Project Summaries

Below are some of the projects currently under investigation in the School:

Modelling of Greenland and Antarctica

CFD for Hypersonic Propulsion

Transport and Stirring in Geophysical Flows: A Dynamical Systems Approach

Chaotic Wave Scattering

Superconductivity in ZrZn2

Quantum Oscillations in the Superconducting State of MgB2

Ab Initio Molecular Dynamics of Hydrothermal Solutions

Saturation of Electrostatic Instability in Two-Species Plasma

Parallel Solution of Sparse Linear Systems Defined Over GF(p)

Electronic Structure of Weak Itinerant Ferromagnets

CFD for Aeroelastic Simulation

Vortex Interactions at High-Reynolds Number With Meshless Methods


Numerical Modelling of Greenland and Antarctica

Dr. A. Payne, School of Geographical Sciences

The aim is be able to predict the future evolution of the Greenland and Antarctic Ice Sheets. We use a variety of numerical (implicit finite-difference techniques) models to simulate the flow of these ice masses. They typically incorporate a prognostic equation ice thickness evolution (as a consequence of flow divergence and snow accumulation), the 3d advection/diffusion equation for temperature evolution, and a diagnostic equation for stress and velocity regimes. The models use the MPI protocol to run on parallel processors.

CFD for Hypersonic Propulsion

Peter Ess, Department of Aerospace Engineering.

Blunt body generated combustion for supersonic (Mach 5) stoichiometrically premixed air-hydrogen gas flow is studied numerically for different channel blockage ratios in order to find optimum conditions for combustion. Hence, the laminar Navier-Stokes equations with detailed chemical reaction schemes are used, yielding a very stiff system of equations. For this, an implicit multi-block finite-volume code is being developed and used to simulate such gas flows in steady-state or time-accurate manner on single or multi-processor computers.

Transport and Stirring in Geophysical Flows: A Dynamical Systems Approach

Dr. Ana Maria Mancho, Dr. D. Small, Prof. S. Wiggins, School of Mathematics, Department of Applied Mathematics

Dynamical systems theory offers an ideal mathematical framework in which to study Lagrangian transport and mixing issues in geophysical flows. A code which simulates a 3 layer quasigeostrophic model of the ocean, is used to generate data where many ideas of transport are tested and developed. Different regimes are reached by changing the input parameters. These data are analyzed with several python codes which identificate stagnation points (SP) and calculate distinguished hyperbolic trajectories (DHT) near those points. The results are prepared to be visualized with TECPLOT. In order to calculate manifolds associated with the DHT we are analyzing effectiveness of different numerical interpolation schemes for tracking particle trajectories in non time periodic finite vector fields.

Chaotic Wave Scattering

Dr. Ishio Hiromu, School of Mathematics, Department of Applied Mathematics.

I am solving the wave equation for a chaotic open cavity. Discretization of the equation reduces to a linear matrix equation which should be diagonalized to obtain its solution. I iterate the same calculation by changing an external parameter such as wave length etc. Therefore, parallel processing of the calculation is strongly desirable. Because of the complexity of the systems with chaotic geometries, high-performance computers are required to solve the wave equation of particles. We adopt plane wave expansion method, obtain large scale of matrices each element of which has integrals, and calculate the inverse of them. In most of the time, we iterate the same calculation, changing external parameters.

Superconductivity in ZrZn2

Ben Powell, Department of Physics.

ZrZn2 is a recently discovered ferromagnetic superconductor. The coexistence of ferromagnetism and superconductivity is a great surprise as normally superconductivity is destroyed by magnetic fields or magnetic impurities in the crystal. Early results show that for a particular choice of parameters the critical temperature of the superconductor rises as the exchange splitting of the ferromagnet increases. We will study the appropriate Bogoliubov-de Gennes equations for this system. To do this involves diagonalising a matrix at a large number of points in reciprocal space and then integrating over all of the eigenstates independently at each of the points in reciprocal space. This is an intrinsically parallel task. The diagonalisation of the BdG matrices at each point in reciprocal space are independent of one another. Therefore the integration mesh can be sub-divided into independent regions, each handled by a separate processor. Also each different temperature and magnetic field parameter set can be handled independently.

Quantum Oscillations in the Superconducting State of MgB2

Dr. Gilles Santi, Department of Physics.

The discovery of superconductivity in MgB2 with a Tc of 39 K has spawned a tremendous amount of experimental and theoretical work. From the various experiments, the emerging consensus seems to go towards an electron-phonon driven, two-gap superconductor. It is proposed to compute the damping of the quantum oscillations in the superconducting state using the semiclassical theory for superconductors and the result from ab-initio electronic structure calculation. The pseudopotential plane wave electronic structure code is parallelized over the G vectors (the different plane waves).

Ab Initio Molecular Dynamics of Hydrothermal Solutions

Dr. D. Sherman, Earth Sciences.

Hydrothermal and aqueous solutions play a fundamental role in geochemistry. The formation of ore deposits and the transport of metal pollutants in groundwater all depend on the coordination chemistry of metals in aqueous solutions. The determination of aqueous coordination chemistry, especially at high temperature, is difficult. Some of the most fundamental chemical systems in geology are still poorly understood. Ab initio molecular dynamical simulations using the Car Parinello algorithm and density functional theory allow us to probe the coordination chemistry of metals in aqueous solutions as a function of temperature for the first time. The CASTEP code for such calculations has been developed and parallelized by the UK Car Parinello consortium. We have used this code for work at the Manchester computing centre with promising results. For example, we have shown that the chemistry of copper is different from that predicted by extrapolated thermodynamic data. At the same time, our simulations are in perfect agreement with recent spectroscopic results. It is clear that atomistic calculations on ore forming solutions may greatly change our understanding of geochemical processes.

