{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T00:31:59Z","timestamp":1774571519037,"version":"3.50.1"},"reference-count":55,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2019,3,28]],"date-time":"2019-03-28T00:00:00Z","timestamp":1553731200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["NSF CAREER, 1652515, IIS-1320635, DMS-1436591, 1835712"],"award-info":[{"award-number":["NSF CAREER, 1652515, IIS-1320635, DMS-1436591, 1835712"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001711","name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","doi-asserted-by":"publisher","award":["P2TIP2_175859"],"award-info":[{"award-number":["P2TIP2_175859"]}],"id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2019,6,30]]},"abstract":"<jats:p>We introduce an integrated meshing and finite-element method pipeline enabling solution of partial differential equations in the volume enclosed by a boundary representation. We construct a hybrid hexahedral-dominant mesh, which contains a small number of star-shaped polyhedra, and build a set of high-order bases on its elements, combining triquadratic B-splines, triquadratic hexahedra, and harmonic elements. We demonstrate that our approach converges cubically under refinement, while requiring around 50% of the degrees of freedom than a similarly dense hexahedral mesh composed of triquadratic hexahedra. We validate our approach solving Poisson\u2019s equation on a large collection of models, which are automatically processed by our algorithm, only requiring the user to provide boundary conditions on their surface.<\/jats:p>","DOI":"10.1145\/3313797","type":"journal-article","created":{"date-parts":[[2019,3,29]],"date-time":"2019-03-29T12:45:01Z","timestamp":1553863501000},"page":"1-16","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":32,"title":["Poly-Spline Finite-Element Method"],"prefix":"10.1145","volume":"38","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5969-636X","authenticated-orcid":false,"given":"Teseo","family":"Schneider","sequence":"first","affiliation":[{"name":"New York University, New York, NY"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7304-9882","authenticated-orcid":false,"given":"J\u00e9r\u00e9mie","family":"Dumas","sequence":"additional","affiliation":[{"name":"New York University, nTopology, New York, NY"}]},{"given":"Xifeng","family":"Gao","sequence":"additional","affiliation":[{"name":"New York University, Florida State University, Tallahassee, FL"}]},{"given":"Mario","family":"Botsch","sequence":"additional","affiliation":[{"name":"Bielefeld University, Bielefeld, Germany"}]},{"given":"Daniele","family":"Panozzo","sequence":"additional","affiliation":[{"name":"New York University, New York, NY"}]},{"given":"Denis","family":"Zorin","sequence":"additional","affiliation":[{"name":"New York University, New York, NY"}]}],"member":"320","published-online":{"date-parts":[[2019,3,28]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"crossref","unstructured":"Martin Aigner Christoph Heinrich Bert J\u00fcttler Elisabeth Pilgerstorfer Bernd Simeon and Vuong. 2009. Swept Volume Parameterization for Isogeometric Analysis.  Martin Aigner Christoph Heinrich Bert J\u00fcttler Elisabeth Pilgerstorfer Bernd Simeon and Vuong. 2009. Swept Volume Parameterization for Isogeometric Analysis.","DOI":"10.1007\/978-3-642-03596-8_2"},{"key":"e_1_2_2_2_1","doi-asserted-by":"crossref","unstructured":"Douglas Arnold Daniele Boffi and Richard Falk. 2002. Approximation by quadrilateral finite elements. Math. Comput. (2002).  