A comparative study of the consistency of Galerkin-type and least-squares finite element formulations for bifurcation problems

Abstract eng:
The computation of reliable results using finite elements is a major engineering goal. Under the assumption of a linear theory many stable and reliable (standard and mixed) finite elements have been developed. Unfortunately, in the nonlinear regime, e.g. applying these elements in the field of incompressible, hyperelastic materials, problems can occur. A typical example is the detection of stability points using mixed Galerkin finite elements, compare e.g. (Auricchio, Beirão da Veiga, Lovadina,& Reali 2005), (Auricchio, Beirão da Veiga, Lovadina, & Reali 2010) and (Auricchio, Beirão da Veiga, Lovadina, Reali, Taylor, & Wriggers 2013). In this contribution we analyze, amongst others, the consistency of Galerkin-type and least-squares finite element formulations for structural stability problems in the framework of hyperelasticity. Basis for the least-squares element formulation is a div-grad first order system consisting of the equilibrium condition, the constitutive equation and a stress symmetry condition all written in a residual form, compare also (Schwarz, Steeger, & Schröder 2014). The sum of squared L2 (B ) -norms of the residuals leads to the basic functional which has to be minimized, see also (Jiang 1998). For the approximation of the displacement and the stress field we use independent interpolations. In order to show the performance of the proposed method a boundary value problem will be analyzed.

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