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Mathematical Problems in Engineering
Volume 2011, Article ID 503791, 12 pages
http://dx.doi.org/10.1155/2011/503791
Research Article

Fictitious Domain Technique for the Calculation of Time-Periodic Solutions of Scattering Problem

1Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China
2College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 8 July 2010; Accepted 27 December 2010

Academic Editor: Maria do Rosário de Pinho

Copyright © 2011 Ling Rao and Hongquan Chen. 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

The fictitious domain technique is coupled to the improved time-explicit asymptotic method for calculating time-periodic solution 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. We use the Dirac delta function to transport the variational forms of the wave equations 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. The numerical experiments show that the new method performs as well as the method using conventional finite element approximation.

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 [15]). 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 [79], 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 [79]). 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 𝑅2 [1]. Let us consider a scattering body 𝜔 with boundary 𝛾=𝜕𝜔, illuminated by an incident monochromatic wave of period 𝑇 and incidence β. We bound 𝑅𝑛𝜔by an artificial boundary Γ. We denote by Ω the region of 𝑅𝑛 between 𝛾 and Γ (see Figure 1). The scattered field 𝑢 satisfies the following wave equation and boundary conditions: 𝑢𝑡𝑡Δ𝑢=0,in𝑄(=Ω×(0,𝑇)),𝑢=𝑔,on𝜎(=𝛾×(0,𝑇)),𝜕𝑢+𝜕𝑛𝜕𝑢𝜕𝑡=0,onΣ(=Γ×(0,𝑇)),(2.1) where 𝑔=Re[𝑒𝑖𝑘𝑡𝑒𝑖𝑘(𝑥cos𝛽+𝑦sin𝛽)], with𝑖=1,𝑘=2𝜋/𝑇.

503791.fig.001
Figure 1: Computational Domain.

Due to the periodic requirement, 𝑢 also should satisfy 𝑢(0)=𝑢(𝑇),𝑢𝑡(0)=𝑢𝑡(𝑇).(2.2)

Equation (2.1) represent the electric field 𝑢 satisfying the two-dimensional Maxwell equation written in transverse magnetic (TM) form.

3. Exact Controllability and Least-Squares Formulations

Solving problem (2.1)-(2.2) is equivalent to finding a pair {𝑣0,𝑣1} such that 𝑢(0)=𝑣0,𝑢𝑡(0)=𝑣1,𝑢(𝑇)=𝑣0,𝑢𝑡(𝑇)=𝑣1,(3.1) where 𝑢 is the solution of (2.1). Problem (2.1), and (3.1) is an exact controllability problem which can be solved by the following controllability methodology given by [1].

Let 𝐸 is the space containing {𝑣0,𝑣1}𝐸=𝑉𝑔×𝐿2(Ω),(3.2) with 𝑉𝑔={𝜑𝜑𝐻1(Ω),𝜑|𝛾=𝑔(0)}. Least-squares formulations of (2.1), and (3.1) are given by min𝐯𝐸𝐽(𝐯),(3.3) with 1𝐽(𝐯)=2Ω||𝑦(𝑇)𝑣0||2+||𝑦𝑡(𝑇)𝑣1||2𝑣𝑑𝑥,𝐯=0,𝑣1,(3.4) where 𝑦 is the solution of 𝑦𝑡𝑡Δ𝑦=0,in𝑄=(Ω×(0,𝑇)),(3.5)𝑦=𝑔,on𝜎(=𝛾×(0,𝑇)),(3.6)𝜕𝑦+𝜕𝑛𝜕𝑦𝜕𝑡=0,onΣ(=Γ×(0,𝑇)),(3.7)𝑦(0)=𝑣0,𝑦𝑡(0)=𝑣1.(3.8) The problem (3.3)–(3.8) may be solved by the conjugate algorithm [1]. Because this method looks some complicated, we use an alternative improved time-explicit asymptotic algorithm [4] to solve it. This method introduces an auxiliary parameter to control the time-explicit asymptotic iteration, and the auxiliary parameter is updated during the iteration based on the existing or current iterated solution of the wave equation. The algorithm is presented as follows.

