“Beyond developable: computational design and fabrication with auxetic materials”

  • ©Mark Pauly, Mina Konaković-Luković, Keenan Crane, Sofien Bouaziz, Bailin Deng, and Daniel Piker




    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.


    1. Aflalo, Y., Kimmel, R., and Zibulevsky, M. 2013. Conformal mapping with as uniform as possible conformal factor. SIAM Journal on Imaging Sciences 6, 1, 78–101.Google ScholarCross Ref
    2. Bazant, M. Z., and Crowdy, D. 2005. Conformal Mapping Methods for Interfacial Dynamics.Google Scholar
    3. Ben-Chen, M., and Gotsman, C. 2008. Characterizing shape using conformal factors. In Proceedings of the 1st Eurographics Conference on 3D Object Retrieval, 3DOR ’08, 1–8. Google ScholarDigital Library
    4. Ben-Chen, M., Gotsman, C., and Bunin, G. 2008. Conformal flattening by curvature prescription and metric scaling. Computer Graphics Forum 27, 2, 449–458.Google ScholarCross Ref
    5. Bobenko, A. I., and Suris, Y. B. 2008. Discrete Differential Geometry: Integrable Structure. AMS.Google Scholar
    6. Bouaziz, S., Deuss, M., Schwartzburg, Y., Weise, T., and Pauly, M. 2012. Shape-up: Shaping discrete geometry with projections. Comput. Graph. Forum 31, 5, 1657–1667. Google ScholarDigital Library
    7. Bouaziz, S., Martin, S., Liu, T., Kavan, L., and Pauly, M. 2014. Projective dynamics: Fusing constraint projections for fast simulation. ACM Trans. Graph. 33, 4 (July), 154:1–154:11. Google ScholarDigital Library
    8. Caneparo, L. 2014. Digital Fabrication in Architecture, Engineering and Construction. Springer. Google ScholarDigital Library
    9. Cho, Y., Shin, J.-H., Costa, A., Kim, T. A., Kunin, V., Li, J., Lee, S. Y., Yang, S., Han, H. N., Choi, I.-S., and Srolovitz, D. J. 2014. Engineering the shape and structure of materials by fractal cut. Proceedings of the National Academy of Sciences 111, 49, 17390–17395.Google ScholarCross Ref
    10. Cignoni, P., Pietroni, N., Malomo, L., and Scopigno, R. 2014. Field-aligned mesh joinery. ACM Trans. Graph. 33, 1 (Feb.), 11:1–11:12. Google ScholarDigital Library
    11. Crane, K., Pinkall, U., and Schröder, P. 2011. Spin transformations of discrete surfaces. ACM Trans. Graph. 30, 4 (July), 104:1–104:10. Google ScholarDigital Library
    12. Crane, K., Pinkall, U., and Schröder, P. 2013. Robust fairing via conformal curvature flow. ACM Trans. Graph. 32, 4 (July), 61:1–61:10. Google ScholarDigital Library
    13. Desbrun, M., Meyer, M., and Alliez, P. 2002. Intrinsic parameterizations of surface meshes. Computer Graphics Forum 21, 3, 209–218.Google ScholarCross Ref
    14. Deuss, M., Deleuran, A., Bouaziz, S., Deng, B., Piker, D., and Pauly, M. 2015. ShapeOp — a robust and extensible geometric modelling paradigm. In Modelling Behaviour. Springer International Publishing, 505–515.Google Scholar
    15. Evans, K. E., and Alderson, A. 2000. Auxetic materials: Functional materials and structures from lateral thinking! Advanced Materials 12, 9, 617–628.Google Scholar
    16. Feeman, T. G. 2002. Portraits of the Earth: A Mathematician Looks at Maps. American Mathematical Society.Google Scholar
    17. Garg, A., Sageman-Furnas, A. O., Deng, B., Yue, Y., Grinspun, E., Pauly, M., and Wardetzky, M. 2014. Wire mesh design. ACM Trans. Graph. 33, 4 (July), 66:1–66:12. Google ScholarDigital Library
