“Differentiable Stripe Patterns for Inverse Design of Structured Surfaces” by Maestre, Du, Hinchet, Coros and Thomaszewski

  • ©Juan Sebastian Montes Maestre, Yinwei Du, Ronan Hinchet, Stelian Coros, and Bernhard Thomaszewski




    Differentiable Stripe Patterns for Inverse Design of Structured Surfaces

Session/Category Title:   Surfaces, Strips, Lights




    Stripe patterns are ubiquitous in nature and everyday life. While the synthesis of these patterns has been thoroughly studied in the literature, their potential to control the mechanics of structured materials remains largely unexplored. In this work, we introduce Differentiable Stripe Patterns—a computational approach for automated design of physical surfaces structured with stripe-shaped bi-material distributions. Our method builds on the work by Knöppel and colleagues [2015] for generating globally-continuous and equally-spaced stripe patterns. To unlock the full potential of this design space, we propose a gradient-based optimization tool to automatically compute stripe patterns that best approximate macromechanical performance goals. Specifically, we propose a computational model that combines solid shell finite elements with XFEM for accurate and fully-differentiable modeling of elastic bi-material surfaces. To resolve non-uniqueness problems in the original method, we furthermore propose a robust formulation that yields unique and differentiable stripe patterns. We combine these components with equilibrium state derivatives into an end-to-end differentiable pipeline that enables inverse design of mechanical stripe patterns. We demonstrate our method on a diverse set of examples that illustrate the potential of stripe patterns as a design space for structured materials. Our simulation results are experimentally validated on physical prototypes.


