{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T14:19:55Z","timestamp":1785507595189,"version":"3.56.0"},"reference-count":35,"publisher":"Association for Computing Machinery (ACM)","issue":"5","license":[{"start":{"date-parts":[[2008,12,1]],"date-time":"2008-12-01T00:00:00Z","timestamp":1228089600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["NIBIB\/NIH R01EB006615"],"award-info":[{"award-number":["NIBIB\/NIH R01EB006615"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CAREER-0430528GEMT-CompBio-0621999"],"award-info":[{"award-number":["CAREER-0430528GEMT-CompBio-0621999"]}],"id":[{"id":"10.13039\/100000001","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":[[2008,12]]},"abstract":"<jats:p>We propose an efficient scheme for evaluating nonlinear subspace forces (and Jacobians) associated with subspace deformations. The core problem we address is efficient integration of the subspace force density over the 3D spatial domain. Similar to Gaussian quadrature schemes that efficiently integrate functions that lie in particular polynomial subspaces, we propose cubature schemes (multi-dimensional quadrature) optimized for efficient integration of force densities associated with particular subspace deformations, particular materials, and particular geometric domains. We support generic subspace deformation kinematics, and nonlinear hyperelastic materials. For an<jats:italic>r<\/jats:italic>-dimensional deformation subspace with<jats:italic>O<\/jats:italic>(<jats:italic>r<\/jats:italic>) cubature points, our method is able to evaluate sub-space forces at<jats:italic>O<\/jats:italic>(<jats:italic>r<\/jats:italic><jats:sup>2<\/jats:sup>) cost. We also describe composite cubature rules for runtime error estimation. Results are provided for various subspace deformation models, several hyperelastic materials (St.Venant-Kirchhoff, Mooney-Rivlin, Arruda-Boyce), and multi-modal (graphics, haptics, sound) applications. We show dramatically better efficiency than traditional Monte Carlo integration.<\/jats:p>","DOI":"10.1145\/1409060.1409118","type":"journal-article","created":{"date-parts":[[2008,12,3]],"date-time":"2008-12-03T21:56:04Z","timestamp":1228341364000},"page":"1-10","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":157,"title":["Optimizing cubature for efficient integration of subspace deformations"],"prefix":"10.1145","volume":"27","author":[{"given":"Steven S.","family":"An","sequence":"first","affiliation":[{"name":"Cornell University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Theodore","family":"Kim","sequence":"additional","affiliation":[{"name":"Cornell University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Doug L.","family":"James","sequence":"additional","affiliation":[{"name":"Cornell University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2008,12]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"crossref","unstructured":"Baraff D. and Witkin A. 1992. Dynamic simulation of non-penetrating flexible bodies. In Computer Graphics (Proceedings of SIGGRAPH 92) 303--308. 10.1145\/133994.134084 Baraff D. and Witkin A. 1992. Dynamic simulation of non-penetrating flexible bodies. In Computer Graphics (Proceedings of SIGGRAPH 92) 303--308. 10.1145\/133994.134084","DOI":"10.1145\/142920.134084"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073300"},{"key":"e_1_2_2_3_1","volume-title":"Proceedings of ACM SIGGRAPH Symposium on Computer Animation (SCA","author":"Barbi\u010d J.","year":"2007"},{"key":"e_1_2_2_4_1","unstructured":"Barbi\u010d J. 2007. Real-time Reduced Large-Deformation Models and Distributed Contact for Computer Graphics and Haptics. PhD thesis Carnegie Mellon University. Barbi\u010d J. 2007. Real-time Reduced Large-Deformation Models and Distributed Contact for Computer Graphics and Haptics. PhD thesis Carnegie Mellon University."},{"key":"e_1_2_2_5_1","unstructured":"Bathe K.-J. 1996. Finite Element Procedures second ed. Prentice Hall. Bathe K.-J. 1996. Finite Element Procedures second ed. Prentice Hall."},{"key":"e_1_2_2_6_1","doi-asserted-by":"crossref","unstructured":"Bonet J. and Wood R. D. 2008. Nonlinear Continuum Mechanics for Finite Element Analysis second ed. Cambridge University Press New York. Bonet J. and Wood R. D. 2008. Nonlinear Continuum Mechanics for Finite Element Analysis second ed. Cambridge University Press New York.","DOI":"10.1017\/CBO9780511755446"},{"key":"e_1_2_2_7_1","volume-title":"ACM SIGGRAPH Symposium on Computer Animation, 41--48","author":"Capell S."},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2005.13"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.1109\/2945.764872"},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/383259.383262"},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1145\/566654.566578"},{"key":"e_1_2_2_12_1","volume-title":"Proc. of SIGGRAPH 98, Computer Graphics Proceedings, Annual Conference Series, 9--20","author":"Grzeszczuk R."},{"key":"e_1_2_2_13_1","volume-title":"Scientific Computing: An Introductory Survey","author":"Heath M. T.","year":"2002"},{"key":"e_1_2_2_14_1","unstructured":"Hildebrand F. B. 1956. Introduction to Numerical Analysis. McGraw-Hill New York. Hildebrand F. B. 1956. Introduction to Numerical Analysis. McGraw-Hill New York."