@phdthesis{
author={Nabizadeh,Mohammad},
year={2025},
title={Fluid Dynamics: From Geometric Formulations to Structure-Preserving Simulations},
journal={ProQuest Dissertations and Theses},
pages={304},
note={Copyright - Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works; Last updated - 2025-07-23},
abstract={This dissertation develops a suite of geometric and structure-preserving methods for simulating incompressible fluid dynamics, beginning with foundational geometric formulations and culminating in structure-preserving numerical schemes. We first introduce the Covector Fluids formulation, which recasts the Euler equations in terms of the momentum covector—a differential one-form dual to velocity—allowing advection to be expressed as a pullback via the Lie derivative. This geometric viewpoint leads to a discretization that respects Kelvin’s circulation theorem and commutes with pressure projection, eliminating common splitting errors and preserving circulation. While a velocity-based method, it emulates vortex methods without their computational cost, and integrates easily into existing solvers. It captures rich vortex dynamics with significantly reduced numerical dissipation behavior over traditional approaches. Building on this foundation, we propose CO-FLIP (Coadjoint Orbits Fluid Implicit Particles), a high-order, structure-preserving solver in the hybrid Eulerian–Lagrangian framework. By representing the fluid as a modified Hamiltonian system evolving based a novel formulation, we call Discrete Euler equations, CO-FLIP captures the motion of incompressible flow along coadjoint orbits. The method uses local, divergence-free interpolation with exact bidirectional transfer between grid and particles, and it is discretized in time with a symplectic Lie group integrator. It preserves circulation, energy, and weak-form pressure projection, achieving excellent long-term stability and Casimir conservation, even at low resolutions--culminating in a method with exceptional fidelity in capturing vortical structures and complex flow dynamics. Further, to enhance fidelity while retaining structure, we extend CO-FLIP with nonlinear discretization schemes, such as neural networks trained to represent turbulent divergence-free velocity fields. These CO-FLIP algorithms are shown to follow a constrained variational principle, which yields strong solutions confined to a discrete velocity manifold. This principle is naturally interpreted through sub-Riemannian geometry and vakonomic dynamics, where the system evolves along geodesics within constrained distribution. Finally, we address the simulation of flows on unbounded domains using the Kelvin transform, a conformal mapping that converts infinite-domain Poisson problems, among others, into well-posed problems on bounded domains without introducing artificial boundaries. By factoring out known asymptotic behavior, the transformed problem remains smooth and compact, enabling accurate pressure projection for open flows using standard solvers. Together, these contributions demonstrate that encoding the differential-geometric structure of fluid dynamics into numerical schemes yields methods with superior conservation, long-term stability, and enhanced physical realism.},
keywords={Computer graphics; Fluid simulation; Geometric mechanics; Geometry processing; Structure-preserving algorithms; Vakonomic dynamics; Computer science; Mechanics; Fluid mechanics; 0346:Mechanics; 0204:Fluid mechanics; 0984:Computer science},
isbn={9798286494286},
language={English},
url={https://www.proquest.com/dissertations-theses/fluid-dynamics-geometric-formulations-structure/docview/3226972823/se-2},
}