PhD Dissertation Presentation
Title: A Cut Finite Element Immersed Boundary Method and its Application to a Chemotaxis Model
Abstract: Originally developed for numerical analysis of cardiac blood flow, the immersed boundary method introduced in 1977 by Peskin has gained popularity in various computational applications involving a fluid with an immersed elastic structure. Peskin's method solves for the velocity and pressure of the fluid, then uses the local velocity to move the immersed boundary. There has been a recent surge in efforts to further study this method due to its growth in popularity across many applications, including cell mechanics and fluid dynamics. Many advances have been made to better incorporate the forces applied to the fluid by the elastic structure. Specifically, we will consider the finite element approach first introduced by Boffi, Gastaldi, and Heltai.
We will introduce a finite element method for Stokes equations with a massless immersed membrane. We design and implement a high-accuracy cut finite element method (CutFEM) which enables the use of a structured mesh that is not aligned with the immersed membrane. Allowing the membrane to move in the time-dependent Stokes problem, we design a semi-implicit time discretization. We then prove that this semi-implicit scheme is unconditionally energy stable and illustrate this theoretical result with numerical simulations. We improve the first-order temporal accuracy of the semi-implicit algorithm by implementing and analyzing an unconditionally stable Crank-Nicolson scheme. Next, we turn our attention to the steady state Stokes problem to improve computational efficiency. A domain decomposition approach and a Dirichlet-Neumann preconditioner are developed for the two-subdomain problem. Finally, we will use the immersed boundary method to incorporate interior and exterior fluid flow into a local excitation, global inhibition chemotaxis model.
Dr. Marcus Sarkis, WPI (Advisor)
Dr. Roger Lui, WPI (Co-advisor)
Dr. Zhongqiang Zhang, WPI
Dr. Sarah Olson, WPI
Dr. Johnny Guzmán, Brown University
Dr. Blanca Ayuso de Dios, University of Bologna