Theses and Dissertations at Montana State University (MSU)
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Item Particle imaging velocimetry data assimilation using least-square finite element methods(Montana State University - Bozeman, College of Engineering, 2016) Rajaraman, Prathish Kumar; Chairperson, Graduate Committee: Jeffrey Heys; T. A. Manteuffel, M. Belohlavek, E. McMahon and Jeffrey J. Heys were co-authors of the article, 'Echocardiograpic particle imaging velocimetry data assimilation with least square finite element methods' in the journal 'Computers & mathematics with applications' which is contained within this thesis.; G. D. Vo, G. Hansen and Jeffrey J. Heys were co-authors of the article, 'Comparison of continuous and discontinuous finite element methods for parabolic differential equations employing implicit time intergation' submitted to the journal 'International journal of numerical methods for heat & fluid flow' which is contained within this thesis.; T. A. Manteuffel, M. Belohlavek and Jeffrey J. Heys were co-authors of the article, 'Combining existing numerical models with data assimilation using weighted least-squares finite element methods' submitted to the journal 'International journal of numerical methods in biomedical engineering' which is contained within this thesis.Recent advancements in the field of echocardiography have introduced various methods to image blood flow in the heart. Of particular interest is the left ventricle of the heart, which pumps oxygenated blood from the lungs out through the aorta. One method for imaging blood flow is injecting FDA-approved micro-bubbles into the left ventricle, and then, using the motion of the microbubbles and the frame rate of the ultrasound scan, the blood velocity can be calculated. In addition to blood velocity, echocardiologists are also interested in calculating pressure gradients and other flow properties, but this is not currently possible because the velocity data obtained is two-dimensional and contains noise. In order to realize the full potential of microbubbles as a tool for determining the pumping efficiency and health of the LV, three-dimensional velocity data is required. Our goal is to assimilate two-dimensional velocity data from ultrasound experiments into a three-dimensional computer model. In order to achieve this objective, a numerical method is needed that can approximate the solution of a system of differential equations and assimilate an arbitrary number of noisy experimental values at arbitrary points within the domain of interests to provide a "most probable" approximate solution that is accordingly influenced by the experimental data. In this thesis we present two different approaches for data assimilation, the first approach is more computationally expensive, but requires only a single step. The second approach uses a two stage data assimilation technique but is computationally less expensive. The motivation for using the least-squares finite element method approach is that it provides many advantages such as the ability to match the numerical solution more closely to more accurate data and less closely to the less accurate data.Item Weighted least-squares finite element methods for PIV data assimilation(Montana State University - Bozeman, College of Engineering, 2011) Wei, Fei; Chairperson, Graduate Committee: Jeffrey HeysThe ability to diagnose irregular flow patterns clinically in the left ventricle (LV) is currently very challenging. One potential approach for non-invasively measuring blood flow dynamics in the LV is particle image velocimetry (PIV) using microbubbles. To obtain local flow velocity vectors and velocity maps, PIV software calculates displacements of microbubbles over a given time interval, which is typically determined by the actual frame rate. In addition to the PIV, ultrasound images of the left ventricle can be used to determine the wall position as a function of time, and the inflow and outflow fluid velocity during the cardiac cycle. Despite the abundance of data, ultrasound and PIV alone are insufficient for calculating the flow properties of interest to clinicians. Specifically, the pressure gradient and total energy loss are of primary importance, but their calculation requires a full three-dimensional velocity field. Echo-PIV only provides 2D velocity data along a single plane within the LV. Further, numerous technical hurdles prevent three-dimensional ultrasound from having a sufficiently high frame rate (currently approximately 10 frames per second) for 3D PIV analysis. Beyond microbubble imaging in the left ventricle, there are a number of other settings where 2D velocity data is available using PIV, but a full 3D velocity field is desired. This thesis develops a novel methodology to assimilate two-dimensional PIV data into a three-dimensional Computational Fluid Dynamics simulation with moving domains. To illustrate and validate our approach, we tested the approach on three different problems: a flap displaced by a fluid jut; an expanding hemisphere; and an expanding half ellipsoid representing the left ventricle of the heart. To account for the changing shape of the domain in each problem, the CFD mesh was deformed using a pseudo-solid domain mapping technique at each time step. The incorporation of experimental PIV data can help to identify when the imposed boundary conditions are incorrect. This approach can also help to capture effects that are not modeled directly like the impacts of heart valves on the flow of blood into the left ventricle.