Studies On Numerical Solution Of Direct And Inverse Scattering Problems From Obstacle And Open Cavity

Acoustic and electromagnetic scattering theory has played a central role in mathematical physics. Broadly speaking, the direct scattering problem is to determine scattered wave and its asymptotic behavior from a knowledge of the incident wave and the differential equation governing the wave motion; The inverse scattering problem is to determine the shape of the scatterers or some other physical index from a knowledge of the scattered wave or its the asymptotic behavior. A basic inverse problem in classical scattering theory is the inverse obstacle scattering problem by time-harmonic acoustic wave, the researches on it are very important in many applications such as radar detecting, medical imaging and non-destructive obstacle testing through low-frequency wave. The researches on direct and inverse cavity scattering problems are also very important in many applications, such as the radar target stealth technology and rough surface scattering problems. Some effective numerical methods are introduced in this paper to solve the inverse obstacle scattering problem, the direct and inverse cavity scattering problems.â… . An improved hybrid method for obstacle scattering problemsMathematically, the scattering of time-harmonic acoustical waves by an infinite long cylindrical obstacle with a cross section D(?) R² leads to two-dimensional exterior boundary value problems.Definition 1 (Object scattering problem). We consider D (?) R² to be an open bounded obstacle with a C²- smooth boundaryΓ:= (?)D and an unbounded and connected complement. Then, given an incident field uⁱ, the direct obstacle scattering problems is to find the total field u := uⁱ + u^s, where u^s is the scattered field, such that u solves the Helmholtz equation in the exterior of the obstaclewhere k is the wave number, The operator B in (2) defines the boundary condition and is related with the physical properties of the obstacle D. The most frequently occurring boundary conditions are the Dirichlet boundary condition u|r = 0 and the impedance boundary condition (?) with the exterior unit normal vector v toΓand some real-valued impedance functionλ≥0 onΓ.It is known that the scattered field u^s has an asymptotic behavior of the formwhere (?) = x/|x|. The function u_∞is known as the far field pattern of u^s and is analytical on the unite circle S.The inverse scattering problem that we are concerned with is defined as follows:Definition 2 (Inverse obstacle scattering problem). For the obstacle scattering problems defined in Definition 1, given the far-field pattern u_∞on S for only one incident wave uⁱ(x) = e^ikx·d with d∈S, the inverse problem is to determine both the position and shape of the obstacle D without knowledge of the physical properties of the obstacle.The single layer operator and double layer operator are defined bytheir far field operator are defined by where (?) is the fundamental solution to the two-dimensional Helmholtz equation, (?). For simplicity, we use the notation v^s defined by (?) in the following part. Form the Helmholtz representation, the definition of far field and the jump relations, we have With the far field u_∞known, (4) and (5) are the boundary integral equations we will use to solve u^s and v^s onΓ.Theorem 1 If k² is not a Dirichlet eigenvalue for the negative Laplacian in the interior ofΓ, the boundary integral equations (4) and (5) can be rewritten as the following equations:where the operator (?) is a compact linear operator from L²(Γ) to L²(Ω), it is also injective and has dense range.Theorem 1 guarantees the application of the Tikhonov regularization scheme to the solving of the equation (6). Then we can use the equations (6) and (7) to solve the boundary data u^s and v^s onΓ.As we described before, the obstacle scattering problem (1)-(3) may have the Dirichlet boundary condition or the impedance boundary condition. We now introduce a general boundary condition defined bywhere R(u₀) represents the real part of u₀ and S(u₀) represents the imaginary part for some u₀∈C. The general condition has the following relationship with the Dirichlet boundary condition and the impedance boundary condition.Theorem 2 Assume thatΓ:= (?)D is a closed C²- contour with D to be an open bounded obstacle, u∈C²(R²\(?))∩C(R²\(?)), and u≠0 everywhere onΓif u(?)0 onΓ. Then u satisfies the general boundary condition G(u) = 0, onΓ,if and only if u satisfies one of the following boundary conditions:(a) u = 0 onΓ.(b) (?)u/(?)v + iλu = 0 onΓwithλbeing some real-valued function.We conclude from this theorem that the general boundary condition almost exactly includes the Dirichlet boundary condition and the impedance boundary condition. So instead of using a particular boundary condition, we use the general boundary condition to find the boundary of the scatterer. A parametrization of the boundary curve is required for the following analysis. We assume thatwith a 2Ï€periodic C²- smooth function z : R→R² and counter-clockwise orientation such that (?) is injective. Given a C²- smooth field u defined in the neighborhood ofγ, we define the general boundary operator by (?)