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https://dspace.lboro.ac.uk/2134/89
Fri, 18 Apr 2014 17:50:11 GMT2014-04-18T17:50:11ZTime-domain simulations for floating structures
https://dspace.lboro.ac.uk/2134/14475
Title: Time-domain simulations for floating structures
Authors: Fitzgerald, Colm J.
Abstract: In this thesis numerical and analytical investigations of wave-structure interactions are
conducted within the linearised theory of water waves. The primary objective of the
thesis was to develop a numerical time-domain solution method capable of simulating
wave-structure interactions in three-dimensions involving axisymmetric structures.
Although the solution method was developed for three-dimensional problems, many
two-dimensional interactions were also simulated using an existing time-domain solution
method.
The numerical method for obtaining the solution of the time-domain water wave problem
combines a cubic spline boundary element method (BEM) which yields a solution to
the boundary integral equation with a time-stepping algorithm to advance the solution
in time. The assumption regarding the axisymmetric nature of the structural geometry
results in significant simplifications of the governing boundary integral equation and
allows the existing BEM implementation for two-dimensional problems to be used as
the basis for the solution method. The time-advancement algorithm was implemented
such that radiation, scattering and floating body interactions can be simulated.
Despite the focus on the time-domain investigations, the interactions were also considered
in the frequency-domain to complement the time-domain results and for the purposes
of verification. The analytical frequency-domain investigations are particularly
relevant to highly resonant interactions where the response of the fluid and structure
is related to the location of the resonance in the complex frequency plane. The complementary
frequency-domain analysis was utilised in the development of a damped
harmonic oscillator model to approximate the transient fluid motions in resonant scattering
interactions. Passive trapped modes which can be supported by both fixed and
floating structures were discovered in frequency-domain uniqueness investigations in
the water-wave problem for a floating structure and their existence was confirmed in
both two and three dimensions using time-domain excitation simulations. Finally, the
time-domain BEM code was utilised to simulate various wave-structure interactions of
practical interest.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.Thu, 01 Jan 2009 00:00:00 GMThttps://dspace.lboro.ac.uk/2134/144752009-01-01T00:00:00ZEvaluating student engagement with mathematics support
https://dspace.lboro.ac.uk/2134/14435
Title: Evaluating student engagement with mathematics support
Authors: Symonds, Ria
Abstract: This thesis reports the findings of quantitative and qualitative research to evaluate the
effectiveness of mathematics support and to examine the issue of student engagement
in relation to its effectiveness. Usage data regarding Loughborough University's
Mathematics Learning Support Centre was analysed to understand which students
make use of the support and the extent of that usage. It was found that the majority of
students who need mathematics support were not accessing the resources. Rich
contextual data were gathered by interviews and focus groups, which indicated that a
number of barriers had prevented students from initially using the support. However,
whilst some students overcame these barriers to become regular users of the support,
other students (who were 'at risk' of failing the mathematical component of their
courses) did not. Students who were 'at risk' and had not accessed the support lacked
the motivation to engage with mathematics and the available support.
This thesis also evaluated the effectiveness of a proactive support initiative involving
small group teaching. Despite encouraging students to engage with mathematics
support, since they did not have to take the initiative themselves, it was revealed that a
lack of student engagement had had a profound effect on the success of this support.
Qualitative data was analysed to provide an insight as to why students had failed to
engage with the initiative. Constructs of students' attitudes towards mathematics and
their learning approaches were investigated. In particular, it was found that students
who engage with mathematics support are generally well-motivated and cognitively
engaged. These students held generally positive attitudes towards mathematics and
deployed metacognitive learning strategies by regularly monitoring and directing their
learning in order to achieve their high educational aims.
This study ... has implications both for research and practice. From a practical
perspective, it appears that mathematics support has moved from one of remedial
support to one of enhancement. It is recommended that action should be taken to
provide extrinsic motivation to encourage engagement with the support. However,
from a research perspective, a more rigorous investigation of the students' attitudes
and learning approaches and how these constructs relate to their levels of engagement
with mathematics support would be useful. Further information in this area could be
used to provide further quantification of the efficacy of mathematics support.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.Thu, 01 Jan 2009 00:00:00 GMThttps://dspace.lboro.ac.uk/2134/144352009-01-01T00:00:00ZUnitary double products as implementors of bogolubov transformations
https://dspace.lboro.ac.uk/2134/14306
Title: Unitary double products as implementors of bogolubov transformations
Authors: Jones, Paul
Abstract: This thesis is about double product integrals with pseudo rotational generator, and
aims to exhibit them as unitary implementors of Bogolubov transformations. We
further introduce these concepts in this abstract and describe their roles in the
thesis's chapters. The notion of product integral, (simple product integral, not
double) is not a new one, but is unfamiliar to many a mathematician. Product
integrals were first investigated by Volterra in the nineteenth century. Though often
regarded as merely a notation for solutions of differential equations, they provide
a priori a multiplicative analogue of the additive integration theories of Riemann,
Stieltjes and Lebesgue. See Slavik [2007] for a historical overview of the subject.
