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This thesis uses the Immersed Boundary Method (IBM) to simulate the movement of a human heart. The IBM was developed by Charles Peskin in the 70’s to solve Fluid-Structure Interaction models (FSI). The heart is embedded inside a fluid (blood) which moves according to the Navier-Stokes equation. The Navier-Stokes equation is solved by the Spectral Method. Forces on the heart muscle can be divided into two kinds: Active Force and Passive Force. Passive includes the effect of curvature (Peskin’s model), spring model, and the torsional spring (or beam) model. Active force is modeled by the 3-element Hill model, which was used in the 30’s to model skeletal muscle. We performed simulations with different combinations of these four forces. Numerical simulations are performed using MATLAB. We downloaded Peskin’s code from the Internet and modified the Force.m file to include the above four forces. This thesis only considers heart muscle movement in the organ (macroscopic) scale.

State-of-the-art deep learning methods for outlier detection make the assumption that outliers will appear far away from inlier data in the latent space produced by distribution mapping deep networks. However, this assumption fails in practice,because the divergence penalty adopted for this purpose encourages mapping outliers into the same high-probability regions as inliers. To overcome this shortcoming,we introduce a novel deep learning outlier detection method, called Outlier Preserving Distribution Mapping Autoencoder (OP-DMA), which succeeds to map outliers to low probability regions in the latent space of an autoencoder. For this we leverage the insight that outliers are likely to have a higher reconstruction error than inliers. We thus achieve outlier-preserving distribution mapping through weighting the reconstruction error of individual points by the value of a multivariate Gaussian probability density function evaluated at those points. This weighting implies that outliers will result in an overall penalty if they are mapped to low-probability regions. We show that if the global minimum of our newly proposed loss function is achieved,then our OP-DMA maps inliers to regions with a Mahalanobis distance less than \delta, and outliers to regions past this \delta, \delta being the inverse ChiSquared CDF evaluated at 1−\alpha with \alpha the percentage of outliers in the dataset. We evaluated OP-DMA on 11 benchmark real-world datasets and compared its performance against 7 different state-of-the-art outlier detection methods, including ALOCC and MO-GAAL. Our experiments show that OP-DMA outperforms the state-of-the-art methods on 7 of the datasets, and performs second best on 3 of the remaining 4 datasets, while no other method won on more than 1 dataset.

Diffusion imaging is a relatively recent branch of magnetic resonance imaging that produces images of human physiology through diffusion of water molecules within the body. One difficulty in calculating diffusion coefficients, particularly in the brain, is the multitude of natural barriers to water diffusion, such as cell membranes, myelin sheaths, and fiber tracts. These barriers mean that water diffusion is not a homogeneous random process. Due to the complexity of modeling these structures, a simplifying assumption made in some methods of data analysis is that there are no barriers to water diffusion. We develop tools to simulate the diffusion of water in an inhomogeneous medium, which may then be used to test the accuracy of this assumption. The inherent difficulty (and computational cost) of including barriers (e.g., cell membranes) can be lessened by employing the immersed boundary (IB) method to represent these structures without the need for complicated computational grids. The contribution of this thesis is the implementation and validation of an IB method that allows for diffusion across semi-permeable membranes. The method is tested for a square interface aligned with the computational grid by comparing it to a second numerical scheme that uses standard finite differences. We also calculate the rate of convergence for the IB method to assess the numerical accuracy. To demonstrate the flexibility of the IB method to simulate diffusion with any interface shape, we also present simulations for irregular interfaces. / UOIT

