Some Research On Numerical Methods Of Direct And Inverse Obstacle Scattering Problems In The Time Domain

Scattering and inverse scattering problems are important research topics in mathematical physics,which have significant applications in energy,medicine,detection and other fields,these problems have continuously attracted much attention by many researchers.Scattering and inverse scattering problems can be divided into frequency domain problems and time domain problems.The frequency domain problems can describe the spectral characteristics commendably,but lack the ability to capture instantaneous information.The time domain problems are able to describe the transient information well and simulate the more general and non-linear materials.In this thesis,we first consider the time domain inverse scattering problem for acoustic wave,and an inversion method is established to reconstruct the position and shape of the obstacle based on the time domain scattering data.On this basis,the scattering and inverse scattering problems of elastic wave in the time domain are studied,we start with the scattering problem,we establish a mathematical model to describe the scattering phenomenon of elastic wave by analyzing the physical mechanism of elastic wave,and discuss the well-posedness of the scattering problem.Furthermore,based on the scattering problem of elastic wave,we explore how to reconstruct the shape and position information of the obstacle based on the time domain scattering data of elastic wave.To solve the inverse obstacle scattering problem for acoustic wave,we establish a quantitative method of convolution quadrature method combined with the nonlinear integral equation method to reconstruct the position and shape of the obstacle.First,based on the retarded potential theory and the boundary condition,the time domain retarded potential boundary integral equation is proposed.Then,we adopt the convolution quadrature method for time discretization to transform the retarded potential boundary integral equation into a system of decoupled boundary integral equations for Helmholtz equation with complex wave numbers,and complex wave numbers are time dependent.On this basis,by using the iterative idea of the nonlinear integral equation method,a boundary integral equation system composed of field equation and data equation is established,given an approximation for the boundary of the obstacle,one can solve the field equation for the density function and the data equation is linearized with respect to the boundary which requires the Fr?echet derivative of the corresponding operator,at last,the scaled Newton method with Tikhonov regularization is used to solve the update of the boundary,so that a new approximation of the boundary can be obtained at each iteration step,and the shape and position information of the obstacle can be reconstructed.As for the analysis of the forward scattering problem for elastic wave,we give the well-posedness of the scattering problem by using the Galerkin method and energy estimate method.First,using the fact that the wave has a finite speed of propagation in the time domain,a compressed coordinate transformation in polynomial form with high order smoothness is constructed to reduce the scattering problem into an initial boundary value problem in a bounded domain over a finite time interval.Then,based on the Galerkin method with modified subspace,the existence of the solution is given,and the uniqueness and stability estimate of the solution is established by the energy estimate method.A priori estimate with explicit time dependence is also obtained by considering the time domain variational problem directly and using a special auxiliary function.To solve the inverse obstacle scattering problem for elastic wave,we extend the convolution quadrature method combined with the nonlinear integral equation method in acoustic wave case.This extend is non-trivial,the fundamental solution of the Navier equation is in the form of matrix,and the singularity is difficult to be separated,we use the Helmholtz decomposition to transform the boundary value problem of the timedomain Navier equation into a coupled boundary value problem of the wave equation,and then adopt the convolution quadrature method combined with the nonlinear integral equation method for the inverse problem.It is worth mentioning that due to the complexity of the coupled boundary conditions,the numerical calculation is difficult,and the numerical discretization of boundary integral equation and the treatment of singular integral in the algorithm are more complex.We introduce a special decomposition method for the kernel function to realize the above problems in numerical calculation.