Parametric AFEM for Geometric Evolution Equations and Coupled Fluid-Membrane Interaction

Abstract

When lipid molecules are immersed in aqueous environment at a proper concentration they spontaneously aggregate into a bilayer or membrane that forms an encapsulating bag called vesicle. This phenomenon is of interest in biophysics because lipid membranes are ubiquitous in biological systems, and an understanding of vesicles provides an important element to understand real cells. Also lately there has been a lot of activity when different types of lipids are used in the membrane. Doing mathematics in such a complex physical phenomena, as most problems coming from the bio-world, involves cyclic iterations of: modeling and analysis, design of a solving method, its implementation, and validation of the numerical results. In this thesis, motivated by the modeling and simulation of biomembrane shape and behavior, new techniques and tools are developed that allow us to handle large deformations of surface flows and fluid-structure interaction problems using the finite element method (FEM). Most simulations reported in the literature using this method are academic and do not involve large deformation. One of the questions this work is able to address is whether the method can be successfully applied to more realistic applications. The quick answer is not without additional crucial ingredients. To make the method work it is necessary to develop a synergistic set of tools and a proper way for them to interact with each other. They include space refinement/coarsening, smoothing and time adaptivity. Also a method to impose isoperimetric constraints to machine precision is developed. Another use of the computational tools developed for the parametric method is mesh generation. A mesh generation code is developed that has its own unique features not available elsewhere as for example the generation of two and three dimensional meshes compatible for bisection refinement with an underlying coarse macro mesh. A number of interesting simulations using the methods and tools are presented. The simulations are meant first to examine the effect of the various computational tools developed. But also they serve to investigate the nonlinear dynamics under large deformations and discover some illuminating similarities and differences for geometric and coupled membrane-fluid problems.