{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T16:22:40Z","timestamp":1782318160215,"version":"3.54.5"},"reference-count":49,"publisher":"Wiley","license":[{"start":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T00:00:00Z","timestamp":1776124800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T00:00:00Z","timestamp":1776124800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["281466253"],"award-info":[{"award-number":["281466253"]}],"id":[{"id":"10.13039\/501100001659","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>\n                    Newton's Method is widely used to find the solution of complex non\u2010linear simulation problems. To guarantee a descent direction, it is common practice to clamp the negative eigenvalues of each element Hessian prior to assembly\u2014a strategy known as\n                    <jats:italic>Projected Newton<\/jats:italic>\n                    (PN)\u2014but this perturbation often hinders convergence.\n                  <\/jats:p>\n                  <jats:p>In this work, we observe that projecting only a small subset of element Hessians is sufficient to secure a descent direction. Building on this insight, we introduce Progressively Projected Newton (PPN), a novel variant of Newton's Method that uses the current iterate's residual to cheaply determine the subset of element Hessians to project. The benefit is twofold: most eigendecompositions are avoided and the global Hessian remains closer to its original form, reducing the number of Newton iterations.<\/jats:p>\n                  <jats:p>We compare PPN with PN and Project\u2010on\u2010Demand Newton (PDN) in a comprehensive set of experiments covering contact\u2010free and contact\u2010rich deformables, co\u2010dimensional and rigid\u2010body simulations, and a range of time step sizes, tolerances and resolutions. PPN reduces the amount of element projections in dynamic simulations by one order of magnitude while simultaneously improving convergence, consistently being the fastest solver in our benchmark.<\/jats:p>","DOI":"10.1111\/cgf.70386","type":"journal-article","created":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T10:18:21Z","timestamp":1776161901000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Progressively Projected Newton's Method"],"prefix":"10.1111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4651-7542","authenticated-orcid":false,"given":"J. A.","family":"Fern\u00e1ndez\u2010Fern\u00e1ndez","sequence":"first","affiliation":[{"name":"RWTH Aachen University  Aachen Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6818-2953","authenticated-orcid":false,"given":"F.","family":"L\u00f6schner","sequence":"additional","affiliation":[{"name":"RWTH Aachen University  Aachen Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1908-4027","authenticated-orcid":false,"given":"J.","family":"Bender","sequence":"additional","affiliation":[{"name":"RWTH Aachen University  Aachen Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,4,14]]},"reference":[{"key":"e_1_2_9_2_2","doi-asserted-by":"publisher","DOI":"10.1145\/2601097.2601116"},{"key":"e_1_2_9_3_2","doi-asserted-by":"crossref","unstructured":"ChengY. LiuL. FuX.: Eigenvalue blending for projected newton.Computer Graphics Forum(Apr.2025). doi:10.1111\/cgf.70027. 3","DOI":"10.1111\/cgf.70027"},{"key":"e_1_2_9_4_2","doi-asserted-by":"publisher","DOI":"10.1145\/3680528.3687650"},{"key":"e_1_2_9_5_2","doi-asserted-by":"crossref","unstructured":"ChenH. LiuH.-T. D. LevinD. I. ZhengC. JacobsonA.: Stabler neo-hookean simulation: Absolute eigenvalue filtering for projected newton. InSpecial Interest Group on Computer Graphics and Interactive Techniques Conference Conference Papers \u201824(July2024) SIGGRAPH \u201824 ACM pp.1\u201310. doi:10.1145\/3641519.3657433. 3 6","DOI":"10.1145\/3641519.3657433"},{"key":"e_1_2_9_6_2","doi-asserted-by":"publisher","DOI":"10.1145\/3658179"},{"key":"e_1_2_9_7_2","first-page":"1","volume-title":"ACM SIGGRAPH 2023 Conference Proceedings","author":"Chen Y.","year":"2023"},{"key":"e_1_2_9_7_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3588432.3591489. 2","DOI":"10.1145\/3588432.3591489"},{"key":"e_1_2_9_8_2","doi-asserted-by":"crossref","unstructured":"Fern\u00e1ndez-Fern\u00e1ndezJ. A. LangeR. LaibleS. ArrasK. O. BenderJ.: Stark: A unified framework for strongly coupled simulation of rigid and deformable bodies with frictional contact. In2024 IEEE International Conference on Robotics and Automation (ICRA)(2024). doi:10.1109\/ICRA57147.2024.10610574. 5","DOI":"10.1109\/ICRA57147.2024.10610574"},{"key":"e_1_2_9_9_2","doi-asserted-by":"crossref","unstructured":"Fern\u00e1ndez-Fern\u00e1ndezJ. L\u00f6schnerF. WesthofenL. LongvaA. BenderJ.: Symx: Energy-based simulation from symbolic expressions.ACM Trans. Graph. (Sept.2025). Just Accepted. URL:https:\/\/doi.org\/10.1145\/3764928","DOI":"10.1145\/3764928"},{"key":"e_1_2_9_9_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3764928. 