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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. 

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 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. 

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. 

This paper presents a new representation of curve dynamics, with applications to vortex filaments in fluid dynamics. Instead of representing these filaments with explicit curve geometry and Lagrangian equations of motion, we represent curves implicitly with a new co-dimensional 2 level set description. Our implicit representation admits several redundant mathematical degrees of freedom in both the configuration and the dynamics of the curves, which can be tailored specifically to improve numerical robustness, in contrast to naive approaches for implicit curve dynamics that suffer from overwhelming numerical stability problems. Furthermore, we note how these hidden degrees of freedom perfectly map to a Clebsch representation in fluid dynamics. Motivated by these observations, we introduce untwisted level set functions and non-swirling dynamics which successfully regularize sources of numerical instability, particularly in the twisting modes around curve filaments. A consequence is a novel simulation method which produces stable dynamics for large numbers of interacting vortex filaments and effortlessly handles topological changes and re-connection events.