Algorithm 3.1. We have the following steps.Step 1 : (initialization). (1) Given 𝐯={𝑣0,𝑣1}𝐸 as an initial guess.
(2) compute the first periodic solution 𝐲𝑇: solving wave equation problem (3.5)–(3.8) to have solution 𝐲𝑇={𝑦(𝑇),𝑦𝑡(𝑇)}.
(3) compute the second periodic solution 𝐲2𝑇: solving wave equation problem (3.5)–(3.8) to get solution 𝐲2𝑇={𝑦(2𝑇),𝑦𝑡(2𝑇)} with initial condition 𝑦(0)=𝑦(𝑇),𝑦𝑡(0)=𝑦𝑡(𝑇).

Step 2 : (compute 𝛽 and update 𝐯,𝐲𝑇). (1) Compute 𝛽 by𝛽=12𝐽𝐲𝑇𝐽(𝐯)Ω||𝛿𝑇𝑇||𝑦(𝑇)2+||𝛿𝑇𝑇𝑦𝑡||(𝑇)2𝑑𝑥,(3.9) where 𝛿𝑇𝑇𝑦(𝑇)=𝑦(2𝑇)2𝑦(𝑇)+𝑦(0).
(2) Update 𝐯 and 𝐲𝑇 by𝑣𝐯=0+𝛽𝑦(𝑇)𝑣0,𝑣1+𝛽𝑦𝑡(𝑇)𝑣1,𝐲𝑇=𝑦(𝑇)+𝛽(,𝑦𝑦(2𝑇)𝑦(𝑇))𝑇(𝑇)+𝛽𝑦𝑡(2𝑇)𝑦𝑡(.𝑇)(3.10)

Step 3 : (solve wave equation to obtain 𝐲2𝑇). Solve (3.5)–(3.8) for the second periodic solution 𝐲2𝑇={𝑦(2𝑇),𝑦𝑡(2𝑇)} with initial condition 𝑦(0)=𝑦(𝑇),𝑦𝑡(0)=𝑦𝑡(𝑇).
Step 4 (test of the convergence). Compute control function 𝐽(𝐲𝑇). If the value of 𝐽(𝐲𝑇) satisfies a given accuracy, then 𝐯=𝐲𝑇 is taken as final solution, otherwise return to Step 2.

4. Fictitious Domain Method for Solving Wave Equation

Note that the above algorithm needs solve wave equations (3.5)–(3.8). The equivalent variational formulation of (3.5)–(3.7) is Ω𝑦𝑡𝑡𝑧𝑑𝑥+Ω𝑦𝑧𝑑𝑥+Γ𝜕𝑦𝜕𝑡𝑑Γ=0,𝑧𝑉0,𝑦=𝑔,on𝜎,(4.1) where 𝑉0={𝜑𝜑𝐻1(Ω),𝜑|𝛾=0}.

The implementation used in [1] is based on an explicit finite difference scheme in time combined to piecewise linear finite element approximations for the space variables. Time discretization is carried out by a centered second-order difference scheme with time step Δ𝑡=𝑇/𝑁. After time discretization, (4.1) with (3.8) becomes1Δ𝑡2Ω𝑦𝑛+12𝑦𝑛+𝑦𝑛1𝑧𝑑𝑥+Ω𝑦𝑛+11𝑧𝑑𝑥+2Δ𝑡Γ𝑦𝑛+1𝑦𝑛1𝑧𝑑Γ=0,𝑧𝑉0,𝑦𝑛+1𝑡=𝑔𝑛+1𝑦,on𝛾,0=𝑣0,𝑦1𝑦12Δ𝑡=𝑣1.(4.2)

The fully discrete system can be obtained by the corresponding space discretization. Because Ω is irregular, if we directly use fitted meshes of Ω as in [1], we will meet great trouble of constructing meshes and difficulty of computation especially to those shape optimization problems with several scatters. So, we consider the problem (3.5)–(3.8) in the extended rectangular domain 𝐵=𝜔Ω with boundary Γ by the following boundary Lagrangian fictitious domain method. It allows the propagation to be simulated on 𝐵 with uniform meshes. By introducing Lagrangian multipliers to enforce the Dirichlet boundary condition on 𝛾, (3.5)–(3.8) is equivalent to the following variational problem.

