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Recent advances in cell biology and experimental techniques using reconstituted cell extracts have generated significant interest in understanding how geometry and topology influence active fluid dynamics. In this work, we present a comprehensive continuum theory and computational method to explore the dynamics of active nematic fluids on arbitrary surfaces without topological constraints. The fluid velocity and nematic order parameter are represented as the sections of the complex line bundle of a two-manifold. We introduce the Levi–Civita connection and surface curvature form within the framework of complex line bundles. By adopting this geometric approach, we introduce a gauge-invariant discretization method that preserves the continuous local-to-global theorems in differential geometry. We establish a nematic Laplacian on complex functions that can accommodate fractional topological charges through the covariant derivative on the complex nematic representation. We formulate advection of the nematic field based on a unifying definition of the Lie derivative, resulting in a stable geometric semi-Lagrangian (sL) discretization scheme for transport by the flow. In general, the proposed surface-based method offers an efficient and stable means to investigate the influence of local curvature and global topology on the two-dimensional hydrodynamics of active nematic systems. 

The dynamics of evolving fluid films in the viscous Stokes limit is relevant to various applications, such as the modelling of lipid bilayers in cells. While the governing equations were formulated by Scriven (1960), solving for the flow of a deformable viscous surface with arbitrary shape and topology has remained a challenge. In this study, we present a straightforward discrete model based on variational principles to address this long-standing problem. We replace the classical equations, which are expressed with tensor calculus in local coordinates, with a simple coordinate-free, differential-geometric formulation. The formulation provides a fundamental understanding of the underlying mechanics and translates directly to discretization. We construct a discrete analogue of the system using Onsager's variational principle, which, in a smooth context, governs the flow of a viscous medium. In the discrete setting, instead of term-wise discretizing the coordinate-based Stokes equations, we construct a discrete Rayleighian for the system and derive the discrete Stokes equations via the variational principle. This approach results in a stable, structure-preserving variational integrator that solves the system on general manifolds. 

We present a formula for the signed area of a spherical polygon via prequantization. In contrast to the traditional formula based on the Gauss–Bonnet theorem that requires measuring angles, the new formula mimics Green’s theorem and is applicable to a wider range of degenerate spherical curves and polygons. 

Identifying optimal structural designs given loads and constraints is a primary challenge in topology optimization and shape optimization. We propose a novel approach to this problem by finding a minimal tensegrity structure—a network of cables and struts in equilibrium with a given loading force. Through the application of geometric measure theory and compressive sensing techniques, we show that this seemingly difficult graph-theoretic problem can be reduced to a numerically tractable continuous optimization problem. With a light-weight iterative algorithm involving only Fast Fourier Transforms and local algebraic computations, we can generate sparse supporting structures featuring detailed branches, arches, and reinforcement structures that respect the prescribed loading forces and obstacles. 

The demand for a more advanced multivariable calculus has rapidly increased in computer graphics research, such as physical simulation, geometry synthesis, and differentiable rendering. Researchers in computer graphics often have to turn to references outside of graphics research to study identities such as the Reynolds Transport Theorem or the geometric relationship between stress and strain tensors. This course presents a comprehensive introduction to exterior calculus, which covers many of these advanced topics in a geometrically intuitive manner. The course targets anyone who knows undergraduate-level multivariable calculus and linear algebra and assumes no more prerequisites. Contrary to the existing references, which only serve the pure math or engineering communities, we use timely and relevant graphics examples to illustrate the theory of exterior calculus. We also provide accessible explanations to several advanced topics, including continuum mechanics, fluid dynamics, and geometric optimizations. The course is organized into two main sections: a lecture on the core exterior calculus notions and identities with short examples of graphics applications, and a series of mini-lectures on graphics topics using exterior calculus. 

We address the problem of synthesizing physical animations that can loop seamlessly. We formulate a variational approach by deriving a physical law in a periodic time domain. The trajectory of the animation is represented as a parametric closed curve, and the physical law corresponds to minimizing the bending energy of the curve. Compared to traditional keyframe animation approaches, our formulation is constraint-free, which allows us to apply a standard Gauss--Newton solver. We further propose a fast projection method to efficiently generate an initial guess close to the desired animation. Our method can handle a variety of physical cyclic animations, including clothes, soft bodies with collisions, and N-body systems. 

The vorticity-streamfunction formulation for incompressible inviscid fluids is the basis for many fluid simulation methods in computer graphics, including vortex methods, streamfunction solvers, spectral methods, and Monte Carlo methods. We point out that current setups in the vorticity-streamfunction formulation are insufficient at simulating fluids on general non-simply-connected domains. This issue is critical in practice, as obstacles, periodic boundaries, and nonzero genus can all make the fluid domain multiply connected. These scenarios introduce nontrivial cohomology components to the flow in the form of harmonic fields. The dynamics of these harmonic fields have been previously overlooked. In this paper, we derive the missing equations of motion for the fluid cohomology components. We elucidate the physical laws associated with the new equations, and show their importance in reproducing physically correct behaviors of fluid flows on domains with general topology. 

We propose Coadjoint Orbit FLIP (CO-FLIP), a high order accurate, structure preserving fluid simulation method in the hybrid Eulerian-Lagrangian framework. We start with a Hamiltonian formulation of the incompressible Euler Equations, and then, using a local, explicit, and high order divergence free interpolation, construct a modified Hamiltonian system that governs our discrete Euler flow. The resulting discretization, when paired with a geometric time integration scheme, is energy and circulation preserving (formally the flow evolves on a coadjoint orbit) and is similar to the Fluid Implicit Particle (FLIP) method. CO-FLIP enjoys multiple additional properties including that the pressure projection is exact in the weak sense, and the particle-to-grid transfer is an exact inverse of the grid-to-particle interpolation. The method is demonstrated numerically with outstanding stability, energy, and Casimir preservation. We show that the method produces benchmarks and turbulent visual effects even at low grid resolutions.