Koopman-based reduced-order modelling of discrete dislocation dynamics simulations

Abstract

Abstract The plastic behaviour of single crystals emerges from the motion and interaction of crystal dislocations, which may nowadays be simulated with so-called discrete dislocation dynamics (DDD) simulations. Data-driven methods offer the prospect that results from DDD simulations may be directly exploited for model development in crystal plasticity. The current work explores the use of the Koopman operator to create a data-driven reduced-order model that describes the spatio-temporal evolution of dislocations in a face-centred cubic single crystal under uniaxial tensile loading. Spatio-temporal results from DDD simulations are used to extract multiple continuum dislocation dynamics (CDD) state variables beyond the usual measure of total dislocation density for each slip system. This information comes from the dislocation line geometry. We treat the CDD density evolution as a dynamic system and use variants of the (extended) dynamic mode decomposition (eDMD) to derive linear reduced-order models. Our three goals include: (i) investigating whether Koopman frameworks can successfully model CDD evolution from DDD data, (ii) determining the impact of considering higher-order CDD variables on predictive performance, and (iii) testing whether physically informed formulations that maintain translational symmetry can enhance extrapolation capabilities beyond training data in a spatially resolved case. The results of the study show that DMD and its translation-invariant extensions demonstrate strong potential, yet they also expose challenges which should be addressed for a large-scale application of Koopman methods to DDD data.