On wave boundary elements for radiation and scattering problems with piecewise constant impedance

Abstract

Publisher's copyright statement: c 2005 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Additional information:Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. wall width is properly selected, the fundamental resonant frequency of inner triangular PIFA can be coupled with the resonance that is introduced by the V-slot, and consequently a wide-band operation is obtained in the higher frequency band. A large operating bandwidth of 36% has been demonstrated. In addition, the radiation patterns across the entire operating bands are also measured; the antenna gain is about 4 and 4.3 dBi for the lower and higher frequency bands, respectively. REFERENCES On Wave Boundary Elements for Radiation and Scattering Problems With Piecewise Constant ImpedanceEmmanuel Perrey-Debain, Jon Trevelyan, and Peter BettessAbstract-Discrete methods of numerical analysis have been used successfully for decades for the solution of problems involving wave diffraction, etc. However, these methods, including the finite element and boundary element methods, can require a prohibitively large number of elements as the wavelength becomes progressively shorter. In this paper, a new type of interpolation for the wave field is described in which the usual conventional shape functions are modified by the inclusion of a set of plane waves propagating in multiple directions. Including such a plane wave basis in a boundary element formulation is found in this paper to be highly successful. Results are shown for a variety of scattering/radiating problems from convex and nonconvex obstacles on which are prescribed piecewise constant Robin conditions. Notable results include a conclusion that, using this new formulation, only approximately three degrees of freedom per wavelength are required.

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

“…Another option is to use the so-called CHIEF scheme, the basic idea of which is to overdetermine the solution of (3.4) with equations arising from enforcing a version of (3.3) at interior points x ∈ Ω called the CHIEF points. This scheme, proposed in [20], is used, for example, in [18].…”

“…The problem (1.1)-(1.3) has received significant recent attention in the literature [1,18,21]. Standard boundary or finite element approximations suffer from the requirement that the number of degrees of freedom must increase at least linearly with respect to k in order to maintain accuracy.…”

“…(1.1) and (1.3) with the boundary condition γ + u t = 0 on Γ replacing (1.2). Using a boundary element approach, with a hybrid approximation space consisting of the product of plane waves with piecewise polynomials, very efficient schemes have been developed for scattering by smooth obstacles [10] and by convex polygons [6], with in each case the number of degrees of freedom required to achieve a prescribed level of accuracy depending only very mildly on k. In contrast, the only numerical scheme for (1.1)-(1.3) that we are aware of that has been developed specifically for the purpose of efficiency at high frequencies is that in [18], where a circle of piecewise constant impedance is considered. There the approximation space is enriched with plane waves travelling in multiple directions; this reduces the number of degrees of freedom required per wavelength from ten to three (for engineering accuracy), but the cost still grows linearly with k. For a full review of recent developments in integral equation methods for high frequency scattering we refer to [3].…”

A Wavenumber Independent Boundary Element Method for an Acoustic Scattering Problem

“…This idea has been applied to both finite element (e.g., [4,13,41,29,14,43]) and boundary element schemes [26,1,24,44,45,20,43,27,10,46,28]. Promising numerical results are reported, but most of the papers are lacking in mathematical analysis, especially with regard to how any error estimates depend on the wavenumber k. As the present paper follows the same general approach of enriching the approximation space, we survey this body of work in a little more detail.…”

“…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]. The theoretical analysis carried out (e.g., [4]) and computational results (e.g., [43]) confirm that these methods converge very rapidly as the number of plane wave directions used increases.…”