Numerical Methods For Scattering Problems Of Near Field And Unboundted Surfaces

The propagation and scattering of acoustic and electromagnetic waves are funda-mental problems in numerous engineering and scientific fields. However, the numericalsimulation of these phenomena remains a serious challenge. There are still many problemsunsolved, particularly for the problems in unbounded domain and with big wavenumber.The dissertation, divided into three parts, mainly concerns the numerical simulation ofan important experimental mode in near field optics modeling photon scanning tunnelingmicroscope and the well-posedness of scattering by unbounded surfaces.Part one consists of Chapter1and Chapter2. We first briefly formulate the back-ground of acoustic and electromagnetic scattering problems. Then we introduce the funda-mental knowledge about Helmholtz equation and Maxwell equations. At last, we present ashort review of the current techniques based on wave functions containing ultra weak vari-ational formulation, discontinuous enrichment method, least squares method and partitionof unity method.Part two includes Chapter3and Chapter4. We focus on the model of photonscanning tunneling microscope and study the direct scattering problem. We considerthe model shown in Figure1. We denote the point in the plane by x=(x, y)∈R2.By Γ0={x|y=0}, the whole space R2is divided into two parts R2R²=R₊²âˆªR₂∪Γ-0. The corresponding refractive indexes are n+and n respectively,such that n+<n. A sample S with refractive nsis deposited on the substrate Γ0andillustrated from below by time harmonic plane waves ui=exp(iαx+iηy) at an angleθ where α=n k0sin θ and η=n k0cos θ with the free-space wave number k0. Whenθ becomes greater than a critical angle θcr, the total inner reflection happen. Thenevanescent waves appear in R+2. We try to find the scattered field uscaused by theinfluence by the sample.We have to truncate the zone by Lipstchiz artificial boundary Γ with unit out normalν to restrict the problem on a bounded domain,i.e. Γ=. Then we impose absorbing boundary condition on Γ for us with refractive Thus according to the electromagnetic theory of Maxwell, under the TM polarization, the total field u=uref+us satisfy the following boundary value problem: with g=((?)v-ik0n(x)uref, the reference field uref is given by where ut and ur denote the transmitted and reflected waves, respectively. More precisely, with It is known from (2) and (3) that when the incident angle θ is greater than the critical one θcr, i.e. k0n+<|α|, γ(α) is pure imaginary, thus the transmitted field becomes an evanescent wave which propagates along the substrate in the x direction and decays in the y direction.We propose two methods for above problem.â… . The ultra weak variational formulation for near field scatteringWe first partition the domain Ω, into a collection of disjoint elements Ωk,k1,2,…,N, denoted and assume that the mesh is chosen such that for every Ωk with unit outward normal vk,(?)Ωk is Lipschitz boundary and refractive n(x) is constant Ωk on each element Ωk, i.e. nk=n(x)|Ωk. For two elements Ωk and Ωj, we define with normal nk pointing from Ωk to Ωj if Γk,j≠(?).A face of (?)Ωk on the exterior boundary Γ is denoted by Γk=Γ∩(?)Ωk with unit outward normal vk if (?)Ω∩Γ≠(?).The parameter σ defined on the skeleton of the mesh is given as From the definition of σ, it is easily seen that σ>0and σ(x)=k0n(x), x∈F. We denote σk,j=σ|Γk,j and σk=σ|Γk.Hilbert space X is defined by with inner product and the induced norm Here and in the following we denote xk=x|(?)Ωk, fx∈X.Space H is defined by N withWe introduce following operators.NExtension mapping Hk is defined by where ek is the unique solution of the boundary value problemOperator F=(Fk)∈(?)(X) is defined byOperator Π∈(?)(X) is defined byBy Green theorem, we deduce an ultra weak variational formulation equivalent to boundary value problem (1).Theorem1be the solution of (1) and satisfy the regularity hypothesis Define x∈X, such that Then x satisfy following ultra weak variational formulation where Here b∈X is defined, via the Riesz representation theorem, byConversely, if x satisfy (4), then u defined by ((?)vk+iσ)uk=xk solves problem (1). The ultra weak variational formulation can be further written as If denote the dual operator of F by F*∈C(X), we define A=F*H. Then the ultra weak variational formulation (4) is equivalent to the problem: find x∈X, such that where X denotes the unit operator.We use a discrete space Xh (?) X of finite dimension to substitute the space X. Then we obtain the discrete problem: find xh∈Xh such that or equivalently Here Ph denotes the orthogonal