Hydraulic fracturing is a multi-physics multi-scale problem related to natural processes such as the formation of dikes. It also has wide engineering applications such as extraction of unconventional resources, enhanced geothermal energy and carbon capture and storage. Current simulators are highly simplified because of the assumption of homogeneous reservoir. Unconventional reservoirs are heterogeneous owing to the presence of natural fracture network. Because of high computational effort, three-dimensional multi-scale simulations are uncommon, in particular, modelling material as a heterogeneous medium. Lattice Element Method (LEM) is therefore proposed for multi-scale simulation of heterogeneous material. In LEM, material is discretised into cells and their interactions are modelled by lattices, hence a three-dimensional model is simplified to a network of one-dimensional lattice. Normal, shear and rotational springs are used to define the constitutive laws of a lattice. LEM enables desktop computers for simulation of a lattice model that consists of millions of lattices. From simulations, normal springs govern the macroscopic bulk deformation while shear springs govern the macroscopic distortion. There is fluctuation of stresses even under uniform loading which is one of the characteristics of a lattice model. The magnitude increases with the stiffness ratio of shear spring to normal spring. Fracturing process can be modelled by LEM by introducing a microscopic tensile strength and a microscopic shear strength to the lattice properties. The strength parameters can be related to fracture toughness with the length scales of cells. From simulations, the relationships between model parameters and macroscopic parameters that are measurable in experiments are identified. From the simulations of uni-axial tension tests, both the spring stiffness ratio and the applied heterogeneity govern the fracturing process. The heterogeneity increases the ductility at the expense of the reduction on the macroscopic strengths. Different stages of fracturing are identified which are characterised by the model heterogeneity. Heterogeneous models go through the stages of the spatially distributed microscrack formation, the growth of multiple fracture clusters to the dominant fracture propagation. For homogeneous models, one of the microcracks rapidly propagates and becomes a dominant fracture with the absence of intermediate stages. From the uni-axial compression test simulations, the peak compressive stress is reached at the onset of the microscopic shear crack formation. Ductility is governed by the stiffness reduction ratio of a lattice in closed fractured stage to its unfractured stage. A novel Dual Lattice Model (DLM) is proposed for hydraulic fracture simulation by coupling a solid lattice model with a fluid lattice model. From DLM simulations of hydraulic fracturing of the classical penny shape crack problem under hydrostatic condition, the heterogeneities from both the fracture asperity and the applied heterogeneity increase the apparent fracture toughness. A semi-analytical solution is derived to consider the effect of fluid viscosity in the elastic deformation regime. Two asymptotes are identified that gives steep pressure gradients near the injection point and near the fracture tip which are also identified in the DLM simulations. Simulations also show three evolving regimes on energy dissipation/transfer mechanisms: the viscosity dominant, the elastic deformation dominant and the mixture of elastic deformation and toughness.

No description available.

The spin-lattice relaxation rates are calculated for a rigid magnetic spin cluster in an elastic medium in the presence of a magnetic eld using the latticerame method. This rate is then compared with both the rate calculated using the quantum mechanical densitymatrix method and with the Glauber dynamics. These calculation results are used in the contribution of various heat baths, such as a phonon bath in various dimensions or a fermionic bath, to transition rates that enter into dynamic Monte Carlo simulations of molecular magnets and nanomagnets.

Doctor of Philosophy / Department of Mathematics / Gerald H. Höhn / We classify the genera of doubly even binary codes, the genera of even lattices, and the genera of rational vertex operator algebras (VOAs) arising from the modular tensor categories (MTCs) of rank up to 4 and central charges up to 16. For the genera of even lattices, there are two types of the genera: code type genera and non code type genera. The number of the code type genera is finite. The genera of the lattices of rank larger than or equal to 17 are non code type. We apply the idea of a vector valued modular form and the representation of the modular group SL[subscript]2(Z) in [Bantay2007] to classify the genera of the VOAs arising from the MTCs of ranks up to 4 and central charges up to 16.

