I will be going over writing representations of both permutation and monomial progenitors, which include 2^{*4} : D_4, 2^(*7) :L_2 (7) as permutation progenitors, and monomial progenitors 7^(*2) :_m S_3 \times 2, 11^{*2} :_m (5:2)^{*}5, 11^{*3} :_m (25:3), 11^{*4} :_m (4 : 5)^{*}5. Also, the images of these different progenitors at both lower and higher fields and orders. \\ We will also do the double coset enumeration of S5 over D6, S6 over 5 : 4, A_5 x A_5 over (5:2)^{*}5, and go on to also do the double coset enumeration over maximal subgroups for larger constructions. We will also do the construction of sporadic group M22 over maximal subgroup A7, and also J1 with the monomial representation 7^(*2) :_m S_3 \times 2 over maximal subgroup PSL(2,11). We will also look at different extension problems of composition factors of different groups, and determine the isomorphism types of each extension.

The effectiveness of the Additive Correction Multigrid (ACM) algorithm, a line-byline Tri-diagonal Matrix Algorithm (TDMA), and simple Gauss-Seidel (GS) iteration in numerical heat transfer analysis is investigated on a conventional single processor computer and on a distributed memory parallel computer. The performance of these methods is studied by solving a two-dimensional, steady heat conduction problem. The execution time of ACM on a single processor is proportional to the number of unknowns to the 1.5 power. This is in contrast to the execution time of the TDMA for which the execution time is proportional to the number of unknowns to the 2.0 power. The GS , TDMA and ACM algorithms are adapted to a model IPSC2 Intel hypercube which has a 32 processing nodes each with 8 MBytes oflocal memory. Because GS is a local method, it has almost perfect speed up, but it also converges more slowly than TDMA, The TDMA, on the other hand, is affected by domain decomposition to a greater extent than GS. As the number of processors used to solve the problem is increased, the execution times for GS and TDMA are essentially equal. Solving the model problem with 32 processors on a 192x192 grid resulted in parallel efficiencies of 95%, 80% and 78% for the GS, TDMA, and ACM algorithms, respectively. Though the parallel efficiency of ACM was the lowest of the three, the parallel ACM algorithm required an order of magnitude less time to solve the model than either parallel GS or parallel TDMA without multigrid.

Numerical analyses of two different geotechnical problems, namely a bridge abutment and a geosynthetic reinforced embankment are presented in this thesis. Settlement, bearing capacity and slope stability are the major factors that need to be considered in the design of a foundation near a sloped fill. In this thesis, the behaviour of a small scale model footing located near the shoulder of a sloped fill was investigated numerically. Single and multiple layers of geogrid were used to reinforce the sloped fill, and their effects on the load-deformation behaviour and bearing capacity of the footing were explored. The analyses showed 80%, 168%, 295% and 375% maximum improvement in the ultimate bearing capacity with 1, 2, 3 and 4 reinforcement layers respectively. This maximum improvement depends on the embedment depths of the reinforcement layers below the foundation and the suggested optimal depths are discussed. Typically, greatest improvement in ultimate bearing capacity with a single layer of reinforcement was obtained when the reinforcement was at a depth between 0.50 and 0.75 times the foundation width. Similarly, highest ultimate bearing capacity with 2 reinforcement layers was predicted when the spacing between them was 1.0 times the width of the foundation. However, higher settlement was estimated at failure for the reinforced sloped fill than the unreinforced one. The second problem investigated was the three-dimensional (3D) analysis of Sackville embankment, a geosynthetic reinforced embankment on soft soil. Previous analyses using two-dimensional (2D) numerical modelling of Sackville embankment indicated potential 3D effects affecting the performance of this embankment. Therefore, 3D analysis incorporating geometric variations of Sackville embankment foundation soil, anisotropic model for fluid flow, mobilization of geotextile stresses in minor direction and the boundary effects (lateral directions) were taken into account in this analysis. The predicted performance of Sackville embankment were compared with the field data and the previously reported 2D analysis results in terms of vertical and horizontal displacements and excess pore pressures in the foundation soil, and geotextile stresses, strains and displacements. Better overall predictions of the Sackville embankment performance was obtained from this 3D analysis than the previous analysis reported in the literature.

