Many an approach to the estimation of magnetic source depths from magnetic data has been investigated over the past half a century. These approaches have been shown to have particular strengths and weaknesses with few implemented on a wide scale, commercial basis. A review of many of the more popular, as well as a few of the more obscure methods, is presented within this work. The history of multi-scale computation, with emphasis on its application to potential fields is summarized. A newly developed depth estimation technique dubbed Multi-Scale Tilt Depth Estimation is offered. The method has been shown to derive suitable depth estimates when applied to modelled data computed from a range of simple synthetic models. Sensitivity of the method to model type, dip, interference and noise has been tested. A number of mitigating strategies to improve and stabilize the method’s performance have been proposed. Results of the successful application of the method to field datasets from the Bushveld Complex and surrounding areas in South Africa are shown. Code to execute the method, written in Matlab is offered in Appendix A. Figures of the application of the method to all synthetic models have been included in Appendix B and C. A portion of this work has been presented at the South African Geophysical Association’s 11th Biennial Technical Meeting and Exhibition in the form of verbal and poster presentations accompanied by a short paper which is included here in Appendix D.

Multiscale (analytical and computational) models have been developed based on a thermodynamic variational principle (TVP) to model sintering and eventual mudcracking in thermal barrier coatings (TBCs) made using the electron beam physical vapour deposition (EB-PVD) process. It is assumed that the sintering occurs by interfacial diffusion at local contacts between columns and driven by changes in interface free energy and elastic stored energy of the coating. The models link diffusional processes at the scale of contacting feathery columns with the macroscopic deformation and sintering response. In service, the columns can come into contact and sinter together. As sintering progresses there is a build up of strain energy in the system which reduces the driving force for sintering and leads to either complete or incomplete sintering of the TBC depending on the magnitude of effective modulus (E) of the coating. By seeding the coating with initial imperfections, different types of behaviour are observed depending on the value of E and the spacing between imperfections. For compliant coatings, the response is insensitive to the presence of imperfections and the coating fully sinters. At higher values of E, strain energy is released by the development of intercolumnar cracks in the coating, which can propagate to the interface with the TGO (thermally grown oxide), deflect into the interface and propagate, leading to spallation of regions of the coating and loss of thermal protection. It is observed that cracks develop at initial imperfections in the structure. The greater the spacing between imperfections the faster the development of cracks at these locations. If a TBC contains a distribution of imperfections there is progressive formation of cracks, with the average spacing decreasing with time, after an initial incubation period. The crack density eventually saturates to a constant value, which depends on the mechanical properties of the TBC. Initially, a crack spacing, CS, in the range 1.5H ≤ CS ≤ 3H has been predicted based on trapezoidal contact models. Here H is the thickness of the coating. Crack spacing predicted using this model is consistent in the lower range of experimentally observed crack spacing. However, axisymmetric contact models predict a crack spacing, CS, in the range 4H ≤ CS ≤ 8H, which is in good agreement with experimentally observed crack spacing range 3H ≤ CS ≤ 10H reported in the literature. Compared to the trapezoidal contact models, axisymmetric contact models more accurately predict the sintering response.

A macroscale Internal State Variable (ISV) constitutive model coupling magnetism effects with thermal, elastic, and damage effects is developed. Previous models for magnetic and mechanical fields included constitutive equations describing their effects on the material system studied independently. Some models explain the mechanisms behind mechanical deformations caused by magnetization changes that are presented in the literature. They mainly focus on the nanoscale level. Other models, describe the behavior of one specific magnet that is mostly a permanent magnet. However permanent magnets are made of rare-earth elements that are subjected to a high supply risk. In attempt to find an alternative to permanent magnets, a mathematical model that captures the physical behavior of magnets is needed, to help develop a tool to create a new permanent magnet. The ISV constitutive model herein describes the macroscale mechanical deformation caused by magnetic fields on ferromagnetic materials, Iron (Fe), Cobalt (Co) and Nickel (Ni) precisely. The ISV model internally coheres the kinematic, thermodynamic, and kinetic relationships of deformation using the evolving histories of internal variables. For the kinematics, a multiplicative decomposition of deformation gradient is employed including a magnetization term, and the Jacobian that represents the conservation of mass and conservation of momentum. The First and Second Law of Thermodynamics are used to constrain the appropriate constitutive relations through the Clausius-Duhem inequality. The kinetic framework employs a stress-strain relationship with a flow rule that couples the thermal, mechanical, and damage terms. To determine the ISVs needed to mimic the behavior of magnetic materials, we conducted various magnetic experiments on three different specimens made of Iron, Nickel and Cobalt. Experiments captured the mechanical deformation of a rod sample when subjected to a magnetic field using the Michelson Interferometer. To study the magnetic hysteresis of Iron, Nickel, and Cobalt, previous literature data were used. It was shown that the magnetization equation modeled the hysteresis of Iron, Nickel, and Cobalt. The magnetostrictive strain equation shows good agreement for Nickel and Cobalt, but further investigation should be done for Iron.

