The PML Method Analysis For Some Scattering Problems In Unbounded Domain

In this paper,we mainly study the theoretical analysis of scattering problems truncated by perfectly matched layer in biperiodic structure and unbounded rough surface elastic medium.We need to solve the scattering field or diffraction field of the scattering problem in unbounded domain.In order to use the classical numerical algorithm,such as the finite element method or finite difference method,to solve the scattering problems,we need to truncate the unbounded physical domain into a bounded computational domain.So it is necessary to introduce some artificial boundary conditions.If usual boundary conditions are imposed on the artificial boundary,large error will rise.So some artificial media with absorption property should be introduced.The media absorbe the outgoing wave,but not pollute the inner region.Thus the scattering problem in truncated region is equivalent to the original scattering problem.This special medium layer is usually called the perfectly matched layer.This method is simple and easy to be implemented,so it is widely used in the calculation of wave propagation.In the analysis of the perfectly matched layer problem in biperiodic elastic structure,we consider two kinds of penetrate medium.Choosing a plane P wave as the incident wave,the total displacement field satisfies Navier equation.For the theoretical analysis,we define a quasi biperiodic function space and the corresponding inner product.With the help of two kinds of potential function,we use Helmholtz decomposition to divide scattering field into a combination of the shear wave and the longitudinal wave.Since both the potential function and scattering field are quasi biperiodic functions,we expand the potential function and the scattering field in Fourier series form,and derive a transparent boundary condition of the scattering problem.Thus the scattering problem is transformed into a boundary value problem.By introducing a complex coordinate extension function,we study truncated perfectly matched layer problem of the original scattering problem.Finally,we discuss the error between the original scattering problem solution and the truncated perfectly matched layer problem solution,and prove the well-posedness and convergence of the solution of the truncated PML problem.For the two-dimensional unbounded rough surface scattering problem,the wave propagation is governed by the two-dimensional Navier equation.Assuming that the rough surface is Lipschitz continuous and satisfies the homogeneous Dirichlet boundary condition.By introducing two scalar potential functions,we use the Helmholtz decomposition to split the elastic displacement field into the superposition of a shear wave and a longitudinal wave.Applying the differential properties of Fourier transform,we give the Fourier transform of the first variable of the scattering field and the potential function.Combining with the boundary conditions,we derive a transparent boundary condition for scattering problem,and obtain the corresponding boundary value problem.For the three dimensional problem,we introduce a scalar potential function and a vector potential function.Denoting the first two variables by a variable,and taking Fourier transform,we can equivalently derive the corresponding boundary value problem.In the 2D and 3D scattering problems,we introduce complex coordinate extension functions,and study the truncated perfectly matched layer problem,which is an approximate to original scattering problem.The analysis of PML for scattering problems in unbounded domain with rough surface is based on the assumption that the variational problem has a unique solution,and satisfies the inf-sup condition.Combining with trace theorem,we obtain the existence of weak solution of perfectly matched layer problem,and prove that the error of solutions between the original scattering problem and the truncated the perfectly matched layer problem exponentially decays with the increase of parameters and thickness of the PML medium.However,this method is not able to exclude resonance at some frequencies,which may caused by the technique are used.