This talk discusses possibilities for establishing high order smoothed finite element methods (high-order S-FEM). In a high-order S-FEM approach, smoothed strains in a smoothing domain are expressed in polynomial functions with linear and high-order terms. In particular, a high order node-based smoothed finite element method (high-order NS-FEM) using triangular elements that can be automatically generated is presented for static, free and forced vibration analyses of solids. We provide first proofs on convergence. Then, two versions of high-order NS-FEM are presented: NS-FEM-1 that uses 1st order (linear) smoothed strain fields, and NS-FEM-2 that use 2nd order  smoothed strain fields. Numerical results will also be presented to demonstrate the performance and unique properties of these high order modes, including (1) close-to-exact stiffness which is softer than the “over-stiff” FEM and stiffer than the “over-soft” standard NS-FEM using constant smoothed strains; (2) ultra-accuracy in term of displacements; (3) no spurious non-zero energy modes, hence temporally stable and working well for dynamic problems. Note that the high-order strain field is achieved without increasing the degrees of freedom (DOFs) compared to the linear FEM model using the same mesh.

Keywords: high-order NS-FEM, strain field reconstruction, node-based smoothing, close-to-exact stiffness, temporally stability.

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[19]G.R. Liu, J. Zhang and K.Y. Lam. A gradient smoothing method (GSM) with directional correction for solid mechanics problems. Computational Mechanics; 41: 457–472. 2008.

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Dr. Liu received PhD from Tohoku University, Japan in 1991. He was a PDF at Northwestern University, USA from 1991-1993. He is currently a Professor and Ohio Eminent Scholar (State Endowed Chair) at the University of Cincinnati. He authored a large number of journal papers and books including two bestsellers: “Mesh Free Method: moving beyond the finite element method” and “Smoothed Particle Hydrodynamics: a Meshfree Particle Methods.”  He is the Editor-in-Chief of the International Journal of Computational Methods, Associate Editor of IPSE and MANO. He is the recipient of numerous awards, including the Singapore Defence Technology Prize, NUS Outstanding University Researcher Award and Best Teacher Award, APACM Computational MechanicsAwards, JSME Computational Mechanics Awards, ASME Ted Belytschko Applied Mechanics Award, and Zienkiewicz Medal from APACM. He is listed as a world top 1% most influential scientist (Highly Cited Researchers) by ThomsonReuters in 2014, 2015 and 2016, with a total ISI citations by others of 13458 and ISI H-index of 68.