Comparative tensor visualisation within the framework of consistent time-stepping schemes

Abstract

Nowadays, the design of so-called consistent time-stepping schemes that basically feature a physically correct time integration, is still a state-of-the-art topic in the area of numerical mechanics. Within the proposed framework for finite elastoplasto-dynamics, the spatial as well as the time discretisation rely both on a Finite Element approach and the resulting algorithmic conservation properties have been shown to be closely related to quadrature formulas that are required for the calculation of time-integrals. Thereby, consistent integration schemes, which allow a superior numerical performance, have been developed based on the introduction of an enhanced algorithmic stress tensor, compare [MMS06]-[MMS07c]. In this contribution, the influence of this consistent stress enhancement, representing a modified time quadrature rule, is analysed for the first time based on the spatial distribution of the tensor-valued difference between the standard quadrature rule, relying on a specific evaluation of the well-known continuum stresses, and the favoured nonstandard quadrature rule, involving the mentioned enhanced algorithmic stresses. This comparative analysis is carried out using several visualisation tools tailored to set apart spatial and temporal patterns that allow to deduce the influence of both step size and material constants on the stress enhancement. The resulting visualisations indeed confirm the physical intuition by pointing out locations where interesting changes happen in the data.