Mixed Finite Element Formulations for Polyconvex Anisotropic Material Formulations

Abstract

Our research activity takes place within the research project GR 3297/4, funded by `Deutsche Forschungsgemeinschaft' (DFG), and aims at a robust simulation method for fiber-reinforced materials in light-weight structures. One goal is to avoid looking-effects in the static and dynamic regime, which occur due to nearly incompressible matrix materials and highly stiff fibers. Therefore, we extend the mixed finite elment formulations, hown in References [1, 2, 3]. In the description of the material behavior, we also use polyconvex strain energy functions [4]. In case of the so-called CoFEM element in Reference [2], the volumetric dilatation and the cofactor of the right Cauchy-Green tensor are approximated independently beside the displacement. In Reference [3], this formulation is extended so, that the right Cauchy-Green tensor of the anisotropic strain energy function is also approximated independently. In this presentation, we also approximate the cofactor and the volumetric dilatation of the anisotropic right Cauchy-Green tensor independently. We analyse the spatial convergence of the new mixed finite elements for hexahedral elements up to a cubic approximation in space. Thereby, we look especially at the different possible combinations of polynomial degrees of the independent mixed variables and the impact of this on the efficiency of the simulation (see Reference [5]). As numerical examples serve the well-known cooks cantilever beam and an axisymmetric pipe. Hereby, the bodies have different materials domains with different material parameters and fiber directions.

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