Douglas Arnold Daniele Boffi and Richard Falk. 2002. Approximation by quadrilateral finite elements. Math. Comput. (2002).","DOI":"10.1090\/S0025-5718-02-01439-4"},{"key":"e_1_2_2_3_1","volume-title":"Thomas J. R. Hughes, and Giancarlo Sangalli.","author":"Bazilevs Yuri","year":"2006","unstructured":"Yuri Bazilevs , L. Beirao da Veiga , J Austin Cottrell , Thomas J. R. Hughes, and Giancarlo Sangalli. 2006 . Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes. Math. Meth. Appl. Sci . (2006). Yuri Bazilevs, L. Beirao da Veiga, J Austin Cottrell, Thomas J. R. Hughes, and Giancarlo Sangalli. 2006. Isogeometric analysis: Approximation, stability and error estimates for h-refined meshes. Math. Meth. Appl. Sci. (2006)."},{"key":"e_1_2_2_4_1","doi-asserted-by":"crossref","unstructured":"L. Beir\u00e3o Da Veiga F. Brezzi A. Cangiani G. Manzini L. D. Marini and A. Russo. 2013. Basic principles of virtual element methods. Math. Meth. Appl. Sci. (2013).  L. Beir\u00e3o Da Veiga F. Brezzi A. Cangiani G. Manzini L. D. Marini and A. Russo. 2013. Basic principles of virtual element methods. Math. Meth. Appl. Sci. (2013).","DOI":"10.1142\/S0218202512500492"},{"key":"e_1_2_2_5_1","volume-title":"Proceedings of the 4th International Meshing Roundtable.","author":"Benzley Steven E.","year":"1995","unstructured":"Steven E. Benzley , Ernest Perry , Karl Merkley , Brett Clark , and Greg Sjaardema . 1995 . A comparison of all hexagonal and all tetrahedral finite element meshes for elastic and elasto-plastic analysis . In Proceedings of the 4th International Meshing Roundtable. Steven E. Benzley, Ernest Perry, Karl Merkley, Brett Clark, and Greg Sjaardema. 1995. A comparison of all hexagonal and all tetrahedral finite element meshes for elastic and elasto-plastic analysis. In Proceedings of the 4th International Meshing Roundtable."},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.4562"},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511618635"},{"key":"e_1_2_2_8_1","unstructured":"R. T. Q. Chen Y. Rubanova J. Bettencourt and D. Duvenaud. 2018. Neural ordinary differential equations. Advances in Neural Information Processing Systems 31 S. Bengio H. Wallach H. Larochelle K. Grauman N. Cesa-Bianchi and R. Garnett (Eds.). Curran Associates Inc. 6571\u20136583. http:\/\/papers.nips.cc\/paper\/7892-neural-ordinary-differential-equations.pdf.  R. T. Q. Chen Y. Rubanova J. Bettencourt and D. Duvenaud. 2018. Neural ordinary differential equations. Advances in Neural Information Processing Systems 31 S. Bengio H. Wallach H. Larochelle K. Grauman N. Cesa-Bianchi and R. Garnett (Eds.). Curran Associates Inc. 6571\u20136583. http:\/\/papers.nips.cc\/paper\/7892-neural-ordinary-differential-equations.pdf."},{"key":"e_1_2_2_9_1","doi-asserted-by":"crossref","unstructured":"A. O. Cifuentes and A. Kalbag. 1992. A performance study of tetrahedral and hexahedral elements in 3-D finite element structural analysis. Finite Elements in Analysis and Design (1992).   A. O. Cifuentes and A. Kalbag. 1992. A performance study of tetrahedral and hexahedral elements in 3-D finite element structural analysis. Finite Elements in Analysis and Design (1992).","DOI":"10.1016\/0168-874X(92)90040-J"},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1002\/9780470749081"},{"key":"e_1_2_2_11_1","doi-asserted-by":"crossref","unstructured":"L. Beir\u00e3o da Veiga A. Buffa D. Cho and G. Sangalli. 2011. Isogeometric analysis using T-splines on two-patch geometries. Comput. Meth. Appl. Mech. Eng. (2011).  L. Beir\u00e3o da Veiga A. Buffa D. Cho and G. Sangalli. 