A Stable Node-based Smoothed Finite Element Method For The Wave Scattering Problem With Transparent Boundary Conditions

The research of the wave scattering has important applications in the fields of radar detection,geological exploration,sonar positioning,nondestructive detection,military and medical imaging.Among them,the scattering of open cavities and obstacles are two important unbounded domain problems.In this paper,the numerical methods of the electromagnetic wave open cavities scattering and the elastic wave obstacles scattering are studied.A variety of transparent boundary conditions(TBC)are derived and combined with the stable node based transparent boundary smooth finite element method(SNS-FEM),SNS-FEM-TBC is proposed to analyze these two kinds of problems.For the electromagnetic wave open cavity scattering problem,the scattering problem of time-harmonic electromagnetic plane wave by an open cavity with perfect conductive boundary is considered.This paper mainly studies the numerical solution of an open cavity scattering in the case of transverse magnetic(TM)polarization.Firstly,the three-dimensional Maxwell equation satisfied by the electromagnetic wave is transformed into two-dimensional transverse magnetic(TM)polarization and transverse electric(TE)polarization,in which both polarization satisfy Helmholtz equation.Then,in order to carry out the numerical solution,two TBCs are derived based on the analytical solution of Helmholtz equation: linear TBC and semicircular TBC.In this way,the unbounded domain problem can be equivalently simplified as a boundary value problem in a bounded domain.Then,SNS-FEM is used to numerically solved the open cavity problem with these two TBCs.Compared with the node-based smooth finite element(NS-FEM),this method uses linear gradient instead of the original constant gradient,and solves the instability of NS-FEM.The gradient can be expressed as a linear function about x and y by Taylor expansion.By approximating the original node based smooth domain to a circular domain,the gradient can be approximately calculated by using four integral points in the domain.Finally,the smooth Galerkin weak formula of SNS-FEM-TBC is derived,and a linear algebraic system for calculating the linear smooth gradient of scattered field is given.A numerical example is used to verify the effectiveness of SNS-FEM-TBC method.For the elastic wave obstacle scattering problem,the scattering of the elastic plane wave by two-dimensional homogeneous isotropic elastic medium is considered.Firstly,because the Navier equation is a vector equation and its direct calculation is more complex,the Navier equation is decomposed into two Helmholtz equations with the coupled boundary conditions by using Helmholtz decomposition and introducing two scalar potential functions.Then,based on the analytical solution of Helmholtz equation on the circular truncated domain,the TBC is deduced,and the unbounded domain problem is transformed into a bounded domain problem.Then the formulas of FEM and SNS-FEM with TBC are derived by the same method.Finally,several numerical examples are used to verify that SNS-FEM-TBC can obtain more stable and accurate solutions than standard FEM-TBC.