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Details of Grant 

EPSRC Reference: EP/X035883/1
Title: Optimal Grain Diagrams: Mathematical Analysis and Algorithms
Principal Investigator: Alpers, Dr A
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematical Sciences
Organisation: University of Liverpool
Scheme: New Investigator Award
Starts: 01 July 2023 Ends: 30 June 2025 Value (£): 234,024
EPSRC Research Topic Classifications:
Algebra & Geometry Numerical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
28 Feb 2023 EPSRC Mathematical Sciences Prioritisation Panel February 2023 Announced
Summary on Grant Application Form
The project aims to enable new materials discovery by advancing the theory and computation of geometric diagram structures describing polycrystals by fully exploiting, for the first time, the recent link to constrained clusterings.

One of the fundamental challenges for new materials discovery is to understand and control the forming of grain structures in metals, alloys, ceramics, and other polycrystals. In practice, grain structures are accessible as grain maps, i.e., as large 2D or 3D images resulting from an imaging process. Analysing geometric features in these data sets (e.g., grain boundaries) is of crucial importance for understanding and predicting materials properties. Detailed time-resolved studies have so far not been possible due the the large number of pixels/voxels involved in the processing.

Grains structures, however, can be approximated by polygons or, even better, by objects with piecewise quadratic boundaries. The computations performed on this geometric level can drastically decrease computation times if these representations involve only small numbers of parameters. We have shown in 2015 that so-called generalised balanced power diagrams, computed by a clustering technique, yield unprecedented good fits to measured grain structures. As we could show recently that the clustering technique can be optimised to involve only a small number of pixes/voxels, it is now timely to develop these and similar grain diagrams into a transformative tool for new materials discovery, enabling, for the first time, time-resolved high-resolution studies.

In particular, the aim of this project is to provide an in-depth mathematical analysis and efficient algorithms for the following three tasks/objectives:

1. For a given noise-free grain map, determine a 'best fitting' grain diagram. This will allow us to represent such maps by few or physically meaningful parameters, laying the foundation for data analysis and simulations of grain forming processes, which was formerly out of reach with present methods.

2. For a given grain map obtained by surface imaging, determine a 'best fitting' grain diagram. This will demonstrate, for the first time, how results from the noise-free case carry over to a relevant noisy case yielding a conceptionally novel method for analysing grain maps.

3. Model and analyse dynamic grain structures via grain diagrams. This scientific groundwork for analysing new dynamic tessellation models will allow us to gain new insights or perspectives on grain forming mechanisms under specific experimental conditions.

The project will combine and advance methods from data science (constrained clustering), optimisation, and convex geometry. Real data will be provided by collaborators on this project. Together, the results from these studies will provide rigorous models, algorithms, and a mathematical analysis that allows to characterize large static and, for the first time, dynamic grain maps from measured parameters.
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Organisation Website: http://www.liv.ac.uk