    18. Gatt, R., Mizzi, L., Azzopardi, J. I., Azzopardi, K. M., Attard, D., Casha, A., Briffa, J., and Grima, J. N. 2015. Hierarchical auxetic mechanical metamaterials. Scientific Reports 5.Google Scholar
    19. Gibson, I., Rosen, D., and Stucker, B. 2015. Additive Manufacturing Technologies: 3D Printing, Rapid Prototyping, and Direct Digital Manufacturing, 2nd ed. Springer.Google Scholar
    20. Gu, X., and Yau, S.-T. 2003. Global conformal surface parameterization. In Proc. SGP, 127–137. Google ScholarDigital Library
    21. Gu, X. D., and Yau, S.-T. 2008. Computational conformal geometry. International Press of Boston, Inc.Google Scholar
    22. Guo, R. 2011. Local rigidity of inversive distance circle packing. Transactions of the American Mathematical Society 363, 9, 4757–4776.Google ScholarCross Ref
    23. Hildebrand, K., Bickel, B., and Alexa, M. 2012. crdbrd: Shape fabrication by sliding planar slices. Computer Graphics Forum 31, 2, 583–592. Google ScholarDigital Library
    24. Iarussi, E., Li, W., and Bousseau, A. 2015. Wrapit: Computer-assisted crafting of wire wrapped jewelry. ACM Trans. Graph. 34, 6 (Oct.), 221:1–221:8. Google ScholarDigital Library
    25. Igarashi, Y., Igarashi, T., and Mitani, J. 2012. Beady: Interactive beadwork design and construction. ACM Trans. Graph. 31, 4 (July), 49:1–49:9. Google ScholarDigital Library
    26. Kharevych, L., Springborn, B., and Schröder, P. 2006. Discrete conformal mappings via circle patterns. ACM Trans. Graph. 25, 2 (Apr.), 412–438. Google ScholarDigital Library
    27. Kilian, M., Flöry, S., Chen, Z., Mitra, N. J., Sheffer, A., and Pottmann, H. 2008. Curved folding. ACM Trans. Graph. 27, 3 (Aug.), 75:1–75:9. Google ScholarDigital Library
    28. Kim, J., Hanna, J. A., Byun, M., Santangelo, C. D., and Hayward, R. C. 2012. Designing responsive buckled surfaces by halftone gel lithography. Science 335, 6073 (Mar.), 1201–1205.Google ScholarCross Ref
    29. Lévy, B., Petitjean, S., Ray, N., and Maillot, J. 2002. Least squares conformal maps for automatic texture atlas generation. ACM Trans. Graph. 21, 3 (July), 362–371. Google ScholarDigital Library
    30. Li, H., Adams, B., Guibas, L. J., and Pauly, M. 2009. Robust single-view geometry and motion reconstruction. ACM Trans. Graph. 28, 5 (Dec.), 175:1–175:10. Google ScholarDigital Library
    31. Li, S., Zeng, W., Zhou, D., Gu, X. D., and Gao, J. 2013. Compact conformal map for greedy routing in wireless mobile sensor networks. In INFOCOM, IEEE, 2409–2417.Google Scholar
    32. Lipman, Y., and Funkhouser, T. 2009. Möbius voting for surface correspondence. ACM Trans. Graph. 28, 3 (July), 72:1–72:12. Google ScholarDigital Library
    33. Lipman, Y. 2012. Bounded distortion mapping spaces for triangular meshes. ACM Trans. Graph. 31, 4 (July), 108:1–108:13. Google ScholarDigital Library
    34. Liu, Y., Pottmann, H., Wallner, J., Yang, Y.-L., and Wang, W. 2006. Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graph. 25, 3 (July), 681–689. Google ScholarDigital Library
    35. Luo, S.-J., Yue, Y., Huang, C.-K., Chung, Y.-H., Imai, S., Nishita, T., and Chen, B.-Y. 2015. Legolization: Optimizing lego designs. ACM Trans. Graph. 34, 6 (Oct.), 222:1–222:12. Google ScholarDigital Library
    36. Mullen, P., Tong, Y., Alliez, P., and Desbrun, M. 2008. Spectral conformal parameterization. Computer Graphics Forum 27, 5, 1487–1494. Google ScholarDigital Library
