Designer Microstructure via Optimal Transport Theory

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 non-destructive testing using ultrasound. For the steel industry application, 'optimal' means the best fit with a user-defined grain size distribution. We will generate the optimal tessellations by developing numerical optimisation methods for minimising functions of Voronoi diagrams. For the non-destructive 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 long-range interaction energy is a Wasserstein distance. These energies arise in many areas including signal compression, data clustering, and energy-driven 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.

This project will have multiple types of impact: industrial, academic, and training.

(i) Impact in the steel industry. Objective 1.1 of my proposal is in collaboration with engineers from the Research and Development department of Tata Steel. The goal is to generate realistic geometric models of the microstructure of polycrystalline steels. These models, known as representative volume elements, will then be used by engineers in Tata Steel in multiscale simulations of steels (computational plasticity and homogenization), with the ultimate goal of improving steel grades (alloys) and steel-forming processes. This is of great importance to the automotive industry; better steels means safer, lighter, and more fuel-efficient cars, and a reduction in carbon dioxide emissions. This objective will also lead to new mathematics, namely new numerical optimisation algorithms for minimising functions of Voronoi diagrams.

(ii) Impact in non-destructive testing. Objective 1.2 forms an important step along the pathway to the development of new ultrasound technology for the detection of flaws in aircraft, oil pipelines, and nuclear plants. Its goal is to develop efficient optimisation algorithms for reconstructing the microstructure of a polycrystalline metal from a collection of ultrasound measurements made on the boundary of the metal. Knowledge of the anisotropic microstructure of a metal, obtained using the results of this project, can then be fed into backscatter imaging algorithms to enhance the detection of flaws. This step along the pathway to impact will be performed in collaboration with the Centre for Ultrasonic Engineering at Strathclyde, where the theory will be compared with experiments performed using ultrasonic arrays.

The mathematical impact of this project will be the development of new numerical methods for computing the derivative of geodesic distances with respect to anisotropic metrics, which has applications in tomography beyond non-destructive testing, namely in medical ultrasound tomography and seismic tomography.

(iii) Academic impact. An interdisciplinary workshop involving mathematicians and engineers will facilitate knowledge exchange between communities working in microstructure modelling, numerical optimal transport theory, computational geometry, and crystallization. In addition to the workshop, the results of this project will be disseminated through the PI's collaborators in the UK, France, the Netherlands and Germany, and at several international conferences.

(iv) Training. A PDRA will be trained in both mathematics and its applications in engineering and industry. Mathematical training will be provided though lectures by the visiting researchers Mark Peletier, Filippo Santambrogio and Tony Mulholland, which will be offered to all the local PhD students and postdocs. The PDRA will also visit and work with engineers in the Tata Steel Research and Development department and the Centre for Ultrasonic Engineering (Strathclyde).