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My PhD in 5 Pictures: NF Morrison

1/24/2022

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NF Morrison, Teaching Fellow

My thesis title: Computations of flows of elastic liquids

What does this mean?

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 Lagrangian finite-element mesh.  Here "finite-element" means that the fluid domain is partitioned into a mesh consisting of a large number of separate “elements” (tetrahedra in this case), and “Lagrangian” means that the vertices of the mesh are considered as material points within the fluid itself, so that the elements deform along with the fluid as it flows (as opposed to an Eulerian mesh with elements remaining fixed in the laboratory frame of reference).  Because a Lagrangian mesh becomes increasingly deformed as the flow evolves, a bespoke adaptive mesh reconnection and improvement algorithm was integrated within the method.  Validation was via comparison to established “benchmark” problems in the 2D or axisymmetric case.  The fully 3D method was then applied to investigate asymmetric phenomena which had not been fully explained previously, for example the viscoelastic drift of sedimenting particles in pipe flow.
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 \(2\varepsilon a\).
2
​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).
3
​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.
4
​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.
5
​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.
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My PhD in 5 Pictures: Eleanor Russell

12/14/2021

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In this series our group members pick five images that each illustrate an aspect of their PhD.

Eleanor Russell, 3rd Year PhD student in Applied Mathematics.

My thesis title: Thermal metamaterials and heat spreaders

What does this mean?
  
Heat spreaders: Our funnel-shaped heat spreader designs act as point-to-plane heat source converters. The material properties of each spreader guide the flow of heat in a spatially uniform manner such that, when forced by a small thermal source at the base, a uniform temperature is achieved across the larger top surface. Anisotropy (directional dependence) and inhomogeneity (spatial dependence) are typically needed to achieve these desired effects. We design and employ thermal metamaterials together with other techniques to obtain the necessary anisotropic and inhomogeneous properties.
  
Metamaterials: Meta is a Greek word meaning beyond, and so the concept of metamaterials refers to materials that are beyond that of the natural world so to speak. The unique, engineered properties of metamaterials means that we can manipulate and control physical fields (such as electromagnetic, acoustic and elastic wave fields) in new and exciting ways - the results of which are unachievable with natural materials alone! In my PhD we focus on the manipulation of thermal fields with thermal metamaterials, the properties of which are determined through transformation theory.

Transformation theory:  In the context of my research, transformation theory is based on the heat equation being form invariant after a spatial transformation. To apply the theory for a given problem we:
  • 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.
1
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.
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.
5
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.
In the image below there is a photo of me helping at Bradford Science Festival earlier this year. The main aim of this weekend was to get primary school children excited about maths and science by explaining some simple concepts about sound. The next image shows a simple maze game that I designed on Scratch to help demonstrate the effects of different thermal metamaterials. The final image shows a photo I took of the Utah State Captiol when I attended a conference in Salt Lake City back in 2019.
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My PhD in 5 Pictures: Will Parnell

9/21/2021

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In this series our group members pick five images that each illustrate an aspect of their PhD.

​Will Parnell, Professor of Applied Mathematics. PhD in Applied Mathematics, University of Manchester, 2004

My thesis title: Homogenization techniques for wave propagation in composite materials


What does this mean?
  • 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.
4
​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.
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    The Mathematics of Waves and Materials group are are a research group in the Department of Mathematics at the University of Manchester.  We work on the theoretical, numerical and experimental aspects of both materials and waves. See our research page for more information and details.

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