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

*(English)*Zbl 1398.74431Summary: 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.

##### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74H45 | Vibrations in dynamical problems in solid mechanics |

74K20 | Plates |

74K25 | Shells |

##### Keywords:

numerical methods; meshfree methods; finite element method (FEM); smoothed finite element method (SFEM); stability; gradient smoothing; solution bound; vibration##### Software:

Mfree2D
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\textit{Z.-Q. Zhang} and \textit{G. R. Liu}, Comput. Mech. 46, No. 2, 229--246 (2010; Zbl 1398.74431)

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##### References:

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