{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T16:26:32Z","timestamp":1787156792431,"version":"3.56.0"},"reference-count":37,"publisher":"Wiley","license":[{"start":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T00:00:00Z","timestamp":1787097600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"},{"start":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T00:00:00Z","timestamp":1787097600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/501100001711","name":"Swiss National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100003351","name":"SNF","doi-asserted-by":"publisher","award":["200021_236469"],"award-info":[{"award-number":["200021_236469"]}],"id":[{"id":"10.13039\/100003351","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Computer Graphics Forum"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Structured sheet materials such as 3D\u2010printed rod networks, multi\u2010material thin shells, and multi\u2010layer laminates exhibit diverse mechanical behaviors. To avoid the computational burden of native\u2010scale simulations, data\u2010driven homogenization offers a promising alternative. This process involves probing a representative patch of material\u2014a unit cell\u2014with a set of stretching and bending tests subject to periodic boundary conditions. Because macro\u2010scale bending moments are not directly available from native\u2010scale simulations, existing methods exclusively rely on elastic energy data. Unfortunately, using only elastic energy from uniaxial tests is not sufficient for capturing the full moment\u2010curvature relationship, and imposing biaxial curvature states would necessarily break periodicity. We present a moment\u2010based homogenization method that infers curvature coupling using only uniaxial bending tests. Our method computes macro\u2010scale bending moments from native\u2010scale simulations for a wide range of mechanical models. To this end, we translate internal deformations into elastic stresses and then integrate these stresses through the thickness and across the unit\u2010cell patch. We use the resulting homogenized bending moments along with energy data to fit neural bending energy density functions. We demonstrate our method on a diverse set of materials, including multi\u2010material shells, rod networks, and multi\u2010layer sheets. Our results show improved accuracy compared to existing approaches and realistic double\u2010curvature behavior when applied to larger samples.<\/jats:p>","DOI":"10.1111\/cgf.70513","type":"journal-article","created":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T15:42:22Z","timestamp":1787154142000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Taking a Moment to Characterize the Bending Response of Thin Sheet Materials"],"prefix":"10.1111","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-0340-746X","authenticated-orcid":false,"given":"P.","family":"Xie","sequence":"first","affiliation":[{"name":"ISTA  Austria"},{"name":"ETH Z\u00fcrich  Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-9758-8861","authenticated-orcid":false,"given":"J. S. Montes","family":"Maestre","sequence":"additional","affiliation":[{"name":"ETH Z\u00fcrich  Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6604-4784","authenticated-orcid":false,"given":"S.","family":"Coros","sequence":"additional","affiliation":[{"name":"ETH Z\u00fcrich  Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8086-7664","authenticated-orcid":false,"given":"B.","family":"Thomaszewski","sequence":"additional","affiliation":[{"name":"ETH Z\u00fcrich  Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,8,19]]},"reference":[{"issue":"4","key":"e_1_2_7_2_2","doi-asserted-by":"crossref","DOI":"10.1145\/1778765.1778853","article-title":"Discrete viscous threads","volume":"29","author":"Bergou M.","year":"2010","journal-title":"ACM Trans. Graph."},{"issue":"3","key":"e_1_2_7_3_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/1531326.1531395","article-title":"Capture and modeling of non-linear heterogeneous soft tissue","volume":"28","author":"Bickel B.","year":"2009","journal-title":"ACM transactions on graphics (TOG)"},{"issue":"4","key":"e_1_2_7_4_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/1778765.1778800","article-title":"Design and fabrication of materials with desired deformation behavior","volume":"29","author":"Bickel B.","year":"2010","journal-title":"ACM Transactions on Graphics (TOG)"},{"key":"e_1_2_7_5_2","volume-title":"ACM SIGGRAPH 2008 Papers","author":"Bergou M.","year":"2008"},{"key":"e_1_2_7_6_2","unstructured":"FergusonZ. et al.:IPC Toolkit 2020. URL:https:\/\/github.com\/ipc-sim\/ipc-toolkit. 