Extensions of the theory of product integrals to multiplicative versions of Ito and
especially quantum Ito calculus were first studied by Hudson, Ion and Parthasarathy
in the 1980's, Hudson et al. [1982]. The first developments of double product integrals
was a theory of an algebraic kind developed by Hudson and Pulmannova
motivated by the study of the solution of the quantum Yang-Baxter equation by
the construction of quantum groups, see Hudson and Pulmaanova [2005]. This
was a purely algebraic theory based on formal power series in a formal parameter.
However, there also exists a developing analytic theory of double product integral.
This thesis contributes to this analytic theory. The first papers in that direction are
Hudson [2005b] and Hudson and Jones [2012]. Other motivations include quantum
extension of Girsanov's theorem and hence a quantum version of the Black-Scholes
model in finance. They may also provide a general model for causal interactions in
noisy environments in quantum physics. From a different direction "causal" double
products, (see Hudson [2005b]), have become of interest in connection with quantum
versions of the Levy area, and in particular quantum Levy area formula (Hudson
[2011] and Chen and Hudson [2013]) for its characteristic function. There is a close
association of causal double products with the double products of rectangular type
(Hudson and Jones [2012] pp 3). For this reason it is of interest to study "forwardforward"
rectangular double products.
In the first chapter we give our notation which will be used in the following chapters
and we introduce some simple double products and show heuristically that they
are the solution of two different quantum stochastic differential equations. For each
example the order in which the products are taken is shown to be unimportant;
either calculation gives the same answer. This is in fact a consequence of the so
called multiplicative Fubini Theorem Hudson and Pulmaanova [2005].
In Chapter two we formally introduce the notion of product integral as a solution
of two particular quantum stochastic differential equations.
In Chapter three we introduce the Fock representation of the canonical commutation
relations, and discuss the Stone-von Neumann uniqueness theorem. We define
the notion of Bogolubov transformation (often called a symplectic automorphism,
see Parthasarathy [1992] for example), implementation of these transformations by
an implementor (a unitary operator) and introduce Shale's theorem which will be
relevant to the following chapters. For an alternative coverage of Shale's Theorem,
symplectic automorphism and their implementors see Derezinski [2003].
In Chapter four we study double product integrals of the pseudo rotational type.
This is in contrast to double product integrals of the rotational type that have been
studied in (Hudson and Jones [2012] and Hudson [2005b]). The notation of the
product integral is suggestive of a natural discretisation scheme where the infinitesimals
are replaced by discrete increments i.e. discretised creation and annihilation
operators of quantum mechanics. Because of a weak commutativity condition, between
the discretised creation and annihilation operators corresponding on different
subintervals of R, the order of the factors of the product are unimportant (Hudson
[2005a]), and hence the discrete product is well defined; we call this result the discrete
multiplicative Fubini Theorem. It is also the case that the order in which the
products are taken in the continuous (non-discretised case) does not matter (Hudson
[2005a], Hudson and Jones [2012]). The resulting discrete double product is shown
to be the implementor (a unitary operator) of a Bogolubov transformation acting
on discretised creation and annihilation operators (Bogolubov transformations are
invertible real linear operators on a Hilbert space that preserve the imaginary part
of the inner product, but here we may regard them equivalently as liner transformations
acting directly on creation and annihilations operators but preserving adjointness
and commutation relations). Unitary operators on the same Hilbert space
are a subgroup of the group of Bogolubov transformations. Essentially Bogolubov
transformations are used to construct new canonical pairs from old ones (In the
literature Bogolubov transformations are often called symplectic automorphisms).
The aforementioned Bogolubov transformation (acting on the discretised creation
and annihilation operators) can be embedded into the space L2(R+) L2(R+) and
limits can be taken resulting in a limiting Bogolubov transformation in the space
L2(R+) L2(R+). It has also been shown that the resulting family of Bogolubov
transformation has three important properties, namely bi-evolution, shift covariance
and time-reversal covariance, see (Hudson [2007]) for a detailed description of these
properties.