The understanding of the masticatory apparatus including its functional and structural relationship with other components of the cranium increasingly requires an interdisciplinary approach. Recently, "traditional biological sciences" such as anatomy, comparative biology, anthropology and evolution have increasingly meshed with elements from other domains, such as mechanical engineering and material sciences, which has resulted in new and exciting paradigms to be explored. This is particularly true in the field of craniofacial biomechanics yet there are still many unexplored issues and numerous questions that remain unanswered. Numerical modelling in general and Finite Element Analysis (FEA) in particular, represent a numerical experimental procedure to generate such information. Originally derived from the field of structural engineering, FEA has steadily permeated its way into craniofacial biomechanics and has proven itself as a most useful scientific tool. The present study introduces an engineering-based workframe for applying FEA to craniofacial biomechanical research in a comprehensive manner to cover the entire analytical spectrum, from developing questions to providing their solutions. The study is composed of two major experimental parts addressing both the linear elastic and the non-linear behaviour of some biomaterials encountered in the craniofacial arena. In the first part I analysed mandibular biomechanics using linear elastic models while in the second part I used nonlinear discrete models to determine the optimal elastic properties of the cervical restorative materials. Modern humans have a number of anatomical features that set us apart from our ancestors. Amongst these perhaps the most striking is the emergence of a protruding chin, otherwise absent in other archaic humans and hominids. While it has been shown that the chin has its embryological origins in the postnatal remodelling of bone in the area around the mandibular symphysis which produces the midline keel in the form of an inverted �T� the functional significance of this novel evolutionary feature is still obscure. It is accepted that the mandible is optimally designed for resisting masticatory stress, whereby optimal is seen as maximual strength at the lowest biological cost. Here, I tested the currently most accepted theory, namely that the chin provides mechanical resistance to the mandible during mastication. In other words, I tested the hypothesis that a chinned mandible would be stiffer and hence experience lower strains when compared to a non-chinned counterpart under identical loadings. My functional analysis consisted firstly of three simple models which reproduce a simian shelf, a flat and a chinned symphysis, loaded using two unidirectional loadcases (torsion and wishboning) to represent a distortion similar to that which occurs in the mandible during mastication. Secondly, I developed complex geometrical models which incorporated the cortical bone, medullary bone and teeth. The models were then analysed using the same loadcases as those used for the first theoretical models. Additionally, I incorporated the coronal bending and also a coupled loadcase which simulated the complex deformation of the mandible during biting. The aim here was to test the hypothesis that the presence of a chin changed the strain pattern in the mastication-loaded mandible. The results were then interpreted using Frost�s mechanostat theory which relates in a more precise manner the mechanical loading environment to the adaptive response of the bone. My results showed that the calculated strain values for both the chinned and flat mandibles were within the normal bone maintenance levels of the mechanostat during molar biting. In other words, variation in bone strain magnitude across the mandible, which should differ between the chinned and the non-chinned mandibles if the hypothetical mechanical role of the chin is true, is similar in both forms. I concluded that the development of the human chin is thus unrelated to the functional demands placed upon it by mastication. I suggested a new functional demand associated with pronounced tongue activity during speech. I hypothesise that it is the resistance to stresses induced by strong, repetitive contractions of the tongue and perioral musculature during, phonation that shaped the modern human chin. I tested my hypothesis by loading the symphyseal region with two principal nonmasticatory, muscle systems; firstly, the tongue and secondly the peri-oral muscular curtain, anterior to the symphysis. My results suggested that the flat, non-chinned symphysis when subjected to speech-related genioglossal movements will undergo adaptive changes which would result in an optimised (chinned) shape, such as that found in the modern human symphysis. These results thus offer a new foundation to an old hypothesis and a solution to the longstanding controversy over the origin of the human chin. I conclude that forces generated by speech rather than those generated by mastication, shaped the chin in anatomically modern humans. Prompted by an earlier observation I further investigated the apparent cross-over distribution of strains on the mandibular corpora during mastication. In doing so, I tested the hypothesis that this cross-over may be linked with another particular anatomical feature of the mandible that of the postcanine cortical asymmetry, which appears to be stereotypical among anthropoids. The results of my study hence suggest that strain patterns within the human mandible are more complex than previously thought. Not only do strains differ between lingual and buccal aspects of working and non-working sides, but they also differ within these areas (i.e. from alveolus to corpus, to lower border regions). I conclude that postcanine cortical asymmetry may be a retained evolutionary trait rather than the result of masticatory biomechanics. In the second section of the thesis I introduced a different analysis regime which allows the prediction of fracture initiation and propagation. In this part I analysed the mechanics underlying the failure of the restorations placed in non-carious cervical lesions and suggested changes in the material properties of the restorations used to treat them. Non-carious cervical lesions (NCCL) include those entities characterised by the cervical loss of hard dental tissue that occurs in the absence of any carious process. To distinguish between lesions that occur due to excessive occlusal load and other non-carious cervical lesion (i.e. erosion and abrasion) the clinical term "abfraction" has been adopted. Although a common clinical issue, failure of restoration placed in these lesions has not been subjected to a rigorous biomechanical analysis. To determine which of the material�s parameters should be changed and to what extent, I employed a combined numerical approach. Here I introduced a novel approach in simulating the cracking of restorative materials and tooth tissues which is based on a simpler material formulation and can be used in an advanced nonlinear numerical analysis. The material model I used allows automatic crack insertion and growth and also uniquely accounts for the microdamage which precedes the instalment of macroscopic cracks. The first step was to balance the factors that may affect failure employing a linear analysis with a stress-based approach to failure. Here, the aim was to investigate the influence of lesion shape and depth as well as the direction of occlusal loading on the mechanical response of the cervical glass-ionomer cements restoration in a lower first premolar. This analysis showed that the direction of loading was the major contributor to the failure of the restoration. The next step was to apply this fracture model to the restorations of the NCCL in order to verify if the material is able to accurately simulate the location and type of mechanical failure. The data for this problem, i.e. the geometry and the loadcase were derived from the conclusions of linear analysis, that is I chose the "worst case scenario" as the upper boundary of material endurance. My results showed that under the action of para-functional loadings the GIC failed on the cervical margin. I also showed that prior to fracture the restorative material undergoes strain softening, which in turn introduces damage and weakens the materials involved. After successfully testing the proposed model, the final step was to determine which material properties and restorative techniques would be most reliable under given biomechanical conditions. The present work relied on the hypothesis that a more flexible material would partially buffer the local stress concentration and hence reduce the likelihood of mechanical failure of the restoration. My study, a first of its kind, proposes a radical approach to address the problems of material improvement, namely: numerical-based material optimisation engineering. That is, I aimed to identify the "most favourable" selection of elastic modulus or E value for the restorative material, which will allow it to survive under the unfavourable occlusal loading conditions that may prevail. Two filling techniques were considered; firstly a single bulk material, namely glass-ionomer (GIC) and secondly a layered technique. The latter consisted of a layer of GIC supporting a composite bulk restorative. I chose two thicknesses for the GIC layer, 50 and 150 microns. My results showed that the restorative materials currently used in cervical non-carious lesions are largely unsuitable in terms of resistance to fracture of the restoration mostly because of their relative high stiffness irrespective of the filling technique. The best results are obtained for a bulk filling with a 1GPa elastic modulus material case in which the tensile stresses are about 50% of the failure limit. This approach in determining the mechanical properties of the restorative is novel and unique so far in the dental literature. The direct benefit of this study was the improvement of the restorative material, as it can be engineered to withstand the conditions identified as major cause of failure. This is consonant with the call for new materials better tailored for some specific needs.

