Creating macroscale effective interfaces encapsulating microstructural physics

这项提案寻求为一项为期三年的综合性研究计划提供资金，该计划旨在研究在宏观尺度上模拟微观界面现象的方法。一个跨越许多学科的基本问题是：给定一个微观结构的界面，它能否被完全在宏观尺度上提出的有效的“平均”边界条件所取代。如果是这样的话，它能准确地再现微观结构所产生的物理效果吗？这种有效的边界条件能否严格地推导出来，而不是以某种特别的方式导出？这样做的限制是什么？该提案旨在回答这些问题，目的是能够准确和有效地预测在三个明显不相关的领域中的复杂物理行为：表面瑞利-布洛赫波的波传播和来自设计者结构化表面的波的反射，微图案表面相变的统计力学，以及通过结构域的扩散建模。这些领域都共享一个复杂的结构界面，我们提出的通用总体数学方法将导致封装占主导地位的微尺度物理的有效边界条件；这将代表着这些领域中的每一个领域的相当大的进步。主要的数学方法将以齐化理论为基础，利用长度上的差异从多个尺度分析中创建渐近。均化通常用于散体材料有短期波动和溶液在长尺度上变化时，其用于界面的探索要少得多。重要的是，我们还致力于将均匀化理论的有效范围从长波、准静态区域扩大到可以与微观结构相同尺度变化的区域。这项分析工作将得到详细的数值模拟的补充，这些数值模拟将用于验证所开发的界面模型的有效性。这项工作将由伦敦帝国理工学院数学系的一个团队承担，他们的技能和优势互补：Pavliotis(均化理论，随机过程)、Parry(统计力学，相变)和Craster(波传播，均化，流体力学)。

This proposal seeks funding for a comprehensive three year researchprogram into methodologies for modeling microscale interfacial phenomena on the macroscale level. A fundamental question stretching across many disciplines is: Given a microstructured interface can it be replaced by an effective ``averaged'' boundary condition entirely posed upon a macroscale. If so, can it accurately reproduce the physical effects created by the microstructure? Can this effective boundary condition be derived rigorously, rather than in some ad-hoc fashion, and what are the limitations in so doing? The proposal aims to answer these questions, with the goal of being able to accurately and efficiently predict complex physical behaviour in three apparently unconnected fields: in wave propagation for surface Rayleigh-Bloch waves and for the reflection of waves from designer structured surfaces, in the statistical mechanics of phase transitions on micropatterned surfaces, and in modeling diffusions through structured domains. These fields all share a complex structured interface and the generic overarching Mathematical approach we propose will lead to effective boundary conditions encapsulating the dominant microscale Physics; this will represent a considerable advance in each of these areas. The primary Mathematical approach will be based around Homogenization theory utilizing the discrepancy in lengthscales to create asymptotics from multiple scales analysis. Homogenization is conventionally used when the bulk material has short-scale fluctuations and the solution varies on a long-scale, its use for interfaces is much less well explored. Importantly we also aim to enhance the range of validity of homogenization theory away from long-wave, quasi-static, regimes to ones that can vary on the same scale as the microstructure. This analytical work will be complemented by detailed numerical simulations that will act to verify the efficacy of the developed interfacial models. The work will be undertaken by a team from the Mathematics Department at Imperial College London with complementary skills and strengths: Pavliotis (Homogenization theory, stochastic processes), Parry (Statistical mechanics, phase transitions) and Craster (Wave propagation, homogenization, fluid mechanics).