Interior Penalty Discontinuous Galerkin Method Solving Time-domain Fluid-solid Interaction Problem

In this paper,we consider the interaction between a bounded penetrable elastic body and a compressible inviscid fluid,in which the elastic body is coupled with the fluid through transmission boundary conditions.The incent acoustic wave is scattered by the elastic body,to determine the resulting scattering wave is the so-called fluid-solid interaction problem.This problem is one of the central importance in detecting and identifying submerged objects and deserves to be studied widely.The problem is mathematically formulated as an initial-boundary transmission problem.However,most of the investigations study typical fluid-structure interaction problems under the time-harmonic setting.The time-domain fluid-solid interaction problem we considered is that the governing equation in unbounded fluid region is the acoustic wave equation and the displacement field of the elastic body satisfies the elastic wave equation,the two wave equations are coupled together by the transmission boundary condition on fluid-solid interface.Since we discuss the unbounded domain,the acoustic scattering field or the elastic wave displacement field has to satisfy certain radiation condition in order to ensure the proper wave performance.Artificial boundary is introduced so that the original unbounded problem is transformed into a bounded region problem.We prove that the solution of the transformed problem exists and is unique and give the variational form of the problem.A symmetric interior penalty discontinuous Galerkin method is presented for the transformed problem.The resulting stiffness matrix is symmetric positive definite and the mass matrix is essentially diagonal;hence,this method leads to fully explicit time integration when coupled with an explicit time stepping scheme.Several prosperities of the semi-bilinear form are derived.the error in the energy norm is shown to converge with the optimal ordermin{,}()s kO h with respect to the mesh size h,the polynomial degree k,and the regularity exponent s of the continuous solution.Numerical results confirm the expected convergence rate illustrate the versatility of the method.