Caustics are places where the light intensity diverges, and where the wave front has a singularity. We use a self-similar description to derive the detailed spatial structure of a cusp singularity, from where caustic lines originate. We also study singularities of higher order, which have their own, uniquely three-dimensional structure. We use this insight to study shock formation in classical compressible Euler dynamics. The spatial structure of these shocks is that of a caustic, and is described by the same similarity equation.

# Industrial and Applied Mathematics Seminar

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

J. M. Foster 1 , N. E. Courtier 2 , S. E. J. O’Kane 3 , J. M. Cave 3 , R. Niemann 4 , N. Phung 5 , A. Abate 5 , P. J. Cameron 4 , A. B. Walker 3 & G. Richardson 2 .

1 School of Mathematics & Physics, University of Portsmouth, UK. {jamie.michael.foster@gmail.com}

2 School of Mathematics, University of Southampton, UK.

3 School of Physics, University of Bath, UK.

4 School of Chemistry, University of Bath, UK.

5 Helmholtz-Zentrum Berlin, Germany.

Metal halide perovskite has emerged as a highly promising photovoltaic material. Perovskite-based solar cells now exhibit power conversion efficiencies exceeding 22%; higher than that of market-leading multi-crystalline silicon, and comparable to the Shockley-Queisser limit of around 33% (the maximum obtainable efficiency for a single junction solar cell). In addition to fast electronic phenomena, occurring on timescales of nanoseconds, they also exhibit much slower dynamics on the timescales of several seconds and up to a day. One well-documented example of this is the ‘anomalous’ hysteresis observed in current-voltage scans where the applied voltage is varied whilst the output current is measured. There is now a consensus that this is caused by the motion of ions in the perovskite material affecting the internal electric field and in turn the electronic transport.

We will discuss the formulation of a drift-diffusion model for the coupled electronic and ionic transport in a perovskite solar cell as well as its systematic simplification via the method of matched asymptotic expansions. We will use the resulting reduced model to give a cogent explanation for some experimental observations including, (i) the apparent disappearance of current-voltage hysteresis for certain device architectures, and (ii) the slow fading of performance under illumination during the day and subsequent recovery in the dark overnight. Finally, we suggest ways in which materials and geometry can be chosen to reduce charge carrier recombination and improve device performance.

Variational methods have been extensively used in the past decades to rigorously derive nonlinear models in the description of thin elastic films. In this context, natural growth or differential swelling-shrinking lead to models where an elastic body aims at reaching a space-dependent metric. We will describe the effect of such, generically incompatible, prestrain metrics on the singular limits' bidimensional models. We will discuss metrics that vary across the specimen in both the midplate and the thin (transversal) directions. We will also cover the case of the oscillatory prestrain, exhibit its relation to the non-oscillatory case via identifying the effective metrics, and discuss the role of the Riemann curvature tensor in the limiting models.

Boundary layers control the transport of momentum, heat, solutes and other quantities between walls and the bulk of a flow. The Prandtl-Blasius boundary layer was the first quantitative example of a flow profile near a wall and could be derived by an asymptotic expansion of the Navier-Stokes equation. For higher flow speeds we have scaling arguments and models, but no derivation from the Navier-Stokes equation. The analysis of exact coherent structures in plane Couette flow reveals ingredients of such a more rigorous description of boundary layers. I will describe how exact coherent structures can be scaled to obtain self-similar structures on ever smaller scales as the Reynolds number increases.

A quasilinear approximation allows to combine the structures self-consistently to form boundary layers. Going beyond the quasilinear approximation will then open up new approaches for controlling and manipulating boundary layers.

Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of partial differential equations, harmonic, stochastic and statistical analysis, and optimisation. Starting with a discussion on the intrinsic structure of images and their mathematical representation, in this talk we will learn about variational models for image analysis and their connection to partial differential equations, and go all the way to the challenges of their mathematical analysis as well as the hurdles for solving these - typically non-smooth - models computationally. The talk is furnished with applications of the introduced models to image de-noising, motion estimation and segmentation, as well as their use in biomedical image reconstruction such as it appears in magnetic resonance imaging.

Transformation theory has long been known to be a mechanism for the design of metamaterials. It gives rise to the required properties of the material in order to direct waves in the manner desired. This talk will focus on the mathematical theory underpinning the design of acoustic and elastodynamic metamaterials based on transformation theory and aspects of the experimental confirmation of these designs. In the acoustics context it is well-known that the governing equations are transformation invariant and therefore a whole range of microstructural options are available for design, although designing materials that can harness incoming acoustic energy in air is difficult due to the usual sharp impedance contrast between air and the metamaterial in question. In the elastodynamic context matters become even worse in the sense that the governing equations are not transformation invariant and therefore we generally require a whole new class of materials.

In the acoustics context we will describe a new microstructure that consists of rigid rods that is (i) closely impedance matched to air and (ii) slows down sound in air. This is shown to be useful in a number of configurations and in particular it can be employed to half the resonant frequency of the standard quarter-wavelength resonator (or alternatively it can half the size of the resonator for a specified resonant frequency) [1].

In the elastodynamics context we will show that although the equations are not transformation invariant one can employ the theory of waves in pre-stressed hyperelastic materials in order to create natural elastodynamic metamaterials whose inhomogeneous anisotropic material properties are generated naturally by an appropriate pre-stress. In particular it is shown that a certain class of hyperelastic materials exhibit this so-called “invariance property” permitting the creation of e.g. hyperelastic cloaks [2,3] and invariant metamaterials. This has significant consequences for the design of e.g. phononic media: it is a well-known and frequently exploited fact that pre-stress and large deformation of hyperelastic materials modifies the linear elastic wave speed in the deformed medium. In the context of periodic materials this renders materials whose dynamic properties are “tunable” under pre-stress and in particular this permits tunable band gaps in periodic media [4]. However the invariant hyperelastic materials described above can be employed in order to design a class of phononic media whose band-gaps are invariant to deformation [5]. We also describe the concept of an elastodynamic ground cloak created via pre-stress [6].

[1] Rowley, W.D., Parnell, W.J., Abrahams, I.D., Voisey, S.R. and Etaix, N. (2018) “Deepening subwavelength acoustic resonance via metamaterials with universal broadband elliptical microstructure”. Applied Physics Letters 112, 251902.

[2] Parnell, W.J. (2012) “Nonlinear pre-stress for cloaking from antiplane elastic waves”. Proc Roy Soc A 468 (2138) 563-580.

[3] Norris, A.N. and Parnell, W.J. (2012) “Hyperelastic cloaking theory: transformation elasticity with pre-stressed solids”. Proc Roy Soc A 468 (2146) 2881-2903 [4] Bertoldi, K. and Boyce, M.C. (2008) “Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures”. Phys Rev B 77, 052105.

[5] Zhang, P. and Parnell, W.J. (2017) “Soft phononic crystals with deformation-independent band gaps” Proc Roy Soc A 473, 20160865.

[6] Zhang, P. and Parnell, W.J. (2018) “Hyperelastic antiplane ground cloaking” J Acoust Soc America 143 (5)