On stabilization of the node based smoothed finite element method for free vibration problems

The instability can be clearly shown as spurious non-zero energy modes in free vibration analysis. In this paper, we present a stabilization of the node-based smoothed finite element method (SN-FEM) that is stable (no spurious non-zero energy modes) and more effective than the standard finite element method (FEM). Three numerical illustrations are given to evince the high reliability of the proposed formulation. | Vietnam Journal of Mechanics, VAST, Vol. 32, No. 3 (2010), pp. 167 – 181 ON STABILIZATION OF THE NODE-BASED SMOOTHED FINITE ELEMENT METHOD FOR FREE VIBRATION PROBLEMS Bui Xuan Thang1 , Nguyen Xuan Hung1,2 , Ngo Thanh Phong1 1 University of Science - VNU - HCM 2 Ton Duc Thang University HCM Abstract. The node-based smoothed finite element method (NS-FEM) has been recently proposed by Liu et al to enhance the computational effect for solid mechanics problems. However, it is evident that the NS-FEM behaves “overly-soft” and so it may lead to instability for dynamic problems. The instability can be clearly shown as spurious non-zero energy modes in free vibration analysis. In this paper, we present a stabilization of the node-based smoothed finite element method (SN-FEM) that is stable (no spurious non-zero energy modes) and more effective than the standard finite element method (FEM). Three numerical illustrations are given to evince the high reliability of the proposed formulation. 1. INTRODUCTION Chen et al. [2] proposed the strain smoothing technique to achieve a stabilization in the nodal integrated meshfree methods. Then it is applied to the natural element method in [18]. This technique has been developed into a generalized smoothing technique by Liu et al. [5]. The generalized smoothing technique allows displacement functions to be discontinuous and forms the theoretical foundation for the linear conforming point interpolation method (LC-PIM) [9]. In addition, Liu et al. also introduced the linearly conforming radial point interpolation method (LC-RPIM) [4], the element-based smoothed finite element method (SFEM) [3, 10, 11, 14], the edge-based smoothed finite element method (ES-FEM) [6], and the node-based smoothed finite element method (NS-FEM) [8]. Concerning on the NS-FEM, the domain discretization is still based on element as same as the standard FEM. However, the formulation of the system stiffness matrix is appreciated by smoothing domain associated .

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