A Unified Simplicial Model for Mixed-Dimensional and Non-Manifold Deformable Elastic Objects

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected] present a unified method to simulate deformable elastic bodies consisting of mixed-dimensional components represented with potentially non-manifold simplicial meshes. Building on well-known simplicial rod, shell, and solid models for elastic continua, we categorize and define a comprehensive palette expressing all possible constraints and elastic energies for stiff and flexible connections between the 1D, 2D, and 3D components of a single conforming simplicial mesh. This palette consists of three categories: point connections, in which simplices meet at a single vertex around which they may twist and bend; curve connections in which simplices share an edge around which they may rotate (bend) relative to one another; and surface connections, in which a shell is embedded on or into a solid. To define elastic behaviors across non-manifold point connections, we adapt and apply parallel transport concepts from elastic rods. To address discontinuous forces that would otherwise arise when large accumulated relative rotations wrap around in the space of angles, we develop an incremental angle-update strategy. Our method provides a conceptually simple, flexible, and highly expressive framework for designing complex elastic objects, by modeling the geometry with a single simplicial mesh and decorating its elements with appropriate physical models (rod, shell, solid) and connection types (point, curve, surface). We demonstrate a diverse set of possible interactions achievable with our method, through technical and application examples, including scenes featuring complex aquatic creatures, children's toys, and umbrellas.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (RGPIN-04360-2014

Double Bubbles Sans Toil and Trouble: Discrete Circulation-Preserving Vortex Sheets for Soap Films and Foams

© ACM, 2015. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in Da, F., Batty, C., Wojtan, C., & Grinspun, E. (2015). Double Bubbles Sans Toil and Trouble: Discrete Circulation-Preserving Vortex Sheets for Soap Films and Foams. Acm Transactions on Graphics, 34(4), 149. https://doi.org/10.1145/2767003Simulating the delightful dynamics of soap films, bubbles, and foams has traditionally required the use of a fully three-dimensional many-phase Navier-Stokes solver, even though their visual appearance is completely dominated by the thin liquid surface. We depart from earlier work on soap bubbles and foams by noting that their dynamics are naturally described by a Lagrangian vortex sheet model in which circulation is the primary variable. This leads us to derive a novel circulation-preserving surface-only discretization of foam dynamics driven by surface tension on a non-manifold triangle mesh. We represent the surface using a mesh-based multimaterial surface tracker which supports complex bubble topology changes, and evolve the surface according to the ambient air flow induced by a scalar circulation field stored on the mesh. Surface tension forces give rise to a simple update rule for circulation, even at non-manifold Plateau borders, based on a discrete measure of signed scalar mean curvature. We further incorporate vertex constraints to enable the interaction of soap films with wires. The result is a method that is at once simple, robust, and efficient, yet able to capture an array of soap films behaviors including foam rearrangement, catenoid collapse, blowing bubbles, and double bubbles being pulled apart.This work was supported in part by the NSF (Grant IIS-1319483),ERC (Grant ERC-2014-StG-638176), NSERC (Grant RGPIN-04360-2014), Adobe, and Intel

Simulating liquids on dynamically warping grids

We introduce dynamically warping grids for adaptive liquid simulation. Our primary contributions are a strategy for dynamically deforming regular grids over the course of a simulation and a method for efficiently utilizing these deforming grids for liquid simulation. Prior work has shown that unstructured grids are very effective for adaptive fluid simulations. However, unstructured grids often lead to complicated implementations and a poor cache hit rate due to inconsistent memory access. Regular grids, on the other hand, provide a fast, fixed memory access pattern and straightforward implementation. Our method combines the advantages of both: we leverage the simplicity of regular grids while still achieving practical and controllable spatial adaptivity. We demonstrate that our method enables adaptive simulations that are fast, flexible, and robust to null-space issues. At the same time, our method is simple to implement and takes advantage of existing highly-tuned algorithms

