{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T22:15:03Z","timestamp":1776204903128,"version":"3.50.1"},"reference-count":71,"publisher":"Association for Computing Machinery (ACM)","issue":"1","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"crossref","award":["281466253; 411281008"],"award-info":[{"award-number":["281466253; 411281008"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2026,2,28]]},"abstract":"<jats:p>Optimization time integrators are effective at solving complex multi-physics problems including deformable solids with non-linear material models, contact with friction, strain limiting, and so on. For challenging problems, Newton-type optimizers are often used, which necessitates first- and second-order derivatives of the global non-linear objective function. Manually differentiating, implementing, testing, optimizing, and maintaining the resulting code is extremely time-consuming, error-prone, and precludes quick changes to the model, even when using tools that assist with parts of such pipeline.<\/jats:p>\n          <jats:p>\n            We present SymX,\n            <jats:xref ref-type=\"fn\">\n              <jats:sup>1<\/jats:sup>\n            <\/jats:xref>\n            an open source framework that computes the required derivatives of the different energy contributions by symbolic differentiation, generates optimized code, compiles it on-the-fly, and performs the global assembly. The user only has to provide the symbolic expression of each energy for a single representative element in its corresponding discretization and our system will determine the assembled derivatives for the whole simulation. We demonstrate the versatility of SymX in complex simulations featuring different non-linear materials, high-order finite elements, rigid body systems, adaptive discretizations, frictional contact, and coupling of multiple interacting physical systems.\n          <\/jats:p>\n          <jats:p>SymX\u2019s derivatives offer performance on par with SymPy, an established off-the-shelf symbolic engine, and produces simulations at least one order of magnitude faster than TinyAD, an alternative state-of-the-art integral solution.<\/jats:p>","DOI":"10.1145\/3764928","type":"journal-article","created":{"date-parts":[[2025,9,2]],"date-time":"2025-09-02T10:48:34Z","timestamp":1756810114000},"page":"1-19","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["SymX: Energy-based Simulation from Symbolic Expressions"],"prefix":"10.1145","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4651-7542","authenticated-orcid":false,"given":"Jos\u00e9 Antonio","family":"Fern\u00e1ndez-Fern\u00e1ndez","sequence":"first","affiliation":[{"name":"RWTH Aachen University","place":["Aachen, Germany"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6818-2953","authenticated-orcid":false,"given":"Fabian","family":"L\u00f6schner","sequence":"additional","affiliation":[{"name":"RWTH Aachen University","place":["Aachen, Germany"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4427-2377","authenticated-orcid":false,"given":"Lukas","family":"Westhofen","sequence":"additional","affiliation":[{"name":"RWTH Aachen University","place":["Aachen, Germany"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6665-8302","authenticated-orcid":false,"given":"Andreas","family":"Longva","sequence":"additional","affiliation":[{"name":"RWTH Aachen University","place":["Aachen, Germany"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1908-4027","authenticated-orcid":false,"given":"Jan","family":"Bender","sequence":"additional","affiliation":[{"name":"RWTH Aachen University","place":["Aachen, Germany"]}]}],"member":"320","published-online":{"date-parts":[[2025,10,7]]},"reference":[{"key":"e_1_3_4_2_1","first-page":"265","volume-title":"Proceedings of the 12th USENIX Conference on Operating Systems Design and Implementation.","author":"Abadi Mart\u00edn","year":"2016","unstructured":"Mart\u00edn Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, et\u00a0al. 2016. TensorFlow: A system for large-scale machine learning. In Proceedings of the 12th USENIX Conference on Operating Systems Design and Implementation.USENIX Association, USA, 265\u2013283."