As computer generated special effects play an increasingly integral role in the development of films and other media, simulating granular material continues to be a challenging and resource intensive process. Solutions tend to be pieced together in order to address the complex and different behaviors of granular flow. As such, these solutions tend to be brittle, overly specific, and unnatural. With the introduction of a holistic 3-phase granular simulation, we can now create a reliable and adaptable granular simulation.Our solution improves upon this hybrid solution by addressing the issue of particle flow and correcting interpenetration amongst particles while maintaining the efficiency of the overall simulation. We achieve this by projecting particles onto a 2D manifold and implementing density correction using a quadratic solver. Particle updates are projected back into 3D to spread the particles apart on each frame.Keywords:

We present a numerical model of Bridgman crystal growth. Pseudo-incompressibility constraint is used to handle jumps in density during phase change. ALE formulation is employed to account for moving parts of the system. Field equations and movement of material interfaces are decoupled in fractional step manner. Naviér-Stokes problem is extended to solid phase where no flow is enforced by Darcy-like forcing. Latent heat of phase change is added to effective heat capacity as approximate Dirac-$\delta$. Backward Euler discretization in space and P2/P1/P1 in space are used. Transient and stationary solutions are being found and compared to temperatures measured directly inside a steady system. Influence of pull-rates on growth process and shape of phase interface are being examined. Powered by TCPDF (www.tcpdf.org)

The research described in this thesis began with a single long-term objective: modelling of the vertebral artery during chiropractic manipulation of the cervical spine. Although chiropractic treatment has become prevalent, the possible correlation between neck manipulation and subsequent stroke in patients has been the subject of debate without resolution. Past research has been qualitative or statistical, whereas resolution demands a fundamental understanding of the associated mechanics.<p> Analysis in the thesis begins with a study of the anatomy and properties pertinent to the chiropractic problem. This indicates that the complexity of the problem will necessitate a long-term multidisciplinary effort including a nonlinear finite element formulation effective in analysing image data for soft tissue modelled as nearly incompressible. This leads to an assessment of existing finite element methods and the conclusion that new equation solving techniques are needed to ensure numerical stability.<p> Three techniques for effectively eliminating the source of numerical instability are developed and demonstrated with the aid of original finite element codes. Two of the methods are derived as modifications of matrix decomposition algorithms, while the third method constitutes a new finite element formulation. In addition, the understanding gained in developing these methods is used to produce a theorem for assessing a different but related problem: deformation of a nearly incompressible material subjected to a single concentrated force. Throughout the thesis, an interdisciplinary path from chiropractic problem to numerical algorithms is outlined, and results are in the form of mathematical proofs and derivations of both existing and new methods.

The research described in this thesis began with a single long-term objective: modelling of the vertebral artery during chiropractic manipulation of the cervical spine. Although chiropractic treatment has become prevalent, the possible correlation between neck manipulation and subsequent stroke in patients has been the subject of debate without resolution. Past research has been qualitative or statistical, whereas resolution demands a fundamental understanding of the associated mechanics.<p> Analysis in the thesis begins with a study of the anatomy and properties pertinent to the chiropractic problem. This indicates that the complexity of the problem will necessitate a long-term multidisciplinary effort including a nonlinear finite element formulation effective in analysing image data for soft tissue modelled as nearly incompressible. This leads to an assessment of existing finite element methods and the conclusion that new equation solving techniques are needed to ensure numerical stability.<p> Three techniques for effectively eliminating the source of numerical instability are developed and demonstrated with the aid of original finite element codes. Two of the methods are derived as modifications of matrix decomposition algorithms, while the third method constitutes a new finite element formulation. In addition, the understanding gained in developing these methods is used to produce a theorem for assessing a different but related problem: deformation of a nearly incompressible material subjected to a single concentrated force. Throughout the thesis, an interdisciplinary path from chiropractic problem to numerical algorithms is outlined, and results are in the form of mathematical proofs and derivations of both existing and new methods.