projector from X onto Xh. By the properties of A, we have following result.Theorem2The discrete problem (8) is uniquely solvable.In practice, We use solutions of the homogeneous Helniholtz equation to construct the discrete space Xh. On each element Ωk, we choose functions ek,l l=1,2,…,P(P∈N), Which are nonzero solutions of the homogeneous Helniholtz equation, i.e but zero on the other elements, i.e. ek,l=0, on Ω,j for j≠k. We denote the space spanned by ek,l byNext, we define zk,l by Then discrete space Xh is constructed from zk,l, preciselyIn our numerical simulations, we use p=2q+1(q>1, q∈N) plane waves on Ωk with barycenter of the form For simplicity, the directions of the wave vectors dk,l of these plane waves have been chosen equidistributed in the plane,Since the emergence of the evanescent waves, We also use them on Ωk to construct base functions for capturing numerically the characteristics of its small-scale feature, of whereFurthermore, we analyze the error estimates and convergence of the problem.By [u]Γk,l and [(?)vu]Γk,l we denote the jump of u and its derivative (?)vu across the interface Γk,l respectively, i.e. We also define the corresponding normsWe have following result about a(·,·).Theorem3For x∈X, define u=ε(x), then it holds (13)By the duality technique, we prove the following estimate.Theorem4Let x∈and xh∈Xh be the solution of (4) and (8), respectively. If denote u=Γ(x) and uh=ε(xh), we have The constant C is independent of u and h but relies on a.According to above results, we prove the estimate concerning interpolation error. Theorem5Let x∈X and xh∈Xh be the solution of (4) and (8), respectively. If denote u=ε(x) and uh=ε(xh), we have The constant C is independent of u and h but relies on σ.Thus we prove the final error estimate together with the the approximation properties of the space Vh.Theorem6be the solution of (1) and (8), respec tively. If denote uh=ε(xh), then for q≥K we have And the constant C is independent of u and h but relies on σFinally, we present several numerical examples to demonstrate the convergence and effectiveness of the method introduced above. The results also show that the method is fast and accurate.â…¡. The least square method for near field scatteringWe first partition the domain Ω, and use the same notations by the way in previous section. Then on each element Ωk, we choose P smooth functions φk,l l=1,2,…,P which are nonzero solutions of homogeneous Helmholtz equation Δu+k02nk2u=0on Ωk and vanish outside Ωk. We further denote the space spanned by φk,l as φk,l(k1,2,…,N, l=1,2,…, P). Then we approximate the solution u of (1) by ua defined by ua∈Vh where the coefficients alk need to be determined. Moreover, we define the functional with a∈CM unknown (M=NP).The solution of the least squares problem gives the coefficients of ua and then ua is the numerical solution of our problem.In practice, the candidate for the basis functions φk,l are plane waves and evanescent waves.We prove a simple duality result that provides the basic error estimate for the method.Theorem7Let u be the solution of (1) and ua∈Vh has the form like (17) with coeffi-cients defined by (19). ThenIn our numerical simulations, we use p=2q+1(q≥1, q∈N) plane waves onΩk with barycenter xk=(xk,yk), of the form For simplicity, the directions of the wave vectors dk,l of these plane waves have been chosen equidistributed in the plane,Since the emergence of the evanescent waves, We also use them onΩk to construct base functions for capturing numerically the characteristics of its small-scale feature, of the form where forWe denote the space spanned by ek,l byThen by the approximation properties of the space Vh, we have Theorem8be the solution of (1) and ua∈Vh has the form like (17) with coefficients defined by (19). Moreover we denote k=k0max nk. Then forq≥K we have with The con-stant C is independent of u and h but relies on aFinally, we present several numerical examples to demonstrate the convergence and effectiveness of the method introduced above. The results also show that the method is fast and accurate.Part three consists of Chapter5. We consider another Helmholtz boundary value problem, which models scattering of time harmonic waves by unbounded sound soft sur-faces. We analyze the well-posedness of the problem and give a priori estimate for two cases in which the medium above the surfaces are anisotropic and inhomogeneous respectively.Suppose D is a connected open set such that for some constants f-<f+, it holds where en denotes the unit vectorin the direction of xn. Let F=(?)D and Sa=D\Ua for a≥f+Hilbert space Va is defined by with a≥f+, on which we impose the wave