Advanced Discretizations and Domain Decomposition Algorithms for Multiphysics Couplings of Fluid Flows and Solid Mechanics

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

A computational framework will be developed for modeling coupled physical processes. It will be applied to geoscience and biomedical problems of societal importance. The research will investigate coupled subsurface and surface flows to model interactions between contaminated aquifers, rivers, lakes, and wetlands. The project will model flows in fractured and deformable reservoirs to provide predictive simulations of hydraulic fracturing and carbon sequestration, including surface subsidence, wellbore collapse, and fault activation. The project will further model flow in arteries, accounting for flow within the arterial wall. This has an effect on the blood velocity in the lumen and the speed of the pressure wave, as well as low density lipoproteins (LDL) transport and drugs filtered into the tissue. The research will lead to the development of simulation tools that advance drug delivery as well prevention, detection, and therapy of cardiovascular diseases such as atherosclerosis. The objective of this project is mathematical and computational modeling of multiphysics systems of coupled flow and mechanics problems with multiscale input parameters. The simulation domain is decomposed into a union of subdomains, each one associated with a physical, mathematical, and numerical model. Physically meaningful interface conditions are imposed on the discrete level via mortar finite elements or penalty methods. The formulation provides great flexibility for multiphysics and multinumerics couplings. Furthermore, when combined with coarse scale mortar elements, it provides a multiscale approximation and an efficient way to solve the coarse grid problem in parallel. The project will develop 1) Mathematically rigorous and physically meaningful multiphysics models; 2) Robust, accurate and efficient multiscale discretization techniques; 3) Efficient parallel domain decomposition solvers and preconditioners; 4) Efficient non-iterative time-partitioned algorithms. Two main components of the proposed research are A) mixed elasticity formulations and discretizations, and their coupling with mixed flow discretizations in the multiphysics framework; B) space-time multidomain variational formulations and discretizations allowing for different time stepping in different subdomains. A computational framework will be developed and applied to geoscience and biomedical problems. The research will develop variational formulations of Partial differential Equations systems coupling free and porous media fluid flows with deformations of the porous solids. Free fluid models such as Stokes, Brinkman, or Navier-Stokes equations will be coupled through physically meaningful interface conditions with Darcy flow. Regions with Darcy flow through deformable porous media will be modeled by the Biot system of poroelasticity. Nonlinear models for non Newtonian fluids, as well as reduced-dimension fracture models will also be investigated. An emphasis will be placed on mixed elasticity formulations coupled with mixed Stokes and Darcy formulations. The PI will study well-posedness of the variational formulations and will develop stable and accurate discretizations. Novel cell-centered mixed finite element methods for elasticity and poroelasticity will be investigated. The essential-type interface conditions will be imposed on a coarse scale via mortar finite elements. The PI will carry out stability and a priori multiscale error analysis. the PI will develop efficient parallel non-overlapping domain decomposition algorithms for the solution of the resulting algebraic systems by reducing the global problem to a coarse scale interface problem. The PI will analyze the condition number of the interface operator and will develop efficient preconditioners for speeding up the interface iteration. The PI will also study penalty methods, such as the Nitsche's coupling method, to impose interface conditions, resulting in loosely coupled formulations amendable to efficient non-iterative time-partitioned algorithms. The PI will study the stability and accuracy of the methods, as well as their properties as preconditioners for monolithic schemes. The PI will further develop multidomain space-time variational formulations and discretizations for the multiphysics models, coupling spatial non-overlapping domain decomposition methods with Galerkin-type approximations in time, allowing for different time steps associated with different regions and different types of physics. 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.