The stretch forming process is frequently used in the automotive industry (outer pan- els, inner panels, stiffeners etc.), the packaging industry and household appliances sector, to manufacture complicated shapes and curvatures. However it requires accurate prediction of tool geometries and manufacturing parameters to avoid the currently used trial and error approach. Metal forming is also associated with cer- tain defects like local thinning, wrinkling, tearing etc. Avoiding such defects and prediction of spring back presumably requires a thorough understanding of the de- formation mechanics and material behavior beyond the elastic range. In the stretch forming operation, material essentially passes through the elastic, yield point and plastic states. Elastic behavior can be explained based on classical theory of elasticity wherein linear trend of infinitesimal deformation is expressed by generalized Hooke’s law. In the plastic range, the theory is based on certain exper- imental observations of the macroscopic behavior of metals in the uniform state of combined stresses. Experimentally observed results are idealized into mathematical formulation to describe the complex behavior of metals under combined state of stress. These formulations are based on some assumptions like material behavior is time independent, strain rate effects could be neglected, hysteresis loop and Bauschinger effects which arise from the non-uniformity of the microscopic scale could be disregarded etc. The thermal effects are neglected and material is assumed to be isotropic. Supposedly because of these assumptions existing theory of plastic- ity does not accurately predict the phenomenon of stretch forming occurring during plastic deformation. Theories are being developed like that of Rao and Shrinivasa [2002], which consider stresses during deformation as resistance due to shape change, volume change, rate of shape change and rate of volume change. Such theories need variation of material parameters like bulk modulus (K), shear modulus (G), bulk viscosity (µ’) and shear viscosity (µ) as deformation progreses. Therefore uni-axial tension exper- iments have been conducted to find out the strains at the corresponding loads. Mild steel and aluminum have been chosen for the experiments. Chemical and physical properties of the materials are chosen such that they are very similar to those used in the automotive industry for stretch forming. A procedure is developed using uni-axial tension test results to calculate the material parameters for the entire range of material deformation. For mild steel, bulk modulus and shear modulus decrease and become almost zero as the material deforms from elastic to transition region. After transition zone, both moduli increase and then decrease as material deforms in the strain-hardening region. For aluminum both bulk and shear moduli decrease non-linearly as material deforms from elastic to plastic region. The behavior of bulk modulus and shear modulus are consistent with the stress-strain behavior of the materials. For mild steel as well as aluminum, the bulk and shear viscosities are positive in the elastic region and in the large deformation region the values are small compared to elastic region. We can separate the various stresses, hydrostatic, deviatoric and viscous stresses, associated with (µ) and (µ’) and contribution of each to the total stresses can be obtained. It is observed that contribution from the viscous stresses is as high as 5 % when the material is subjected to large strain rate tests. The strain rate in stretch forming operation may be different from the strain rate at which the material parameters are calculated. Knowing the material para- meters at one strain rate, the stress-strain curves at different strain rates can be predicted. The repeatability of computation of the material parameters and contributions from the viscous and non-viscous stresses for large deformation has been ascertained by using different test samples. The material parameters obtained from one set of samples have been applied to different samples and experimental versus predicted stresses have been found to match fairly well. A lot more work needs to be done to reach the goal of accurately predicting the behavior during stretch forming. Test data on different materials need to be generated and the new theories need to be validated for compression as well as loading and unloading cases.

Ziel dieser Arbeit ist die Simulation der Deformation von Bauteilen, welche aus nahezu inkompressiblem Material bestehen. Dabei soll sich das Material sowohl linear als auch nichtlinear elastisch verhalten können. Zusätzlich soll die Belastung des Bauteils beliebig gewählt werden können, das heißt, es sollen kleine als auch große Deformationen möglich sein.

<p>Within this study, low order finite elements were applied to problems with (near) incompressible material behaviour. Solutions were obtained for creep, using transient and dynamic iterative solvers with volumetric strain enhancement algorithms, as well as a flow solution obtained using the fractional step method. To enhance creep algorithm performance, a radial return procedure was implemented.</p> <p>Preliminary results show that the fraction step method and dynamic iterative solver implementing dynamic relaxation provided adequate results, while other methods required improvement. Volumetric strain enhancement was insufficient to correct pressure drift when using transient analysis. The fractional step method was able to provide good results, but is sensitive to time step and initial stress field.</p> <p>A thorough evaluation of convergence criteria was conducted through tracking of norms and errors. The trend of norms was used to evaluate the number of iterations required to reach steady-state. The solution acquired from the method of successive approximations was improved and quality pressure plots were obtained, in contrast to the experience from the preliminary analysis.</p> <p>An analysis of the Barnes ice cap was conducted to verify formulation performance in the context of a real problem. Dynamic relaxation provided results closest to the measured trend and values.</p> / Master of Applied Science (MASc)

