We propose a novel gauge fluid solver that evolves Clebsch wave functions on particle flow maps (PFMs). The key insight underlying our work is that particle flow maps exhibit superior performance in transporting point elements—such as Clebsch components—compared to line and surface elements, which were the focus of previous methods relying on impulse and vortex gauge variables for flow maps. Our Clebsch PFM method incorporates three main contributions: a novel gauge transformation enabling accurate transport of wave functions on particle flow maps, an enhanced velocity reconstruction method for coarse grids, and a PFM-based simulation framework designed to better preserve fine-scale flow structures. We validate the Clebsch PFM method through a wide range of benchmark tests and simulation examples, ranging from leapfrogging vortex rings and vortex reconnections to Kelvin-Helmholtz instabilities, demonstrating that our method outperforms its impulse- or vortex-based counterparts on particle flow maps, particularly in preserving and evolving small-scale features.

This paper introduces a two-phase interfacial fluid model based on the impulse variable to capture complex vorticity-interface interactions. Our key idea is to leverage bidirectional flow map theory to enhance the transport accuracy of both vorticity and interfaces simultaneously and address their coupling within a unified Eulerian framework. At the heart of our framework is an impulse ghost fluid method to solve the two-phase incompressible fluid characterized by its interfacial dynamics. To deal with the history-dependent jump of gauge variables across a dynamic interface, we develop a novel path integral formula empowered by spatiotemporal buffers to convert the history-dependent jump condition into a geometry-dependent jump condition when projecting impulse to velocity. We demonstrate the efficacy of our approach in simulating and visualizing several interface-vorticity interaction problems with cross-phase vortical evolution, including interfacial whirlpool, vortex ring reflection, and leapfrogging bubble rings.

We propose a novel Particle Flow Map (PFM) method to enable accurate long-range advection for incompressible fluid simulation. The foundation of our method is the observation that a particle trajectory generated in a forward simulation naturally embodies a perfect flow map. Centered on this concept, we have developed an Eulerian-Lagrangian framework comprising four essential components: Lagrangian particles for a natural and precise representation of bidirectional flow maps; a dual-scale map representation to accommodate the mapping of various flow quantities; a particle-to-grid interpolation scheme for accurate quantity transfer from particles to grid nodes; and a hybrid impulse-based solver to enforce incompressibility on the grid. The efficacy of PFM has been demonstrated through various simulation scenarios, highlighting the evolution of complex vortical structures and the details of turbulent flows. Notably, compared to NFM, PFM reduces computing time by up to 49 times and memory consumption by up to 41%, while enhancing vorticity preservation as evidenced in various tests like leapfrog, vortex tube, and turbulent flow.

We propose a new incompressible Navier-Stokes solver based on the impulse gauge transformation. The mathematical model of our approach draws from the impulse-velocity formulation of Navier-Stokes equations, which evolves the fluid impulse as an auxiliary variable of the system that can be projected to obtain the incompressible flow velocities at the end of each time step. We solve the impulse-form equations numerically on a Cartesian grid. At the heart of our simulation algorithm is a novel model to treat the impulse stretching and a harmonic boundary treatment to incorporate the surface tension effects accurately. We also build an impulse PIC/FLIP solver to support free-surface fluid simulation. Our impulse solver can naturally produce rich vortical flow details without artificial enhancements. We showcase this feature by using our solver to facilitate a wide range of fluid simulation tasks including smoke, liquid, and surface-tension flow. In addition, we discuss a convenient mechanism in our framework to control the scale and strength of the turbulent effects of fluid.

We present the Moving Eulerian-Lagrangian Particles (MELP), a novel mesh-free method for simulating incompressible fluid on thin films and foams. Employing a bi-layer particle structure, MELP jointly simulates detailed, vigorous flow and large surface deformation at high stability and efficiency. In addition, we design multi-MELP: a mechanism that facilitates the physically-based interaction between multiple MELP systems, to simulate bubble clusters and foams with non-manifold topological evolution. We showcase the efficacy of our method with a broad range of challenging thin film phenomena, including the Rayleigh-Taylor instability across double-bubbles, foam fragmentation with rim surface tension, recovery of the Plateau borders, Newton black films, as well as cyclones on bubble clusters.

