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Modified Rayleigh conjecture method and its applications. (English) Zbl 1143.35344

Summary: The Rayleigh conjecture about convergence up to the boundary of the series representing the scattered field in the exterior of an obstacle \(D\) is widely used by engineers in applications. However this conjecture is false for some obstacles. AGR introduced the Modified Rayleigh Conjecture (MRC), which is an exact mathematical result. In this paper we present the theoretical basis for the MRC method for 2D and 3D obstacle scattering problems, for static problems, and for scattering by periodic structures. We also present successful numerical algorithms based on the MRC for various scattering problems. The MRC method is easy to implement for both simple and complex geometries. It is shown to be a viable alternative for other obstacle scattering methods. Various direct and inverse scattering problems require finding global minima of functions of several variables. The Stability Index Method (SIM) combines stochastic and deterministic method to accomplish such a minimization.

MSC:

35P25 Scattering theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
78A40 Waves and radiation in optics and electromagnetic theory

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