The aim was to design and validate a bespoke computational method for three-dimensional simulation of elastic liquids, and use it to study phenomena which are hard to simulate. The broader field is CFD (computational fluid dynamics), but with a particular focus on classes of non-Newtonian fluid for which the governing equations are complicated and problematic to solve via the standard approaches deployed in commercial software. The method involved a

1

The main geometry under investigation was that of a rigid spherical particle sedimenting under gravity in an infinite cylindrical vertical pipe. In Stokes flow, the sphere falls vertically with no sideways drift, due to reversibility. In Newtonian flow with inertia, there is a sideways migration to a particular location (the Segré-Silberberg effect), whereas in viscoelastic flow there is a sideways drift towards the pipe wall, sometimes referred to as “negative lift.” The traditional benchmark problem sets the sphere’s radius (\(a\)) to be half that of the pipe (\(2a\)), with the sphere falling along the pipe’s axis, and the asymmetric problem (as shown) has the sphere initially offset by \(2\varepsilon a\).

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The mesh reconnection algorithm was based primarily on properties of the Delaunay triangulation, which for any given set of vertices has the property that the circumsphere of each tetrahedron contains no vertex. In practice, this involves considering local “flips” between the two configurations shown. On the left there are two tetrahedra sharing a common face, \(ABC\). The “2-3 flip” removes this face and adds a new edge \(DE\), forming instead three tetrahedra as shown on the right. The “3-2 flip” is the same but in reverse. Combining these with additional heuristic measures, the computational method was able to preserve mesh quality (in the sense of reducing the largest dihedral angle).

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Delaunay triangulation is optimal in 2D in the sense of maximizing the smallest angle in any triangle, but this property does not hold in 3D. Consequently the Delaunay mesh in 3D can include some tetrahedral shapes with “bad” aspect-ratios for flow simulation purposes, known as “slivers.” A sliver is a very flat tetrahedron without any particularly short edges. In the example shown, the four vertices are nearly coplanar. On the left, a sliver is shown from two perspectives; on the right it is shown within its circumsphere. The avoidance of slivers is considered a major goal of high-quality 3D mesh generation, and effective remedies involve assigning artificial weights to vertices, or adding/removing vertices in the neighbourhood of each sliver.

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To investigate dynamic viscoelastic particle drift, it is necessary to consider the build-up of polymer stretch within the flow (in the cross-sectional images, yellow is highly stretched, dark blue is unstretched, and between each image the sphere has fallen by one diameter). At early times the polymer becomes highly stretched in the narrow gap first, and the sphere initially drifts inwards towards the axis. As the material in the wider gap becomes highly stretched, there is a transition to an outward drift, and this dominates at later times. On the far side of the pipe there is a build up of stress due to shearing by the back flow through the wider gap between the sphere’s wake and the far wall.

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Using an argument which considers the radial pressure gradient due to normal stresses, in both the narrow gap and the wider gap on either side of the sphere, an analytical approximation for the ratio of outward drift velocity to vertical falling velocity was established. In the case of a FENE-CR fluid this is approximately \(\frac{2c}{3\pi K_S} \min{(1, \mathrm{We})}\), where \(K_S\) is the Stokes drag coefficient, \(c\) is the polymer concentration, and \(\mathrm{We}\) is the Weissenberg number for the flow. This approximation is linear in concentration (for the dilute case), and provides an accurate agreement to the numerical results (until \(c\) becomes large), constituting a qualitative and quantitative explanation of viscoelastic particle drift.

We welcomed three new group members this October. Elena Medvedeva joined us as a first year PhD student, supervised by Anastasia Kisil and co-supervised by Raphael Assier. Elena is working on the investigation of discrete diffraction problems, and understanding the links with their continuous counterparts. We also have two new PDRAs working with Will Parnell: Dr Daniel Sy-Ngoc Nguyen and Dr Marie Touboul. Daniel is working on the mathematical modelling of nanoreinforced syntactic foams, and Marie on novel resonant microstructures for elastodynamic metamaterials.

Congratulations to Dr Marianthi Moschou, who was formally awarded her PhD in 2021, and well done to Erik Garcia-Neefjes and Cheuk-Him Yeung, who have submitted their theses and are awaiting their viva examinations.

Raphael Assier was promoted to Reader in August: Many congratulations Raphael! This year, Raphael was a member of the jury selecting the finalists of the IMA Lighthill-Thwaites prize in Applied Mathematics. Raphael was the first winner of the biennial prize, in 2011.

Raphael Assier was promoted to Reader in August: Many congratulations Raphael! This year, Raphael was a member of the jury selecting the finalists of the IMA Lighthill-Thwaites prize in Applied Mathematics. Raphael was the first winner of the biennial prize, in 2011.

