Applied and Computational Mathematics Research Group
Our research in the area of applied and computational mathematics is informed by problems at the interface with physical sciences, biological sciences and engineering and there are many productive interdisciplinary collaborations within Cardiff University and further afield.
Our national and international collaborators include research groups at Imperial, Oxford, Cambridge, Warwick, Swansea, Canberra, Curtin (Perth), Perth, Delft, Northwestern, Ljubljana and Montréal.
The group hosts the Cardiff University Student Chapter of the Society for Industrial and Applied Mathematics and the Institute of Mathematics and its Applications (SIAM-IMA Student Chapter) which encompasses postgraduate students and faculty members from across the University who are interested in mathematics or scientific computing and their real-world applications.
The main areas of research within the current group are:
Theoretical and applied fluid mechanics
Free-surface flows, dynamics of liquid films and jets, hydrodynamic stability theory, laminar-turbulent transition mechanisms, boundary-layer and wake flow instabilities, boundary layer flow control, viscoelastic flows, bubble dynamics, constitutive modelling of polymeric liquids.
Mathematics and mechanics of solids
Nonlinear elasticity, contact problems, limit states analysis, constitutive modelling in materials science.
The development of mathematical, computational and statistical methods to address biological and medical problems.
Inverse problems in materials modelling, homogenisation and the mechanics of composites.
Numerical analysis and scientific computing
The development of algorithms for the numerical solution of partial differential applications.
Head of Group
Honorary Distinguished Professor
- +44 (0)29 2087 4827
Senior Lecturer in Applied Mathematics
- +44 (0)29 2087 5570
Lecturer in Applied Mathematics
- +44 (0)29 2087 5116
All seminars are held at 14:10 in Room M/2.06, Senghennydd Road, Cardiff unless stated otherwise.
Programme organiser and contact: Dr Usama Kadri
3 October 2017
Dr. Davide Crivelli
Detecting damage with waves: an overview of structural health monitoring research in Engineering
Acoustic emission is a promising technique for monitoring damage on structures. It provides a non-invasive, real-time and passive system for detecting cracks and defects in a range of structures as they happen. Applications range from civil structures, such as bridges and railways; rotating machinery; aerospace, such as impact and damage detection in advanced composite structures.
We face multiple challenges however when the amount of data from an Acoustic Emission system has to be interpreted in a reliable way. Wave propagation, source characterization, interaction with damage and sensor characteristics make traditional signal interpretation and source localization a challenging task.
This talk will give a general overview of acoustic emission for structural health monitoring. It will present a series of case studies, describing how Cardiff research has tackled the main challenges of signal localization and interpretation, and how the research has helped bringing the technology closer to commercial applications. It will also present a collaboration with Maths stemmed from acoustic emission research and applied to low frequency acoustic gravity waves in the search of MH370.
10 October 2017
Dr. Thomas Woolley
Patterns, cellular movement and brain tumors
I present three pieces of work that illustrate the power of mathematics as a tool for understanding biology. Although the applications appear to be disparate the underlying mathematics is very similar.
I begin by looking at theoretical and experimental pattern formation, with emphasis on whisker formation in mice. Here, reaction-diffusion equations are used to provide insights into how the wavelength of the whiskers are controlled.
Next, I consider the phenomena of blebbing cells. Initially, I use a diffusion equation to understand the motion of muscle stem cells and illustrate how old cells fundamentally move differently to old cells. This is then extended to include solid mechanics, which allows us to link the structural properties of the cell to their motion.
Finally, reaction-diffusion equations are used to understand the formation of brain tumours. Critically, the cells move at different speeds in white and grey matter, including this information can lead to very different migration patterns of the tumours.
18 October 2017
Stability of Oscillatory Rotating Disk Boundary Layers
The rotating disk boundary layer has long been considered as providing an archetypal model for studying the stability of three-dimensional boundary-layer flows, and the crossflow inflexion point instability mechanism is common to both the rotating disk boundary layer and the flow over a swept wing. Thus the investigation of strategies for controlling the behaviour of disturbances that develop in the rotating disk flow may prove to be helpful for the identification and assessment of aerodynamical technologies that have the potential to maintain laminar flow over swept wings.
We will consider the changes in the stability behaviour that arise when the rotating disk base-flow configuration is altered by imposing a periodic modulation in the rotation rate of the disk surface. Thomas et. al. [Proc. R. Soc. A (2011) 467:2643-2662] have previously demonstrated that Tollmien-Schlichting waves can be stabilised when a similarly induced Stokes layer is conjoined to a plane channel flow.
