Designer Microstructure via Optimal Transport Theory

规则的几何镶嵌出现在自然界的许多地方。六边形无处不在，从蜂巢到巨人堤道的六边形玄武岩柱。不规则多边形的镶嵌在自然界中也可以观察到，例如在长颈鹿的皮肤上。Voronoi图是一种重要的不规则多边形镶嵌。例如，城市的Voronoi镶嵌可以通过超市的位置生成；如果我们假设每个人都去最近的商店，那么超市的“集水区”将城市镶嵌起来，结果是它们形成了一个多边形镶嵌，称为沃罗诺伊图。类似Voronoi图的模式在很多地方都令人惊讶地出现：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佳镶嵌（Voronoi图）。“最优”的定义取决于应用。在项目的第一部分，镶嵌代表金属中的颗粒，这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，“最优”是指与用户定义的粒度分布的最佳拟合。我们将通过开发用于最小化Voronoi图函数的数值优化方法来生成最佳的镶嵌。对于无损检测应用，“最佳”是指最适合超声测量。在这种情况下，将通过开发非均匀介质层析成像的数值方法来产生最佳的镶嵌。在这个项目的第二部分，镶嵌是Voronoi区域的最优定位问题。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。更精确地说，我们的目标是证明一类非局部粒子系统的结晶结果，其中远程相互作用能是沃瑟斯坦距离。这些能量出现在许多领域，包括信号压缩、数据聚类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。尽管实验证据表明，许多粒子系统，如金属中的原子，具有周期性基态，但只有少数严格的数学结果。我们的方法将结合变异演算和最优运输理论的工具。这一领域的任何严格进展都将具有挑战性和重大意义。该项目涉及数学家、工程师和钢铁行业，并将对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的结合来实现。

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.