Many partial differential equations (PDEs) admit conserved quantities like mass or energy. Those quantities are often essential to establish well-posed results. When approximating a PDE by a finite-difference scheme, it is natural to ask whether related discretized quantities remain conserved under the scheme. Such conservation may establish the stability of the numerical scheme. We present an algorithm for checking the preservation of a polynomial quantity under a polynomial finite-difference scheme. In our algorithm, schemes can be explicit or implicit, have higher-order time and space derivatives, and an arbitrary number of variables. Additionally, we present an algorithm for, given a scheme, finding conserved quantities. We illustrate our algorithm by studying several finite-difference schemes.

<p>We consider existence and stability of breather solutions to discrete nonlinear Schrodinger (dNLS) and discrete Klein-Gordon (dKG) equations near the anti-continuum limit, the limit of the zero coupling constant. For sufficiently small coupling, discrete breathers can be uniquely extended from the anti-continuum limit where they consist of periodic oscillations on excited sites separated by "holes" (sites at rest).</p> <p>In the anti-continuum limit, the dNLS equation linearized about its discrete breather has a spectrum consisting of the zero eigenvalue of finite multiplicity and purely imaginary eigenvalues of infinite multiplicities. Splitting of the zero eigenvalue into stable and unstable eigenvalues near the anti-continuum limit was examined in the literature earlier. The eigenvalues of infinite multiplicity split into bands of continuous spectrum, which, as observed in numerical experiments, may in turn produce internal modes, additional eigenvalues on the imaginary axis. Using resolvent analysis and perturbation methods, we prove that no internal modes bifurcate from the continuous spectrum of the dNLS equation with small coupling.</p> <p>Linear stability of small-amplitude discrete breathers in the weakly-coupled KG lattice was considered in a number of papers. Most of these papers, however, do not consider stability of discrete breathers which have "holes" in the anti-continuum limit. We use perturbation methods for Floquet multipliers and analysis of tail-to-tail interactions between excited sites to develop a general criterion on linear stability of multi-site breathers in the KG lattice near the anti-continuum limit. Our criterion is not restricted to small-amplitude oscillations and it allows discrete breathers to have "holes" in the anti-continuum limit.</p> / Doctor of Philosophy (PhD)

Master of Science / Department of Grain Science and Industry / Kingsly Ambrose / It is a well-known phenomenon that the break-release, particle size and size distribution of wheat milling are functions of machine operational parameters and grain properties. Due to the non-uniformity in characteristics and properties of wheat kernel, the kernel physical and mechanical properties may affect the size reduction process. The discrete element method (DEM) is a numerical modeling technique that can be used to study and understand the effect of physical and mechanical properties of a material on processing. The overall objective of this study is to develop a DEM model of the 1st break wheat milling process. In this study, different physical and mechanical properties of wheat mill streams were determined for using as the input parameters in DEM model development. The particle size and size distribution (PSD), true, bulk and tapped density, young’s modulus, coefficient of static and rolling friction, and coefficient of restitution were measured for wheat kernel, 1st break and flour from hard red winter (HRW), hard red spring (HRS), and soft red winter (SRW) wheat. Overall moisture content was found to have a greater significant effect on the physical properties i.e. density and PSD of the mill streams than material properties i.e. Young’s modulus, coefficients of friction and coefficient of restitution. The DEM model of 1st break wheat milling was developed using both single and multi-sphere approaches. The single sphere approach simulated the size reduction of a spherical cluster of bonded particles with mono-sized particles. The model was simulated for hard red winter (HRW) wheat milling at 16% moisture levels and validated using lab scale milling trials giving a PSD of 437.73 m with a percent deviation of prediction of 235.37. The deviation of prediction was reduced to 192.29 with a mean PSD of 371.52 m by conducting sensitivity analysis by modifying the shear modulus and coefficient of restitution values. In the multi-sphere approach, a bonded cluster resembling a wheat kernel in shape and size was used to simulate the milling process. The model predicted a 1st break PSD of 412.65 µm which had a deviation of 185.89 from lab scale and 156.78 from plant scale milling. The model could however satisfactorily predict the variation in PSD from 1st break milling with moisture content with reasonable accuracy. Future capabilities using the model include performing additional sensitivity analysis to understand the effect of other mechanical properties of wheat on the 1st break PSD. It can also be used to improve the 1st break release during wheat milling.

