EPSRC Reference: 
EP/R013527/1 
Title: 
Designer Microstructure via Optimal Transport Theory 
Principal Investigator: 
Bourne, Dr D 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematical Sciences 
Organisation: 
Durham, University of 
Scheme: 
First Grant  Revised 2009 
Starts: 
01 January 2018 
Ends: 
30 September 2018 
Value (£): 
101,152

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Continuum Mechanics 
Mathematical Analysis 


EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Regular geometric tessellations arise in many places in nature. Hexagons are everywhere, from beehives to the hexagonal basalt columns at Giant's Causeway . Tessellations by irregular polygons are also observed in nature, for example on a giraffe's skin. Voronoi diagrams are an important type of irregular polygonal tessellation. For example, a Voronoi tessellation of a city can be generated by the locations of supermarkets; if we assume that each person travels to their closest store, then the 'catchment areas' of the supermarkets tessellate the city, and it turns out that they form a polygonal tessellation, called a Voronoi diagram . Patterns resembling Voronoi diagrams arise in surprisingly many places: biological cells, soap bubbles, and the microstructure of metals.
The goal of this mathematical research project is to develop rigorous numerical and analytical methods for generating optimal tessellations (Voronoi diagrams). The definition of 'optimal' depends on the application.
In the first part of the project the tessellations represent grains in metals, which are microscopic regions in a metal with the same crystal structure and orientation, and we consider applications in the steel industry and in nondestructive testing using ultrasound. For the steel industry application, 'optimal' means the best fit with a userdefined grain size distribution. We will generate the optimal tessellations by developing numerical optimisation methods for minimising functions of Voronoi diagrams. For the nondestructive testing application, 'optimal' means the best fit with ultrasound measurements. In this case the optimal tessellations will be generated by developing numerical methods for tomography in heterogeneous media.
In the second part of this project the tessellations are the Voronoi regions for an optimal location problem. Our goal is to show that certain systems of particles tend to arrange in regular, periodic patterns. To be more precise, our goal is to prove crystallization results for a class of nonlocal particle systems, where the longrange interaction energy is a Wasserstein distance. These energies arise in many areas including signal compression, data clustering, and energydriven pattern formation. The challenge of proving that particle systems have periodic ground states is known as the crystallization conjecture. Despite experimental evidence that many particle systems, such as atoms in metals, have periodic ground states, there are only a handful of rigorous mathematical results. Our approach will combine tools from the calculus of variations and optimal transport theory. Any rigorous progress in this field will be challenging and significant.
This project involves mathematicians, engineers, and the steel industry and will lead to impact in all three areas. This can only be achieved via a combination of rigorous analytical and numerical optimisation methods.

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