Saturation of electrostatic instability in two-species plasma

Dr. R.R. Kerswell, School of Mathematics

A classical problem in plasma theory is how electrostatic instabilities develop in the nonlinear regime. Traditional approaches treat the single species Vlasov equation and predict a trapping scaling whereby the saturation amplitude scales as gamma to the power 2 where gamma is the distance of a controlling parameter from criticality. However, in fact, a plasma really consists of two species - electrons and heavy ions. Analysis extended to include this second species suggests that in this new situation nonlinearity becomes important when the amplitude is O(gamma to the power 5 over 2). The project is fleshing out this theory for the 2-species Vlasov system and making quantitative comparisons with numerical solutions of the full equations. These solutions possess fine structure at particle resonances (critical layers) and hence are computationally expensive to capture.

Ref: Balmforth, N.J. & Kerswell R.R. "Saturation of electrostatic instability in two-species plasma" J. Plasma Phys. 68, 87-117, 2002.

Parallel Solution of Sparse Linear Systems Defined Over GF(p)

Dr. D Page, Department of Computer Science

The security of modern public key cryptography is usually based on the presumed hardness of problems such as integer factorisation or the finding discrete logarithms. As such, cryptosystems are generally parameterised so that it is computationally infeasible for an attacker to solve said problems and hence undo the security of the system. The Number Field Sieve (NFS) and Function Field Sieve (FFS) are two methods that are able to attack the problems of integer factorisation and finding discrete logarithms respectively, in a much more efficient way than using a naive, brute force approach. These methods operate in two phases. First, the sieving phase collects a large number of relations that represent small pieces of information about the problem being attacked. These relations are combined into a large linear system, whose solution in the second, matrix phase, allows efficient calculation of answers to the initial problem.

Although solution of linear systems is a well studied problem, and is often disregarded as trivial in theoretical discussion of the NFS and FFS, the matrix phase can present a practical barrier to the sizes of problem that can be attacked. Since the linear systems are generally very large, with perhaps hundreds of thousands of unknowns and equations, fast and scalable solutions are vital if one is to make progress. Traditional iterative methods for solving such systems are generally preferred due to the size and sparseness of the linear systems in questions: implementing such methods, especially in parallel, remains an under-investigated challenge. As part of ongoing research into Identity Based Encryption (IBE), our work investigates efficient implementations of the matrix phase, using the Beowulf cluster for parallel solution of such linear systems defined over GF(p), where p is a large prime. This in turn allows sound reasoning about the real-world security properties of cryptosystems attacked using the FFS or NFS.

Electronic Structure of Weak Itinerant Ferromagnets

Dr. S. Dugdale, Department of Physics

Following the discovery of the coexistence of weak itinerant ferromagnetism with superconductivity in UGe2, ZrZn2 and URhGe, there has been a great deal of interest in understanding their electronic structure. These materials are likely to have exotic pairing mechanisms, with magnetic fluctutations being a likely candidate. Particularly when combined with experimental investigations, first-principles calculations of this electronic structure can provide significant insight into this behaviour.

This project aims at determining the fully-relativistic electronic structure of a number of nearly and weakly magnetic systems, with the objective of relating their electronic structure to the presence or absence of superconductivity.

CFD for Aeroelastic Simulation

Dr. C.B. Allen, Department of Aerospace Engineering

Numerical simulation of aeroelastic response involves time-synchronised simulation of unsteady fluid flows and structural response, over long integration times, to compute the time-dependent response of flexible structures. Furthermore, numerical dissipation present in all CFD codes is grid density dependent and, hence, to capture the essential physics of these flows, large meshes are required, in the region of 0.5-5.0 million points for a fixed-wing case. Parallel processing is, therefore, essential for these flows, both to reduce run-times and distribute the required memory.

Rotor blade flows are significantly more expensive to compute than fixed-wing cases, as the vortical wake shed by each blades impinges on the other blades. Hence, very fine meshes are required over much of the domain, to capture the wake in sufficient detail for it to have the correct influence on blade loading. An unsteady, parallel, finite-volume solver has been developed, and hover and forward flight simulations have so far been performed using upto 20 million points.

The long term objective is the simulation of the complete vehicle, i.e. rotor disk plus fuselage and tail rotor, but it is estimated that to capture the main rotor wake in sufficient detail to compute the effect of impingement on the tail rotor will require 100 million+ points.

Vortex Interactions at High-Reynolds Number With Meshless Methods.

Dr. L.A. Barba, Department of Applied Mathematics

Computing the interaction of viscous vortices using traditional CFD methods is severely hindered by numerical diffusion. In general, the vortices diffuse too rapidly to properly capture the details of their interaction with each other or with structures. To deal with this problem, a fully meshless method has been developed, which is characterized by non-diffusive truncation errors. It is a new formulation of the vortex particle method, using the core spreading scheme for viscous effects, and a meshless spatial adaption technique based on radial basis function (RBF) interpolation. Numerical experiments have demonstrated increased accuracy in comparison with the standard approach of remeshing with high-order kernels. The method has been implemented in parallel using the PETSc library.

Applications in vortex interactions so far include the relaxation of a perturbed monopole, exhibiting under non-linear effects a nonsymmetric quasi-steady state (rotating tripole), and the early interaction of two co-rotating vortices.

A poster recently presented to the ICTAM-Warsaw is avilable.