},{"key":"e_1_2_2_15_1","unstructured":"Hoberock J. and Jia Y. 2008. Chapter 12: High-Quality Ambient Occlusion. In GPU Gems 3 H. Nguyen Ed. Addison-Wesley 257--274. Hoberock J. and Jia Y. 2008. Chapter 12: High-Quality Ambient Occlusion. In GPU Gems 3 H. Nguyen Ed. Addison-Wesley 257--274."},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/882262.882359"},{"key":"e_1_2_2_17_1","volume-title":"Proceedings of SIGGRAPH 99, Computer Graphics Proceedings, Annual Conference Series, 65--72","author":"James D. L."},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1145\/566654.566621"},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073206"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141911.1141983"},{"key":"e_1_2_2_21_1","volume-title":"Flexible Simulation of Deformable Models Using Discontinuous Galerkin FEM. In ACM SIGGRAPH\/Eurographics Symposium on Computer Animation, 105--115","author":"Kaufmann P."},{"key":"e_1_2_2_22_1","volume-title":"EigenSkin: Real Time Large Deformation Character Skinning in Hardware. In ACM SIGGRAPH Symposium on Computer Animation, 153--160","author":"Kry P. G."},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1002\/nme.167"},{"key":"e_1_2_2_24_1","unstructured":"Lawson C. L. and Hanson R. J. 1974. Solving Least Square Problems. Prentice Hall Englewood Cliffs NJ. Lawson C. L. and Hanson R. J. 1974. Solving Least Square Problems. Prentice Hall Englewood Cliffs NJ."},{"key":"e_1_2_2_25_1","volume-title":"Proceedings of ACM SIGGRAPH 2000, Computer Graphics Proceedings, Annual Conference Series, 165--172","author":"Lewis J. P."},{"key":"e_1_2_2_26_1","volume-title":"Proc. Intl. Symp. Medical Simulation (ISMS","volume":"3078","author":"Liu Y.","year":"2004"},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276469"},{"key":"e_1_2_2_28_1","unstructured":"Nelles O. 2000. Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models. Springer Verlag December. Nelles O. 2000. Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models. Springer Verlag December."},{"key":"e_1_2_2_29_1","volume-title":"Proceedings of ACM SIGGRAPH 2001, Computer Graphics Proceedings, Annual Conference Series, 529--536","author":"O'Brien J. F."},{"key":"e_1_2_2_30_1","doi-asserted-by":"crossref","unstructured":"Pentland A. and Williams J. 1989. Good vibrations: Modal dynamics for graphics and animation. In Computer Graphics (Proceedings of SIGGRAPH 89) 215--222. 10.1145\/74333.74355 Pentland A. and Williams J. 1989. Good vibrations: Modal dynamics for graphics and animation. In Computer Graphics (Proceedings of SIGGRAPH 89) 215--222. 10.1145\/74333.74355","DOI":"10.1145\/74333.74355"},{"key":"e_1_2_2_31_1","volume-title":"C: The Art of Scientific Computing","author":"Press W. H.","year":"1992"},{"key":"e_1_2_2_32_1","doi-asserted-by":"crossref","unstructured":"Terzopoulos D. and Witkin A. 1988. Physically Based Models with Rigid and Deformable Components. IEEE Computer Graphics &amp; Applications 8 6 (Nov.) 41--51. 10.1109\/38.20317 Terzopoulos D. and Witkin A. 1988. Physically Based Models with Rigid and Deformable Components. IEEE Computer Graphics &amp; Applications 8 6 (Nov.) 41--51. 10.1109\/38.20317","DOI":"10.1109\/38.20317"},{"key":"e_1_2_2_33_1","doi-asserted-by":"crossref","unstructured":"Terzopoulos D. Platt J. Barr A. and Fleischer K. 1987. Elastically Deformable Models. In Computer Graphics (Proceedings of SIGGRAPH 87) 205--214. 10.1145\/37401.37427 Terzopoulos D. Platt J. Barr A. and Fleischer K. 1987. Elastically Deformable Models. In Computer Graphics (Proceedings of SIGGRAPH 87) 205--214. 10.1145\/37401.37427","DOI":"10.1145\/37402.37427"},{"key":"e_1_2_2_34_1","doi-asserted-by":"crossref","unstructured":"Vakakis A. 2001. Normal modes and localization in nonlinear systems. Kluwer Academic Publishers. Vakakis A. 2001. Normal modes and localization in nonlinear systems. Kluwer Academic Publishers.","DOI":"10.1007\/978-94-017-2452-4"},{"key":"e_1_2_2_35_1","doi-asserted-by":"crossref","unstructured":"Witkin A. and Welch W. 1990. Fast Animation and Control of Nonrigid Structures. In Computer Graphics (Proceedings of SIGGRAPH 90) 243--252. 10.1145\/97879.565650 Witkin A. and Welch W. 1990. Fast Animation and Control of Nonrigid Structures. In Computer Graphics (Proceedings of SIGGRAPH 90) 243--252. 10.1145\/97879.565650","DOI":"10.1145\/97879.565650"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1409060.1409118","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1409060.1409118","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T14:47:21Z","timestamp":1750258041000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1409060.1409118"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,12]]},"references-count":35,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2008,12]]}},"alternative-id":["10.1145\/1409060.1409118"],"URL":"https:\/\/doi.org\/10.1145\/1409060.1409118","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,12]]},"assertion":[{"value":"2008-12-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}