Discretization of the Amplitude Phase-Field Crystal Equation
The study of polycrystalline materials requires theoretical and computational techniques that allow multiscale investigations. The amplitude expansion of the phase field crystal model (APFC) allows the description of crystal lattice properties on diffusive timescales by focusing on continuous fields varying on length scales larger than the atomic spacing. Thus, it allows the simulation of large systems while preserving details of the crystal lattice.
The goal of the thesis is to develop a finite element discretization to solve these equations. In particular, the project considers a real space approach using the finite element method within the Dune/AMDiS framework with mesh adaptivity based on the local rotation of the (poly)crystal.
The project is divided into the following subtasks:
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Study the Cahn-Hilliard example code from the AMDiS repository as a structurally similar equation, compare [2]. Follow also the implementation in a BaseProblem given in the amdis-extensions repository.
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Extend the example for the complex-valued case as a preparation for the APFC equation of a single amplitude and then implement the full APFC system as a combination of multiple amplitudes, following [1, page 2-3]. In the implementation, use a CouplingBaseProblem.
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Introduce a local mesh adaptivity, by the criterion introduced in [1, page 5].
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Study the performance difference between a structured grid and an unstructured adaptively refined grid. What is necessary to gain the same accuracy? How to measure accuracy?
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Use the PETSc backend of AMDiS to parallelize the problem solver. Use the fgmres+bjacobi setup for the parallel solution of the linear system, compare [1, page 7]. Compare again the performance of a structured against an unstructured grid.
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Study the growth of a single crystal in a melt, and the growth of a polycrystal. Begin with the 2d triangular lattice. If time permits, also test the 3d FCC lattice.
Literature
[1] Simon Praetorius, Marco Salvalaglio, and Axel Voigt, An efficient numerical framework for the amplitude expansion of the phase-field crystal model, 2019, Modelling Simul. Mater. Sci. Eng. 27 044004, [arXiv] [doi]
[2] Petia Boyanova, Minh Do-Quang, and Maya Neytcheva, Solution methods for the Cahn-Hilliard equation discretized by conforming and non-conforming finite elements, 2011, Tech. Report, Uppsala University, [link]