The vast majority of the research efforts in finite capacity scheduling over the past several years has focused on the generation of precise and almost exact measures for the working schedule presupposing complete information and a deterministic environment. During execution, however, production may be the subject of considerable variability, which may lead to frequent schedule interruptions. Production scheduling mechanisms are developed based on centralised control architecture in which all of the knowledge base and databases are modelled at the same location. This control architecture has difficulty in handling complex manufacturing systems that require knowledge and data at different locations. Adopting biological control principles refers to the process where a schedule is developed prior to the start of the processing after considering all the parameters involved at a resource involved and updated accordingly as the process executes. This research reviews the best practices in gene transcription and translation control methods and adopts these principles in the development of an autonomous finite capacity scheduling control logic aimed at reducing excessive use of manual input in planning tasks. With autonomous decision-making functionality, finite capacity scheduling will as much as practicably possible be able to respond autonomously to schedule disruptions by deployment of proactive scheduling procedures that may be used to revise or re-optimize the schedule when unexpected events occur. The novelty of this work is the ability of production resources to autonomously take decisions and the same way decisions are taken by autonomous entities in the process of gene transcription and translation. The idea has been implemented by the integration of simulation and modelling techniques with Taguchi analysis to investigate the contributions of finite capacity scheduling factors, and determination of the ‘what if’ scenarios encountered due to the existence of variability in production processes. The control logic adopts the induction rules as used in gene expression control mechanisms, studied in biological systems. Scheduling factors are identified to that effect and are investigated to find their effects on selected performance measurements for each resource in used. How they are used to deal with variability in the process is one major objective for this research as it is because of the variability that autonomous decision making becomes of interest. Although different scheduling techniques have been applied and are successful in production planning and control, the results obtained from the inclusion of the autonomous finite capacity scheduling control logic has proved that significant improvement can still be achieved.

Many powders and particulate solids are stored and handled in large quantities across various industries. These solids often encounter handling and storage difficulties that are caused by the material cohesion. The cohesive strength of a bulk material is a function of its past consolidation stress. For example, high material cohesive strength as a result from high storage stresses in a silo can cause ratholing problems during discharge. Therefore, it is essential to consider the stress-history dependence when evaluating such handling behaviour. In recent years the Discrete Element Method (DEM) has been used extensively to study the complex behaviour of granular materials. Whilst extensive DEM studies have been performed on cohesionless solids, much less work exists on modelling of cohesive solids. The commonly used DEM models to model adhesion such as the JKR, DMT and linear cohesion models have been shown to have difficulty in predicting the stress-history dependent behaviour for cohesive solids. DEM modelling of cohesive solid at individual particle level is very challenging. To apply the model at single particle level accurately would require one to determine the model parameters at particle level and consider the enormous complexity of interfacial interaction. Additionally it is computationally prohibitive to model each and every individual particle and cohesion arising from several different phenomena. In this study an adhesive elasto-plastic contact model for the mesoscopic discrete element method (DEM) with three dimensional non-spherical particles is proposed with the aim of achieving quantitative predictions of cohesive powder flowability. Simulations have been performed for uniaxial consolidation followed by unconfined compression to failure using this model. Additionally, the scaling laws necessary to produce scale independent predictions for cohesionless and cohesive solids was also investigated. The influence of DEM input parameters and model implementation have been explored to study the effect of particle (meso-scale) properties on the bulk behaviour in uniaxial test simulation. The DEM model calibration was achieved using the Edinburgh Powder Tester (EPT) – an extended uniaxial tester to measure flowability of bulk solids. The EPT produced highly repeatable flowability measurements and was shown to be a good candidate for DEM model calibration. The implemented contact model has been shown to be capable of predicting the experimental flow function (unconfined compressive strength versus the prior consolidation stress) for a limestone powder which has been selected as a reference solid in the Europe wide PARDEM research network. Contact plasticity in the model is shown to affect the flowability significantly and is thus essential for producing satisfactory computations of the behaviour of a cohesive granular material. The model predicted a linear relationship between a normalized unconfined compressive strength and the product of coordination number and solid fraction. Significantly, it has been found that contribution of adhesive force to the limiting friction has a significant effect on bulk unconfined strength. Failure to include the adhesive contribution in the calculation of the frictional resistance may lead to under-prediction of unconfined strength and incorrect failure mode. The results provide new insights and propose a micromechanical based measure for characterising the strength and flowability of cohesive granular materials. Scaling of DEM input parameters in a 3D simulation of the loading regimes in a uniaxial test indicated that whilst both normal and tangential contact stiffness (loading, unloading, and load dependent) scales linearly with radius of the particle, the adhesive forces scales with the square of the radius of the particles. This is a first step towards a mesoscopic representation of a cohesive powder that is phenomenological based to produce the key bulk characteristics of a granular solid and the results indicate that it has potential to gain considerable computational advantage for large scale DEM simulations. The contact model parameters explored include particle contact normal loading stiffness, tangential stiffness, and contact friction coefficient. The DEM model implementation parameters included numerical time step, strain rate, and boundary condition. Many useful observations have been made with significant implications for the relative importance of the DEM input parameters. Finally the calibration procedure was applied to a spray dried detergent powder and the simulation results are compared to whole spectrum of loading regime in a uniaxial experiment. The experimental and simulation results were found to be in reasonable agreement for the flow function and compression behaviour.

