Efficient Liquid Animation: New Discretizations for Spatially Adaptive Liquid Viscosity and Reduced-Model Two-Phase Bubbles and Inviscid Liquids

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

Nonstandard Finite Element Methods

[no abstract available

Incompressible Lagrangian fluid flow with thermal coupling

In this monograph is presented a method for the solution of an incompressible viscous fluid flow with heat transfer and solidification usin a fully Lagrangian description on the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation.Postprint (published version

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on 2-7 Apr. 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection shows its rapid trend to further diversity and depth

Simulation of pore-scale flow using finite element-methods

I present a new finite element (FE) simulation method to simulate pore-scale flow. Within the pore-space, I solve a simplified form of the incompressible Navier-Stoke’s equation, yielding the velocity field in a two-step solution approach. First, Poisson’s equation is solved with homogeneous boundary conditions, and then the pore pressure is computed and the velocity field obtained for no slip conditions at the grain boundaries. From the computed velocity field I estimate the effective permeability of porous media samples characterized by thin section micrographs, micro-CT scans and synthetically generated grain packings. This two-step process is much simpler than solving the full Navier Stokes equation and therefore provides the opportunity to study pore geometries with hundreds of thousands of pores in a computationally more cost effective manner than solving the full Navier-Stoke’s equation. My numerical model is verified with an analytical solution and validated on samples whose permeabilities and porosities had been measured in laboratory experiments (Akanji and Matthai, 2010). Comparisons were also made with Stokes solver, published experimental, approximate and exact permeability data. Starting with a numerically constructed synthetic grain packings, I also investigated the extent to which the details of pore micro-structure affect the hydraulic permeability (Garcia et al., 2009). I then estimate the hydraulic anisotropy of unconsolidated granular packings. With the future aim to simulate multiphase flow within the pore-space, I also compute the radii and derive capillary pressure from the Young-Laplace equation (Akanji and Matthai,2010

Asynchronous Stabilisation and Assembly Techniques for Additive Multigrid

Multigrid solvers are among the best solvers in the world, but once applied in the real world there are issues they must overcome. Many multigrid phases exhibit low concurrency. Mesh and matrix assembly are challenging to parallelise and introduce algorithmic latency. Dynamically adaptive codes exacerbate these issues. Multigrid codes require the computation of a cascade of matrices and dynamic adaptivity means these matrices are recomputed throughout the solve. Existing methods to compute the matrices are expensive and delay the solve. Non- trivial material parameters further increase the cost of accurate equation integration. We propose to assemble all matrix equations as stencils in a delayed element-wise fashion. Early multigrid iterations use cheap geometric approximations and more accurate updated stencil integrations are computed in parallel with the multigrid cycles. New stencil integrations are evaluated lazily and asynchronously fed to the solver once they become available. They do not delay multigrid iterations. We deploy stencil integrations as parallel tasks that are picked up by cores that would otherwise be idle. Coarse grid solves in multiplicative multigrid also exhibit limited concurrency. Small coarse mesh sizes correspond to small computational workload and require costly synchronisation steps. This acts as a bottleneck and delays solver iterations. Additive multigrid avoids this restriction, but becomes unstable for non-trivial material parameters as additive coarse grid levels tend to overcorrect. This leads to oscillations. We propose a new additive variant, adAFAC-x, with a stabilisation parameter that damps coarse grid corrections to remove oscillations. Per-level we solve an additional equation that produces an auxiliary correction. The auxiliary correction can be computed additively to the rest of the solve and uses ideas similar to smoothed aggregation multigrid to anticipate overcorrections. Pipelining techniques allow adAFAC-x to be written using single-touch semantics on a dynamically adaptive mesh

Modeling multiphase flow in porous medium systems at multiple scales

Problems involving multiphase flow and transport in porous media arise in a number of scientific and engineering applications including oil reservoir engineering and groundwater remediation. The inherent complexity of multiphase systems and the marked heterogeneity over multiple spatial scales result in significant challenges to the fundamental understanding of the multiphase flow and transport processes. For many decades, multiphase flow has been modeled using the traditional approach based on mass conservation and the generalized Darcy's law. The traditional approach, however, is subject to model errors and numerical errors. The focus of this dissertation research is to improve models of flow and transport in porous medium systems using numerical modeling approaches for a range of scales including pore scale and continuum scale. A major part of this research examines the deficiency of Darcy's relationship and its extension to multiphase flow using the lattice-Boltzmann (LB) approach. This study investigates the conventional relative permeability saturation relation for systems consisting of water and non-aqueous phase liquid (NAPL). In addition, it also examines the generalized formulation accounting for the interfacial momentum transfer and lends additional support to the hypothesis that interfacial area is a critical variable in multiphase porous medium systems. Another major part of the research involves developing efficient and robust numerical techniques to improve the solution approach for existing models. In particular, a local discontinuous Galerkin (LDG) spatial discretization method is developed in combination with a robust and established variable order, variable step-size temporal integration approach to solve Richards' equation (RE). Effective spatial adaptive LDG methods are also developed to further enhance the efficiency. The resulting simulator with both spatial and temporal adaption has demonstrated good performance for a series of problems modeled by RE