Temporal stabilization of the node-based smoothed finite element method and solution bound of linear elastostatics and vibration problems

Abstract

A stabilization procedure for curing temporal instability of node-based smoothed finite element method (NS-FEM) is proposed for dynamic problems using linear triangular element. A stabilization term is added into the smoothed potential energy functional of the original NS-FEM, consisting of squared-residual of equilibrium equation. A gradient smoothing operation on second order derivatives is applied to relax the requirement of shape function, so that the squared-residual can be evaluated using linear elements. Numerical examples demonstrate that stabilization parameter can "tune" NS-FEM from being "overly soft" to "overly stiff", so that eigenvalue solutions can be stabilized. Numerical tests provide an empirical value range of stabilization parameter, within which the stabilized NS-FEM can still produce upper bound solutions in strain energy to the exact solution of force-driven elastostatics problems, as well as lower bound natural frequencies for free vibration problems.

A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates

“…In this paper, we exploit several interesting properties of the NS-FEM for analyzing plates. The NS-FEM has been already extended to perform adaptive analysis [52], linear elastostatics and vibration 2D solid problems [53]. Also, alpha finite element methods (αFEM) have been recently proposed have been recently proposed as an alternative to the NS-FEM and shown to significantly improve the results obtained by conventional and smoothed FEM techniques, at the cost of the introduction of a problem-dependent parameter α [54][55][56].…”

A finite element method based on C0‐continuous assumed gradients

“…Notice the remarkably well behaving unstable nodally integrated elements C3D_8N_NI and C3D_4N_NI and the partially stabilized nodally integrated hexahedron C3D_8N_1I with one integration point in the interior. As being noted by several authors [5,22,40], the tetrahedron C3D_4N_NI exhibits an overly soft behavior, i.e. the displacement converges from too large values.…”

“…The properties of the solution are equivalent to other NI approaches. Also, instabilities were reported in [5] which manifest in spurious low-energy modes.…”

“…In order to provide a simple expression for the variations of P, the constitutive law is replaced by a linear-elastic material. The scaling parameter is determined from = l c /E with parameter , characteristic length l c and elastic modulus E. The same penalty has been used to stabilize NS-FEM in [5].…”

Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods

“…However, ES-FEM usually produces lower bound to the exact solution in strain energy. A stabilization technique for the NS-FEM has been recently proposed by adding the square-residual of the equilibrium equation over the problem domain regulated by a stabilization parameter to the smoothed potential energy functional [42]. Furthermore, -FEM can also eliminate the spurious modes of NS-FEM model by choosing a proper scaling factor and still be soft enough so as to maintain the upper-bound property.…”