Find {𝑦,𝜆}𝐻1(𝐵)×𝐿2(𝛾), such that 𝐵𝑦𝑡𝑡𝑧𝑑𝑥+𝐵𝑦𝑧𝑑𝑥+Γ𝜕𝑦𝜕𝑡𝑑Γ+𝛾𝜆𝑧𝑑𝛾=0,𝑧𝐻1(𝐵),𝛾𝜇(𝑦𝑔)𝑑𝛾=0,𝜇𝐿2(𝛾),𝑦(0)=𝑣0,𝑦𝑡(0)=𝑣1.(4.3)

Let Δ𝑡=𝑇/𝑁, discretize (4.3) with respect to time with 𝑦0=𝑣0,𝑦0𝑦1Δ𝑡=𝑣1,(4.4) for 𝑛=0,1,,𝑁, we compute 𝑦𝑛+1, 𝜆𝑛+1 via the solution of1Δ𝑡2𝐵𝑦𝑛+12𝑦𝑛+𝑦𝑛1𝑧𝑑𝑥+𝐵𝑦𝑛+1𝑧𝑑𝑥2Δ𝑡Γ𝑦𝑛+1𝑦𝑛1𝑧𝑑Γ+𝛾𝜆𝑛+1𝑧𝑑𝛾=0,𝑧𝐻1(𝐵),(4.5)𝛾𝜇𝑦𝑛+1𝑔𝑛+1𝑑𝛾=0,𝜇𝐿2(𝛾).(4.6) Below, we consider conjugate gradient method for solving (4.5) and (4.6).

For given 𝑦𝑛, 𝑦𝑛1, define linear functional 𝑓 on 𝐻1(𝐵)1𝑓(𝑧)=Δ𝑡2𝐵2𝑦𝑛+𝑦𝑛1𝑧𝑑𝑥+𝐵𝑦𝑛1𝑧𝑑𝑥2Δ𝑡Γ𝑦𝑛1𝑧𝑑Γ,𝑧𝐻1(𝐵).(4.7) Let1𝑎(𝑤,𝑧)=Δ𝑡2𝐵1𝑤𝑧𝑑𝑥+2Δ𝑡Γ𝑤𝑧𝑑Γ,𝑤,𝑧𝐻1(𝐵).(4.8) Suppose 𝑧0 satisfies 𝑎𝑧0,𝑧+𝑓(𝑧)=0,𝑧𝐻1(𝐵).(4.9) Then, (4.5) is 𝑎𝑦𝑛+1𝑧0+,𝑧𝛾𝜆𝑛+1𝑧𝑑𝛾=0,𝑧𝐻1(𝐵).(4.10) Define 𝐴𝐿1/2(𝛾)𝐿1/2(𝛾),𝐴𝜇=𝑦𝜇|𝛾, for all 𝜇𝐿2(𝛾), where 𝑦𝜇 satisfies 𝑎𝑦𝜇+,𝑧𝛾𝜇𝑧𝑑𝛾=0,𝑧𝐻1(𝐵).(4.11) Let , denote scalar product in 𝐿2(𝛾), then 𝑎𝑦𝜇,𝑦𝜇+𝜇,𝐴𝜇=0𝜇,𝜇𝐿2(𝛾).(4.12)𝐴 is symmetric and positive definite. Then, in 𝐿2(𝛾) (4.5) (or (4.10)) becomes 𝐴𝜆𝑛+1=𝑦𝑛+1𝑧0|𝛾.(4.13) By (4.6), 𝑦𝑛+1|𝛾=𝑔𝑛+1.(4.14)

Then, 𝐴𝜆𝑛+1=𝑔𝑛+1𝑧0|𝛾.(4.15) Its variational form is 𝐴𝜆𝑛+1=𝑧,𝜇0|𝛾𝑔𝑛+1,𝜇,𝜇𝐿2(𝛾).(4.16)

A conjugate gradient algorithm for the solution 𝜆𝑛+1 of (4.16) is given by the following.

Step 1 : (initialization). (1) Give initial value 𝜆0𝐿2(𝛾) and a real number 𝜀>0 small enough.
(2) Find 𝑢0𝐻1(𝐵) such that 𝑎𝑢0,𝑧+𝑓(𝑧)+𝛾𝜆0𝑧𝑑𝛾=0,𝑧𝐻1(𝐵),(4.17) that is, 1Δ𝑡2𝐵𝑢02𝑦𝑛+𝑦𝑛1𝑧𝑑𝑥+𝐵𝑦𝑛+1𝑧𝑑𝑥2Δ𝑡Γ𝑢0𝑦𝑛1𝑧𝑑Γ+𝛾𝜆0𝑧𝑑𝛾=0,𝑧𝐻1(𝐵).(4.18)
(3) Calculate 𝑑0𝐿2(𝛾) by 𝛾𝑑0𝜇𝑑𝛾=𝛾𝑔𝑛+1𝑢0𝜇𝑑𝛾,𝜇𝐿2(𝛾).(4.19)
(4) Set 𝑤0=𝑑0.

