Direct And Inverse Interior Scattering Problems In Elastic Cavity

The direct and inverse scattering problems in an elastic cavity are studied in this paper.It mainly consists of two parts:numerical solution of the direct problem;theoretical analysis and numerical simulation of the inverse problem.For the direct interior scattering problem,we apply the Finite Element method to solve the direct problem.A corresponding variational form is established firstly.By analyzing the spatial discretization,numerical integration and boundary condition processing,we give a complete numerical formulation and calculation process.Finally,several numerical examples are presented in the different spatial discretization cases of triangulation and quadrilateral subdivision,which complete the numerical simulation.Numerical results show the feasibility and efficiency of our algorithm.From the error results,it also can be seen that the L~?convergence order of linear finite element under norm and L~2 norm are close to the second-order convergence,while the convergence order underH~1 semi-norm is close to the first-order convergence.The first part deals with the ill-posedness by constructing the scattered field from the elastic field on the interior measurement curve based on the potential theory.Then the nonlinearity is solved by transferring the original problem to a minimization problem,i.e.,reconstructing the boundary as the closed curve,on which the boundary condition for the total field can be satisfied in the sense of L~2 norm.In this paper,the feasibility of the decomposition method is analyzed theoretically,and some numerical examples are given to verify the rationality and effectiveness of the method.In the numerical example,four different elastic cavities are reconstructed using the scattering field due to incident dipole of only S-wave or P-wave.The numerical results show that the horizontal or vertical direction will produce some interference information,but the interference information does not form a closed curve.Considering that this is an internal scattering problem and the approximate size of the cavity is known,a closed curve of reasonable size can be selected as the shape reconstruction result of the cavity.