DTE AICCOMAS 2025

Physically Informed Koopman Analysis of Evolving Dislocation Data

  • Hochrainer, Thomas (GRAZ UNIVERSITY OF TECHNOLOGY)
  • Heininger, Bernhard (GRAZ UNIVERSITY OF TECHNOLOGY)
  • Heinemann, Niko (GRAZ UNIVERSITY OF TECHNOLOGY)
  • Kar, Gurudas (GRAZ UNIVERSITY OF TECHNOLOGY)

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Plasticity in crystalline materials is the result of the motion and interaction of dislocations. Continuum Dislocation Dynamics (CDD), offers a promising approach for an efficient continuum-level description of crystal plasticity, reducing the computational complexity compared to Discrete Dislocation Dynamics (DDD). Based on the recent availability of large scale discrete DDD simulations, modern data driven methods hold the promise to learn CDD equations directly from evolving DDD data. Koopman theory provides a method to analyse the evolution of non-linear systems by methods from linear algebra. The idea is, that the non-linear evolution in phase space yields an equivalent linear but infinite dimensional operator on functions of phase space coordinates. In contrast to other machine learning methods, obtaining this so-called Koopman operator requires relative small data sets. This operator may be analysed for eigenvalues and eigenfunctions which yield salient spatio-temporal information about the system. We present a physically informed method for analysing DDD data, which respects the translational symmetry of the spatially periodic data [2] and symmetries between the slip system. Considering all these symmetries strongly reduces the amount of required data. With these methods we obtain the Koopman operator of a driven dislocation system as a combination of an `autonomous’ Koopman operator, describing the relaxation of dislocation systems, and a loading dependent Koopman operator. We analyse the obtained spatiotemporal modes and discuss the scale dependence of the determined operator. REFERENCES [1] T. Hochrainer. Multipole expansion of continuum dislocation dynamics in terms of alignment tensors. Philosophical Magazine, 95(12):1321–1367, 2015. [2] T. Hochrainer, G. Kar; Approximation of translation invariant Koopman operators for coupled non-linear systems. Chaos 1 August 2024; 34 (8): 083119.