Differential operators on sketches via alpha contours

A vector sketch is a popular and natural geometry representation depicting a 2D shape. When viewed from afar, the disconnected vector strokes of a sketch and the empty space around them visually merge into positive space and negative space, respectively. Positive and negative spaces are the key elements in the composition of a sketch and define what we perceive as the shape. Nevertheless, the notion of positive or negative space is mathematically ambiguous: While the strokes unambiguously indicate the interior or boundary of a 2D shape, the empty space may or may not belong to the shape’s exterior. For standard discrete geometry representations, such as meshes or point clouds, some of the most robust pipelines rely on discretizations of differential operators, such as Laplace-Beltrami. Such discretizations are not available for vector sketches; defining them may enable numerous applications of classical methods on vector sketches. However, to do so, one needs to define the positive space of a vector sketch, or the sketch shape. Even though extracting this 2D sketch shape is mathematically ambiguous, we propose a robust algorithm, Alpha Contours, constructing its conservative estimate: a 2D shape containing all the input strokes, which lie in its interior or on its boundary, and aligning tightly to a sketch. This allows us to define popular differential operators on vector sketches, such as Laplacian and Steklov operators. We demonstrate that our construction enables robust tools for vector sketches, such as As-Rigid-As-Possible sketch deformation and functional maps between sketches, as well as solving partial differential equations on a vector sketch

Surface-Only Liquids

© ACM, 2016. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in Da, F., Hahn, D., Batty, C., Wojtan, C., & Grinspun, E. (2016). Surface-Only Liquids. Acm Transactions on Graphics, 35(4), 78. https://doi.org/10.1145/2897824.2925899We propose a novel surface-only technique for simulating incompressible, inviscid and uniform-density liquids with surface tension in three dimensions. The liquid surface is captured by a triangle mesh on which a Lagrangian velocity field is stored. Because advection of the velocity field may violate the incompressibility condition, we devise an orthogonal projection technique to remove the divergence while requiring the evaluation of only two boundary integrals. The forces of surface tension, gravity, and solid contact are all treated by a boundary element solve, allowing us to perform detailed simulations of a wide range of liquid phenomena, including waterbells, droplet and jet collisions, fluid chains, and crown splashes.European Research Council, National Science Foundation, Natural Sciences and Engineering Research Council of Canad

Variational Stokes: A Unified Pressure-viscosity Solver for Accurate Viscous Liquids

© ACM, 2017. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in Larionov, E., Batty, C., & Bridson, R. (2017). Variational Stokes: A Unified Pressure-viscosity Solver for Accurate Viscous Liquids. ACM Trans. Graph., 36(4), 101:1–101:11. https://doi.org/10.1145/3072959.3073628We propose a novel unsteady Stokes solver for coupled viscous and pressure forces in grid-based liquid animation which yields greater accuracy and visual realism than previously achieved. Modern fluid simulators treat viscosity and pressure in separate solver stages, which reduces accuracy and yields incorrect free surface behavior. Our proposed implicit variational formulation of the Stokes problem leads to a symmetric positive definite linear system that gives properly coupled forces, provides unconditional stability, and treats difficult boundary conditions naturally through simple volume weights. Surface tension and moving solid boundaries are also easily incorporated. Qualitatively, we show that our method recovers the characteristic rope coiling instability of viscous liquids and preserves fine surface details, while previous grid-based schemes do not. Quantitatively, we demonstrate that our method is convergent through grid refinement studies on analytical problems in two dimensions. We conclude by offering practical guidelines for choosing an appropriate viscous solver, based on the scenario to be animated and the computational costs of different methods.Natural Sciences and Engineering Research Council of Canad

