Modeling, discretizations, and solution strategies for multiphysics problems

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

The goal of this project is to advance the state-of-the-art in modeling and computation of multiphysics systems that model the physical interactions between two or more media, such as couplings of fluid flows, rigid or deformable porous media, and elastic structures. Typical examples are coupling of free fluid and porous media flows, fluid-structure interaction, and fluid-poroelastic structure interaction (FPSI). The developed methods will be employed for several biomedical and geoscience applications. Biomedical applications include investigation of non-Newtonian and poroelastic effects in arterial flows on important clinical markers such as wall shear stress and relative residence time, modeling LDL transport and drug delivery in blood flows, as well as flows in the eye and the brain. Geoscience applications include tracing organic and inorganic contaminants in coupled surface-subsurface hydrological systems, predicting hydrocarbon extraction in hydraulic fracturing, geothermal energy production, and modeling the effect of proppant particles in injected polymers on the fracture width and flow patterns. While focused on FPSI, the developments in this project will be applicable to modeling and computation of a wide class of multiphysics problems with a broad range of applications. The project consists of a comprehensive program for mathematical and computational modeling of multiphysics problems that includes 1) development and analysis of new mathematical models, 2) design and analysis of stable, accurate, and robust structure-preserving numerical methods, 3) development and analysis of efficient time-splitting and multiscale domain decomposition algorithms for the solution of the resulting algebraic problems, and 4) applications to the geosciences and biomedicine. Variational formulations of new fluid--poroelastic structure interaction (FPSI) models based on Navier-Stokes - Biot couplings will be developed, extending current model capabilities to flows with higher Reynolds numbers. Fully coupled nonlinear FPSI-transport models, including miscible displacement models with concentration-dependent fluid viscosity, stress-dependent diffusion, and non-isothermal models will also be studied. Novel discretization techniques will be investigated for the numerical approximation of the FPSI models. The focus will be on dual mixed and total pressure discretizations with local conservation of mass and momentum, accurate approximations with continuous normal components for velocities and stresses, and robustness with respect to physical parameters. These 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. Efficient multiscale domain decomposition and time-splitting algorithms will be developed for the solution of the resulting algebraic systems. The domain decomposition 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 and local time-stepping. The convergence of the space-time coarse-scale mortar interface iteration will be studied by analyzing the spectrum of the interface operator. Iterative and non-iterative time-splitting methods will also be investigated. 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.