Designer Microstructure via Optimal Transport Theory

自然界中的许多地方都会出现规则的几何镶嵌。从蜂巢到巨人堤道上的六角玄武岩柱子，六边形随处可见。自然界中也可以观察到不规则多边形的镶嵌，例如在长颈鹿的皮肤上。Voronoi图是不规则多边形镶嵌的一种重要类型。例如，一个城市的Voronoi镶嵌可以由超市的位置来生成；如果我们假设每个人都去离他们最近的商店，那么超市的‘集水区’就会对城市进行镶嵌，结果是它们形成了一个多边形镶嵌，称为Voronoi图。类似Voronoi图的图案出现在令人惊讶的许多地方：生物细胞、肥皂泡和金属的微观结构。这个数学研究项目的目标是开发严格的数值和分析方法来生成最优的镶嵌(Voronoi图)。“最佳”的定义取决于应用。在项目的第一部分中，镶嵌表示金属中的颗粒，这些颗粒是金属中具有相同晶体结构和取向的微观区域，我们考虑在钢铁工业和超声波无损检测中的应用。对于钢铁工业应用，“最佳”意味着最符合用户定义的晶粒度分布。我们将通过开发数值优化方法来最小化Voronoi图的函数来生成最优镶嵌。对于无损检测应用，“最佳”意味着与超声波测量的最佳匹配。在这种情况下，将通过开发非均匀介质中层析成像的数值方法来生成最优镶嵌。在本项目的第二部分中，镶嵌是用于最优位置问题的Voronoi区域。我们的目标是证明某些粒子系统倾向于以规则的、周期性的模式排列。更准确地说，我们的目标是证明一类非局域粒子系统的结晶结果，其中长程相互作用能是Wasserstein距离。这些能量出现在许多领域，包括信号压缩、数据集群和能量驱动的模式形成。证明粒子系统具有周期性基态的挑战被称为结晶猜想。尽管有实验证据表明，许多粒子系统，如金属中的原子，都有周期性的基态，但只有少数几个严格的数学结果。我们的方法将结合变分和最优运输理论中的工具。这一领域的任何严格进展都将是具有挑战性和重大意义的。这个项目涉及数学家、工程师和钢铁行业，并将在这三个领域产生影响。这只能通过严格的分析和数值优化方法相结合来实现。

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.