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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 85
Edited by: B.H.V. Topping
Paper 38

Smooth Finite Element Methods

S. Bordas1, N.X. Hung2, T. Rabczuk3 and N.D. Hung4

1Civil Engineering Department, University of Glasgow, United Kingdom
2Department of Mathematics and Informatics, University of Natural Sciences, VNU-HCM, Vietnam
3Mechanical Engineering Department, University of Canterbury, Christchurch, New Zealand
4LTAS, Division of Fracture Mechanics, University of Liège, Belgium

Full Bibliographic Reference for this paper
S. Bordas, N.X. Hung, T. Rabczuk, N.D. Hung, "Smooth Finite Element Methods", in B.H.V. Topping, (Editor), "Proceedings of the Fifteenth UK Conference of the Association of Computational Mechanics in Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 38, 2007. doi:10.4203/ccp.85.38
Keywords: stabilised conforming nodal integration, FEM, plates, shells, fluid-flow, elasto-plasticity, distorted meshes, polygonal meshes, locking, incompressibility.

We present an improvement of the finite element method which is almost insensitive to mesh distortion and suited to polygonal meshes. The technique was coined the Smooth Finite Element Method (SFEM) by Liu et al. [1]. We show that some versions of the method completely suppress locking in incompressible elasticity and elasto-plasticity, plates and shells and that it constitutes a remarkable link between displacement and equilibrium finite elements. [6,5,2,3,4].

We proved in the above papers that solutions yielded by the proposed method are in a space bounded by the standard, finite element solution (infinite number of subcells) and a quasi-equilibrium finite element solution (one single subcell). We show elsewhere the equivalence of the one-subcell element with a quasi-equilibrium finite element, leading to a global a posteriori error estimate.

We apply the method to compressible and incompressible linear elasticity problems. The method can always achieve higher accuracy and convergence rates than the standard finite element method, especially in the presence of incompressibility, singularities or distorted meshes, for a slightly smaller computational cost.

It is shown numerically that the one-cell smoothed four-noded quadrilateral finite element has a convergence rate of 2.0 in the energy norm for problems with smooth solutions, which is remarkable. For problems with rough solutions, this element always converges faster than the standard finite element, and is free of volumetric locking without any modification of integration scheme.

G.R. Liu, K.Y. Dai, and T.T. Nguyen. "A smoothed finite element for mechanics problems", Computational Mechanics, 2006. doi:10.1007/s00466-006-0075-4
H. Nguyen-Xuan, S. Bordas, and H. Nguyen-Dang. Addressing volumetric locking by selective integration in finite elements with stabilized conforming nodal integration. Journal of Mechanics of Materials and Structures (JoMMS), submitted, 2007.
H. Nguyen-Xuan, S. Bordas, and H. Nguyen-Dang. Locking-free mindlin-reissner plates: a mixed interpolation with smoothed curvatures. International Journal for Numerical Methods in Engineering, submitted, 2007.
H. Nguyen-Xuan, S. Bordas, and H. Nguyen-Dang. Shell elements: severe mesh distortion, locking. International Journal for Numerical Methods in Engineering, submitted, 2007.
H. Nguyen-Xuan, S. Bordas, and H. Nguyen-Dang. Smooth finite elements: connecting displacement and equilibrium finite elements, applications to elasto-plasticity. Computer Modelling in Engineering and Sciences, submitted, 2007.
H. Nguyen-Xuan, S. Bordas, and H. Nguyen-Dang. Smooth finite elements: incompressibility, accuracy, superconvergence, singularities and selective integration. International Journal for Numerical Methods in Engineering, in press, 2007.

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