Current work encompasses three distinct investigatory approaches. Linearised direct numerical simulations have been conducted, using the vorticity-based methods that were first adopted by Davies & Carpenter [J. Comput. Phys (2001) 172:119-165]. These simulations are complemented by a local in time linear stability analysis, that is made possible by imposing an artificial frozen base-flow approximation. This localised analysis is deployed together with a more exact global treatment based upon Floquet theory, which avoids the need for any simplification of the temporal dependency of the base-flow.
24 October 2017
Václav Klika, PhD
On modelling self-organisation in real systems
Nowadays models for self-organisation are being used in systems with a great degree of complexity and across disciplines. We show that the widely used Turing model is sensitive to inputs, type of domain growth, but also to the precision of model formulation itself. Hence a great care is needed when applying Turing's model for self-organisation to real problems. For this purpose we consider derivation of evolution equations within non-equilibrium thermodynamic to identify physically relevant formulations. Only then we subject these models to a detailed mathematical analysis. We offer possible extensions of the concept of self-organisation to more general situations and discuss its physical interpretation.
The essence and importance of these ideas is illustrated on the reaction-diffusion-advection system, where we indicate that such a system should be preferred from both physical and mathematical viewpoint. Further we point to the indispensable role of physical viewpoint during relevant model formulations. Using the non-equilibrium thermodynamic framework physically consistent extensions of Turing model are revealed as well as functional constraints for present parameters.
7 November 2017
Richard Hewitt, PhD
Localised streaks in a Blasius boundary layer
Streaks are common feature of perturbed boundary-layer flows. They play a central role in transient growth mechanisms and are a building block of exact coherent structures. Most theoretical work has focused on streaks that are periodic in the spanwise direction, but in this work we consider a single spatially localised streak embedded into a Blasius boundary layer. For small streak amplitudes, we show the perturbation can be described in terms of a set of eigenmodes that correspond to an isolated streak/roll structure. These modes are new, and arise from a bi-global eigenvalue calculation; they decay algebraically downstream and may be viewed as the natural three-dimensional extension of the two-dimensional Libby \& Fox (1963, JFM vol. 17) solutions. Despite their bi-global nature, we show that a subset of these eigenmodes is fundamentally related to both the Libby & Fox solutions, and those presented by Luchini (1996, JFM vol. 327), as derived for (spanwise) periodic disturbances at small spanwise wavenumber. This surprising connection is made by an analysis of the far-field decay of the bi-global state. We also address the downstream development of nonlinear streaks, confirming that the aforementioned eigenmodes are recovered as the streak/roll decays downstream. Some comparisons are made with available experimental data.
14 November 2017
John A. Mackenzie, PhD
An Adaptive Moving Mesh Method for Geometric Evolutions Laws and Bulk-Surface PDEs: Application to a Model of Cell Migration and Chemotaxis
In this talk I will consider the adaptive numerical solution of curve-shortening flow with a driving force. An adaptive moving mesh approach is used to distribute the mesh points in the tangential direction. This ensures that the resulting meshes evolve smoothly in time and are well adjusted to resolve areas of high curvature. Experiments will be presented to highlight the improvement in accuracy obtained using the new method in comparison with uniform arc-length mesh distributions. We will also discuss the use of the evolving adaptive curve mesh in the adaptive generation of bulk meshes for the solution of bulk-surface PDEs in time dependent domains.
The main motivation for developing these computational tools is the modelling of single cell migration and chemotaxis. Chemoattractant gradients are usually considered in terms of sources and sinks that are independent of the chemotactic cell. However, recent interest has focused on “self-generated” gradients, in which cell populations create their own local gradients as they move. Here we consider the interplay between chemoattractants and single cells. To achieve this we model the breakdown of extracellular attractants by membrane-bound enzymes. Model equations are parameterised using published estimates from Dictyostelium cells chemotaxing towards cyclic AMP. We find that individual cells can substantially modulate their local attractant field under physiologically appropriate conditions of attractant and enzymes. This means the attractant concentration perceived by receptors can be a small fraction of the ambient concentration. This allows efficient chemotaxis in chemoattractant concentrations that would be saturating without local breakdown.
21 November 2017
Diffraction of hydroelastic waves by a circular cylinder
Linear problem of wave diffraction is studied for a circular cylinder mounted at the sea bed and piercing the fluid surface which is covered by ice plate of infinite extent. The water depth is constant. The ice plate is modeled by a thin elastic plate of constant thickness clamped to the surface of the cylinder. One-dimensional incident hydroelastic wave of small amplitude propagates towards the cylinder. and is diffracted on the cylinder. Deflection of the ice plate and the bending stresses in it are determined by two methods: (a) using the integral Weber transform in radial direction, (b) using the vertical modes for the fluid of constant depth with the rigid bottom and elastic upper boundary. The solution by the second method is straightforward but we cannot prove that the solution is complete because the properties of the vertical modes are not known yet. The solution by the Weber transform is more complicated but this solution is unique. In this talk we will show that these two solutions are identical. This result justifies the method of the vertical modes in the hydroelastic wave diffraction problems.
