Mathematical Problems in Engineering

Volume 2015, Article ID 524345, 10 pages

http://dx.doi.org/10.1155/2015/524345

## A Least-Squares FEM for the Direct and Inverse Rectangular Cavity Scattering Problem

^{1}Department of Mathematics, Dalian Maritime University, Dalian 116026, China^{2}School of Mathematics, Jilin University, Changchun 130012, China

Received 29 September 2014; Revised 14 February 2015; Accepted 14 February 2015

Academic Editor: Stefano Lenci

Copyright © 2015 Enxi Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is concerned with the scattering problem of a rectangular cavity. We solve this problem by a least-squares nonpolynomial finite element method. In the method, we use Fourier-Bessel functions to capture the behaviors of the total field around corners. And the scattered field towards infinity is represented by a combination of half-space Green functions. Then we analyze the convergence and give an error estimate of the method. By coupling the least-squares nonpolynomial finite element method and the Newton method, we proposed an algorithm for the inverse scattering problem. Numerical experiments are presented to show the effectiveness of our method.

#### 1. Introduction

In recent years the scattering theory in electromagnetic cavity problem is very popular in mathematical physics and plays an important role in practical applications, for example, the radar detecting. Radar cross section (RCS) is a measure of the detectability of a target by a radar system and the RCS by a cavity is usually significant in overall RCS of an object. So the accurate prediction of the RCS of a cavity, that is, the electromagnetic field scattered by a cavity, attracts many scientists’ interests. The cavity scattering theory can also be used to detect cracks or holes in metallic surfaces such as the aircraft wings. These cracks or holes may be invisible to a visual inspection but detectable by electromagnetic waves, so the understanding of scattering theory of cavity can help aerodynamicist to design and check the aircraft wings. In reality the engineers often use electromagnetic waves to check the material. Then the inverse scattering problem of a cavity can serve as a mathematical model.

For the scattering problem of an open cavity embedded in an infinite ground plane, Ammari et al. in [1, 2] reformulated the problem into a bounded domain via a variational approach and gave the existence and uniqueness results in two and three dimensions. In [3], Ammari et al. investigated the integral equations method to solve this problem and gave the existence and uniqueness results of the solution to the corresponding integral equations. Other numerical methods used to solve the scattering problem of open cavity are time-domain finite difference methods [4], finite element methods [5, 6], several hybrid methods [7–9], and so forth. Recently, in [10] Bao et al. analyzed the stability of the scattering from a large rectangular cavity and their stability estimates provided the explicit dependence on the high wave number and the depth of the cavity. Although the result in [10] is not optimal, it is essential for conducting convergence analysis of numerical methods. In [11] Li et al. also presented some stability estimates with the explicit dependency of wave number for the open cavity problem. And they proposed a Legendre spectral Galerkin method for the scattering problem of rectangular cavity. About the inverse scattering problem of cavity, we refer to [12, 13] and the references therein. In [13] the author recovered the shape of the cavity from the scattered field given in the aperture and gave the conditions needed for proving the uniqueness of this inverse problem. In [12], Feng and Ma investigated the inverse problem of determining the shape of the open cavity from the information of the far field patterns of scattered field. The results on the uniqueness and the local stability of the inverse problem in the 2-dimensional TM (transverse magnetic) polarization were also proved in the paper. Lately, Bao et al. studied the direct and inverse problem of open cavity in TM and TE (transverse electric) polarization. In [14] they proposed a method of symmetric coupling of finite element and boundary integral equations to solve the direct problem and proved the existence and uniqueness of weak solutions. As for inverse problem, the domain derivatives of the field with respect to the shape of the cavity were derived and the uniqueness and local stability results were established.

In [15], the authors proposed a nonpolynomial finite element method for the scattering problem of a polygonal obstacle. This scattering problem is defined in the whole except the obstacle domain. In this paper, we consider the scattering problem of a rectangular open cavity. This scattering problem is defined in the whole upper half-plane and cavity domain which embedded in the lower half-plane. Inspired by [15], we solve this scattering problem of a rectangular cavity by the least-squares nonpolynomial finite element method. In case of the obstacle scattering problem, the scattering field towards infinity is represented by a combination of free-space Green functions, but in case of our open cavity scattering problem, the scattering field towards infinity is represented by a combination of half-space Green functions. Furthermore, in [15] the authors only gave the approximation properties of finite element space. In our paper we give the error estimate using the approximation properties in [15] and dual technique in [16]. The difference between [16] and our paper is that we solve the Helmholtz equation in semi-infinite domain and the authors in [16] were concerned with the Helmholtz problem in bounded domain.

In [12, 14] the existence, uniqueness, and local stability of inverse scattering problem for open cavity are proved. Here, the local stability means when the Hausdorff distance between two cavities is not too big, this distance can be controlled by the difference of the scattered fields for these two cavities. But in the above two papers, the authors did not present a numerical method for inverse cavity problem. In our paper we propose a Newton method coupling with the least-squares finite element method to solve the inverse scattering problem of open cavity. In order to implement the Newton method we should get the domain derivative, that is, the derivative of scattered field about the domain parameters. Paper [17] gave the general theory of domain derivative for boundary value problem. References [18, 19] presented the differential equation which the domain derivative satisfies in inverse obstacle problem. With the help of these results we give the domain derivative of the inverse scattering problem for rectangular cavity and use it in the numerical algorithm.

In this paper, we consider the scattering problem of an open cavity with a rectangular cross section and assume that the cross section of the rectangular cavity is under = 0. Let be a rectangular domain given by , where , , and . Denote the upper half-plane by , and define ; see Figure 1. By this definition, a rectangular cavity can be determined by three parameters , , and , where is the left tip of the opening aperture, is the width of the cavity, and is the depth of the cavity.