interface and is a goodalternative to classical techniques, such as finite differencing.
TentPitcher: A Meshing Algorithm For SpaceTime Discontinuous Galerkin Methods
 IN PROC. 9TH INT’L. MESHING ROUNDTABLE
, 2000
"... Spacetime discontinuous Galerkin (DG) methods provide a solution for a wide variety of numerical problems such as inviscid Berger's equation and elastodynamic analysis. Recent research shows that in order to solve a DG system using an elementbyelement procedure, the spacetime mesh has to sa ..."
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Cited by 12 (2 self)
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Spacetime discontinuous Galerkin (DG) methods provide a solution for a wide variety of numerical problems such as inviscid Berger's equation and elastodynamic analysis. Recent research shows that in order to solve a DG system using an elementbyelement procedure, the spacetime mesh has to satisfy a cone constraint, i.e. that the faces of the mesh can not be steeper in the time direction than a specified angle function ff(). Whenever there is a face that violates the cone constraint, the elements at the face must be coupled in the solution. In this paper we consider the problem of generating a simplicial spacetime mesh where the size of each group of elements that need to be coupled is bounded by a constant number k. We present an algorithm for generating such meshes which is valid for any nD\ThetaTIME domain (n is a natural number). The k in the algorithm is based on a node degree in a ndimensional space domain mesh.
Midfrequency vibroacoustic modelling: challenges and potential solutions
 In Proceedings of ISMA 2002
, 2002
"... At present, the main numerical modelling techniques for acoustic and (coupled) vibroacoustic analysis are based on element based techniques, such as the finite element and boundary element method. Due to the huge computational efforts, the use of these deterministic techniques is practically restri ..."
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Cited by 10 (5 self)
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At present, the main numerical modelling techniques for acoustic and (coupled) vibroacoustic analysis are based on element based techniques, such as the finite element and boundary element method. Due to the huge computational efforts, the use of these deterministic techniques is practically restricted to lowfrequency applications. For highfrequency modelling, some alternative, probabilistic techniques such as SEA have been developed. However, there is still a wide midfrequency range, for which no adequate and mature prediction techniques are available at the moment. In this frequency range, the computational efforts of conventional element based techniques become prohibitively large, while the basic assumptions of the probabilistic techniques are not yet valid. In recent years, a vast amount of research has been initiated in a quest for an adequate solution for the current midfrequency problem. This paper discusses the various methodologies that are being explored in this perspective. The main focus of this paper lies on the methodology that looks for deterministic techniques with an enhanced convergence rate and computational efficiency compared to the conventional element based methods in order to shift the practical frequency limitation towards the midfrequency range. In this respect, special attention is paid to the wave based prediction technique for (coupled) vibroacoustic analysis that is being developed at the KULeuven Noise and Vibration Research group. The method is based on an indirect Trefftz approach. Various recent validations have revealed the beneficial convergence rate of this novel technique, thereby exhibiting its potential to comply with the midfrequency modelling challenge. 1.
A MatrixFree Interpretation Of The NonLocal DirichletToNeumann Radiation Boundary Condition
 International Journal for Numerical Methods in Engineering
, 1995
"... This communication describes an efficient implementation of the nonlocal DirichlettoNeumann (DtN) radiation boundary condition which arises in the solution of exterior problems in acoustics. Exterior problems in acoustics involve unbounded fluid domains whose finite element solution requires the ..."
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Cited by 8 (2 self)
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This communication describes an efficient implementation of the nonlocal DirichlettoNeumann (DtN) radiation boundary condition which arises in the solution of exterior problems in acoustics. Exterior problems in acoustics involve unbounded fluid domains whose finite element solution requires the introduction of a truncation boundary in order to obtain a finite computational domain. The nonlocal DtN condition is an exact nonreflecting boundary condition which is imposed on this truncation boundary. Unfortunately, the discretization of the nonlocal DtN boundary condition results in a dense, fully populated matrix whose storage and factorization become increasingly expensive. We describe here a matrixfree interpretation of the nonlocal DtN map suitable for iterative solution methods, which allows the use of this exact boundary condition without any storage penalties related to its nonlocal nature. key words: acoustics; radiation boundary conditions; nonlocal operators; matrixfr...
Parallel Preconditioning Based on hHierarchical Finite Elements with Application to Acoustics
 Stanford University
, 1995
"... In this paper we investigate a multilevel preconditioning approach based on the hversion of the hierarchical finite element method. Finite element formulations that employ hierarchical shape functions yield better conditioned matrix problems than formulations based on the usual Lagrange basis funct ..."
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Cited by 8 (4 self)
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In this paper we investigate a multilevel preconditioning approach based on the hversion of the hierarchical finite element method. Finite element formulations that employ hierarchical shape functions yield better conditioned matrix problems than formulations based on the usual Lagrange basis functions. This improved conditioning leads to a faster rate of convergence when gradienttype iterative solution methods are used. However, global interactions between shape functions in the multilevel hhierarchic basis result in significant computational complexity of hhierarchic formulations. To avoid such shortcomings, we perform all finite element computations in the nodal basis, and employ projections between the nodal and hierarchical bases to construct a preconditioning operator. We apply the preconditioning approach for the iterative solution of linear systems arising from finite element discretization of the exterior Helmholtz equation. Numerical results are presented to examine conve...