A rapid boundary perturbation algorithm for scattering by families of rough surfaces

Abstract

a b s t r a c tIn this paper we describe a novel algorithm for the computation of scattering returns by families of rough surfaces. This algorithm makes explicit use of the fact that some scattering profiles of engineering interest (e.g., traveling ocean waves) come in branches parameterized analytically by a bifurcation quantity. Our approach delivers recursions which not only can be implemented to yield a rapid, robust and high-order numerical scheme, but also give a new proof of analyticity of scattering quantities with respect to the bifurcation parameter of the surface family. The real advantage of this new approach is that it computes, in one step, the scattered field for all possible members of the family of surfaces. By contrast, other state-of-the-art schemes must restart when the returns from a new surface are desired, so that the cost of our new approach is greatly advantaged when the number of samples of the family reaches even modest values. Numerical results which verify the accuracy of our approach and demonstrate their utility in computing grating efficiencies scattered by traveling surface ocean waves are presented.

A Boundary Perturbation Method for Vector Electromagnetic Scattering from Families of Doubly Periodic Gratings

“…This work generalizes the recent contribution of one of the authors [6] from the case of the scattering of acoustic waves in two dimensions by a rough interface (governed by the scalar Helmholtz equation), to not only three dimensions, but also the full vector electromagnetic Maxwell equations. Each of these generalizations provides its own set of new challenges including vast new memory and computational time requirements, and the need to deal with the vector Helmholtz equations coupled by the surface boundary conditions.…”

“…As we have seen in previous publications [6][7][8][9] a well-chosen change of variables can have extremely beneficial effects for both the analysis and performance of Boundary Perturbation methods. As before we make the following "domain flattening" change of variables (the C-method in electromagnetics [2] or σ -coordinates in oceanography [13]):…”

“…this condition at infinity can be stated rigorously in the near-field (at z = a, a > |g| L ∞ ) by means of the Rayleigh expansion [6]:…”

“…For us the advantage of the factorization (2.9) is that while the boundary data for E at z = g contains somewhat complicated, explicit, dependence upon the boundary shape g, the factored boundary data is much simpler and thus more amenable to our techniques. It is not difficult to show [6] that the change of dependent variable (2.9) transforms the governing equations (2.7) to…”

“…Finally, a Boundary Perturbation algorithm, resulting in a high-order Fourier collocation/Chebyshev tau/Taylor method, is applied resulting in a highly accurate and robust computational approach. This change of variables coupled to a Boundary Perturbation approach (the method of "Transformed Field Expansions"-TFE) has been utilized in a number of contexts recently, including Laplace's equation [7], the scalar Helmholtz equation [6,9], and the classical water wave problem [10] with great success. As we find later, this success is reproduced here resulting in a compelling algorithm for the scenario we describe.…”