3","DOI":"10.1145\/3764928"},{"key":"e_1_2_9_10_2","doi-asserted-by":"publisher","DOI":"10.1145\/2980179.2980231"},{"key":"e_1_2_9_11_2","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459802"},{"key":"e_1_2_9_12_2","doi-asserted-by":"publisher","DOI":"10.1109\/tvcg.2015.2459687"},{"key":"e_1_2_9_13_2","doi-asserted-by":"publisher","DOI":"10.1145\/3643028"},{"key":"e_1_2_9_14_2","doi-asserted-by":"crossref","unstructured":"HolzD. JeskeS. R. L\u00f6schnerF. BenderJ. YangY. AndrewsS.: Multiphysics simulation methods in computer graphics.Computer Graphics Forum(2025). doi:10.1111\/cgf.70082. 2","DOI":"10.1111\/cgf.70082"},{"key":"e_1_2_9_15_2","doi-asserted-by":"publisher","DOI":"10.1145\/3687986"},{"key":"e_1_2_9_16_2","doi-asserted-by":"publisher","DOI":"10.1145\/3306346.3323014"},{"key":"e_1_2_9_17_2","doi-asserted-by":"crossref","unstructured":"KimT. EberleD.: Dynamic deformables: implementation and production practicalities (now with code!). InACM SIGGRAPH 2022 Courses(Aug.2022) SIGGRAPH '22 ACM pp.1\u2013259. doi:10.1145\/3532720.3535628. 3","DOI":"10.1145\/3532720.3535628"},{"key":"e_1_2_9_18_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.14111"},{"key":"e_1_2_9_19_2","unstructured":"KharevychL. YangW. TongY. KansoE. MarsdenJ. E. Schr\u00f6derP. DesbrunM.: Geometric variational integrators for computer animation. InProceedings of the 2006 ACM SIGGRAPH\/Eurographics Symposium on Computer Animation(2006) SCA '06 p.43\u201351. doi:10.2312\/SCA\/SCA06\/043-051. 2"},{"key":"e_1_2_9_20_2","doi-asserted-by":"publisher","DOI":"10.1145\/3072959.2990496"},{"key":"e_1_2_9_21_2","doi-asserted-by":"publisher","DOI":"10.1145\/2508363.2508406"},{"key":"e_1_2_9_22_2","doi-asserted-by":"publisher","DOI":"10.1145\/3550454.3555507"},{"key":"e_1_2_9_23_2","doi-asserted-by":"publisher","DOI":"10.1145\/3606922"},{"key":"e_1_2_9_23_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3606922. 2","DOI":"10.1145\/3606922"},{"key":"e_1_2_9_24_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.15183"},{"key":"e_1_2_9_25_2","doi-asserted-by":"publisher","DOI":"10.1145\/3386569.3392425"},{"key":"e_1_2_9_26_2","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459767"},{"issue":"4","key":"e_1_2_9_27_2","first-page":"67:1","article-title":"Affine body dynamics: fast, stable and intersection-free simulation of stiff materials","volume":"41","author":"Lan L.","year":"2022","journal-title":"ACM Transactions on Graphics"},{"key":"e_1_2_9_27_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3528223.3530064. 2","DOI":"10.1145\/3528223.3530064"},{"key":"e_1_2_9_28_2","unstructured":"LongvaA. L\u00f6schnerF. Fern\u00e1ndez-Fern\u00e1ndezJ. A. LarionovE. AscherU. M. BenderJ.:Pitfalls of projection: A study of newton-type solvers for incremental potentials 2023. arXiv Preprint. arXiv:2311.14526. 3"},{"key":"e_1_2_9_29_2","volume-title":"ACM SIGGRAPH 2024 Conference Papers","author":"Li X.","year":"2024"},{"key":"e_1_2_9_29_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3641519.3657449. 2","DOI":"10.1145\/3641519.3657449"},{"key":"e_1_2_9_30_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.14104"},{"key":"e_1_2_9_31_2","doi-asserted-by":"publisher","DOI":"10.1145\/2010324.1964967"},{"key":"e_1_2_9_32_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-40065-5"},{"key":"e_1_2_9_33_2","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2017.2730875"},{"key":"e_1_2_9_34_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0045-7825(98)00230-8"},{"key":"e_1_2_9_35_2","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.14607"},{"key":"e_1_2_9_36_2","doi-asserted-by":"publisher","DOI":"10.1145\/3180491"},{"key":"e_1_2_9_37_2","doi-asserted-by":"publisher","DOI":"10.1145\/3241041"},{"key":"e_1_2_9_38_2","doi-asserted-by":"publisher","DOI":"10.1145\/3606934"},{"key":"e_1_2_9_39_2","doi-asserted-by":"publisher","DOI":"10.1145\/3072959.3073618"},{"key":"e_1_2_9_40_2","doi-asserted-by":"publisher","DOI":"10.2312\/SCA\/SCA05\/181-190"},{"key":"e_1_2_9_41_2","doi-asserted-by":"publisher","DOI":"10.1145\/3606929"},{"key":"e_1_2_9_42_2","doi-asserted-by":"publisher","DOI":"10.1145\/2980179.2980236"},{"key":"e_1_2_9_43_2","doi-asserted-by":"publisher","DOI":"10.1145\/3618372"},{"issue":"4","key":"e_1_2_9_44_2","first-page":"122:1","article-title":"A Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling","volume":"42","author":"Xie T.","year":"2023","journal-title":"ACM Transactions on Graphics"},{"key":"e_1_2_9_44_3","doi-asserted-by":"crossref","unstructured":"doi:10.1145\/3592115. 2","DOI":"10.1145\/3592115"}],"container-title":["Computer Graphics Forum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/cgf.70386","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1111\/cgf.70386","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/cgf.70386","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T10:18:26Z","timestamp":1776161906000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1111\/cgf.70386"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,14]]},"references-count":49,"alternative-id":["10.1111\/cgf.70386"],"URL":"https:\/\/doi.org\/10.1111\/cgf.70386","archive":["Portico"],"relation":{},"ISSN":["0167-7055","1467-8659"],"issn-type":[{"value":"0167-7055","type":"print"},{"value":"1467-8659","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,14]]},"assertion":[{"value":"2026-04-14","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e70386"}}