Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed

Laghrouche, Omar; Bettess, P.; Perrey‐Debain, Emmanuel; Trevelyan, J.

doi:10.1016/j.cma.2003.12.074

Cited by 59 publications

(64 citation statements)

References 21 publications

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“…Seguindo o procedimento usual [1,4], tem-se que a forma fraca da equação de Helmholtz em cada regiãoé dada por…”

Section: Formulaçãounclassified

“…As restrições ao longo da interface Γ são garantidas através do ML e finalmente encontra-se o sistema resultante do MEFG, como em [2,4].…”

Section: Formulaçãounclassified

“…Para isto, foi desenvolvido uma metodologia que permite encontrar os nós e pesos de Gauss que estão melhores distribuídos no elemento de referência triangular. Considerando que neste trabalho o MEFG será aplicado para resolver um problema com mudança de meio, optou-se em utilizar o Multiplicador de Lagrange (ML) para garantir a continuidade entre os diferentes materiais [2,4]. No que segue, será apresentado a formulação fraca de um problema de dispersão ou propagação eletromagnética onde o domínioé formado por duas regiões de diferentes materiais.…”

Section: Introductionunclassified

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Quadratura de Gauss de Alta Ordem Adaptativa no Método dos Elementos Finitos Generalizados

Facco¹,

Bastos²,

Moura³

et al. 2018

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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“…Seguindo o procedimento usual [1,4], tem-se que a forma fraca da equação de Helmholtz em cada regiãoé dada por…”

Section: Formulaçãounclassified

“…As restrições ao longo da interface Γ são garantidas através do ML e finalmente encontra-se o sistema resultante do MEFG, como em [2,4].…”

Section: Formulaçãounclassified

Section: Introductionunclassified

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Quadratura de Gauss de Alta Ordem Adaptativa no Método dos Elementos Finitos Generalizados

Facco¹,

Bastos²,

Moura³

et al. 2018

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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“…Most commonly the approximation space consists of standard finite element basis functions multiplied by plane waves traveling in a large number of directions, approximately uniformly distributed on the unit circle (in 2D) or sphere (in 3D). This is the approach in the generalized finite element method of Babuška and Melenk [4], the ultra weak variational formulation of Cessenat and Després [13,14], and the least squares method of Monk and Wang [41]; see also [43,32,37]. In the boundary element context this approach is used in the microlocal discretization method of de La Bourdonnaye et al [26,27] and in the work of Perrey-Debain et al [44,45,43,46].…”

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confidence: 99%

A Wavenumber Independent Boundary Element Method for an Acoustic Scattering Problem

Langdon¹,

Chandler-Wilde²

2006

SIAM J. Numer. Anal.

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Abstract. In this paper we consider the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data, a problem which models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we propose a novel Galerkin boundary element method, using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that, on each element, the approximation space contains polynomials (of degree ν) multiplied by traces of plane waves on the boundary. We prove stability and convergence and show that the error in computing the total acoustic field is O(N −(ν+1) log 1/2 N ), where the number of degrees of freedom is proportional to N log N . This error estimate is independent of the wavenumber, and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity.

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“…Yet, the both methods can be extended for problems with jumps in material parameters (see e.g. [17] and [13]). In the PUFEM, this is done using the Lagrange multipliers whereas in UWVF the extension is needs only minor modifications to the form used in this study.…”

mentioning

confidence: 99%

Comparison of two wave element methods for the Helmholtz problem

Huttunen

Gamallo

Astley

2008

Commun. Numer. Meth. Engng.

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In comparison with low‐order finite element methods (FEMs), the use of oscillatory basis functions has been shown to reduce the computational complexity associated with the numerical approximation of Helmholtz problems at high wave numbers. We compare two different wave element methods for the 2D Helmholtz problems. The methods chosen for this study are the partition of unity FEM (PUFEM) and the ultra‐weak variational formulation (UWVF). In both methods, the local approximation of wave field is computed using a set of plane waves for constructing the basis functions. However, the methods are based on different variational formulations; the PUFEM basis also includes a polynomial component, whereas the UWVF basis consists purely of plane waves. As model problems we investigate propagating and evanescent wave modes in a duct with rigid walls and singular eigenmodes in an L‐shaped domain. Results show a good performance of both methods for the modes in the duct, but only a satisfactory accuracy was obtained in the case of the singular field. On the other hand, both the methods can suffer from the ill‐conditioning of the resulting matrix system. Copyright © 2008 John Wiley & Sons, Ltd.

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Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed

Cited by 59 publications

References 21 publications

Quadratura de Gauss de Alta Ordem Adaptativa no Método dos Elementos Finitos Generalizados

Quadratura de Gauss de Alta Ordem Adaptativa no Método dos Elementos Finitos Generalizados

A Wavenumber Independent Boundary Element Method for an Acoustic Scattering Problem

Comparison of two wave element methods for the Helmholtz problem

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