A Smoothed Finite Element Method With TBC For The Elastic Wave Scattering By An Obstacle

The obstacle scattering problem refers to the influence of the obstacle on the incident wave.It has been widely used in various engineering fields,such as monitoring of cable accidents by electromagnetic waves,detecting of underwater oil by radar and sonar and exploring of crust and geological structure by elastic waves.Due to the elastic wave can obtain more accurate solution than acoustic wave and electromagnetic wave,the elastic wave scattering problem has attracted more and more attention.However,the obstacle shape of elastic wave is usually complex and so the elastic wave obstacle scattering problem is difficult to solve accurately.Therefore,a new numerical method based on W~2form--smoothing finite element method(S-FEM)is proposed to solve the elastic wave scattering problem.This method combines the advantages of finite element method(FEM)and meshless method,and has the characteristics of high accuracy,fast convergence and insensitive to mesh quality.This method has been applied in many engineering problems and has obtained good results.The common S-FEM model includes:a cell-based S-FEM(CS-FEM),a node-based S-FEM(NS-FEM)and an edge-based S-FEM(ES-FEM),etc.This paper mainly uses CS-FEM and ES-FEM to solve the elastic wave scattering problem in two-dimensional isotropic and uniform elastic medium.The essence of the problem is a boundary value problem(Boundary-Value Problem,BVP)defined in the unbounded region,which satisfies the Navier equation.To solve numerically,the transparent boundary condition(Transparent Boundary Conditions,TBC)technique with high accuracy is applied to truncate the unbounded region into bounded region.Since the derivation of the TBC formula of the Navier equation is complicated,finally the Navier equation is decomposed into two Helmholtz equations with coupling boundary conditions by using the Helmholtz decomposition.Then the TBC satisfying the coupling equation is deduced in detail.In this way,we obtain the BVP in the bounded region.Meanwhile,based on FEM background mesh--quadrilateral mesh(Q4)and triangular mesh(T3),cell-based smoothing domain(CS-D)and edge-based smoothing domain(ES-D)are established.Further,the S-FEM formula of coupling Helmholtz equation with TBC is derived.Using CS-FEM and ES-FEM,the discrete system equations are obtained.The FEM,CS-FEM and ES-FEM are applied to study the effect of angular frequency and mesh quality on the solution of Helmholtz equations with coupled boundary conditions of the elastic wave obstacle scattering problems by four numerical examples.Meanwhile,the numerical solution of the scattered field of the Navier equation is got by using the solution of the Helmholtz equation.In addition,the convergence rate of three methods with mesh size is analysed.The results show that the ES-FEM can obtain more stable and accurate solutions and has faster convergence rate for the elastic wave obstacle scattering problem compared with the traditional FEM and CS-FEM.