Weighted least-squares finite element methods for PIV data assimilation
The 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.