Improved Node-based Smoothed Finite Element For Obstacle Scattering Problem

In recent decades,obstacle scattering has been widely used in engineering and military fields.In this paper,frequency domain elastic wave and time domain acoustic obstacle scattering problems are considered in the two-dimensional.Firstly,aiming at the frequency domain elastic wave obstacle scattering problem,the scattered field satisfying Navier equation is studied,which is a vector field consisting of coupled transverse and longitudinal waves.By applying the perfectly matched layer(PML)technique to this scattering problem,the external boundary value problem on the bounded domain is established.In this paper,a stable node-based smoothed finite element method with PML(SNS-FEM-PML)is proposed to solve this scattering problem.In this algorithm,the stability term is constructed by the Taylor expansion of the gradient to cure the instability of the original the node-based smoothed finite element method.The stability term contains the linear variation of the gradient with respect to x and y,which is calculated using four integration points in the equivalent circle of the node-based smoothed domain(N-SD).Further,a smoothed Galerkin weak forms of the SNS-FEM-PML model is derived and a linear equation system of Navier equations is constructed.Besides,the softening effect and convergence of the SNS-FEM model are proved theoretically,and the validity of the SNS-FEM-PML is verified by numerical examples.The results show that the convergence order of L~2and H~1errors of the SNS-FEM-PML model is consistent with the theory of FEM,and the relative error convergence rate is higher than that of the FEM.Next,the scattering problem is transformed into solving the two scalar potential equations by using the Helmholtz decomposition.The PML and transparent boundary conditions(TBC)are introduced to truncate the problem domain.In this paper,a node-based smoothed finite element method with linear gradient fields(NS-FEM-L)is proposed to solve the scalar potential functions.The basic idea of method is to construct the linear gradient using polynomials in the N-SD and use three linearly independent weight functions to solve the unknown coefficients of the smoothed linear gradient function.Further,based on the weakened weak formulation,a system of linear equation for the smoothed gradient of NS-FEM-L with PML and TBC are established.Numerical examples show that the method has higher accuracy and stability,and the modified stiffness matrices make NS-FEM-L-PML and NS-FEM-L-TBC with desirable stiffness.Finally,for the time domain acoustic obstacle scattering problem,the Laplace transform is used to transform the time-domain acoustic wave equation into a modified Helmholtz equation on the Laplace domain.The solution of acoustic scattered field in the external domain is constructed by modified Bessel function and Fourier series.Based on Dt N mapping,non-reflecting boundary conditions(NRBC)is established to transform the scattering problem into a boundary value problem in a bounded domain.In this numerical algorithm,the Newmark method is used for time discretization and combined with the NS-FEM-L for spatial discretization,and the computation of the NRBC convolution term is derived in detail.Numerical results show that the proposed method is effective and convergent.