Study Of Numerical Computation Of Some Scattering Problems With Optimal Perfectly Matched Layer Technique

Scattering and inverse scattering theory of sound wave and electromagnetic wave plays an important role in mathematical physics. Usually, direct cattering problem is to find scattered field or far field of the scattered field with given incident wave and scatterer; However, inverse scattering problem is to find boundary of the scatterer or physical parameter with given scattered field or its far field. There are integral equation method, finite element method and PML method for solving direct scattering problems, and iterative techniques, analytic continuation methods and sampling method for inverse scattering problems. The theory and numerical methods of scattering problems and inverse problems are still arresting in the research of applied mathematics and computational mathematics. In this paper, some scattering problems with optimal PML technique are discussed. What we have done is as follows:â… . The optimal PML technique for scattering problem of bounded scatter1. The optimal PML technique for time-harmonic electromagnetic scattering problem of bounded scatterLet us consider the following problem:where u = uⁱ + u^s is the total field, uⁱ is the incident field, u^s is the scattering field. D (?) R² is a bounded domain with boundaryΓ_D. g∈H^-1/2(Γ_D), n is the unit outer normal toΓ_D. We assume the wave number k∈R is a constant. We introduce the variational form for(1)-(3)([11]): Given g∈H^-1/2(Γ_D), find u∈H¹(Ω_R) such thatwherewhere the operator T :H^1/2(Γ_R)→H^1/2(Γ_R) is definded as follow:The geometry of the scattering with truncated opt PML is as follow:whereΩ^PML = {x∈R²: R <|x|<ρ). Let us assumewhere where m≥2 is a constant andε₀ is a little positive constant.We prove the following theorem:Theorem 1. ifε₀≤(?), then Im(?).and we also obtain that the solution of PML problem (?) satisfieswhere (?). We introduce the variational form for PML problem(9): Given g∈H^-1/2(Γ_D), find (?)∈H¹(Ω_R) such thatwhereWe arrive the following theorem:Theorem 2.where C > 0 is a constant independent ofε₀ and M is a positive constant.So we arrive the error estimation between the PML solution and scattering problem solution:Theorem 3. PML problem(9)has a unique solution ifε₀ is little enough, and we prove the following result where C > 0 is a constant independent ofε₀ and M is a positive constant. The theorem 3 shows that the solution of PML truncated problem converges exponentially to the scattering solution ifε₀ is little enough. In the OPT PML technique the thickness of PML layer is free. Numerical experiments illustrate that the solution of PML truncated problem is independent of the thickness of PML layer. It offers us a new calculation method. We can improve the error by choosing a properε₀ with the same thickness of PML layer.2. A Uniaxial Optimal Perfectly Matched Layer Method for Time-harmonic Scattering Problemswhere D (?) R² is a bounded domain with Lipschitz boundaryΓ_D, g∈H^-1/2(Γ_D) is determined by the incoming wave, and n is the unit outer normal vector ofΓ_D,Ï„= |x|. We assume that k∈R is a constant. Let D be contained in the interior of the rectangular domain B₁ = {x∈R² : |x₁| < L₁/2, |x₂| < L₂/2}. And letΓ₁ = (?)B₁, n₁ be the unit outer normal vector ofΓ₁. For given f∈H^1/2(Γ₁), similar to [9] we introduce the Dirichlet-to-Neumann operator T:H^1/2(Γ₁)→H^-1/2(Γ₁)whereξis the solution of the following problemProblem (13) has a unique solution (?)(cf. e.g. [1]). The scattering problem (12) inΩ₁ = B₁\(?) can be formulated the following weak formulation (cf. e.g. [1]): Given g∈H^-1/2(Γ_D), find u∈H¹(Ω₁) such thatwhere a : H¹(Ω₁)×H¹(Ω₁)→C is the sesquilinear form:and (·,·)_ΓD stands for the inner product on L²(Γ_D). The existence of a unique solution of the scattering problem (14) is known (cf. e.g. [1],[82]). Then the general theory in Babuska - Aziz implies that there exists a constantμ> 0 such that the following inf-sup condition is satisfiedLet B₂ = {x∈R² :|x₁| < L₁/2 + d₁, |x₂| < L₂/2 + d₂} be the rectangle which contains B₁ and letΓ₂ = (?)B₂.We introduce integrable PML parametersγ_j(x_j) such thatγ_j = 1 for |x_j|≤(?) andγ_j(x_j) = 1 + (?)σ_j(x_j) for |x_j| >(?), j = 1,2, (?)≤|t|≤(?) + d_j, andσ_j(t) = 0 as 0≤|t|≤(?). Then we can define the complex stretched coordinate (?) by σ_j(t) is definded bywith small positive constantε₀ and m≥2.Theorem 4. Ifε₀≤(?) then (?).The PML solution (?) inΩ₂ = B₂\(?) is defined as the solution of the following systemwhere A=diag(γ₂(x₂)/γ₁(x₁),γ₁(x₁)/γ₂(x₂)).Similarly, we defind the Dirichlet-to-Neumann operator (?).So we obtain the following theorem:Theorem 5. For any f∈H^1/2(Γ₁), we havewhere C > 0 is a constant independent of k,ε₀.Theorem 6. For sufficiently smallε₀ > 0, the PML problem (19) has a unique solution (?)