Beyond developable: computational design and fabrication with auxetic materials

We present a computational method for interactive 3D design and rationalization of surfaces via auxetic materials, i.e., flat flexible material that can stretch uniformly up to a certain extent. A key motivation for studying such material is that one can approximate doubly-curved surfaces (such as the sphere) using only flat pieces, making it attractive for fabrication. We physically realize surfaces by introducing cuts into approximately inextensible material such as sheet metal, plastic, or leather. The cutting pattern is modeled as a regular triangular linkage that yields hexagonal openings of spatially-varying radius when stretched. In the same way that isometry is fundamental to modeling developable surfaces, we leverage conformal geometry to understand auxetic design. In particular, we compute a global conformal map with bounded scale factor to initialize an otherwise intractable non-linear optimization. We demonstrate that this global approach can handle non-trivial topology and non-local dependencies inherent in auxetic material. Design studies and physical prototypes are used to illustrate a wide range of possible applications

Design of Geometric Molecular Bonds

An example of a nonspecific molecular bond is the affinity of any positive charge for any negative charge (like-unlike), or of nonpolar material for itself when in aqueous solution (like-like). This contrasts specific bonds such as the affinity of the DNA base A for T, but not for C, G, or another A. Recent experimental breakthroughs in DNA nanotechnology demonstrate that a particular nonspecific like-like bond ("blunt-end DNA stacking" that occurs between the ends of any pair of DNA double-helices) can be used to create specific "macrobonds" by careful geometric arrangement of many nonspecific blunt ends, motivating the need for sets of macrobonds that are orthogonal: two macrobonds not intended to bind should have relatively low binding strength, even when misaligned. To address this need, we introduce geometric orthogonal codes that abstractly model the engineered DNA macrobonds as two-dimensional binary codewords. While motivated by completely different applications, geometric orthogonal codes share similar features to the optical orthogonal codes studied by Chung, Salehi, and Wei. The main technical difference is the importance of 2D geometry in defining codeword orthogonality.Comment: Accepted to appear in IEEE Transactions on Molecular, Biological, and Multi-Scale Communication

Origami fold as algebraic graph rewriting

AbstractWe formalize paper fold (origami) by graph rewriting. Origami construction is abstractly described by a rewriting system (O,↬), where O is the set of abstract origamis and ↬ is a binary relation on O, that models fold. An abstract origami is a structure (Π,∽,≻), where Π is a set of faces constituting an origami, and ∽ and ≻ are binary relations on Π, each representing adjacency and superposition relations between the faces.We then address representation and transformation of abstract origamis and further reasoning about the construction for computational purposes. We present a labeled hypergraph of origami and define fold as algebraic graph transformation. The algebraic graph-theoretic formalism enables us to reason about origami in two separate domains of discourse, i.e. pure combinatorial domain where symbolic computation plays the main role and geometrical domain R×R. We detail the program language for the algebraic graph rewriting and graph rewriting algorithms for the fold, and show how fold is expressed by a set of graph rewrite rules

A Survey of Developable Surfaces: From Shape Modeling to Manufacturing

Developable surfaces are commonly observed in various applications such as architecture, product design, manufacturing, and mechanical materials, as well as in the development of tangible interaction and deformable robots, with the characteristics of easy-to-product, low-cost, transport-friendly, and deformable. Transforming shapes into developable surfaces is a complex and comprehensive task, which forms a variety of methods of segmentation, unfolding, and manufacturing for shapes with different geometry and topology, resulting in the complexity of developable surfaces. In this paper, we reviewed relevant methods and techniques for the study of developable surfaces, characterize them with our proposed pipeline, and categorize them based on digital modeling, physical modeling, interaction, and application. Through the analysis to the relevant literature, we also discussed some of the research challenges and future research opportunities.Comment: 20 pages, 24 figures, Author submitted manuscrip

Nanoscale Structure and Elasticity of Pillared DNA Nanotubes

We present an atomistic model of pillared DNA nanotubes (DNTs) and their elastic properties which will facilitate further studies of these nanotubes in several important nanotechnological and biological applications. In particular, we introduce a computational design to create an atomistic model of a 6-helix DNT (6HB) along with its two variants, 6HB flanked symmetrically by two double helical DNA pillars (6HB+2) and 6HB flanked symmetrically by three double helical DNA pillars (6HB+3). Analysis of 200 ns all-atom simulation trajectories in the presence of explicit water and ions shows that these structures are stable and well behaved in all three geometries. Hydrogen bonding is well maintained for all variants of 6HB DNTs. We calculate the persistence length of these nanotubes from their equilibrium bend angle distributions. The values of persistence length are ~10 {\mu}m, which is 2 orders of magnitude larger than that of dsDNA. We also find a gradual increase of persistence length with an increasing number of pillars, in quantitative agreement with previous experimental findings. To have a quantitative understanding of the stretch modulus of these tubes we carried out nonequilibrium Steered Molecular Dynamics (SMD). The linear part of the force extension plot gives stretch modulus in the range of 6500 pN for 6HB without pillars which increases to 11,000 pN for tubes with three pillars. The values of the stretch modulus calculated from contour length distributions obtained from equilibrium MD simulations are similar to those obtained from nonequilibrium SMD simulations. The addition of pillars makes these DNTs very rigid.Comment: Published in ACS Nan

