Theory and Application of Metamaterials - 2025 entry

In this module, the analytical and numerical modelling of metamaterials will be investigated. You will gain deep mathematical insight into the mathematical modelling of periodic and quasi-periodic systems through solving the wave equation in electromagnetism, acoustics, and elasticity using  complex analysis, Bloch theory, and  matrix methods. You will gain an appreciation of  the underpinning theory of numerical methods including finite element and finite difference simulations.  Following on from this, you will become proficient in using the powerful  transfer matrix method, and gain experience using commercial software through a series of lab-based modelling sessions.  

To give you the tools required to solve the wave equation in a wide range of physical contexts, applying this to design composite materials that can fulfil a given design specification.  The aim is to give you the ability to analyse the problem both mathematically and numerically using commercial finite element software, both of which are extremely valuable for future employment.

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge

Discipline Specific Skills and Knowledge

Personal and Key Transferable / Employment Skills and Knowledge

• Introduction to metamaterials: manipulating waves in electromagnetism, acoustics, and elasticity. Examples of invisibility cloaking, advanced audio performance, and earthquake protection (1hr).
• The ubiquity of waves: wave equations across physics, the importance of constitutive relations in wave equations, and their relation to microscopic structure (2hrs).
• Analytical tools: fundamental solution of the wave equation:  Complex analysis, Green’s functions, and approximations (the Helmholtz equation) (3hrs).
• Waves in periodic structures: Lattices, the reciprocal lattice and the Brillouin Zone; the importance of symmetry and Bloch’s theorem (3hrs).
• Dispersive waves: Analysis of group velocity and computing band structures: The Plane Wave Expansion Method (3hrs).
• Modelling acoustic metamaterials using the Transfer Matrix Method (TMM): Introduction to the TMM, wave propagation in closed acoustic systems, the Helmholtz Resonator, thermoviscous acoustics, and inverse design of acoustic metamaterials (4hrs).
• The finite difference and finite element methods: introduction and examples (2hr) 
• Application to real world problems: case study with example applications of above techniques to a design problem (run by commercial partner COMSOL Multiphysics) (4hr) + (LAB SESSION I)

Deferral – if you have been deferred for any assessment you will be expected to submit the relevant assessment. The mark given for a re-assessment taken as a result of deferral will not be capped and will be treated as it would be if it were your first attempt at the assessment.

Referral – if you have failed the module overall (i.e. a final overall module mark of less than 50%) you will be expected to submit the relevant assessment. The mark given for a re-assessment taken as a result of referral will be capped at 50%.

Basic reading:

• Wave Propagation in Periodic Structures (Brillouin)
• Electromagnetism (Griffiths)
• Wave Propagation in Elastic Solids (Achenbach)
• The Physics of Vibrations and Waves (Pain)
• Acoustic Waves in Periodic Structures, Metamaterials, and Porous Media (Jiménez, Umnova & Groby)

Web-based and electronic resources:

• ELE

Reading list for this module: