Discrete Differential Geometry Lab

Andrew Sageman-Furnas

I am currently a Postdoc at the Technical University of Berlin. I received my PhD in October 2017 from the Discrete Differential Geometry Lab at the University of Goettingen.

Research interests

Discrete Differential Geometry, Structured and Unstructured Materials / Textiles, Computer Graphics & Fabrication.

Net: aosafu (at) mathematik.uni-goettingen.de


Refereed Articles
circular networks
Topology determines force distributions in one-dimensional random spring networks
Knut M. Heidemann, Andrew O. Sageman-Furnas, Abhinav Sharma, Florian Rehfeldt, Christoph F. Schmidt, and Max Wardetzky. Physical Review E 97, 022306, 2018.
Abstract: 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 \emph{network topology} is a crucial determinant of force distributions in elastic spring networks.

circular networks
Topology counts: force distributions in circular spring networks
Knut M. Heidemann, Andrew O. Sageman-Furnas, Abhinav Sharma, Florian Rehfeldt, Christoph F. Schmidt, and Max Wardetzky. Physical Review Letters 120, 068001, 2018.
Abstract: 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.

Form finding in elastic gridshells
Changyeob Baek, Andrew O. Sageman-Furnas, Mohammad K. Jawed and Pedro M. Reis. Proceedings of the National Academy of Sciences (USA) 2018 January, 115 (1) 75-80. https://doi.org/10.1073/pnas.1713841115.
Abstract: 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.

A 2x2 Lax representation, associated family, and Baecklund transformation for circular K-nets
Tim Hoffmann and Andrew O. Sageman-Furnas. Discrete and Computational Geometry, 2016; doi: 10.1007/s00454-016-9802-6.
Abstract: We present a 2x2 Lax representation for discrete circular nets of constant negative Gauss 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 Gauss 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.
[accepted pdf] [doi] [arXiv]

quad nets
A discrete parametrized surface theory in R^3
Tim Hoffmann, Andrew O. Sageman-Furnas, and Max Wardetzky. International Math Research Notices, 2016; doi: 10.1093/imrn/rnw015.
Abstract: We propose a discrete surface theory in R^3 that unites the most prevalent versions of discrete special parametrizations. This theory encapsulates a large class of discrete surfaces given by a Lax representation and, in particular, the one-parameter associated families of constant curvature surfaces. The theory is not restricted to integrable geometries, but extends to a general surface theory.
[pdf] [doi] [arXiv]

Meltables: Fabrication of Complex 3D Curves by Melting
Andrew O. Sageman-Furnas, Nobuyuki Umetani, and Ryan Schmidt. 2015 Conference Proceedings: ACM SIGGRAPH Asia. 4 pages, 2015.
Abstract: We propose a novel approach to fabricating complex 3D shapes via physical deformation of simpler shapes. Our focus is on objects composed of a set of planar beams and joints, where the joints are thin parts of the object which temporarily become living hinges when heated, close to a fixed angle defined by the local shape, and then become rigid when cooled. We call this class of objects Meltables. We present a novel algorithm that computes an optimal joint sequence which approximates a 3D spline curve while satisfying fabrication constraints. This technique is used in an interactive Meltables design tool. We demonstrate a variety of Meltables, fabricated with both 3D-printing and standard PVC piping. [pdf] [video] [project page]
Wire Mesh
Wire Mesh Design
Akash Garg, Andrew O. Sageman-Furnas, Bailin Deng, Yonghao Yue, Eitan Grinspun, Mark Pauly, and Max Wardetzky. ACM Transactions on Graphics 33:4, pp. 66:1–66:12, 2014.
Abstract: 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. [low-res pdf] [high-res pdf] [video] [fabrication video]
The Sphereprint: An approach to quantifying the conformability of flexible materials
Andrew O. Sageman-Furnas, Parikshit Goswami, Govind Menon, and Stephen J Russell. Textile Research Journal vol. 84:8, pp. 793–807, 2014.
Abstract: 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. [pdf] [doi]

Research Reports and Course Notes

Oberwolfach Reports
Towards a curvature theory for general quad meshes
Andrew O. Sageman-Furnas (joint work with Tim Hoffmann and Max Wardetzky), Oberwolfach Reports No. 13/2015.
Abstract: We present a curvature theory for general nonplanar quad meshes (a discrete analogue of smooth parametrized surfaces). As in the smooth setting, the resulting curvatures can be understood both in terms of a Steiner-type offset formula (extending the curvature theory for quad meshes with planar quadrilaterals) and in terms of the first and second fundamental forms. Our curvature theory equips the nonplanar quad meshes that correspond to discrete analogues of surfaces of constant curvature (constructed purely by algebraic means) with the appropriate curvatures.

SGP 2015
SGP 2015 COURSE SLIDES on Variational Time Integrators
Andrew O. Sageman-Furnas
Abstract: These slides are a brief introduction to Variational Time Integrators as presented during the Graudate School at the 2015 Symposium for Geometry Processing in Graz, Austria. Please contact me with corrections or comments.
[keynote] [pdf] [pptx]

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