Mathematical foundations of metamaterials: homogenisation, dissipation and operator theory

超材料的数学基础:均质化、耗散和算子理论

基本信息

  • 批准号:
    EP/L018802/2
  • 负责人:
  • 金额:
    $ 90.48万
  • 依托单位:
  • 依托单位国家:
    英国
  • 项目类别:
    Fellowship
  • 财政年份:
    2014
  • 资助国家:
    英国
  • 起止时间:
    2014 至 无数据
  • 项目状态:
    已结题

项目摘要

In order to understand how various physical media behave under specific conditions, for example: a) how the earth surface is deformed during an earthquake, or b) what image resolution one can achieve with a fibre-optic endoscope, mathematicians write differential equations (DEs) and study analytical properties of their solutions, which are then interpreted to draw conclusions about real-life objects. Part of this activity involves analysis of DEs that additionally depend on a parameter. In the above examples this parameter could be: a) the ratio of the size of the rock-forming crystals to the thickness of layers of rock in the ground, orb) the thickness-to-length ratio of the endoscope. The project will develop a new approach to the analysis of solutions of parameter-dependent DEs, based on recent achievements in the mathematical theory of ``operators'' and their spectra, which could roughly be thought of as the sets of the operator ``values''. In general, the spectrum of an operator is found in the two-dimensional-plane of ``complex'' numbers. However, for many operators representing DEs, the spectrum is a subset of a straight line in this plane. It turns out that considering such an operator as a member of a wider operator family, whose spectra are not necessarily situated on the same line, brings about a lot of technical benefits, in a similar way analysis in the complex plane helps understanding real numbers. In the last 50 years or so, many elegant mathematical results about operators (and about DEs as their particular case) have been obtained by using this analogy. We will exploit these results in order to improve our understanding of the behaviour of families of DEs. As a particular source of such families we will study equations representing composites, i.e. media that have several simpler constituent parts. Many objects around us are composites, for example, wood, porous rocks, foams, bubbly liquids, reinforced resins, polycrystal metals. Mathematical statements that we aim at will provide new information about such real-life objects concerning, for example, their acoustic properties, or the way in which they interact with an electromagnetic field. From the physics point of view, members of this wider operator family admit some dissipation (i.e. loss of energy) in comparison to the original ``loss-free'' setup. The project will provide a general mathematical framework for such dissipative extensions in the case of DEs describing composites, yielding a new analytic approach to the study of their effective properties.
为了了解各种物理介质在特定条件下的行为,例如:a)地震期间地球表面是如何变形的,或者b)用光纤内窥镜可以获得什么样的图像分辨率,数学家写出微分方程(DEs)并研究其解的解析性质,然后对其进行解释,得出关于现实生活中物体的结论。该活动的一部分涉及对依赖于参数的de的分析。在上面的例子中,这个参数可以是:a)形成岩石的晶体的大小与地面岩层的厚度之比,b)内窥镜的厚度与长度之比。该项目将根据“算子”及其谱的数学理论的最新成果,开发一种新的方法来分析参数相关微分方程的解,“算子”及其谱大致可以被认为是算子“值”的集合。一般来说,算符的谱是在复数的二维平面上找到的。然而,对于许多表示DEs的算子来说,频谱是这个平面中直线的子集。事实证明,考虑这样一个算子作为一个更广泛算子族的成员,其谱不一定位于同一条线上,带来了很多技术上的好处,以类似的方式,在复平面上的分析有助于理解实数。在过去50年左右的时间里,通过使用这种类比获得了许多关于运算符(以及作为其特殊情况的DEs)的优雅数学结果。我们将利用这些结果来提高我们对DEs族行为的理解。作为此类族的特殊来源,我们将研究表示复合材料的方程,即具有几个更简单组成部分的介质。我们周围的许多物体都是复合材料,例如,木材、多孔岩石、泡沫、气泡液体、增强树脂、多晶金属。我们的目标是数学陈述将提供关于这些现实生活中的物体的新信息,例如,它们的声学特性,或者它们与电磁场相互作用的方式。从物理学的角度来看,与原来的“无损耗”设置相比,这个更宽的算子族的成员承认一些耗散(即能量损失)。该项目将为描述复合材料的DEs的这种耗散扩展提供一个一般的数学框架,为研究其有效性质提供一种新的分析方法。

项目成果

期刊论文数量(10)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
High contrast homogenisation in nonlinear elasticity under small loads
小载荷下非线性弹性的高对比度均匀化
  • DOI:
    10.3233/asy-171430
  • 发表时间:
    2017
  • 期刊:
  • 影响因子:
    1.4
  • 作者:
    Cherdantsev M
  • 通讯作者:
    Cherdantsev M
Spectral and Evolution Analysis of Composite Elastic Plates with High Contrast
高对比度复合弹性板的光谱和演化分析
  • DOI:
    10.1007/s10659-022-09958-5
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    2
  • 作者:
    Bužancic M
  • 通讯作者:
    Bužancic M
Extreme localisation of eigenfunctions to one-dimensional high-contrast periodic problems with a defect
具有缺陷的一维高对比度周期性问题的特征函数的极端局域化
  • DOI:
    10.48550/arxiv.1702.03538
  • 发表时间:
    2017
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Cherdantsev M
  • 通讯作者:
    Cherdantsev M
Bending of thin periodic plates
薄周期板的弯曲
Norm-resolvent convergence for Neumann Laplacians on manifolds thinning to graphs
诺依曼拉普拉斯算子在流形细化到图上的范数求解收敛
  • DOI:
    10.48550/arxiv.2205.04397
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Cherednichenko K
  • 通讯作者:
    Cherednichenko K
{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ monograph.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ sciAawards.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ conferencePapers.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ patent.updateTime }}