L'industrie de la microélectronique est en perpétuelle évolution, surtout concernant la diminution des dimensions des composants. Ainsi, pour les nouvelles générations de mémoires non volatiles Flash, le poly-silicium de la grille flottante pourrait être remplacé par des nanoplots discrets de silicium. L'élaboration de ces nanoplots par LPCVD (Low Pressure Chemical Vapor Deposition) à partir de silane SiH4 sur un substrat amorphe SiO2 demeure l'une des voies privilégiées par l'industrie. Le fonctionnement de ce type de mémoires est fortement dépendant des conditions de synthèse des nanoplots de silicium. Ce travail de cette thèse visait donc à améliorer la compréhension des mécanismes de nucléation et de croissance en jeu. Nous avons étudié les premiers instants de la nucléation en chimie quantique, grâce à l'utilisation de la théorie DFT, en considérant l'oxyde de silicium comme surface de dépôt. Des lois cinétiques intrinsèques ont été tirées de ces résultats DFT et elles ont été implémentées dans un modèle de simulation à l'échelle du procédé industriel, sur la base du code de CFD Fluent. Pour la nucléation, il est apparu que seul le silylène, SiH2, peut se chimisorber à la surface du substrat. De plus, sa faible concentration et la première désorption de H2, qui est très lente, expliquent le temps d'incubation. Pour la croissance, le caractère auto-catalytique des dépôts a été expliqué par la contribution très forte du silane au dépôt dès la seconde chimisorption. L'étape limitant la croissance est clairement la désorption de H2. La réalisation d'essais expérimentaux et la comparaison avec le modèle multi-échelles issu de notre travail a permis d'expliquer pourquoi les cinétiques classiques de la littérature surestiment la vitesse de dépôt des nanoplots. Il est aussi apparu que la vitesse de dépôt du silicium sur des nanoplots en croissance est plus forte que celle d'un film de silicium continu « épais ». La prise en compte des sites de chimisorption lors des premiers instants et la description détaillée de la désorption de H2 sont des paramètres clés pour rendre compte du comportement des dépôts de nanoplots de silicium. / The need of high integrated systems of the everyday life involves a permanent evolution of the microelectronic industry. Integrated circuits involving non volatile Flash memories are good examples of these trends. In this technology, the poly-silicon floating gate could be replaced by a discrete trap floating gate in which discrete traps are made up of silicon nanodots. The synthesis of nanodots by LPCVD (Low Pressure Chemical Vapor Deposition) from silane SiH4 on SiO2 surfaces remains one of the most promising ways of industrial synthesis. Despite a huge experimental effort, fundamental understanding of the key mechanisms of nanodots nucleation and growth remains elusive. Here we find the main objectives of the thesis. For nucleation, our main results reveal that only silylene SiH2 is involved in the very first steps of nucleation. The incubation time experimentally observed can be explained by the low SiH2 concentration and the first slow H2 desorption process. For growth, silane is the main responsible for deposition, which explains the autocatalytic behaviour of silicon deposition. The growth limiting step is clearly the H2 desorption process. Comparisons between experimental and multiscale modelling allow to explain why classical kinetics of the literature overestimate nanodots deposition rate. We have found that the silicon deposition rate is higher on nanometer silicon dots than on a continuous silicon film. Key parameters to conveniently model nanodots deposition are good descriptions of the first chemisorption sites and of the H2 desorption process.

