SUMMARYAnalogously to the classical return-mapping algorithm, so-called variational constitutive updates are numerical methods allowing to compute the unknown state variables such as the plastic strains and the stresses for material models showing an irreversible mechanical response. In sharp contrast to standard approaches in computational inelasticity, the state variables follow naturally and jointly from energy minimization in case of variational constitutive updates. This leads to significant advantages from a numerical, mathematical as well as from a physical point of view. However, while the classical return-mapping algorithm has been being developed for several decades and thus, it has already reached a certain maturity, variational constitutive updates have drawn attention only relatively recently. This is particularly manifested in the numerical performance of such algorithms. Within the present paper, the numerical efficiency of variational constitutive updates is critically analyzed. It will be shown that a naive approximation of the flow rule causes a singular Hessian within the respective Newton-Raphson scheme. However, by developing a novel parameterization of the flow rule, an efficient algorithm is derived. Its performance is carefully compared to that of the classical return-mapping scheme. This comparison clearly shows that the novel variationally consistent implementation is, at least, as efficient as the classical return-mapping algorithm.

SUMMARYMultiscale approaches based on homogenization theory provide a suitable framework to incorporate information associated with a small scale (microscale) into the considered large scale (macroscopic) problem. In this connection, the present paper proposes a novel computationally efficient hybrid homogenization method. Its backbone is a variationally consistent FE 2 approach in which every aspect is governed by energy minimization. Particularly, scale bridging is realized by the canonical principle of energy equivalence. Since a direct implementation of the aforementioned variationally consistent FE 2 approach is numerically extensive, an efficient approximation based on Ritz's method is advocated. By doing so, the material parameters defining an effective macroscopic material model capturing the underlying microstructure can be efficiently computed. Furthermore, the variational scale bridging principle provides some guidance to choose a suitable family of macroscopic material models. Comparisons between the results predicted by the novel hybrid homogenization method and full field finite element simulations show that the novel method is indeed very promising for multiscale analyses.

The numerical efficiency of so-called variational constitutive updates for finite strain plasticity theory is analyzed. These updates compute the unknowns such as the plastic strains by minimizing an appropriate functional. Within the present paper, different parameterizations of the flow rule are utilized within the variational constitutive update scheme. It is shown that comparing to the return-mapping algorithm, the variational updates require significantly less iteration steps and thus, is numerically highly efficient.