Granular materials are in abundance in nature and are estimated to constitute over 75% of all raw materials passing through the industry. Granular or particulate solids are thus of considerable interest to many industrial sectors and research communities, where many unsolved challenges still remain. This thesis investigates the micro- and macro-phenomena in densely packed particulate systems by means of the Discrete Element Method (DEM), which is a numerical tool for analysing the internal complexities of granular material as the mechanical interactions are considered at the grain scale. It presents an alternative approach to phenomenological continuum approaches when studying localisation problems and finite deformation problems in granular materials. In order to develop a comprehensive theoretical understanding of particulate matter and to form a sound base to improve industrial processes, it is desirable to study the mechanical behaviour of granular solids subject to a variety of loading conditions. In this thesis, three loading actions were explored in detail, which are biaxial compression, rigid object penetration and progressive formation of granular piles. The roles of particle shape and contact friction in each of these loading scenarios were investigated. The resulting packing structures were compared and studied to provide a micromechanical insight into the development of contact force network which governs the collective response. The interparticle contact forces and displacements were then used to evaluate the equivalent continuum stress and strain components thus providing the link between micro- and macroscopic descriptions. The information collected from the evolution of strong contact network illustrates the underlying mechanism of force transmission and propagation. DEM simulations presented in this thesis demonstrate strong capability in predicting the bulk behaviour as well as capturing local phenomenon occurring in the system. The research first simulates a testing environment of biaxial compression in DEM, in which the phenomenon of strain localisation was investigated, with special attention given to the interpretation of underlying failure mechanism. Several key micromechanical quantities of interest were extracted to understand the bifurcation instability, such as force chains, contact orientation, particle rotation and void ratio. In the simulation of progressive formation of granular piles, a counterintuitive pressure profile with a significant pressure dip under the apex was predicted for three models under certain conditions. Both particle shape and preparation history were shown to be important in the resulting pressure distribution. During the rigid body penetration into a granular sample, the contact forces were used to evaluate the equivalent continuum stress components. Significant stress concentration was developed around the punch base which further led to successive collapse and reformation of force chains. Taking the advantage of micromechanical analysis at particle scale, two distinct bearing failure mechanisms were identified as the penetration proceeded. To further quantify the nature of strain mobilisation leading to failure, Particle Image Velocimetry (PIV) was employed to measure the deformation over small strain interval in association with shear band propagation in the biaxial test and deformation pattern in the footing test. The captured images from DEM simulation and laboratory experiments were evaluated through PIV correlation. This optical measuring technique is able to yield a significant improvement in the accuracy and spatial resolution of the displacement field over highly strained and localised regions. Finally, a series of equivalent DEM simulations were also conducted and compared with the physical footing experiments, with the objective of evaluating the capability of DEM in producing satisfactory predictions.

