Discrete Elastic Rods Research Articles

Shape-retaining beam-like morphing structures via localized snap through

In this study, we present a concept of morphing structure – featuring an arch mounted on a compliant base – that can be reconfigured via snap-through buckling and leverages bistability to retain its morphed shape. We show that one-dimensional arrays of such units yield beam-like structures that, upon localized snapping, can attain multiple, morphologically distinct stable shapes. Units are modeled using discrete elastic rods, a reduced-order formulation for beams and beam structures, and the results are validated via experiments. We leverage our model to understand the influence of the geometrical design parameters on the response and final shape of a unit. We then show that the morphed shapes of arrays of units can be predicted by concatenating results of simulations on single units, and leverage this idea to inverse-design structures that can be snapped into target stable shapes. Ultimately, our work suggests an up-scalable way to create shape-retaining morphing structures with target stable shapes.

Mechanical Response of Fisherman’s Knots During Tightening

Abstract The fisherman’s knot, renowned for its strength and reliability, finds applications in engineering and medicine. However, a comprehensive understanding of its mechanics remains limited in scientific literature. In this paper, we present a systematic study of the tightening behavior of the fisherman’s knot through a combined approach of tabletop experiments and discrete elastic rods simulations. Our experimental setup involves gradually applying tension to the two ends of the fisherman’s knot until it fractures. We observed a correlation between the knot’s material properties and its behavior during tightening, leading up to fracture. The tightening process of the fisherman’s knot exhibits distinct “sliding” or “stretching” motions, influenced by factors such as friction and elastic stiffness. Furthermore, the failure modes of the knot (material fracture and topological failure) are determined by an interplay between elastic stiffness, friction, and initial conditions. This study sheds light on the underlying mechanics of the fisherman’s knot and provides insight into its behavior during the tightening process, contributing to the broader understanding of the mechanics of knots in practical applications.

Towards Realtime

This paper introduces a hair simulator optimized for real-time applications, including console and cloud gaming, avatar live-streaming, and metaverse environments. We view the collisions between strands as a mechanism to preserve the overall volume of the hair and adopt explicit Material Point Method (MPM) to resolve the strand-strand collision. For simulating single-strand behavior, a semi-implicit Discrete Elastic Rods (DER) model is used. We build upon a highly efficient GPU MPM framework recently presented by Fei et al. [2021b] and propose several schemes to largely improve the performance of building and solving the semi-implicit DER systems on GPU. We demonstrate the efficiency of our pipeline by a few practical scenes that achieve up to 260 frames-per-second (FPS) with more than two thousand simulated strands on Nvidia GeForce RTX 3080.

Bifurcations and Stability Analysis of Elastic Slender Structures Using Static Discrete Elastic Rods Method

Abstract Discrete elastic rods (DER) method provides a computationally efficient means of simulating the nonlinear dynamics of one-dimensional slender structures. However, this dynamic-based framework can only provide first-order stable equilibrium configuration when combined with the dynamic relaxation method, while the unstable equilibria and potential critical points (i.e., bifurcation and fold point) cannot be obtained, which are important for understanding the bifurcation and stability landscape of slender bodies. Our approach modifies the existing DER technique from dynamic simulation to a static framework and computes eigenvalues and eigenvectors of the tangential stiffness matrix after each load incremental step for bifurcation and stability analysis. This treatment can capture both stable and unstable equilibrium modes, critical points, and trace solution curves. Three representative types of structures—beams, strips, and gridshells—are used as demonstrations to show the effectiveness of the modified numerical framework, which provides a robust tool for unveiling the bifurcation and multistable behaviors of slender structures.

Snap Buckling in Overhand Knots

Abstract When an overhand knot tied in an elastic rod is tightened, it can undergo a sudden change in shape through snap buckling. In this article, we use a combination of discrete differential geometry (DDG)-based simulations and tabletop experiments to explore the onset of buckling as a function of knot topology, rod geometry, and friction. In our setup, two open ends of an overhand knot are slowly pulled apart, which leads to snap buckling in the knot loop. We call this phenomenon “inversion” since the loop appears to move dramatically from one side of the knot to the other. This inversion occurs due to the coupling of elastic energy between the braid (the portion of the knot in self-contact) and the loop (the portion of the knot with two ends connected to the braid). A numerical framework is implemented that combines discrete elastic rods with a constraint-based method for frictional contact to explore inversion in overhand knots. The numerical simulation robustly captures inversion in the knot and is found to be in good agreement with experimental results. In order to gain physical insight into the inversion process, we also develop a simplified model of the knot that does not require simulation of self-contact, which allows us to visualize the bifurcation that results in snap buckling.

