A novel Galerkin-like weakform and a superconvergent alpha finite element method (S αFEM) 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 α. A novel Galerkin-like weakform derived from the Hellinger–Reissner variational principle is proposed for establ...

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Veröffentlicht in:Journal of computational physics Jg. 228; H. 11; S. 4055 - 4087
Hauptverfasser: Liu, G.R., Nguyen-Xuan, H., Nguyen-Thoi, T., Xu, X.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Kidlington Elsevier Inc 20.06.2009
Elsevier
Schlagworte:
Alpha finite element method ( αFEM)
Computational techniques
Exact sciences and technology
Finite element method (FEM)
Mathematical methods in physics
Meshfree methods
Node-based smoothed finite element method (NS-FEM)
Numerical methods
Physics
Solution bounds
Strain construction methods
Superconvergence
Meshfree methods
Node-based smoothed finite element method (NS-FEM)
Solution bounds
Alpha finite element method ( αFEM)
Numerical methods
Strain construction methods
Finite element method (FEM)
Superconvergence
Lower bound
Singularity
Digital simulation
Alpha finite element method (αFEM)
Elasticity
Numerical method
Linear model
Bounded solution
Calculation methods
Exact solution
Finite element method
Upper bound
Variational principle
Convergence rate
Calculation
ISSN:0021-9991, 1090-2716
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Zusammenfassung: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 α. 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 αFEM) is then formulated by using the constructed strain field and the Galerkin-like weakform for solid mechanics problems. The implementation of the S αFEM is straightforward and no additional parameters are used. We prove theoretically and show numerically that the S αFEM 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 αFEM can always produce both lower and upper bounds to the exact solution in the energy norm for all elasticity problems by properly choosing an α. In addition, a preferable- α approach has also been devised to produce very accurate solutions for both displacement and energy norms and a superconvergent rate in the energy error norm. Furthermore, a model-based selective scheme is proposed to formulate a combined S αFEM/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 αFEM.
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ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2009.02.017