2011. Isogeometric analysis using T-splines on two-patch geometries. Comput. Meth. Appl. Mech. Eng. (2011).","DOI":"10.1016\/j.cma.2011.02.005"},{"key":"e_1_2_2_12_1","volume-title":"The nonconforming virtual element method","author":"de Dios Blanca Ayuso","year":"2016","unstructured":"Blanca Ayuso de Dios , Konstantin Lipnikov , and Gianmarco Manzini . 2016. The nonconforming virtual element method . ESAIM : Mathematical Modelling and Numerical Analysis ( 2016 ). Blanca Ayuso de Dios, Konstantin Lipnikov, and Gianmarco Manzini. 2016. The nonconforming virtual element method. ESAIM: Mathematical Modelling and Numerical Analysis (2016)."},{"key":"e_1_2_2_13_1","volume-title":"Evans","author":"Engvall Luke","year":"2017","unstructured":"Luke Engvall and John A . Evans . 2017 . Isogeometric unstructured tetrahedral and mixed-element Bernstein-Bezier discretizations. Comput. Meth. Appl. Mech. Eng . (2017). Luke Engvall and John A. Evans. 2017. Isogeometric unstructured tetrahedral and mixed-element Bernstein-Bezier discretizations. Comput. Meth. Appl. Mech. Eng. (2017)."},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1145\/2897824.2925957"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.5555\/1122905.1648459"},{"key":"e_1_2_2_16_1","doi-asserted-by":"crossref","unstructured":"Richard Franke. 1979. A Critical Comparison of Some Methods for Interpolation of Scattered Data.  Richard Franke. 1979. A Critical Comparison of Some Methods for Interpolation of Scattered Data.","DOI":"10.21236\/ADA081688"},{"key":"e_1_2_2_17_1","volume-title":"Efficient volumetric PolyCube-map construction. Comput. Graph. Forum","author":"Fu Xiaoming","year":"2016","unstructured":"Xiaoming Fu , Chongyang Bai , and Yang Liu . 2016. Efficient volumetric PolyCube-map construction. Comput. Graph. Forum ( 2016 ). Xiaoming Fu, Chongyang Bai, and Yang Liu. 2016. Efficient volumetric PolyCube-map construction. Comput. Graph. Forum (2016)."},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1145\/3072959.3073676"},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/3130800.3130848"},{"key":"e_1_2_2_20_1","volume-title":"All-hex mesh generation via volumetric PolyCube deformation. Comput. Graph. Forum","author":"Gregson James","year":"2011","unstructured":"James Gregson , Alla Sheffer , and Eugene Zhang . 2011. All-hex mesh generation via volumetric PolyCube deformation. Comput. Graph. Forum ( 2011 ). James Gregson, Alla Sheffer, and Eugene Zhang. 2011. All-hex mesh generation via volumetric PolyCube deformation. Comput. Graph. Forum (2011)."},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","DOI":"10.5555\/1731309.1731339"},{"key":"e_1_2_2_22_1","doi-asserted-by":"publisher","DOI":"10.1145\/2602141"},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/2070781.2024177"},{"key":"e_1_2_2_24_1","volume-title":"NURBS, exact geometry and mesh refinement. Comput. Meth. Appl. Mech. Eng.","author":"Hughes Thomas J. R.","year":"2005","unstructured":"Thomas J. R. Hughes , John A. Cottrell , and Yuri Bazilevs . 2005. Isogeometric analysis: CAD, finite elements , NURBS, exact geometry and mesh refinement. Comput. Meth. Appl. Mech. Eng. ( 2005 ). Thomas J. R. Hughes, John A. Cottrell, and Yuri Bazilevs. 2005. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Meth. Appl. Mech. Eng. (2005)."},{"key":"e_1_2_2_25_1","unstructured":"Thomas J. R. Hughes. 2000. The Finite Element Method. Linear Static and Dynamic Finite Element Analysis.  Thomas J. R. Hughes. 2000. The Finite Element Method. Linear Static and Dynamic Finite Element Analysis."},{"key":"e_1_2_2_26_1","volume-title":"Soni","author":"Ito Yasushi","year":"2009","unstructured":"Yasushi Ito , Alan M. Shih , and Bharat K . Soni . 