Trabecular bone has a complex and porous microstructure. This study develops approaches to determine the mechanical behaviour of this material at the macroscopic level through the use of homogenisation-based multiscale methods using micro-finite element simulations. In homogenisation-based finite element methods, a simulation involving a representative volume element of the microstructure of the considered material is performed with a specific set of boundary conditions. The macroscopic stresses and strains are retrieved as averaged quantities defined over this domain. Most of the homogenisation-based work on trabecular bone has been performed to study its macroscopic elastic regime, and therefore define its constant macroscopic stiffness tensor. The rod and plate-shaped microstructure of trabecular bone can be precisely identified with advanced scanning tools, such as micro-computed tomography devices. Taking into account the size requirements to achieve a certain independence of boundary conditions for trabecular bone in a homogenisation-based multiscale setting, the resulting stack of images can have around ten million solid voxels after binarisation. Although a completely linear finite element simulation with such a large system may be feasible with commercial packages (with the proper time and memory requirements), it is not possible to perform a nonlinear simulation for such a mesh in a reasonable time frame, and the amount of required memory may not be available. A highly scalable parallel driver program which solves finite strain elastoplastic systems was developed within the framework of the existing parallel code ParaFEM. This code was used throughout this study to evaluate the yield and post-yield properties of trabecular bone. It was run on cutting edge high performance computing platforms (BlueGene/Q at the Hartree Centre, Science and Technology Facilities Council; and ARCHER, UK National Supercomputing Service, at Edinburgh Parallel Computing Centre). Micro-finite element simulations require definition of properties at the microscopic scale and it is unclear how these properties affect the macroscopic response. This study examines the effect of compressive hydrostatic yield at the microscopic scale on the macroscopic behaviour. Two different microscopic yield criteria, one permitting yielding at compressive hydrostatic stresses and the other not, were considered. A large number of load cases were examined. It was found that these two microscopic yield criteria only influence macroscopic yield behaviour in load scenarios which are compression-dominated; for other load cases, macroscopic response is insensitive to the choice of the microscopic yield criterion, provided it has an appropriate strength asymmetry. Also, in compression-dominated load cases, high density bone is much more sensitive as it is more like a continuum, resulting in the microscopic properties being more directly upscaled. Only a few previous studies have employed homogenisation to evaluate the macroscopic yield criterion of trabecular bone. However, they either used a simplified microscopic yield surface or examined only a small number of load cases. A thorough multiaxial evaluation of the macroscopic yield surface was performed by applying a wide range of loading scenarios (160 load cases) on trabecular bone samples. Closed-form yield surfaces with different symmetries (isotropy, orthotropy and full anisotropy) were fitted to the numerically obtained macroscopic yield points in strain space, and the fitting errors were evaluated in detail for different subsets of load cases. Although orthotropy and full anisotropy showed the smallest fitting errors, they were not significantly superior to the isotropic fit. Thus, isotropy in strain space presents itself as the most suitable option due to the simplicity of its implementation. The study showed that fitting errors do depend on the chosen set of load cases and that shear load cases are extremely important as it was found that even for these highly aligned samples, trabecular bone presents some degree of shear asymmetry, i.e. different strength in clockwise and counter-clockwise shear directions. There have been no previous attempts to evaluate the post-yield behaviour of trabecular bone through homogenisation-based studies on detailed micro-finite element trabecular bone meshes. A damage and plasticity constitutive law for the microscale based on existing data in the literature was considered. A homogenisation-based multiscale approach was used to evaluate the hardening and stiffness reduction at the macroscale when uniaxial load scenarios are applied to trabecular bone samples, for a small range of plastic strain Euclidean norms. Results show that damage progression at the macroscale for trabecular bone is not isotropic, which is contrary to what has been assumed previously, and that both the evolution of the yield surface and damage are different for tension, compression and shear. Nonetheless, they can be correlated with plastic strain Euclidean norms by using linear relationships. It was also observed that macroscopic damage in a specific load case affects differently the on-axis orthotropic stiffness and the off-axis orthotropic stiffness components. The findings of this study will permit the use of a more rigorous definition of the post-elastic macroscopic behaviour of trabecular bone in finite element settings.