},{"key":"e_1_3_4_3_1","doi-asserted-by":"publisher","DOI":"10.11588\/ans.2015.100.20553"},{"key":"e_1_3_4_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/s12532-018-0139-4"},{"key":"e_1_3_4_5_1","first-page":"172","volume-title":"Proceedings of the ACM SIGGRAPH 2022 Courses.","author":"Andrews Sheldon","year":"2022","unstructured":"Sheldon Andrews, Kenny Erleben, and Zachary Ferguson. 2022. Contact and friction simulation for computer graphics. In Proceedings of the ACM SIGGRAPH 2022 Courses.Association for Computing Machinery, New York, NY, USA, 172 pages. DOI:10.1145\/3532720.3535640"},{"key":"e_1_3_4_6_1","article-title":"Exact corotational linear fem stiffness matrix","author":"Barbic Jernej","year":"2012","unstructured":"Jernej Barbic. 2012. Exact corotational linear fem stiffness matrix. USC, Los Angeles, CA, USA (2012).","journal-title":"USC, Los Angeles, CA, USA"},{"key":"e_1_3_4_7_1","doi-asserted-by":"publisher","unstructured":"Jan Bender Kenny Erleben and Jeff Trinkle. 2014. Interactive simulation of rigid body dynamics in computer graphics. Comput. Graph. Forum 33 1 (February 2014) 246\u2013270. DOI:10.1111\/cgf.12272","DOI":"10.1111\/cgf.12272"},{"key":"e_1_3_4_8_1","doi-asserted-by":"publisher","DOI":"10.2312\/SGP\/SGP06\/227-230"},{"key":"e_1_3_4_9_1","doi-asserted-by":"publisher","DOI":"10.1145\/2892632"},{"key":"e_1_3_4_10_1","doi-asserted-by":"publisher","DOI":"10.1109\/38.486688"},{"key":"e_1_3_4_11_1","doi-asserted-by":"publisher","DOI":"10.1145\/2601097.2601116"},{"key":"e_1_3_4_12_1","doi-asserted-by":"publisher","DOI":"10.1145\/3272127.3275011"},{"key":"e_1_3_4_13_1","doi-asserted-by":"publisher","DOI":"10.1145\/1778765.1778775"},{"key":"e_1_3_4_14_1","doi-asserted-by":"publisher","DOI":"10.1145\/3528223.3530076"},{"key":"e_1_3_4_15_1","doi-asserted-by":"publisher","unstructured":"Deshana Desai Etai Shuchatowitz Zhongshi Jiang Teseo Schneider and Daniele Panozzo. 2022. ACORNS: An easy-to-use code generator for gradients and Hessians. SoftwareX 17 (2022) 100901. DOI:10.1016\/j.softx.2021.100901","DOI":"10.1016\/j.softx.2021.100901"},{"key":"e_1_3_4_16_1","first-page":"12","volume-title":"Proceedings of the 2011 International Conference for High Performance Computing, Networking, Storage and Analysis.","author":"DeVito Zachary","year":"2011","unstructured":"Zachary DeVito, Niels Joubert, Francisco Palacios, Stephen Oakley, Montserrat Medina, Mike Barrientos, Erich Elsen, Frank Ham, Alex Aiken, Karthik Duraisamy, et\u00a0al. 2011. Liszt: A domain specific language for building portable mesh-based PDE solvers. In Proceedings of the 2011 International Conference for High Performance Computing, Networking, Storage and Analysis.Association for Computing Machinery, New York, NY, USA, 12 pages. DOI:10.1145\/2063384.2063396"},{"key":"e_1_3_4_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/3132188"},{"key":"e_1_3_4_18_1","doi-asserted-by":"publisher","DOI":"10.1145\/3153420"},{"key":"e_1_3_4_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459802"},{"key":"e_1_3_4_20_1","doi-asserted-by":"publisher","unstructured":"Jos\u00e9 Antonio Fern\u00e1ndez-Fern\u00e1ndez Ryan Goldade Ladislav Kavan Jan Bender and Philipp Herholz. 2025. Interactive facial animation: Enhancing facial rigs with real-time shell and contact simulation. Proc. ACM Comput. Graph. Interact. Tech. 8 4 (August 2025). DOI:10.1145\/3747860","DOI":"10.1145\/3747860"},{"key":"e_1_3_4_21_1","first-page":"16888","volume-title":"Proceedings of the 2024 IEEE International Conference on Robotics and Automation","author":"Fern\u00e1ndez-Fern\u00e1ndez Jos\u00e9 Antonio","year":"2024","unstructured":"Jos\u00e9 Antonio Fern\u00e1ndez-Fern\u00e1ndez, Ralph Lange, Stefan Laible, Kai O. Arras, and Jan Bender. 2024. STARK: A unified framework for strongly coupled simulation of rigid and deformable bodies with frictional contact. In Proceedings of the 2024 IEEE International Conference on Robotics and Automation. 