    1. Katia Bertoldi, Vincenzo Vitelli, Johan Christensen, and Martin Van Hecke. 2017. Flexible mechanical metamaterials. Nature Reviews Materials 2, 11 (2017), 1–11.
    2. Gaurav Bharaj, David I. W. Levin, James Tompkin, Yun Fei, Hanspeter Pfister, Wojciech Matusik, and Changxi Zheng. 2015. Computational Design of Metallophone Contact Sounds. ACM Trans. Graph. 34, 6, Article 223 (Oct. 2015), 13 pages.
    3. Bernd Bickel, Moritz Bächer, Miguel A. Otaduy, Hyunho Richard Lee, Hanspeter Pfister, Markus Gross, and Wojciech Matusik. 2010. Design and Fabrication of Materials with Desired Deformation Behavior. In ACM SIGGRAPH 2010 Papers (Los Angeles, California) (SIGGRAPH ’10). Association for Computing Machinery, New York, NY, USA, Article 63, 10 pages.
    4. Narasimha Boddeti, Yunlong Tang, Kurt Maute, David W. Rosen, and Martin L. Dunn. 2020. Optimal design and manufacture of variable stiffness laminated continuous fiber reinforced composites. Scientific Reports 10 (12 2020). Issue 1.
    5. R. Bridson, S. Marino, and R. Fedkiw. 2005. Simulation of Clothing with Folds and Wrinkles. In ACM SIGGRAPH 2005 Courses (Los Angeles, California) (SIGGRAPH ’05). Association for Computing Machinery, New York, NY, USA, 3–es.
    6. Desai Chen, David I. W. Levin, Wojciech Matusik, and Danny M. Kaufman. 2017. Dynamics-Aware Numerical Coarsening for Fabrication Design. ACM Trans. Graph. 36, 4, Article 84 (jul 2017), 15 pages.
    7. Honglin Chen, Hsueh-TI Derek Liu, Alec Jacobson, and David I. W. Levin. 2020. Chordal Decomposition for Spectral Coarsening. ACM Trans. Graph. 39, 6, Article 265 (nov 2020), 16 pages.
    8. Hsiao-Yu Chen, Arnav Sastry, Wim M. van Rees, and Etienne Vouga. 2018. Physical Simulation of Environmentally Induced Thin Shell Deformation. ACM Trans. Graph. 37, 4, Article 146 (jul 2018), 13 pages.
    9. Tian Chen, Julian Panetta, Max Schnaubelt, and Mark Pauly. 2021. Bistable auxetic surface structures. ACM Transactions on Graphics 40 (7 2021). Issue 4.
    10. Yu Ju (Edwin) Chen, David I. W. Levin, Danny Kaufmann, Uri Ascher, and Dinesh K. Pai. 2019. EigenFit for Consistent Elastodynamic Simulation across Mesh Resolution. In Proceedings of the 18th Annual ACM SIGGRAPH/Eurographics Symposium on Computer Animation (Los Angeles, California) (SCA ’19). Association for Computing Machinery, New York, NY, USA, Article 5, 13 pages.
    11. Keenan Crane, Mathieu Desbrun, and Peter Schröder. 2010. Trivial Connections on Discrete Surfaces. Computer Graphics Forum (SGP) 29, 5 (2010), 1525–1533.
    12. Akash Garg, Eitan Grinspun, Max Wardetzky, and Denis Zorin. 2007. Cubic Shells. In Proceedings of the 2007 ACM SIGGRAPH/Eurographics Symposium on Computer Animation (San Diego, California) (SCA ’07). Eurographics Association, Goslar, DEU, 91–98.
    13. Yotam Gingold, Adrian Secord, Jefferson Y Han, Eitan Grinspun, and Denis Zorin. 2004. A discrete model for inelastic deformation of thin shells. In ACM SIGGRAPH/Eurographics symposium on computer animation. Citeseer.
    14. Aldair E. Gongora, Kelsey L. Snapp, Richard Pang, Thomas M. Tiano, Kristofer G. Reyes, Emily Whiting, Timothy J. Lawton, Elise F. Morgan, and Keith A. Brown. 2022. Designing lattices for impact protection using transfer learning. Matter 5, 9 (2022), 2829–2846.
    15. Eitan Grinspun, Anil N Hirani, Mathieu Desbrun, and Peter Schröder. 2003. Discrete shells. In Proceedings of the 2003 ACM SIGGRAPH/Eurographics symposium on Computer animation. Eurographics Association, 62–67.
    16. Ruslan Guseinov, Eder Miguel, and Bernd Bickel. 2017. CurveUps: Shaping Objects from Flat Plates with Tension-Actuated Curvature. ACM Trans. Graph. 36, 4, Article 64 (July 2017), 12 pages.
    17. Christian Hafner, Christian Schumacher, Espen Knoop, Thomas Auzinger, Bernd Bickel, and Moritz Bächer. 2019. X-CAD: Optimizing CAD Models with Extended Finite Elements. ACM Trans. Graph. 38, 6, Article 157 (Nov. 2019), 15 pages.
    18. R. Hauptmann and K. Schweizerhof. 1998. A systematic development of ‘solid-shell’ element formulations for linear and non-linear analyses employing only displacement degrees of freedom. Internat. J. Numer. Methods Engrg. 42, 1 (1998), 49–69. <49::AID-NME349>3.0.CO;2-2
    19. Gerhard A. Holzapfel and Thomas C. Gasser. 2001. A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computational aspects and applications. Computer Methods in Applied Mechanics and Engineering 190, 34 (2001), 4379–4403.
