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

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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.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Centre for Advanced Computations in Engineering Science, National University of Singapore, 9 Engineering Drive 3, Singapore, 117576, Singapore

    Zhi-Qian Zhang & G. R. Liu

  2. Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineering Drive 3, Singapore, 117576, Singapore

    Zhi-Qian Zhang & G. R. Liu

Authors
  1. Zhi-Qian Zhang
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  2. G. R. Liu
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Correspondence to Zhi-Qian Zhang.

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Zhang, ZQ., Liu, G.R. Temporal stabilization of the node-based smoothed finite element method and solution bound of linear elastostatics and vibration problems. Comput Mech 46, 229–246 (2010). https://doi.org/10.1007/s00466-009-0420-5

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  • DOI: https://doi.org/10.1007/s00466-009-0420-5

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