Cette étude s'intéresse au vieillissement thermique d'élastomères synthétiques, amorphes, vulcanisés et additionnés de charges (noir de carbone). Sur la base d'une même formulation matériau, on étudie plus particulièrement, l'impact du procédé initial (conditions de vulcanisation), les conséquences de l'évolution physico-chimique des matériaux sur le comportement mécanique et l'influence d'un chargement mécanique permanent durant le vieillissement. Des caractérisations mécaniques (essais cycliques, relaxations par paliers, essais de compressibilité) et physico-chimiques (suivi des variation de dimension et de masse, essais de gonflement dans un solvant) sont réalisées afin de quantifier l'impact du vieillissement. Le phénomène dominant étant une augmentation de la densité de réticulation (maturation des ponts polysulfures en ponts disulfures ou monosulfures). D'une manière générale la partie expérimentale a permis de formuler un certain nombre d'hypothèses (isotropie, insensibilité de certaines caractéristiques physiques aux vieillissement, etc.) qui ont guidé le développement d'un modèle multiphysique. Ce modèle s'appuie sur une approche thermo-chimio-mécanique formulée dans le cadre de la thermodynamique des processus irréversibles avec introduction de variables internes afin de traduire d'une part les non-linéarités de comportement de ce type de matériau (grandes déformations, viscoélasticité non-linéaire et effet Payne), et d'autre part de décrire l'évolution physico-chimique du réseau macromoléculaire (qui dépend de la température et de l'état mécanique). Cette modélisation a permis d'introduire un couplage réciproque entre les états physico-chimique et mécanique / This study mainly concerns the thermal aging phenomenon in amorphous synthetic rubbers, initially vulcanized and filled with carbon blacks. On the basis of one material formulation, we study: the impact of the process (vulcanization condition), the influence of the chemo-physical evolution of the cross-linked network on the mechanical behavior and the influence of a permanent mechanical load during aging. Mechanical characterizations (cyclic, relaxation and hydrostatic tests) and chemo-physical ones (variation of mass and volume, swelling in solvent) are realized in order to quantify the impact of ageing. The main phenomena observed is an increase of the crosslink density (maturation of polysulfides to monosulfide or disulfide crosslinks). From a general point of view, we can formulate many hypothesis from the experimental characterizations (isotropy, non-dependence of some physical properties on ageing, etc.). For the modeling, we have adopted a themo-chemo-mechanical approach that is based on the thermodynamics of irreversible processes and the introduction of internal variables in order to phenomelogically describe on one hand the nonlinear mechanical behavior at finite strain (nonlinear viscoelasticty, Payne effect, etc.) and on the other hand the chemo-physical evolution of the macromolecular network (which depends on temperature and mechanical state). This approach has permitted to introduce a reciprocal coupling between chemo-physical and mechanical states

For years, the modelling of the brain tissue has been widely investigated. Indeed, a biomechanical model of the brain would find applications in fields like image-guided neurosurgery or safety system design (helmets, car manufacturing, security fences design,...). In the present thesis, we propose an original, fractional calculus-based, constitutive equation of the brain tissue. Different types of experiments (unconfined compression, relaxation and cyclic tests) were carried out in order to characterize the mechanical behaviour of the brain tissue. The use of the digital image correlation enabled us, for the first time to our best knowledge, to prove the incompressibility of the brain tissue, a widespread assumption. The constitutive model was calibrated and validated using these experiments. The agreement between the model and the experimental results is very satisfactory. The model reproduces with only a few percent error the stress-strain curves for the compression tests at three different loading rates covering two orders of magnitude as well as the behaviour in the relaxation and cyclic tests. La modélisation du tissu cérébral fait l'objet de nombreuses recherches depuis quelques années. En effet, un modèle biomécanique du cerveau trouverait des applications dans le domaine de la neurochirurgie guidée par l'image et chez les fabricants de système de sécurité (casques, automobiles, barrières de sécurité,...). Dans cette thèse, nous proposons un modèle d'équation constitutive original pour le tissu cérébral basé sur l'utilisation des dérivées d'ordre réel arbitraire (aussi connues sous le terme de dérivées fractionnaires). Divers types d'essais (compression, relaxation et essais cycliques) ont également été réalisés afin de caractériser le comportement mécanique du tissu cérébral. L'utilisation de la corrélation d'images numériques a permis, pour la première fois à notre connaissance, de démontrer expérimentalement son incompressibilité, justifiant ainsi une hypothèse couramment rencontrée. La loi constitutive développée a été calibrée et validée à partir de ces essais. L'accord entre le modèle et les résultats expérimentaux est très satisfaisant en ce qui concerne le comportement en compression à trois vitesse de déformation différentes s'étalant sur deux ordres de grandeur. Il en va de même pour les comportements en relaxation et lors des essais cycliques.

This thesis investigates the numerical simulation of three-dimensional, mechanical deformation problems in the context of large deformations. The main focus lies on the prediction of non-linearly elastic, incompressible material. Based on the equilibrium of forces, we present the weak formulation of the large deformation problem. The discrete version can be derived by using linearisation techniques and an adaptive mixed finite element method. This problem turns out to be a saddle point problem that can, among other methods, be solved via the Bramble-Pasciak conjugate gradient method or the minimal residual algorithm. With some modifications the resulting simulation can be improved but we also address remaining limitations. Some numerical examples show the capability of the final FEM software. In addition, we briefly discuss the special case of linear elasticity with small deformations. Here we directly derive a linear weak formulation with a saddle point structure and apply the adaptive mixed finite element method. It is shown that the presented findings can also be used to treat the nearly incompressible case.