Step 2. For all 𝑘>0, calculate 𝜆𝑘+1, 𝑑𝑘+1, 𝑤𝑘+1 from 𝜆𝑘, 𝑑𝑘, 𝑤𝑘.
(1) Find 𝑢𝑘𝐻1(𝐵) such that 𝑎𝑢𝑘+,𝑧𝛾𝑤𝑘𝑧𝑑𝛾=0,𝑧𝐻1(𝐵),(4.20) that is, 1Δ𝑡2𝐵𝑢𝑘1𝑧𝑑𝑥+2Δ𝑡Γ𝑢𝑘𝑧𝑑Γ+𝛾𝑤𝑘𝑧𝑑𝛾=0,𝑧𝐻1(𝐵).(4.21)
(2) Calculate 𝜌𝑘:𝜌𝑘=𝛾|𝑑𝑘|2𝑑𝛾/𝛾𝑢𝑘𝑤𝑘𝑑𝛾.
(3) Calculate 𝜆𝑘+1: 𝜆𝑘+1=𝜆𝑘𝜌𝑘𝑤𝑘.
(4) Calculate the new gradient 𝑑𝑘+1𝐿2(𝛾) by 𝛾𝑑𝑘+1𝜇𝑑𝛾=𝛾𝑑𝑘𝜇𝑑𝛾+𝜌𝑘𝛾𝑢𝑘𝜇𝑑𝛾,𝜇𝐿2(𝛾).(4.22)

Step 3 (test of the convergence). If 𝑑𝑘+1𝐿2(𝛾)/𝑑0𝐿2(𝛾)𝜀, then take𝜆𝑛+1=𝜆𝑘+1 and solve (4.10) for the corresponding solution 𝑦𝑛+1, take 𝑦𝑛+1 as the final solution; else, compute 𝛾𝑘 by 𝛾𝑘=𝑑𝑘+1𝐿2(𝛾)𝑑𝑘𝐿2(𝛾),(4.23) and update 𝑤𝑘 by 𝑤𝑘+1=𝑑𝑘+𝛾𝑘𝑤𝑘.(4.24) Set 𝑘=𝑘+1, return to Step 2.

5. Improving the Computation Procedure of the Space Discretizations

Conventionally, we solve (4.18) and (4.21) by the finite element method (see [79]). In the computation procedure of the finite element discretizations, the mesh of the extended domain is regular, but the boundary is irregular. We will meet the trouble of computing the boundary integrals which leads to complex set operations like intersection and subtraction between irregular boundary 𝛾 and regular mesh of 𝐵. In order to avoid these difficulties and solve (4.18) and (4.21) more efficiently, we use the Dirac delta function to improve the computation procedure of the discretizations. We discuss this method as follows.

We construct a regular Eulerian mesh on 𝐵𝐵𝑘=𝑥𝑖𝑗𝑥𝑖𝑗=𝑥0+𝑖,𝑦0+𝑗,0𝑖,𝑗𝐼,(5.1) where is the mesh width (for convenience, kept the same both in 𝑥- and in 𝑦-directions). Assume that the configuration of the simple closed curve 𝛾 is given in a parametric form (𝑠),0𝑠𝐿. The discretization of the boundary 𝛾 employs a Lagrangian mesh, represented as a finite collection of Lagrangian points {𝑋𝑘,0𝑘𝑀} apart from each other by a distance Δ𝑠, usually taken as being /2. Let 𝛿() be a Dirac delta function. In the following calculation procedure, 𝛿 is approximated by the distribution function 𝛿. The choice here is given by the product 𝛿(𝑥)=𝑑𝑥1𝑑𝑥2,(5.2) where 𝑥=(𝑥1,𝑥2) and 𝑑 is defined by 𝑑(𝑧)=0.251+cos𝜋𝑧2,|𝑧|2,0,|𝑧|>2.(5.3) Using the above Dirac delta function, we can transport the variational form (4.18) to the difference form. We write 𝛾𝜆0𝑧𝑑𝛾 in (4.18) as the following form: 𝛾𝜆0𝑧𝑑𝛾=𝐻1(𝐵)𝐿0,𝑧𝐻1(𝐵),(5.4) where 𝐿0(𝑥)=𝐿0𝜆0(𝑠)𝛿(𝑥𝑋(𝑠))𝑑𝑠,𝑥𝐵,(5.5) that is, 𝜆0 calculated over the Lagrangian points are distributed over the Eulerian points. Thus, we can write (4.18) in the difference form as follows: 𝑢02𝑦𝑛+𝑦𝑛1Δ𝑡2Δ𝑦𝑛+𝐿0=0,in𝐵,𝜕𝑦𝑛+𝑢𝜕𝑛0𝑦𝑛12Δ𝑡=0,onΓ.(5.6) Thus, the solution of (4.18) is 𝑢0=2𝑦𝑛𝑦𝑛1+Δ𝑡2Δ𝑦𝑛𝐿0𝑢,in𝐵,0=𝑦𝑛12Δ𝑡𝜕𝑦𝑛𝜕𝑛,onΓ.(5.7) The discrete form of (5.5) is 𝐿0𝑥𝑖𝑗=𝑘𝜆0𝑘𝛿𝑥𝑖𝑗𝑋𝑘Δ𝑠,𝑥𝑖𝑗𝐵.(5.8) So, we can obtain 𝑢0(𝑥𝑖𝑗) for all 𝑥𝑖𝑗𝐵.