Systematic Variation of Friction of Rods

Abstract The mechanical response of a knot tied in elastic rods strongly depends on the frictional force due to rod–rod contact. The behavior of a knot can be qualitatively different based on the frictional coefficient of the elastic rod. Systematic variation of friction during rod–rod contact is a crucial component of any experimental design to uncover the underlying ingredients behind the mechanics of knots. In this paper, we demonstrate a novel process of controlling the friction of a continuous rod by adhering non-spherical inorganic micro-particles. Polymeric binder is used to deliver the particles as asperities over the rod substrate and by controlling their size and distribution the coefficient of friction of the rod is determined. In parallel, numerical simulations with the discrete elastic rods algorithm are used to reproduce the experimental observations. Tabletop experiments are performed where overhand knots with a variety of unknotting numbers are pulled tight. The force–extension curve of these experiments shows that the proposed process can successfully tune the friction between rods.

Design and Closed‐Loop Motion Planning of an Untethered Swimming Soft Robot Using 2D Discrete Elastic Rods Simulations

Despite tremendous progress in the development of untethered soft robots in recent years, existing systems lack the mobility, model‐based control, and motion planning capabilities of their piecewise rigid counterparts. As in conventional robotic systems, the development of versatile locomotion of soft robots is aided by the integration of hardware design and control with modeling tools that account for their unique mechanics and environmental interactions. Here, a framework for physics‐based modeling, motion planning, and control of a fully untethered swimming soft robot is introduced. This framework enables offline co‐design in the simulation of robot parameters and gaits to produce effective open‐loop behaviors and enables closed‐loop planning over motion primitives for feedback control of a frog‐inspired soft robot testbed. This pipeline uses a discrete elastic rods (DERs) physics engine that discretizes the soft robot as many stretchable and bendable rods. On hardware, an untethered aquatic soft robot that performs frog‐like rowing behaviors is engineered. Hardware validation verifies that the simulation has sufficient accuracy to find the best candidates for sets of parameters offline. The simulator is then used to generate a trajectory library of the robot's motion in simulation that is used in real‐time closed‐loop path following experiments on hardware.

Buckling instability and dynamic response of a planar gridshell under thermal load

We introduce a discrete model to predict the buckling instability and the vibration performance of an elastic gridshell which experiences a thermal load. With both thermal stress and temperature-dependent material property considered, a discrete framework for the simulation of hollow grid is developed based on a well-established Discrete Elastic Rods (DER) method and a specific mapping technique. It is found that the critical temperatures which induce a structural instability are identical for single rod system and multiple rods network, regardless of rod density. Meanwhile, its natural frequency linearly enhances with the growth of rod number. Moreover, the increase of environmental temperature will result in that the resonant peak of displacement response moves toward a lower frequency. Interestingly, we found that the natural frequency of gridshell is closer to the beam model, while its displacement response distribution shows similarity to plate theory. The free vibration frequencies of one-dimensional (1D) beam and two-dimensional (2D) plate with a simply-supported boundary condition under thermal effects are also derived analytically for cross-validation. We hope our findings could provide a fundamental insight in structural dynamics and further facilitate the designs in mechanical and aerospace engineering, e.g., deployable structures and smart metamaterials.

Loki

We introduce Loki, a new framework for robust simulation of fluid, rigid, and deformable objects with non-compromising fidelity on any single element, and capabilities for coupling and representation transitions across multiple elements. Loki adapts multiple best-in-class solvers into a unified framework driven by a declarative state machine where users declare 'what' is simulated but not 'when,' so an automatic scheduling system takes care of mixing any combination of objects. This leads to intuitive setups for coupled simulations such as hair in the wind or objects transitioning from one representation to another, for example bulk water FLIP particles to SPH spray particles to volumetric mist. We also provide a consistent treatment for components used in several domains, such as unified collision and attachment constraints across 1D, 2D, 3D deforming and rigid objects. Distribution over MPI, custom linear equation solvers, and aggressive application of sparse techniques keep performance within production requirements. We demonstrate a variety of solvers within the framework and their interactions, including FLIPstyle liquids, spatially adaptive volumetric fluids, SPH, MPM, and mesh-based solids, including but not limited to discrete elastic rods, elastons, and FEM with state-of-the-art constitutive models. Our framework has proven powerful and intuitive enough for voluntary artist adoption and has delivered creature and FX simulations for multiple major movie productions in the preceding four years.