Kirill Cherednichenko其他文献

Dynamic Coexistence of Sexual and Asexual Invasion Fronts in a System of Integro-Difference Equations
  • DOI:
    10.1007/s11538-009-9416-8
  • 发表时间:
    2009-04-22
  • 期刊:
  • 影响因子:
    2.200
  • 作者:
    Claudia Carrillo;Kirill Cherednichenko;Nicholas Britton;Michael Mogie
  • 通讯作者:
    Michael Mogie
Bloch-wave homogenization for spectral asymptotic analysis of the periodic Maxwell operator
用于周期性麦克斯韦算子谱渐近分析的布洛赫波均质化
  • DOI:
    10.1080/17455030701551930
  • 发表时间:
    2007
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Kirill Cherednichenko;S. Guenneau
  • 通讯作者:
    S. Guenneau
Finite Difference Time Domain Method For Grating Structures
光栅结构的时域有限差分法
  • DOI:
  • 发表时间:
    2019
  • 期刊:
  • 影响因子:
    0
  • 作者:
    T. Antonakakis;Fadi Baïda;A. Belkhir;Kirill Cherednichenko;S. Cooper;Richard V. Craster;G. Demésy;John DeSanto;Gérard Granet;Evgeny Popov
  • 通讯作者:
    Evgeny Popov

Kirill Cherednichenko的其他文献

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

{{ truncateString('Kirill Cherednichenko', 18)}}的其他基金

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
  • 财政年份:
    2021
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Research Grant
Mathematical foundations of metamaterials: homogenisation, dissipation and operator theory
超材料的数学基础:均质化、耗散和算子理论
  • 批准号:
    EP/L018802/1
  • 财政年份:
    2014
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Fellowship
The mathematical analysis and applications of a new class of high-contrast phononic band-gap composite media
一类新型高对比度声子带隙复合介质的数学分析及应用
  • 批准号:
    EP/I018662/1
  • 财政年份:
    2011
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Research Grant
Variational convergence for nonlinear high-contrast homogenisation problems
非线性高对比度均匀化问题的变分收敛
  • 批准号:
    EP/F03797X/1
  • 财政年份:
    2008
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Research Grant

相似海外基金

Mathematical Foundations of Intelligence: An "Erlangen Programme" for AI
智能的数学基础:人工智能的“埃尔兰根计划”
  • 批准号:
    EP/Y028872/1
  • 财政年份:
    2024
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Research Grant
SAFER - Secure Foundations: Verified Systems Software Above Full-Scale Integrated Semantics
SAFER - 安全基础:高于全面集成语义的经过验证的系统软件
  • 批准号:
    EP/Y035976/1
  • 财政年份:
    2024
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Research Grant
Statistical Foundations for Detecting Anomalous Structure in Stream Settings (DASS)
检测流设置中的异常结构的统计基础 (DASS)
  • 批准号:
    EP/Z531327/1
  • 财政年份:
    2024
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Research Grant
Social Foundations of Cryptography
密码学的社会基础
  • 批准号:
    EP/X017524/1
  • 财政年份:
    2024
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Research Grant
Collaborative Research: AF: Medium: Foundations of Oblivious Reconfigurable Networks
合作研究:AF:媒介:遗忘可重构网络的基础
  • 批准号:
    2402851
  • 财政年份:
    2024
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Continuing Grant
Conference: Theory and Foundations of Statistics in the Era of Big Data
会议:大数据时代的统计学理论与基础
  • 批准号:
    2403813
  • 财政年份:
    2024
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Standard Grant
CAREER: Statistical foundations of particle tracking and trajectory inference
职业:粒子跟踪和轨迹推断的统计基础
  • 批准号:
    2339829
  • 财政年份:
    2024
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Continuing Grant
CAREER: Architectural Foundations for Practical Privacy-Preserving Computation
职业:实用隐私保护计算的架构基础
  • 批准号:
    2340137
  • 财政年份:
    2024
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Continuing Grant
CAREER: Foundations, Algorithms, and Tools for Browser Invalidation
职业:浏览器失效的基础、算法和工具
  • 批准号:
    2340192
  • 财政年份:
    2024
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Continuing Grant
CAREER: Foundations of semi-infinite and equilibrium constrained optimization
职业:半无限和平衡约束优化的基础
  • 批准号:
    2340858
  • 财政年份:
    2024
  • 资助金额:
    $ 90.48万
  • 项目类别:
    Continuing Grant
{{ showInfoDetail.title }}

作者:{{ showInfoDetail.author }}

知道了