Mathematics

Applied Mathematics Seminars 

Autumn 2017

All seminars (unless otherwise stated) will take place on Tuesdays at 3.00pm in Room D103 (25 Gordon Street). See the map for further details. There will be tea afterwards in Mathematics Room 606 (25 Gordon Street). If you require any more information on the Applied seminars please contact Prof Jean-Marc Vanden-Broeck (e-mail: j.vanden-broeck AT ucl.ac.uk or tel: 020-7679-2835) or Prof ilia Kamotski (e-mail: i.kamotski AT ucl.ac.uk or tel: 020-7679-3937).

10 October 2017

Speaker: Dr Ewelina Zatorska (UCL)

Title: Transport of congestion in two-phase compressible/incompressible flow
Abstract:

Can the fluid equations be used to model pedestrian motion or traffic?  In this talk, I will present the compressible-incompressible two phase system describing the flow in the free and in the congested regimes.  I will show how to approximate such system by the compressible Navier-Stokes equations with singular pressure for the fixed barrier densities, together with some recent developments for the barrier densities varying in the space and time. At the end, I will present the numerical results showing that our macroscopic system captures some features characteristic for microscopic models of collective behaviour.

This is a talk based on several papers in collaboration with: D. Bresch, C. Perrin, P. Degond, P. Minkowski, and L. Navoret

17 October 2017

Speaker: Dr Stefan Frei (UCL)

Title: A fully Eulerian approach for fluid-structure interactions with large displacements and contact
Abstract:

This talk is concerned with the simulation of fluid-structure interaction problems with large solid displacements up to contact of different solids or a solid with a wall. To be able to deal  with the topology changes in the fluid domain when it comes to contact, we use a monolithic fully Eulerian approach. The model consists of a hyperelastic material law for the solid and the incompressible Navier-Stokes equations for the fluid part, both of them being formulated in Eulerian coordinates. We remark that when it comes to contact, it is however questionable whether the Navier-Stokes equations still represent a sufficiently accurate description of the underlying physics.

The second part of this talk deals with the finite element approximation of the monolithic Eulerian system that poses several  numerical challenges, as the solid and the interface move over mesh  lines and the solution is not smooth across the interface.  We present accurate discretisation schemes in space and time  based on the concept to resolve the interface locally within the discretisation.

Finally, we apply the numerical framework to simulate plaque growth in arteries and to the problem  of a bouncing ball including contact with the ground  and bouncing down some stairs.

24 October 2017

Speaker: Prof. Demetrios Papageorgiou (Imperial College London)

Title: Nonlinear stability of viscous multilayer flows
Abstract:

Shear flows of immiscible viscous fluids in a stratified arrangement can become linearly unstable even when single phase flows under the same conditions are linearly stable at all Reynolds numbers. A good example is plane Couette and plane Poiseuille flow. When more than one fluid is present a new interfacial mode exists that can be unstable at any non-zero Reynolds number (and in fact even at zero Reynolds number if there are enough layers in the configuration). In this talk I will present nonlinear stability results for both two-layer Couette flow and three-layer Couette-Poiseuille flow. In the former case, a new nonlinear theory will be presented that gives very good agreement with available experiments as well as direct numerical simulations that we have also undertaken. (The theory has also been extended to viscoelastic flows and time permitting a brief summary will be given.) In the second part I will consider three layers so that there are two fluid-fluid interfaces. Asymptotic analysis will be used to derive parameter-free coupled nonlinear evolution equations for the two interfaces. Numerical results will be presented and also a new theory that enables us to predict the nonlinear stability of the flow in the sense that we can identify the structure and initial amplitudes of perturbations that will cause the linearly stable flow to "transition" to non-uniform nonlinear states. The mechanism behind this centres on the form of the nonlinear flux functions of the PDEs and in particular hyperbolic to elliptic transitions in space-time. Even though the theory has been carried out at low or even zero Reynolds numbers, it is equally applicable to inviscid flows found in geophysical and oceanographic applications.

31 October 2017

Speaker: Dr Emilian Parau (University of East Anglia)

Title: Nonlinear hydroelastic waves under an ice cover and related flows
Abstract:

Computations of nonlinear hydroelastic waves travelling at the surface of an ideal fluid covered by a thin ice plate are presented. The continuous ice-plate model is based on the special Cosserat theory of hyperelastic shells satisfying Kirchhoff's hypothesis. Forced and solitary waves are computed using boundary integral methods and their evolution in time is analysed using a pseudospectral method based on FFT and in the expansion of the Dirichlet-Neumann operator.  The fluid is either of constant density of stratified and two-dimensional or three-dimensional problems are considered. When the ice-plate is fragmented, a new model is proposed by allowing the coefficient of the flexural rigidity to vary spatially.  The attenuation of solitary waves is studied by using time-dependent simulations.

7 November 2017

NO SEMINAR - READING WEEK

14 November 2017

Please see the Departmental Colloquia webpage

21 November 2017

Speaker: Dr Iain Smears (UCL)

Title: Numerical methods for stochastic optimal control problems
Abstract:

Stochastic optimal control theory is essentially concerned with how to make decisions for influencing stochastic dynamical systems, and finds applications in many diverse areas, such as aerospatial engineering, finance, and energy, to mention only a few. Mathematically, these are modelled as optimisation problems of a given objective functional subject to a Stochastic Differential Equation (SDE) where the dynamics are influenced by the control function that is to be determined. A strategy for handling these optimal control problems involves solving certain related deterministic Partial Differential Equations (PDE), namely the associated Hamilton-Jacobi-Bellman (HJB) equations. A distinctive feature of these second-order elliptic and parabolic PDE is that they are fully nonlinear, in the sense that all partial derivatives up to second-order appear nonlinearly.

In this talk, we will give an overview of current research into numerical methods for approximating the solutions of these PDE. In particular, we will explain how the nonlinearity of the equations leads to significant challenges for designing and analysing the convergence of numerical approximations. We will then explain how the previously open problem of constructing and proving convergence of a higher-order accurate method for these problems was recently treated for an important subclass of these PDE, namely those equations that obey a structural property called the Cordes condition, which originates from the analysis of nondivergence form elliptic equations with discontinuous coefficients. We will conclude with some numerical computations showing the improved computational efficiency and accuracy of these methods.

28 November 2017

Speaker: Dr Jurriaan Gillissen (UCL)

Title: Hydrodynamic Sensing of the Size, Shape and Elasticity of Liposomes
Abstract:

Liposomes are ~50 nm, lipid bilayer containers, that (among other things) transport materials between cells and between organelles, by endocytosis (pinch-off) and exocytosis (fusion). These processes depend on bilayer elasticity k and here we develop a method to experimentally measure k. The method involves adsorbing liposomes on an oscillating quartz crystal and monitoring the associated changes in the resonance frequencies. These changes reflect the hydrodynamic forces on the oscillating liposomes, and thereby indirectly contain information on the liposome shape and the associated elasticity k.  We solve this inverse problem using computational fluid dynamics and find k consistent with literature values.

5 December 2017

NO SEMINAR

12 December 2017

Please see the Departmental Colloquia webpage