On the subject of judging panels, Anastasia Kisil was one of five judges for the Royal Society Insight Investment Science Book Prize in 2021. The panel was chaired by leading immunologist, presenter and writer, Professor Luke O’Neill FRS, and the winning book, announced in November, was Entangled Life: How Fungi Make Our Worlds, Change Our Minds and Shape Our Futures (Bodley Head), by biologist and writer Merlin Sheldrake. You can read about the Sheldrake's book here. |

Anastasia’s European Commission H2020 MCSA-RISE award,

There were several new publications from the group in 2021. Highlights include work on *Geometrical and mechanical characterisation of hollow thermoplastic microspheres for syntactic foam applications,* by Matthew Curd, Neil Morrison, Zeshan Yousaf and Will Parnell along with with Michael Smith from the University of Cambridge and Parmesh Gajjar from the Henry Royce Institute here at Manchester [1].

In May, work by Raphael Assier with Andrey Shanin from Moscow State University:*Vertex Green’s Functions of a Quarter-Plane: Links Between the Functional Equation, Additive Crossing and Lamé Functions*, appeared in The Quarterly Journal of Mechanics and Applied Mathematics [2]. More recently, in October a review article on the Wiener-Hopf technique by Anastasia Kisil with I. David Abrahams, Gennady Mishuris and Sergei V. Rogosin, was published in Proceedings of the Royal Society A [3]. More publications from 2021 can be found in our ongoing list of academic publications.

In May, work by Raphael Assier with Andrey Shanin from Moscow State University:

In June, Matt Nethercote spoke online at Days on Diffraction 2021. Days on Diffraction is an annual conference organised jointly by St. Petersburg Department of V.A., Steklov Institute of Mathematics, Russian Academy of Sciences, Euler International Mathematical Institute, and St. Petersburg State University. Matt’s talk was entitled

Raphael Assier gave a 3 lectures series:

As in 2020, our engagement outputs were almost entirely online in 2021. Despite this, we enjoyed a varied programme of activities. 166 secondary pupils took part in our online workshop 'The Great Maths Hunt', which looks at the hidden maths behind everyday situations. We loved hearing the pupils’ ideas! We created Slinky Science activities and experiment guides for Glasgow Science Festival, and Waves and Sound experiments for the University of Manchester’s online community festival. In October, PhD student Eleanor Russell created some fun online games for Science X, the Science and Engineering Faculty’s annual engagement event. Eleanor’s games explore the science and maths behind

Also in October we attended our first in-person event since the beginning of the pandemic. PhD students Tom White and Eleanor Russell, and PE Manager Naomi Curati attended Bradford Science Festival, and used everyday objects to explore the question “how big is sound?” More than 500 visitors joined us over two days, and feedback was extremely positive.

We completed our A to Z of the Maths of Waves and Materials blog series in August, with Tom White contributing our final entry,

Frontiers for Young Minds is hosting a special collection:

We continued our collaboration with local illustrator John Cooper to create another explainer video in 2021, this time about the concept of Neutral Inclusions.

We look forward to another busy year in 2022. You can find a list of upcoming events that group members are involved in here, and various external conferences taking place in 2022 here. Teaching fellow Marianthi Moschou is organising an interdisciplinary conference for STEM undergraduates:

[1] M.E. Curd, N.F. Morrison, M.J.A. Smith, P. Gajjar, Z. Yousaf, W.J. Parnell **(2021) ***Geometrical and mechanical characterisation of hollow thermoplastic microspheres for syntactic foam applications* **Composites Part B: Engineering** 223 108952

[2] R.C. Assier and A.V. Shanin**(2021) ***Vertex Green’s functions of a quarter-plan e. Links between the functional equation, additive crossing and Lamé functions.* **Q.J. Mech. Appl. Math.,** 74(3):251-295

[3] A.V. Kisil, I. David Abrahams, G. Mishuris and S.V. Rogosin**(2021) ***The Wiener–Hopf technique, its generalizations and applications: constructive and approximate methods*

**Proc. R. Soc**. A 477(2254):20210533**, **DOI: 10.1098/rspa.2021.0533

[4] W. J. Parnell, W. Rowley and N. Curati N (2021)*Tackling Noise Pollution With Slow Sound*. **Front. Young Minds**. 9:703592. DOI: 10.3389/frym.2021.703592

]]>[2] R.C. Assier and A.V. Shanin

[3] A.V. Kisil, I. David Abrahams, G. Mishuris and S.V. Rogosin

[4] W. J. Parnell, W. Rowley and N. Curati N (2021)

- Consider what is needed in terms of manipulation.
- Apply an appropriate transformation between a virtual and physical space.
- Measure how much this transformation stretches (or compresses) the virtual space using the Jacobian matrix. (See A-to-Z)
- Use the Jacobian matrix to determine the properties of the thermal metamaterial that achieves the desired manipulation.