Angiogenesis is the process by which new blood vessels develop from existing vessels. Angiogenesis is important in a number of conditions such as embryogenesis, wound healing and cancer. It has been modelled phenomenologically at the macroscale, using the well-known 'snail-trail' approach in which trailing endothelial cells follow the paths of other, leading endothelial cells. In this thesis, we systematically determine the collective behaviour of endothelial cells from their behaviour at the cell-level during corneal angiogenesis. We formulate an agent-based model, based on the snail-trail process, to describe the behaviour of individual cells. We incorporate cell motility through biased random walks, and include processes which produce (branching) and annihilate (anastomosis) cells to represent sprout and loop formation. We use the transition probabilities associated with the discrete model and a mean-field approximation to systematically derive a system of non-linear partial differential equations (PDEs) of population behaviour that impose physically realistic density restrictions, and are structurally different from existing snail-trail models. We use this framework to evaluate the validity of a classical snail-trail model and elucidate implicit assumptions. We then extend our framework to explicitly account for cell volume. This generates non-linear PDE models which vary in complexity depending on the extent of volume exclusion incorporated on the microscale. By comparing discrete and continuum models, we assess the extent to which continuum models, including the classical snail-trail model, account for single and multi-species exclusion processes. We also distinguish macroscale exclusion effects introduced by each cell species. Finally, we compare the predictive power of different continuum models. In summary, we develop a microscale to macroscale framework for angiogenesis based on the snail-trail process, which provides a systematic way of deriving population behaviour from individual cell behaviour and can be extended to account for more realistic and/or detailed cell interactions.

In this thesis we study a parameter identification problem for a stationary diffusion equation posed in heterogeneous media. This problem is closely related to the Calderón problem with anisotropic conductivities. The anisotropic case is particularly difficult and is ill-posed both in regards to uniqueness of solution and stability on the data. Since the present problem is posed in heterogeneous media, we can take advantage of multiscale modelling and the tools of homogenization theory in the study of the inverse problem, unlike the original Calderón problem. We investigate the possibilities of combining the theory of the Calderón problem with homogenization theory in order to obtain a well-posed parameter identification. We find that homogenization theory indeed can be used to make progress towards a well-posed identification of the diffusion coefficient. The success of the method is, however, dependent both on the precise structure of the heterogeneous media and on the modelling of the measurements in the invese problem framework. We have in mind a particular problem formulation which is motivated by an experiment to determine effective coefficients of materials used in food packaging. This experiment comes with a set of requirements on both the heterogeneous media and on the method for making measurements that, unfortunately, are in conflict with the currently available results for well-posedness. We study also an optimization approach to solving the inverse problem under these application specific requirements. Some progress towards well-posedness of the optimization problem is made by proving existence of minimizer, again with homogenization theory playing a key role in obtaining the result. In a proof-of-concept computational study this optimization approach is implemented and compared to two other optimization problems. For the two tested heterogeneous media, the only optimization method that manages to identify reasonably well the diffusion coefficient is the one which makes use of homogenization theory.

Cancer is a complex illness that affects millions of people every year. Amongst the most frequently encountered variants of this illness are solid tumours. The growth of solid tumours depends on a large number of factors such as oxygen concentration, cell reproduction, cell movement, cell death, and vascular environment. The aim of this thesis is to provide further insight in the interconnections between these factors by means of numerical simulations. We present a multiscale model for tumor growth by coupling a microscopic, agent-based model for normal and tumor cells with macroscopic mean-field models for oxygen and extracellular concentrations. We assume the cell movement to be dominated by Brownian motion. The temporal and spatial evolution of the oxygen concentration is governed by a reaction-diffusion equation that mimics a balance law.To complement this macroscopic oxygen evolution with microscopic information, we propose a lattice-free approach that extends the vascular distribution of oxygen. We employ a Markov chain to estimate the sprout probability of new vessels. The extension of the new vessels is modeled by enhancing the agent-based cell model with chemotactic sensitivity. Our results include finite-volume discretizations of the resulting partial differential equations and suitable approaches to approximate the stochastic differential equations governing the agent-based motion. We provide a simulation framework that evaluates the effect of the various parameters on, for instance, the spread of oxygen. We also show results of numerical experiments where we allow new vessels to sprout, i.e. we explore angiogenesis. In the case of a static vasculature, we simulate the full multiscale model using a coupled stochastic/deterministic discretization approach which is able to reduce variance at least for a chosen computable indicator, leading to improved efficiency and a potential increased reliability on models of this type.