    20. Caigui Jiang, Florian Rist, Hui Wang, Johannes Wallner, and Helmut Pottmann. 2022. Shape-morphing mechanical metamaterials. Computer-Aided Design 143 (2022), 103146.
    21. David Jourdan, Skouras Mélina, Romero Victor, Etienne Vouga, and Adrien Bousseau. 2022a. Simulation of printed-on-fabric assemblies. Association for Computing Machinery (ACM), 1–11.
    22. David Jourdan, Mélina Skouras, Etienne Vouga, and Adrien Bousseau. 2022b. Computational Design of Self-Actuated Surfaces by Printing Plastic Ribbons on Stretched Fabric. Computer Graphics Forum 41 (5 2022), 493–506. Issue 2.
    23. David Jourdan, Mélina Skouras, Etienne Vouga, Adrien Bousseau, D Jourdan, M Skouras, E Vouga, and A Bousseau. 2021. Printing-on-Fabric Meta-Material for Self-Shaping Architectural Models. AAG 2020-Advances in Architectural Geometry., 19 pages. https://hal.inria.fr/hal-02925036
    24. Peter Kaufmann, Sebastian Martin, Mario Botsch, Eitan Grinspun, and Markus Gross. 2009. Enrichment Textures for Detailed Cutting of Shells. ACM Trans. Graph. 28, 3, Article 50 (jul 2009), 10 pages.
    25. Lily Kharevych, Patrick Mullen, Houman Owhadi, and Mathieu Desbrun. 2009. Numerical coarsening of inhomogeneous elastic materials. ACM Transactions on graphics (TOG) 28, 3 (2009), 1–8.
    26. Felix Knöppel, Keenan Crane, Ulrich Pinkall, and Peter Schröder. 2015. Stripe patterns on surfaces. ACM Transactions on Graphics 34. Issue 4.
    27. Yeongbin Ko and Phill Seung Lee. 2017. A 6-node triangular solid-shell element for linear and nonlinear analysis. Internat. J. Numer. Methods Engrg. 111 (9 2017), 1203–1230. Issue 13.
    28. Mina Konaković-Luković, Julian Panetta, Keenan Crane, and Mark Pauly. 2018. Rapid Deployment of Curved Surfaces via Programmable Auxetics. ACM Trans. Graph. 37, 4, Article 106 (July 2018), 13 pages.
    29. Dan Koschier, Jan Bender, and Nils Thuerey. 2017. Robust extended finite elements for complex cuting of deformables. ACM Transactions on Graphics 36. Issue 4.
    30. Kurt Leimer and Przemyslaw Musialski. 2020. Reduced-Order Simulation of Flexible Meta-Materials. Proceedings – SCF 2020: ACM Symposium on Computational Fabrication.
    31. Minchen Li, Zachary Ferguson, Teseo Schneider, Timothy R Langlois, Denis Zorin, Daniele Panozzo, Chenfanfu Jiang, and Danny M Kaufman. 2020. Incremental potential contact: intersection-and inversion-free, large-deformation dynamics. ACM Trans. Graph. 39, 4 (2020), 49.
    32. Hsueh-Ti Derek Liu, Alec Jacobson, and Maks Ovsjanikov. 2019. Spectral Coarsening of Geometric Operators. ACM Trans. Graph. 38, 4, Article 105 (jul 2019), 13 pages.
    33. Zhenyuan Liu, Jingyu Hu, Hao Xu, Peng Song, Ran Zhang, Bernd Bickel, and Chi-Wing Fu. 2022. Worst-Case Rigidity Analysis and Optimization for Assemblies with Mechanical Joints. Computer Graphics Forum 41, 2 (2022).
    34. Luigi Malomo, Jesús Pérez, Emmanuel Iarussi, Nico Pietroni, Eder Miguel, Paolo Cignoni, and Bernd Bickel. 2018. FlexMaps: Computational Design of Flat Flexible Shells for Shaping 3D Objects. ACM Trans. Graph. 37, 6, Article 241 (Dec. 2018), 14 pages.
    35. Jonàs Martínez, Jérémie Dumas, and Sylvain Lefebvre. 2016. Procedural voronoi foams for additive manufacturing. ACM Transactions on Graphics (TOG) 35, 4 (2016), 1–12.
    36. Jonàs Martínez, Mélina Skouras, Christian Schumacher, Samuel Hornus, Sylvain Lefebvre, and Bernhard Thomaszewski. 2019. Star-shaped metrics for mechanical metamaterial design. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1–13.
    37. Jonàs Martínez, Haichuan Song, Jérémie Dumas, and Sylvain Lefebvre. 2017. Orthotropic k-nearest foams for additive manufacturing. ACM Transactions on Graphics (TOG) 36, 4 (2017), 1–12.
    38. Nicolas Moës, Mathieu Cloirec, Patrice Cartraud, and J-F Remacle. 2003. A computational approach to handle complex microstructure geometries. Computer methods in applied mechanics and engineering 192, 28–30 (2003), 3163–3177.
    39. Juan Montes, Bernhard Thomaszewski, Sudhir Mudur, and Tiberiu Popa. 2020. Computational Design of Skintight Clothing. ACM Trans. Graph. 39, 4, Article 105 (jul 2020), 12 pages.
    40. Ella Moore, Ioannis Karamouzas, and Michael Porter. 2018. Precision control of tensile properties in fabric for computational fabrication. Proceedings – SCF 2018: ACM Symposium on Computational Fabrication.
    41. Przemyslaw Musialski, Christian Hafner, Florian Rist, Michael Birsak, Michael Wimmer, and Leif Kobbelt. 2016. Non-linear Shape Optimization Using Local Subspace Projections. ACM Trans. Graph. 35, 4, Article 87 (July 2016), 13 pages.