<p> Atherosclerosis generally occurs near the branching in the arteries where there tends to be flow irregularities. A build up of fatty deposits (plaque) occurs in the blood vessel in such regions making it to lose its elasticity. Such hardening of the arteries and the narrowing of the lumen can cause severe atheromas and even high blood pressure and blockage of the vessels. It is observed in North America that nearly 47% of the deaths are caused due to cardiovascular diseases and hence determination of such regions becomes very critical and can be very beneficial if done at an earlier stage. In this thesis, we present: an approach to model the pulsating flow of blood through such an atherosclerosis affected region of the artery using finite element method and further discuss the statistical model used to implement the optimization techniques to estimate the region of maximum rigidity. Here within we present a numerical and non-invasive approach to predict such regions. The computational modeling is carried out under two categories: a. The mathematical model and b. The statistical model. </p> <p> The mathematical model which is the forward model, comprises of the artery and the cardiac muscle as hyperelastic material modeled with the neo-hookean model and the three dimensional Navier-Stokes equations solve for the blood flowing through it. We perform fluid dynamic analysis for the blood flowing through the vessel to compute the velocity at different time instances and mechanical analysis to compute the deformation of the artery which is a function of the elasticity of the vessel. The two models are interconnected to each other by boundary conditions as the normal component of the surface force provides the coupling between the two models. The shear modulus represents a measure of the elasticity of the vessel. We use linear spatial basis functions to model the shear modulus which spatially varies along the geometry of the vessel thus we have a region of atherosclerosis and the geometry shows the stenosis. The change in the shear modulus affects the velocity of blood through the vessel. </p> <p> In the statistical model, we propose an inverse computational model for estimating the elasticity profile of the arterial wall where we implement the inverse modeling approach to estimate the maximum shear modulus which helps us to predict the region of atherosclerosis. The velocity and the deformation obtained for a particular shear modulus from our COMSOL forward model provide the realistic simulated measurements that are made noisy by introduction of white Gaussian noise with different SNR and we try to estimate the shear modulus that minimizes the error-function. We use COMSOL with MATLAB for simultaneous iterative computations of velocity and deformation measurements by running the optimization code. We estimate these unknown parameters using optimization algorithm that minimizes the cost function of our model. For our estimation we use the least squares estimator and we derive the maximum likelihood estimator. The unconstrained optimization is carried out with Neider Mead Simplex Method and the Trust Region Method which uses only the function evaluations to find the minimum: making it a very robust algorithm and very efficient for problems that are nonlinear or have a number of discontinuities. Our preliminary results demonstrate significant change in velocity of the blood and occurrence of vortices in the region of less elasticity and the tendency of the artery to deform minimum in the hardened less elastic region. Our estimation results show that the parameters are identifiable. The mean square error of the estimate as a function of SNR shows accuracy of the estimation. </p> / Thesis / Master of Applied Science (MASc)

Gas-solid flow is fundamental to many industrial processes. Extensive experimental and numerical studies have been devoted to understand the interphase momentum and heat transfer in these systems. Most of the studies have focused on spherical particle shapes, however, in most natural and industrial processes, the particle shape is seldom spherical. In fact, particle shape is one of the important parameters that can have a significant impact on momentum, heat and mass transfer, which are fundamental to all processes. In this study particle-resolved simulations are performed to study momentum and heat transfer in flow through a fixed random assembly of ellipsoidal particles with sphericity of 0.887. The incompressible Navier-Stokes equations are solved using the Immersed Boundary Method (IBM). A Framework for generating particle assembly is developed using physics engine PhysX. High-order boundary conditions are developed for immersed boundary method to resolve the heat transfer in the vicinity of fluid/particle boundary with better accuracy. A complete framework using particle-resolved simulation study assembly of particles with any shape is developed. The drag force of spherical particles and ellipsoid particles are investigated. Available correlations are evaluated based on simulation results and recommendations are made regarding the best combinations. The heat transfer in assembly of ellipsoidal particle is investigated, and a correlation is proposed for the particle shape studied. The lift force, lateral force and torque of ellipsoid particles in assembly and their variations are quantitatively presented and it is shown that under certain conditions these forces and torques cannot be neglected as is done in the larger literature. / Ph. D.

