The Diffuse Interface Method and Applications to Coupled Systems in Fluid Dynamics

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Systems involving moving domains in which a fluid interacts with a neighboring region are often modeled using partial differential equations with coupling conditions that hold across a common interface. Such systems occur in medicine (liver perfusion, lymphatic circulation, the closing and opening of heart valves), geomechanics (fracture propagation, coupling between the surface and groundwater flows), and other applications. Numerical simulations of such systems are based on discrete approximations to the governing equations. To accurately describe the dynamics, interface tracking methods, in which the nodes of the computational mesh are aligned with the parametric representation of the interface, are often used. However, interface-tracking methods rapidly become difficult to apply when domain deformations are large. To prevent numerical failures, computationally expensive mesh regenerations or similar techniques are necessary when mesh elements become highly skewed. The diffuse interface method is an alternative strategy based on a fixed mesh approach. For this method, the model is reformulated using a phase-field function that smoothly transitions from zero in one region to one in the other region. The computational mesh nodes do not have to be aligned with the interface, whose location is now captured using the phase-field function. This approach is useful even when the domain does not change in time, or in cases where the interface between the two regions is difficult to determine exactly, or when the geometry of the interface is complex. However, the diffuse interface method introduces an additional error at the interface, which needs to be carefully controlled. This project aims to establish mathematical foundations for application of the diffuse interface method in fluid dynamics. The techniques developed in this work are expected to be applicable to other coupled systems involving fluids and poroelastic and/or elastic structures as well. The project includes training of graduate students through involvement in the research. This project focuses on the development of mathematical theory and numerical methods for the diffuse interface method applied to coupled systems in fluid dynamics. This will be achieved by studying a hierarchy of coupled flow models, including the fluid-porous medium interaction, fluid-poroelastic structure interaction, and fluid-elastic structure interaction. For each model, the work entails proving the well-posedness of the underlying diffuse interface problem, showing the convergence of the diffuse interface model to the corresponding sharp interface model as the width of the interfacial layer goes to zero, and calculating the rate of convergence including the modeling error and the approximation error of the discrete solution based on the finite element method. The analysis will be performed using weighted Sobolev spaces. Numerical methods will be developed and implemented for each model. 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.