Bioprosthetic heart valves (BHVs) are the most popular artificial replacements for diseased valves that mimic the structure of native valves. However, the life span of BHVs remains limited to 10-15 years, and the mechanisms that underlie BHVs failure remain poorly understood. Therefore, developing a unifying mathematical framework which captures material damage phenomena in the fluid-structure interaction environment would be extremely valuable for studying BHVs failure. Specifically, in this framework the computational domain is composed of three subregions: the fluid (blood) , the fracture structure (damaged BHVs) modeled by the recently developed nonlocal (peridynamics) theory, and the undamaged thin structure (undamaged BHVs). These three subregions are numerically coupled to each other with proper interface boundary conditions.

In this talk, I will introduce two coupling problems and the corresponding numerical methods in this multiscale/multiphysics framework. In the first problem the coupling strategy for fluid and thin structure is investigated. This problem presents unique challenge due to the large deformation of BHV leaflets, which causes dramatic changes in the fluid subdomain geometry and difficulties on the traditional conforming coupling methods. To overcome the challenge, the immersogeometric method was developed where the fluid and thin structure are discretized separately and coupled through penalty forces. To ensure the capability of the developed method in modeling BHVs, we have provided theoretical error estimates and validated this method by comparing the numerical results with experiments. The second part focuses on developing a fluid—peridynamics coupling framework to capture the fluid-induced material damage. In the peridynamic and other nonlocal models the loading boundary conditions should be defined in a nonlocal way, while in fluid—structure interfaces the hydrodynamic loadings from the fluid side are typically provided on a sharp co-dimension one surface. To overcome this challenge, we have proposed a new nonlocal Neumann-type boundary condition which provides an approximation of physical boundary conditions on a sharp surface. Based on this nonlocal boundary condition, we havedeveloped a stable and asymptotically compatible fluid—peridynamics coupling framework without overlapping regions.

Dr. Yue Yu is an associate professor of Applied Mathematics at Lehigh University which is located at Bethlehem, Pennsylvania, USA. Dr. Yu received her PhD in Applied mathematics from Brown University in 2014, and went to Harvard University as a postdoctoral fellow. She joined the faculty of Lehigh University on fall 2014. She received a CAREER award from the National Science Foundation on 2018. Dr. Yu is generally interested in theoretical and computational issues related to multiscale and multiphysics problems. Her recent work is focused on developing a systematic unifying mathematical framework and computational tools for coupling the blood flow and the damaged soft tissues.