27 November 2017
Dr Eldad Avital
Challenges in Fluid-Structure Interaction from Renewable Energy Application to Bio-Fluids
In the first part of the talk we will briefly describe a recently developed computational methodology coupling fluid-flow simulation using finite volume with structural dynamics simulation using a combined finite-discrete element method. Applications in water flow as of sediments and hydro-kinetic turbines will be discussed, followed by looking at recent designs for hydrogen fusion power and the challenges. Bio-fluid applications of red blood cell flow and upper renal system will also be illustrated. In the second part of the talk we will discuss our aerodynamic method for blade design calling for continuous surface curvature. It will be computationally and experimentally shown to increase aerodynamic efficiency, particularly at high incidence where stall delay can be achieved. Reduction of tonal noise for low Reynolds number blade sections will also be addressed.
12 December 2017
Constitutive modelling of myocardium and other fibre-reinforced soft tissues
Nonlinear solid mechanics is used to model normal function and pathological conditions in soft tissues. In many cases, the mechanical behaviour of tissues can be adequately represented using an idealised elastic material. For instance, myocardium can be regarded as a passive (non-linear) hyperelastic solid with pronounced anisotropic properties due to its complex microstructure.
In the first part of the talk we examine basic ideas utilised in structural and semi-structural constitutive models, discuss the multiplicative decomposition framework for growth and remodeling (G&R), and highlight some challenges in using these elements.
The second part of the talk is dedicated to the Generalised Structure Tensor (GST) approach, which is used to formulate constitutive models for anisotropic fibre-reinforced composited with fibre distribution or dispersion [Gasser et al. JRS’06]. The GST approach has been so far successfully applied to models based on invariants I4 and I5, which capture the effect of deformation on each fibre family in isolation. We extend the GST approach to models based on the invariant I8, which couples two fibre families. Using the Holzapfel-Ogden model for myocardium, we demonstrate that accounting for fibre dispersion in the I8 term can have a significant effect on the predicted material response and may also reduce material symmetry.
19 December 2017
Prof. Till Bretschneider
Image-based modelling of cell dynamics
Modern live-cell fluorescence microscopy enables us to visualise dynamic cellular processes in unprecedented detail. I will present ongoing research projects which are concerned with bringing together i) image analysis methods for tracking cells and their movements as well as quantifying spatio-temporal patterns of fluorescently labelled cellular constituents, and ii) mathematical models to investigate regulatory mechanisms of cellular biochemistry and mechanics.
16 January 2018
Professor Frederic Dias
What makes ocean waves go rogue in the real world?
The study of extreme ocean waves is a rapidly expanding area of research worldwide. Although much work in this area is based on modeling and experiments in controlled wave tanks, the starting point of all studies is wave observation in the natural world. During this talk, we will provide some evidence of extreme wave events, describe the main mechanisms for their generation and conclude with what we believe makes ocean waves go rogue in the real world.
23 January 2018
A mechanistic approach to skin biophysics learning from mathematical and computational models
Besides the brain, no other organ of the human body plays such a central role in our everyday biological and social life than the skin. After all, this interface is the first line of defence of our body against the external environment and acts as a physical interface. It controls many types of exchanges between our inner and outside worlds which take the form of mechanical, thermal, biological, chemical and electromagnetic processes. Moreover, the skin tells a story about our health, age, past traumas, emotions, ethnicity, and our social and physical environments. Considering the place of the skin in our life and its multiple physiological functions, understanding its complex physiology and biophysics in health, disease and trauma has become, particularly in the last two decades, a broad multidisciplinary research arena.
To unravel some of the secrets of such a complex organ new experimental, imaging and computational techniques are needed and novel mechanistic theories explaining particular mechanobiological processes need to be formulated and put to the test. Developing and exploiting such an integrated framework underpin many aspects of our research which aims to understand the interplay between the microstructural and material properties of the skin, particularly as they evolve over the life course. As mounting evidence suggests, the skin microstructure can play a critical role in how macroscopic deformations are modulated at the microscopic level. These structural mechanisms are also at the heart of skin tribology by being part of, and conditioning mechanical load transmission and the nature of surface physics interactions. Skin biophysics is therefore fundamental to many industrial sectors from biomedical devices, personal care and cosmetic products to vehicle safety, textile, sport equipment, wearable electronics and tactile surfaces.
In this talk, I will present some of the latest modelling approaches we develop to gain a mechanistic understanding of the interplay between the material and structural properties of the skin, and ultimately, to exploit this knowledge for a variety of clinical and industrial applications. Examples will include computational contact homogenisation procedures to study skin friction, constitutive modelling of skin ageing and analysis of skin surface instabilities to understand mechanisms of wrinkle formation