Perfectly Matched Layer Method for the Wave Scattering Problem by a Step-Like Surface

A fact that is often overlooked in inverse scattering theory is that there are many inverse scattering problems and the ones chosen for study are frequently done so on the basis of mathematical convenience rather than physical reality. Of course in some cases mathematical convenience and physical reality coincide and there is no problem. However in many other situations this is not the case. For example the two most commonly studied inverse scattering problems in acoustics are, given the far-field pattern of the scattered wave at a fixed frequency, 1) to determine the shape of a sound-soft or sound-hard obstacle and 2) to determine the “index of refraction” of a penetrable obstacle under the assumption that the pressure is smoothly varying across the boundary of the obstacle [3]. These problems only make sense if it is known a priori that the object is in fact sound-soft or sound-hard or, if the object is penetrable, that the pressure is smoothly varying, i.e. that certain physical properties of the object are known as apriori. If multi-frequency far-field data is assumed known then further assumptions are needed on the dispersion properties of the scattering object. From this point of view, the first problem in inverse scattering theory is to find out what can be determined about the scattering object if nothing at all is known apriori about its physical composition or dispersive properties. This is the problem we will address ourselves to in this paper.

In this paper, we propose a uniaxial perfectly matched layer (PML) method for solving the time-harmonic scattering problems in two-layered media. The exterior region of the scatterer is divided into two half spaces by an infinite plane, on two sides of which the wave number takes different values. We surround the computational domain where the scattering field is interested by a PML with the uniaxial medium property. By imposing homogeneous boundary condition on the outer boundary of the PML, we show that the solution of the PML problem converges exponentially to the solution of the original scattering problem in the computational domain as either the PML absorbing coefficient or the thickness of the PML tends to infinity.

The uniaxial perfectly matched layer (PML) method uses a rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in dealing with problems involving anisotropic scatterers. In this paper we first derive the uniaxial PML method for solving the time-domain scattering problem based on the Laplace transform and complex coordinate stretching in the frequency domain. We prove the long-time stability of the initial-boundary value problem of the uniaxial PML system for piecewise constant medium properties and show the exponential convergence of the time-domain uniaxial PML method. Our analysis shows that for fixed PML absorbing medium properties, any error of the time-domain PML method can be achieved by enlarging the thickness of the PML layer as $\ln T$ for large $T>0$. Numerical experiments are included to illustrate the efficiency of the PML method.

The 2D problem of scattering the plane monochromatic electromagnetic wave by a perfectly conducting circular cylinder embedded in a dielectric dispersive half-space with losses is considered. The asymptotic relations for the scattered field are obtained. The possibility of applying the results obtained for determination of the scattered parameters (the depth, the radius of surface curvature at the bright point, the orientation) is pointed out.

There had been two approaches for visualising the tissues within an area of interest in the body. One involves the steps of acquiring shadowgraphs using a highly transmissive electromagnetic field, which is widely used in technologies such as X-ray computed tomography and nuclear magnetic resonance, and then constructing three-dimensional (3D) images from multiple shadowgraphs using computational calculation. The other involves converging highly scatterable waves using a lens and connecting data reflected from convergent points to construct images. The former method requires penetrability with no degree of freedom in wave selection, whereas the latter method does not enable visualisation of the background behind the reflecting object, and consequently obtain only surficial images. In this context, we addressed the inverse scattering problem, which had remained unsolved in the history of applied mathematics field, aiming to develop a universal methodology that outperforms the above two principles for ‘viewing objects’. Our efforts have been fruitful, as we established a scattering field theory that made it possible to solve this problem analytically. This theory proposes that 3D images can be precisely constructed by mathematically converging any points inside an object using measured data of scattered waves. This paper introduces not only the significance of the scattering field theory and the measurement principles but also a wide variety of cases in which the breast cancer tissue in the breast was visualised three-dimensionally using microwaves.

This chapter contains sections titled: Diffraction at a Canonical Conic Surface Scattering at a Disk Scattering at Cones: Focal Field Bodies of Revolution with Nonzero Gaussian Curvature: Backscattered Focal Fields Bodies of Revolution with Nonzero Gaussian Curvature: Axially Symmetric Bistatic Scattering Problems

Wave Scattering Problems for Selected Structures

In the Wiener-Hopf analysis of wave scattering problems related to the modified Wiener-Hopf geometries, solutions often contain infinite branch-cut integrals with unknown integrands and/or infinite series with unknown coefficients. It is therefore required to evaluate these infinite integrals and series appropriately in order for the analysis to be complete. In this paper, we shall investigate the solutions arising in the boundary value problems for the modified Wiener-Hopf geometries, and develop efficient methods for obtaining approximate solutions based on a rigorous asymptotics. Applications of the results to specific scattering problems are discussed. Some numerical results on the radar cross section (RCS) are also presented.

The Unimoment Method for Elastic Wave Scattering Problems