Session R56: Discrete Structures: Geometry, Mechanics, Graphics, and Computation II

Geometric defects, weak forces, and self-similar buckling in non-Euclidean elastic sheets

Non-Euclidean elastic sheets (like lettuce, flowers, and sea slugs) exhibit extreme properties including an inherent floppiness, which we argue is governed by and may, in turn, be quantified by non-smooth geometric defects. The presence and interaction of these localized defects in hyperbolic sheets may be modelled and explored with rough solutions via discrete differential geometry (DDG) based on taking a singular limit that assumes a no-stretching constraint. Non-smooth geometric defects are identified by a skeleton of intersecting asymptotic curves which, along with geometric data along the curves, enables constructing complex morphologies with intricate wrinkling shapes. By computing various energies on discretized elements, numerical simulations using this novel DDG framework reveal the significant impact of non-smooth geometric defects on elastic energy as well as the non-negligible role of weak forces, i.e., effects other than stretching or bending, and associated scaling laws. Ultimately, these modeling techniques have the potential to explain real-world sheets and enable the control/design of thin soft structures.   

Discrete PolyCube Surface Flows

Discrete flows have been of particular interest to researchers in discrete differential geometry, computational geometry, and computer graphics, due to their connection to the Plateau problem in mathematics, mechanical behavior of various shells and membranes, and their application to shape sciene problems such as surface fairing, parameterization, collision modeling, registration, and interpolation. We study a new flow, formulated as a variational problem on certain bundles of quotient spaces of rotations over the surface, and describe algorithms to discretize and simulate it. The flow has the following property: stationary surfaces of the flow are polycubes, polyhedra whose facets meet only at right angles. Spheres flow to cubes, and more generally, surface features sharpen or flatten. Moreover the flow is intrinsic to the surface itself, and does not require a choice of preferred planes or directions in the ambient space. This flow has potential applications in crystallograpy, as well as simplification, discretization, and hexahedralization of shapes.   

Spatial and Network Effects in Distributed System Design

Designing a modern complex system requires keeping track of the interplay of the system's logical topology, spatial arrangement, and functionality. Existing frameworks mostly focus on how one of these aspects influences others in a single direction, rather than keeping track of the mutually deterministic nature of design elements. We demonstrate how to determine mutual influences of topology and spatial constraints on each other for a whole ensemble of possible system arrangements. We cast this problem in the modern graphical language of tensor networks, which facilitates computation and allows for extracting a variety of ensemble observables. We demonstrate the power of the approach on a model system routing problem from Naval Engineering, however the method is easily generalizable to other problems.   

Physics-inspired optimization for quadratic unconstrained problems using a digital annealer

The Fujitsu Digital Annealer is designed to solve fully connected quadratic unconstrained binary optimization (QUBO) problems. It is implemented on application-specific CMOS hardware and currently solves problems of up to 1024 variables using a parallel-trial algorithm based on simulated annealing. We compare the performance of the Digital Annealer to simulated annealing and parallel tempering Monte Carlo with isoenergetic cluster moves on different spin-glass problems. Our results show that the Digital Annealer currently exhibits a time-to-solution speedup of roughly two orders of magnitude for fully connected spin-glass problems, over the single-core implementations of simulated annealing and parallel tempering Monte Carlo used in this study. This, however, is not the case for sparse two-dimensional problems. We also benchmarked an early implementation of the Parallel Tempering Digital Annealer. Our results suggest an improved scaling over the other algorithms for fully-connected problems of average difficulty with bimodal disorder. The use of specialized hardware paired with fast algorithms thus allows for a more detailed study of statistical physics Hamiltonians in the near future.   

Deployable and Mechanical Properties of An Origami Spring Structure

Origami has recently been studied intensively by physicists, mathematicians, and engineers due to its deployability, versatility and low cost in potential engineering applications. Examples include self-deployable solar arrays in space and self-folded origami inspired robots. To understand deployable properties of an origami structure, models of an origami “spring” were investigated. Origami spring structures may be useful in robotics and structural designs as their length can vary, while maintaining structural rigidity. Mathematical methods based on the geometry of these spring structures are used to predict ideal maximum height, formulate a relationship between the angle of rotation and the spring height, and to obtain a Poisson’s ratio of the spring structure. We show that the Poisson’s ratio is not constant and is scalable with respect to different geometric parameters. Elastic modulus of the spring structures are also predicted using models based on Hooke’s law and the stationary principle. By studying origami spring models, we discovered interesting and potentially practical engineering designing insights that are embedded in this smart structure.   

Discrete Geometric Simulation of Elastic Ribbons

We report a discrete differential geometry based numerical simulation for elastic ribbons. Ribbon is a mechanical structure whose three dimensions are very different: length >> width >> thickness. In our framework, we use the elastic energy form, with two essential geometric constraints, of a one-dimensional model of the ribbon [Dias and Audoly, J. Elast. 2015]. As non-zero natural curvature in both in-plane (geodesic curvature) and out-of-plane directions have been considered, this model allows a unified treatment of various types of ribbons, e.g. annular and rectangular. This continuous model is discretized into a mass-spring system in a manner similar to well established Discrete Elastic Rods algorithm for simulation of elastic rods. A system of discrete equations of motions are developed that can be solved implicitly using Newton's method. In parallel with simulations, we perform experiments with several test cases, e.g. large deformation of elastic ribbons under gravity, coupling of twisting and bending in rectangular ribbons, shape of Mobius strips, and buckling in annular ribbons. Quantitative comparison between experiments and simulations validates the correctness of our numerical method.   