Heterotic target space dualities with line bundle cohomology

Die vorliegende Dissertation befasst sich mit verschiedenen Aspekten und Techniken zur Konstruktion von String-Modellen. In diesem Kontext ist es nötig die Topologie von Calabi-Yau Mannigfaltigkeiten zu verstehen, da diese ausschlaggebend für die Nullmodenstruktur des entsprechenden Differenzialoperators und damit für das Teilchenspektrum der kompaktifizierten Niederenergietheorie ist. Für diejenigen Calabi-Yau Räume, die als Unterräume torischer Varietäten definiert werden, sind alle topologischen Größen in der Kohomololgie von Linienbündeln über der entsprechenden torischen Varietät verschlüsselt. Aus diesem Grund umfasst ein Teil dieser Dissertation die Entwicklung eines effizienten Algorithmus’ für ihre Berechnung. Nach der mathematischen Vorbereitung widmen wir uns der Herleitung und dem Beweis des auf diese Weise entstandenen mathematischen Theorems. Wir untersuchen zudem eine Verallgemeinerung auf Räume, die durch das Herausteilen einer Zn-Symmetrie konstruiert werden. Anschließend demonstrieren wir die zahlreichen Anwendungen dieser Methoden zur Konstruktion von String-Modellen. Außerdem finden wir einen Zusammenhang zwischen Kohomologiegruppen von Linienbündeln und getwisteten Sektoren von Landau-Ginzburg Modellen. Als nächstes nutzen wir die entwickelten Methoden um so genannte Zielraum Dualitäten zwischen heterotischen Modellen zu untersuchen. Diese Modelle weisen eine asymmetrische (0,2)-Weltflächensupersymmetrie auf und können über geeichte lineare Sigma-Modelle formuliert werden, in welchen sie eine Phasenstruktur ausbilden. Es lässt sich nun zeigen, dass die Phasenräume verschiedener physikalischer Modelle durch nicht-geometrische Phasen miteinander verbunden sind, was eine hochgradig nicht-triviale Dualität der entsprechenden Geometrien implizieren könnte. Unser Beitrag ist nun die Untersuchung der hierdurch verbundenen und daher potentiell dualen Modelle. Wir entwickeln ein Verfahren, welches die Konstruktion aller dualer Modelle zu einem beliebigen (0,2) Modell erlaubt und finden Evidenz dafür, dass es sich hierbei um eine echte Dualität und nicht bloß um einen Übergang verschiedener physikalischer Modelle ineinander handelt. In diesem Kontext untersuchen wir verschiedenste Szenarien, u.A. Modelle mit den Eichgruppen E6, SO(10) und SU(5), sowie mit Kompaktifizierungsräumen der Kodimension eins und zwei. In einer Untersuchung der Stringlandschaft werden dazu über 80.000 Räume auf diese Dualität untersucht

Novel Paradigms in Physics-Based Animation: Pointwise Divergence-Free Fluid Advection and Mixed-Dimensional Elastic Object Simulation

This thesis explores important but so far less studied aspects of physics-based animation: a simulation method for mixed-dimensional and/or non-manifold elastic objects, and a pointwise divergence-free velocity interpolation method applied to fluid simulation. Considering the popularity of single-type models e.g., hair, cloths, soft bodies, etc., in deformable body simulations, more complicated coupled models have gained less attention in graphics research, despite their relative ubiquity in daily life. This thesis presents a unified method to simulate such models: elastic bodies consisting of mixed-dimensional components represented with potentially non-manifold simplicial meshes. Building on well-known simplicial rod, shell, and solid models, this thesis categorizes and defines a comprehensive palette expressing all possible constraints and elastic energies for stiff and flexible connections between the 1D, 2D, and 3D components of a single conforming simplicial mesh. For fluid animation, this thesis proposes a novel methodology to enhance grid-based fluid animation with pointwise divergence-free velocity interpolation. Unlike previous methods which interpolate discrete velocity values directly for advection, this thesis proposes using intermediate steps involving vector potentials: first build a discrete vector potential field, interpolate these values to form a pointwise potential, and apply the continuous curl to recover a pointwise divergence-free flow field. Particles under these pointwise divergence-free flows exhibit significantly better particle distributions than divergent flows over time. To accelerate the use of vector potentials, this thesis proposes an efficient method that provides boundary-satisfying and smooth discrete potential fields on uniform and cut-cell grids. This thesis also introduces an improved ramping strategy for the “Curl-Noise” method of Bridson et al. (2007), which enforces exact no-normal-flow on the exterior domain boundaries and solid surfaces. The ramping method in the thesis effectively reduces the incidence of particles colliding with obstacles or creating erroneous gaps around the obstacles, while significantly alleviating the artifacts the original ramping strategy produces