Abstract: Signed distance functions (SDFs) are essential for many problems across graphics and vision. But while it is easy to compute the SDF of a closed shape free of defects, it is difficult to robustly compute the SDF of a shape that has been corrupted by holes, noise, or self-intersections. This talk presents the signed heat method for solving this problem: given "bad" geometry, the method gives a good SDF of the unknown uncorrupted geometry, without having to explicitly reconstruct the shape. The method hence simultaneously solves a difficult surface reconstruction problem while computing signed distance. Furthering this theme, this talk presents a unified framework for both reconstruction and distance computation that shows (1) the robustness of the signed heat method is hard to beat, and (2) broader implications for "robustifying" problems beyond distance computation.

Abstract: Consider an electrically conducting fluid (i.e., a plasma) that is initially at rest. If we allow the fluid to relax to a minimum energy state subject to the ideal magnetohydrodynamics (MHD) equations, can we understand the relaxation process or predict the relaxed state? In this talk, inspired by this so-called "magnetic relaxation problem," we explore surprising connections between ideal MHD and conformal geometry, geodesibility of vector fields as well as optimal transport. We also discuss a novel numerical approach to the relaxation of discrete plasma filaments that emerges from these insights.

Abstract: Many tasks in computer graphics, including fluid animation, are typically addressed by discretizing the entire domain and solving large, globally coupled systems. The successes of pointwise Monte Carlo methods in rendering and geometry processing raise the question: can Monte Carlo methods also benefit physics-based simulations?
In this talk, I will present my research aimed at answering this question. Over the past few years, I have developed Monte Carlo fluid simulation methods using vorticity- and velocity-based formulations. A significant part of this research includes advancements in Monte Carlo PDE solvers. These new methods offer flexibility in using substep solvers and can handle complex boundaries without relying on a cut-cell or conforming mesh discretization that traditional approaches require. I will also discuss insights into how our methods relate to existing techniques in computer graphics and conclude by highlighting several open research directions.

Abstract: This talk will cover two recent projects: In the first half I will discuss solid knitting, a new fabrication method which we recently proposed for making dense, volumetric objects from stacked layers of knit fabric, and will give an overview of the solid knitting machine which we built to automate the process. The layer-by-layer approach of solid knitting is inspired by 3D printing, but because they layers are held together purely by the topological stitch structure---without requiring any sort of adhesive---solid knit objects can easily be unraveled to recover their constituent yarn. In the second half, I will discuss Harnack tracing, a new ray tracing method for visualizing level sets of harmonic functions. Harmonic functions appear ubiquitously in applications from surface reconstruction to architectural geometry, but usually cannot be visualized directly using existing techniques. I will show how our method can be used to visualize smooth surfaces directly from point clouds (via Poisson surface reconstruction) or polygon soup (via generalized winding numbers) without linear solves or mesh extraction, to visualize nonplanar polygons (possibly with holes), surfaces from architectural geometry, and key mathematical objects including knots, links, spherical harmonics, and Riemann surfaces.

Abstract: In the world of interactive computer graphics— spanning video games, virtual reality training, and more— physics simulation plays a fundamental role. However, achieving high-quality, responsive simulations in this context comes with a set of complex technical hurdles. This talk delves into the pursuit of simulations that are not just fast and responsive, but also how the richness of real-time simulations can be ameliorated without sacrificing performance and with minimal changes to existing simulation pipelines. I will explore some of our recent advancements on improving the stability of constrained rigid body simulations, a mainstay of interactive graphics, and additionally showcase new developments on enhancing realism, including advanced friction models, impact and collision responses, and more!