A l'heure actuelle, les nouvelles contraintes de la réglementation thermique en vigueur ne cessent de s'adapter au contexte économique global pour lequel la recherche d'une efficacité énergétique dans le bâtiment est devenue incontournable. Pour répondre à ces défis, des Matériaux intelligents à Changement de Phase (MCP) ont fait leur apparition sur le marché de la construction. Grâce à leur capacité de stockage de l'énergie, les MCP sont de plus en plus associés aux matériaux de construction classiques (béton, plâtre, etc.) afin d'améliorer leur inertie thermique et apporter un meilleur confort aux usagers. Pour ce faire, les propriétés thermophysiques intrinsèques aux MCP doivent être suffisamment maitrisées afin de pouvoir contrôler les propriétés du produit composite final. Dans ce contexte, cette thèse est une contribution ayant pour objectif de développer des méthodologies spécifiques pour une meilleure caractérisation des MCP et des béton-MCP. Une panoplie d'approches expérimentales a été présentée pour l'identification des propriétés thermophysiques des MCP et pour identifier l'effet d'incorporation et de l'endommagement de ces matériaux sur les propriétés thermiques et mécaniques de béton. Plusieurs modèles d'homogénéisation ont été utilisés afin de prédire le comportement thermique des bétons-MCP en utilisant les propriétés thermiques moyennées obtenues expérimentalement. Une étude probabiliste paramétrique a été menée afin de prendre en compte les incertitudes liées à la dispersion aléatoire des mesures expérimentales de propriétés thermiques du béton-MCP. Les résultats issus des essais expérimentaux ont été intégrés dans le cadre d'une étude numérique par la Méthode des Volumes finis (MVF) afin d'étudier le mécanisme de transfert de chaleur à travers une paroi en béton-MCP / The thermal policies have been kept to fit the new economic in a global context particularly in terms of buildings energy efficiency. To meet these challenges, different technologies have been used such as the Phase Change Materials (PCMs) which have the ability to store and release energy. PCMs are generally used with conventional building materials in order to improve their thermal inertia and provide better comfort to users. To enhance the properties of the final composite, the PCMs thermo-physical properties must be sufficiently controlled. In this context, this thesis is a contribution aimed to develop specific methodologies for better characterization of PCM and PCM-concrete. Different experimental approaches will be presented for the identification of PCMs thermophysical properties and to identify the effect of the incorporation and the damage of these materials on the thermal and mechanical properties of concrete. A multiscale modelling considering the average of experimental thermal properties was applied to predict the thermal behaviour of PCMs-concrete. A probabilistic study of experimental uncertainties will be also conducted to assess the level of confidence of the impact of PCM on the thermodynamic properties of PCM-concrete. A numerical study was conducted using experimental data to study the heat transfer through a PCM-concrete wall

This thesis is concerned with the numerical simulation of high Reynolds number, three-dimensional, incompressible flows in open domains. Many problems treated in Computational Fluid Dynamics (CFD) occur in free space: e.g., external aerodynamics past vehicles, bluff bodies or aircraft; shear flows such as shear layers or jets. In observing all these flows, we can remark that they are often unsteady, appear chaotic with the presence of a large range of eddies, and are mainly dominated by convection. For years, it was shown that Lagrangian Vortex Element Methods (VEM) are particularly well appropriate for simulating such flows. In VEM, two approaches are classically used for solving the Poisson equation. The first one is the Biot-Savart approach where the Poisson equation is solved using the Green's function approach. The unbounded domain is thus implicitly taken into account. In that case, Parallel Fast Multipole (PFM) solvers are usually used. The second approach is the Vortex-In-Cell (VIC) method where the Poisson equation is solved on a grid using fast grid solvers. This requires to impose boundary conditions or to assume periodicity. An important difference is that fast grid solvers are much faster than fast multipole solvers. We here combine these two approaches by taking the advantages of each one and, eventually, we obtain an efficient VIC-PFM method to solve incompressible flows in open domain. The major interest of this combination is its computational efficiency: compared to the PFM solver used alone, the VIC-PFM combination is 15 to 20 times faster. The second major advantage is the possibility to run Large Eddy Simulations (LES) at high Reynolds number. Indeed, as a part of the operations are done in an Eulerian way (i.e. on the VIC grid), all the existing subgrid scale (SGS) models used in classical Eulerian codes, including the recent "multiscale" models, can be easily implemented.