Designer Microstructure via Optimal Transport Theory

通过最佳传输理论设计微观结构

• 批准号: EP/R013527/1
• 负责人: David Bourne
• 金额: $ 12.89万
• 依托单位: Durham University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FR013527%2F1
• 关键词: Designer; Microstructure; via; Optimal; Transport

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.

规则的几何镶嵌出现在自然界的许多地方。从蜂巢到巨人堤道上的六角形玄武岩柱，到处都是障碍物。不规则多边形的镶嵌也在自然界中观察到，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细分可以由超市的位置生成;如果我们假设每个人都去最近的商店，那么超市的“集水区”会细分城市，结果是它们形成了一个多边形细分，称为Voronoi图。类似于Voronoi图的图案出现在许多地方：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佳的镶嵌（Voronoi图）。“最佳”的定义取决于应用。在项目的第一部分中，镶嵌表示金属中的晶粒，这些晶粒是具有相同晶体结构和取向的金属中的微观区域，我们考虑了在钢铁行业和使用超声波的无损检测中的应用。对于钢铁行业应用，“最佳”意味着与用户定义的粒度分布最佳拟合。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，最佳的镶嵌将产生的数值方法层析成像在非均匀media.In本项目的第二部分的镶嵌是Voronoi区域的最佳定位问题。我们的目标是证明某些粒子系统倾向于以规则的周期性模式排列。更准确地说，我们的目标是证明结晶结果的一类非局部粒子系统，其中的长程相互作用能是一个Wasserstein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。尽管有实验证据表明，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。我们的方法将结合联合收割机工具，从变分法和最优运输理论。在这一领域的任何严格的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，工程师和钢铁工业，并将在所有三个领域产生影响。这只能通过严格的分析和数值优化方法相结合来实现。

项目成果

An inverse problem for Voronoi diagrams: A simplified model of non-destructive testing with ultrasonic arrays

Voronoi 图的反问题：超声阵列无损检测的简化模型

• DOI: 10.1002/mma.6977
• 发表时间: 2020
• 期刊: Mathematical Methods in the Applied Sciences
• 影响因子: 2.9
• 作者: Bourne D

Semi-discrete optimal transport methods for the semi-geostrophic equations

半地转方程的半离散最优输运方法

• 发表时间: 2021
• 作者: Bourne D

Laguerre tessellations and polycrystalline microstructures: a fast algorithm for generating grains of given volumes

• DOI: 10.1080/14786435.2020.1790053
• 发表时间: 2020-07-24
• 期刊: PHILOSOPHICAL MAGAZINE
• 影响因子: 1.6
• 作者: Bourne, D. P.;Kok, P. J. J.;Spanjer, W. D. T.
• 通讯作者: Spanjer, W. D. T.

Controlling Fragment Competition on Pathways to Addressable Self-Assembly.

控制可寻址自组装途径上的片段竞争。

• DOI: 10.1021/acs.jpcb.8b08096
• 发表时间: 2018
• 期刊: The journal of physical chemistry. B
• 作者: Madge J

Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems

• DOI: 10.1007/s00220-021-04216-6
• 发表时间: 2020-12
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: D. Bourne;R. Cristoferi