    37. Pérez, J., Thomaszewski, B., Coros, S., Bickel, B., Canabal, J. A., Sumner, R., and Otaduy, M. A. 2015. Design and fabrication of flexible rod meshes. ACM Trans. Graph. 34, 4 (July), 138:1–138:12. Google ScholarDigital Library
    38. Pottmann, H., Eigensatz, M., Vaxman, A., and Wallner, J. 2015. Architectural geometry. Computers & Graphics 47, 145–164. Google ScholarDigital Library
    39. Rafsanjani, A., and Pasini, D. 2016. Multistable compliant auxetic metamaterials inspired by geometric patterns in Islamic arts. APS March Meeting.Google Scholar
    40. Resch, R. D. 1973. The topological design of sculptural and architectural systems. In Proceedings of the June 4-8, 1973, National Computer Conference and Exposition, ACM, New York, NY, USA, AFIPS ’73, 643–650. Google ScholarDigital Library
    41. Röhrig, T., Sechelmann, S., Kycia, A., and Fleischmann, M. 2014. Surface panelization using periodic conformal maps. In Advances in Architectural Geometry 2014. Springer International Publishing, 199–214.Google Scholar
    42. Schiftner, A., Höbinger, M., Wallner, J., and Pottmann, H. 2009. Packing circles and spheres on surfaces. ACM Trans. Graph. 28, 5 (Dec.), 139:1–139:8. Google ScholarDigital Library
    43. Schwartzburg, Y., and Pauly, M. 2013. Fabrication-aware design with intersecting planar pieces. Computer Graphics Forum 32, 2, 317–326.Google ScholarCross Ref
    44. Skouras, M., Thomaszewski, B., Kaufmann, P., Garg, A., Bickel, B., Grinspun, E., and Gross, M. 2014. Designing inflatable structures. ACM Trans. Graph. 33, 4 (July), 63:1–63:10. Google ScholarDigital Library
    45. Song, P., Fu, C.-W., Goswami, P., Zheng, J., Mitra, N. J., and Cohen-Or, D. 2013. Reciprocal frame structures made easy. ACM Trans. Graph. 32, 4 (July), 94:1–94:13. Google ScholarDigital Library
    46. Springborn, B., Schröder, P., and Pinkall, U. 2008. Conformal equivalence of triangle meshes. ACM Trans. Graph. 27, 3 (Aug.), 77:1–77:11. Google ScholarDigital Library
    47. Stephenson, K. 2003. Circle packing: a mathematical tale. Notices of the AMS 50, 11, 1376–1388.Google Scholar
    48. Sullivan, J. M. 2011. Conformal tiling on a torus. In Proceedings of Bridges 2011: Mathematics, Music, Art, Architecture, Culture, Tessellations Publishing, R. Sarhangi and C. H. Séquin, Eds., 593–596.Google Scholar
    49. Tachi, T. 2010. Origamizing polyhedral surfaces. IEEE Trans. Vis. Comput. Graph. 16, 2, 298–311. Google ScholarDigital Library
    50. Tachi, T. 2013. Freeform origami tessellations by generalizing Resch’s patterns. Journal of Mechanical Design 135, 11.Google ScholarCross Ref
    51. Tang, C., Bo, P., Wallner, J., and Pottmann, H. 2016. Interactive design of developable surfaces. ACM Trans. Graph. 35, 2 (Jan.), 12:1–12:12. Google ScholarDigital Library
    52. Testuz, R., Schwartzburg, Y., and Pauly, M. 2013. Automatic generation of constructable brick sculptures. In Proc. Eurographics.Google Scholar
    53. Vaxman, A., Müller, C., and Weber, O. 2015. Conformal mesh deformations with Möbius transformations. ACM Trans. Graph. 34, 4 (July), 55:1–55:11. Google ScholarDigital Library
    54. Weber, O., Myles, A., and Zorin, D. 2012. Computing extremal quasiconformal maps. Computer Graphics Forum 31, 5, 1679–1689. Google ScholarDigital Library
    55. Zimmer, H., and Kobbelt, L. 2014. Zometool rationalization of freeform surfaces. Visualization and Computer Graphics, IEEE Transactions on 20, 10 (Oct), 1461–1473.Google Scholar

ACM Digital Library Publication:

Overview Page: