The Evolution of Cellular Structures via Curvature Flow

Abstract

This dissertation explores cellular structures that evolve over time, primarily through curvature flow. This models <italic>coarsening</italic> in isotropic polycrystalline materials, an energy-minimizing process in which small cells gradually disappear and the average cell size increases.

Chapter 1 is an informal introduction and is accessible to general readers without a particular mathematical or scientific background. In it we introduce cell structures, curvature flow, steady states, and universal steady states. We use many examples to motivate questions we consider in later chapters.

Chapter 2 investigates one-dimensional cell structures that evolve through a variety of evolution equations. We show that many cell structures evolve towards universal steady states that depend only on the evolution equations and that are generally independent of initial conditions. Chapter 3 considers two-dimensional systems that evolve via curvature flow. We describe a simulation method, present analysis of its numerical accuracy, and provide a large set of results from simulations. Chapter 4 investigates three-dimensional systems that evolve via mean curvature flow. We describe a simulation method, present analysis of its numerical accuracy, and provide large amounts of data from simulations. We also introduce methods of characterizing the combinatorial structure of individual cells.

Chapter 5 compares and contrasts the curvature-flow evolution of two- and three-dimensional cell structures; we also consider properties of two-dimensional cross-sections of three-dimensional structures which have evolved through curvature flow. We conclude in Chapter 6 with a list of directions for further exploration. Appendix A describes a linear measure called the mean width, which plays a crucial role in describing systems which evolve through mean curvature flow. Appendices B and C describe technical details of the two- and three-dimensional simulation methods.