Reducing Dispersion of Linear Triangular Elements for the Helmholtz Equation

Harari, Isaac; Nogueira, Carnot L.

doi:10.1061/(asce)0733-9399(2002)128:3(351)

Cited by 45 publications

(30 citation statements)

References 28 publications

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“…We remark that, from a general point of view, the direction 22.5 • is not far from being optimal to design a reliable GLS scheme for some densities n = 10 and 20. This is in accordance with the results derived by Harari and Nogueira in Reference [40]. As a consequence, from now on, we will consider this almost optimal direction in the GLS scheme.…”

Section: The Sound-hard Circular Cylindersupporting

confidence: 91%

Numerical accuracy of a Padé-type non-reflecting boundary condition for the finite element solution of acoustic scattering problems at high-frequency

Kechroud

Antoine

Soulaïmani

2005

Int. J. Numer. Meth. Engng

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SUMMARYThe present text deals with the numerical solution of two-dimensional high-frequency acoustic scattering problems using a new high-order and asymptotic Padé-type artificial boundary condition. The Padé-type condition is easy-to-implement in a Galerkin least-squares (iterative) finite element solver for arbitrarily convex-shaped boundaries. The method accuracy is investigated for different model problems and for the scattering problem by a submarine-shaped scatterer. As a result, relatively small computational domains, optimized according to the shape of the scatterer, can be considered while yielding accurate computations for high-frequencies.

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Section: The Sound-hard Circular Cylindersupporting

confidence: 91%

Numerical accuracy of a Padé-type non-reflecting boundary condition for the finite element solution of acoustic scattering problems at high-frequency

Kechroud

Antoine

Soulaïmani

2005

Int. J. Numer. Meth. Engng

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“…Nevertheless, because of the approximation involved it is not expected that the proportionality values in the numerical-spectral solution be equal to those of the exact-spectral solution. This in fact is usually referred to as dispersion and has been studied by many authors [26,27,[43][44][45][46][47][48]. Now a large mesh with repeatable pattern is considered.…”

Section: Finite Element Analysismentioning

confidence: 99%

“…We note that vectors k( 1 ) are normalized with respect to their lengths. In Equation (27) coefficients c k( 1 ) are to be determined from boundary conditions at x = 0 or y = 0.…”

Section: Remarkmentioning

confidence: 99%

“…The number of terms in the inner summation in (27) is dependent on the number of k( 1 ) for which 1 and 2 satisfy (24) and may vary when 1 takes different values.…”

Section: Remarkmentioning

confidence: 99%

“…However, with the proposed method one can still use remedies for reducing the dispersion effect. For such a purpose the reader can use methods like generalized finite element method [24], Galerkin least-squares FEM [25][26][27], partition of unity method [28][29][30][31][32][33], residual-free bubbles [34], the ultra weak variational method [35], subgrid modelling [36], least-squares method [37], residual-based FEM [38], or discontinuous Galerkin method proposed by Farhat et al [39,40]. Noteworthy is that most of the remedies mentioned focus on Helmholtz problems.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic solution of unbounded domains using finite element method: discrete Green's functions in frequency domain

Boroomand

Mossaiby

2006

Numerical Meth Engineering

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SUMMARYIn this paper we present a new approach for finite element solution of time-harmonic wave problems on unbounded domains. As representatives of the wave problems, discrete Green's functions are evaluated in finite element sense. The finite element mesh is considered to be of repeatable pattern (cell) constructed in rectangular co-ordinates. The system of FE equations is therefore reduced to a set of well-known dispersion equations by using a spectral solution approach. The spectral wave bases are constructed directly from the FE dispersion equations. Radiation condition is satisfied by selecting the wave bases so that the wave information is transmitted in appropriate directions at the cell level. Dirichlet/Neumann boundary conditions are defined at the edges of a quadrant of the main domain while using the axes of symmetry of the problem. A new discrete transformation method, recently proposed by the authors, is used to satisfy the boundary conditions. Comprehensive studies are made for showing the validity, accuracy and convergence of the solutions. The results of the benchmark problems indicate that the proposed method can be used to evaluate discrete Green's functions whose analytical forms are not available.

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Acoustics

Thompson

Pinsky

2004

Encyclopedia of Computational Mechanics

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State‐of‐the‐art computational methods for linear acoustics are reviewed. The equations of linear acoustics are summarized and then transformed to the frequency domain for time‐harmonic waves governed by the Helmholtz equation. Two major current challenges in the field are specifically addressed: numerical dispersion errors that arise in the approximation of short unresolved waves, polluting resolved scales and requiring a large computational effort, and the effective treatment of unbounded domains by domain‐based methods. A discussion of the indefinite sesquilinear forms in the corresponding weak form are summarized. A priori error estimates, including both dispersion (phase error) and global pollution effects for moderate to large wave numbers in finite element methods, are discussed. Stabilized and other wave‐based discretization methods are reviewed. Domain‐based methods for modeling exterior domains are described including Dirichlet‐to‐Neumann (DtN) methods, absorbing boundary conditions, infinite elements, and the perfectly matched layer (PML). Efficient equation‐solving methods for the resulting complex‐symmetric (non‐Hermitian) matrix systems are discussed including parallel iterative methods and domain decomposition methods including the FETI‐H method. Numerical methods for direct solution of the acoustic wave equation in the time domain are reviewed.

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Reducing Dispersion of Linear Triangular Elements for the Helmholtz Equation

Cited by 45 publications

References 28 publications

Numerical accuracy of a Padé-type non-reflecting boundary condition for the finite element solution of acoustic scattering problems at high-frequency

Numerical accuracy of a Padé-type non-reflecting boundary condition for the finite element solution of acoustic scattering problems at high-frequency

Dynamic solution of unbounded domains using finite element method: discrete Green's functions in frequency domain

Acoustics

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