Quantitative tools for upscaling the micro-geometry of resonant media

用于放大谐振介质微观几何形状的定量工具

• 批准号: EP/V013025/1
• 负责人: Kirill Cherednichenko
• 金额: $ 40.12万
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV013025%2F1
• 关键词: Quantitative; tools; upscaling; micro; geometry

In many contexts of physics, engineering, and materials science, one is faced with the task of understanding and quantifying the behaviour of a mixture ("composite") of a large number of individual material components, which could have, for example, acoustic or electromagnetic properties. To address this task, many analytical and computational approaches have been developed, working well under some conditions on the geometric and material properties of the components of the composite but providing poor approximations when these conditions are relaxed. The challenge to widen the range of media amenable to analytical approaches has driven the wider subject area of mathematical homogenisation, where new methods have to be developed for dealing with problems that do not fall into the existing frameworks. One of the contexts where new approaches are called for are composite media with components that "resonate" with a wave motion taking place in the medium, that is those situations when the wavelength (wave speed times the temporal period) in some of the constituent elements of the medium is comparable to the actual size of these elements. The existing approximations, based on a standard "homogenisation" rationale, are unable to capture the essential features of interactions between the medium and the wave motion in such cases, as the related averaging techniques work under the assumption of smallness between the size of heterogeneity and associated wavelength. The project will provide a quantitative framework for electromagnetic composites where some of the components exhibits resonant properties, this leading to non-classical behaviour on the macro-scale. We will study the case of conducting (for example, metallic) inclusions in a dielectric medium, which is important for obtaining an insight into the behaviour of composite materials with unusual, counter-intuitive properties (such as "negative" refraction), the key ingredient of many active devices and functional materials. Such materials have recently been the focus of ground-breaking physics experiments (for example, those involving "split-ring" resonators), although little is known about their dispersive properties (i.e. the dependence of the wave-speed on the frequency). Although qualitative progress has been made in recent years, quantitative mathematical formulations for composites with conducting components are beyond the reach of the existing methods of mathematical theory of homogenisation (used to analyse the overall properties of composites), due to the intrinsic dispersive nature of conductors. Inspired by recent advances in applications of operator theory to the asymptotic analysis of boundary-value problems of mathematical physics and the subsequent quantitative results for composites with arbitrary micro-geometries, we will develop a mathematical framework for inhomogeneous periodic media with frequency-dependent interface conditions and will use it for an explicit derivation of effective properties of composites with conducting inclusions, with a sharp control of the approximation error. We will then provide quantitatively sharp formulae for frequency dispersion in photonic crystals with conducting inclusions, opening up new avenues for on-demand metallic photonic fibre design with explicit control of bandgap propagation for a wide range of frequencies.

在物理学、工程学和材料科学的许多背景下，人们面临着理解和量化大量单个材料成分的混合物(“复合材料”)的行为的任务，这些材料成分可能具有例如声学或电磁性质。为了解决这一问题，已经开发了许多分析和计算方法，在某些条件下对复合材料组件的几何和材料特性很好地工作，但当这些条件放宽时，提供的近似较差。扩大服从分析方法的媒体范围的挑战，推动了数学同质化这一更广泛的学科领域，在这个领域，必须开发新的方法来处理不属于现有框架的问题。需要新方法的环境之一是具有与在介质中发生的波动共振的分量的复合介质，即当介质的一些组成元素中的波长(波速乘以时间周期)与这些元素的实际大小相当时的情况。现有的基于标准的“均一化”原理的近似，不能捕捉在这种情况下介质和波动之间相互作用的基本特征，因为相关的平均技术是在非均质的尺寸和相关的波长之间的小的假设下工作的。该项目将为电磁复合材料提供一个量化框架，其中一些组件显示出共振特性，这导致了宏观尺度上的非经典行为。我们将研究介电介质中导电(例如，金属)夹杂物的情况，这对于深入了解具有不寻常的、反直觉的性质(如“负”折射)的复合材料的行为是重要的，这是许多有源器件和功能材料的关键成分。这类材料最近一直是开创性物理实验(例如，涉及“裂环”谐振器的材料)的焦点，尽管人们对它们的色散特性(即波速与频率的关系)知之甚少。尽管近年来取得了质的进展，但由于导体的固有色散性质，含有导电成分的复合材料的定量数学公式超出了现有的均化数学理论方法(用于分析复合材料的整体性能)的范围。在算符理论应用于数学物理边值问题的渐近分析和任意微几何复合材料的定量结果方面的最新进展的启发下，我们将建立一个具有频率相关界面条件的非均匀周期介质的数学框架，并用它来显式推导具有导电夹杂的复合材料的有效性质，并精确地控制逼近误差。然后，我们将提供具有导电夹杂的光子晶体中频率色散的定量精确公式，为按需金属光子光纤设计开辟了新的途径，并在广泛的频率范围内明确控制带隙传播。

项目成果

Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties. II

临界对比偏微分方程的有效行为：微共振、频率转换和时间色散特性。

• DOI: 10.48550/arxiv.2307.01125
• 发表时间: 2023
• 作者: Cherednichenko K

Asymptotic analysis of operator families and applications to resonant media

算子族的渐近分析及其在谐振介质中的应用

• DOI: 10.48550/arxiv.2204.01199
• 发表时间: 2022
• 作者: Cherednichenko K

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• DOI: 10.4213/mzm13447
• 发表时间: 2022
• 期刊: ?????????????? ???????
• 作者: Kiselev A

Operator-Norm Resolvent Asymptotic Analysis of Continuous Media with High-Contrast Inclusions

具有高对比度包含的连续介质的算子范数求解渐近分析

• DOI: 10.1134/s0001434622030051
• 发表时间: 2022
• 期刊: Mathematical Notes
• 影响因子: 0.6
• 作者: Kiselev A

From Complex Analysis to Operator Theory: A Panorama - In Memory of Sergey Naboko

从复分析到算子理论：全景——纪念谢尔盖·纳博科

• DOI: 10.1007/978-3-031-31139-0_12
• 发表时间: 2023
• 作者: Cherednichenko K