Efficiency and accuracy of a co-simulation may considerable be increased by using a variable communication time grid. Therefore, an error estimator for controlling the macro-step size is required. Here, error estimators are derived and investigated for explicit and implicit co-simulation approaches. The manuscript focuses on mechanical co-simulation models. The basic results may, however, also be applied to arbitrary, non-mechanical co-simulation models.
The approaches used to calculate the fatigue life of components must inevitably consider multiaxial stresses. Compared to proportional loading, the calculation of nonproportional loading is particularly challenging, especially since different materials exhibit the effects of nonproportional hardening and shifts in fatigue life. In this paper, the critical plane approach of scaled normal stresses, first proposed by Gaier and Dannbauer and later published in a modified version by Riess et al., is investigated in detail. It is shown that, on the one hand, compatibilities exist or can be established with known proportional strength criteria that can account for the varying ductility of different materials. Furthermore, it is demonstrated that the scaled normal stress approach can be formulated in such a way that different strength criteria can be used therein. As an example, the generally formulated approach for scaled normal stresses is applied to test results from ductile cast iron material EN-GJS-500-14. Different correction factors accounting for nonproportional loading are investigated. Through appropriate parameterization of one of the studied corrections, proportional and nonproportional test results were observed to fall within one common scatter band.
We consider implicit co-simulation and solver-coupling methods, where different subsystems are coupled in time domain in a weak sense. Within such weak coupling approaches, a macro-time grid is introduced. Between the macro-time points, the subsystems are integrated independently. The subsystems only exchange information at the macro-time points. To describe the connection between the subsystems, coupling variables have to be defined. For many implicit co-simulation and solver-coupling approaches an Interface-Jacobian is required. The Interface-Jacobian describes, how certain subsystem state variables at the interface depend on the coupling variables. Concretely, the Interface-Jacobian contains partial derivatives of the state variables of the coupling bodies with respect to the coupling variables. Usually, these partial derivatives are calculated numerically by means of a finite difference approach. A calculation of the coupling gradients based on finite differences may entail problems with respect to the proper choice of the perturbation parameters and may therefore cause problems due to ill-conditioning. A second drawback is that additional subsystem integrations with perturbed coupling variables have to be carried out. In this manuscript, analytical approximation formulas for the Interface-Jacobian are derived, which may be used alternatively to numerically calculated gradients based on finite differences. Applying these approximation formulas, numerical problems with ill-conditioning can be circumvented. Moreover, efficiency of the implementation may be increased, since parallel simulations with perturbed coupling variables can be omitted. The derived approximation formulas converge to the exact gradients for small macro-step sizes.
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