2009 . Octree-based reasonable-quality hexahedral mesh generation using a new set of refinement templates. Int. J. Numer. Methods Eng . (2009). Yasushi Ito, Alan M. Shih, and Bharat K. Soni. 2009. Octree-based reasonable-quality hexahedral mesh generation using a new set of refinement templates. Int. J. Numer. Methods Eng. (2009)."},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2013.250"},{"key":"e_1_2_2_28_1","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276466"},{"key":"e_1_2_2_29_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073229"},{"key":"e_1_2_2_30_1","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2018.00269"},{"key":"e_1_2_2_31_1","doi-asserted-by":"crossref","unstructured":"Na Lei Xiaopeng Zheng Jian Jiang Yu-Yao Lin and David Xianfeng Gu. 2017. Quadrilateral and hexahedral mesh generation based on surface foliation theory. Comput. Meth. Appl. Mech. Eng. (2017).  Na Lei Xiaopeng Zheng Jian Jiang Yu-Yao Lin and David Xianfeng Gu. 2017. Quadrilateral and hexahedral mesh generation based on surface foliation theory. Comput. Meth. Appl. Mech. Eng. (2017).","DOI":"10.1016\/j.cma.2016.09.044"},{"key":"e_1_2_2_32_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2012.177"},{"key":"e_1_2_2_33_1","doi-asserted-by":"publisher","DOI":"10.1145\/2366145.2366196"},{"key":"e_1_2_2_34_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2013.07.031"},{"key":"e_1_2_2_35_1","doi-asserted-by":"publisher","DOI":"10.1145\/2508363.2508388"},{"key":"e_1_2_2_36_1","doi-asserted-by":"crossref","unstructured":"Gianmarco Manzini Alessandro Russo and N. Sukumar. 2014. New perspectives on polygonal and polyhedral finite element methods. Math. Meth. Appl. Sci. (2014).  Gianmarco Manzini Alessandro Russo and N. Sukumar. 2014. New perspectives on polygonal and polyhedral finite element methods. Math. Meth. Appl. Sci. (2014).","DOI":"10.1142\/S0218202514400065"},{"key":"e_1_2_2_37_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-04319-2_5"},{"key":"e_1_2_2_39_1","doi-asserted-by":"publisher","DOI":"10.5555\/1731309.1731340"},{"key":"e_1_2_2_40_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cag.2010.03.011"},{"key":"e_1_2_2_41_1","volume-title":"CubeCover - Parameterization of 3D volumes. Comput. Graph. Forum","author":"Nieser Matthias","year":"2011","unstructured":"Matthias Nieser , Ulrich Reitebuch , and Konrad Polthier . 2011. CubeCover - Parameterization of 3D volumes. Comput. Graph. Forum ( 2011 ). Matthias Nieser, Ulrich Reitebuch, and Konrad Polthier. 2011. CubeCover - Parameterization of 3D volumes. Comput. Graph. Forum (2011)."},{"key":"e_1_2_2_42_1","volume-title":"Owen and Sunil Saigal","author":"Steven","year":"2000","unstructured":"Steven J. Owen and Sunil Saigal . 2000 . H-Morph: An indirect approach to advancing front hex meshing. Int. J. Numer. Methods Eng . (2000). Steven J. Owen and Sunil Saigal. 2000. H-Morph: An indirect approach to advancing front hex meshing. Int. J. Numer. Methods Eng. (2000)."},{"key":"e_1_2_2_43_1","doi-asserted-by":"publisher","DOI":"10.1145\/2766937"},{"key":"e_1_2_2_44_1","doi-asserted-by":"publisher","DOI":"10.1109\/MCSE.2014.53"},{"key":"e_1_2_2_45_1","doi-asserted-by":"publisher","DOI":"10.1137\/130908737"},{"key":"e_1_2_2_46_1","doi-asserted-by":"publisher","DOI":"10.1145\/1015706.1015715"},{"key":"e_1_2_2_47_1","volume-title":"Johnson","author":"Shepherd Jason F.","year":"2008","unstructured":"Jason F. Shepherd and Chris R . Johnson . 2008 . Hexahedral mesh generation constraints. Eng. Comput . (2008). Jason F. Shepherd and Chris R. Johnson. 2008. Hexahedral mesh generation constraints. Eng. Comput. (2008)."