An Embedded Surface Mesh Method for Simulating Mechanical Heart Valves Using Kriging Interpolation

Mechanical heart valve replacements play a crucial role in treating valvular heart diseases, offering numerous advantages that make them a preferred choice in specific clinical scenarios. Unlike bioprosthetic valves, rigid mechanical heart valves provide exceptional durability, making them a compelling option to consider for long-term treatment. In this context, our objective is to demonstrate the capability of our numerical method in accurately simulating the behavior of a rigid mechanical heart valve. To achieve this goal, we validate our simulations by comparing them against experimental results of such a device within an artificial rigid aorta. From the physical point of view, this experimental setup requires the solution of two distinct yet interrelated physics: fluid dynamics and the behavior of solid components. Moreover, from the implementation perspective, a parallel implementation is an absolute necessity to precisely reproduce the behavior of this experimental model. For the numerical coupling strategy, we implement an embedded boundary method. This approach involves discretizing the fluid using a non-body-conforming mesh, where the boundaries of the rigid bodies are embedded within the fluid mesh. The force exerted by the fluid on a body is determined from the residual of the momentum equations, while the body's velocity is imposed as a boundary condition in the fluid. To enhance the accuracy of our approximation, we utilize a universal kriging interpolation to impose the rigid body velocity. The fluid's physics are described by the incompressible Navier-Stokes equations, stabilized using a variational multiscale finite element method, and solved through a fractional step-like scheme at the algebraic level. The incompressible Navier-Stokes solver adopts a parallel approach based on a master-worker strategy. Regarding the rigid valves, their motions are determined by the Newton-Euler equations. While the fluid mesh is partitioned to solve the fluid physics in a distributed memory parallel environment, all the data of the valves is stored and shared inside each one of these partitions. Thus, the movement of the valves has to be determined for each partition. This feature has a minimal impact on the total computation time since the valve data is relatively small compared to the fluid data. In addition, computational geometry tools have been employed to accurately track the boundaries of the bodies within the fluid mesh as quickly as possible.