From pharmaceutical powders to agricultural grains, a great proportion of the materials handled in industrial situations are granular or particulate in nature. The variety of stesses that the matierals may experience and the resulting bulk behaviours may be complex. In agricultural engineering, a better understanding into agricultural processes such as seeding, harvesting, transporting and storing will help to improve the handling of agricultural grains with optimised solutions. A detailed understanding of a granular system is crucial when attempting to model a system, whether it is on a micro (particle) or macro (bulk) scale. As numerical capabilities are ever increasing, the Discrete Element Method (DEM) is becoming an increasingly popular numerical technique for computing the behaviour of discrete particels for both industrial and scientific applications. A look into the literature shows a lack of validation of what DEM can predict, specifically with respect to bulk behaviour. In addition, when validation studies are conducted, discrepancies between bulk responses in physical tests and numerical predictions using measured particles properties may arise. The aire of this research is to develop a methodology to calibrate DEM models for agricultural grains using data meaured in bulk physical tests. The methodology will have a wider application to granular solids in general and will advance understanding in the area of DEM model calibration. A contrasting set of granular materials were used to develop the methodology including 3 inorganic solids (single and paired glass beads, and polyethylene terephthalate pellets) and two organic materials (black eyes beans and black kidney beans). The developed methodology consists of three steps: 1. The development of bulk physical tests to measure the bulk responses that will be used to calibrate the DEM models, 2. The creation of the numerical dataset that will describe how the DEM input parameters influence the bulk responses , and 3. The optimisation of the DEM parameters using a searching algorithm and the results from Step 1 and 2. Two laboratory devices were developed to provide calibration data for the proposed methodology: a rotating drum and an confined compression test. These devices were chosen as they can produce bulk responses that are repeatable and easy to quantify, as well as generate discriminating results in numerical simulations when DEM parameters are varied. The bulk response determined from the rotating drum device was the dynamic angle of repose Ør formed when the granular material in a 40% filled drum is rotating at a speed of 7 rpm. the confined compression apparatus was used to determine the bulk stiffness of a system by monitoring the change in void ratio from the stress applied during a loading and unloading cycle. The gradient of the loading and unloadng curves termed λ and κ respectively were chosen as the bulk responses to calibrate the DEM models. The experimental results revealed that the dynamic Ør was significantly influences by the particle aspect ration and boundary conditions. The stiffness parameters were found to be predominantly influences by the initial packing arrangement. The numerical dataset describing how the DEM input parameters influence the numerical bulk responses was created by simulating the bulk physical tests, varying selected DEM parameters and monitoring the effects on bulk parameters. To limit the number of simulations required, design of experiment (DOE) methods were used to determine a reduced factorial matrix of simulations. In additions, an extensive parametric investigation on the non-optimised parameters as well as a scaling sensitivity study was carried out. The final step in determining the optimised parameters is to use a searching algorithm to infer the DEM parameters based on the numerical dataset and used the experimental results as calibration data. To perform a comparative study, tow searching algorithms were explored: the first was a simple method based on Microsoft Excel's Solver algorithm coupled with a weighted inverse distance method. The second made used of the statistical analysis program Statistica. It was shown that the Excel Solver algorithm is simpler and quicker to use but for the present first implementation, could only perform an optimisation based on two bulk responses. Statistica required the creation of a staistical model based on the numerical dataset before using the profiling and desirability searching technique, but was able to optimise the parameter using all three bulk responses. A verification and validation of the optimisation methodology was conducted using the optimised parameters for the black eyed beans. A verification was cnducted by simulating the two calibration experiments using the optimsed parameters and comparing these with the experiments. In addition, a validation was peformed by predicting the response of ta shallow footing penetration on a bed of black eyed beans. It was found that DEM simulations using optimised parameters predicted vertical stress on the footing during penetration to an acceptable degree of accuracy for industrial applications (<10%) at penetration depths up to 30mm.

This thesis presents a new bonded particle model that accurately predicts the wideranging behaviour of cementitious materials. There is an increasing use of the Discrete Element Method (DEM) to study the behaviour of cementitious materials such as concrete and rock; the chief advantage of the DEM over continuum-based techniques is that it does not predetermine where cracking and fragmentation initiate and propagate, since the system is naturally discontinuous. The DEM’s ability to produce realistic representations of cementitious materials depends largely on the implementation of an inter-particle bonded-contact model. A new bonded-contact model is proposed, based on the Timoshenko beam theory which considers axial, shear and bending behaviour of inter-particle bonds. The developed model was implemented in the commercial EDEM code, in which a thorough verification procedure was conducted. A full parametric study then considered the uni-axial loading of a concrete cylinder; the influence of the input parameters on the bulk response was used to produce a calibrated model that has been shown to be capable of producing realistic predictions of a wide range of behaviour seen in cementitious materials. The model provides useful insights into the microscopic phenomena that result in the bulk loading responses observed for cementitious materials such as concrete. The new model was used to simulate the loading of a number of deformable structural elements including beams, frames, plates and rings; the numerical results produced by the model provided a close match to theoretical solutions.

