A data-driven discrete elastic rod model for shells and solids

Les structures en forme de tige sont omniprésentes dans le monde aujourd'hui. Désormais, prédire avec précision leur comportement pour l'ingénierie et les environnements virtuels est indispensable pour de nombreuses industries, notamment l'infographie, l'animation par ordinateur et la conception informatique. Dans ce mémoire, nous explorons un nouveau modèle de calcul pour les tiges élastiques qui exploite les données de simulation pour reproduire les effets de coque et de solide présents dans les tiges qui brisent les hypothèses de la théorie classique de la tige de Kirchhoff, présentant ainsi une voie d'amélioration possible pour de nombreux états de l'art techniques. Notre approche consiste à prendre un ensemble de données de simulations à partir de solides volumétriques ou de coques pour former un nouveau modèle d'énergie définie positive polynomiale d'ordre élevé pour une tige élastique. Cette nouvelle énergie élargit la gamme des comportements des matériaux qui peuvent être modélisés pour la tige, permettant ainsi de capturer une plus large gamme de phénomènes. Afin de proposer et tester ce modèle, nous concevons un pipeline expérimental pour tester les limites de la théorie linéaire des tiges et étudier les géométries d'interface entre les cas coque à tige et volume à coque pour observer les effets d'un modèle de matériau non linéaire et une section transversale non elliptique dans la déformation de la tige. Nous étudions également la relation entre la courbure de la tige et la déformation de la section transversale et la courbure pour introduire une modification sur le terme de flexion de l'énergie. Cela nous permet de reproduire à la fois le comportement de flexion asymétrique présent dans les poutres volumétriques minces et les poutres à coque avec des sections transversales non convexes. Des suggestions pour de nouvelles améliorations des modèles et des techniques expérimentales sont également données.Rod-like structures are ubiquitous in the world today. Henceforth accurately predicting their behavior for engineering and virtual environments are indispensable for many industries including computer graphics, computer animation, and computational design. In this thesis we explore a new computational model for elastic rods that leverages simulation data to reproduce shell and solid-like effects present in rods that break the assumptions of the classical Kirchhoff rod theory, thus presenting a possible improvement avenue to many states-of-the-art techniques. Our approach consists of taking a data set of simulations from both volumetric solids or shells to train a novel high-order polynomial positive-definite energy model for an elastic rod. This new energy increases the range of material behaviors that can be modeled for the rod, thus allowing for a larger range of phenomena to be captured. In order to propose and test this model, we design an experimental pipeline to test the limits of the linear theory of rods and investigate the interface geometries between the Shell-Rod and Volume-Shell cases to observe the effects of a nonlinear material model and a non-elliptical cross-section in the rod deformation. We also investigate the relation between rod curvature and deformation of the cross-section and curvature to introduce a modification on the bending term of the energy. This allows us to reproduce both the asymmetric bending behavior present in thin volumetric solid and shell beams with non-convex cross-sections. Suggestions for further improvements in models and experimental techniques are also given

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

The multiplicative deformation split for shells with application to growth, chemical swelling, thermoelasticity, viscoelasticity and elastoplasticity

This work presents a general unified theory for coupled nonlinear elastic and inelastic deformations of curved thin shells. The coupling is based on a multiplicative decomposition of the surface deformation gradient. The kinematics of this decomposition is examined in detail. In particular, the dependency of various kinematical quantities, such as area change and curvature, on the elastic and inelastic strains is discussed. This is essential for the development of general constitutive models. In order to fully explore the coupling between elastic and different inelastic deformations, the surface balance laws for mass, momentum, energy and entropy are examined in the context of the multiplicative decomposition. Based on the second law of thermodynamics, the general constitutive relations are then derived. Two cases are considered: Independent inelastic strains, and inelastic strains that are functions of temperature and concentration. The constitutive relations are illustrated by several nonlinear examples on growth, chemical swelling, thermoelasticity, viscoelasticity and elastoplasticity of shells. The formulation is fully expressed in curvilinear coordinates leading to compact and elegant expressions for the kinematics, balance laws and constitutive relations

Multiple equilibrium states of a curved-sided hexagram: Part II-Transitions between states

Curved-sided hexagrams with multiple equilibrium states have great potential in engineering applications such as foldable architectures, deployable aerospace structures, and shape-morphing soft robots. In Part I, the classical stability criterion based on energy variation was used to study the elastic stability of the curved-sided hexagram and identify the natural curvature range for stability of each state for circular and rectangular rod cross-sections. Here, we combine a multi-segment Kirchhoff rod model, finite element simulations, and experiments to investigate the transitions between four basic equilibrium states of the curved-sided hexagram. The four equilibrium states, namely the star hexagram, the daisy hexagram, the 3-loop line, and the 3-loop "8", carry uniform bending moments in their initial states, and the magnitudes of these moments depend on the natural curvatures and their initial curvatures. Transitions between these equilibrium states are triggered by applying bending loads at their corners or edges. It is found that transitions between the stable equilibrium states of the curved-sided hexagram are influenced by both the natural curvature and the loading position. Within a specific natural curvature range, the star hexagram, the daisy hexagram, and the 3-loop "8" can transform among one another by bending at different positions. Based on these findings, we identify the natural curvature range and loading conditions to achieve transition among these three equilibrium states plus a folded 3-loop line state for one specific ring having a rectangular cross-section. The results obtained in this part also validate the elastic stability range of the four equilibrium states of the curved-sided hexagram in Part I. We envision that the present work could provide a new perspective for the design of multi-functional deployable and foldable structures

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

Compliant morphing structures from twisted bulk metallic glass ribbons

In this work, we investigate the use of pre-twisted metallic ribbons as building blocks for shape-changing structures. We manufacture these elements by twisting initially flat ribbons about their (lengthwise) centroidal axis into a helicoidal geometry, then thermoforming them to make this configuration a stress-free reference state. The helicoidal shape allows the ribbon to have preferred bending directions that vary throughout its length. These bending directions serve as compliant joints and enable several deployed and stowed configurations that are unachievable without pre-twist, provided that compaction does not induce material failure. We fabricate these ribbons using a bulk metallic glass (BMG), for its exceptional elasticity and thermoforming attributes. Combining numerical simulations, an analytical model based on shell theory and torsional experiments, we analyze the finite-twisting mechanics of various ribbon geometries. We find that, in ribbons with undulated edges, the twisting deformations can be better localized onto desired regions prior to thermoforming. Finally, we join together multiple ribbons to create deployable systems. Our work proposes a framework for creating fully metallic, yet compliant structures that may find application as elements for space structures and compliant robots