Solid-fluid two-phase flows, where the solid volume fraction is large either by geometry or by population (as in slurry flows), are ubiquitous in nature and industry. The interaction between the fluid and the suspended solids, in such flows, are too strongly coupled rendering the assumption of a single-way interaction (flow influences particle motion alone but not vice-versa) invalid and inaccurate. Most commercial flow solvers do not account for twoway interactions between fluid and immersed solids. The current state-of-art is restricted to two-way coupling between spherical particles (of very small diameters, such that the particlediameter to the characteristic flow domain length scale ratio is less than 0.01) and flow. These solvers are not suitable for solving several industrial slurry flow problems such as those of hydrates which is crucial to the oil-gas industry and rheology of slurries, flows in highly constrained geometries like microchannels or sessile drops that are laden with micro-PIV beads at concentrations significant for two-way interactions to become prominent. It is therefore necessary to develop direct numerical simulation flow solvers employing rigorous two-way coupling in order to accurately characterise the flow profiles between large immersed solids and fluid. It is necessary that such a solution takes into account the full 3D governing equations of flow (Navier-Stokes and continuity equations), solid translation (Newton’s second law) and solid rotation (equation of angular momentum) while simultaneously enabling interaction at every time step between the forces in the fluid and solid domains. This thesis concerns with development and rigorous validation of a 3D solid-fluid solver based on a novel variant of immersed-boundary method (IBM). The solver takes into account full two-way fluid-solid interaction with 6 degrees-of-freedom (6DOF). The solid motion solver is seamlessly integrated into the Gerris flow solver hence called Gerris Immersed Solid Solver (GISS). The IBM developed treats both fluid and solid in the manner of “fluid fraction” such that any number of immersed solids of arbitrary geometry can be realised. Our IBM method also allows transient local mesh adaption in the fluid domain around the moving solid boundary, thereby avoiding problems caused by the mesh skewness (as seen in common mesh-adaption algorithms) and significantly improves the simulation efficiency. The solver is rigorously validated at levels of increasing complexity against theory and experiment at low to moderate flow Reynolds number. At low Reynolds numbers (Re 1) these include: the drag force and terminal settling velocities of spherical bodies (validating translational degrees of freedom), Jeffrey’s orbits tracked by elliptical solids under shear flow (validating rotational and translational degrees of freedom) and hydrodynamic interaction between a solid and wall. Studies are also carried out to understand hydrodynamic interaction between multiple solid bodies under shear flow. It is found that initial distance between bodies is crucial towards the nature of hydrodynamic interaction between them: at a distance smaller than a critical value the solid bodies cluster together (hydrodynamic attraction) and at a distance greater than this value the solid bodies travel away from each other (hydrodynamic repulsion). At moderately high flow rates (Re O(100)), the solver is validated against migratory motion of an eccentrically placed solid sphere in Poisuelle flow. Under inviscid conditions (at very high Reynolds number) the solver is validated against chaotic motion of an asymmetric solid body. These validations not only give us confidence but also demonstrate the versatility of the GISS towards tackling complex solid-fluid flows. This work demonstrates the first important step towards ultra-high resolution direct numerical simulations of solid-fluid flows. The GISS will be available as opensource code from February 2015.

In this dissertation, Immersed Boundary Method (IBM) is evaluated in ribbed duct geometries to show the potential of simulating complex geometry with a simple structured grid. IBM is first investigated in well-accepted benchmark cases: channel flow and pipe flow with circular cross-section. IBM captures all the flow features with very good accuracy in these two cases. Then a two side ribbed duct geometry is test using IBM at Reynolds number of 20,000 under fully developed assumption. The IBM results agrees well with body conforming grid predictions. A one side ribbed duct geometry is also tested at a bulk Reynolds number of 1.5⨉10⁴. Three cases have been examined for this geometry: a stationary case; a case of positive rotation at a rotation number (Ro=ΩDₕ/U) of 0.3 (destabilizing); and a case of negative rotation at Ro= -0.3 (stabilizing). Time averaged mean, turbulent quantities are presented, together with heat transfer. The overall good agreement between IBM, BCG and experimental results suggests that IBM is a promising method to apply to complex blade geometries. Due to the disadvantage of IBM that it requires large amount of cells to resolve the boundary near the immersed surface, wall modeled LES (WMLES) is evaluated in the final part of this thesis. WMLES is used for simulating turbulent flow in a developing staggered ribbed U-bend duct. Three cases have been tested at a bulk Reynolds number of 10⁵: a stationary case; a positive rotation case at a rotation number Ro=0.2; and a negative rotation case at Ro=-0.2. Coriolis force effects are included in the calculation to evaluate the wall model under the influence of these effects which are known to affect shear layer turbulence production on the leading and trailing sides of the duct. Wall model LES prediction shows good agreement with experimental data. / Master of Science

