A novel Galerkin-like weakform and a superconvergent alpha finite element method (S alpha FEM) for mechanics problems using triangular meshes
A carefully designed procedure is presented to modify the piecewise constant strain field of linear triangular FEM models, and to reconstruct a strain field with an adjustable parameter a. A novel Galerkin-like weakform derived from the Hellinger-Reissner variational principle is proposed for establishing the discretized system equations. The new weak form is very simple, possesses the same good properties of the standard Galerkin weakform, and works particularly well for strain construction methods. A superconvergent alpha finite element method (S alpha TEM) is then formulated by using the constructed strain field and the Galerkin-like weakform for solid mechanics problems. The implementation of the S alpha FEM is straightforward and no additional parameters are used. We prove theoretically and show numerically that the S alpha FEM always achieves more accurate and higher convergence rate than the standard FEM of triangular elements (T3) and even more accurate than the four-node quadrilateral elements (Q4) when the same sets of nodes are used. The S alpha FEM can always produce both lower and upper bounds to the exact solution in the energy norm for all elasticity problems by properly choosing an alpha. In addition, a preferable-alpha approach has also been devised to produce very accurate solutions for both displacement and energy norms and alpha superconvergent rate in the energy error norm. Furthermore, a model-based selective scheme is proposed to formulate a combined S alpha FEM/NS-FEM model that handily overcomes the volumetric locking problems. Intensive numerical studies including singularity problems have been conducted to confirm the theory and properties of the S alpha FEM. (C) 2009 Elsevier Inc. All rights reserved.
superconvergence strain construction methods numerical methods meshfree methods finite element method (fem) alpha finite element method (alpha fem) node-based smoothed finite element method (ns-fem) solution bounds