Scattering by quasi-symmetric pipes

Carley, Michael

doi:10.1121/1.2159432

Cited by 7 publications

(4 citation statements)

References 17 publications

(13 reference statements)

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“…The Green's functions for the Laplace equation will have a logarithmic singularity in the planar and axisymmetric case (where the Green's function is proportional to an elliptic integral) and also in the case of an asymmetric problem in an axisymmetric domain [3,6]. In each case, discretization of the boundary of the domain and use of a collocation method give rise to integrals of the form…”

Section: Introduction Boundary Integral Methods Employing Hypersingumentioning

confidence: 99%

“…Introduction. Boundary integral methods employing hypersingular integrals have become increasingly popular over recent decades with applications in potential problems such as acoustics [6] and fracture mechanics [13]. Hypersingular integrals arise naturally when it is required to compute the field quantities in such problems, for example, the potential and its gradients, and when specialized techniques for the avoidance of "interior resonance" are used, such as that of Burton and Miller [5].…”

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confidence: 99%

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Numerical Quadratures for Singular and Hypersingular Integrals in Boundary Element Methods

Carley¹

2007

SIAM J. Sci. Comput.

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Abstract.A method is developed for the computation of the weights and nodes of a numerical quadrature which integrates functions containing singularities up to order 1/x 2 , without the requirement to know the coefficients of the singularities exactly. The work is motivated by the need to evaluate such integrals on boundary elements in potential problems and is a simplification of a previously published method, but with the advantage of handling singularities at the endpoints of the integral. The numerical performance of the method is demonstrated by application to an integral containing logarithmic, first, and second order singularities, characteristic of the problems encountered in integrating a Green's function in boundary element problems. It is found that the quadrature is accurate to 11-12 decimal places when computed in double precision. 1. Introduction. Boundary integral methods employing hypersingular integrals have become increasingly popular over recent decades with applications in potential problems such as acoustics [6] and fracture mechanics [13]. Hypersingular integrals arise naturally when it is required to compute the field quantities in such problems, for example, the potential and its gradients, and when specialized techniques for the avoidance of "interior resonance" are used, such as that of Burton and Miller [5]. When such integrals arise, they require special numerical treatment and cannot be evaluated using the standard tools of Gaussian quadrature. Instead, approaches tailored to the problem must be used, which can add considerable complexity to the code. This paper introduces a method for the design of numerical quadrature rules which evaluate hypersingular integrals without requiring a detailed analysis of the integrand. The method is based on that of Kolm and Rokhlin [12], who developed a procedure for the design of such rules, but is considerably simplified by the use of Brandão's approach to finite part integrals [4] and is extended to the case of integrals with endpoint singularities-essential if such a scheme is to be used with boundary elements.To fix ideas, we assume that we are dealing with a two-dimensional potential problem, such as the Laplace or Helmholtz equation:

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Section: Introduction Boundary Integral Methods Employing Hypersingumentioning

confidence: 99%

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confidence: 99%

Numerical Quadratures for Singular and Hypersingular Integrals in Boundary Element Methods

Carley¹

2007

SIAM J. Sci. Comput.

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“…The boundary integral methods (BEM) are a powerful technique for 3-D problems in acoustics and electromagnetics [1][2][3][4]. However, for radiating of revolution structures, the axisymmetric BEM formulation becomes necessary [5][6][7][8][9][10][11]. The aim of this work is to develop an axisymmetric integral variational formulation derived from the three-dimensional variational formulation for the radiation problems of thin bodies of revolution.…”

Section: Introductionmentioning

confidence: 99%

Acoustic Radiation of Axisymmetric Thin Bodies by Integral Variational Method

Beldi

Zarrouk

2012

AMR

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In this paper, a variational formulation by integral equations for the study of acoustic radiation by thin axisymmetric bodies is developed. This new approach derives from the three-dimensional variational formulation. It is based on the Fourier decomposition with respect to the angle of revolution. The three-dimensional problem is reduced to the resolution of several two-dimensional problems. Thus, by construction, the obtained axisymmetric variational equation is prepared to the numerical calculations because it avoids the regularisation of the double normal derivative of modal Green’s function. As for the Fourier coefficients of the singular part of Green’s function and its normal derivative, they are evaluated precisely by the same recurrence relation expressed in terms of the complete elliptic integrals. In addition, the axisymmetric free term derived from the 3-D solid angle, is given by a new expression. Numerical results clearly demonstrate the accuracy of this approach to predict the acoustic fields particularly on corners.

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“…In the case of the Helmholtz equation, the singular behaviour of the Green's function will be related to the Green's function of the corresponding Laplace equation [8, for example]. The Green's functions for the Laplace equation will have a logarithmic singularity in the planar and axisymmetric case (where the Green's function is proportional to an elliptic integral) and also in the case of an asymmetric problem in an axisymmetric domain [3,5]. The Green's function for the two-dimensional Laplace equation is G = log |x − x 1 |.…”

Section: Introductionmentioning

confidence: 99%

Numerical Quadratures for Near-Singular and Near-Hypersingular Integrals in Boundary Element Methods

Carley¹

2009

Mathematical Proceedings of the Royal Irish Academy

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A method of deriving quadrature rules has been developed which gives nodes and weights for a Gaussian-type rule which integrates functions of the form:without having to explicitly analyze the singularities of f (x, y, t) or separate it into its components. The method extends previous work on a similar technique for the evaluation of Cauchy principal value or Hadamard finite part integrals, in the case when y ≡ 0. The method is tested by evaluating standard reference integrals and its error is found to be comparable to machine precision in the best case.

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Scattering by quasi-symmetric pipes

Cited by 7 publications

References 17 publications

Numerical Quadratures for Singular and Hypersingular Integrals in Boundary Element Methods

Numerical Quadratures for Singular and Hypersingular Integrals in Boundary Element Methods

Acoustic Radiation of Axisymmetric Thin Bodies by Integral Variational Method

Numerical Quadratures for Near-Singular and Near-Hypersingular Integrals in Boundary Element Methods

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