We propose a novel Clebsch method to simulate the free-surface vortical flow. At the center of our approach lies a level-set method enhanced by a wave-function correction scheme and a wave-function extrapolation algorithm to tackle the Clebsch method's numerical instabilities near a dynamic interface. By combining the Clebsch wave function's expressiveness in representing vortical structures and the level-set function's ability on tracking interfacial dynamics, we can model complex vortex-interface interaction problems that exhibit rich free-surface flow details on a Cartesian grid. We showcase the efficacy of our approach by simulating a wide range of new free-surface flow phenomena that were impractical for previous methods, including horseshoe vortex, sink vortex, bubble rings, and free-surface wake vortices.

We propose a particle-based method to simulate thin-film fluid that jointly facilitates aggressive surface deformation and vigorous tangential flows. We build our dynamics model from the surface tension driven Navier-Stokes equation with the dimensionality reduced using the asymptotic lubrication theory and customize a set of differential operators based on the weakly compressible Smoothed Particle Hydrodynamics (SPH) for evolving pointset surfaces. The key insight is that the compressible nature of SPH, which is unfavorable in its typical usage, is helpful in our application to co-evolve the thickness, calculate the surface tension, and enforce the fluid incompressibility on a thin film. In this way, we are able to two-way couple the surface deformation with the in-plane flows in a physically based manner. We can simulate complex vortical swirls, fingering effects due to Rayleigh-Taylor instability, capillary waves, Newton's interference fringes, and the Marangoni effect on liberally deforming surfaces by presenting both realistic visual results and numerical validations. The particle-based nature of our system also enables it to conveniently handle topology changes and codimension transitions, allowing us to marry the thin-film simulation with a wide gamut of 3D phenomena, such as pinch-off of unstable catenoids, dripping under gravity, merging of droplets, as well as bubble rupture.

We propose a novel Lagrangian geometric representation using segment clouds to simulate incompressible fluid exhibiting strong anisotropic vortical features. The central component of our approach is a cloud of discrete segments enhanced by a set of local segment reseeding operations to facilitate both the geometrical evolution and the topological updates of vortical flow. We build a vortex dynamics solver with the support for dynamic solid boundaries based on discrete segment primitives. We demonstrate the efficacy of our approach by simulating a broad range of challenging flow phenomena, such as reconnection of non-closed vortex tubes and vortex shedding behind a rotating object.

We propose a novel gauge fluid solver based on Clebsch wave functions to solve incompressible fluid equations. Our method combines the expressive power of Clebsch wave functions to represent coherent vortical structures and the generality of gauge methods to accommodate a broad array of fluid phenomena. By evolving a transformed wave function as the system's gauge variable enhanced by an additional projection step to enforce pressure jumps on the free boundaries, our method can significantly improve the vorticity generation and preservation ability for a broad range of gaseous and liquid phenomena. Our approach can be easily implemented by modifying a standard grid-based fluid simulator. It can be used to solve various fluid dynamics, including complex vortex filament dynamics, fluids with different obstacles, and surface-tension flow.

We introduce Sparse Symplectically Integrated Neural Networks (SSINNs), a novel model for learning Hamiltonian dynamical systems from data. SSINNs combine fourth-order symplectic integration with a learned parameterization of the Hamiltonian obtained using sparse regression through a mathematically elegant function space. This allows for interpretable models that incorporate symplectic inductive biases and have low memory requirements. We evaluate SSINNs on four classical Hamiltonian dynamical problems: the Hénon-Heiles system, nonlinearly coupled oscillators, a multi-particle mass-spring system, and a pendulum system. Our results demonstrate promise in both system prediction and conservation of energy, often outperforming the current state-of-the-art black-box prediction techniques by an order of magnitude. Further, SSINNs successfully converge to true governing equations from highly limited and noisy data, demonstrating potential applicability in the discovery of new physical governing equations.

Silicon-based CMOS technology can be scaled well into the nanometer regime. High-performance, planar, ultrathin-body devices fabricated on silicon-on-insulator substrates have been demonstrated down to 15-nm gate lengths. We have also introduced the FinFET, a double-gate device structure that is relatively simple to fabricate and can be scaled to gate lengths below 10 nm. In this paper, some of the key elements of these technologies are described, including sublithographic patterning, the effects of crystal orientation and roughness on carrier mobility, gate work function engineering, circuit performance, and sensitivity to process-induced variations.