Mathemateg a Ffiseg / Mathematics & Physics
http://hdl.handle.net/2160/14
Sat, 23 Jul 2016 04:19:02 GMT2016-07-23T04:19:02ZControl of open quantum systems
http://hdl.handle.net/2160/43312
Control of open quantum systems
Arenz, Christian
Known as decoherence, the unavoidable interaction of a quantum system with its
surrounding environment is usually considered to be detrimental for quantum information
processing. In this thesis the coherent, open loop control of such open
systems is studied. Concepts from quantum control theory and the theory of open
quantum system are adopted in order to fight decoherence and implement quantum
gates in a noiseless manner. In particular, Lie algebraic methods and numerical
optimization tools are used to investigate the control properties of a single spin interacting
with a spin environment. We show that, independent of the size of the
environment, every unitary transformation can be implemented on the system spin
through a single control field. We proceed by investigating dynamical decoupling,
a method to suppress the interactions with the environment, for finite- and for infinite
dimensional systems. We prove that every finite dimensional system can be
protected from decoherence, even if the environment is infinite dimensional, whereas
for noise described by a Lindblad master equation dynamical decoupling will never
succeed. This will lead to a new method to distinguish decoherence from intrinsic
noise terms. We further prove that not every infinite dimensional system can be
protected from decoherence through dynamical decoupling. Afterwards we investigate
dynamical decoupling of systems that are described by quadratic Hamiltonians,
showing that such interactions can always be suppressed with two simple operations.
In the last part we investigate the coherent control of a Lindblad master equation.
We show that a strong noise process exhibiting a decoherence free subspace can
substantially increase the number of unitary operations that can be implemented,
allowing us to fully control parts of the system. Afterwards we develop a scheme to
make Hamiltonians and Lindbladians commutative by adding an auxiliary system.
The old, possibly non-commutative dynamics, is recovered through a non-selective
measurement.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/2160/433122016-01-01T00:00:00ZA computational study of some rheological influences on the "splashing experiment"
http://hdl.handle.net/2160/43292
A computational study of some rheological influences on the "splashing experiment"
Tome, M. F.; McKee, S.; Walters, Ken
In various attempts to relate the behaviour of highly-elastic liquids in complex flows to their rheometrical behaviour, obvious candidates for study have been the variation of shear viscosity with shear rate, the two normal stress differences N(1) and N(2), especially N(1), the extensional viscosity, and the dynamic moduli G' and G ''. In this paper, we shall confine attention to 'constant-viscosity' Boger fluids, and, accordingly, we shall limit attention to N(1), eta(E), G' and G ''. We shall concentrate on the "splashing" problem (particularly that which arises when a liquid drop falls onto the free surface of the same liquid). Modern numerical techniques are employed to provide the theoretical predictions. We show that high eta(E) can certainly reduce the height of the so-called Worthington jet, thus confirming earlier suggestions, but other rheometrical influences (steady and transient) can also have a role to play and the overall picture may not be as clear as it was once envisaged. We argue that this is due in the main to the fact that splashing is a manifestly unsteady flow. To confirm this proposition, we obtain numerical simulations for the linear Jeffreys model. (C) 2010 Elsevier B.V. All rights reserved.
Fri, 01 Oct 2010 00:00:00 GMThttp://hdl.handle.net/2160/432922010-10-01T00:00:00ZWeight function approach to studying perfect and imperfect interfaces in anisotropic and piezoelectric bimaterials
http://hdl.handle.net/2160/42808
Weight function approach to studying perfect and imperfect interfaces in anisotropic and piezoelectric bimaterials
Pryce, Lewis
The focus of the thesis is interfacial crack problems in anisotropic
and piezoelectric bimaterials. We seek to solve a variety of problems
using weight function techniques and singular integral equations.
We begin by studying a dynamic crack along a perfectly bonded
interface in an anisotropic bimaterial. Using a weight function de-
rived from a mirrored problem it is possible to derive important ma-
terial parameters which govern the crack propagation. Following this
a static crack is considered. However, in this case the materials are not
bonded perfectly, an imperfect interface is present instead. A method
is derived where singular integral equations for the imperfect interface
problem are derived through use of perfect interface weight functions.