(z) := G(u(z)), that isWith the boundary operators(?) can be represented by (?) and (?),The Frechet differentiation of the general boundary operator (?) will be considered in the following theorem.Theorem 3 (?) : C²[0,2Ï€]→C[0,2Ï€] is Frechet differentiable and the derivative is given byin [0, 2Ï€], where the Frechet derivative (?) and (?) are given by(3.2.17), (?) and (?) are given by (3.2.18). If u satisfies theHelmholtz equation, (?) and (?) are given by (3.2.19).For simplicity, we assume that the obstacle boundaryΓand its approximate curves can be parameterized in the formwhere r(t) is 2Ï€periodic positive C² functions. The improved hybrid method consists of two steps. Given an approximationΓ_n of the obstacle boundaryΓwith parametrization z_n of the form (13). In the first step, one deals with the ill-posedness. We solve the boundary integral equations (6) and (7) withΓreplaced byΓ_n to obtain (?) and (?) , then we obtain an approximation to the total field trace and its exterior normal trace (?) and (?) . In the second step, one deals with the nonlinearity. u still represents the total field of obstacle scattering problem (1)-(3), and (?) is the general boundary operator defined in (10). For reconstructing the shape of the obstacle boundary, we only need to find the closed curve, on which the total field satisfies the general boundary condition G(u) = 0. In other words, we need to find an update parametrization z_n + h satisfying (?)(z_n + h) = 0. Therefore, as in the classic Newton method, we solve the linearized equationwith respect to the shift h to get the update boundary contourΓ_n+1 with the parametrization z_n+1 = z_n + h. The Frechet derivative (?) is given in Theorem 3 and the solution of (14) needs the data (?) and (?) obtained in the first step. We conclude that the improved hybrid method consists of repeating the two steps iteratively until some stopping criteria are fulfilled.In the numerical experiments, we use the improved hybrid method to reconstruct the triangle obstacle and peanut obstacle with different boundary conditions. The numerical experiments show that the scatterers can be well reconstructed, even for the cases with noise data. Particularly, for the impedance boundary condition, the unknown impedance function can also be reconstructed.â…¡. A PML method for cavity scattering problems LetΩ(?) R² be the cross-section of a z-invariant infinite trough domainΣ, letΩ₀ (?) R² being the cavity with its fillings protrude above the ground plane. Denote S as the cavity wall,Γthe cavity aperture so that (?)Ω₀ = S∪Γ.Γ^c := (?)Ω\S is the infinite ground plane. We denote R₊² the half space above.The cavity scattering problem can be decomposed into two fundamental polarizations: TM polarization and TE polarization.Definition 3 (Cavity scattering problem, TM). Given a incident field uⁱ, let u^r be the reflected field satisfyingthe scattering problem is to find the scattered field u^s∈C²(Ω)∩C(Ω) such that the total field u := uⁱ + u^r + u^s solveswhere k is the wave number,ε_r is the relative electric permittivity,ε_r = 1 inΩ\(?)₀, Rε_r≥α> 0, Sε_r≥0 inΩ₀,andε_r∈L^∞(Ω). u satisfies the continuity conditions onΓ:Definition 3 (Cavity scattering problem, TE). Given a incident field uⁱ, let u^r be the reflected field satisfyingthe scattering problem is to find the scattered field u^s∈C²(Ω)∩C(Ω) such that the total field u := uⁱ + u^r +u^s solvesu satisfies the continuity conditions onΓ: Let B_r := {x : |x|≤r} be the circle with radius r and R₊² be the upper half-plane.Γ_r := (?)B_r∩R₊²,Ω_r :=Ω∩B_r, where the radius r can be large enough. For the cavity scattering problems in TM and TE polarizations, the following theorem shows that each of them has a unique variational solution.Theorem 4 The cavity scattering problem (TM) has a unique variational solution uin (?). The cavity scattering problem (TE) has a uniquevariational solution u in W :=H¹(Ω_R).The solution domainΩand its boundaryΓ^c∪S are both infinite, it makes the solution of the cavity scattering problems (TM) and (TE) difficult. As shown in Figure 2, we will introduce a perfectly matched layer(PML layer) B_PML := {x∈R₊² : R < |x| <ρ} to reduce the original cavity scattering problems defined in the infinite domain to problems in finite domainΩ_ρ.Letα(r) = 1+iσ(r) be the model medium property of PML layer withσ∈C(R),σ≥0 , andσ= 0, r≤R. LetDefine the matrix A = A(r,θ) in polar coordinates by Corresponding to the scattered field u^s of the cavity scattering problem (TE), the PML scattered field (?)is the solution of the following PML problem:where (?).Corresponding to the scattered field u^s of the cavity scattering problem (TM), the PML scattered field (?) is the solution of the following PML problem:where s(x) := (k² -ε_rk²)(uⁱ + u^r), x∈Ω_ρ, and q(x) := -(uⁱ + u^r), x∈(?)Ω_ρ\Γ_ρ.Instead of solving cavity scattering problems (TE) and (TM), We solve their PML problems (18) and (19) by finite element method. The following theorem shows that the PML scattered field (?) is the approximation of the scattered field u^s, for the PML scattered field (?) converging exponentially to the scattered field u^s as the thickness of the PML layer tends to infinity.We make two assumptions:(H1)σ=(?), for some constantσ₀ > 0 and some integer m≥1;(H2) The PML problem (P₀) defined in (4.2.9) has a unique variational solution. For sufficiently largeσ₀, (H2) can be proved.Theorem 5 For sufficiently largeσ₀, the PML problem (18) of the cavity scattering problem (TE) has a unique variational solution (?)