The work presented in this thesis focuses on improving the computational efficiency when simulating viscous liquids and air bubbles immersed in liquids by designing new discretizations to focus computational effort in regions that meaningfully contribute to creating realistic motion. For example, when simulating air bubbles rising through a liquid, the entire bubble volume is traditionally simulated despite the bubble’s interior being visually unimportant. We propose our constraint bubbles model to avoid simulating the interior of the bubble volume by reformulating the usual incompressibility constraint throughout a bubble volume as a constraint over only the bubble’s surface. Our constraint method achieves qualitatively similar results compared to a two-phase simulation ground-truth for bubbles with low densities (e.g., air bubbles in water). For bubbles with higher densities,
we propose our novel affine regions to model the bubble’s entire velocity field with a single affine vector field. We demonstrate that affine regions can correctly achieve hydrostatic equilibrium for bubble densities that match the surrounding liquid and correctly sink for higher densities. Finally, we introduce a tiled approach to subdivide large-scale affine regions into smaller subregions. Using this strategy, we are able to accelerate single-phase free surface flow simulations, offering a novel approach to adaptively enforce incompressibility in free surface liquids without complex data structures.
While pressure forces are often the bottleneck for inviscid fluid simulations, viscosity can impose orders of magnitude greater computational costs. We observed that viscous liquids require high simulation resolution at the surface to capture detailed viscous buckling and rotational motion but, because viscosity dampens relative motion, do not require the same resolution in the liquid’s interior. We therefore propose a novel adaptive method to solve free surface viscosity equations by discretizing the variational finite difference approach of Batty and Bridson (2008) on an octree grid. Our key insight is that the variational method
guarantees a symmetric positive definite linear system by construction, allowing the use of fast numerical solvers like the Conjugate Gradients method. By coarsening simulation grid cells inside the liquid volume, we rapidly reduce the degrees-of-freedom in the viscosity linear system up to a factor of 7.7x and achieve performance improvements for the linear solve between 3.8x and 9.4x compared to a regular grid equivalent. The results of our adaptive method closely match an equivalent regular grid for common scenarios such as: rotation and bending, buckling and folding, and solid-liquid interactions
