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.
G.R. Liu and S.S. Quek. The finite element method: a practical course. Butterworth Heinemann, Oxford, 2003.
O.C. Zienkiewicz and R.L. Tayor. The finite element method, fifth edition. Butterworth Heinemann, Oxford, 2000.
Hughes and T.J. R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall. 1987.
J.S. Chen, C.T. Wu, S. Yoon and Y. You. A stabilized conforming nodal integration for Galerkin meshfree method. International Journal for Numerical Methods in Engineering. 50:435-466. 2000.
G.R. Liu. A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods. International Journal of Computational Methods. 5(2): 199-236. 2008.
G.R. Liu, G.Y. Zhang, K.Y. Dai, Y.Y. Wang, Z.H. Zhong, G.Y. Li and X. Han. A linearly conforming point interpolation method (LC-PIM) for 2D mechanics problems. International Journal of Computational Methods. 2(4): 645-665. 2005.
G.Y. Zhang, G.R. Liu, Y.Y. Wang, H.T. Huang, Z.H. Zhong, G.Y. Li and X. Han. A linearly conforming point interpolation method (LC-PIM) for three dimensional elasticity problems. International Journal for Numerical Methods in Engineering. 72: 1524-1543. 2007.
G.R. Liu and G.Y Zhang. Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM). International Journal for Numerical Methods in Engineering. 74: 1128-1161. 2008.
G.R. Liu and N.T. Trung. Smoothed finite element methods[M]. CRC press, 2016.
Liu G R, Nguyen-Thoi T, Nguyen-Xuan H, et al. A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems[J]. Computers & structures, 2009, 87(1-2): 14-26.
Liu G R, Chen L, Nguyen‐Thoi T, et al. A novel singular node‐based smoothed finite element method (NS‐FEM) for upper bound solutions of fracture problems[J]. International Journal for Numerical Methods in Engineering, 2010, 83(11): 1466-1497.
G.R. Liu, X. Xu, G.Y. Zhang and T. Nguyen-Thoi. A superconvergent point interpolation method (SC-PIM) with piecewise linear strain field using triangular mesh. International Journal of Numerical Methods in Engineering. 2009. 77:1439-1467.
G.Y. Zhang, G.R. Liu and X. Xu. A strain-constructed point interpolation method and strain field construction schemes for solid mechanics problems using triangular mesh. Applied Mathematics and Computation. 2012. 219: 2067-2086.
K.Y. Dai and G.R. Liu. Free and forced vibration analysis using the smoothed finite element method (SFEM). Journal of Sound and Vibration. 301:803-820. 2007.
G.R. Liu, T. Nguyen-Thoi and K.Y. Lam. An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids. Journal of Sound and Vibration. 320:1100-1130. 2009.
Y. Li, G.R. Liu and J.H. Yue. A Novel Node-based Smoothed Radial Point Interpolation Method for 2D and 3D Solid Mechanics Problems. Computers and Structures. 196: 157-172. 2018.
G.R. Liu and G.Y. Zhang. Smoothed Point Integration Methods: G Space theory and Weakened Weak Forms; World Scientific Publishing Co. Pte. Ltd.
G.R. Liu and M.B. Liu. Smoothed particle hydrodynamics: a meshfree practical method. World Scientific, Singapore. 2003.
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.
G.R. Liu and X.G. Xu. A gradient smoothing method (GSM) for fluid dynamics problems. International Journal for Numerical Methods in Fluids; 58: 1101–1133. 2008.
G.R. Liu. Meshfree Methods: Moving Beyond the Finite Element Method, 2nd Edition. Taylor & Francis/CRC Press, Boca Raton, FL. 2009.
G.R. Liu. A Novel Pick-Out Theory and Technique for Constructing the Smoothed Derivatives of Functions for Numerical Methods[J]. International Journal of Computational Methods,15(3). 2018.
Y.T. Gu and G.R. Liu. A meshless local Petrov-Galerkin (MPLG) method for free and vibration analyses for solids. Computational mechanics. 27:188-198. 2001.
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.