Keywords: mixed finite element, volume bubble, incompatible modes, enhanced strains.

In Mahnken et al. [1] and Mahnken and Caylak [2] area and volume bubble functions for the stabilization of tetrahedral elements are introduced, where linear interpolation functions for the displacement field and linear interpolation functions for the pressure field are used. Results for linear elastic and physically nonlinear problems was presented in [1,2]. This paper concentrates on stabilization of mixed triangular elements in the linear elastic regime. In particular we compare the stabilization effect with volume bubble functions for the method of incompatible modes and the enhanced strain method.

In the numerical example firstly a verification of the patch test is obtained. Furthermore, Cook's membrane problem in linear elasticity is investigated. We consider plane strain conditions for the compressible behavior with Poisson's ratio nu = 0.33 and for the incompressible behavior with nu = 0.4999. In the numerical results we avoid volume locking and drastically damp the stress oscillation. Regarding mesh refinement, both formulations, the method of incompatible modes and the enhanced strain method render the results of the reference solution. Furthermore, the results show that the incompatible modes element T1P1IM2ST converges slightly faster than the enhanced strain element T1P1ES2ST.

Additionally we consider the stress distribution sigma₁₁ along the clamped edge of the Cook's membrane. The results of T1P1IM2ST and T1P1ES2ST are compared with the reference solution Q1R, reduced integrated hexahedral element. It can be seen, that the incompatible mode version T1P1IM2ST provides slightly better results than the enhanced strain version T1P1ES2ST. In both cases for incompatible modes and enhanced strains, we observe no oscillation in the stress distribution.

A work considering area bubble functions for stabilization of the mixed triangular elements is in preparation and will be submitted next. Furthermore, the results in this paper are concerned to physically and geometrically linear problems. Therefore, future developments will be directed to physically and geometrically nonlinearities.

1

R. Mahnken, I. Caylak, G. Laschet, "Two Mixed Finite Element Formulations with Area Bubble Functions for Tetrahedral Elements", Computer Methods in Applied Mechanics and Engineering, 197/9-12:1147-1165, 2008. doi:10.1016/j.cma.2007.10.007

2

R. Mahnken, I. Caylak, "Stabilization of bi-linear mixed finite elements for tetrahedral with enhanced interpolations using volume and area bubble functions", Int. J. Num. Meths. Eng., 75:377-413, 2008. doi:10.1016/j.cma.2007.10.007

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