Novel Ideas and Analysis for Interface and Fluid-Structure Interaction Problems and Applications

North Carolina State University, Raleigh NC

Investigators

Abstract

Studying interface and fluid-structure interaction problems, such as those involving oil and water mixtures, gas bubbles, ice and water interfaces, tumor growth, or cell deformation, has many practical applications. It is often costly to carry out experiments on such systems, and computational simulation provides an alternative for study of these challenging problems. The objectives of this research project are to model such systems and to design efficient and practical computational algorithms to simulate, solve, and control those problems. Newly developed augmented immersed interface methods will be applied to several important applications in multi-phase flows and fluid-structure interactions. The investigator will develop and disseminate software packages implementing these methods. Graduate students will be involved in the research. This project concerns the development and analysis of some new ideas for interface and fluid-structure interaction problems based on structured meshes. The methods under development will be supported by rigorous mathematical analysis and numerical experiments. Applications include optimal control of interface problems, fluid-structure interactions in modeling tissue mechanics coupled with cell biology, and tip propagation of a crack. The methods are based on structured meshes such as Cartesian meshes that are not necessarily aligned with the interface in two and three dimensions. The projects include: (1) new augmented methods for a fluid structure interaction between a fluid flow modeled by Stokes or Navier-Stokes equations and a porous medium modeled by the Darcy's law; (2) a new computational framework for accurate gradient computation on a boundary or interface with accurate solution globally; (3) a new second-order symmetric, consistent, and parameter-free immersed finite element method; (4) a new SVD-free augmented IFE method for elliptic interface problems with non-homogeneous jump conditions; (5) applications of the new methods for cell deformation in different layers (porous media, Stokes flow, Navier-Stokes flow) and scales; and (6)numerical simulations of crack propagation based on the Mumford-Shah minimizer. This project will have positive effect on education by attracting graduate students and postdoctoral researchers to conduct research in this area. Some components of the project will be designed as undergraduate projects. The project will produce software useful to computational science involving discontinuities and singularities, free-boundary/moving interfaces, multi-phase and multi-physics, and irregular domain problems.