Mechanics Modeling of Non-rigid Origami: From Qualitative to Quantitative Accuracy

Origami, the ancient art of paper folding, has recently evolved into a design and fabrication framework for various engineering systems at vastly different scales: from large-scale deployable airframes to mesoscale biomedical devices to small-scale DNA machines. The increasingly diverse applications of origami require a better understanding of the fundamental mechanics and dynamics induced by folding. Therefore, formulating a high-fidelity simulation model for origami is crucial, especially when large amplitude deformation/rotation exists during folding. The currently available origami simulation models can be categorized into three branches: rigid-facet models, bar-hinge models, and finite element models. The first branch of models assumes that the origami facets are rigid panels and creases behaving like hinges. It is a powerful tool for kinematics analysis without unnecessary complexities. On the other hand, the bar-hinge models have become widely used for simulating nonrigid-foldable origamis. The basic idea of these models is to place stretchable bar elements along the creases and across facet diagonals, discretizing the continuous origami into a pin-jointed truss frame system. Therefore, one can analyze facet deformations, including in-plane shearing, out-of-plane bending, and twisting. Moreover, more complex crease deformations can also be captured by adding appropriate components to the bar-hinge models. Because of their simplicity and modeling capability, the bar-hinge models have been utilized with many successes in analyzing the global deformation of non-rigid origami and uncovering its mechanical principles. However, one can only achieve qualitatively accurate predictions of the bar-hinge models compared to the physical experiments, especially when complex deformation exists during origami folding. The third branch, finite element models, does not impose explicit simplification on the facet deformation using shell elements. It can accurately analyze the deformation modes of origami structures; however, their disadvantages are also evident. On the one hand, it requires a time-consuming cycle for both modeling and computing, including pre-processing and post-processing. On the other hand, the traditional shell element might experience convergence issues when large and dynamic rotations occur, as commonly observed in origami systems. This thesis investigates the mechanics modeling of non-rigid origami and proposes a new dynamic model based on Absolute Nodal Coordinate Formulation (ANCF hereafter). Firstly, we discuss the accuracy of the widely used bar-hinge model through a case study on the multi-stability behavior in a non-rigid stacked Miura-origami structure. The model successfully investigates the underpinning principles of the multi-stability behavior in non-rigid origami and finds the existence of asymmetric energy barriers for extension and compression by tailoring its crease stiffness and facet bending stiffness. This interesting phenomenon can be exploited to create a mechanical diode. Experiment results confirm the existence of asymmetric multi-stability; however, the model\u27s prediction is only qualitatively verified due to its assumption of discrete lattices. In the next part, we develop a new origami mechanics model based on ANCF, a powerful modeling tool for the nonlinear dynamic simulation of multibody systems with large rotation and deformation. The new model treats origami as ANCF thin plate elements rotating around compliant creases, and the so-called torsional spring damper connectors are developed and utilized to simulate crease folding. Finally, its modeling accuracy is experimentally validated through two case studies, including motion analysis of simple fold mechanism and dynamic deployment of Miura-ori structures. The new origami simulation model can be used to quantitatively predict the dynamic responses of non-rigid origami with complex deformations. It can help deepen our knowledge of folding-induced mechanics and dynamics and broaden the application of origami in science and engineering

Active Self-Assembly of Algorithmic Shapes and Patterns in Polylogarithmic Time

We describe a computational model for studying the complexity of self-assembled structures with active molecular components. Our model captures notions of growth and movement ubiquitous in biological systems. The model is inspired by biology's fantastic ability to assemble biomolecules that form systems with complicated structure and dynamics, from molecular motors that walk on rigid tracks and proteins that dynamically alter the structure of the cell during mitosis, to embryonic development where large-scale complicated organisms efficiently grow from a single cell. Using this active self-assembly model, we show how to efficiently self-assemble shapes and patterns from simple monomers. For example, we show how to grow a line of monomers in time and number of monomer states that is merely logarithmic in the length of the line. Our main results show how to grow arbitrary connected two-dimensional geometric shapes and patterns in expected time that is polylogarithmic in the size of the shape, plus roughly the time required to run a Turing machine deciding whether or not a given pixel is in the shape. We do this while keeping the number of monomer types logarithmic in shape size, plus those monomers required by the Kolmogorov complexity of the shape or pattern. This work thus highlights the efficiency advantages of active self-assembly over passive self-assembly and motivates experimental effort to construct general-purpose active molecular self-assembly systems