The main objective of this research was to model the airflow paths through grain bulks and predict the resistance to airflow. The discrete element method (DEM) was used to simulate the pore structures of grain bulks. A commercial software package PFC3D (Particle Flow Code in Three Dimension) was used to develop the DEM model. In the model, soybeans kernels were considered as spherical particles. Based on simulated positions (coordinates) and radii of individual particles, the characteristics of airflow paths (path width, tortuosity, turning angles, etc.) in the vertical and horizontal directions of the grain bed were calculated and compared. The discrete element method was also used to simulate particle packing in porous beds subjected to vertical vibration. Based on the simulated spatial arrangement of particles, the effect of vibration on critical pore structure parameters (porosity, tortuosity, pore throat width) was quantified. A pore-scale flow branching model was developed to predict the resistance to airflow through the grain bulks. Delaunay tessellation was also used to develop a pore network model to predict airflow resistance. Experiments were conducted to measure the resistance to airflow to validate the models. It was found that the discrete element models developed using PFC3D was capable of predicting the pore structures of grain bulks, which provided a base for geometrically constructing airflow paths through the pore space between particles. The tortuosity for the widest and narrowest airflow paths predicted based on the discrete element model was in good agreement with the experimental data reported in the literature. Both pore-scale models (branched path and network) proposed in this study for predicting airflow resistance (pressure drop) through grain bulks appeared promising. The predicted pressure drop by the branched path model was slightly (<12%) lower than the experimental value, but almost identical to that recommended by ASABE Standard. The predicted pressure drop by the network model was also lower than the measured value (2.20 vs. 2.44 Pa), but very close to that recommended by ASABE Standard (2.20 vs. 2.28Pa). / February 2017

During the twentieth century, particle physics developed into a cornerstone of modern physics, culminating in the Standard Model. Even though this theory has proved to be of extraordinary power, it is still incomplete in several respects. It is our aim in this bachelor thesis to discuss some possible theories beyond the Standard Model, the main focus being on Grand Unified Theories, while also taking a look at attempts of further unication via discrete family symmetry. At the heart of all these theories lies the concept of local gauge invariance, which is introduced as a fundamental principle, followed by an overview of the Standard Model itself. No theory has so far managed to unify all elementary particles and their interactions, but some interesting features are highlighted. We also give a hint at some possible paths to go in the future in the quest for a unication in particle physics. / Under 1900-talet utvecklades partikelfysiken till en av de fundamentala teorierna inom fysiken, och kom att sammanfattas i den s.k. Standardmodellen. Även om denna modell rönt exceptionella framgånger vad gäller beskrivningen av elementarpartiklar och deras växelverkan, är den fortfarande ofullständig på flera sätt. Syftet med denna kandidatuppsats är att diskutera möjliga teorier bortom Standardmodellen såsom Storförenande Teorier och diskreta familjesymmetrier vars avsikt är att koppla samman de tre familjerna av fermioner i Standardmodellen. Men först introduceras idén om lokal gaugeinvarians, vilken ligger till grund for dessa teorier, varpå en översikt av Standardmodellen följer. Ingen teori har ännu lyckats ge en helt tillfredsställande bild av elementarpartiklar och deras interaktion, men en del intressanta egenskaper hos föreslagna teorier belyses i denna uppsats. Slutligen ges en del spekulativa förslag på väger att gå i framtida försök till föreningar inom partikelfysiken.