Thesis (MSc)--University of Stellenbosch, 2011. / ENGLISH ABSTRACT: The nonlinear LULU smoothers excel at removing impulsive noise from sequences and possess a variety of theoretical properties that make it possible to perform a so-called Discrete Pulse Transform, which is a novel multiresolution analysis technique that decomposes a sequence into resolution levels with a large amount of structure, analogous to a Discrete Wavelet Transform. We explore the use of a one-dimensional Discrete Pulse Transform as the central element in a digital image compressor. We depend crucially on the ability of space-filling scanning orders to map the two-dimensional image data to one dimension, sacrificing as little image structure as possible. Both lossless and lossy image compression are considered, leading to five new image compression schemes that give promising results when compared to state-of-the-art image compressors. / AFRIKAANSE OPSOMMING: Die nielineêre LULU gladstrykers verwyder impulsiewe geraas baie goed uit rye en besit verskeie teoretiese eienskappe wat dit moontlik maak om ’n sogenoemde Diskrete Puls Transform uit te voer; ’n nuwe multiresolusie analise tegniek wat ’n ry opbreek in ’n versameling resolusie vlakke wat ’n groot hoeveelheid struktuur bevat, soortgelyk tot ’n Diskrete Golfie Transform. Ons ondersoek of ’n eendimensionele Diskrete Puls Transform as die sentrale element in ’n digitale beeld kompressor gebruik kan word. Ons is afhanklik van ruimtevullende skandeer ordes om die tweedimensionele beelddata om te skakel na een dimensie, sonder om te veel beeld struktuur te verloor. Vyf nuwe beeld kompressie skemas word bespreek. Hierdie skemas lewer belowende resultate wanneer dit met die beste hedendaagse beeld kompressors vergelyk word.

The crux of elliptic curve cryptography, a popular mechanism for securing data, is an asymmetric problem. The elliptic curve discrete logarithm problem, as it is called, is hoped to be generally hard in one direction but not the other, and it is this asymmetry that makes it secure. This paper describes the mathematics (and some of the computer science) necessary to understand and compute an attack on the elliptic curve discrete logarithm problem that works in a special case. The algorithm, proposed by Nigel Smart, renders the elliptic curve discrete logarithm problem easy in both directions for elliptic curves of so-called "trace one." The implication is that these curves can never be used securely for cryptographic purposes. In addition, it calls for further investigation into whether or not the problem is hard in general.

This thesis is a blend of neutron transport theory and numerical analysis. We start with the study of the problem of the Mika/Case eigenexpansion used in the solution process of the homogeneous one-speed Boltzmann neutron transport equation with anisotropic scattering for plane symmetry. The anisotropic scattering is expressed as a finite Legendre series in which the coefficients are the ``scattering coefficients'. This eigenexpansion consists of a discrete spectrum of eigenvalues with its corresponding eigenfunctions and the continuous spectrum [-1,+1] with its corresponding eigendistributions. In the general case where the anisotropic scattering can be of any (finite) order, multiple discrete eigenvalues exist and these have to be located to have the complete spectrum. We have devised a stable and robust method that locates all these discrete eigenvalues. The method is a two-step process: first the number of discrete eigenvalues is calculated and this is followed by the calculation of the discrete eigenvalues themselves, now being able to count them down and make sure none are forgotten. During our numerical experiments, we came across what we called near-singular eigenvalues: discrete eigenvalues that are located extremely close to the continuum and hence lead to near-singular behaviour in the eigenfunction. Our solution method has been adapted and allows for the automatic detection of such a near-singular eigenvalue. For the elements of the continuous spectrum [-1,+1], there is no non-zero function satisfying the associated eigenequation but there is a non-zero distribution that does satisfy it. It is not feasible to compute a distribution as such but one can evaluate integrals in which this distribution appears. The continuum part of the eigenexpansion can hence only be characterised by its (angular) moments. Accurate and fast numerical quadrature is needed to evaluate these integrals. Several quadrature methods have been evaluated on a representative test function. The eigenexpansion was proved to be orthogonal and complete and hence can be used to represent the infinite medium Green's function. The latter is the building block of the Boundary Sources Method, an integral solution method for the neutron transport equation. Using angular and angular/spatial moments of the Green's function, it is possible to solve with high accuracy slab problems. We have written a one-dimensional slab code implementing this Boundary Sources Method allowing for media with arbitrary order anisotropic scattering. Our results are very good and the code can be considered as a benchmark code for others. As a final application, we have used our code to study the discrete spectrum of a well-known scattering kernel in radiative transfer, the Henyey-Greenstein kernel. This kernel has one free parameter which is used to fit the kernel to experimental data. Since the kernel is a continuous function, a finite Legendre approximation needs to be adopted. Depending on the free parameter, the approximation order and the number of secondaries per collision, the number of discrete eigenvalues ranges from two to thirty and even more. Bounds for the minimum approximation order are derived for different requirements on the approximation: non-negativity, an absolute and relative error tolerance.

