Random Matrix Theory for Homogenization of Composites

复合材料均匀化的随机矩阵理论

• 批准号: 1715680
• 负责人: Kenneth Golden
• 金额: $ 35.38万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-15 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1715680&HistoricalAwards=false
• 关键词: Random; Matrix; Theory; Homogenization; Composites

1715680Golden Composite materials are highly valued in engineering and manufacturing for their unique physical and chemical properties that are superior to those of conventional products. Science and industry are continually searching for composites that are stronger, lighter, and less expensive than their traditional counterparts. These structured media -- or metamaterials -- may exhibit increased electrical conductivity, enhanced thermal or acoustic insulation properties, more reliable durability, or even seemingly unattainable properties such as invisibility. Specific examples include lighter wings and fuselage materials for aircraft, body armor for the military, artificial joints used in orthopedic surgery, and professional sporting equipment. Composites also appear throughout the natural world -- in human and animal bodies, and in most components found within and on the surface of the Earth. Examples include bone, lungs, porous rocks containing oil and gas, agricultural soils, and sea ice. The mathematical theory of homogenization for composite materials has been developed to explain observed effective properties of existing composites, which can then be used to predict and discover new composites with less need for costly experimentation. Recently the investigators discovered an unexpected mathematical parallel between homogenization for composites and the Anderson theory of the metal-insulator transition, for which Anderson shared the Nobel prize. In this project the investigators develop new methods for studying composites based on this parallel, bringing the powerful ideas of the Anderson transition to bear on a broad range of problems in the theory of composites. Graduate and undergraduate students participate in the work of the project. In this project the spectral theory of homogenization for transport in composite media is investigated through the lens of random matrix theory. A powerful approach to homogenization problems is the analytic continuation method, which encodes information about the microstructure of the composite through the spectral measure of a self-adjoint random operator governing classical transport in the medium. Random matrix theory naturally arises by considering finite discrete models of composites, which allows the spectral measure, and thus the macroscopic behavior of the composite, to be computed in terms of the eigenvectors and eigenvalues of the random matrices. Surprisingly, as a percolation threshold is approached, these eigenvalues and eigenvectors display strikingly similar behavior to what is observed in Anderson transitions in condensed matter, optics, acoustics, and water waves. This unexpected connection enables the investigators to develop new methods of analysis and computation for homogenization of two-phase composites and related systems such as polycrystals and advection-diffusion processes. Moreover, their approach ties together previously unrelated fields of random matrix theory and homogenization, opening up new avenues for investigation and application. Graduate and undergraduate students participate in the work of the project.

1715680黄金复合材料因其独特的物理和化学性能优于传统产品，在工程和制造业中受到高度重视。科学和工业一直在寻找比传统材料更坚固、更轻、更便宜的复合材料。这些结构介质（或超材料）可能会表现出更高的导电性、更强的隔热或隔音性能、更可靠的耐用性，甚至是看似无法实现的性能，如不可见性。具体的例子包括用于飞机的轻型机翼和机身材料、用于军事的防弹衣、用于骨科手术的人工关节以及专业运动器材。复合材料也出现在整个自然世界中——在人类和动物的身体中，以及在地球内部和表面上发现的大多数部件中。例如骨头、肺、含有石油和天然气的多孔岩石、农业土壤和海冰。复合材料均匀化的数学理论已经发展到解释现有复合材料观察到的有效性能，然后可以用来预测和发现新的复合材料，而不需要昂贵的实验。最近，研究人员在复合材料的均质化和安德森的金属-绝缘体跃迁理论之间意外地发现了数学上的相似之处，安德森也因此获得了诺贝尔奖。在这个项目中，研究人员开发了基于这种并行研究复合材料的新方法，将安德森过渡的强大思想应用于复合材料理论中的广泛问题。研究生和本科生参与该项目的工作。本课题从随机矩阵理论的角度研究了复合介质中输运均匀化的光谱理论。解析延拓法是解决均匀化问题的一种有效方法，该方法通过控制介质中经典输运的自伴随随机算子的谱测量来编码复合材料微观结构的信息。随机矩阵理论是通过考虑复合材料的有限离散模型而自然产生的，它允许光谱测量，从而复合材料的宏观行为，可以根据随机矩阵的特征向量和特征值来计算。令人惊讶的是，当渗透阈值接近时，这些特征值和特征向量与凝聚态物质、光学、声学和水波中的安德森跃迁中观察到的行为惊人地相似。这种意想不到的联系使研究人员能够开发新的分析和计算方法，用于两相复合材料和相关系统，如多晶和平流扩散过程的均质化。此外，他们的方法将以前不相关的随机矩阵理论和均质化领域联系在一起，为研究和应用开辟了新的途径。研究生和本科生参与该项目的工作。

项目成果

Model reduction for fractional elliptic problems using Kato's formula

• DOI: 10.3934/mcrf.2021004
• 发表时间: 2019-04
• 期刊: Mathematical Control & Related Fields
• 影响因子: 1.2
• 作者: H. Dinh;Harbir Antil;Yanlai Chen;E. Cherkaev;A. Narayan

Busemann functions and semi-infinite O’Connell–Yor polymers

Busemann 函数和半无限 OConnell 聚合物

• DOI: 10.3150/19-bej1177
• 发表时间: 2020
• 期刊: Bernoulli
• 影响因子: 1.5
• 作者: Alberts, Tom;Rassoul-Agha, Firas;Simper, Mackenzie
• 通讯作者: Simper, Mackenzie

Forward and inverse homogenization of the electromagnetic properties of a quasiperiodic composite

准周期复合材料电磁特性的正向和逆向均匀化

• DOI: 10.23919/ursi-emts.2019.8931468
• 发表时间: 2019
• 期刊: 2019 URSI International Symposium on Electromagnetic Theory (EMTS
• 作者: Cherkaev, Elena;Guenneau, Sebastien;Wellander, Niklas
• 通讯作者: Wellander, Niklas

Order to disorder in quasiperiodic composites

• DOI: 10.1038/s42005-022-00898-z
• 发表时间: 2022-06
• 期刊: Communications Physics
• 影响因子: 5.5
• 作者: D. Morison;N. B. Murphy;E. Cherkaev;K. Golden

Wave-Driven Assembly of Quasiperiodic Patterns of Particles

粒子准周期模式的波驱动组装

• DOI: 10.1103/physrevlett.126.145501
• 发表时间: 2021
• 期刊: Physical Review Letters
• 影响因子: 8.6
• 作者: Cherkaev, Elena;Guevara Vasquez, Fernando;Mauck, China;Prisbrey, Milo;Raeymaekers, Bart
• 通讯作者: Raeymaekers, Bart