number κ0>0dependent scalar product (u,v)va=fsa ((?)u·(?)v+κ02uv)dx, and induced norm||u||Va. For s∈R, Hs(Γa) denotes the Sobolev space with norm where we identify Fa with Rn-1and (?) is Fourier transformation.I. For nonhomogeneous medium caseWhen the medium above unbounded surface is anisotropic, time harmonic acoustic waves scattering by the sound soft surface is formulated by following boundary value problem:given a source g∈L2(D), supported in SH for some H≥f+, we seek to find scattered field u:D→C, such that u|Sa∈Va,k0for every a>f+, and in a distributional sense.. We assume that κ satisfy with constants κ+>0, κ0>0.As part of the boundary value problem, we apply following radiation condition known as angular spectrum representation where x=(x1,X2,…,xn-1)∈Rn-1(n=2,3) and FH(ξ) denotes the Fourier transfor-mation of FH=u|ΓH. This radiation condition shows that above the rough surface and the support of g, the solution can be represented in integral form as a superposition of upward traveling and evanescent plane waves.For a>H>f+, there exist continuous embedding, i.e. trace operatorsWe introduce the sesquilinear form b:VH×VH→C defined by Here (·,·) denotes the scalar product of L2(SH), T is the Dirichlet-to-Neumann operator on ΓH defined by where Mz is the multiplying byThen we can formulate the variational problem:find u∈VH such thatWe next prove the equivalence between the boundary value problem and the varia-tional problem. Theorem9If u is a solution of the boundary value problem (26),(28) then u|SH satisfies the variational problem (30). Conversely, if u solves the variational problem (30), let FH=γ-u, and the definition of u is extended to D by (28), for x∈UH, then the extended function satisfies the boundary value problem (26),(28), with g extended by zero and κ extended by taking the value κ0from SH to D.Thus we can obtain the well-posedness of boundary value problem through that of variational one.Theorem10Suppose D satisfies the assumption (25), and wave number κ satisfies (27). Then variational problem (30) has a unique solution u∈VH such thatWe reduce the proof of above theorem to establish following priori estimation. Theorem11Suppose D satisfies the assumption (25), and wave number κ satisfies (27). Assume g∈L2(SH) and u∈VH satisfy ThenWe need first to establish a Rellich identity for the solution of Helmholtz equation.Theorem12Suppose with If u∈H1(SH) is a solution of the variational problem (5.10), then it holdsFurther, we use the Rellich identity to derive a priori estimate of the solution of the variational problem for the case that Γ is smooth. Theorem13Suppose Γ={(x,xn)\xn=f(x), x∈Rn-1} with∈(Rn-1), D satisfies the assumption (25), wave number κ satisfies (27). Assume g∈L2(Sh) and u∈Vh satisfy ThenBy the results on approximation of nonsmooth boundary domains by smooth bound-ary domains, we prove Theorem11thus Theorem10.Finally, we consider the corresponding finite element discrete problem and analyze the error estimation.â…¡. For anisotropic medium caseOur method can also be extended to the case that there is anisotropic medium above the unbounded surface. The time harmonic acoustic waves scattering by the sound soft surface is formulated by following boundary value problem: given a source gï¿¡L2(D), supported in Sh for some H≥f+, we seek to find scattered field u: D→C, such that u|sa∈Va for every a≥f+, and in a distributional sense. Here κ0is the wave number and A=(aij)∈Cn×n has the special structure with a symmetric submatrix aij∈L∞(Sh) and constant an>0. Moreover, we assume that and A|ΓH=a02â…  with constant a0>0and unit matrix â… . We also suppose that spectral norm||A||2is bounded (||?||2is the Euclid norm of matrices), A is elliptic and non-positive, i.e. there is constant c0>0such that As part of the boundary value problem, we apply following radiation condition known as angular spectrum representation where Fh(ξ) is the Fourier transformation of FH=μ|ΓH.We introduce the sesquilinear form VH×VH→C defined by with the Dirichlet-to-Neumann operator T1on ΓH defined by where M1is the operator multiplying byWe then prove that the boundary value problem (35),(37)is equivalent to the follow-ing variational problem:find μ∈VH such thatThus we can obtain the well-posedness of boundary value problem through that of variational one.Following the previous process for nonhomogeneous case, we establish the Rellich type identity. By the results on approximation of nonsmooth boundary domains by smooth boundary domains, we finally have the following result. Theorem14Suppose D satisfy (25), then variational problem (39) has unique solution μ∈VH such that with K0=κ0(H-f₎.