Natural frequencies of pre-buckled rods and gridshells

A discrete differential geometry (DDG)-based method is proposed to numerically study the natural frequencies of elastic rods and gridshells in their post-buckling configurations. A fully implicit numerical framework is developed based on Discrete Elastic Rods (DER) algorithm, in order to characterize the mechanical behaviors of an elastic gridshell comprised of multiple rods. When their footprints are constrained along a shrinking trajectory, the slender structures as well as their constructed network would experience a geometrically nonlinear instability and deform into an out-of-plane configuration. By checking the eigenvalues and eigenvectors of the mass and stiffness matrix, the linear vibration near the post-buckling equilibria are characterized through a numerical approach. Exploiting the efficiency and the robustness of the developed discrete method, a systematic parameter sweep is performed to quantify the vibration frequency of pre-buckled gridshells with respect to the number of rods and the pre-compressed distance. It is found that the natural frequency for both pre-deformed rods and gridshells would linearly decrease as the enlargement of compressive distance, even though the geometrically nonlinear deformations have been taken into account. Moreover, the vibration frequency almost linearly rises when the number of rods in a gridshell becomes larger. These findings could provide a fundamental insight in revealing more complex structural dynamics and facilitating the designs of buckling-induced assembly in some man-made systems, e.g., avoidance of the resonance in soft electronics.

Ballooning in spiders using multiple silk threads

In this paper, three-dimensional numerical simulations of ballooning in spiders using multiple silk threads are performed using the discrete elastic rods method. The ballooning of spiders is hypothesized to be caused by the presence of the negative electric charge of the spider silk threads and the positive electric potential field in the Earth's atmosphere. The numerical model presented here is first validated against experimental data from the open literature. After which, two cases are examined, in the first it is assumed that the electric charge is uniformly distributed along the threads while in the second, the electric charge is located at the thread tip. It is shown that the normalized terminal ballooning velocity, i.e., the velocity at which the spiders balloon after they reach steady-state, decrease linearly with the normalized lift force, especially for the tip located charge case. For the uniform electric charge case, this velocity shows a slightly weaker dependence on the normalized lift force. Moreover, it is shown in both cases that the normalized terminal ballooning velocity has no dependence on the normalized elastic bending stiffness of the threads and on the normalized viscous forces. Finally, the multithread bending process shows a three-dimensional conical sheet. Here we show that this behavior is caused by the Coulomb repelling forces owing to the threads electric charge which leads to dispersing the threads apart and thus avoid entanglement.

A bottom‐up optimization method for inverse design of two‐dimensional clamped‐free elastic rods

AbstractRod‐like structures, such as DNA, climbing plants, and cables, pervade the nature and our daily life and also belong to a frequently encountered engineering problem, which usually assume a deformed shape based on the competition between elasticity (stretching, bending, twisting) and external forces (e.g., gravity). These structures often undergo geometrically nonlinear deformation under its own weight. Computing the natural (undeformed) shape of a rod from the observed deformed configuration is a nonlinear inverse problem. Here we introduce a numerical method to solve this problem in case planar (2D) rods, that is, beams, under a clamped‐free boundary condition. A simulation model is developed based on the discrete elastic rods (DER) algorithm to address the forward deformation process from natural to deformed shapes under gravity. Based on the internal mechanical link among the nodes on the discretized elastic rod, a bottom‐up optimization method is proposed in this article for an efficient inverse solution. Associated with it, a global searching algorithm is developed to search for the optimal bending curvature along the arc length of the rod. The method is numerically validated for accuracy and sheds light on the inverse design of rod‐like structures for various applications, for example, soft robots and tails of animated characters.

