| In this thesis,we mainly consider the scattering problems of two kinds of complex two-layered cavities.The first kind is the scattering problem of the two-layered cavity with conduction boundary conditions.The internal boundary of the cavity is inverted by using the factorization method based on the near field information,and numerical examples are given.The second kind is the scattering problem of the twolayered cavity with mixed conduction boundary conditions.The internal boundary of the cavity is inverted by using linear sampling method based on near-field information,and the conductivity of the physical coating on the internal boundary is calculated.The thesis consists of five chapters:In Chapter 1,we summarize the background of the related problems and state the main results of the present thesis.We also give some preliminary results used in the whole thesis.In Chapter 2,we study the well-posedness of direct scattering problem in a twolayered cavity with generalized boundary conditions on the external boundary and conduction boundary conditions on the internal boundary in Rd where d=2,3.We prove the well-posedness of direct scattering problem by using variational method and boundary integral equation method,respectively.The variational method has less requirements on the smoothness of the parameters and the boundary of the region,but it cannot give the representation of the solution.The boundary integral equation method requires that the parameters in the problem be constants,but it does give the integral representation of solution to the problem,which brings convenience to the numerical simulation of the internal boundary of the inversion cavity and its physical properties.In Chapter 3,we study the uniqueness of the inverse scattering problem of two-layered cavity with conduction boundary.Uniqueness is the primary problem in the study of the inverse problem.It is a theoretical guarantee for an effective numerical method and is extremely challenging.With the help of a well-posed internal conduction problem with source terms,and a prior estimate of the conduction problem with boundary values of Lp(1<p<2),by the reciprocity relation with respect to the near field,the uniqueness of the internal boundary and conductivity of the double-layer cavity is obtained by contradictory method.In Chapter 4,we study the theoretical basis of the reconstruction of the internal boundary of two-layered cavity with conduction boundary by factorization method,and give the related numerical simulations.The factorization method is the most representative method of the qualitative method to study inverse problems,and its key is the decomposition of operators and the verification of the range identity theorem.In this chapter,the internal boundary of the cavity has conduction boundary conditions w-u=f,?w/?v-λ?u/?v+ηω=h,on Γ1,where w is the scattering field,u is the penetration field,A>0 is the transmission coefficient,and the conductivity η satisfies R(η)≥0 and J(η)>0.We describe exactly the kernel space of the boundary valued operator G,obtain the decomposition N=-GM*G*,and verify that the decomposition of N satisfies the identity theorem of range.The feasibility of the decomposition method is verified by numerical simulation.In Chapter 5,we study the theoretical basis of reconstructing the internal boundary of the two-layered cavity with mixed boundary by linear sampling method and give the approximate calculation of the conductivity of the boundary coating.The biggest advantage of the linear sampling method is that it does not need to know the relevant prior information of the scatterer.In this chapter,the internal boundary of the cavity has mixed boundary conditions w-u=f on Γ,?w/?v-λ?u/?v=g on Γ1,?w/?v-λ?u/?v+ηω=h on Γ2.We prove the well-posedness of the direct scattering problem with the above boundary conditions by the variational method,and verify that the decomposition of the near-field operator satisfies the properties required by the linear sampling method,and give the calculation method of the conductivity Γ. |
