Finite Element Method For Fluid-solid Interaction Problems With Periodically Distributed Bounded Obstacles

The fluid-solid interaction problems(FSI)investigate the interaction between the changes of deformable solids under the action of the fluid field and the effect upon the fluid field by the deformation or displacement of the solids.The problems have a wide range of applications and research value in many scientific fields.In mathematics,we various fluid-solid interaction problems as partial differential equation boundary value problems,using appropriate numerical methods to solve.In this paper,we study a class of two-dimensional fluid-solid interaction scattering problems with periodically distributed bounded obstacles.The mathematical model of the problem describes the scattering of incident acoustic waves by periodically bounded solids embedded in the fluid.We use the finite element method to solve this class of fluid-solid interaction models.Without loss of generality,we assume the obstacles are periodically distributed in the horizontal direction.We can only consider the fluid-solid interaction problem in a single periodic cell.However,the fluid region is still unbounded in the vertical direction.We will apply the Dirichlet-to-Neumann(DtN)maps to define the artificial boundary conditions.The DtN map,regarded as one of absorbing boundary conditions,is a common tool for defining transport boundary conditions,which exploits the propagation nature of acoustic waves to establish a relationship between the understood Dirichlet and Neumann values at artificial boundaries.Then,the finite element method is employed to get the numerical solutions.We first describe the mathematical model of the fluid-solid interaction problems.Then we introduce an artificial boundary method for dealing with unbounded regions.The key to the method is to introduce an artificial boundary condition defined by DTN mapping on the artificial boundary,which translates the problem to the bounded region.We introduce the DtN operator based on the Fourier series and its properties,and use it to build a variational form on the truncated domain.Then by using analytic Fredholm theory,we analyse the existence and uniqueness of weak solutions to the variational problem.Since in practical numerical calculations,we need to truncate the exact DtN operator,then the original variational problem becomes a modified variational problem.But the error arising from truncation affects the numerical results,we give truncation error estimates for the DtN operator and obtain an error estimate that decays exponentially and then prove the uniqueness of the existence of weak solutions to the modified variational problem.Next,we solve the variational problem using the finite element method to obtain the Galerkin form of the variational problem.Then we establish a priori error estimates for the finite element solutions on the Sobolev spaces V1 and H0,respectively.We obtain that the error is O(h)in V1 and O(h2)in H0.Finally,we give two numerical arithmetic examples to verify the correctness of the theoretical results.