These properties are often anisotropic and inhomogeneous. We approximate this behaviour with layered, composite materials whilst also considering interfacial effects.

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Illustration of the transformation process used to design metamaterials. Here we show an example of a ground cloak transformation where a virtual space with coordinates \(\mathbf{x}’=[x’,y’]\) is compressed upwards to form the physical space with coordinates \(\mathbf{x}=[x,y]\). This mapping from \(\mathbf{x}'\) to \(\mathbf{x}\) creates a triangular void in the physical space. We measure how much this transformation compresses each point in the virtual space using the Jacobian, denoted by \(\mathbf{F}\). From this we can determine the necessary properties for the metamaterial regions, for example, in my PhD we focus on steady-state thermal conduction and so we only need to engineer the thermal conductivity, which is given by

\begin{equation}

\mathbf{k} = \dfrac{\mathbf{F}k'\mathbf{F}^T}{\det\mathbf{F}}\tag{1} ,

\end{equation}

where \(k’\) is the conductivity in the virtual domain. When the conductivity in the transformed, metamaterial regions satisfy (1), any object placed within the void is undetectable from the external field – making it invisible! The effects of a traditional ground cloak are simulated in image 2. In my PhD I use a modified version of this mapping to utilize it in a new, heat spreading context. A simulation of this application is provided in the image 3.

\begin{equation}

\mathbf{k} = \dfrac{\mathbf{F}k'\mathbf{F}^T}{\det\mathbf{F}}\tag{1} ,

\end{equation}

where \(k’\) is the conductivity in the virtual domain. When the conductivity in the transformed, metamaterial regions satisfy (1), any object placed within the void is undetectable from the external field – making it invisible! The effects of a traditional ground cloak are simulated in image 2. In my PhD I use a modified version of this mapping to utilize it in a new, heat spreading context. A simulation of this application is provided in the image 3.

2

Simulation of a traditional ground cloak application:

a) The temperature field for a given material and set of boundary conditions. Black lines represent isotherms. b) Same material and boundary conditions, but with part of the domain removed. This triangular void causes disturbances, referred to as perturbations, to the temperature field. We can use these perturbations to detect the apple. c) Same material and boundary conditions, but with triangular void surrounded by an anisotropic ground cloak. The unique properties inside the ground cloak counteract the perturbations caused by the void. As a result, we can no longer detect the apple – making it invisible! |

3

Simulation of a modified ground cloak mapping being used in a new, heat spreading context.

a) The temperature field through a funnel-shaped design when a natural, isotropic material is used. White lines represent isotherms and show that a uniform temperature is not naturally achieved across the top surface when the funnel is forced by a heat source at its base. b) The temperature field through a heat spreader designed using a modified ground cloak. The design has three components: an upper isotropic component that is unaffected by the mapping; and two lower metamaterial layers that counteract the perturbations caused by the funnel-shaped geometry. The unique properties in the metamaterial layers guide the flow of heat through the spreader in spatially uniform manner, leading to uniform temperature across the top surface, as desired. |

Ground cloaks are a particularly nice example of thermal metamaterials as their properties are anisotropic, but not homogeneous. Therefore we can accurately approximate them with a laminated design, which can be seen in image 4.

4

The realisation process for the metamaterial in image 3(b), which can be approximated with a rotated laminate.

a) Illustration of the composite design consisting of alternating layers aligned with the principal axes of the system. The thickness and conductivity of each layer are determined through effective medium theory. b) Simulation of this design using 5 bilayers. The temperature field here approximates that of the metamaterial simulation in image 3(b). This design better represents the true metamaterial as we increase the number of bilayers. A sufficiently large number of layers would eliminate temperature variation across the top surface completely. |

Although not shown here, we have extended this model (and the other models in our research) to include interfacial effects. These effects can dominate in multi-layered designs such as laminates.

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Although PhD students obviously spend most of their time researching, this is not the only thing we do! A PhD can involve many responsibilities, for example, during my PhD I have spent time:

- Being a teaching assistant for undergraduate courses.
- Attending conferences to present work and network with other researchers in my field.
- Creating online material for outreach and widening participation – like writing blog posts and A-to-Z entries for this website!
- Helping at science festivals to promote the research focuses of the Mathematics of Waves and Materials group.

