Finite Element Solution of the Helmholtz Equation with High Wave Number Part II: The h-p Version of the FEM

Ihlenburg, Frank; Babuška, Ivo

doi:10.1137/s0036142994272337

Cited by 350 publications

(308 citation statements)

References 15 publications

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“…With suitable boundary conditions the Helmholtz equation (5) can be solved using a variety of numerical methods [10][11][12][13][14] . The accuracy of the numerical solution from the Helmholtz equation depends significantly on the wavenumber, κ (κ=ω/c).…”

Section: Simulationsmentioning

confidence: 99%

Optimisation of 20 kHz sonoreactor geometry on the basis of numerical simulation of local ultrasonic intensity and qualitative comparison with experimental results

Klíma

Frías-Ferrer

González-Garcı́a

et al. 2007

Ultrasonics Sonochemistry

128

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The intensity distribution of the ultrasonic energy is, after the frequency, the most significant parameter to characterize ultrasonic fields in any sonochemical experiment.Whereas in the case of low intensity ultrasound the measurement of intensity and its distribution is well solved, in the case of high intensity (when cavitation takes place) the measurement is much more complicated. That is why the predicting the acoustic pressure distribution within the cell is desirable.A numerical solution of the wave equation gave the distribution of intensity within the cell. The calculations together with experimental verification have shown that the whole reactor behaves like a resonator and the energy distribution depends strongly on its shape.The agreement between computational simulations and experiments allowed optimisation of the shape of the sonochemical reactor. The optimal geometry resulted in a 2 strong increase in intensity along a large part of the cell. The advantages of such optimised geometry are (i) the ultrasonic power necessary for obtaining cavitation is low, (ii) low power delivered to the system results in only weak heating; consequently no cooling is necessary and (iii) the "active volume" is large, i.e. the fraction of the reactor volume with high intensity is large and is not limited to a vicinity close to the horn tip.

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Section: Simulationsmentioning

confidence: 99%

Optimisation of 20 kHz sonoreactor geometry on the basis of numerical simulation of local ultrasonic intensity and qualitative comparison with experimental results

Klíma

Frías-Ferrer

González-Garcı́a

et al. 2007

Ultrasonics Sonochemistry

128

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“…We note that error estimates for finite element approximations to the Helmholtz equation (2.1) with high wave numbers were derived in [8,18,19] for the 1-D case, and in [9,6] for 2-D cases and in [12] for the 1-D Bessel equation reduced from a 3-D spherical domain, respectively.…”

Section: Model Equation and A Priori Estimatesmentioning

confidence: 99%

“…The first test function is the usual one. As in [7], the key step is to choose a suitable second test function which enables us to obtain a priori estimates without using the Green's functions as in [8,18,19]. In the following proof, ε j > 0, 1 ≤ j ≤ 5, are adjustable real numbers.…”

Section: A Priori Estimatesmentioning

confidence: 99%

“…Remark 3.2. For n = 1, an error estimate of the same order as in (3.9) was derived in [19] for the hp FEM under the condition kh 1. Our estimate is valid without any restriction on k and N and is bounded by a weaker weighted seminorm.…”

Section: Formula (39) Follows From Lemmas 31 and 32 And (37a)mentioning

confidence: 99%

“…The Galerkin finite element method (FEM) for (1.2) in the one-dimensional case was first carried out in [8], where the well-posedness and error estimates of the Galerkin FEM were established under the condition k 2 h 1 using the Green's function and an argument due to Schatz [21]. A refined analysis for (1.2) in the one-dimensional case was performed in [18] (resp., [19]) for the h version (resp., hp version) of FEM, where the well-posedness and error estimates were established under the condition kh 1 using the discrete Green's functions. The proofs in these works rely heavily on the use of explicit forms of continuous and/or discrete Green's functions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Spectral Approximation of the Helmholtz Equation with High Wave Numbers

Shen¹,

Wang

2005

SIAM J. Numer. Anal.

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Abstract.A complete error analysis is performed for the spectral-Galerkin approximation of a model Helmholtz equation with high wave numbers. The analysis presented in this paper does not rely on the explicit knowledge of continuous/discrete Green's functions and does not require any mesh condition to be satisfied. Furthermore, new error estimates are also established for multidimensional radial and spherical symmetric domains. Illustrative numerical results in agreement with the theoretical analysis are presented.

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Influence of the pollution on the admissible field error estimation for FE solutions of the Helmholtz equation

Bouillard

1999

Int. J. Numer. Meth. Engng.

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SUMMARYThe a posteriori error estimation in constitutive law has already been extensively developed and applied to "nite element solutions of structural analysis problems. The paper presents an extension of this estimator to problems governed by the Helmholtz equation (e.g. acoustic problems) that we have already partially reported, this paper containing informations about the construction of the admissible "elds for acoustics. Moreover, it has been proven that the upper bound property of this estimator applied to elasticity problems (the error in constitutive law bounds from above the exact error in energy norm) does not generally apply to acoustic formulations due to the presence of the speci"c pollution error. The numerical investigations of the present paper con"rm that the upper bound property of this type of estimator is veri"ed only in the case of low (non-dimensional) wave numbers while it is violated for high wave numbers due to the pollution e!ect.

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Finite Element Solution of the Helmholtz Equation with High Wave Number Part II: The h-p Version of the FEM

Cited by 350 publications

References 15 publications

Optimisation of 20 kHz sonoreactor geometry on the basis of numerical simulation of local ultrasonic intensity and qualitative comparison with experimental results

Optimisation of 20 kHz sonoreactor geometry on the basis of numerical simulation of local ultrasonic intensity and qualitative comparison with experimental results

Spectral Approximation of the Helmholtz Equation with High Wave Numbers

Influence of the pollution on the admissible field error estimation for FE solutions of the Helmholtz equation

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