The weight functions are then extended to fracture in piezoelectric
bimaterials which allows equivalent integral equations to be derived
relating the mechanical and electrical elds. In past literature a num-
ber of results have been found which can only be used when consider-
ing a symmetric load system on the crack faces. All of the problems
considered here have asymmetric loading.
Firstly, a steady-state formulation is used to derive asymptotic
coe cients of the crack displacement and interfacial tractions for a
dynamic crack along a perfect interface. The method can be used to
nd many asymptotic coe cients but the one of most importance here
is the stress intensity factor which therefore enables the calculation
of energy release rate at the crack tip. As an example an orthotropic
bimaterial with two di erent loading con gurations is used to examine
the importance of crack speed and load asymmetry on the properties
of the crack propagation.
iv
We proceed to study imperfect interface conditions for an anisotropic
bimaterial. Usually when looking at such a problem it is necessary to
derive new weight functions which correspond to the imperfect inter-
face. An innovative method which makes use of the Betti formula and
existing weight functions for the analogous perfect interface problem
is derived. This procedure is used to obtain singular integral equations
which relate the crack loading, which is assumed to be known, to the
displacement jump over both the crack and interface and tractions
along the bonded area between the materials. Examples of the results
obtained through solving the integral equations numerically are given.
Finally, we extend the weight functions used previously in the the-
sis to a piezoelectric setting. The general form of the weight function
for any piezoelectric bimaterial is given before two speci c examples
are studied in depth. The examples are chosen in such a way to illus-
trate the e ect that the poling direction of the bimaterial can have on
both the mechanical and electrical elds. For both examples explicit
expressions are derived for the weight functions which are then used
to derive singular integral equations which can be used to study the
e ect of both mechanical loading and electrical charges being applied
to the crack faces. To nish we present some examples for both poling
directions to illustrate the use of the derived equations.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/2160/428082015-01-01T00:00:00ZMathematical and numerical modeling of hydraulic fractures for non-Newtonian fluids
http://hdl.handle.net/2160/42786
Mathematical and numerical modeling of hydraulic fractures for non-Newtonian fluids
Perkowska, Monika
The aim of this thesis is to construct an accurate and effective numerical algorithm to solve a problem of hydraulic fracturing for non--Newtonian fluids of power--law type rheology. Also, to improve the existing semi--analytical solutions using the recent advances in the area.
The theoretical background is presented, along with the practical applications and physical processes driving the fracture growth. Equations used to mathematically formulate the problem are derived. A full mathematical description is supplemented by boundary and initial conditions. Basic 1D models are presented for various crack propagation regimes.
Mechanisms used extensively throughout this work are thoroughly analysed, amongst which are: (i) numerical integration, (ii) regularisation techniques, (iii) implementation of boundary conditions, (iv) utilisation of appropriate dependent variables, (v) asymptotic behaviour of solution, (vi) appropriate meshing strategy. Each of these methods is described in detail and investigated.
A universal particle velocity based algorithm for simulating hydraulic fractures is proposed. The computations are based on two dependent variables: the crack opening and the reduced particle velocity. The application of the latter facilitates utilisation of the local condition of Stefan type (speed equation) to trace the fracture front. The condition is given in a general explicit form which relates the crack propagation speed (and the crack length) to the solution tip asymptotics.
A number of analytical benchmark solutions are derived. They are employed to validate the computational accuracy of the proposed algorithm. Moreover, the performance of the numerical scheme is tested against other solutions available in the literature.
Following the analysis of the performance of the algorithm, new improved approximations are provided for each model. They can be used as benchmark solutions for testing other algorithms.
The extensive computations prove that the numerical scheme is stable and efficient. It provides accurate results for various hydraulic fracturing models and regimes, as well as fracturing fluids. The algorithm works equally well for time--independent and transient versions of the problem. The utilisation of a modular structure and the adaptive character of its basic blocks result in a flexible numerical scheme.
The numerical code has been provided as a supplementary attachment to the electronic version of this thesis, and is also available on request from the author.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/2160/427862016-01-01T00:00:00Z