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是电磁环境中的一个这样的例子，然而波导在声学或压电设置中也很有趣。一个典型的波导模型将包括一个pde系统，波塞冬复合畴与一些薄结构；通常人们认为波导是一个被挤压成三维的二维“横截面”结构。由于域的不规则性和控制方程的性质，这样的系统往往计算量很大。利用均质化理论和光谱分析的最新结果，该项目旨在开发分析工具，这将使这种波导问题的处理更有效。波导问题的“横截面结构”通常是周期性的，并且（通常通过分离变量）将3D问题简化为横截面结构上的一系列2D问题，通过沿波导的传播常数进行参数化。考虑到这一点，该项目将：(A)随着薄结构变得越来越精细，在适当的频率范围内研究2d问题的频谱。(B)推导出有效的二维“奇异结构”问题，并对其谱进行分析，得到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，奇异结构问题比薄结构问题更适合于分析方法，后者提供了一个模型，可以将几何和材料参数与传播波的特性联系起来。我们最初的重点是将(B)中的问题从(A)中推导出来。这将涉及到典型的例子，如标量波动方程，以获得熟悉的数学技术和对象将被使用。当移动到更复杂的偏微分方程系统时，例如由麦克斯韦方程或弹性方程控制的波导，这将是参考指南。因此，该项目的短期目标是形式化从薄结构波问题(A)推导有效的单结构问题(B)的过程；选择物理和工程中出现的问题。这些问题的频谱分析将由原始薄结构问题的数值计算来执行和支持，以证明奇异结构近似。长期目标将侧重于将单一结构作为一种材料内含物进行处理。作为一个例子，在光子设置中，具有介电包体的金属材料从介电-介电包体诱导出不同的边界条件（因此波和光谱）。这反过来又提出了应该为相应的单结构包含施加条件的问题。该项目还应以这种方式研究电磁学、弹性学和压电学中的选择波传播问题。光纤（电磁）和压电材料是巴斯大学物理系和工程系的活跃研究课题，因此这提供了一系列需要考虑的基础问题。总之，该项目的目标是：1)开发分析工具，以制定近似薄结构（波传播）问题的奇异结构问题。2)使用这些工具推导（选择）电磁波、弹性和压电背景下波导中的波传播模型。3)对这些模型进行分析，寻找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.