A High Frequency Boundary Element Method for Scattering by Convex Polygons with Impedance Boundary Conditions

Chandler-Wilde, Simon N.; Langdon, Stephen; Mokgolele, M.

doi:10.4208/cicp.231209.040111s

Cited by 22 publications

(31 citation statements)

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“…Applying this idea within a boundary element method (BEM) context is particularly attractive since in this case one need only understand the high frequency behaviour on the boundary of the scatterer, rather than throughout the whole propagation domain. Computational methods implementing this approach have been applied successfully to problems of scattering by both smooth [8,21,22,26] and non-smooth [15,20,2,29,16] convex scatterers, the latter building on previous work on hybrid h-version BEM methods for the special problem of acoustic scattering in a half-space with impedance boundary conditions [28].…”

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A High Frequency $hp$ Boundary Element Method for Scattering by Convex Polygons

Hewett¹,

Langdon²,

Melenk³

2013

SIAM J. Numer. Anal.

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Abstract. In this paper we propose and analyse a hybrid hp boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.Key words. High frequency scattering, Boundary Element Method, hp-method AMS subject classifications. 65N12, 65N38, 65R201. Introduction. Conventional numerical schemes for time-harmonic acoustic scattering problems, with piecewise polynomial approximation spaces, become prohibitively expensive in the high frequency regime where the scatterer is large compared to the wavelength of the incident wave. For two-dimensional problems the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to frequency. On the other hand, approximation via high frequency asymptotics alone is often insufficiently accurate when the frequency lies within ranges of practical interest. These issues are very well understood; see e.g. [9,10,37,30,12] and the many references therein.The problem of 'bridging the gap' between conventional numerical methods and fully asymptotic approaches has received a great deal of attention in recent years. Significant progress has been made in developing numerical methods which can achieve a prescribed level of accuracy at high frequencies with fewer degrees of freedom than conventional approaches. A key idea underpinning much recent work is to express the scattered field as a sum of products of known oscillatory functions, selected using knowledge of the high frequency asymptotics, with slowly oscillating amplitude functions, and to approximate just the amplitudes by piecewise polynomials (we call this the hybrid numerical-asymptotic approach). Applying this idea within a boundary element method (BEM) context is particularly attractive since in this case one need only understand the high frequency behaviour on the boundary of the scatterer, rather than throughout the whole propagation domain. Computational methods implementing this approach have been applied successfully to problems of scattering by both smooth [8, 21, 22, 26] and non-smooth [15, 20, 2, 29, 16] convex scatterers, the latter building on previous work on hybrid h-version BEM methods for the special problem of acoustic scat...

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confidence: 99%

A High Frequency $hp$ Boundary Element Method for Scattering by Convex Polygons

Hewett¹,

Langdon²,

Melenk³

2013

SIAM J. Numer. Anal.

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“…In this paper, we have described how to implement the boundary element method developed in 4 for high frequency scattering by convex polygons with impedance boundary conditions. We explaine in detail how to solve the resulting system of linear equation.…”

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“…7 Various options exist to overcome this difficulty. 4 We begin by defining some notations, see Fig. 1.…”

Section: Boundary Integral Equation Methodsmentioning

confidence: 99%

“…(2) -(4) can be reformulated into boundary integral equation by applying Green's rep- resentation theorems [7.12 and 9.6] 6 and Green's second theorem [theorem 4.4], 6 (see also 4 for details), which leads to standard boundary integral equation for unknown u t :…”

Section: Boundary Integral Equation Methodsmentioning

confidence: 99%

“…In Section 6 we choose an example for our numerical experiment and present some results, whereas most of them can be found in. 4,5 We discuss our conclusion and some recommendations in Section 7.…”

Section: Introductionmentioning

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Implementation of a Boundary Element Method for High Frequency Scattering by Convex Polygons with Impedance Boundary Conditions

Mokgolele¹

2016

IJAV

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Many acoustic and electromagnetic wave scattering problems can be formulated as the Helmholtz equation. Standard finite and boundary element method solution of these problems becomes expensive, as the frequency of incident wave increases. On going research has been devoted to finding methods that do not loose robustness when the wave number increases. Recently, Chandler-Wilde et al. have proposed a novel Galerkin boundary element method to solve the problem of acoustic scattering by a convex polygon with impedance boundary conditions. They applied approximation spaces consisting of piecewise polynomials supported on a graded mesh with smaller elements adjacent to the corners of the polygon and multiplied by plane wave basis functions. They demonstrated via rigorous error analysis that was supported by numerical experiments that the number of degrees of freedom required to achieve a prescribed level of accuracy need only grow logarithmically as frequency increases. In this paper, we discuss issues related to detail implementation of their numerical method.

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Measuring the effect of rounding the corners of scattering structures

Markowskei

Smith

2017

Radio Sci.

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In studying acoustic or electromagnetic wave diffraction, the choice of an appropriate canonical structure to model the dominant features of a scattering scenario can be very illuminating. A common approach used when dealing with domains with corners is to round the corners, producing a smooth surface, eliminating the singularities introduced by the corners. In order to quantify the effect of corner rounding, this paper examines the diffraction from cylindrical scatterers which possess corners, that is, points at which the normal changes discontinuously. We develop a numerical method for the scattering of a plane wave normally incident on such cylindrical structures with soft, hard, or impedance loaded boundary conditions. We then examine the difference between various test structures with corners and with the corners rounded to assess the impact on near‐ and far‐field scattering, as a function of the radius of curvature in the vicinity of the rounded corner point. We then examine the nature of the differences in the far field between the cornered and rounded scatterers as well as the effect on the differences as the frequency of the plane wave increases and obtain precise quantitative estimates for the rate of convergence of the maximum difference between the far‐field solutions as the radius of curvature of the rounded scatterer approaches zero.

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A High Frequency Boundary Element Method for Scattering by Convex Polygons with Impedance Boundary Conditions

Cited by 22 publications

References 17 publications

A High Frequency $hp$ Boundary Element Method for Scattering by Convex Polygons

A High Frequency $hp$ Boundary Element Method for Scattering by Convex Polygons

Implementation of a Boundary Element Method for High Frequency Scattering by Convex Polygons with Impedance Boundary Conditions

Measuring the effect of rounding the corners of scattering structures

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