DSpace Collection:https://dspace.lboro.ac.uk/2134/46422016-07-24T22:37:01Z2016-07-24T22:37:01ZSafety system design optimisationPattison, Rachel L.https://dspace.lboro.ac.uk/2134/220192016-07-19T15:22:16Z2000-01-01T00:00:00ZTitle: Safety system design optimisation
Authors: Pattison, Rachel L.
Abstract: This thesis investigates the efficiency of a design optimisation scheme that is
appropriate for systems which require a high likelihood of functioning on demand.
Traditional approaches to the design of safety critical systems follow the preliminary
design, analysis, appraisal and redesign stages until what is regarded as an acceptable
design is achieved. For safety systems whose failure could result in loss of life it is
imperative that the best use of the available resources is made and a system which is
optimal, not just adequate, is produced.
The object of the design optimisation problem is to minimise system unavailability
through manipulation of the design variables, such that limitations placed on them by
constraints are not violated.
Commonly, with mathematical optimisation problem; there will be an explicit
objective function which defines how the characteristic to be minimised is related to
the variables. As regards the safety system problem, an explicit objective function
cannot be formulated, and as such, system performance is assessed using the fault tree
method. By the use of house events a single fault tree is constructed to represent the
failure causes of each potential design to overcome the time consuming task of
constructing a fault tree for each design investigated during the optimisation
procedure. Once the fault tree has been constructed for the design in question it is
converted to a BDD for analysis.
A genetic algorithm is first employed to perform the system optimisation, where the
practicality of this approach is demonstrated initially through application to a High-Integrity
Protection System (HIPS) and subsequently a more complex Firewater
Deluge System (FDS).
An alternative optimisation scheme achieves the final design specification by solving
a sequence of optimisation problems. Each of these problems are defined by
assuming some form of the objective function and specifying a sub-region of the
design space over which this function will be representative of the system
unavailability.
The thesis concludes with attention to various optimisation techniques, which possess
features able to address difficulties in the optimisation of safety critical systems.
Specifically, consideration is given to the use of a statistically designed experiment
and a logical search approach.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2000-01-01T00:00:00ZNumerical and analytical study of the convective Cahn-Hilliard equationAlesemi, Mesharihttps://dspace.lboro.ac.uk/2134/217742016-07-05T10:05:43Z2016-01-01T00:00:00ZTitle: Numerical and analytical study of the convective Cahn-Hilliard equation
Authors: Alesemi, Meshari
Abstract: We consider the convective Cahn-Hilliard equation that is used as a model of coarsening dynamics
in driven systems and that in two spatial dimensions (x; y) has the form
ut + Duux + r2(u u3 + r2u) = 0:
Here t denotes time, u = u(x; y; t) is the order parameter and D is the parameter measuring
the strength of driving. We primarily consider the case of one spatial dimension, when there is
no y-dependence. For the case of no driving, when D = 0, the standard Cahn-Hilliard equation
is recovered, and it is known that solutions to this equation are characterised by an initial
stage of phase separation into regions of one phase surrounded by the other phase (i.e., clusters
or droplets/holes or islands are obtained) followed by the coarsening process, where the
average size of the clusters grows in time and the number of the clusters decreases. Moreover,
two main coarsening mechanisms have been identified in the literature, namely, coarsening due
to volume and translational modes. On the other hand, for the case of strong driving, when
D ! 1, the well-known Kuramoto-Sivashinsky equation is recovered, solutions of which are
characterised by complicated chaotic oscillations in both space and time. The primary aim of
the present thesis is to perform a detailed and systematic investigation of the transitions in the
solutions of the convective Cahn-Hilliard equation for a wide range of parameter values as the
driving-force parameter is increased, and, in particular, to understand in detail how the coarsening
dynamics is affected by driving. We find that one of the coarsening modes is stabilised
at relatively small values of D, and the type of the unstable coarsening mode may change as
D increases. In addition, we find that there may be intervals in the driving-force parameter D
where coarsening is completely stabilised. On the other hand, there may be intervals where twomode
solutions are unstable and the solutions can evolve, for example, into one-droplet/hole
solutions, symmetry-broken two-droplet/hole solutions or time-periodic solutions. We present
detailed stability diagrams for 2-mode solutions in the parameter planes and corroborate our
findings by time-dependent simulations. Finally, we present preliminary results for the case of
the (convective) Cahn-Hilliard equation in two spatial dimensions.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2016-01-01T00:00:00ZComputation of scattering matrices and resonances for waveguidesRoddick, Greghttps://dspace.lboro.ac.uk/2134/217622016-07-05T10:34:59Z2016-01-01T00:00:00ZTitle: Computation of scattering matrices and resonances for waveguides
Authors: Roddick, Greg
Abstract: Waveguides in Euclidian space are piecewise path connected subsets of R^n that can be written as the union of a compact domain with boundary and their cylindrical ends. The compact and non-compact parts share a common boundary. This boundary is assumed to
be Lipschitz, piecewise smooth and piecewise path connected. The ends can be thought of as the cartesian product of the boundary with the positive real half-line. A notable feature of Euclidian waveguides is that the scattering matrix admits a meromorphic continuation to a certain Riemann surface with a countably infinite number of leaves [2], which we will
describe in detail and deal with. In order to construct this meromorphic continuation,
one usually first constructs a meromorphic continuation of the resolvent for the Laplace
operator. In order to do this, we will use a well known glueing construction (see for example [5]), which we adapt to waveguides. The construction makes use of the meromorphic Fredholm theorem and the fact that the resolvent for the Neumann Laplace operator on the ends of the waveguide can be easily computed as an integral kernel. The resolvent can then be used to construct generalised eigenfunctions and, from them, the scattering matrix.Being in possession of the scattering matrix allows us to calculate resonances; poles of
the scattering matrix. We are able to do this using a combination of numerical contour integration and Newton s method.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2016-01-01T00:00:00ZBehaviour of Eigenfunction Subsequences for Delta-Perturbed 2D Quantum SystemsNewman, Adamhttps://dspace.lboro.ac.uk/2134/215682016-06-25T07:51:04Z2016-01-01T00:00:00ZTitle: Behaviour of Eigenfunction Subsequences for Delta-Perturbed 2D Quantum Systems
Authors: Newman, Adam
Abstract: We consider a quantum system whose unperturbed form consists of a self-adjoint $-\Delta$ operator on a 2-dimensional compact Riemannian manifold, which may or may not have a boundary. Then as a perturbation, we add a delta potential / point scatterer at some select point $p$. The perturbed self-adjoint operator is constructed rigorously by means of self-adjoint extension theory. We also consider a corresponding classical dynamical system on the cotangent/cosphere bundle, consisting of geodesic flow on the manifold, with specular reflection if there is a boundary.
