Informal Applied and Numerical Analysis Seminars:

Math finance -as the name suggests- has a primary interest in economic valuations under uncertainty. However, there are also real-world quantities driven by the same underpinning economic uncertainties. These quantities include event probabilities and expected values, which are directly useful in planning and optimising real-world operations. By considering these [forgotten?] quantities, this talk shows: 1) How to calculate these quantities, from a combination of models and data, 2) That economic valuations are not necessarily useful as proxies for real-world event probabilities, 3) That event probabilities and valuations are quantities of joint interest in real-world policy and regulation. In the context of the utility industries, we show that a joint optimisation strategy for both the operator and the regulator leads to a Nash equilibrium.

We consider a cylindrical vessel containing two stably stratified fluids of large viscosity contrast. When the vessel is oscillated sinusoidally about its vertical axis an interfacial instability can develop, during which regularly spaced viscous fingers penetrate the less viscous fluid. Motivated by Shyh and Munson's (J. Fluid Eng., 1986) experiments, Miller (D. Phil Thesis, Oxford University, 1986) modelled the flow in the more viscous layer by the Stokes equations and treated the flow in the other layer as an irrotational inviscid liquid. Recently, Juel and Talib (J. Fluid Mech., 2011, submitted) have demonstrated experimentally that the Miller model provides good predictions of the stability thresholds provided the capillary length is much larger than the thickness of the Stokes layer in the less viscous fluid. In systems in which this is not the case, the inviscid approximation becomes less appropriate. This motivates the current study in which a fully viscous model is used in order to improve the predictions of the stability thresholds. We start by solving the axisymmetric free surface Navier-Stokes equations numerically, using the finite element method. We then examine the growth of small-amplitude non-axisymmetric perturbations to this axisymmetric base state. This allows us to determine the parameters beyond which a given azimuthal mode becomes unstable.

The notion of a fractal set and of fractal geometry has been with us since Benoit Mandelbrot’s pioneering work that started in the late 1960s and early 1970s. The study of fractals has evolved from calculating the fractal dimensions of mathematical and natural structures in the 1970s and 1980s, to the study of wave propagation in fractal structures from the 1990s to today. Although there has been a huge amount of effort in pushing forward this field (on Google Scholar there are around half a million articles containing the word `fractal') the use of fractal ideas in technology is still in its infancy. In this talk I will talk about one very recent and ongoing programme of work to pull through these ideas into the world of ultrasound technology. Motivated by nature's ultrasound users, and from the needs of the end users in medicine and in industry, I will derive a theoretical model that clearly shows the potential of these ideas. The theoretical model uses the discrete lattice counterpart of the Sierpinski gasket and renormalisation to arrive at analytical expressions for the main operating characteristics of the ultrasound transducer device. I will finish by discussing the prototype devices that are in the process of being built and tested.

Dislocations are mobile line defects in crystalline lattices that are important because dislocation motion is the dominant mechanism of plasticity in crystalline metals. Dislocations move in response to shear stresses and the distortion of the lattice means that each dislocation also sets up its own stress field throughout the crystal. Hence, there are systems of mutual repulsion and attraction between dislocations. Given the high density of dislocations found in metals, it becomes an interesting problem to determine the configurations that a large number of dislocations can take at equilibrium.

It has long been known that a simple pile-up of dislocations against an obstacle can be modelled by replacing the discrete dislocations with a continuum density. However, this continuum approximation breaks down near either end of the pile-up. Moreover, the obvious approach that works with 'normal' dislocations fails when applied to many other configurations, including arrays of dislocation dipoles. In this talk, I will discuss some techniques of discrete-to-continuum asymptotics, and demonstrate how they can be used to obtain appropriate mathematical descriptions of dislocation configurations that are still effectively one-dimensional, such as a pile-up of dislocation dipoles against a lock or a pile-up of dislocation walls. The method can easily be generalised to problems outside dislocation theory, and I will also comment on the theoretical challenges and on other opportunities to describe systems of large numbers of interacting particles.

Operational models are used to forecast the prevailing ocean conditions for offshore activities around the polar regions.
This includes predictions of the wave activity present and also the extent and concentration of the expanses of floating sea ice that characterise the polar seas. However, currently no interactions of these two fields are accounted for.

This is an important omission because a) the waves exert forces on the ice that are capable of causing fracture, consequently affecting the behaviour of the ice; and b) the presence of the ice reduces the intensity of the waves with distance, thus confining their destructive impact to a finite region from the ice edge.

In this talk, methods of modelling the wave attenuation phenomenon, based on scattering theory will be described, along with a discussion on the issues associated with assimilating wave/ice interactions into the overall operational model.

Representative volume elements (RVEs) from porous or cellular solids can often be too large for numerical or experimental determination of effective elastic constants. Volume elements which are smaller than the RVE can be useful in extracting apparent elastic stiffness tensors which provide bounds on the homogenized elastic stiffness tensor. Here, we make efficient use of boundary element analysis to compute the volume averages of stress and strain needed for such an analysis. For boundary conditions which satisfy the Hill criterion, we demonstrate the extraction of apparent elastic stiffness tensors using a symmetric Galerkin boundary element method. We apply the analysis method to two examples of a porous ceramic. Finally, we extract the eigenvalues of the fabric tensor for the example problems and provide predictions on the apparent elastic stiffnesses as a function of solid volume fraction.