In Victoria, Australia, brown coal is utilised as a major source of energy for the power generation industry. Victorian and South Australian brown coals have a very high moisture content and therefore, the efficiencies of power generation in traditional pulverised fuel fired furnaces are low. Fluidised beds offer a number of advantages over conventional furnaces, leading to improvements in efficiency and environmental impact. A disadvantage with implementing fluidised bed technology is the issue of scale-up. Fluidised bed behaviour can alter significantly with changes in scale, because of their strong dependence on the bed hydrodynamics. Hence, there is a need to accurately model bed behaviour to ensure that the effect of changes in scale are well understood and will not become costly and time consuming. Computational Fluid Dynamics (CFD) techniques can be applied to fluidised bed systems to gain a better understanding of the hydrodynamic behaviour involved. In the past, numerical models have considered only single particle sizes due to the added complexity of interaction between particles of differing sizes and densities. Industrial fluidised beds typically contain more than one particle size and density, therefore there is a need to develop a numerical model which takes this into account. The aim of this thesis is to develop and validate CFD techniques for modelling the behavior of a gas-solid fluidised bed containing more than one particle size and density. To provide validation data for the numerical model, physical experiments are undertaken on a small two-dimensional bubbling gas-solid fluidised bed. Mixing and segregation behaviour of different materials are investigated. The experiments demonstrate that whilst only a small proportion of the bed consists of different size/density particles, significant changes in bed behaviour are apparent. Changes in bubble rise velocity, bubble size and bubble shape are observed. A number of constitutive equations must be included in the numerical model, including relationships for the momentum transfer between various phases and solids pressure. Different combinations of these constitutive equations are investigated. A new equation for particle-particle interactions is derived and included in a CFD model. The CFD model is validated against both data in the literature and physical experiments. From the validation studies, an optimum equation set is identified. This optimum equation set produces numerical results that closely resemble experimental bed behaviour, thus bringing the goal of solving scale-up problems one step closer. The use of this type of CFD model will ultimately result in timely and cost effective solutions for both the power generation and chemical processing industries.

