Asymptotic and Numerical Analysis of Wave Propagation in Thin-Structure Waveguides

波导是理解许多物理系统的核心，也是当今使用的许多技术的基础。光学和光子晶体纤维就是电磁环境中的一个例子，然而波导在声学或压电环境中也很有趣。一个典型的波导模型将涉及一个系统的偏微分方程，波塞冬复合域与一些薄结构;通常认为一个波导作为一个2D的“横截面“结构，已被挤出到3D。由于域的不规则性和控制方程的性质，这样的系统往往是计算量大。利用均匀化理论和最近的结果从频谱分析，该项目旨在开发分析工具，这将使治疗这样的波导问题更有效。波导问题的“横截面结构”是典型的周期性和（通常通过分离变量）的3D问题减少到一个家庭的2D问题的横截面结构，参数化的传播常数沿波导。考虑到这一点，该项目将：（A）在适当的频率范围内研究二维问题的频谱，薄结构的厚度越来越细。(B)推导出有效的二维“奇异结构”问题并分析其谱，并进行适当的误差分析和收敛结果。我们选择的方法背后的关键思想是，奇异结构问题比其薄结构问题更适合于解析方法，后者提供了一个模型，可以将几何和材料参数与传播波的性质联系起来。我们最初的重点将是从（A）中的问题公式化（B）中的问题。这将涉及典型的例子，如标量波动方程，以获得熟悉的数学技术和对象，将被使用。这将是参考指南时，移动到更复杂的系统的偏微分方程，如波制导由麦克斯韦方程或方程的弹性。因此，该项目的短期目标是将从薄结构波问题（A）导出有效奇异结构问题（B）的过程形式化;选择物理和工程中出现的问题。对这些问题的谱分析将由原来的薄结构问题的数值计算来支持，以证明奇异结构近似的合理性。长期目标将集中在奇异结构作为材料夹杂物的处理上。作为一个例子，在光子设置，金属材料与介电夹杂物诱导不同的边界条件（hencewaves和光谱）从介电-介电夹杂物。这反过来又提出了相应的奇异结构夹杂物应施加的条件的问题。该项目还应着眼于以这种方式从电磁学、弹性和压电学研究选择波传播问题。光纤（电磁学）和压电材料是巴斯物理和工程系积极研究的主题，因此这提供了一系列需要考虑的基本问题。总之，该项目的目标是：1）开发分析工具，以制定近似薄结构的奇异结构问题（波传播）问题。2）使用这些工具来推导电磁学、弹性力学和压电学背景下波导中波传播的（一系列）模型。3）对这些模型进行分析，寻求类似于应用中所需的信息。

Wave guidance is central to understanding many physical systems and underpins much technology in use today.Optical and photonic-crystal fibres are one such example in an electromagnetic context, however wave guidance isalso of interest in acoustic or piezoelectric settings. A typical waveguide model will involve a system of PDEs, posedon composite domains with some thin-structure; typically one thinks of a waveguide as a 2D "cross-sectional"structure that has been extruded into 3D. Due to the irregularity of the domain and nature of the governing equations,such systems tend to be computationally heavy. Using homogenisation theory and recent results from spectralanalysis, the project aims to develop analytical tools, which will make the treatment to such wave-guidance problemsmore efficient.The "cross-sectional structure" of a wave-guidance problem is typically periodic and (usually by separation ofvariables) the 3D problem reduces to a family of 2D problems on the cross-sectional structure, parametrised by thepropagation constant down the waveguide. With this in mind, the project will:(A) Investigate the spectrum of the 2D-problem in an appropriate frequency regime, with the thin-structure becomingincreasingly fine.(B) Derive the effective 2D-"singular-structure" problem and analyse its spectrum, with appropriate error analysis andconvergence results. The key idea behind our chosen approach is that the singular-structure problem is moreamenable to analytical approaches than its thin-structure counterpart, which provides a model that can link geometricand material parameters to the properties of the propagating waves.Our initial focus will be on the formulation of the problems in (B) from those in (A). This will involve archetypicalexamples such as a scalar wave-equation, to gain familiarity with the mathematical techniques and objects that are tobe used. This will be the reference guide when moving to more complex systems of PDEs, such as wav e-guidancegoverned by Maxwell's equations or equations of elasticity. Hence, the short-term objective of the project is toformalise the process of deriving the effective singular-structure problems (B) from thin-structure wave problems (A);selecting problems that arise in physics and engineering. Spectral analysis of these problems will be performed andsupported by numerical computations for the original thin-structure problem, to justify the singular-structureapproximation.Longer-term objectives would focus on the treatment of the singular-structure as a material inclusion. As an examplein the photonic setting, a metallic material with a dielectric inclusion induces different boundary conditions (hencewaves and spectra) from a dielectric-dielectric inclusion. This in turn raises questions as to the conditions that shouldbe imposed for the corresponding singular-structure inclusion. The project should also look to investigate selectwave-propagation problems from electromagnetism, elasticity and piezoelectricity in this manner. Optical fibres(electromagnetism) and piezoelectric materials are subjects of active research in the Physics and Engineeringdepartments in Bath, and so this provides a basis set of problems to consider.In summary, the objectives of the project are:1) Development of analytical tools to formulate singular-structure problems that approximate thin-structure (wavepropagation)problems.2) Using these tools to derive (a selection of) models of wave propagation in waveguides in the contexts ofelectromagnetism, elasticity and piezoelectricity.3) Performing an analysis of these models, seeking information akin to that which would be desired in application.