L'approche la plus courante en Mécanique des Fluides Numérique pour réaliser les simulations d'écoulement cardiovasculaire consiste à utiliser des méthodes numériques Body-fitted. Ces méthodes ont permis d'obtenir des simulations d'écoulement sanguin dans les artères qui sont précises et utiles. Toutefois, la génération du maillage body-fitted est une tâche qui demande beaucoup de temps et d'expertise à l'utilisateur.Les méthodes de Frontières Immergées sont des méthodes numériques alternatives qui ont l'avantage d'être plus simples d'emploi car elles ne requièrent aucune tâche de maillage de la part de l'utilisateur. Le travail présenté ici vise à évaluer le potentiel d'un méthode de Frontières Immergées à réaliser des simulations d'écoulement cardiovasculaire.Ce travail s'attache, dans un premier temps, à décrire les capacités de cette méthode numérique à rendre compte de l'imperméabilité et de la mobilité des parois sur des cas relativement simples mais représentatifs d'écoulements cardiovasculaires. Ensuite, des applications de la méthode à des cas d'écoulement cardiovasculaire plus complexes sont montrées. Il s'agira d'abord d'une simulation de l'écoulement dans un modèle rigide d'artère aorte. Puis, la simulation d'un écoulement à l'intérieur d'un ventricule cardiaque à paroi mobile sera montrée. / The most common approach in Computational Fluid Dynamics(CFD) for simulating blood flow into vessel is to make use of a body-fitted me-thod. This approach has lead to accurate and useful simulations of blood flowinto arteries. However, generation of the body-fitted grid is time consuming andrequires from the user an engineering knowledge.The Immersed Boundary Method has emerged as an alternate method whichdoes not require from the user any grid generation task. Simulations are done on astructured Cartesian grid which can be automatically generated. Here we addressthe question of the capability of an Immersed Boundary Method to cope withcardiovascular flow simulations.In particular, we assess the impermeable and moving properties of the wallwhen using the Immersed Boundary Method on simple but relevant vascular flowcases. Then, we show more complex and realistic cardiovascular flow simulations.The first application consists of blood flow simulation inside an aorta cross model.Then, the simulation of blood flow inside a cardiac ventricle with moving wall isshown.

We present a novel dimensionally split Cartesian cut cell method to compute inviscid, viscous and turbulent flows around rigid geometries. On a cut cell mesh, the existence of arbitrarily small boundary cells severely restricts the stable time step for an explicit numerical scheme. We solve this `small cell problem' when computing solutions for hyperbolic conservation laws by combining wave speed and geometric information to develop a novel stabilised cut cell flux. The convergence and stability of the developed technique are proved for the one-dimensional linear advection equation, while its multi-dimensional numerical performance is investigated through the computation of solutions to a number of test problems for the linear advection and Euler equations. This work was recently published in the Journal of Computational Physics (Gokhale et al., 2018). Subsequently, we develop the method further to be able to compute solutions for the compressible Navier-Stokes equations. The method is globally second order accurate in the L1 norm, fully conservative, and allows the use of time steps determined by the regular grid spacing. We provide a full description of the three-dimensional implementation of the method and evaluate its numerical performance by computing solutions to a wide range of test problems ranging from the nearly incompressible to the highly compressible flow regimes. This work was recently published in the Journal of Computational Physics (Gokhale et al., 2018). It is the first presentation of a dimensionally split cut cell method for the compressible Navier-Stokes equations in the literature. Finally, we also present an extension of the cut cell method to solve high Reynolds number turbulent automotive flows using a wall-modelled Large Eddy Simulation (WMLES) approach. A full description is provided of the coupling between the (implicit) LES solution and an equilibrium wall function on the cut cell mesh. The combined methodology is used to compute results for the turbulent flow over a square cylinder, and for flow over the SAE Notchback and DrivAer reference automotive geometries. We intend to publish the promising results as part of a future publication, which would be the first assessment of a WMLES Cartesian cut cell approach for computing automotive flows to be presented in the literature.