Chapter 2 describes the mathematics of the unperturbed and perturbed quantum systems, as well as outlining the classical dynamical system. Included in the discussion on the delta-perturbed quantum system is consideration concerning the strength of the delta potential. It is reckoned that the delta potential effectively has negative infinitesimal strength.
Chapter 3 continues on with investigations from [KMW10], concerned with perturbed eigenfunctions that approximate to a linear combination of only two "surrounding" unperturbed eigenfunctions. In Thm. 4.4 of [KMW10], conditions are derived under which a sequence of perturbed eigenfunctions exhibits this behaviour in the limit. The approximating pair linear combinations belong to a class of quasimodes constructed within [KMW10]. The aim of Chapter 3 in this thesis is to improve on the result in [KMW10].
In Chapter 3, preliminary results are first derived constituting a broad consideration of the question of when a perturbed eigenfunction subsequence approaches linear combinations of only two surrounding unperturbed eigenfunctions. Afterwards, the central result of this Chapter, namely Thm. 3.4.1, is derived, which serves as an improved version of Thm. 4.4 in [KMW10]. The conditions of this theorem are shown to be weaker than those in [KMW10]. At the same time though, the conclusion does not require the approximating pair linear combinations to be quasimodes contained in the domain of the perturbed operator. Cor. 3.5.2 allows for a transparent comparison between the results of this Chapter and [KMW10].
Chapter 4 deals with the construction of non-singular rank-one perturbations for which the eigenvalues and eigenfunctions approximate those of the delta-perturbed operator. This is approached by means of direct analysis of the construction and formulae for the rank-one-perturbed eigenvalues and eigenfunctions, by comparison that of the delta-perturbed eigenvalues and eigenfunctions. Successful results are derived to this end, the central result being Thm. 4.4.19. This provides conditions on a sequence of non-singular rank-one perturbations, under which all eigenvalues and eigenbasis members within an interval converge to those of the delta-perturbed operator.
Comparisons have also been drawn with previous literature such as [Zor80], [AK00] and [GN12]. These deal with rank-one perturbations approaching the delta potential within the setting of a whole Euclidean space $\mathbb{R}^n$, for example by strong resolvent convergence, and by limiting behaviour of generalised eigenfunctions associated with energies at every $E\in(0,\infty)$. Furthermore in Chapter 4, the suggestion from Chapter 2 that the delta potential has negative infinitessimal strength is further supported, due to the coefficients of the approximating rank-one perturbations being negative and tending to zero. This phenomenon is also in agreement with formulae from [Zor80], [AK00] and [GN12].
Chapter 5 first reviews the correspondence between certain classical dynamics and equidistribution in position space of almost all unperturbed quantum eigenfunctions, as demonstrated for example in [MR12]. Equidistribution in position space of almost all perturbed eigenfunctions, in the case of the 2D rectangular flat torus, is also reviewed. This result comes from [RU12], which is only stated in terms of the "new" perturbed eigenfunctions, which would only be a subset of the full perturbed eigenbasis. Nevertheless, in this Chapter it is explained how it follows that this position space equidistribution result also applies to a full-density subsequence of the full perturbed eigenbasis.
Finally three methods of approach are discussed for attempting to derive this position space equidistribution result in the case of a more general delta-perturbed system whose classical dynamics satisfies the particular key property.
[AK00] S. Albeverio and P. Kurasov: Singular Perturbations of Differential Operators. London Math. Soc. Lecture Note Ser. 271. Cambridge University Press (2000).
[GN12] P. G. Grinevich and R. G. Novikov: Faddeev eigenfunctions for point potentials in two dimensions. Phys. Lett. A 376, 1102 (2012).
[KMW10] J. P. Keating, J. Marklof and B. Winn: Localized eigenfunctions in \v{S}eba billiards. J. Math. Phys. 51, 062101 (2010).
[MR12] J. Marklof and Z. Rudnick: Almost all eigenfunctions of a rational polygon are uniformly distributed. J. Spectr. Theory 2, 107 (2012).
[RU12] Z. Rudnick and H. Uebersch\"{a}r: Statistics of Wave Functions for a Point Scatterer on the Torus. Commun. Math. Phys. 316, 763 (2012).
[Zor80] J. Zorbas: Perturbation of self-adjoint operators by Dirac distributions. J. Math. Phys. 21(4), 840 (1980).
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2016-01-01T00:00:00Z