Smooth finite element methods: Convergence, accuracy and properties
Article first published online: 18 OCT 2007
Copyright © 2007 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Engineering
Volume 74, Issue 2, pages 175–208, 9 April 2008
How to Cite
Nguyen-Xuan, H., Bordas, S. and Nguyen-Dang, H. (2008), Smooth finite element methods: Convergence, accuracy and properties. Int. J. Numer. Meth. Engng., 74: 175–208. doi: 10.1002/nme.2146
- Issue published online: 11 MAR 2008
- Article first published online: 18 OCT 2007
- Manuscript Revised: 4 JUN 2007
- Manuscript Accepted: 4 JUN 2007
- Manuscript Received: 19 JUN 2006
- finite element method;
- stabilized conforming nodal integration;
- strain smoothing;
- mixed variational principle;
- boundary integration;
- distorted meshes;
A stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented. The integration of the stiffness matrix is performed on the boundaries of the finite elements. A rigorous variational framework based on the Hu–Washizu assumed strain variational form is developed.
We prove 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 (a 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. Copyright © 2007 John Wiley & Sons, Ltd.