Form finding in elastic gridshells Baek, Changyeob; Sageman-Furnas, Andrew O.; Jawed, Mohammad K. ...
Proceedings of the National Academy of Sciences - PNAS,
01/2018, Letnik:
115, Številka:
1
Journal Article
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Elastic gridshells comprise an initially planar network of elastic rods that are actuated into a shell-like structure by loading their extremities. The resulting actuated form derives from the ...elastic buckling of the rods subjected to inextensibility. We study elastic gridshells with a focus on the rational design of the final shapes. Our precision desktop experiments exhibit complex geometries, even from seemingly simple initial configurations and actuation processes. The numerical simulations capture this nonintuitive behavior with excellent quantitative agreement, allowing for an exploration of parameter space that reveals multistable states. We then turn to the theory of smooth Chebyshev nets to address the inverse design of hemispherical elastic gridshells. The results suggest that rod inextensibility, not elastic response, dictates the zeroth-order shape of an actuated elastic gridshell. As it turns out, this is the shape of a common household strainer. Therefore, the geometry of Chebyshev nets can be further used to understand elastic gridshells. In particular, we introduce a way to quantify the intrinsic shape of the empty, but enclosed regions, which we then use to rationalize the nonlocal deformation of elastic gridshells to point loading. This justifies the observed difficulty in form finding. Nevertheless, we close with an exploration of concatenating multiple elastic gridshell building blocks.
We present a computational approach for designing
wire meshes
, i.e., freeform surfaces composed of woven wires arranged in a regular grid. To facilitate shape exploration, we map material properties ...of wire meshes to the geometric model of
Chebyshev nets
. This abstraction is exploited to build an efficient optimization scheme. While the theory of Chebyshev nets suggests a highly constrained design space, we show that allowing controlled deviations from the underlying surface provides a rich shape space for design exploration. Our algorithm balances globally coupled material constraints with aesthetic and geometric design objectives that can be specified by the user in an interactive design session. In addition to sculptural art, wire meshes represent an innovative medium for industrial applications including composite materials and architectural façades. We demonstrate the effectiveness of our approach using a variety of digital and physical prototypes with a level of shape complexity unobtainable using previous methods.
Surface curvature both emerges from, and influences the behavior of, living objects at length scales ranging from cell membranes to single cells to tissues and organs. The relevance of surface ...curvature in biology is supported by numerous experimental and theoretical investigations in recent years. In this review, first, a brief introduction to the key ideas of surface curvature in the context of biological systems is given and the challenges that arise when measuring surface curvature are discussed. Giving an overview of the emergence of curvature in biological systems, its significance at different length scales becomes apparent. On the other hand, summarizing current findings also shows that both single cells and entire cell sheets, tissues or organisms respond to curvature by modulating their shape and their migration behavior. Finally, the interplay between the distribution of morphogens or micro‐organisms and the emergence of curvature across length scales is addressed with examples demonstrating these key mechanistic principles of morphogenesis. Overall, this review highlights that curved interfaces are not merely a passive by‐product of the chemical, biological, and mechanical processes but that curvature acts also as a signal that co‐determines these processes.
Curvature as a local descriptor for shape has been revealed to play a fundamental role in the development of biological systems. Advanced 3D characterization methods allow its quantification across time and length scales indicating that cells and tissue growth can cause emergence of curved surfaces but in turn curvature also acts as a trigger for specific biological processes.
Filamentous polymer networks govern the mechanical properties of many biological materials. Force distributions within these networks are typically highly inhomogeneous, and, although the importance ...of force distributions for structural properties is well recognized, they are far from being understood quantitatively. Using a combination of probabilistic and graph-theoretical techniques, we derive force distributions in a model system consisting of ensembles of random linear spring networks on a circle. We show that characteristic quantities, such as the mean and variance of the force supported by individual springs, can be derived explicitly in terms of only two parameters: (i) average connectivity and (ii) number of nodes. Our analysis shows that a classical mean-field approach fails to capture these characteristic quantities correctly. In contrast, we demonstrate that network topology is a crucial determinant of force distributions in an elastic spring network. Our results for 1D linear spring networks readily generalize to arbitrary dimensions.
The Sphereprint is introduced as a means to characterize hemispherical conformability, even when buckling occurs, in a variety of flexible materials such as papers, textiles, nonwovens, films, ...membranes, and biological tissues. Conformability is defined here as the ability to fit a doubly curved surface without folding. Applications of conformability range from the fit of a wound dressing, artificial skin, or wearable electronics around a protuberance such as a knee or elbow to geosynthetics used as reinforcements. Conformability of flexible materials is quantified by two dimensionless quantities derived from the Sphereprint. The Sphereprint ratio summarizes how much of the specimen conforms to a hemisphere under symmetric radial loading. The coefficient of expansion approximates the average stretching of the specimen during deformation, accounting for hysteresis. Both quantities are reproducible and robust, even though a given material folds differently each time it conforms. For demonstration purposes, an implementation of the Sphereprint test methodology was performed on a collection of cellulosic fibrous assemblies. For this example, the Sphereprint ratio ranked the fabrics according to intuition from least to most conformable in the sequence: paper towel, plain weave, satin weave, and single knit jersey. The coefficient of expansion distinguished the single knit jersey from the bark weave fabric, despite them having similar Sphereprint ratios and, as expected, the bark weave stretched less than the single knit jersey did during conformance. This work lays the foundation for engineers to quickly and quantitatively compare the conformance of existing and new flexible materials, no matter their construction.
Networks of elastic fibers are ubiquitous in biological systems and often provide mechanical stability to cells and tissues. Fiber-reinforced materials are also common in technology. An important ...characteristic of such materials is their resistance to failure under load. Rupture occurs when fibers break under excessive force and when that failure propagates. Therefore, it is crucial to understand force distributions. Force distributions within such networks are typically highly inhomogeneous and are not well understood. Here we construct a simple one-dimensional model system with periodic boundary conditions by randomly placing linear springs on a circle. We consider ensembles of such networks that consist of N nodes and have an average degree of connectivity z but vary in topology. Using a graph-theoretical approach that accounts for the full topology of each network in the ensemble, we show that, surprisingly, the force distributions can be fully characterized in terms of the parameters (N,z). Despite the universal properties of such (N,z) ensembles, our analysis further reveals that a classical mean-field approach fails to capture force distributions correctly. We demonstrate that network topology is a crucial determinant of force distributions in elastic spring networks.
Chebyshev nets from commuting PolyVector fields Sageman-Furnas, Andrew O.; Chern, Albert; Ben-Chen, Mirela ...
ACM transactions on graphics,
11/2019, Letnik:
38, Številka:
6
Journal Article
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We propose a method for computing global Chebyshev nets on triangular meshes. We formulate the corresponding global parameterization problem in terms of
commuting
PolyVector fields, and design an ...efficient optimization method to solve it. We compute, for the first time, Chebyshev nets with automatically-placed singularities, and demonstrate the realizability of our approach using real material.
(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We present a ... Lax representation for discrete circular nets of constant negative Gaus curvature. It is tightly linked to ...the 4D consistency of the Lax representation of discrete K-nets (in asymptotic line parametrization). The description gives rise to Baecklund transformations and an associated family. All the members of that family-although no longer circular-can be shown to have constant Gaus curvature as well. Explicit solutions for the Baecklund transformations of the vacuum (in particular Dini's surfaces and breather solutions) and their respective associated families are given.
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