Simon Chandler-Wilde (Brunel. UK)

Integral equation methods for scattering by unbounded rough surfaces

Abstract

Rough surfaces, and problems of rough surface scattering abound in engineering and nature. They occur at a wide range of length scales, for example as small defects in manufacturing processes, for instance in the forming of sheet steel, or as larger undulations, for example water waves on the ocean surface, and variations in height in natural ground surfaces. The study, mathematical modelling, and computer simulation of rough surface scattering has therefore been of interest in diverse areas, including radar imaging, optics, and solid state physics. The basic problem that is studied in all of these contexts is that of a plane or other incident acoustic, elastic, or electromagnetic wave being scattered by a notionally infinite surface which is the graph of a bounded function. That is, the surface has the equation $x_3=f(x_1,x_2)$, where $x_1$, $x_2$, $x_3$ are standard Caretesian coordinates, for some continuous function $f$ satisfying that $c_-\le f(x_1,x_2)\le c_+$, for some constants $c_-$ and $c_+$. Recent reviews, concentrating on computational methods but also discussing approximate analytical methods and aspects of physical experimental technique include [1-4].

By far the most popular computational techniques for rough surface scattering in the engineering literature are integral equation methods, working in the frequency domain (see [3,4]). The study of integral equation methods for time harmonic acoustic and electromagnetic wave scattering has also long been of interest to applied mathematicians and numerical analysts (see [5-7] and the references therein). However, attention has only recently been paid to the many distinctive mathematical and computational features which arise when integral equation methods are employed for scattering by unbounded rough surfaces. These new features include: (i) for some configurations, the rough surface can support surface waves, solutions of the homogeneous problem localised near the unbounded surface; (ii) the possibility of such solutions to the homogeneous problem leads to the failure of standard arguments to prove uniqueness of solution for integral equations and boundary value problems; (iii) the Fredholm alternative is no longer available to establish existence of solution from uniqueness for second kind integral equation formulations, so that new theories of solvability are required; (iv) the standard boundary integral operators of time harmonic wave scattering are no longer bounded operators on any reasonable function space once the surface is unbounded; (v) for computational purposes truncation of the infinite boundary is required, but the stability and convergence of this procedure is unclear; (vi) how do boundary element methods perform on unbounded surfaces, in particular is a small value of $kh$, where $k$ is the wavenumber and $h$ the maximum element size, sufficient for stability and accuracy? (vii) which matrix compression and iterative solution techniques are suitable for rough surface scattering problems?

In this talk we will survey the current practice in the engineering and physics literature with regard to integral equation methods for rough surface scattering. We will also survey the recent work at Brunel, Coventry, Karlsruhe, and Goettingen on resolving a number of the above questions, in particular work on the well-posedness of integral equation and related boundary value problem formulations (e.g. [8-12]), on the approximation of infinite surfaces by finite or periodic surfaces [13,14], on the numerical analysis of boundary element methods on unbounded surfaces [15,16], and on iterative solvers and matrix compression techniques. In this last regard we shall concentrate on considering modifications and numerical analysis of one of the most popular computational methods for large rough surface scattering problems, namely the banded matrix iterative algorithm/canonical grid method (e.g. [17-19]).

[1] J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces, Bristol: Hilger, 1991.

[2] G. Voronovich, Wave Scattering from Rough Surfaces, 2nd edn., Berlin: Springer, 1999.

[3] K. F. Warnick and W.C. Chew, Numerical simulation methods for rough surface scattering. Wave Random Media 11 (2001), R1-R30.

[4] M. Saillard and A. Sentenac, Rigorous solutions for electromagnetic scattering by rough surfaces. Wave Random Media 11 (2001), R103-R137.

[5] D. Colton and R. Kress, Integral Equation methods in Scattering Theory, New York: Wiley, 1983.

[6] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Theory, 2nd edn., Berlin: Springer, 1998.

[7] J.-C. Nedelec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Berlin: Springer, 2001.

[8] S.N. Chandler-Wilde and B. Zhang, A uniqueness result for scattering by infinite rough surfaces, SIAM J. Appl. Math. 58, (1998), 1774-1790.

[9] S.N. Chandler-Wilde and B. Zhang, Electromagnetic scattering by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate, Proc. R. Soc. Lond. A454, (1998), 519-542 .

[10] S.N. Chandler-Wilde and B. Zhang, Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers, SIAM J. Math. Anal. 30, (1999), 559-583.

[11] S.N. Chandler-Wilde, C.R. Ross, C. R. and Zhang, B., Scattering by infinite one-dimensional rough surfaces, Proc. R. Soc. London A455, (1999), 3767-3787.

[12] T. Arens, S.N. Chandler-Wilde, and K.O. Haseloh, Solvability and spectral properties of integral equations on the real line: II. $L^p$ spaces and applications, submitted for publication.

[13] S.N. Chandler-Wilde, B. Zhang, and C.R. Ross, On the solvability of second kind integral equations on the real line, J. Math. Anal. Appl. 245, (2000), 28-51.

[14] A. Meier and S.N. Chandler-Wilde, On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering, Math. Meth. Appl. Sci. 24, (2001), 209-232.

[15] A. Meier, T. Arens, S.N. Chandler-Wilde, and A. Kirsch, A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces, J. Integral Equat. Appl. 12, (2000), 281-321.

[16] S.N. Chandler-Wilde, M. Rahman, and C.R. Ross, A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane, to appear in Numer. Math.

[17] L. Tsang, C.H. Chan, K. Pak, and H. Sangani, Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/Canonical Grid method. IEEE Tran. Ant. Prop. 43 (1995), 851-859.

[18] S.Q. Li, C.H. Chan, L. Tsang, Q. Li, and L. Zhou, Parallel implementation of the sparse-matrix/canonical grid method for the analysis of two-dimensional random rough surfaces (three-dimensional scattering problem) on a Beowulf system. IEEE T. Geosci. Remote 38 (2000), 1600-1608.

[19] J.T. Johnson and R.J. Burkholder, Coupled canonical grid/discrete dipole approach for computing scattering from objects above or below a rough interface. IEEE T. Geosci. Remote 39 (2001), 1214-1220.