University of Manchester

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)

This talk will concern the problem of inference when the posterior measure involves continuous models which require approximation before inference can be performed. Typically one cannot sample from the posterior distribution directly, but can at best only evaluate it, up to a normalizing constant. Therefore one must resort to computationally-intensive inference algorithms in order to construct estimators. These algorithms are typically of Monte Carlo type, and include for example Markov chain Monte Carlo, importance samplers, and sequential Monte Carlo samplers. The multilevel Monte Carlo method provides a way of optimally balancing discretization and sampling error on a hierarchy of approximation levels, such that cost is optimized. Recently this method has been applied to computationally intensive inference. This non-trivial task can be achieved in a variety of ways. This talk will review 3 primary strategies which have been successfully employed to achieve optimal (or canonical) convergence rates – in other words faster convergence than i.i.d. sampling at the finest discretization level. Some of the specific resulting algorithms, and applications, will also be presented.

Tendons are vital connective tissues that anchor muscle to bone to allow the transfer of forces to the skeleton. They exhibit highly non-linear viscoelastic mechanical behaviour that arises due to their complex, hierarchical microstructure, which consists of fibrous subunits made of the protein collagen. Collagen molecules aggregate to form fibrils with diameters of tens to hundreds of nanometres, which in turn assemble into larger fibres called fascicles with diameters of tens to hundreds of microns. In this talk, I will discuss the relationship between the three-dimensional organisation of the fibrils and fascicles and the macroscale mechanical behaviour of the tendon. In particular, I will show that very simple constitutive behaviour at the microscale can give rise to highly non-linear behaviour at the macroscale when combined with geometrical effects.

Blisters form when a thin surface layer of a solid body separates/delaminates from the underlying bulk material over a finite, bounded region. It is ubiquitous in a range of industrial applications, e.g. blister test is applied to assess the strength of adhesion between thin elastic films and their solid substrates, and during natural processes, such as formation and spreading of laccoliths or retinal detachment.

We study a special case of blistering, in which a thin elastic membrane is adhered to the substrate by a thin layer of viscous fluid. In this scenario, the expansion of the newly formed blister by fluid injection occurs via a displacement flow, which peels apart the adhered surfaces through a two-way interaction between flow and deformation. If the injected fluid is less viscous than the fluid already occupying the gap, patterns of short and stubby fingers form on the propagating fluid interface in a radial geometry. This process is regulated by membrane compliance, which if increased delays the onset of fingering to higher flow rates and reduces finger amplitude. We find that the morphological features of the fingers are selected in a simple way by the local geometry of the compliant cell. In contrast, the local geometry itself is determined from a complex fluid–solid interaction, particularly in the case of rectangular blisters. Furthermore, changes to the geometry of the channel cross-section in the latter case lead to a rich variety of possible interfacial patterns. Our experiments provide a link between studies of airway reopening, Saffman-Taylor fingering and printer’s instability.   

In my talk I will explain how to relate 1-dimensional representations of finite W-algebras with multiplicity free primitive ideals of universal enveloping algebras and representations of minimal dimension of the corresponding reduced enveloping algebras (Humphreys' conjecture). I will also mention some open problems in the field.

The vast majority of the World's documented meteorite specimens have been collected from Antarctica. This is due to Antarctica’s ice dynamics, which allows for the significant concentration of meteorites onto ice surfaces known as Meteorite Stranding Zones. However, meteorite collection data shows a significant anomaly exists: the proportion of iron-based meteorites are under-represented compared to those found in the rest of the World. Here I explain that englacial solar warming provides a plausible explanation for this shortfall: as meteorites are transported up towards the surface of the ice they become exposed to increasing amounts of solar radiation, meaning it is possible for meteorites with a high-enough thermal conductivity (such as iron) to reach a depth at which they melt their underlying ice and sink back downwards, offsetting the upwards transportation. An enticing consequence of this mechanism is that a sparse layer of  meteorites lies just beneath the surface of these Meteorite Stranding Zones...

Although Toeplitz matrices are often dense, matrix-vector products with Toeplitz matrices can be quickly performed via circulant embedding and the fast Fourier transform. This makes their solution by preconditioned Krylov subspace methods attractive. 

For a wide class of symmetric Toeplitz matrices, symmetric positive definite circulant preconditioners that cluster eigenvalues have been proposed. MINRES or the conjugate gradient method can be applied to these problems and descriptive convergence theory based on eigenvalues guarantees fast convergence. 

In contrast, although circulant preconditioners have been proposed for nonsymmetric Toeplitz systems, guarantees of fast convergence are generally only available for CG for the normal equations (CGNE). This is somewhat unsatisfactory because CGNE has certain drawbacks, including slow convergence and a larger condition number. In this talk we discuss a simple alternative symmetrization of nonsymmetric Toeplitz matrices, that allows us to use MINRES to solve the resulting linear system. We show how existing circulant preconditioners for nonsymmetric Toeplitz matrices can be straightforwardly adapted to this situation and give convergence estimates similar to those in the symmetric case.

Hazardous geophysical mass flows, such as snow avalanches, debris-flows and pyroclastic flows, often spontaneously develop large particle rich levees that channelize the flow and enhance their run-out. Measurements of the surface velocity near an advancing flow front have been made at the United States Geological Survey (USGS) debris-flow flume, where 10m^3 of water saturated sand and gravel are allowed to flow down an 80m chute onto a run-out pad. In the run-out phase the flow front is approximately invariant in shape and advances at almost constant speed. By tracking the motion of surface tracers and using a simple kinematic model, it was possible to infer bulk motion as incoming material is sheared towards the front, over-run and shouldered to the side. At the heart of the levee formation process is a subtle segregation-mobility feedback effect. Simple models for particle segregation and the depth-averaged motion of granular avalanches are described and one of the first attempts is made to couple these two types of models together. This process proves to be non-trivial, yielding considerable complexity as well as pathologies that require additional physics to be included.