Mathematical and Computational Modeling of Interaction between Fluids and Poroelastic Structures

The objective of this work is modeling and simulation of the interaction between a free viscous fluid and flow in an adjacent deformable porous medium through an interface separating the two regions. Such physical phenomenon occurs in a broad range of applications, including geosciences, biomedical sciences, and industrial design. The project makes advances in the mathematical and computational modeling of this problem, including development and analysis of new mathematical models and numerical methods for their solution.

High performance computational software designed to run on massively parallel computers will be developed and applied for modeling LDL transport and drug delivery in cardiovascular flows and tracing organic and inorganic contaminants in coupled surface-subsurface hydrological systems.

The goal of this project is mathematical and computational modeling of fluid-poroelastic structure interaction (FPSI). The free fluid flow is modeled by the Stokes or the Navier-Stokes equations, while the poroelastic medium is modeled by the Biot system of poroelasticity. The two regions are coupled via dynamic and kinematic interface conditions, including balance of forces, continuity of normal velocity, and no-slip or slip with friction tangential velocity condition.

The project includes development and analysis of 1) new mathematical models; 2) stable, accurate, and robust structure-preserving numerical methods; and 3) efficient multiscale parallel domain decomposition algorithms for the solution of the resulting algebraic problems. In the first main component, variational formulations of new FPSI models, including Navier-Stokes - Biot couplings, use of Brinkman and Forchheimer models, non-Newtonian models, and fully coupled FPSI-transport models will be developed and analyzed.

These new models will extend current model capabilities to flows with higher Reynolds numbers, fluids with non-Newtonian rheology, and tracking species dissolved in the fluid, including the effect of the concentration on the flow field. Existence of model solutions will be established employing results from semigroup theory and monotone operators in Hilbert or Banach space setting, coupled with fixed point arguments.

The second main component involves investigation of novel discretization techniques for the numerical approximation of the FPSI models, focusing on dual mixed discretizations with local conservation of mass, local momentum conservation, accurate approximations with continuous normal components for velocities and stresses, and robustness with respect to physical parameters. Methods of interest include multipoint stress-flux mixed finite element methods and local-stress mimetic finite difference methods that can be reduced to positive definite cell-centered schemes, coupled through mortar finite elements across the interface.

In the third main component, efficient multiscale domain decomposition algorithms for FPSI will be developed and analyzed. The methodology will be based on space-time variational formulations and will allow for multiple subdomains within each region with non-matching grids along subdomain interfaces, as well different time steps in different subdomains.

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.