An integral equation formulation for the diffraction from convex plates and polyhedra

Abstract

A new formulation of the problem of scattering from obstacles with edges is presented. The formulation is based on decomposing the field into geometrical optics and edge diffraction components, as opposed to the usual incident and scattered wavefields used for BEM and FEM methods. A secondary-source model is available for edgediffraction, which is extended to handle multiple diffraction of all orders. We show that the diffraction component can be represented by the solution of an integral equation formulated on pairs of edge points, so that the total field can be evaluated as as sum of geometrical optics and this diffraction component. Numerical experiments demonstrates the validity of the approach for wavenumbers down to 0.Keywords : Scattering, diffraction, integral equations, convex plates, convex polyhedra MSC : Primary : 65R20, Secondary : 45B05, 78A05, 78A45. An integral equation formulation for the diffraction from convex plates and polyhedraAndreas Asheim * U. Peter Svensson † March 5, 2012Abstract A new formulation of the problem of scattering from obstacles with edges is presented. The formulation is based on decomposing the field into geometrical optics and edge diffraction components, as opposed to the usual incident and scattered wavefields used for BEM and FEM methods. A secondary-source model is available for edge-diffraction, which is extended to handle multiple diffraction of all orders. We show that the diffraction component can be represented by the solution of an integral equation formulated on pairs of edge points, so that the total field can be evaluated as as sum of geometrical optics and this diffraction component. Numerical experiments demonstrates the validity of the approach for wavenumbers down to 0.

The diffracted field and its gradient near the edge of a thin screen

“…A recent integral equation formulation, which permits the computation of arbitrarily high orders of diffraction, has been shown to give consistent results down to 0 Hz for convex, rigid scatterers. 1 Low orders of diffraction have been employed also in highly complex geometries, using a time-domain formulation. 2 Also, comparisons between the boundary element method and an edge diffraction based formulation indicate the efficiency of the latter approach for outdoor sound propagation studies.…”

The Wave-Matching Boundary Integral Equation — An energy approach to Galerkin BEM for acoustic wave propagation problems

“…For a smooth boundary, as is considered herein, (22) and (23) can be shown to be non-singular, so can be computed using standard quadrature techniques. Equation (24) includes a stronger singularity, but in practice the matrix will be evaluated using the following statement [41,42], which only contains a weak singularity: (28) This leaves only weak singularities in (21) and (28); these were regularised using the Sato transform [58] of order 5, as has been shown to be optimal for weakly-singular oscillatory kernels [59].…”

“…Typically, most geometric algorithms are only concerned with reflections from planar boundary sections (centre), though it is also possible to handle low-order reflections from simply curved boundary sections, e.g. a circular cylinder (right), and algorithms that compute edge diffraction by truncating the infinite wedge canonical problem (left) also exist [21,22]. Some HNA-BEM algorithms can be considered to follow a similar principal in that they use different oscillatory functions on different boundary sections [13,14,23].…”

Recent advances in building acoustics: An overview of prediction methods and their applications

“…Asheim and Svensson [50] presented a frequency-domain edge source integral equation that efficiently handles the sum of all higher-order diffraction for rigid, external scattering objects, while computing first-order diffraction separately. Antani et al [51] adopted object-space visibility algorithms to improve the performance of finite-edge diffraction computation for geometrical propagation.…”