Collaborative Research: Hybrid Fluid-Structure Interaction Material Point Method with applications to Large Deformation Problems in Hemodynamics

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Heart valve associated issues in the human organism are the cause of cardiac arrest and heart failure, which may have devastating consequences on a person's health and even lead to death. While not necessarily fatal, pathologies associated with leg vein valves can nevertheless cause severe distress to the people affected and have a negative impact on their life with possibly major complications. For the treatment of valve associated diseases, the most common practice nowadays is the replacement of the malfunctioning valve with a prosthetic device. Unfortunately, prosthetic valves have issues with long term durability and post-implantation complications. Given the necessity of improving the design and selection of existing prosthetic valves, computational methodologies are becoming a valuable tool. The nature of blood flow inside a human valve renders the modeling problem considerably challenging from the mathematical and computational standpoints, as multiple physical phenomena mutually interact. Specifically, the major challenges are the large structural displacements experienced by the valve leaflets, while preserving accurate description of the hydrodynamic force at the fluid-solid interface. The focus of this project is on developing new fluid-structure interaction methodologies with specific interest in the case of large deformations. The important insight provided in this project will enable future valve design optimization while avoiding costly empirical design iterations. In addition to the obvious potential impact on society, the proposed project will be useful to many other applications in science and engineering, and also have beneficial impact on the training, education, and careers of junior researchers in an important, exciting, and mathematically, computationally, and societally impactful area of research.This project is about the development, analysis, and implementation of novel computational techniques for the coupling of finite element methods (FEMs) to material point methods (MPMs) in fluid-structure interaction (FSI) problems. The use of different discretization techniques for the study of multiscale and multiphysics problems is a powerful tool for computational simulations. For instance, one-dimensional models are coupled with multi-dimensional models for computational cost reduction, or FEMs are coupled with finite volume methods to exploit the advantages of the algorithmic and mathematical features of these two methods. With the same idea, the coupling of FEM with MPM represents a promising combination, if different deformation regimes occur within the dynamical regime of a physical model. As a matter of fact, the FEM reaches its best accuracy for small deformations whereas the MPM mixed Eulerian-Lagrangian formulation becomes beneficial when large deformations occur. FEM-MPM coupling has, in fact, been studied only by very few authors, including the PIs, and the coupling of an FSI framework with an MPM approach is yet to be explored. The use of the material point methodology would avoid the mesh entanglement issues that plague many existing FSI methods. To design the desired coupling approach, preliminary work is needed. First, the coupling between an MPM solid body immersed in an FEM fluid will be addressed, using benchmark problems from the FSI literature. At the same time, the mechanical properties of a solid body discretized with the mixed FSI-MPM approach will be studied and the accuracy of the method will be investigated using the Taylor bar test in which a cylinder impacts a rigid wall. Then, the knowledge gained from the preparatory work will be used to realize an FSI-MPM coupling methodology for biological valves, with the valve leaflets modeled with the MPM and the blood vessel and blood flow described in an FEM-FSI framework. Appropriate solvers and preconditioners will also be selected and studied because the discretized nonlinear and linear systems will likely be large and highly coupled. Lastly, the FSI-MPM coupling approach will also be applied for the simulations of stented arteries, with the stent described using the MPM. In this way, complex meshing procedure for the stent can be avoided, while capturing its dynamical behavior. The computational techniques developed within the proposed research will be applicable and prove to be invaluable tools for a broad spectrum of applications such as human valve fluid and structural dynamics, aerospace and civil engineering problems, dam breaking, and airfoil design, to name a few. All our findings will be implemented in FEMuS, an open source library written in C++ language, freely downloadable online. Our effort will hopefully contribute to the standardization of novel computational techniques that are currently available only in research software. Nevertheless, researchers from all over the world can potentially access our findings and join us in this effort, with a substantial speed up in the standardization procedure.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.