Immersogeometric fluid–structure interaction analysis of bioprosthetic heart valves
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The purpose of this dissertation is to develop numerical methods for fluid–structure interaction (FSI) analysis that are suitable for modeling and simulating bioprosthetic heart valves (BHVs). BHVs are prosthetic replacements for the valves that regulate blood flow through the heart. BHVs reproduce natural hemodynamic conditions by mimicking the structure of native heart valves: they consist of thin flexible leaflets, passively driven by interaction with surrounding fluid. Current designs frequently require replacement 10–15 years after implantation. Computer simulation may help identify causes of and solutions to durability issues. Despite much previous research into computer simulation of heart valve FSI, inconvenience or inaccuracy of readily available numerical methods have prevented widespread incorporation of FSI into models of heart valve mechanics. Challenges associated with heart valve FSI simulation include large deformations of the region occupied by fluid, with changes of topology as the valve opens and closes, and low mass of the structure relative to the fluid, which necessitates careful treatment of fluid–structure coupling. The presence of large pressure gradients also requires special attention to the treatment of fluid mass conservation. Further, a useful numerical method for studying and improving designs of BHVs should be able to capture variations of valve geometry without requiring major effort to construct geometry-specific discretizations. To meet these challenges, I develop a new numerical approach, combining the immersed boundary concept of capturing fluid–structure interfaces on unfitted discretizations with recent developments in isogeometric analysis (IGA), which directly uses geometrical designs of engineered systems as discrete analysis meshes. In this work, I immerse an isogeometric structure discretization into an unfitted analysis mesh of the fluid subproblem. I refer to the immersion of design geometries into unfitted analysis meshes as immersogeometric analysis. To reliably couple unfitted discretizations of the fluid and structure subproblems, I introduce a new semi-implicit time integration procedure and analyze its stability and convergence in the context of linear model problems. I verify that this analysis extrapolates to the nonlinear setting through numerical experiments and explore the validity of my modeling assumptions by comparing computer simulations with observations from an in vitro experiment.