Codimensional fluid simulation on simplicial complexes [electronic resource]

Many visually interesting fluid phenomena are characterized by thin liquid sheets, narrow filaments, and small droplets. These range from phenomena observed in labs such as fluid polygons, fishbones, and waterbells, to phenomena that exist in everyday life such as soap bubbles, paint, toothpaste, and mayonnaise. It is extremely difficult to simulate these very thin features that are codimension-1 (two-dimensional films), codimension-2 (fluid filaments), and codimension-3 (very small drops) using numerical methods that are intended to represent only codimension-0 volumetric phenomena. This dissertation presents a new Lagrangian computational framework to simulate these codimensional fluid phenomena using non-manifold simplicial complexes. Tetrahedra, triangles, segments, and points are used as natural geometric analogues of fluid volumes, thin films, filaments, and droplets, respectively. We present a new Poisson solver on simplicial complexes for enforcing fluid incompressibility in different codimensions, along with a physically-guided meshing algorithm to provide temporally consistent information for interparticle forces. Our method naturally allows for transitions between codimensions, either from tetrahedra to triangles to segments to points or vice versa, regardless of the simulation resolution. We apply our framework in simulating two different types of codimensional incompressible flows: surface tension-driven phenomena and non-Newtonian phenomena. For the former, we propose a new codimensional surface tension discretization to model surface tension effects on simplical complexes. We introduce a special rim-based surface tension force to model interesting rim phenomena such as thin film retraction and droplets pinching off. We use our framework to simulate various codimensional surface tension-driven phenomena including blowing bubbles and popping bubbles, as well as thin film catenoids, waterbells, fishbones, and fluid polygons. We then extend our computational framework to model various types of non-Newtonian fluids including shear thinning fluids, shear thickening fluids, Bingham plastics, and elastoplastics. We propose a semi-implicit time integration scheme for elasticity, which when combined with a semi-implicit method for variable viscosity alleviates the need for small time steps. Furthermore, we propose an improved treatment of viscosity on the rims of thin fluid sheets that allows us to capture their elusive, visually appealing twisting motion. In order to simulate complex phenomena such as the mixing of colored paint, we adopt a multiple level set framework and propose a discretization on simplicial complexes that facilitates the tracking of material interfaces across codimensions. We demonstrate the efficacy of our approach by simulating a wide variety of non-Newtonian fluid phenomena exhibiting various codimensional features such as paint, cheese, toothpaste, and mayonnaise. In addition to the new Lagrangian framework for interfacial fluid phenomena, this dissertation also presents a new Eulerian grid structure that can dynamically extend its computational domain to create a significantly larger far-field grid. The far-field grid structure preserves almost every computational advantage of uniform grids including cache coherency, regular subdivisions for parallelization, simple data layout, the existence of efficient numerical discretizations and algorithms for solving partial differential equations, etc. This allows fluid simulations to cover large domains that are often infeasible to enclose with sufficient resolution using a uniform grid, while still effectively capturing fine scale details in regions of interest using dynamic adaptivity. We show the efficiency of our grid structure by simulating different types of fluid including smoke, fire, and water in large space.

Advisor: Fedkiw Ronald P.
Iaccarino Gianluca advisor. Thesis advisor
Levis Philip