Integral Equation Methods For Two-Dimensional Scattering Problem And Their Numerical Realization

The two basic problems in classical scattering theory are the scattering of time-harmonic acoustic or electromagnetic waves by a penetrable inhomogeneous medium of compact support and by a bounded impenetrable obstacle.We now consider the scattering of time-harmonic acoustic waves by a bounded impenetrable obstacle.The two-dimensional scattering problem by sound-soft obstacles leads to the following Dirichlet problem.Exterior Dirichlet Problem.Given a continuous function,on (?)D,find a radi-ating solution u∈C2(R2＼D)∩C(R2＼D)to the Helmholtz equation which satisfies the boundary conditionHere,we seek the solution of Helmholtz equation in the form of acoustic single-layer potential with a density (?)∈C((?)D). Then from the basic jump relations and regulalrjty properties of aCOUStic single-layer potential we see that the density is a solution of the integral equation Now we have obtained the boundary integral equation of the scattering problem.For the fundamental solution to the Helmholtz equationΦ(x.y),we deduce that for |x-y|→0.Therefore,the fundamental solution to the Helmholtz equation in two dimensions has the same singular behavior as the fundamental solution of Laplace equation.We assume that the boundary curve (?)D possesses a regular analytic parametric representation of the form in the counterclockwise orientation satisfying[x′1(t)]2+[x′2(t)]2＞0 for all t.Then we transform the boundary integral equation into the parametric form where we have set As the Hilbert space X,we choose X=L2(0,2π).The operators K,K0 and C are defined by for t∈[0,2π]andψ∈L2(0,2π).We observe that K,K0 and C are compact operators in L2(0,2π)since the kernels are weakly singular.Consider the operator equations of the form where Kis a linear compact operator between Banach space X and Y.A regulariza-tion strategy is a family of linear and bounded operators such that i.e.the operator RαK converge pointwise to the identity.Let K:X→Y is the linear bounded compact operator between the Hilbert spaces X and Y.Determine xα∈X that minimizes the Tikhonov functional Then xαis the unique solution of the normal equation The solution xαcan be written in the form xα=Rαy withWe rewrite the operator equation Kx=y in the form x=(I-αK*K)x+αK*y for someα＞0 and iterating this equation,i.e.,computing for m=1,2,….xm has the explicit form xm=Rmy,where the operator Rm:Y→X is defined byLetX,Y be Banach spaces and K:.X→Y be bounded and one-to-one.Furthermore, let Xn (?)X,Yn (?) Y bc finite-dimensional subspace of dimension n and Qn:Y→Yn be a projection operator.For given y∈Y the projection method for solving the equation Kx=y is to solve the equationLet X,Y are Hilbert spaces,and Qn:Y→Yn is the orthogonal projection operator onto Yn.Then equation QnKxn=Qny reduces to the Galerkin equations Choosing Yn= K(Xn) leads the least squares method,its solution xn∈Xn can be characterized byWe choose interpolation operator as the projection operator,then obtain collo-cation method.Let X be Hilbert space and Xn (?) X be finite-dimensional subspaces with dim Xn= n,let a≤t1＜…＜tn≤b be the collocation points.Then the collocation equations for Kx = y is defined byIn this thesis we make numerical experiments with all the methods we intro-duced and observed the expected result.Then we compute the far field pattern of the scattering field by the numerical solutions of the boundary integral equation.