8"},{"key":"e_1_2_7_7_2","volume-title":"ACM SIGGRAPH 2024 Conference Papers","author":"Feng X.","year":"2024"},{"issue":"3","key":"e_1_2_7_8_2","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1111\/j.1467-8659.2006.00974.x","article-title":"Computing discrete shape operators on general meshes","volume":"25","author":"Grinspun E.","year":"2006","journal-title":"Computer Graphics Forum"},{"key":"e_1_2_7_9_2","first-page":"62","volume-title":"Proceedings of the 2003 ACM SIGGRAPH\/Eurographics Symposium on Computer Animation","author":"Grinspun E.","year":"2003"},{"key":"e_1_2_7_10_2","unstructured":"GuennebaudG. JacobB. et al.:Eigen v3.http:\/\/eigen.tuxfamily.org 2010. 8"},{"issue":"11","key":"e_1_2_7_11_2","doi-asserted-by":"crossref","first-page":"1309","DOI":"10.1002\/nme.2579","article-title":"Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities","volume":"79","author":"Geuzaine C.","year":"2009","journal-title":"International Journal for Numerical Methods in Engineering"},{"key":"e_1_2_7_12_2","first-page":"65","volume-title":"Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques","author":"James D. L.","year":"1999"},{"key":"e_1_2_7_13_2","unstructured":"JacobsonA. PanozzoD. et al.:libigl: A simple C++ geometry processing library 2018.https:\/\/libigl.github.io\/. 8"},{"issue":"3","key":"e_1_2_7_14_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/1531326.1531357","article-title":"Numerical coarsening of inhomogeneous elastic materials","volume":"28","author":"Kharevych L.","year":"2009","journal-title":"ACM Transactions on graphics (TOG)"},{"key":"e_1_2_7_15_2","first-page":"41","volume-title":"Proceedings of the ACM SIGGRAPH\/Eurographics Symposium on Computer Animation","author":"Li Y.","year":"2015"},{"issue":"6","key":"e_1_2_7_16_2","article-title":"Neural metamaterial networks for nonlinear material design","volume":"42","author":"Li Y.","year":"2023","journal-title":"ACM Trans. Graph."},{"issue":"2","key":"e_1_2_7_17_2","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1111\/j.1467-8659.2012.03031.x","article-title":"Data-driven estimation of cloth simulation models","volume":"31","author":"Miguel E.","year":"2012","journal-title":"Computer Graphics Forum"},{"issue":"4","key":"e_1_2_7_18_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2897824.2925922","article-title":"Procedural voronoi foams for additive manufacturing","volume":"35","author":"Mart\u00ednez J.","year":"2016","journal-title":"ACM Transactions on Graphics (TOG)"},{"issue":"30","key":"e_1_2_7_19_2","doi-asserted-by":"crossref","first-page":"2008082","DOI":"10.1002\/adma.202008082","article-title":"Curvature induced by deflection in thick meta-plates","volume":"33","author":"Mirzaali M. J.","year":"2021","journal-title":"Advanced Materials"},{"issue":"4","key":"e_1_2_7_20_2","doi-asserted-by":"crossref","DOI":"10.1145\/3592114","article-title":"Differentiable stripe patterns for inverse design of structured surfaces","volume":"42","author":"Montes Maestre J. S.","year":"2023","journal-title":"ACM Trans. Graph."},{"issue":"4","key":"e_1_2_7_21_2","doi-asserted-by":"crossref","DOI":"10.1145\/3658175","article-title":"Flexscale: Modeling and characterization of flexible scaled sheets","volume":"43","author":"Montes Maestre J. S.","year":"2024","journal-title":"ACM Trans. Graph."},{"issue":"4","key":"e_1_2_7_22_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3072959.3073638","article-title":"Orthotropic k-nearest foams for additive manufacturing","volume":"36","author":"Mart\u00ednez J.","year":"2017","journal-title":"ACM Transactions on Graphics (TOG)"},{"issue":"4","key":"e_1_2_7_23_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3306346.3322989","article-title":"Star-shaped metrics for mechanical metamaterial design","volume":"38","author":"Mart\u00ednez J.","year":"2019","journal-title":"ACM Transactions on Graphics (TOG)"},{"issue":"4","key":"e_1_2_7_24_2","doi-asserted-by":"crossref","DOI":"10.1145\/3072959.3073636","article-title":"A computational design tool for compliant mechanisms","volume":"36","author":"Megaro V.","year":"2017","journal-title":"ACM Trans. Graph."