Homogenisation of high-contrast random composites

About the project:

The School of Mathematics at Cardiff University is delighted to offer a fully funded studentship starting in October 2022.

 Project Summary:

The problem of effective properties of heterogeneous materials (composites) goes back to the works of Maxwell, Rayleigh, and Einstein. Today understanding the properties of composites and search for novel materials is at the forefront of research in industry and science.

One of the most exciting directions in this area is the analysis of high-contrast composites whose constituent components exhibit large difference between their physical characteristics. One can think of small balls (inclusions) made of one material dispersed either periodically or randomly in the base material, so that the characteristic distance between the inclusions is also small. Such composites often possess unusual properties not readily available in nature or traditional materials. Notably, in the context of electromagnetic waves, one of such properties is the band-gap spectrum, which allows for the design of optical materials (photonic crystals) for controlling and manipulating the flow of light.

All this makes the study of such problems especially attractive and rewarding, yet rigorous mathematical analysis of high-contrast composites is often very challenging. Although the high-contrast periodic setting is well explored theoretically, the manufacturing process of periodic composites faces two major challenges: making them with enough precision and designing processes that can robustly mass-produce the material. Therefore, it is of paramount importance to study the situation when the geometrical structure of the composite in question is not exactly periodic, or even ‘’completely’’ random.

The project will be focussed on one the open problems in this newly emerging branch of Homogenisation Theory of PDEs. For example:

-       quantitative analysis of the deviation of effective properties of a composite whose structure is not periodic (e.g., through the spatial distribution or shapes or material characteristics of inclusions) from the properties of a perfectly periodic composite;

or

-       extending the existing results on high-contrast random composites to the case of electromagnetic equations of Maxwell.

Throughout the project, the PhD student will acquire:

·       The ability to identify, investigate and analyse critically relevant information;

·       A deep knowledge in mathematical analysis of partial differential equations, spectral theory and probability;

·       Advanced written and spoken communication skills.

Supervisory meetings will be regular, to guarantee that work progresses as planned and the PhD thesis is completed on time, and to monitor and seek to improve the student’s overall academic development and general wellbeing.

The work will further be assessed annually through a progress review conducted by the project supervisors and one other faculty member.

To expand their mathematical knowledge beyond that offered by their specific project, in addition to research work, in the first two years, the student will attend a total of 100 hours of postgraduate lecture courses run by MAGIC (Mathematics Access Grid Instruction and Collaboration), which is jointly organised remotely by 20 UK Universities. Assignments to the MAGIC courses will be assessed by the course lecturers.

For their general intellectual and personal development, the student will also have numerous opportunities to participate in research seminars, workshops, and national or international conferences where they will be encouraged to present their work to various audiences.

At the School of Mathematics, the successful candidate will become member of a vibrant postgraduate research community, which hosts the SIAM-IMA Student Chapter at Cardiff University, formed by the Society for Industrial and Applied Mathematics and supported by the Institute of Mathematics and its Applications. They will be provided with a desk and a computer, and access to state-of-the-art facilities in the School’s purpose-built new home, Abacws, where the School moved in September 2021.

Cardiff School of Mathematics is committed to establishing an intellectual community based on diversity, fairness, and inclusion, and applications from women and other minorities are particularly encouraged.

How to apply:

Applicants should apply through the Cardiff University online application portal, SIMS. Applicants should select Doctor of Philosophy in Mathematics, with a start date of October 2022. In the research proposal section of your application, please specify the project title “Homogenisation of high-contrast random composites” and supervisor/Lead supervisor “Mikhail Cherdantsev ”. 

In the funding section, please specify that you are applying for advertised funding.

Academic criteria:

 A 1st or upper 2nd class UK Honours degree (or equivalent) and/or a Master’s degree is required in mathematics or a related subject. 

Please upload the following supporting documents on SIMS :

·        Curriculum vitae

·        A personal statement

·        Two completed references

·        Degree certificates and transcripts

DEADLINE FOR APPLICATIONS

 27 May 2022

Eligibility:

This studentship is available to home students.

Cardiff University is committed to supporting and promoting equality and diversity and to creating an inclusive environment for all. We welcome applications from all members of the global community irrespective of age, disability, sex, gender identity, gender reassignment, marital or civil partnership status, pregnancy or maternity, race, religion or belief and sexual orientation.

Assessment:

Applicants are reminded to submit all relevant documents (transcripts, supporting statement, etc) by the deadline. Due to the volume of applications received, incomplete applications will not be considered. 

Short-listed applicants will be invited to interview w/c shortly after the application deadline.

As part of the interview process, applicants will be asked to answer a series of questions by a panel of academics. 

Interviews are expected to take place remotely via Zoom/Teams on within one month after the application deadline.