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A novel Galerkin-like weakform and a superconvergent alpha finite element method (S\(\alpha \)FEM) for mechanics problems using triangular meshes. (English) Zbl 1273.74542
Summary: 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 \(\alpha \). 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\(\alpha \)FEM) 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 a 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.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74G15 Numerical approximation of solutions of equilibrium problems in solid mechanics
Software:
XFEM
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