Asymptotic and Numerical Analysis of Wave Propagation in Thin-Structure Waveguides
Lead Research Organisation:
University of Bath
Department Name: Mathematical Sciences
Abstract
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 is
also of interest in acoustic or piezoelectric settings. A typical waveguide model will involve a system of PDEs, posed
on 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 spectral
analysis, the project aims to develop analytical tools, which will make the treatment to such wave-guidance problems
more efficient.
The "cross-sectional structure" of a wave-guidance problem is typically periodic and (usually by separation of
variables) the 3D problem reduces to a family of 2D problems on the cross-sectional structure, parametrised by the
propagation 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 becoming
increasingly fine.
(B) Derive the effective 2D-"singular-structure" problem and analyse its spectrum, with appropriate error analysis and
convergence results. The key idea behind our chosen approach is that the singular-structure problem is more
amenable to analytical approaches than its thin-structure counterpart, which provides a model that can link geometric
and 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 archetypical
examples such as a scalar wave-equation, to gain familiarity with the mathematical techniques and objects that are to
be used. This will be the reference guide when moving to more complex systems of PDEs, such as wav e-guidance
governed by Maxwell's equations or equations of elasticity. Hence, the short-term objective of the project is to
formalise 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 and
supported by numerical computations for the original thin-structure problem, to justify the singular-structure
approximation.
Longer-term objectives would focus on the treatment of the singular-structure as a material inclusion. As an example
in the photonic setting, a metallic material with a dielectric inclusion induces different boundary conditions (hence
waves and spectra) from a dielectric-dielectric inclusion. This in turn raises questions as to the conditions that should
be imposed for the corresponding singular-structure inclusion. The project should also look to investigate select
wave-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 Engineering
departments 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 of
electromagnetism, elasticity and piezoelectricity.
3) Performing an analysis of these models, seeking information akin to that which would be desired in application.
Optical and photonic-crystal fibres are one such example in an electromagnetic context, however wave guidance is
also of interest in acoustic or piezoelectric settings. A typical waveguide model will involve a system of PDEs, posed
on 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 spectral
analysis, the project aims to develop analytical tools, which will make the treatment to such wave-guidance problems
more efficient.
The "cross-sectional structure" of a wave-guidance problem is typically periodic and (usually by separation of
variables) the 3D problem reduces to a family of 2D problems on the cross-sectional structure, parametrised by the
propagation 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 becoming
increasingly fine.
(B) Derive the effective 2D-"singular-structure" problem and analyse its spectrum, with appropriate error analysis and
convergence results. The key idea behind our chosen approach is that the singular-structure problem is more
amenable to analytical approaches than its thin-structure counterpart, which provides a model that can link geometric
and 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 archetypical
examples such as a scalar wave-equation, to gain familiarity with the mathematical techniques and objects that are to
be used. This will be the reference guide when moving to more complex systems of PDEs, such as wav e-guidance
governed by Maxwell's equations or equations of elasticity. Hence, the short-term objective of the project is to
formalise 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 and
supported by numerical computations for the original thin-structure problem, to justify the singular-structure
approximation.
Longer-term objectives would focus on the treatment of the singular-structure as a material inclusion. As an example
in the photonic setting, a metallic material with a dielectric inclusion induces different boundary conditions (hence
waves and spectra) from a dielectric-dielectric inclusion. This in turn raises questions as to the conditions that should
be imposed for the corresponding singular-structure inclusion. The project should also look to investigate select
wave-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 Engineering
departments 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 of
electromagnetism, elasticity and piezoelectricity.
3) Performing an analysis of these models, seeking information akin to that which would be desired in application.
Planned Impact
The impact of the SAMBa CDT will occur principally through the following two pathways:
1. Direct engagement with industrial partners, leading to PhD projects that are collaborative with industry, and that are focussed on topics with direct industrial impact.
2. The production of PhD graduates with
(a) the mathematical, statistical and computational technical skill sets that have been identified as in crucial demand both by EPSRC and by our industrial partners, coupled to
(b) extensive experience of industrial collaboration.
The underlying opportunity that SAMBa provides is to train graduates to have the ability to combine complex models with 'big data'. Such people will be uniquely equipped to deliver impact: whether they continue with academic careers or move directly to posts in industry, through quantitative modelling, they will provide the information that gives UK businesses competitive advantages. Our industrial partners make it clear to us that competitiveness in the energy, manufacturing, service, retail and financial sectors is increasingly dependent on who can best and most quickly analyse the huge datasets made available by the present information revolution.