*Composite material*. A composite material is a material that is produced from two or more constituents, which usually have quite distinct properties. This usually gives rise to a material that has enhanced properties, e.g. unusually stiff or conductive. Composites typically have a microstructure which can be as small as hundreds of nanometres.*Wave propagation*. By “waves” here we mean any wave that propagates through a composite material, so it could be an acoustic, elastic or electromagnetic wave. Although the physics of these waves are quite different, the mathematics governing how we describe them is remarkably similar and so it means that methods developed for one type of wave can be used to describe another type with remarkable ease.*Homogenization technique*. Mathematically, homogenization is the study of differential equations with rapidly oscillating coefficients. This is important because given that composites consist of at least two different constituents, the properties of the material change with space and often over a very small length scale as noted above. Mathematically, this manifests itself as a differential equation that governs some physical property (e.g. acoustic pressure) with coefficients that vary rapidly in space, corresponding to e.g. density or bulk modulus.

The homogenisation technique makes sense of what the waves “see” or “feel” when they propagate through the composite material in question, normally working best in the case when the wave’s wavelength is much longer than the variation in microstructure. The techniques allow one to determine the “effective material properties” of the composite medium. I focussed mainly on elastic properties in my PhD.

Hover over the images to read the captions, or see below for plain text version.

1

Illustrating the homogenization process. Typically when dealing with materials on an everyday level we assume the “continuum hypothesis” so that the atomic and molecular scales of the medium are smeared out (as illustrated on the right of this image) and we can therefore usefully assign macroscopic properties such as density, viscosity, stiffness, etc. to the medium. Homogenization goes a step further and smears out the microstructure of a composite medium at a larger length scale. Here we illustrate a particulate composite, where inclusions are distributed inside a host matrix medium. The microstructural length scale is “q” and homogenization techniques work well when the macroscopic length scale is much bigger than the microstructural one. The inclusions could be introduced to provide improved conductivity or stiffness whilst reducing weight for example. Homogenization is useful because it means that we can assign a single effective property to the material in question, instead of one that varies in space.

2

Visualisation of the microstructure of a specific type of composite material using X-ray Computed Tomography. These are cross-sections (a-c) and a 3d reconstruction (d) of a syntactic foam. This is a material consisting of thin spherical shells (the circles/spheres here) distributed inside a matrix, often a polymer. These materials are lightweight and stiff and frequently assist in absorbing sound.

3

My PhD focussed mainly on how *elastic waves *propagate inside *fibre-reinforced* composites where the fibres are very long. Here their length is along the x3 axis. In such fibrous materials the microstructure (the fibres) can be considered to be characterised by a repeating periodic cell in the x1-x2 plane. The effective elastic properties of the material can then determined by applying homogenisation techniques. (At least) three different types of elastic waves can be considered to propagate – here illustrated as SH, SV and P waves.

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When fibres are placed on a hexagonal lattice, as illustrated here, the response when waves propagate in the x1-x2 plane is *isotropic*, so that the wave speed is independent of direction of propagation. Here we illustrate this by determining the associated effective “antiplane” shear modulus of the material as a function of the size (radius r) of the fibre. The fibre is ten times as soft as the matrix, so that the more fibre added, the softer the composite becomes (the effective shear modulus decreases). I considered a new approach to the implementation of the so-called *asymptotic homogenization scheme, *which is plotted here in orange. It can be seen that the results agree well with existing methods and energy bounds.

5

I also derived a completely new method, known as the *integral equation method* of homogenisation. This is an extremely efficient new technique, which can be used straightforwardly to model composites where the inclusions have rather arbitrary cross-section and also to geometries where fibres are placed on unusual lattices. Here we illustrate results when fibres are located on a *rectangular lattice. *This induces *anisotropy*, so that the elastic waves propagate with different speeds in different directions. This is illustrated here by plotting the associated effective shear modulus when waves propagate in the x1 and x2 directions. Our new integral equation method is plotted in comparison to the asymptotic method here, showing good agreement in both cases.

Several MWM group members have delivered online talks since the last update. In November, Matt Nethercote contributed to the ICMS virtual seminar series on Waves in Complex Continua, with his talk, “Edge Diffraction of Acoustic Waves by Periodic Composite Metamaterials: The Hollow Wedge”. You can find a recording of Matt’s seminar with all the other "Wavinars" here. You will also find a talk given in the summer by PhD student Erik Garcia-Neefjes, "Thermo-Visco-Elastic effects in Wave Propagation".