The objective of this thesis is to develop a more efficient solver for a large system of linear equations arising from finite element discretization on unstructured tetrahedral meshes for a scalar elliptic partial differential equation of second order for pressure in a commercial computational fluid dynamics (CFD) simulation. Segregated solution methods (or pressure correction type methods) are a widely used approach to obtain solutions of Navier-Stokes equations during numerical simulation by many commercial CFD codes. At each time step, these simulations usually require the approximate solution of a series of scalar equations for velocity, pressure and temperature. Even if the simulation does not require high-accuracy approximations, the large systems of linear equations for pressure may not be efficiently solved. The matrices of these systems of linear equations of real-life industry problems often strongly violate weak diagonal dominance and the numerical simulation often requires solutions of very large systems with over a few hundred thousands degrees of freedom. These conditions produce very ill-conditioned systems of linear equations. Therefore, it is very difficult to solve such systems of linear equations efficiently using most currently available common iterative solvers. A survey of solvers for systems of linear equations was undertaken to determine the preferred solution methodology. An algebraic multigrid preconditioned conjugate gradient (AMGPCG) method solver was chosen for these problems. This solver uses the algebraic multigrid (AMG) cycle as a preconditioner for the conjugate gradient (CG) method. The disadvantages of the conventional AMG method are an expensive setup time and large memory requirements, particularly for three dimensional problems. The disadvantage of an expensive setup time needs to be overcome because the simulation usually requires only low-accuracy approximations for pressure. Also it is important to overcome the disadvantage of the large memory requirements for use in commercial software. In this work, an efficient AMGPCG solver is developed by overcoming the disadvantages of the conventional AMG method. The robustness of AMGPCG is shown theoretically so that the solver is always convergent. Optimum or close to optimum rates of convergence behavior for the solver are shown numerically so that the number of necessary iterations to obtain the estimated solution is approximately independent of mesh resolution. Furthermore, numerical experiments of solving pressure for some industry problems were carried out and compared with other efficient solvers including a fast commercial AMGPCG solver (SAMG, release 20b1). It was found that the developed AMGPCG solver was the fastest among these solvers for solving these problems and its algorithm has been numerically proven to be efficient. In addition, the memory requirement is at an acceptable level for commercial CFD codes.

Mes travaux de thèse portent sur le développement de schémas numériques d'ordre élevé en temps et en espace pour la simulation de la propagation des ondes. Nous avons proposé de discrétiser dans un premier temps l'équation des ondes par rapport au temps, en utilisant une technique de type équation modifiée. Puis nous avons utilisé une méthode d'éléments finis de type Galerkine discontinue pour la discrétisation en espace. En modifiant l'ordre de la discrétisation, nous avons construit des schémas tout aussi précis que ceux déjà existants pour un coût de mise en oeuvre très intéressant. Après avoir validé numériquement la nouvelle méthode, nous nous sommes intéressés à sa stabilité ainsi qu'à son adaptivité en temps et en espace. Pour arriver à cela, nous avons dû faire une étude précise de la stabilité de la méthode de Galerkine discontinue et nous avons proposé des améliorations à cette technique entraînant des gains de temps significatifs.