During their training as part of SAMBa, these students will have already gained experience of industrial collaboration, through their PhD projects and/or the Integrated Think Tanks (ITTs) that we propose, that will give all SAMBa students opportunities to develop these transferable skills. PhD projects that involve industrial collaboration, whether arising from ITTs or not, will themselves deliver economic and social benefits to UK through the private companies and public sector organisations with which SAMBa will collaborate.
We emphasise that Bath is at the forefront of knowledge transfer (KT) activities of the kind needed to translate our research into impact. Our KT agenda has recently been supported by KT Accounts and Impact Acceleration Accounts from EPSRC (£4.9M in total) and a current HEFCE HEIF allocation of £2.4M. Bath is at the forefront of UK activity in KTPs, having completed 150 and currently holding 16 KTP contracts worth around £2.5M.
The SAMBa ITTs are an exciting new mechanism through which we will actively look for opportunities to turn industrial links into research partnerships, supported in the design of these projects by the substantial experience available across the University.
More widely, we envisage impact stemming from a range of other activities within SAMBa:
- We will look to feed the results of projects involving ecological or epidemiological data directly into environmental and public health policy. We have done this successfully many times and have three REF Case Studies describing work of this nature.
- Students will be encouraged to make statistical tools available as open source software. This will promote dissemination of their research results, particularly beyond academia. There is plenty of recent evidence that such packages are taken up and used.
- Students will discuss how to use new media to promote the public understanding of science, for example contributing to projects such as Wikipedia.
- Students will be encouraged to engage in at least one outreach activity. Bath is well known for its varied, and EPSRC-supported, public engagement activities that include Royal Institution Masterclasses, coaching the UK Mathematics Olympiad team, and reaching 50 000 people in ten days with an exhibit at the Royal Society's 350th Anniversary Summer Exhibition in 2010.
1. Direct engagement with industrial partners, leading to PhD projects that are collaborative with industry, and that are focussed on topics with direct industrial impact.
2. The production of PhD graduates with
(a) the mathematical, statistical and computational technical skill sets that have been identified as in crucial demand both by EPSRC and by our industrial partners, coupled to
(b) extensive experience of industrial collaboration.
The underlying opportunity that SAMBa provides is to train graduates to have the ability to combine complex models with 'big data'. Such people will be uniquely equipped to deliver impact: whether they continue with academic careers or move directly to posts in industry, through quantitative modelling, they will provide the information that gives UK businesses competitive advantages. Our industrial partners make it clear to us that competitiveness in the energy, manufacturing, service, retail and financial sectors is increasingly dependent on who can best and most quickly analyse the huge datasets made available by the present information revolution.
During their training as part of SAMBa, these students will have already gained experience of industrial collaboration, through their PhD projects and/or the Integrated Think Tanks (ITTs) that we propose, that will give all SAMBa students opportunities to develop these transferable skills. PhD projects that involve industrial collaboration, whether arising from ITTs or not, will themselves deliver economic and social benefits to UK through the private companies and public sector organisations with which SAMBa will collaborate.
We emphasise that Bath is at the forefront of knowledge transfer (KT) activities of the kind needed to translate our research into impact. Our KT agenda has recently been supported by KT Accounts and Impact Acceleration Accounts from EPSRC (£4.9M in total) and a current HEFCE HEIF allocation of £2.4M. Bath is at the forefront of UK activity in KTPs, having completed 150 and currently holding 16 KTP contracts worth around £2.5M.
The SAMBa ITTs are an exciting new mechanism through which we will actively look for opportunities to turn industrial links into research partnerships, supported in the design of these projects by the substantial experience available across the University.
More widely, we envisage impact stemming from a range of other activities within SAMBa:
- We will look to feed the results of projects involving ecological or epidemiological data directly into environmental and public health policy. We have done this successfully many times and have three REF Case Studies describing work of this nature.
- Students will be encouraged to make statistical tools available as open source software. This will promote dissemination of their research results, particularly beyond academia. There is plenty of recent evidence that such packages are taken up and used.
- Students will discuss how to use new media to promote the public understanding of science, for example contributing to projects such as Wikipedia.
- Students will be encouraged to engage in at least one outreach activity. Bath is well known for its varied, and EPSRC-supported, public engagement activities that include Royal Institution Masterclasses, coaching the UK Mathematics Olympiad team, and reaching 50 000 people in ten days with an exhibit at the Royal Society's 350th Anniversary Summer Exhibition in 2010.
Organisations
People |
ORCID iD |
| Kirill Cherednichenko (Primary Supervisor) | |
| William GRAHAM (Student) |