Simple deformation measures for discrete elastic rods and ribbons

The discrete elastic rod method (Bergou et al. 2008 ACM Trans. Graph . 27 , 63:1–63:12. ( doi:10.1145/1360612.1360662 )) is a numerical method for simulating slender elastic bodies. It works by representing the centreline as a polygonal chain, attaching two perpendicular directors to each segment and defining discrete stretching, bending and twisting deformation measures and a discrete strain energy. Here, we investigate an alternative formulation of this model based on a simpler definition of the discrete deformation measures. Both formulations are equally consistent with the continuous rod model. Simple formulae for the first and second gradients of the discrete deformation measures are derived, making it easy to calculate the Hessian of the discrete strain energy. A few numerical illustrations are given. The approach is also extended to inextensible ribbons described by the Wunderlich model, and both the developability constraint and the dependence of the energy on the strain gradients are handled naturally.

Numerical Exploration on Snap Buckling of a Pre-Stressed Hemispherical Gridshell

AbstractMotivated by the observations of snap-through phenomena in pre-stressed strips and curved shells, we numerically investigate the snapping of a pre-buckled hemispherical gridshell under apex load indentation. Our experimentally validated numerical framework on elastic gridshell simulation combines two components: (i) discrete elastic rods method, for the geometrically nonlinear description of one-dimensional rods, and (ii) a naive penalty-based energy functional, to perform the non-deviation condition between two rods at joint. An initially planar grid of slender rods can be actuated into a three-dimensional hemispherical shape by loading its extremities through a prescribed path, known as buckling-induced assembly; next, this pre-buckled structure can suddenly change its bending direction at some threshold points when compressing its apex to the other side. We find that the hemispherical gridshell can undergo snap-through buckling through two different paths based on two different apex loading conditions. The structural rigidity increases as the number of rods in the gridshell structure becomes denser, which emphasizes the mechanically nonlocal property in hollow grids, in contrast to the local response of continuum shells. The findings may bridge the gap among rods, grids, knits, and shells, for a fundamental understanding of a group of thin elastic structures, and inspire the design of novel micro-electro-mechanical systems and functional metamaterials.

Numerical simulation of bundling of helical elastic rods in a viscous fluid

We introduce a numerical framework to study the fluid-structure interaction between two helical filaments rotating under low Reynolds number condition, motivated by the propulsion of bacteria using helical flagella. Our numerical framework couples the elasticity of the thin filaments, nonlocal hydrodynamic loading, and the contact between multiple elastic rods. Each of these three ingredients is respectively modeled by the Discrete Elastic Rods method (for a geometrically nonlinear description of soft filaments), Regularized Stokeslet Segments method (for the nonlocal drag force in a viscous fluid), and non-penetration condition between rod segments. Two helical rods rotating side by side attract each other and become closer because of their hydrodynamic interplay in a viscous environment. Depending on the initial distance between the two and their rotational frequency, the two filaments can come in physical contact. Exploiting the efficiency and robustness of the simulator, we perform a systematic parameter sweep to quantify the bundling behavior. The findings may shed light on the physics of the bio-locomotion of microorganisms and inspire the design of novel biomimetic soft robots.

Dynamic computation of a tether-net system capturing a space target via discrete elastic rods and an energy-conserving integrator

The tether-net system is a promising technique for active debris removal. The capture process of a tether-net system for debris exhibits the nonlinear dynamics of very high dimensions and requires the dynamic simulation of high accuracy and efficiency in the design stage. This paper presents an integrated approach of both method of discrete elastic rods and algorithm of energy-conserving integration for the above purpose. The study shows that the method of discrete elastic rods is more suitable for modeling the extremely flexible threads of a woven net than the slender beam elements with C1 continuity. Besides, the energy-conserving integrator embedded in the present approach well conserves the total energy, and does not produce any numerical dissipations. The paper gives three numerical examples, including the capturing process of a hexagonal tether-net system for a space target, to demonstrate the advantages of the present approach, as well as the influence of some initial control parameters on the deployment phase of the tether-net system.

