SPAM - Science, Pizza And Mathematics

Predictive Mesoscale Modeling: A Phase-Field Crystal Approach

Presenter: Maik Punke

Abstract: This talk presents a predictive multiscale modeling framework based on the phase-field crystal (PFC) equation. Through various model extensions, a set of partial differential equations is developed whose numerical solutions enable a theoretical analysis of several physical phenomena at the mesoscale. Further, through an advanced numerical framework, time and length scales can be resolved that were previously not accessible with PFC simulation techniques.

The PFC model is based on a coarse-grained density formulation that can be derived from statistical mechanics and was originally developed as a free-energy functional for reproducing periodic density patterns while allowing for diffusive mesoscale dynamics. The current thesis aims at tackling open problems in computational materials science by generalizing the original PFC model equation via a tractable simulation framework. A focus is set on elastic and plastic relaxation phenomena capturing different spatial scales while evolving on diffusive dynamics. Crystalline defects and so-called grain boundaries, i.e., interfaces between individual crystalline grains with different orientations, play a crucial role for structural evolution processes.

The current thesis analyzes grain boundaries including an intrinsic asymmetry of the underlying grain boundary structure. Experimental indications for a directional grain boundary migration under oscillating driving forces are verified and physically interpreted via the developed PFC simulation framework. Similarly, the considered PFC simulation approach contributed to developing a theoretical understanding of how plasticity impacts microstructural evolution processes, verifying and strengthening the experimental evidence. Further, the current thesis proposes an extension of the classical PFC evolution equation under non-isothermal conditions consistent with the fundamental laws of thermodynamics and being formulated within a computationally
manageable partial differential equation structure that guarantees realistic simulations at the mesoscale.

The final part of the dissertation is dedicated to the methodological development of numerical PFC simulations to reach large system dimensions beyond the previously accessible time and length scales which are, in general, limited by three primary constraining factors. On the one hand, the PFC density formulation requires a fine spatial resolution which dictates the reachable system sizes when exploiting numerical discretization techniques. Further, a high numerical accuracy of the discretized PFC evolution process can only be guaranteed when a relatively small time stepping size is considered. Finally, the underlying hardware components restrict the overall computational capabilities due to their memory and performance limitations. Within the last part of this thesis, all three limiting factors are tackled and discretization as well as implementation routines are developed which enable substantial improvements in the accessible time and length scales. Based on the so-called Fourier pseudo-spectral method, a robust hybrid model formulation based on an amplitude expansion of the PFC model is developed which allows for local high-resolution density updates in regions of interest and a coarse-grained description elsewhere. Next, existing time integration strategies for the PFC equation are significantly accelerated by generalizing a numerical stabilizing routine. Eventually, the first numerical PFC solver exploiting multiple graphics processing units (GPUs) is presented, clearly outperforming a single GPU implementation as well as state-of-the-art solvers running on hundreds of central processing units (CPUs). The final part of this thesis builds the foundation for future PFC studies reaching beyond the previously accessible mesoscale. Overall, this thesis makes a substantial
contribution to the development of a predictive simulation framework for multiscale modeling based on the PFC approach.

TBA