Numerical Simulations for Physics-based Training of Robots for Manipulation of Flexible Rods

Robotic manipulation of flexible ropes has wide ranging application from advanced manufacturing to robot-assisted surgery. We report a physics-based scheme to deploy elastic ropes along a prescribed trajectory with a collaborative robot. A numerical simulation tool, based on the Discrete Elastic Rods method, is developed for the modeling of elastic rods including contact and friction. Given the stochastic nature of the friction between the rope and the ground, avoiding friction is a key to repeatability of experiments with the robot. Exploiting the robustness and efficiency of the computational framework, we generate training data in the numerical simulator. This data is used to plan the optimal path for the robotic arm such that friction is minimized during deployment of ropes on a given trajectory. Compared with physics-blind methods that require empirical training by humans for every single rope, our proposed scheme remains valid for any elastic rope regardless of the geometric and material properties. Moreover, vast amount of data can be produced from the simulator in a few hours on a contemporary CPU to train a general neural network with high accuracy.   

A Discrete Geometric Approach to Simulation of Soft Multi-limbed Robots

Because they are primarily composed of mechanically compliant and deformable materials, soft limbed robots can navigate through unstructured terrain and confined spaces without dependency on highly articulated motion and sensing. However, design and control requires a painstaking trial and error process owing to the absence of an accurate and efficient simulation and modeling tools. Here, we present a numerical simulation tool for limbed soft robots inspired by a discrete differential geometry-based computational framework that can run faster than real-time on a single thread of a contemporary desktop processor. The simulation incorporates an implicit method to account for the elasticity of the structure, contact with an uneven surface, and Coulombic friction between the soft robot limbs and ground. To validate the simulation, we build a novel, shape memory alloy driven star-shaped soft robot comprised of seven compliant limbs that can cyclically change shape through electrical Joule heating and passive cooling. Our experiments and simulations show reasonable quantitative agreement and indicate the potential role of this discrete geometric approach as a computational framework in predictive simulations for soft robot design and control.   

Topological Characterization of Meta-stable States in Weakly Non-linear Diffusion Processes on Networks

Complex networks are ubiquitous.
Many dynamical processes on networks, including spreading processes such as SIR and SIS, are generally weakly non-linear and the core of the process consists of a diffusion process.
However, while the diffusion may have a trivial fixed point where the probability distributions over all nodes becomes uniform or proportional to their degree, the nonlinearity generally results in the trivial fixed point becoming unstable, yielding non-trivial outcomes.
Especially in real networks with modular and other internal structures, characterizing these outcomes requires numerical simulations.
Here we report discovering a deep connection between these solutions and symmetry groups.
In particular, we consider a multi-dimensional weakly non-linear dynamical process on a network and show that it is equivalent to a force-directed layout problem.
We then show how the topology of the embedding space and its symmetry group can be used to characterize the low-energy solutions, which turn out to be topological in nature.
Often the trivial fixed-point of the dynamics, where the field over all nodes collapses to zero, is unstable due to constraints arising from non-linear interactions among nodes, resulting in topological low-energy states becoming the dominant.   

Connecting discrete to the continuum: continuum-level simulation of shear-banding in metallic glasses on transforming grids with Lees-Edwards boundary conditons

We simulate a three-dimensional continuum-level elasto-plastic model of a bulk metallic glass based on the shear transformation zone (STZ) theory of amorphous plasticity. The simulation utilizes a new projection method valid in the quasi-static limit based on a mathematical correspondence between the Navier-Stokes equations for incompressible fluid flow and the equations of quasi-static hypoelastoplasticity. We emphasize a variation of the method based on a coordinate transformation that permits the use of Lees-Edwards boundary conditions at the continuum scale, enabling direct comparisons between continuum and discrete simulation. We consider several interesting numerical examples, including simple and pure shear boundary conditions imposed in the transformed frame.   

Maxwell Force Shaping of Dielectric Liquid Films on Curved Conducting Substrates

The dynamical behavior of thin dielectric films on curved substrates is critically important to a range of processes fundamental to the coating industry, micro-lithography and biological flows. Substrate curvature can strongly affect film shape and stability, especially when the local film thickness couples to an external field. For thin dielectric films on planar domains, accurate solutions can be obtained by exploiting the gradient flow structure of the governing equation and appealing to the Helmholtz minimum dissipation principle. Here we show how this minimization principle can be extended to include thin films on curved substrates wherein the local film thickness is actively coupled to an external electric field and whose response is only mitigated by capillary forces. Accurate approximate solutions are obtained by invoking a variational principle and restricting trial solutions to polynomial functions in the direction normal to the substrate. We demonstrate this approach for a thin dielectric film coating a cylindrical conductor using a boundary/finite element method. We find that this solution method offers keen physical insight into allowable film configurations not accessible to planar geometries.   

Cross-Influence of Thermodynamic Driving Forces in Confined Environment

The second law of thermodynamics dictates that under a certain condition, the cross-influences of thermodynamic driving forces (tdf) must be balanced. For a galvanic cell, it is equivalent to the well-known Nernst equation; for a double-layer supercapacitor, it is consistent with the classic Gouy-Chapman model. In our recent experiment on confined large pivalate ions in carbon nanopores, however, it was measured that the cross-influences of the electromotive force and the chemical potential were different from each other by an order of magnitude. We attribute this remarkable phenomenon to the confinement effect of the electrode inner surfaces, which forbids the formation of diffuse layer. We argue that in general, in a low-dimensional environment, in the large dimension(s), the laws of classic statistical physics can be applied; but in the small dimension(s) wherein two tdf interact, the governing equations can be distinct. With this unique mechanism, the second law of thermodynamics may break down, in a dissimilar manner to “Maxwell’s demon”. The concept of unbalanced cross-influence of tdf is further examined through a theoretical analysis on a model system comprising of randomly moving elastic particles restricted in a two-dimensional transition zone in a gravitational field.