Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case

Abstract

In this paper, we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology, based on integral-equations, for the numerical solution of scattering problems at high frequencies. In more detail, here we extend the two-dimensional results in part I of this work to fully threedimensional geometries. As in the former case, our concern here is the determination of the rate of convergence of the multiple-scattering iterations for a collection of three-dimensional convex obstacles that are inherent in the aforementioned high-frequency schemes. To this end, we follow a similar strategy to that we devised in part 123 374 A. Anand et al.I: first, we recast the (iterated, Neumann) multiple-scattering series in the form of a sum of periodic orbits (of increasing period) corresponding to multiple reflections that periodically bounce off a series of scattering sub-structures; then, we proceed to derive a high-frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically; and, finally, we analyze this recurrence to provide an explicit rate of convergence associated with each orbit. While the procedure is analogous to its two-dimensional counterpart, the actual analysis is significantly more involved and, perhaps more interestingly, it uncovers new phenomena that cannot be distinguished in two-dimensional configurations (e.g. the further dependence of the convergence rate on the relative orientation of interacting structures). As in the two-dimensional case, and beyond their intrinsic interest, we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple-scattering series and, thus, to provide significant savings in computational times.

Boundary integral methods in high frequency scattering

“…Each orbit corresponds to reflections off a fixed set of scatterers, and this allows the convergence rate of the Neumann series to be estimated, for sufficiently high frequency and permits the formulation of methods for accelerating its convergence. The most recent work in this direction [5] extended the analysis to the three dimensional case, where additional considerations on the relative orientation of the scattering bodies come into play.…”

“…In [31,32,5] the emphasis is on the convergence of the Neumann series, and assumes the robust solution of the integral equations arising at each iteration. Thus the proof of the k−robustness of the overall algorithm remains a challenging open problem.…”

Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems

“…Bruno et al [12] presented an approach with complexity independent of wavelength by restricting the interval over which boundary integrals are performed to small regions in the immediate vicinity of stationary points; Langdon and Chandler-Wilde [13] have shown that this approach is suitable for polygonal scatterers; Domínguez et al [14] demonstrated that, for problems of asymptotically large wavenumbers, the required number of degrees of freedom increases only with O(k 1/9 ), for a fixed error bound; Anand et al [15] extended this approach for problems of multiple scatterers.…”

Novel basis functions for the partition of unity boundary element method for Helmholtz problems

“…This varying function can then be obtained by approximating it about the boundary of the scatterer using a boundary element scheme. Bruno et al [7] have shown the complexity of this approach to be independent of the wavenumber, presenting results for scatterers of dimension 10 6 λ; Langdon and Chandler-Wilde [8] have shown that this approach is suitable for polygonal scatterers; Domínguez et al [9] demonstrated that, to maintain a fixed error bound for problems of asymptotically large wavenumbers, the required number of degrees of freedom increases only with Ok 1/9 ; Anand et al [10] have extended the approach for problems of multiple scatterers.…”