},{"key":"e_1_2_2_48_1","volume-title":"Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator.","author":"Shewchuk Jonathan Richard","year":"1996","unstructured":"Jonathan Richard Shewchuk . 1996 . Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator. Jonathan Richard Shewchuk. 1996. Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator."},{"key":"e_1_2_2_49_1","doi-asserted-by":"publisher","DOI":"10.1145\/2629697"},{"key":"e_1_2_2_50_1","doi-asserted-by":"publisher","DOI":"10.1145\/2930662"},{"key":"e_1_2_2_51_1","volume-title":"Blacker","author":"Staten Matthew L.","year":"2005","unstructured":"Matthew L. Staten , Steven J. Owen , and Ted D . Blacker . 2005 . Unconstrained Paving 8 Plastering : A New Idea for All Hexahedral Mesh Generation . Matthew L. Staten, Steven J. Owen, and Ted D. Blacker. 2005. Unconstrained Paving 8 Plastering: A New Idea for All Hexahedral Mesh Generation."},{"key":"e_1_2_2_52_1","doi-asserted-by":"publisher","DOI":"10.1145\/1015706.1015810"},{"key":"e_1_2_2_53_1","volume-title":"Hughes","author":"Toshniwal Deepesh","year":"2017","unstructured":"Deepesh Toshniwal , Hendrik Speleers , Ren\u00e9 R. Hiemstra , and Thomas J. R . Hughes . 2017 . Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis. Comput. Meth. Appl. Mech. Eng . (2017). Deepesh Toshniwal, Hendrik Speleers, Ren\u00e9 R. Hiemstra, and Thomas J. R. Hughes. 2017. Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis. Comput. Meth. Appl. Mech. Eng. (2017)."},{"key":"e_1_2_2_54_1","volume-title":"Deepesh Toshniwal, Hendrik Speleers, Xin Li, Carla Manni, John A. Evans, and Thomas J. R. Hughes.","author":"Wei Xiaodong","year":"2018","unstructured":"Xiaodong Wei , Yongjie Jessica Zhang , Deepesh Toshniwal, Hendrik Speleers, Xin Li, Carla Manni, John A. Evans, and Thomas J. R. Hughes. 2018 . Blended B-spline construction on unstructured quadrilateral and hexahedral meshes with optimal convergence rates in isogeometric analysis. Comput. Meth. Appl. Mech. Eng . (2018). Xiaodong Wei, Yongjie Jessica Zhang, Deepesh Toshniwal, Hendrik Speleers, Xin Li, Carla Manni, John A. Evans, and Thomas J. R. Hughes. 2018. Blended B-spline construction on unstructured quadrilateral and hexahedral meshes with optimal convergence rates in isogeometric analysis. Comput. Meth. Appl. Mech. Eng. (2018)."},{"key":"e_1_2_2_55_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.754"},{"key":"e_1_2_2_56_1","doi-asserted-by":"crossref","unstructured":"Y. J. Zhang X. Liang and Guoliang Xu. 2013. A robust 2-refinement algorithm in octree or rhombic dodecahedral tree based all-hexahedral mesh generation. Comput. Meth. Appl. Mech. Eng. (2013).  Y. J. Zhang X. Liang and Guoliang Xu. 2013. A robust 2-refinement algorithm in octree or rhombic dodecahedral tree based all-hexahedral mesh generation. Comput. Meth. Appl. Mech. Eng. (2013).","DOI":"10.1007\/978-3-642-33573-0_10"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3313797","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3313797","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3313797","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T23:54:33Z","timestamp":1750204473000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3313797"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,28]]},"references-count":55,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,6,30]]}},"alternative-id":["10.1145\/3313797"],"URL":"https:\/\/doi.org\/10.1145\/3313797","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,28]]},"assertion":[{"value":"2018-06-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2019-02-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2019-03-28","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}