Application Of Integral Equation Method For Direct And Inverse Acoustic Scattering Problerms

Scattering theory has developed rapidly over the 20th Century,and has been usedin many aJ'eas,such as industry,medical,aerospace,Earth explorer,military and SO onIn this paper,vce introduced integral equation method for 2-D acoustic obstacle directand inverse scattering problems2-D sound--soft obstacle scattering problem can be described as the following ex--terior Dirichlet problemDeFinition 1(Exterior Dirichlet Problem)．Given a continuous function f on OD,finda radiation solution to the Helmholtz equationwhich satisfies the boundav condition Choose a combined acoustic double--and single--layer potentialas an approximation to the solution From the jump relations vce can get a boundaryintegral equationwhere S and K is single--mad double--layer operators respectively．If we get the solution(?)to this equation,then the farfield patternis given by Assume that the boundary curve OD possesses a regular analytic and 2π一periodicparametric representation ofthe formin the counterclockwise orientation satisfying for all t Then w-ededuced the parametric boundary integral equationHere the kernel K has logarithmic singularities at To overcome this singularities,we splitthe kernel intoChoose an equidistant set ofknots and use the quadra-and the trapezoidal ruleto approach the integral,we can get Nystr6m methodLet xXn and Yn be Baaaach spaces and T:be bounded and one-to-one Let X and be n-dimensional subspaces and be a projectionoperator．For given Y∈Y．the projection method for solving the equation Tx—Y is tosolve the equationLet the projection operator be interpolation operatoc w-e can get collocation method:Let X be a Banach space and We define collocation points by nd give n-dimensional subspace Then collocation method is tosolve the collocation equationsLet the projection operator be orthogonal projection,vce can get Galerkin method:LetX and Y be pre-Hilbert spaces and and be n-dimensional subspacesThen Galerkin method is to solve the equationsWe make numerical examples with the method introduced above and obtain the expectresultThe inverse acoustic obstacle scattering problem can be described as the followingproblemDefinition 2．The inverse problem is to determine the support ofthe scatterer D by thegiveninformationoffarfieldpatternAnimportantmethodto solveinverse scatteringproblemsislinear samplingmethodDefinethefaxfield equationbywhereDefinethefaxfield operator F:byDefine the Herglotz wavefunction with kernel byTheorem 1．Assume that D is simply connected,k2 is neither a Dirichlet or Neumanneigenvalue and that the radiation solution ofHelmholtz equation satifies Dirichlet or Neumann or impedance boundary condition then for every∈>0 andYo∈D thereexists af．notion such that--andan wavefunction with kernel g(；Yo),Thetheorem above show-sthatllg becomes unbounded asYo∈Dtendsto x∈dD So the suppoort of the scatterer D can be determined by this property．How-ever,in this theory only the case of Yo∈D is discussed For Yo《D．there are thefollowing theoremsTheorem 2．Assume that OD is Lipschitz and k。is not a Dirichlet eigenvaluefo,一△in D Then if is theg far field operator corresponding to the problem for Dirichletboundary conditions,we have that(1)ifyo∈D,then for every∈>0 there exists a solutiong inequality such that where is the Herglotz wavefunction with kernel (2)if then for every∈>0 and (?)>0 there exists a solution such that where Herglotz wavefunction with kernel Theorem 3·Assume that IS Lidschitz,and having a Lidschitz dissection Then矿F isthefarfieldoperator correspondingtotheproblemsfo,mixed boundary conditions．z e we havethat(1) then for every∈>0 there exists a solution of the inequality such that where is the Herglotz wavefunction with kernel (2) then for every∈>0 and d>0 there exists a solution inequality such that where is the Herglotz wavefunction with kernelWe define the operator which maps the boundary data ontothe far field pattern It can be proofed that in essence,using a regubxization method tosolvethefarfield equationis the same with using a regularization method to solve Theorem 4．Assume that there does not exista Herglotz wavefunnction which is a Dirich—let eigenfunction of—A in D to k2 and compact operator F is injective and normal,andits singular system is where Let q be aregularizationfilterfor F Define the regularization method Ra for F byThen defined regularization methodforGFor the linear sampling method,vce have the following corollary．Corollary 5．Let be a positive sequence with and bedifinedasintheorem 4 and setThen we have will belargewhen and Furthermore．for∈>0 there exists where C is a constantonlydepending on D,∈and (2)We present several numerical examples by using factorization method,and recon-struct the boundary ofthe scatterer．...