We study various quotient maps between frames which are defined by stipulating that they satisfy certain conditions on the cozero parts of their domains and codomains. By way of example, we mention that C-quotient and C -quotient maps (as defined by Ball and Walters- Wayland [7]) are typical of the types of homomorphisms we consider in the initial parts of the thesis. To be little more precise, we study uplifting quotient maps, C1- and C2-quotient maps and show that these quotient maps possess some properties akin to those of a C-quotient maps. The study also focuses on R - and G - quotient maps and show, amongst other things, that these quotient maps coincide with the well known C - quotient maps in mildly normal frames. We also study quasi-F frames and give a ring-theoretic characterization that L is quasi-F precisely when the ring RL is quasi-B´ezout. We also show that quasi-F frames are preserved and reflected by dense coz-onto R -quotient maps. We characterize normality and some of its weaker forms in terms of some of these quotient maps. Normality is characterized in terms of uplifting quotient maps, -normally separated frames in terms of C1-quotient maps and mild normality in terms of R - and G -quotient maps. Finally we define cozero complemented frames and show that they are preserved and reflected by dense z#- quotient maps. We end by giving ring-theoretic characterizations of these frames. / Mathematical Science / D. Phil. (Mathematics)

We present a calculation of vector and axial-vector form factors for each of the octet hyperon semi-leptonic transition matrix elements by using the techniques of lattice QCD where simulations were performed with Nf = 2 + 1 flavours of dynamical O(a)-improved Wilson fermions. We also study the electromagnetic form factors, axial charges and other properties of octet baryons. Errors due to extrapolation to zero transferred momentum are reduced by applying a twist to the boundary conditions on the lattice. Our form factor results compare favourably with experiment and other lattice QCD determinations. By considering an expansion about the SU(3)-flavour symmetric limit we seek to investigate and quantify the symmetry breaking effects in these matrix elements due to the mass splitting between the strange and light quarks. We find good agreement with the Ademollo-Gatto theorem for the vector form factor, a measurable amount of breaking in the axial-vector form factor and significant effects in the weak magnetism form factor. Knowledge of the parameterisation of SU(3)-flavour symmetry breaking allows for a series of constrained fits to be made to the form factor results which are used to arrive at a 'baryonic' estimation of the Cabibbo-Kobayashi-Maskawa matrix element |Vus|.

Thesis (PhD (Mathematical Sciences))--Stellenbosch University, 2008 / Please refer to full text for abstract.

This report combines a series of papers to trace progression in the area of type-2 topoisomerase research. First, Deibler et al. show that knotted DNA is harmful to cells. Knots can block both transcription and replication, and can also act as a catalyst for mutation. Despite the fact that type-2 topoisomerases perform the important functions of unknotting and decatenating DNA, the mechanism by which they accomplish this task is still unknown. Buck and Zechiedrich propose a model in which the enzyme uses local geometry to infer global topology, and thus where to perform segment passage in order to obtain the desired results. In two articles, Liu et al. evaluate this theory quantitatively for the decatenation and unknotting problems. In both cases it is shown that the presence of certain juxtapositions is strongly correlated with global topology. This correlation is not enough, however, and Liu et al. go on to show that when segment passage operations designed to mimic type-2 topoisomerase action are performed at hooked juxtapositions, the overwhelming tendency is towards unknotting and decatenation. / text

No description available.

We calculate the strange quark content of the nucleon, <N|ss|N> − <0|ss|0> using a novel method with the MILC lattice QCD gauge ensembles. The strangeness of the nucleon is related to the interaction cross section between dark matter and ordinary nuclear matter (e.g. in detectors) in many models. Previous results for this quantity suffered from uncontrolled systematic errors and/or large statistical uncertainties. The first result using our methods was the first modern calculation of the strangeness of the nucleon[76] with good control of systematic errors and reasonably small statistical errors, greatly reducing the uncertainty in dark matter detection cross sections. A refinement of this method allows for further reduction of statistical error. On the MILC Asqtad data, we obtain <N|ss|N> = 0.637(55)(stat)(74)(sys). The results obtained from this method are consistent with those obtained from other commonly-used methods applied to the MILC data. We also calculate the disconnected part of the pion-nucleon sigma term and the intrinsic charm of the nucleon using this method. The intrinsic charm has large statistical errors but is consistent with a perturbative calculation.