No description available.

Across industry the majority of raw materials handled are particulate in nature, ranging in size and properties from aggregates to powders. The stress regimes experienced by the granular solids vary and the exhibited bulk behaviours can be complex and unexpected. The prevalence of granular solids makes them an area of interest for industry and researchers alike as many challenges still remain, such as dealing with complex cohesive behaviour in materials, which often gives rise to handling difficulties. Storage and transportation are an important part of the process chain for industries where particulate solids are commonplace. Failure to properly account for the cohesive nature of a particulate solid can be costly as it can easily lead to blockages in a silo such as ratholing or arching near the outlet during discharge. The cohesive strength of a bulk material depends on the consolidation stress it has experienced. As a result, the stress history in the material leading up to a handling scenario needs to be considered when evaluating its handling behaviour. The Discrete Element Method (DEM) has been extensively used to simulate the behaviour of granular materials, however the majority of the focus has been on noncohesive systems. For cohesive solids, it is crucial that the stress history dependent behaviour is adequately captured. Many of the contact models commonly used in DEM simulations to simulate cohesive granular materials such as the JKR model or liquid bridge models are elastic in nature and may not capture the stress history dependent behaviour observed in cohesive particulate solids. A comprehensive study on the effect of cohesion arising from the addition of moisture on the behaviour of two types of LKAB iron ore fines (KPBO and KPRS) has been carried out. The addition of moisture to the sample has been found to have a significant effect on both kinds of fines. KPRS fines were found to have a much higher unconfined strength and flow function at higher moisture contents, and also show a greater increase in cohesion with the addition of moisture, while at moisture contents of less than 2% the KPBO fines demonstrate higher unconfined yield strength. The KPBO fines were also found to achieve a significantly looser initial packing at much lower moisture content when compared to the KPRS fines. The lateral pressure ratio has also been evaluated. In this study a mesoscopic adhesive contact model that accounts for contact plasticity and stress history dependency in the bulk solid, the Edinburgh Elasto-Plastic Adhesion (EEPA) mode, has been presented and mathematically verified. A parametric study of the DEM contact model parameters was conducted to gain a deeper understating of the effect of input parameters on the simulated cohesive bulk behaviour. The EEPA contact model has been used to predict an experimental flow function of KPRS iron ore fines. The contact model has demonstrated the ability to capture the stress history dependent behaviour that exists in cohesive granular solids. The DEM simulations provide a very close match to the experimental flow functions, with the predicted unconfined strengths found to be within the standard deviations of the experimental results. Investigations into the failure mode predicted by the DEM simulations show that the samples are failing from the development of shear planes similar to those observed experimentally. The effect of increasing levels of adhesion has been explored for a flat bottomed silo where the level of adhesion has been varied. The DEM simulations were found to capture the major phenomena occurring in silo discharge including the various flow zones associated with a flat bottomed silo. Funnel flow, the effective transition and mass flow which are associated with a mixed flow pattern were observed in the model silo. The location of the effective transition height was identified: above this was mass flow. The velocity determined from the discharge rate was found to be in excellent agreement with the velocity profiles found in the zones of mass flow. A high velocity core flow zone was observed above the outlet where velocities were greater than 1.25 times the mass flow velocity, VMF. The level of adhesion in the silo was found to affect the discharge rate - a reduced flow rate was found until the eventual blockage of the silo at a high level of adhesion was found. As the level of adhesion increased the probability of arching also increased, and the formation of intermittent arching behaviour was noted in the cases with higher levels of adhesion in the system. The development of both temporary and permanent cohesive arches over the silo outlet were also observed with stopped flow from the silo.