Adaptive Finite Element DtN Methods For Transmission Scattering Problems

The Scattering problems about the acoustic,elastic and electromagnetic waves have a wide range of applications in science and engineering.There are many difficulties in numerically solving scattering problems,such as the unboundedness of the computational region,the oscillation of the solution and the complexity of the scatterer structure.By using the a posteriori error estimate,adaptive finite element DtN methods are proposed to solve different transmission scattering problems,such as orthotropic medium scattering problems,bounded obstacle acoustic-elastic interaction scattering problems and periodic structure acoustic-elastic interaction scattering problems.The specific research contents are as follows.Firstly,for the orthotropic medium scattering problem,we propose an adaptive finite DtN method,which can both adaptively select the truncation parameters of the DtN operator,and adaptively refine the mesh.Based on the Dirichlet-to Neumann(DtN)operator,an exact transparent boundary condition is introduced and the model is formulated as a bounded boundary value problem.By a new duality argument technique,an a posteriori error estimate is derived for the finite element method with the truncated DtN boundary operator.The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN boundary operator,where the latter decays exponentially with respect to the truncation parameter.Based on the a posteriori error estimate,an adaptive finite element algorithm is proposed for solving the orthotropic medium scattering problem,where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are chosen through the finite element discretization error.Numerical experiments are presented to demonstrate the effectiveness of the proposed method.Secondly,for the bounded obstacle acoustic-elastic interaction scattering problem,we give the a posteriori error analysis and the corresponding adaptive finite element DtN algorithm.When the acoustic wave encounters an elastic obstacle,it not only scatters on the surface of the elastic object,but also excites the elastic wave inside the elastic object.In order to ensure the continuity of the velocity normal component and the traction force,it is necessary to impose complex kinematic and dynamic interface conditions at the fluid-solid interface.We design the corresponding error indicators based on the properties of acoustic and elastic waves in different regions,and give the corresponding a posteriori error estimate and the adaptive algorithm.Numerical examples demonstrate the computational efficiency of the adaptive refinement compared with the uniform refinement.Thirdly,for the periodic structure acoustic-elastic interaction scattering problem,we propose an adaptive finite DtN method,which can both adaptively select the truncation parameters of two DtN operators,and adaptively refine the mesh.In this case,both the acoustic wave scattering field and the elastic wave transmitted field satisfy the corresponding Rayleigh radiation condition.Therefore,it is necessary to introduce the DtN operators of both acoustic wave and elastic wave and define the artificial boundary conditions.In addition,to ensure the continuity of the velocity normal component and the traction force,we need to impose both kinematic and kinetic interface conditions at the fluid-solid interface.Due to its periodic structure,we can restrict the problem into one period.In practical computation,it is necessary to truncate two DtN operators at the same time.Using the dual argument method,we give the a posteriori error estimate,which includes both the finite element discrete error and the truncation error of the DtN operators.Based on the a posteriori error,we design an adaptive finite element algorithm.Several typical numerical experiments are given to verify the effectiveness of the algorithm.After each mesh refinement in these numerical experiments,we adjust the nodes at the periodic boundaries to ensure the quasi-periodicity of the numerical solution.