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A novel alpha finite element method (\(\alpha \)FEM) for exact solution to mechanics problems using triangular and tetrahedral elements. (English) Zbl 1194.74433
Summary: The paper presents an alpha finite element method (\(\alpha \)FEM) for computing nearly exact solution in energy norm for mechanics problems using meshes that can be generated automatically for arbitrarily complicated domains. Three-node triangular (\(\alpha \)FEM-T3) and four-node tetrahedral (\(\alpha \)FEM-T4) elements with a scale factor \(\alpha \) are formulated for two-dimensional (2D) and three-dimensional (3D) problems, respectively. The essential idea of the method is the use of a scale factor \(\alpha \in [0,1]\) to obtain a combined model of the standard fully compatible model of the FEM and a quasi-equilibrium model of the node-based smoothed FEM (N-SFEM). This novel combination of the FEM and N-SFEM makes the best use of the upper bound property of the N-SFEM and the lower bound property of the standard FEM. Using meshes with the same aspect ratio, a unified approach has been proposed to obtain a nearly exact solution in strain energy for linear problems. The proposed elements are also applied to improve the accuracy of the solution of nonlinear problems of large deformation. Numerical results for 2D (using \(\alpha \)FEM-T3) and 3D (using \(\alpha \)FEM-T4) problems confirm that the present method gives the much more accurate solution comparing to both the standard FEM and the N-SFEM with the same number of degrees of freedom and similar computational efforts for both linear and nonlinear problems.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
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