Les travaux présentés dans ce mémoire concernent des équations aux dérivées partielles issues de la physique des plasmas ou de la mécanique des fluides. En amont, ils comportent une partie importante de modélisation : approximation de systèmes hyperboliques oscillants, dérivation de systèmes de types Zakharov, modèle Hele-Shaw, modèle de micelles géantes. En aval, ils abordent un certain nombre de problèmes théoriques : existence et unicité des solutions, stabilité/instabilité des ondes solitaires, controle optimal, estimations d'erreur et convergence de modèles. Ils explorent aussi des méthodes numériques de résolution qui ont toutes pour point commun d'utiliser des méthodes de volumes finis ou différences finies sur des grilles cartésiennes en utilisant éventuellement des méthodes de pénalisation.

La modélisation des réacteurs nucléaires repose sur la résolution de l'équation de Boltzmann linéaire. Nous nous sommes intéressés à la résolution spatiale de la forme stationnaire de cette équation. Après discrétisation en énergie et en angle, l'équation hyperbolique est résolue numériquement par la méthode des éléments finis discontinus. Le solveur MINARET utilise cette méthode sur un maillage triangulaire non structuré afin de pouvoir traiter des géométries complexes (comprenant entre autres des arcs de cercle). Cependant, l'utilisation d'arêtes droites introduit une approximation de la géométrie. Autoriser l'existence d'arêtes courbes permet de coller parfaitement à la géométrie, et dans certains cas de diminuer le nombre de triangles du maillage. L'objectif principal de cette thèse est l'étude d'éléments finis sur des triangles possédant un ou plusieurs bords courbes. Le choix des fonctions de base est un des points importants pour ce type d'éléments finis. Un résultat de convergence a été obtenu sous réserve que les triangles courbes ne soient pas trop éloignés des triangles droits associés. D'autre part, un solveur courbe a été développé pour traiter des triangles avec un, deux ou trois bords courbes. Une autre partie de ce travail porte sur l'accélération de la convergence des calculs. En effet, la résolution du problème est itérative et peut, dans certains cas, converger très lentement. Une méthode d'accélération dite DSA (Diffusion Synthetic Acceleration) permet de diminuer le nombre d'itérations et le temps de calcul : un calcul de diffusion est ajouté à chaque itération. L'opérateur de diffusion est un préconditionneur de l'opérateur de transport. La DSA a été mise en oeuvre en utilisant une technique issue des méthodes de pénalisation intérieure. Une analyse de Fourier en 1D et 2D permet d'évaluer l'accélération dans le cas de milieux infinis périodiques et de vérifier la stabilité du schéma lorsque de fortes hétérogénéités existent.

Dans cette thèse, nous avons étudié la simulation numérique des phénomènes d'interaction fluide-structure entre un fluide compressible et une structure déformable. En particulier, nous nous sommes intéressés au couplage par une approche partitionnée entre une méthode de Volumes Finis pour résoudre les équations de la mécanique des fl uides compressibles et une méthode d'Éléments discrets pour le solide, capable de prendre en compte la fissuration. La revue des méthodes existantes de domaines fictifs ainsi que des algorithmes partitionnés couramment utilisés pour le couplage conduit à choisir une méthode de frontières immergées conservative et un schéma de couplage explicite. Il est établi que la méthode d'Éléments Discrets utilisée permet de retrouver le comportement macroscopique du matériau et que le schéma symplectique employé assure la préservation de l'énergie du solide. Puis nous avons développé un algorithme de couplage explicite entre un fluide compressible non-visqueux et un solide indéformable. Nous avons montré des propriétés de conservation exacte de masse, de quantité de mouvement et d'énergie du système ainsi que de consistance du schéma de couplage. Cet algorithme a été étendu au couplage avec un solide déformable, sous la forme d'un schéma semi-implicite. Cette méthode a été appliquée à l'étude de problèmes d'écoulements non-visqueux autour de structures mobiles : les comparaisons avec des résultats numériques et expérimentaux existants démontrent la très bonne précision de notre méthode.