Fictitious domain technique for the calculation of time-periodic solutions of scattering problem.(Research Article)(Report)

1. Introduction

Recently, aircraft design for military application has focused more and more attention on using stealth technologies. It is important to realize Rader stealth through reducing the intensity of scattering signals of Rader in stealth design. Theoretically, the stealth characteristics such as Radar Cross-Section (RCS) for a given aerodynamic body can be obtained by solving the fundamental electromagnetic Maxwell equations. The control method based on exact controllability has been successfully used in computing the time-periodic solutions of scattered fields by multibody reflectors (see [1-5]). An improved time-explicit asymptotic method is afforded through introducing an auxiliary parameter for solving the exact controllability problem of scattering waves [4].

Fictitious domain methods are efficient methods for the solutions of viscous flow problems with moving boundaries [6].In [7-9], fictitious domain method is combined with controllability method to compute time-periodic solution of wave equation, which is proved to be equivalent to the Maxwell equation in two dimensions for the TM mode. A motivation for using fictitious domain method is that it allows the propagation to be simulated on an obstacle free computational region with uniform meshes. In our paper, the fictitious domain technique is coupled to the improved time-explicit asymptotic method for calculating time-periodic solutions of wave equation. Conventionally, the practical implementation of fictitious domain method relies on finite difference time discretizations schemes and finite element approximation. Our new method applies finite difference approximations in space instead of conventional finite element approximation (see [7-9]). We use the Dirac delta function to transport the variational form of the wave equation to the differential form and then solve it by finite difference schemes. Our method is relatively easier to code and requires fewer computational operations than conventional finite element method does.

In Section 2, the formulation of the Scattering problem is presented. In Section 3, we introduce exact controllability problem of the Scattering problem and the corresponding improved time-explicit algorithm. In Section 4, we use fictitious domain method to solve the equivalent variational problem of the relevant time discretization of wave equations. In Section 5, we use the Dirac delta function to improve the computation procedure of the space discretization equations. Finally, the results of numerical experiments and conclusion are presented in Sections 6 and 7.

2. Formulation of the Scattering Problem

We will discuss the scattering of monochromatic incident waves by perfectly conducting obstacle in [R.sup.2] [1]. Let us consider a scattering body [omega] with boundary [gamma] = [partial derivative][omega], illuminated by an incident monochromatic wave of period T and incidence [beta]. We bound [R.sub.n] \ [omega] by an artificial boundary [GAMMA]. We denote by [OMEGA] the region of [R.sup.n] between [gamma] and [GAMMA] (see Figure 1). The scattered field u satisfies the following wave …