In the same way, let𝑊𝑘(𝑥)=𝐿0𝑤𝑘(𝑠)𝛿(𝑥𝑋(𝑠))𝑑𝑠,𝑥𝐵.(5.9) Then, (4.21) also can be written in the difference form as follows: 𝑢𝑘=Δ𝑡2𝑊𝑘,in𝐵,𝑢𝑘=0,onΓ.(5.10) Calculate 𝑊𝑘𝑥𝑖𝑗=𝑚𝑤𝑘𝑚𝛿𝑥𝑖𝑗𝑋𝑚Δ𝑠,𝑥𝑖𝑗𝐵.(5.11) Then, we can get 𝑢𝑘(𝑥𝑖𝑗), for all 𝑥𝑖𝑗𝐵.

Thus 𝑢𝑘|𝛾=𝑢𝑘(𝑋(𝑠))=𝐵𝑢𝑘(𝑥)𝛿(𝑥𝑋(𝑠))𝑑𝑥,0𝑠𝐿.(5.12) Its discrete form is 𝑢𝑘𝑚=𝑖𝑗𝑢𝑘𝑥𝑖𝑗𝛿𝑥𝑖𝑗𝑋𝑚2,1𝑚𝑀.(5.13) And by (4.22), we have 𝑑𝑘+1=𝑑𝑘+𝜌𝑘𝑢𝑘|𝛾.(5.14)

It can be seen from the above discretization process that most of the calculations are done over the Lagrangian points and the neighboring Eulerian points of the boundary 𝛾. The solutions of (4.18) and (4.21) are given explicitly by (5.7) and (5.10). And we only need do the evaluation in (5.8), (5.11), and (5.13) to obtain the solutions of (4.18) and (4.21). So, our method is easier to code and requires fewer computational operations than conventional finite element method (see [79]).

6. Numerical Experiments

In order to validate the methods discussed in the above sections, we apply our algorithm to simulate the scattering of planar monochromatic incident waves by a perfectly conducting obstacle. The obstacle is a Semiopen rectangular cavity; the internal dimensions of the cavity are 4𝜆×1.4𝜆, and the thickness of the wall is 0.2𝜆 as shown in Figure 2. Wavelength 𝜆=0.25 m and incidence of illuminating waves is 0°. The corresponding scattered fields and convergence histories of control function 𝐽 are shown in Figures 3 and 4. Figures 3 and 4 show that our method performs as well as the method discussed in [79] does where fictitious domain method and obstacle fitted meshes were used.

503791.fig.002
Figure 2: Semiopen rectangular cavity.
503791.fig.003
Figure 3: Contours of the scattered field.
503791.fig.004
Figure 4: Convergence histories.

7. Conclusions

In this paper, the fictitious domain technique is coupled to the improved time-explicit asymptotic method for calculating the time-periodic solutions of wave equations. It allows the propagation to be simulated on an obstacle free computational region with uniform meshes. One of the main advantages of the fictitious domain approach is that it is well suited to those shape optimization problems with several scatters that minimize, for example, a Rader Cross Section. We use the Dirac delta function to improve the computation procedure of space discretizations. Numerical experiments invalidate that our algorithms are efficient and easy to implement alternative to more classical wave equation solvers.

Acknowledgment

This research is partially supported by Natural Science Foundation of China (no. 10671092).

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