Yingying Ren | EPFL Graph Search

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Computational Exploration of Multistable Elastic Knots

Mark Pauly, Yingying Ren, Michele Vidulis

We present an algorithmic approach to discover, study, and design multistable elastic knots. Elastic knots are physical realizations of closed curves embedded in 3-space. When endowed with the material thickness and bending resistance of a physical wire, these knots settle into equilibrium states that balance the forces induced by elastic deformation and self-contacts of the wire. In general, elastic knots can have many distinct equilibrium states, i.e. they are multistable mechanical systems. We propose a computational pipeline that combines randomized spatial sampling and physics simulation to efficiently find stable equilibrium states of elastic knots. Leveraging results from knot theory, we run our pipeline on thousands of different topological knot types to create an extensive data set of multistable knots. By applying a series of filters to this data, we discover new transformable knots with interesting geometric and physical properties. A further analysis across knot types reveals geometric and topological patterns, yielding constructive principles that generalize beyond the currently tabulated knot types. We show how multistable elastic knots can be used to design novel deployable structures and engaging recreational puzzles. Several physical prototypes at different scales highlight these applications and validate our simulation.

2023

From kirigami to hydrogels: a tutorial on designing conformally transformable surfaces

Tian Chen, Yingying Ren, Yue Wang

Elastic surfaces that morph between multiple geometrical configurations are of significant engineering value, with applications ranging from the deployment of space-based PV arrays, the erection of temporary shelters, and the realization of flexible display systems, to understanding the encapsulation and release of viral RNAs. In general, ensuring that a shape with a planar rest configuration can deploy into a target 3D shape is a nontrivial problem. Moreover, it is difficult to physically realize the local deformations necessary to achieve such global transformation. Here, we give a tutorial on applying conformal mapping to rationalize the geometrical deformation of several microstructure designs. A conformal map is a function that locally preserves angles and shapes but not lengths: some regions are scaled (enlarged or shrunk) more than others. To transform a planar surface to 3D, we implement the uniform local scalings as mechanical deformations. Numerous natural and architected material systems exhibit such behavior, including kirigami, origami, hydrogel, linkage mechanisms, and fabric membranes. The design and fabrication of conformally transformable surfaces is a transdisciplinary challenge involving insights from advanced manufacturing, computational design, material science, and mechanics. By recognizing that many material systems exhibit isotropic deformation, we hope to inspire researchers to adopt conformal mapping in designing next-generation surface-based engineering systems.

2022

Umbrella Meshes: Elastic Mechanisms for Freeform Shape Deployment

Tian Chen, Florin Isvoranu, Uday Kusupati, Mark Pauly, Davide Pellis, Yingying Ren

We present a computational inverse design framework for a new class of volumetric deployable structures that have compact rest states and deploy into bending-active 3D target surfaces. Umbrella meshes consist of elastic beams, rigid plates, and hinge joints that can be directly printed or assembled in a zero-energy fabrication state. During deployment, as the elastic beams of varying heights rotate from vertical to horizontal configurations, the entire structure transforms from a compact block into a target curved surface. Umbrella Meshes encode both intrinsic and extrinsic curvature of the target surface and in principle are free from the area expansion ratio bounds of past auxetic material systems. We build a reduced physics-based simulation framework to accurately and efficiently model the complex interaction between the elastically deforming components. To determine the mesh topology and optimal shape parameters for approximating a given target surface, we propose an inverse design optimization algorithm initialized with conformal flattening. Our algorithm minimizes the structure's strain energy in its deployed state and optimizes actuation forces so that the final deployed structure is in stable equilibrium close to the desired surface with few or no external constraints. We validate our approach by fabricating a series of physical models at various scales using different manufacturing techniques.

2022