Divergence-Free Hybridizable Discontinuous Galerkin Methods for the Incompressible Navier-Stokes Equations on Moving Domains and Their Application to Fluid-Structure Interaction

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Fluid-structure interaction (FSI) problems can occur in many fields of engineering, and are a crucial consideration in the design of many systems, such as stability and response of aircraft wings in the aerospace industry, flow of blood through arteries in biomedical applications, and response of bridges and tall buildings to winds in civil engineering. These problems are often too complex to solve analytically and so they have to be analyzed by means of numerical simulations. Efficient, accurate, and robust numerical methods to solve FSI problems are of fundamental importance to these applications. In this project, the PI will concentrate his efforts on the development, analysis, and implementation of high-order strongly mass-conserving finite element methods for incompressible flow problems on moving domains and their application to FSI problems. This project provides research training opportunities for graduate students.The PI will (1) develop a novel divergence-free hybridizable discontinuous Galerkin (div-free-HDG) scheme for the incompressible Navier-Stokes (INS) equations on moving domains using the Arbitrary Lagrangian-Eulerian (ALE) framework, (2) extend the ALE-div-free-HDG fluid solver to moving-domain FSI problems that model the interaction of incompressible fluids with compressible or incompressible hyperelastic structures. The div-free-HDG scheme for INS on fixed meshes enjoys features such as higher-order accuracy, global and local conservation properties, energy-stability, pressure robustness, minimal amount of numerical dissipation and computational efficiency. The extension of the div-free-HDG scheme to moving domain problems within the ALE framework poses further theoretical and numerical challenges due to the divergence-conforming velocity finite element space used therein. The PI will use a novel alternative formulation of the div-free-HDG scheme, which does not require the divergence conformity of the velocity finite element space, that can be efficiently adapted to the ALE framework for moving domain INS and FSI problems. This work will be the first systematic study of ALE strongly mass-conserving finite element methods for moving domain INS and FSI problems. Rigorous stability analysis will be derived for the methods and their convergence properties will be investigated. Various strategies to further improve the efficiency of these schemes will be studied.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.