},{"issue":"4","key":"e_1_2_7_25_2","doi-asserted-by":"crossref","DOI":"10.1145\/3450626.3459789","article-title":"Computational inverse design of surface-based inflatables","volume":"40","author":"Panetta J.","year":"2021","journal-title":"ACM Trans. Graph."},{"issue":"4","key":"e_1_2_7_26_2","doi-asserted-by":"crossref","DOI":"10.1145\/3306346.3323040","article-title":"X-shells: a new class of deployable beam structures","volume":"38","author":"Panetta J.","year":"2019","journal-title":"ACM Trans. Graph."},{"issue":"4","key":"e_1_2_7_27_2","article-title":"The human touch: measuring contact with real human soft tissues","volume":"37","author":"Pai D. K.","year":"2018","journal-title":"ACM Trans. Graph."},{"issue":"4","key":"e_1_2_7_28_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2766937","article-title":"Elastic textures for additive fabrication","volume":"34","author":"Panetta J.","year":"2015","journal-title":"ACM Transactions on Graphics (TOG)"},{"issue":"4","key":"e_1_2_7_29_2","article-title":"Computational homogenization for inverse design of surface-based inflatables","volume":"43","author":"Ren Y.","year":"2024","journal-title":"ACM Trans. Graph."},{"issue":"4","key":"e_1_2_7_30_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2766926","article-title":"Microstructures to control elasticity in 3d printing","volume":"34","author":"Schumacher C.","year":"2015","journal-title":"ACM Transactions on Graphics (Tog)"},{"issue":"4","key":"e_1_2_7_31_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3197517.3201278","article-title":"Mechanical characterization of structured sheet materials","volume":"37","author":"Schumacher C.","year":"2018","journal-title":"ACM Transactions on Graphics (TOG)"},{"issue":"4","key":"e_1_2_7_32_2","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1145\/3386569.3392412","article-title":"Homogenized yarn-level cloth","volume":"39","author":"Sperl G.","year":"2020","journal-title":"ACM Trans. Graph."},{"issue":"4","key":"e_1_2_7_33_2","doi-asserted-by":"crossref","DOI":"10.1145\/3592112","article-title":"Beyond chainmail: Computational modeling of discrete interlocking materials","volume":"42","author":"Tang P.","year":"2023","journal-title":"ACM Trans. Graph."},{"key":"e_1_2_7_34_2","volume-title":"Proceedings of the Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Papers","author":"Tang P.","year":"2025"},{"issue":"4","key":"e_1_2_7_35_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2010324.1964966","article-title":"Data-driven elastic models for cloth: modeling and measurement","volume":"30","author":"Wang H.","year":"2011","journal-title":"ACM transactions on graphics (TOG)"},{"issue":"6","key":"e_1_2_7_36_2","doi-asserted-by":"crossref","DOI":"10.1145\/3618396","article-title":"Computational design of flexible planar microstructures","volume":"42","author":"Zhang Z.","year":"2023","journal-title":"ACM Trans. Graph."},{"issue":"4","key":"e_1_2_7_37_2","doi-asserted-by":"crossref","DOI":"10.1145\/2897824.2925888","article-title":"Designing structurally-sound ornamental curve networks","volume":"35","author":"Zehnder J.","year":"2016","journal-title":"ACM Trans. Graph."},{"issue":"6","key":"e_1_2_7_38_2","doi-asserted-by":"crossref","DOI":"10.1145\/3763296","article-title":"Closed-form construction of voronoi diagrams with star-shaped metrics","volume":"44","author":"Zhou H.","year":"2025","journal-title":"ACM Trans. Graph."}],"container-title":["Computer Graphics Forum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/cgf.70513","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1111\/cgf.70513","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/cgf.70513","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T15:42:27Z","timestamp":1787154147000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1111\/cgf.70513"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,19]]},"references-count":37,"alternative-id":["10.1111\/cgf.70513"],"URL":"https:\/\/doi.org\/10.1111\/cgf.70513","archive":["Portico"],"relation":{},"ISSN":["0167-7055","1467-8659"],"issn-type":[{"value":"0167-7055","type":"print"},{"value":"1467-8659","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,19]]},"assertion":[{"value":"2026-08-19","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e70513"}}