Computing Electromagnetic Scattering To Open Cavities By A Symmetric Coupling Of Finite Element And Boundary Integral Method

The study of direct and inverse cavity scattering problem for electromag-netic waves is very important both in theory and applications in the recent years. Given the incident field, the direct problem is to determine the field distribution from the known shape of the cavity; while the inverse problem is to determine the shape of the cavity from the measurement of the field on an artificial boundary enclosing the cavity.Consider the scattering of a time-harmonic electromagnetic plane wave by an arbitrarily shaped and filled cavity embedded in a perfect electrical-ly conducting infinite ground plane. A method of symmetric coupling of finite element and boundary integral equations is presented for the solu-tions of electromagnetic scattering in both transverse electric and magnetic polarization cases. In this paper, both the direct and inverse scattering problems are discussed based on a symmetric coupling method. Variational formulations for the direct scattering problem are presented, existence and uniqueness of weak solutions are studied, and the domain derivatives of the field with respect to the cavity shape are derived. Uniqueness and local stability results are established in terms of the inverse problem. This paper concentrates on the TE polarization first. Let an incoming plane wave uj = exp(iαx1 - iβx2) be incident on the perfect electrically conducting surfaceΓg∪S from above, whereα-κ0 sinθ,β=κ0 cosθ,θ∈(-π/2,π/2) is the angle of incidence with respect to the positive x2-axis, andκ0 =ω(?) is the wavenumber of the free space.The scattered field satisfiesΔus +κ2us = -(κ2 -κ02)uref aboveΓg∪S, (1) us = -uref onΓg∪S. (2) In addition, the scattered field is required to satisfy the radiation conditionThe following lemma is concerned with the uniqueness of the direct scat-tering problem.Lemma 1 The direct problem (1)-(3) has at most one solution.For a bounded regionΩin R2 with boundary (?)Ω,Γ(?)Ω, Hs(Ω) and Hs(Γ) will denote the usual Sobolev spaces with norm‖·‖Hs(Ω) and‖·‖Hs(Γ), respectively. Define the following spaces: L2(Γ) := {u|Γ: u∈L2((?)Ω)}, H1/2(Γ):={u|Γ:u∈H1/2((?)Ω)}, H1/2(Γ) := {u∈H1/2(Γ) : supp u (?)Γ}. It can be proved, inΩ, the following problemsΔu +κ2u = 0 aboveΓg∪S, (4) u = 0 onΓg∪S. (5) ▽·(κ-2▽u) +u = 0 aboveΓg U S. (6) (?)nu = 0 onΓg U S. (7) have an equivalent variational form: Find u∈Hs1(Ω) = {u∈H1(Ω) : u = 0 on S} such that whereφis the normal derivative of the total field u onΓ. i.e.,φ= (?)nu, and〈·,·〉denotes the duality between H-1/2(Γ) and H1/2(Γ).To study the boundary integral equations, the paper introduce the single-layer potential operator Vte, the hypersinguiar integral operator Dte, the double-layer potential operator KTE and its adjoint operator KTE*, which are defined asThe integral equation for the total field on the. boundaryΓis: Using those operators,the equations above can be written as where f= uref , g = (?)nuref , and I is the identity operator.The equations as the following consist of the variational formulation for the symmetric coupling of the finite clement and boundary integral methods for the direct cavity scattering problem. andUsing the results for the integral operators,Lemma 2 The single-layer potential operators VTE is compact from H-1/2(Γ) into H1/2(Γ), the double-layer potential operator KTE and its adjoint KTE* are compact from H1/2(Γ) into H1/2(Γ) and from H-1/2(Γ) into H-1/2(Γ). respectively, and the hypersingular integral operator DTE is compact from H1/2(Γ) into H-1/2(Γ). Furthermore. the sinle-layer potential operator VTE the and hypersingular integral operator Dte are coercive in H-1/2(Γ) and H1/2(Γ), i.e., there exit compact operators V0 and D0 such that Re[〈VTEφ,φ〉+〈V0φ,φj〉]≥C‖φ‖H-1/2(Γ) for allφ∈H-1/2(Γ), Re[〈DTEu,u〉+〈D0u,u〉]≥C‖uH1/2(Γ) for all u∈H1/2(Γ). Denote VTE = HS1(Ω)×H-1/2(Γ) and the norm in VTE is naturally defined for any u = [u,φ]∈VTE Based on the results above, for the direct problem,Theorem 1 The variational problem (13)-(14) admits a unique solution [u,φ]∈VTEThe uniqueness of the inverse cavity scattering problem: to determine the cavity wall S from the total field u measured onΓTheorem 2 Let [uj,φj] be the solution of (11)-(12) inΩj with (?)Ωj =Γ∪Sj for j = 1, 2. If u1 = u2 onΓ, then S1 = S2.Based on this variational form, the paper have the following result for the domain derivative.Theorem 3 Let [u,φ) be the solution of (13)-(14) inΩand n the outward normal to S. Then the domain derivative can be expressed as M'TE(S, p) = [u'|Γ,φ'], where [u',φ']∈H1(Ω)×H-1/2(Γ) is the weak solution of the following boundary value problem u' = -(p·n)φon S, (18) whereφ' = (?)nu' onΓ.In applications, it is impossible to make exact measurements. Stability is crucial in the practical reconstruction of cavity walls since it contains neces-sary information to determine to what extent the data can be trusted.The paper have the following local stability result.Theorem 4 If p∈C2(S,R) and h > 0 is sufficiently small, then d(Ωh,Ω)≤C‖uh - u‖H1/2(Γ), (19) where C is a positive constant independent of h.For the direct and inverse problems for the TM polarization case. The paper will state some parallel results for the direct problem and show the differentiability of the field with respect to the cavity shape and prove a local stability result.Consider the same problem geometry as that in TE polarization case, let an incoming plane wave ui = exp(iαx1- iβx2) be incident on the perfect electrically conducting surfaceΓg∪S from above.It is easy to verify that the scattered field satisfies▽·(κ-2▽us) + us = -▽·[(κ-2 -κ0-2)▽uref] aboveΓg∪S. (?)nus = -(?)nuref onΓg∪S. In addition, the scattered field is required to satisfy the radiation conditionThe following theorem is an analogue of Theorem 1.Theorem 5 The variational problem (4) admits a unique solution [u,φ]∈Vtm .For the uniqueness, domain derivative, and the local stability. It is briefly proved the uniqueness but elaborate the domain derivative and the local stability since the arguments vary quite a lot from those for the TE case.Theorem 6 Let [uj,φj] be the solution of (4) inΩj with (?)Ωj =Γ∪Sj for j = 1. 2. If u1 = u2 onΓ. then S1 = S2.Theorem 7 Let [u,φ] be the solution of (4) inΩand n the out-ward-normal to S. Then the domain derivative can be expressed as M'TE (S,p) = [u'|Γ·φ'), where = [u'φ'}]∈Vtm is the weak solution of the following boundary value problem▽·(κ-2▽u') + u' = 0 inΩ, (20) whereφ' = (?)nu' onΓ.Theorem 8 If p∈C2(S, R) and h > 0 is sufficiently small, then d(ΩhΩ)≤C‖uh - u‖H1/2(Γ) (24) where C is a constant independent of h,Ωh, is the region bounded by Sh andΓ, Sh:x + hp(x)n,p∈C2(S.R).