Anastasia Kisil spoke in the RCMM Wave Scattering and Solid Mechanics online seminar series in December. Her talk, entitled “Edges and Point Scatterers: a simple model for a metamaterial with edges” covered collaborative work with Raphael Assier and Matt Nethercote, and is available to view here. Also in December, Will Parnell spoke about “Elastostatic cloaking, low frequency elastic wave transparency and neutral inclusions” at the ICMS Continuum Mechanics virtual seminar series in December, and online at the Laboratoire de Mécanique et d’Acoustique (Marseille) in October. You can watch the ICMS talk here.

In January, Raphael Assier gave a 3 lecture series to an American Mathematical Society (AMS) Mathematics Research Community (MRC) entitled “An applied perspective on multidimensional complex analysis”, hosted by Harvard University.

Click here to find a list of virtual seminar series.

In January, Raphael Assier gave a 3 lecture series to an American Mathematical Society (AMS) Mathematics Research Community (MRC) entitled “An applied perspective on multidimensional complex analysis”, hosted by Harvard University.

Click here to find a list of virtual seminar series.

Papers from academic year 2019-2020 were discussed in October’s update, but some new publications have been released since then. Most recently in January, several examples from Raphael Assier appeared, including "A Surprising Observation in the Quarter-Plane Diffraction Problem", with I. D. Abrahams in SIAM J. Appl. Math, and "Analytical continuation of two-dimensional wave fields" with Andrey Shanin in Proc. R. Soc. A. An ongoing list can be found here.

Anastasia Kisil was part of a multi-institutional group that were successful in securing funding from European Commission H2020 MCSA-RISE for "EffectFact: Effective Factorisation techniques for matrix-functions". This is a joint venture between several European Institutions in the area of factorisation techniques, Wiener-Hopf and Riemann-Hilbert problems and related numerical techniques, beginning in September 2021. Anastasia, along with Anna Zemlyanova (Kansas State University), Gennady Mishuris (Aberystwyth University) and Xun Huang (Peking University) made a successful application to the BIRS Scientific Board to host a conference in the Institute for Advanced Study in Mathematics (IASM) in Hangzhou, China. "Cross-Fertilisation of ideas from the Riemann–Hilbert Technique and the Wiener–Hopf Method" will be held in September 2022.

William Parnell’s EPSRC award for “The Princess and the Pea: Mathematical Design of Neutral Inclusions and their Fabrication” began this month. This project will explore *neutral inclusions (NIs)*: material inclusions with coatings designed to ensure that stress fields in the host material are unperturbed upon loading, as if the inclusion were absent. NIs have the potential to reduce material failures due to stress concentration, and enable lighter, stronger materials.

Online Engagement. In-person events are likely to be further postponed for the several months, and we are continuing to find ways to engage online. Our Virtual Postcards offer visual snapshots of our research themes, and our ongoing series, the A-Z of the Mathematics of Waves and Materials, takes an alphabetical journey through some of the key concepts behind our work. This week we reached the letter I, with a piece on inclusions, written by Neil Morrison, so there are plenty more updates to come! | |

PhD students Eleanor Russell and Tom White recently featured in the New Scientist Jobs "A Day in the Life" series, where they describe a typical day as a Mathematics PhD student, and discuss the best and worst parts of PhD life. Students interested in a career in Applied Mathematics can explore our Meet a Mathematician series, or visit our YouTube channel.

March brings British Science week, and we are looking forward to engaging with local high school pupils with our virtual workshop “The Great Maths Hunt”, which will take a look at everyday life and challenge students to answer: “Where is the maths?” Students will uncover the hidden mathematical research behind everyday things, and find out who mathematicians work with, how they solve problems, and where a career in maths may take them.

March brings British Science week, and we are looking forward to engaging with local high school pupils with our virtual workshop “The Great Maths Hunt”, which will take a look at everyday life and challenge students to answer: “Where is the maths?” Students will uncover the hidden mathematical research behind everyday things, and find out who mathematicians work with, how they solve problems, and where a career in maths may take them.

From March, we look forward to working with the Metamaterials Network’s Outreach and Education forum, led by Anja Roeding (University of Exeter), and Raphael Assier. The group will coordinate outreach activities in the field of metamaterials across several UK institutions.

Finally, during this extended International Year of Sound, the journal Frontiers for Young minds is preparing a special collection entitled “A World of Sound”. Frontiers for Young Minds is an open access scientific journal written for and reviewed by children. Naomi Curati is one of the collection editors, along with other members of the UK Acoustics Network. We look forward to reading the collection!

]]>Finally, during this extended International Year of Sound, the journal Frontiers for Young minds is preparing a special collection entitled “A World of Sound”. Frontiers for Young Minds is an open access scientific journal written for and reviewed by children. Naomi Curati is one of the collection editors, along with other members of the UK Acoustics Network. We look forward to reading the collection!