Subsequently we show rigorously that this transformation really is a Bogolubov
transformation. We remark that these transformations are Hilbert-Schmidt perturbations
of the identity map and satisfy a criterion specified by Shale's theorem. By
Shale's theorem we then know that each Bogolubov transformation is implemented
in the Fock representation of the CCR. We also compute the constituent kernels
of the integral operators making up the Hilbert-Schmidt operators involved in the
Bogolubov transformations, and show that the order in which the approximating
discrete products are taken has no bearing on the final Bogolubov transformation
got by the limiting procedure, as would be expected from the multiplicative Fubini
Theorem.
In Chapter five we generalise the canonical form of the double product studied in
Chapter four by the use of gauge transformations. We show that all the theory
of Chapter four carries over to these generalised double product integrals. This
is because there is unitary equivalence between the Bogolubov transformation got
from the generalised canonical form of the double product and the corresponding
original one.
In Chapter six we make progress towards showing that a system of implementors
of this family of Bogolubov transformations can be found which inherits properties
of the original family such as being a bi-evolution and being covariant under shifts.
We make use of Shales theorem (Parthasarathy [1992] and Derezinski [2003]). More
specifically, Shale's theorem ensures that each Bogolubov transformation of our
system is implemented by a unitary operator which is unique to with multiplicaiton
by a scalar of modulus 1. We expect that there is a unique system of implementors,
which is a bi-evolution, shift covariant, and time reversal covariant (i.e. which
inherits the properties of the corresponding system of Bogolubov transformation).
This is partly on-going research. We also expect the implementor of the Bogolubov
transformation to be the original double product. In Evans [1988], Evan's showed
that the the implementor of a Bogolubov transformation in the simple product case
is indeed the simple product. If given more time it might be possible to adapt
Evan's result to the double product case.
In Hudson et al. [1984] the three properties of bi-evolution, shift covariant, and time
reversal covariant (in only one degree of freedom) were used to show uniqueness of
the so called time-orthogonal unitary dilation" in the case of non-Fock quantum
stochastic calculus. The double" case in this thesis is much harder. I show using
the bi-evolutionarity and shift-covariance that there exists an unique system of implementors
up to multiplication by a family of scalars
{ e^{i(b-a)(t-s)} }_{(0<a<b;0<<st)}
for some real parameter. It is expected that the parameter can be uniquely specified
using the time reversal property. It is not clear, as of yet, how this should be
done, even though it is quite simple in the similar case of Hudson et al. [1984].
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.Tue, 01 Jan 2013 00:00:00 GMThttps://dspace.lboro.ac.uk/2134/143062013-01-01T00:00:00ZTranscritical flow of a stratified fluid over topography: analysis of the forced Gardner equation
https://dspace.lboro.ac.uk/2134/14279
Title: Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation
Authors: Kamchatnov, A.M.; Kuo, Y.-H.; Lin, T.-C.; Horng, T.-L.; Gou, S.-C.; Clift, R.; El, G.A.; Grimshaw, Roger H.J.
Abstract: Transcritical flow of a stratified fluid past a broad localised topographic obstacle is studied
analytically in the framework of the forced extended Kortewegâ€“de Vries (eKdV), or
Gardner, equation. We consider both possible signs for the cubic nonlinear term in the
Gardner equation corresponding to different fluid density stratification profiles. We identify
the range of the input parameters: the oncoming flow speed (the Froude number)
and the topographic amplitude, for which the obstacle supports a stationary localised
hydraulic transition from the subcritical flow upstream to the supercritical flow downstream.
Such a localised transcritical flow is resolved back into the equilibrium flow
state away from the obstacle with the aid of unsteady coherent nonlinear wave structures
propagating upstream and downstream. Along with the regular, cnoidal undular
bores occurring in the analogous problem for the single-layer flow modeled by the forced
KdV equation, the transcritical internal wave flows support a diverse family of upstream
and downstream wave structures, including kinks, rarefaction waves, classical undular
bores, reversed and trigonometric undular bores, which we describe using the recent
development of the nonlinear modulation theory for the (unforced) Gardner equation.
The predictions of the developed analytic construction are confirmed by direct numerical
simulations of the forced Gardner equation for a broad range of input parameters.
Description: This paper was accepted for publication in the Journal of Fluid Mechanics and the definitive version is available at: http://dx.doi.org/10.1017/jfm.2013.556Tue, 01 Jan 2013 00:00:00 GMThttps://dspace.lboro.ac.uk/2134/142792013-01-01T00:00:00Z