∈H¹(Ω₂), moreover, we have the following error estimatewhereγ=(?),m≥2, L = max(L₁, L₂), C > 0 is independentof k,d₁,d₂,ε₀.Theorem 6 indicates that the solution of the optimal PML problem converges exponentially to the solution of the original scattering problem. Since the convergence is insensitive to the thickness of the PML layer for sufficiently smallε₀, the total computational costs can be saved by choosing the thinner PML layer. â…¡. The optimal PML technique for scattering problem of unbounded scatter1. The optimal PML technique for electromagnetic scattering problem of cavitiesLet us assumeΩ(?) R² is an open cavity, its boundary is S₁,S₂, so we have (?)Ω= S₁∪S₂. R₊² = {(x, y) : y≥0} is the upper half-plane,μ₁,ε₁ is the permittivity and permeability in the cavity. The infinite ground plane excluding the cavity opening is denoted asΓ. Denote U = R₊²\Ωas the infinte domain upon the cavity. We assume there is free space in the domain U, and its permittivity and permeability isμ₀,ε₀. Denote (Eⁱ,Hⁱ) and (E^s,H^s) as the incident electromagnetic wave and scattering field. We consider the PML technique for the following polarization. For the TM polarization, the total filed u satisfies:For the TE polarization, the total filed u satisfies:and u^s satisfies Sommerfeld radiation condition:We introduce the PML layerΩ^PML =Ω₁\Ω₂, where (?), let (?). we choosewhereσ_x,σ_y is a unbounded absorbing function, and in the PML layer they satisfySo we arrive at u^PML is the solution of the following equations:where h(x) = (?), uⁱ denotes the incident field, u^r denotes the reflect field. It is shown that When the cavity opening is very large the amount of calculations is saved and the operation speed is improved by using rectangular perfectly matched layer instead of semicircle. Numerical experiments demonstrate that the optimal PML method is efficient and accurate to solve cavity scattering problems. 2. Rough surface scattering problemFor x = (x₁,...,x_n)∈Rⁿ(n = 2,3), we let (?) = (x₁,...,x_n-1) so that x = {(?),x_n). For H∈R let U_H := {x : x_n > H} andΓ_H := {x : x_n = H}. Let D (?) Rⁿ be a connected open set such that for some constants f_- < f₊ it holds thatThe rough surface scattering problem is: given function g, find u such thatwhere (?), with f at least bounded and continuous.Φis the fundamental solution of the Helmholtz equation. The formulation(29) is called upward propagating radiation condition(UPRC), its Fourier transform can be seen in [54]. The variational problem of (27)-(29)is: find u∈V_H such thatwhere where (?) is a DtN operator, M_z is the operation of multiplying by:There exists a constantγ> 0 such that b(v,φ) satisfies the following inf-sup condition [54].Above S_H we introduce an another layer S_H^L for some L > 0 where the PML is located given by (?). The virtual width of PML layer is denoted (?) and given bywhereσis an absorbing function in PML layer. The truncated PML problem is:The variational problem of (35) is: find u_p∈V_H such thatwherewhere (?) is a DtN operator, (?) is the operation of multiplyingby:There is an error between T and T_p:lemma 7. If we assumeα= Re(k(?)),β= Im(k(?)), then we have whereFrom above research we prove the convergence result as following:Theorem 8. Ifγ> C_U(α,β) andγis the inf-sup constant, the truncated PML problem (35) has a unique solution u_p∈V_H, and the following estimate is satisfiedWe take n = 2 and truncate the computational domain in x direction as well as y direction. Due to the complexity of convergence of two directions truncation, we do not study in this paper. Numerical experiments demonstrate that the opt PML method is efficient to solve the rough surface scattering problem.3. The impedance boundary value problemConsider the following impedance boundary value problem : In the case of an incident plane wave, the incident field uⁱ(x) is given by uⁱ(x) = (?), where d=(d₁,d₂)=(sinθ₀-cosθ₀) andθ₀∈(?) is the angle of incidence. The reflected or scattered part of the wave field is defined by u = u^t -uⁱ where u^t is the total field. U = {(x₁, x₂)∈R² : x₂ > 0} denotes the upper half-plane and its boundary isΓ= {(x₁,0) : x₁∈R}. The scattering field u satisfies:For H≥0, let U_H = {x:x₁≥R,x₂ > H] andΓ_H = {{x₁,H) : x₁≥R}. n is the normal toΓdirected out of U,Φis the standard fundamental solution of the Helmholtz equation, the relative surface admittance is given byWe truncate the domain as followingwhere the band domain (?) is the PML layer. We analy the DtN operator similar to the rough surface scattering problem, and arrive at the following result: Theorem 9. Ifγ> C_U(α,β),γis inf-sup constant then the truncated PML problem has a unique solution u_p∈V_H, andWe truncate the computational domain in x direction as well as y direction. Due to the complexity of convergence of two directions truncation, we do not study in this paper. Numerical experiments demonstrate that the opt PML method is efficient to solve the impedance boundary scattering problem.Due to the complexity of the scattering problems and the inverse scattering problems, there are still many problems to solve. How to improve on the method for computing is our aim for further work.