Simulating Viscous Incompressible Fluids with Embedded Boundary Finite Difference Methods

The behaviour of liquids and gases ranks among the most familiar and yet complex physical phenomena commonly encountered in daily life. To create a seamless approximation of the real world, it is clear that we must be able to accurately simulate fluids. However, a crucial element of what makes fluid behaviour so complex and compelling is its interactions with its surroundings. To simulate the motion of a fluid we cannot consider the Navier-Stokes equations in isolation; we must also examine the boundary conditions at the point where the fluid meets the world. Enforcing these boundary conditions has traditionally been a source of tremendous difficulty. Cartesian grid-based methods typically approximate the world in an unrealistic, axis-aligned block representation, while conforming mesh methods frequently suffer from poor mesh quality and expensive mesh generation. This thesis examines the use of embedded boundary finite difference methods to alleviate these shortcomings by providing a degree of sub-grid information that enables more efficient, flexible, accurate, and realistic simulations. The first key contribution of the thesis is the use of a variational approach to derive novel embedded boundary finite difference methods for fluids, by exploiting the concept of natural boundary conditions. This idea is applied first to animate the interaction between incompressible fluids and irregularly shaped dynamic rigid bodies. I then apply a similar technique to properly handle viscous free surfaces, enabling realistic buckling and coiling in viscous flows. Lastly, I unify these ideas