Past

Semidefinite programming (SDP) techniques have been extremely successful

in many practical engineering design questions. In several of these

applications, the problem structure is invariant under the action of

some symmetry group, and this property is naturally inherited by the

underlying optimization. A natural question, therefore, is how to

exploit this information for faster, better conditioned, and more

reliable algorithms. To this effect, we study the associative algebra

associated with a given SDP, and show the striking advantages of a

careful use of symmetries. The results are motivated and illustrated

through applications of SDP and sum of squares techniques from networked

control theory, analysis and design of Markov chains, and quantum

information theory.

In St John's College.

Oxford Life Sciences Modelling Colloquia Series

Nonconvex minimisation problems are encountered in many applications such as phase transitions in solids (1) or liquids but also in optimal design tasks (2) or micromagnetism (3). In contrast to rubber-type elastic materials and many other variational problems in continuum mechanics, the minimal energy may be not attained. In the sense of (Sobolev) functions, the non-rank-one convex minimisation problem (M) is ill-posed: As illustrated in the introduction of FERM, the gradients of infimising sequences are enforced to develop finer and finer oscillations called microstructures. Some macroscopic or effective quantities, however, are well-posed and the target of an efficient numerical treatment. The presentation proposes adaptive mesh-refining algorithms for the finite element method for the effective equations (R), i.e. the macroscopic problem obtained from relaxation theory. For some class of convexified model problems, a~priori and a~posteriori error control is available with an reliability-efficiency gap. Nevertheless, convergence of some adaptive finite element schemes is guaranteed. Applications involve model situations for (1), (2), and (3) where the relaxation is provided by a simple convexification.

We consider a queuing system with three queues (nodes) and one server.

The arrival and service rates at each node are such that the system overall

is overloaded, while no individual node is. The service discipline is the

following: once the server is at node j, it stays there until it serves all

customers in the queue.

After this, the server moves to the "more expensive" of the two

queues.

We will show that a.s. there will be a periodicity in the order of

services, as suggested by the behavior of the corresponding

dynamical systems; we also study the cases (of measure 0) when the

dynamical system is chaotic, and prove that then the stochastic one

cannot be periodic either.

Discontinuous Galerkin methods (DG) use trial and test functions that are continuous within

elements and discontinuous at element boundaries. Although DG methods have been invented

in the early 1970s they have become very popular only recently.

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DG methods are very attractive for flow and transport problems in porous media since they

can be used to solve hyperbolic as well as elliptic/parabolic problems, (potentially) offer

high-order convergence combined with local mass balance and can be applied to unstructured,

non-matching grids.

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In this talk we present a discontinuous Galerkin method based on the non-symmetric interior

penalty formulation introduced by Wheeler and Rivi\`{e}re for an elliptic equation coupled to

a nonlinear parabolic/hyperbolic equation. The equations cover models for groundwater flow and

solute transport as well as two-phase flow in porous media.

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We show that the method is comparable in efficiency with the mixed finite element method for

elliptic problems with discontinuous coefficients. In the case of two-phase flow the method

can outperform standard finite volume schemes by a factor of ten for a five-spot problem and

also for problems with dominating capillary pressure.

The standard airframe industry tool for flutter analysis is based

on linear potential predictions of the aerodynamics. Despite the

limitations of the modelling this is even true in the transonic

range. There has been a heavy research effort in the past decade to

use CFD to generate the aerodynamics for flutter simulations, to

improve the reliability of predictions and thereby reduce the risk

and cost of flight testing. The first part of the talk will describe

efforts at Glasgow to couple CFD with structural codes to produce

a time domain simulation and an example calculation will be described for

the BAE SYSTEMS Hawk aircraft.

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A drawback with time domain simulations is that unsteady CFD is still

costly and parametric searches to determine stability through the

growth or decay of responses can quickly become impractical. This has

motivated another active research effort in developing ways of

encapsulating the CFD level aerodynamic predictions in models which

are more affordable for routine application. A number of these

approaches are being developed (eg POD, system identification...)

but none have as yet reached maturity. At Glasgow effort has been

put into developing a method based on the behaviour of the

eigenspectrum of the discrete operator Jacobian, using Hopf

Bifurcation conditions to formulate an augmented system of

steady state equations which can be used to calculate flutter speeds

directly. The talk will give the first three dimensional example

of such a calculation.

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For background reports on these topics see

http://www.aero.gla.ac.uk/Research/CFD/projects/aeroelastics/pubs/menu…