{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T00:57:58Z","timestamp":1772499478691,"version":"3.50.1"},"reference-count":19,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["OIA-1946231"],"award-info":[{"award-number":["OIA-1946231"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The Miura ori is a very classical origami pattern used in numerous applications in engineering.\nA study of the shapes that surfaces using this pattern can assume is still lacking.\nA constrained nonlinear partial differential equation (PDE) that models the possible shapes that a periodic Miura tessellation can take in the homogenization limit has been established recently and solved only in specific cases.\nIn this paper, the existence and uniqueness of a solution to the unconstrained PDE is proved for general Dirichlet boundary conditions.\nThen an <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>H<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0259_ineq_0001.png\"\/>\n                        <jats:tex-math>H^{2}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-conforming discretization is introduced to approximate the solution of the PDE coupled to a Newton method to solve the associated discrete problem.\nA convergence proof for the method is given as well as a convergence rate.\nFinally, numerical experiments show the robustness of the method and that nontrivial shapes can be achieved using periodic Miura tessellations.<\/jats:p>","DOI":"10.1515\/cmam-2022-0259","type":"journal-article","created":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T16:41:16Z","timestamp":1685464876000},"page":"85-100","source":"Crossref","is-referenced-by-count":4,"title":["\ud835\udc3b<sup>2<\/sup>-Conformal Approximation of Miura Surfaces"],"prefix":"10.1515","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4022-1257","authenticated-orcid":false,"given":"Fr\u00e9d\u00e9ric","family":"Marazzato","sequence":"first","affiliation":[{"name":"Department of Mathematics , Louisiana State University , Baton Rouge , LA 70803 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2023,5,31]]},"reference":[{"key":"2024010712054096737_j_cmam-2022-0259_ref_001","unstructured":"R. A. Adams and J. J. F. Fournier,\nSobolev Spaces,\nElsevier\/Academic, Amsterdam, 2003."},{"key":"2024010712054096737_j_cmam-2022-0259_ref_002","unstructured":"H. A. Akitaya, E. D. Demaine, T. Horiyama, T. C. Hull, J. S. Ku and T. Tachi,\nRigid foldability is NP-hard,\nJ. Comput. Geom. 11 (2020), no. 1, 93\u2013124."},{"key":"2024010712054096737_j_cmam-2022-0259_ref_003","doi-asserted-by":"crossref","unstructured":"K. Bell,\nA refined triangular plate bending finite element,\nInternat. J. Numer. Methods Engrg. 1 (1969), no. 1, 101\u2013122.","DOI":"10.1002\/nme.1620010108"},{"key":"2024010712054096737_j_cmam-2022-0259_ref_004","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2024010712054096737_j_cmam-2022-0259_ref_005","doi-asserted-by":"crossref","unstructured":"H. Brezis,\nFunctional Analysis, Sobolev Spaces and Partial Differential Equations,\nUniversitext,\nSpringer, New York, 2011.","DOI":"10.1007\/978-0-387-70914-7"},{"key":"2024010712054096737_j_cmam-2022-0259_ref_006","unstructured":"A. Ern and J.-L. Guermond,\nTheory and Practice of Finite Elements,\nAppl. Math. Sci. 159,\nSpringer, New York, 2013."},{"key":"2024010712054096737_j_cmam-2022-0259_ref_007","unstructured":"D. Gilbarg and N. S. Trudinger,\nElliptic Partial Differential Equations of Second Order,\nGrundlehren Math. Wiss. 224,\nSpringer, Berlin, 2015."},{"key":"2024010712054096737_j_cmam-2022-0259_ref_008","doi-asserted-by":"crossref","unstructured":"P. Grisvard,\nElliptic Problems in Nonsmooth Domains,\nClass. Appl. Math. 69,\nSociety for Industrial and Applied Mathematics, Philadelphia, 2011.","DOI":"10.1137\/1.9781611972030"},{"key":"2024010712054096737_j_cmam-2022-0259_ref_009","unstructured":"E. Hernandez, D. Hartl and D. Lagoudas,\nActive Origami,\nSpringer, Cham, 2019."},{"key":"2024010712054096737_j_cmam-2022-0259_ref_010","unstructured":"R. J. Lang,\nOrigami\n                  \n                     \n                        \n                           \n                              \n                              4\n                           \n                        \n                        \n                        {}^{4}\n                     \n                  \n               ,\nCRC Press, Boca Raton, 2009."},{"key":"2024010712054096737_j_cmam-2022-0259_ref_011","unstructured":"A. Leb\u00e9e, L. Monasse and H. Nassar,\nFitting surfaces with the Miura tessellation,\n7th International Meeting on Origami in Science, Mathematics and Education (7OSME). Volume 4,\nTarquin, Hertfordshire (2018), 811\u2013811."},{"key":"2024010712054096737_j_cmam-2022-0259_ref_012","doi-asserted-by":"crossref","unstructured":"S. Liu, G. Lu, Y. Chen and Y. W. Leong,\nDeformation of the Miura-ori patterned sheet,\nInt. J. Mech. Sci. 99 (2015), 130\u2013142.","DOI":"10.1016\/j.ijmecsci.2015.05.009"},{"key":"2024010712054096737_j_cmam-2022-0259_ref_013","unstructured":"K. Miura,\nProposition of pseudo-cylindrical concave polyhedral shells,\nISAS Report 34 (1969), no. 9, 141\u2013163."},{"key":"2024010712054096737_j_cmam-2022-0259_ref_014","doi-asserted-by":"crossref","unstructured":"H. Nassar, A. Leb\u00e9e and L. Monasse,\nCurvature, metric and parametrization of origami tessellations: Theory and application to the eggbox pattern,\nProc. A. 473 (2017), no. 2197, Article ID 20160705.","DOI":"10.1098\/rspa.2016.0705"},{"key":"2024010712054096737_j_cmam-2022-0259_ref_015","doi-asserted-by":"crossref","unstructured":"F. Rathgeber, D. A. Ham, L. Mitchell, M. Lange, F. Luporini, A. T. T. McRae, G.-T. Bercea, G. R. Markall and P. H. J. Kelly,\nFiredrake: Automating the finite element method by composing abstractions,\nACM Trans. Math. 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Lett. 110 (2013), no. 21, Article ID 215501.","DOI":"10.1103\/PhysRevLett.110.215501"},{"key":"2024010712054096737_j_cmam-2022-0259_ref_019","doi-asserted-by":"crossref","unstructured":"A. L. Wickeler and H. E. Naguib,\nNovel origami-inspired metamaterials: Design, mechanical testing and finite element modelling,\nMater. & Design 186 (2020), Article ID 108242.","DOI":"10.1016\/j.matdes.2019.108242"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0259\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0259\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,7]],"date-time":"2024-01-07T12:07:24Z","timestamp":1704629244000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0259\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,31]]},"references-count":19,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,5,23]]},"published-print":{"date-parts":[[2024,1,1]]}},"alternative-id":["10.1515\/cmam-2022-0259"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0259","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,31]]}}}