Computational Crystal Plasticity Homogenization using Empirically Corrected Cluster Cubature (E3C) Hyper-Reduction

Abstract

The computational homogenization of elastoplastic polycrystals is a challenging task due to the huge number of grains required, their complicated interactions and due to the complexity of crystal plasticity models per se. Despite a few successes of reduced order models, mean field and simplified homogenization approaches often remain the preferred choice. In this work, a recently proposed hyper-reduction method (called E3C) for projection-based Reduced Order Models (pROMs) is applied to the problem of computational homogenization of geometrically linearly deforming elastoplastic polycrystals. The main novelty lies in the identification of reduced modes (the 'E3C-modes'), which replace the strain modes of the reduced-order model, leading to a significantly smaller number of integration points. The peculiarity, which distinguishes the method from more conventional hyper-reduction techniques, is that the E3C integration points are not taken from the set of FE integration points. Instead, they can be interpreted as generalized integration points in strain space which are trained such as to satisfy an orthogonality condition, which ensures that the hyper-reduced model matches the equilibrium states and macroscopic stresses of full-field model data as accurately as possible. In addition, the number of grains is reduced, preserving the main features of the original texture of the finite element model. Two macroscopic engineering parts (untextured and textured) are simulated, illustrating the performance of the method in three-dimensional two-scale applications involving hundreds of thousands macroscopic degrees of freedom and millions of grains with computing times in the order of hours (cumulated online and offline effort) on standard laptop hardware.

Figures (13)

• Figure 1: Sketch of the overall computational homogenization scheme.
• Figure 2: Tentative visualization of the E3C-modes $\tilde{\hbox{\boldmath $\hbox{\boldmath $\mathcal{E}$}$}}_k$ (columns of matrix $\tilde{\hbox{\boldmath $\hbox{\boldmath $\mathcal{E}$}$}}$) for the special case of seven E3C-grains ($N_{\rm g}=7$). Black lines: undeformed grains, colored: deformed grains. The grains are disconnected due to the incompatibility of the grain-wise constant strains.
• Figure 3: Plate with a hole with microstructure, consisting of 99 grains. The meshed region of the plate has dimensions 10$\times$5$\times$1 mm$^3$. The {111} pole figure (bottom left) illustrates a nearly isotropic crystal orientation distribution, where the dots are scaled according to the grain sizes.
• Figure 4: Left: Cost function evolution for 20 modes and 20 E3C integration points. The first few iterations are carried out by the conjugate gradient method ('CG'), which was observed to show slower convergence than the subsequently used Levenberg Marquardt algorithm. The maximum iteration number was set to 70. Center: {111} pole figure illustrating the 20 grain orientations obtained by the E3C cost function minimization. Right: Clusters obtained by k-means for retraining.
• Figure 5: Final state of the 'plate with a hole' two-scale simulation (deformations scaled by factor 5) with an online wall-clock time of $\sim$2.5 min (without reconstruction). The results were obtained after clustered training and show the microscopic deformations (FEM reconstructions showing the accumulated plastic slip $\gamma^{\rm acc}$) and corresponding macroscopic stress-strain curves (in MPa) for several macro elements. The red curves show the E3C-predictions, while the corresponding FEM-results, indicated in blue, are hardly visible due to the good match with the E3C-results. The $\sigma_{xx}/\varepsilon_{xx}$-curves (tension) are those in the first quadrant, while the lateral responses (contraction) can be found in the third quadrant. The shear components are small in magnitude, but are also included in the diagrams for completeness.