The theory of auction has become an active research area spanning multiple disciplines such as economics, finance, marketing and management science. But a close examination of it reveals that most of the existing studies deal with ascending (i.e., English) auctions in which it is assumed that the bid increments are continuous. There is a clear lack of research on optimal descending (i.e., Dutch) auction design with discrete bid levels. This dissertation aims to fill this void by considering single-unit, open-bid, first price Dutch auctions in which the bid levels are restricted to a finite set of values, the number of bidders may be certain or uncertain, and a secret reserve price may be present or absent. These types of auctions are most attractive for selling products that are perishable (e.g., flowers) or whose value decreases with time (e.g., air flight seats and concert tickets) (Carare and Rothkopf, 2005). I began by conducting a comprehensive survey of the current literature to identify the key dimensions of an auction model. I then zeroed in on the particular combination of parameters that characterize the Dutch auctions of interest. As a significant departure from the traditional methods employed by applied economists and game theorists, a novel approach is taken by formulating the auctioning problem as a constrained mathematical program and applying standard nonlinear optimization techniques to solve it. In each of the basic Dutch auction model and its two extensions, interesting properties possessed by the optimal bid levels and the auctioneer's maximum expected revenue are uncovered. Numerical examples are provided to illustrate the major propositions where appropriate. The superiority of the optimal strategy recommended in this study over two commonly-used heuristic procedures for setting bid levels is also demonstrated both theoretically and empirically. Finally, economic as well as managerial implications of the findings reported in this dissertation research are discussed.

Supersymmetry is a theoretical model of particle physics that posits a symmetry between bosons and fermions. Supersymmetry proposes the existence of particles that we have not yet observed and through them, offers a more unified view of the universe. In the same way Feynman Diagrams represent Feynman Integrals describing subatomic particle behaviour, supersymmetry algebras can be represented by graphs called adinkras. In addition to being motivated by physics, these graphs are highly structured and mathematically interesting. No one has looked at the Jacobians of these graphs before, so we attempt to characterize them in this thesis. We compute Jacobians through the 11-cube, but do not discover any significant discernible patterns. We then dedicate the rest of our work to generalizing the notion of the Jacobian, specifically to be sensitive to edge directions. We conclude with a conjecture describing the form of the directed Jacobian of the directed $n$-topology. We hope for this work to be useful for theoretical particle physics and for graph theory in general.

In the setting of tropical mathematics, geometric objects are rich with inherent combinatorial structure. For example, each polynomial $p(x,y)$ in the tropical setting corresponds to a tropical curve; these tropical curves correspond to unbounded graphs embedded in $\R^2$. Each of these graphs is dual to a particular subdivision of its Newton polytope; we classify tropical curves by combinatorial type based on these corresponding subdivisions. In this thesis, we aim to gain an understanding of the likeliness of the combinatorial type of a randomly chosen tropical curve by using methods from polytope geometry. We focus on tropical curves corresponding to quadratics, but we hope to expand our exploration to higher degree polynomials.

In their recent paper ``Edge-transitive products," Hammack, Imrich, and Klavzar showed that the direct product of connected, non-bipartite graphs is edge-transitive if and only if both factors are edge-transitive, and at least one is arc-transitive. However, little is known when the product is bipartite. This thesis extends this result (in part) for the case of bipartite graphs using a new technique called "stacking." For R-thin, connected, bipartite graphs A and B, we show that A x B is arc-transitive if and only if A and B are both arc-transitive. Further, we show A x B is edge-transitive only if at least one of A, B is also edge-transitive, and give evidence that strongly suggests that in fact both factors must be edge-transitive.

Soil disturbance and cutting force are two of the most common performance indicators for soil-engaging tools. In this study the interaction of two soil-engaging tools (a disc opener for fertilizer banding and a hoe opener from an air drill) with soil were modeled using Particle Flow Code in Three Dimensions (PFC3D), a discrete element modeling software. When comparing the disc model to the experiment results, the relative error was 11% for the average soil throw, 1.9% for the average draft force, and 51% for the average vertical force. Results from the soil-hoe model showed a relative error of 15% between the simulated soil throw and the measured one. In conclusion, both the soil-disc and soil-hoe models could simulate the selected soil dynamic properties (except for the vertical forces of the disc opener) with a reasonably good accuracy, considering the highly variable nature of the soil. / October 2016