∈W inΩ_R, the PML problem (19) of the cavity scattering problem (TM) has a unique variational solution (?)∈V inΩ_R, where (?). Moreover, we have the estimatewhere (?) is a constant independent of k,ρandσ₀, (?)≤1. â…¢. A hybrid method for inverse cavity scattering problemAs shown in Figure 3, S is the cavity wall of the open cavityΩ₀,Γis the opening of the cavity, (?) is the infinite plane excluding the cavity opening,Ω:= R₊²âˆªÎ©₀∪Γ. We assume that the cavity in R² is formed by a infinite trough having the same geometric shape along x₃, and we assume that the medium is homogeneous in and outside the cavity and the ground is perfectly conducting. The cavity scattering problem is to find the total field u := uⁱ + u^r + u^s satisfyingwhere k > 0 is the wave number, uⁱ,u^r and u^s are incident field, reflected field and scattered field, respectively. uⁱ and u^r satisfy (15).We consider the following inverse cavity scattering problem:Definition 5 (Inverse cavity scattering problem). Given the cavity openingΓ, and given the scattered field u^s onΓfor only one incident wave uⁱ(x) = e^iks·d with d∈S, the inverse problem is to determine both the position and shape of the cavity wall S.The hybrid method for inverse cavity scattering problem is a hybrid between a integral equations method for Cauchy problem and a linearization method. So we will introduce the integral equations method for Cauchy problem firstly.Definition 6 (Cauchy problem). Let D be a bounded domain with a boundary (?)D :=Γ'∪S' of class C², The Cauchy problem is to recover both Dirichlet and Neumann data on S' from the knowledge of these data onΓ' for the Helmholtz equation. In the domain D, let v(x) satisfies the Helmholtz equationΔv + k²v = 0. Prom the Helmholtz representation and the jump relations, we haveGiven the Dirichlet and Neumann data (?) onΓ', we will use the integral equations (21) to solve the the Dirichlet and Neumann data (?) on S'. We define some operators and functions byWith these operators and functions, we can rewrite the integral equations (21) asTheorem 6 Given (?), that is, given (g₁,g₂), the integral equations (22)has at most one pair of solutions (U, V) := (?). If -1/2 is not the eigenvalue of the double layer operator K_S', the operator B₂: C(S')→C(S') is bounded invertible. So (22) is equivalent towhere A₁ - B₁B₂^-1 A₂ is a compact linear operator from C(S') to C(Γ').Theorem 6 ensures the validity of using Tikhonov regularization to solve (23) by Theorem 4.13 of [12]. So we can solve (23) to obtain (?) with (?) known.It will be shown in the following part that the scattered field in the upper half-plane R₊² can be obtained by the scattered field on the cavity openingΓ.We denote by y^* := (y₁, -y₂) the reflection of y := (y₁,y₂) about {(y₁,y₂) : y₂ = 0}, Then G(x,y) :=Φ(x, y) -Φ(x, y^*) is the Green's function of the half-plane Helmholtz operator with Dirichlet boundary condition. Let u^s(x) be the scattered field of cavity scattering problem (20). Prom Helmholtz representation, the Green's second theorem, and u^s(x) and G(x, y) both satisfying the Sommerfeld radiation condition, we haveEquation (24) shows clearly that we can obtain the scattered field and also the corresponding normal derivative on any segmentΓ' in the upper half-plane R₊² by the scattered field on the cavity openingΓ.We assume that the cavity opening isΓ:= {(x₁,0); |x₁|≤l}, we also assume that the cavity wall S and its approximate curves can be parameterized bywhere z₁ and z₂ are both injective and C² functions, and z₂ is negative. We are now in a position to present the hybrid method. The hybrid method can be split into two steps. In the first step, given an approximation S' of the cavity wall S with parametrization z of the form (25), as shown in Figure 4, we can always find a arcΓ' in the upper half-plane R₊² such that S'∪Γ' is a closed curve of class C². By (24), (?) and (?) can be obtained with the scattered field u^s on the cavity openingΓknown. So we can use the integral equations (22) to solve (?) and (?), then weobtain an approximation to the total field trace (?) and its exterior normal trace (?). In the second step, we still denote the total field of cavity scattering problem by u, and we define a operator F by F : z→u (?) z. For reconstructing the shape of the cavity wall, we only need to find the curve, on which the total field u = 0. In other words, we need to find an update parametrization z + h satisfying F(z + h) = 0. Therefore, as in the classic Newton method, we solve the linearized equation F(z) + F'(z)h = 0with respect to the shift h. In the next theorem, we will characterize the Frechet derivative F', before we do it, we have to conclude that the hybrid method consists of repeating the two steps iteratively until some stopping criteria are fulfilled.Theorem 7 The operator F : C²([Ï€,2Ï€])→C([Ï€,2Ï€]) is Frechet differentiable and the Frechet derivative is given by...