    42. William Neveu, Ivan Puhachov, Bernhard Thomaszewski, and Mikhail Bessmeltsev. 2022. Stability-Aware Simplification of Curve Networks. In ACM SIGGRAPH 2022 Conference Proceedings (Vancouver, BC, Canada) (SIGGRAPH ’22). Association for Computing Machinery, New York, NY, USA, Article 20, 9 pages.
    43. Julian Panetta, Florin Isvoranu, Tian Chen, Emmanuel Siéfert, Benoît Roman, and Mark Pauly. 2021. Computational inverse design of surface-based inflatables. ACM Transactions on Graphics 40 (7 2021). Issue 4.
    44. J. Panetta, M. Konakoviundefined-Lukoviundefined, F. Isvoranu, E. Bouleau, and M. Pauly. 2019. X-Shells: A New Class of Deployable Beam Structures. ACM Trans. Graph. 38, 4, Article 83 (July 2019), 15 pages.
    45. Julian Panetta, Abtin Rahimian, and Denis Zorin. 2017. Worst-case stress relief for microstructures. ACM Transactions on Graphics (TOG) 36, 4 (2017), 1–16.
    46. Julian Panetta, Qingnan Zhou, Luigi Malomo, Nico Pietroni, Paolo Cignoni, and Denis Zorin. 2015. Elastic textures for additive fabrication. ACM Transactions on Graphics (TOG) 34, 4 (2015), 1–12.
    47. Jesús Pérez, Miguel A. Otaduy, and Bernhard Thomaszewski. 2017. Computational Design and Automated Fabrication of Kirchhoff-plateau Surfaces. ACM Trans. Graph. 36, 4, Article 62 (July 2017), 12 pages.
    48. Stefan Pillwein and Przemyslaw Musialski. 2021. Generalized deployable elastic geodesic grids. ACM Transactions on Graphics 40 (12 2021). Issue 6.
    49. Abhinit Sati, Ioannis Karamouzas, and Victor Zordan. 2021. DIGISEW: Anisotropic Stitching for Variable Stretch in Textiles. Association for Computing Machinery, New York, NY, USA.
    50. Christian Schumacher, Bernd Bickel, Jan Rys, Steve Marschner, Chiara Daraio, and Markus Gross. 2015. Microstructures to control elasticity in 3D printing. ACM Transactions on Graphics (Tog) 34, 4 (2015), 1–13.
    51. Christian Schumacher, Steve Marschner, Markus Gross, and Bernhard Thomaszewski. 2018. Mechanical Characterization of Structured Sheet Materials. ACM Trans. Graph. 37, 4, Article 148 (July 2018), 15 pages.
    52. Mélina Skouras, Bernhard Thomaszewski, Peter Kaufmann, Akash Garg, Bernd Bickel, Eitan Grinspun, and Markus Gross. 2014. Designing Inflatable Structures. 33, 4 (2014).
    53. Krister Svanberg. 1995. A globally convergent version of MMA without linesearch. In Proceedings of the first world congress of structural and multidisciplinary optimization, Vol. 28. Goslar, Germany, 9–16.
    54. Bernhard Thomaszewski, Stelian Coros, Damien Gauge, Vittorio Megaro, Eitan Grinspun, and Markus Gross. 2014. Computational Design of Linkage-Based Characters. ACM Trans. Graph. 33, 4, Article 64 (July 2014), 9 pages.
    55. Davi Colli Tozoni, Jérémie Dumas, Zhongshi Jiang, Julian Panetta, Daniele Panozzo, and Denis Zorin. 2020. A low-parametric rhombic microstructure family for irregular lattices. ACM Transactions on Graphics (TOG) 39, 4 (2020), 101–1.
    56. Thibault Tricard, Vincent Tavernier, Cédric Zanni, Jonàs Martínez, Pierre-Alexandre Hugron, Fabrice Neyret, and Sylvain Lefebvre. 2020. Freely orientable microstructures for designing deformable 3D prints. ACM Trans. Graph. 39, 6 (2020), 211–1.
    57. Nobuyuki Umetani, Jun Mitani, and Takeo Igarashi. 2010. Designing Custom-made Metallophone with Concurrent Eigenanalysis. In Proceedings of the International Conference on New Interfaces for Musical Expression. Zenodo, 26–30.
    58. Velko Vechev, Juan Zarate, Bernhard Thomaszewski, and Otmar Hilliges. 2022. Computational Design of Kinesthetic Garments. Computer Graphics Forum (2022).
    59. Yi Yang, Katherine Vella, and Douglas P. Holmes. 2021. Grasping with kirigami shells. Science Robotics 6, 54 (2021), eabd6426. arXiv:https://www.science.org/doi/pdf/10.1126/scirobotics.abd6426
    60. Jonas Zehnder, Stelian Coros, and Bernhard Thomaszewski. 2016. Designing Structurally-Sound Ornamental Curve Networks. ACM Trans. Graph. 35, 4, Article 99 (July 2016), 10 pages.
    61. Jonas Zehnder, Espen Knoop, Moritz Bächer, and Bernhard Thomaszewski. 2017. Metasilicone: design and fabrication of composite silicone with desired mechanical properties. ACM Transactions on Graphics (TOG) 36, 6 (2017), 1–13.
    62. Bo Zhu, Mélina Skouras, Desai Chen, and Wojciech Matusik. 2017. Two-scale topology optimization with microstructures. ACM Transactions on Graphics (TOG) 36, 4 (2017), 1.

ACM Digital Library Publication:

Overview Page: