Homogenisation of Multiscale Partial Differential Equations from Physics

物理学多尺度偏微分方程的齐次化

• 批准号: 2087445
• 金额: --
• 依托单位: University College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 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=studentship-2087445
• 关键词: Homogenisation; Multiscale; Partial; Differential; Equations

Research areas: Continuum mechanics, Mathematical analysis, Mathematical physics and Material Engineering - compositesThe project will focus on advancing a general theory of ''non-classical'' homogenisation for partial differential equations (PDEs) coming from Physics, typically with a ''strong'' interaction between the scales. One general scenario relates to strongly heterogeneous media, which may in particular be electromagnetic, acoustic or elastic, where the contrast between the constituent components is high. As a result, some component may respond to an applied dynamic frequency as resonators while others will essentially behave as in a quasi-static regime amenable to a classical homogenisation. Adopting appropriate scaling results in a high-contrast homogenisation problem for relevant PDE, where the two small parameters of the contrast and of the microscale of the heterogeneity are critically scaled. The nature of such a coupling between the scales leads to a number of interesting physical effects at the macroscale. These include frequency bandgaps, high dispersion, wave localisation, etc. The project will be aiming at advancing mathematical understanding of these effects, which would require development of non-classical tools of multi-scale asymptotic analysis of PDEs. Precise topics are envisaged to evolve, but some specific topics may include: analysing the effect of ''coupled resonances'' in high-contrast periodic media with a pre-resonant propagating frequency; investigating the effect of randomness of high-contrast resonators on wave localisation due to trapping by the resonators with statistically distributed eigenfrequencies; analysing the effects of directional vs frequency filtering properties for more general patterns of asymptotic degeneracies including high anisotropies and particularly for vector problems such as in electromagnetism and elastodynamics; high-contrast lattice materials; homogenisation of small-size resonators via capacity-type tools. One additional intriguing challenge is to attempt to develop a homogenisation theory approach to Einstein's equations of General Relativity.

研究领域：连续介质力学、数学分析、数学物理和材料工程 - 复合材料该项目将专注于推进来自物理学的偏微分方程（PDE）的“非经典”均质化一般理论，通常在尺度之间具有“强”相互作用。一种一般情况涉及强异质介质，特别是电磁、声学或弹性介质，其中组成成分之间的对比度很高。因此，一些组件可以作为谐振器响应所施加的动态频率，而其他组件则基本上表现为适合经典均质化的准静态状态。采用适当的缩放会导致相关偏微分方程的高对比度均质化问题，其中对比度和异质性微尺度的两个小参数被严格缩放。尺度之间这种耦合的性质导致了宏观尺度上许多有趣的物理效应。其中包括频带隙、高色散、波局域化等。该项目旨在推进对这些效应的数学理解，这需要开发偏微分方程多尺度渐近分析的非经典工具。预计会发展出精确的主题，但一些具体主题可能包括：分析具有预谐振传播频率的高对比度周期性介质中“耦合谐振”的影响；研究由于具有统计分布特征频率的谐振器的捕获而导致高对比度谐振器的随机性对波定位的影响；分析方向与频率滤波特性对渐近简并的更一般模式（包括高各向异性）的影响，特别是电磁学和弹性动力学等矢量问题；高对比度晶格材料；通过容量型工具对小尺寸谐振器进行均质化。另一个有趣的挑战是尝试开发一种用于爱因斯坦广义相对论方程的均质化理论方法。