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

点击翻译按钮获取中文摘要

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.