Implicit Contact Model for Discrete Elastic Rods in Knot Tying

Abstract Rod–rod contact is critical in simulating knots and tangles. To simulate contact, typically a contact force is applied to enforce nonpenetration condition. This force is often applied explicitly (Euler forward). At every time-step in a dynamic simulation, the equations of motions are solved over and over again until the right amount of contact force successfully imposes the nonpenetration condition. There are two drawbacks: (1) Explicit implementation brings numerical convergence issues. (2) Solving equations of motion iteratively to find this right contact force slows down the simulation. In this article, we propose a simple, efficient, and fully implicit contact model with high convergence properties. This model is shown to be capable of taking large time-steps without forfeiting accuracy during knot tying simulations when compared to previous methods. We introduce a new contact potential, based on a smoothed segment–segment distance function, that is an analytic function of the four endpoints of the two contacting edges. Since this contact potential is differentiable, we can incorporate its force (negative gradient of the energy) and Jacobian (negative Hessian of the energy) into the elastic rod simulation.

Control of Uniflagellar Soft Robots at Low Reynolds Number Using Buckling Instability

Abstract In this paper, we analyze the inverse dynamics and control of a bacteria-inspired uniflagellar robot in a fluid medium at low Reynolds number. Inspired by the mechanism behind the locomotion of flagellated bacteria, we consider a robot comprising a flagellum—a flexible helical filament—connected to a spherical head. The flagellum rotates about the head at a controlled angular velocity and generates a propulsive force that moves the robot forward. When the angular velocity exceeds a threshold value, the hydrodynamic force exerted by the fluid can cause the soft flagellum to buckle, characterized by a dramatic change in its shape. In this computational study, a fluid–structure interaction model that combines Discrete Elastic Rods algorithm with Lighthill's Slender Body Theory is employed to simulate the locomotion and deformation of the robot. We demonstrate that the robot can follow a prescribed path in three-dimensional space by exploiting buckling of the flagellum. The control scheme involves only a single (binary) scalar input—the angular velocity of the flagellum. By triggering the buckling instability at the right moment, the robot can follow the path in three-dimensional space. We also show that the complexity of the dynamics of the helical filament can be captured using a deep neural network, from which we identify the input–output functional relationship between the control input and the trajectory of the robot. Furthermore, our study underscores the potential role of buckling in the locomotion of natural bacteria.

Numerical Method for Direct Solution to Form-Finding Problem in Convex Gridshell

AbstractElastic gridshell is a class of net-like structure formed by an ensemble of elastically deforming rods coupled through joints, such that the structure can cover large areas with low self-weight and allow for a variety of aesthetic configurations. Gridshells, also known as X-shells or Cosserat Nets, are a planar grid of elastic rods in its undeformed configuration. The end points of the rods are constrained and positioned on a closed curve—the final boundary—to actuate the structure into a 3D shape. Here, we report a discrete differential geometry-based numerical framework to study the geometrically nonlinear deformation of gridshell structures, accounting for non-trivial bending-twisting coupling at the joints. The form-finding problem of obtaining the undeformed planar configuration given the target convex 3D topology is then investigated. For the forward (2D to 3D) physically based simulation, we decompose the gridshell structure into multiple one-dimensional elastic rods and simulate their deformation by the well-established discrete elastic rods (DER) algorithm. A simple penalty energy between rods and linkages is used to simulate the coupling between two rods at the joints. For the inverse problem associated with form-finding (3D to 2D), we introduce a contact-based algorithm between the elastic gridshell and a rigid 3D surface, where the rigid surface describes the target shape of the gridshell upon actuation. This technique removes the need of several forward simulations associated with conventional optimization algorithms and provides a direct solution to the inverse problem. Several examples—hemispherical cap, paraboloid, and hemi-ellipsoid—are used to show the effectiveness of the inverse design process.

Discrete Elasto-Plastic Rods

We report a discrete differential geometry-based numerical framework to simulate the rate-independent, elasto-plastic behavior of one dimensional rod-like structures. Our numerical tool first discretizes the rod centerline into a number of nodes and edges in a manner similar to the well-established Discrete Elastic Rods (DER) method – a fast geometrically exact simulation for elastic filaments. The cross section of each vertex is next meshed into multiple fiber-like elements, such that the nonlinear constitutive relation of each element can be individually described by an increment-based associated flow rule and updated through return mapping algorithm. Equations of motion numerically obtained from energy variation at each degree of freedom are integrated through an implicit Euler time marching scheme, for its unconditional convergence and numerical stability. For quantitative comparison, we derive the analytical solutions for several simple cases, and a good match between the analytical solution and the numerical results in the geometrically linear regime indicates the accuracy of our discrete model. Our simulation can seamlessly handle the nonlinearity from both geometric and material sides, which is often not amenable to an analytical approach.