Abstract: Our world is bound by unbreakable physical laws that dictate all our interactions within it: from how we perceive touch, to how we plan and build large-scale infrastructure projects. A fully interactive, physically accurate virtual world would not only allow us to reproduce these interactions cheaply and efficiently, it also allows for the possibility of breaking these laws, widening the scope of what humans can design, build and create. This promises foundational changes in artistic, scientific and engineering disciplines, and has been a long-standing scientific goal since the invention of the first computer displays.
Unfortunately, the road to physical models for virtual interaction is paved with contradictions.
The model needs to be fast enough for interaction, but also accurate enough to model the complexity of various real-world phenomena. The model needs to be realistic enough to inform real-world decisions, while also allowing for non-physical user-interactions which are a central appeal of virtual environments.
Throughout this talk I will outline research I’ve done through my PhD so far, which traverses the treacherous path to interactive virtual physical worlds. In particular I target virtual worlds driven primarily by soft-body physics, which are traditionally neglected in standard fast physics simulation engines. I will show various avenues we can bring reliable soft-body physics to the real-world regime, while also providing design and control interfaces for such soft-body physical worlds.

Abstract: Incompressible fluids enjoy particle relabeling symmetry, giving rise to Kelvin's circulation conservation. For the past two decades, researchers have actively developed methods mimicking this continuous fluid property in discretized simulations. From a computational fluid dynamics (CFD) viewpoint, these discrete models resemble finite difference or finite volume schemes, which unfortunately have limited stability conditions despite the alluring benefits of structure preservations. Thus, non-structure-preserving (but more stable) semi-Lagrangian, fluid-implicit-particle (FLIP), and particle-in-cell (PIC) schemes are still the dominating numerical methods in applied fluid simulations. We propose a new approach to structure-preserving discrete fluids. This approach results in discrete models that resemble the mainstream FLIP or PIC methods but with much more geometric structures. By incorporating isogeometric analysis techniques (i.e. mimetic interpolation), a discrete divergence-free grid velocity interpolates into a smooth divergence-free vector field. Additionally, instead of only moving the positions of the particles by this interpolated velocity field, we act symplectomorphically on both the positions and the momenta by this field. This symplectomorphic group action, therefore, induces a momentum map from the particles' position-momentum space to the dual space of velocities. We argue that this canonical map should be taken as the particle-to-grid information transfer. Since momentum maps preserve the Poisson structure, Hamiltonian flows of particles will map into a coadjoint orbit in the dual Lie algebra of divergence-free velocities. In particular, our method preserves Casimirs, such as 2D enstrophy and 3D helicity. We call our method Coadjoint Orbit FLIP (CO-FLIP), a high-order accurate, structure-preserving fluid simulation method in the hybrid Eulerian-Lagrangian framework. We showcase that traditional simulation methods benefit from structure-preserving techniques by producing higher fidelity vortical structures without the need to have prohibitively high-resolution computation grids.

Abstract: Numerical solutions of partial differential equations (PDEs) are widely used in engineering, especially for modelling phenomena like elastic deformations or fluid simulations. The finite element method (FEM) is the most commonly used technique for discretizing PDEs because of its versatility and range of available (commercial) implementations.
Typically, the PDE solver treats meshing and basis construction as separate problems. However, the basis construction may make assumptions that lead to challenging requirements for meshing software. This can be a significant issue for applications that require fully automatic, robust processing of large collections of meshes or when the PDE solver needs to change the mesh.
We present recent advancements that have led to a unified pipeline that considers meshing and element design as a single challenge. This approach enables a PDE solver that can handle simulations on thousands of domains without requiring parameter tuning.

Abstract: The Laplace Beltrami operator is one of the essential tools in geometric processing. It allows us to solve numerous partial differential equations on discrete surface meshes, which is a fundamental building block in many computer graphics applications. However, discrete Laplacians are typically limited to standard elements like triangles or quadrilaterals, which severely constrains the tessellation of the mesh. This talk presents an easy yet efficient strategy to generalize the Laplace Beltrami and its closely related gradient and divergence operators to more general meshes. Furthermore, we discuss how minimizing the trace affects the spectrum of the polygon Laplacian and how to use this information to improve the operator's robustness and accuracy.