16888\u201316894. DOI:10.1109\/ICRA57147.2024.10610574"},{"key":"e_1_3_4_22_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2015.2459687"},{"key":"e_1_3_4_23_1","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898717761","volume-title":"Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation (2nd. ed.)","author":"Griewank Andreas","year":"2008","unstructured":"Andreas Griewank and Andrea Walther. 2008. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation (2nd. ed.). Society for Industrial and Applied Mathematics, USA."},{"key":"e_1_3_4_24_1","doi-asserted-by":"publisher","DOI":"10.1145\/2450153.2450158"},{"issue":"3","key":"e_1_3_4_25_1","first-page":"251","article-title":"New development in FreeFem++","volume":"20","author":"Hecht F.","year":"2012","unstructured":"F. Hecht. 2012. New development in FreeFem++. Journal of Numerical Mathematics 20, 3-4 (2012), 251\u2013265. Retrieved from https:\/\/freefem.org\/","journal-title":"Journal of Numerical Mathematics"},{"key":"e_1_3_4_26_1","doi-asserted-by":"publisher","DOI":"10.1145\/3687986"},{"key":"e_1_3_4_27_1","doi-asserted-by":"publisher","DOI":"10.1145\/3520484"},{"key":"e_1_3_4_28_1","doi-asserted-by":"publisher","unstructured":"Daniel Holz Stefan Rhys Jeske Fabian L\u00f6schner Jan Bender Yin Yang and Sheldon Andrews. 2025. Multiphysics simulation methods in computer graphics. Computer Graphics Forum 44 2 (2025) e70082. DOI:10.1111\/cgf.70082","DOI":"10.1111\/cgf.70082"},{"key":"e_1_3_4_29_1","doi-asserted-by":"publisher","DOI":"10.1145\/3355089.3356506"},{"key":"e_1_3_4_30_1","doi-asserted-by":"publisher","DOI":"10.1145\/3528223.3530099"},{"key":"e_1_3_4_31_1","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459755"},{"key":"e_1_3_4_32_1","first-page":"43","volume-title":"Proceedings of the 2006 ACM SIGGRAPH\/Eurographics Symposium on Computer Animation.","author":"Kharevych L.","year":"2006","unstructured":"L. Kharevych, Weiwei Yang, Y. Tong, E. Kanso, J. E. Marsden, P. Schr\u00f6der, and M. Desbrun. 2006. Geometric, variational integrators for computer animation. In Proceedings of the 2006 ACM SIGGRAPH\/Eurographics Symposium on Computer Animation.Eurographics Association, Goslar, DEU, 43\u201351. DOI:10.2312\/SCA\/SCA06\/043-051"},{"key":"e_1_3_4_33_1","doi-asserted-by":"publisher","DOI":"10.1145\/3306346.3323014"},{"key":"e_1_3_4_34_1","doi-asserted-by":"publisher","DOI":"10.1145\/2866569"},{"key":"e_1_3_4_35_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13520"},{"key":"e_1_3_4_36_1","doi-asserted-by":"publisher","DOI":"10.1145\/3528223.3530064"},{"key":"e_1_3_4_37_1","doi-asserted-by":"publisher","DOI":"10.1145\/3386569.3392425"},{"key":"e_1_3_4_38_1","doi-asserted-by":"publisher","DOI":"10.1145\/3306346.3322951"},{"key":"e_1_3_4_39_1","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459767"},{"key":"e_1_3_4_40_1","doi-asserted-by":"publisher","DOI":"10.1145\/3550454.3555507"},{"key":"e_1_3_4_41_1","doi-asserted-by":"publisher","unstructured":"Cody J. Permann Derek R. Gaston David Andrs\u0306 Robert W. Carlsen Fande Kong Alexander D. Lindsay Jason M. Miller John W. Peterson Andrew E. Slaughter Roy H. Stogner and Richard C. Martineau. 2020. MOOSE: Enabling massively parallel multiphysics simulation. SoftwareX 11 (January 2020) 100430. DOI:10.1016\/j.softx.2020.100430","DOI":"10.1016\/j.softx.2020.100430"},{"key":"e_1_3_4_42_1","doi-asserted-by":"publisher","DOI":"10.1145\/3355089.3356493"},{"key":"e_1_3_4_43_1","doi-asserted-by":"publisher","DOI":"10.1145\/3606922"},{"key":"e_1_3_4_44_1","doi-asserted-by":"publisher","unstructured":"F. L\u00f6schner J. A. Fern\u00e1ndez-Fern\u00e1ndez S. R. Jeske and J. Bender. 2024. Curved three-director cosserat shells with strong coupling. Computer Graphics Forum 43 8 (2024) e15183. DOI:10.1111\/cgf.15183","DOI":"10.1111\/cgf.15183"},{"key":"e_1_3_4_45_1","volume-title":"Proceedings of the ACM SIGGRAPH\/Eurographics Symposium on Computer Animation.","author":"Macklin M.","year":"2020","unstructured":"M. Macklin, K. Erleben, M. M\u00fcller, N. Chentanez, S. Jeschke, and T. Y. Kim. 2020. Primal\/dual descent methods for dynamics. In Proceedings of the ACM SIGGRAPH\/Eurographics Symposium on Computer Animation.Eurographics Association, Goslar, DEU, Article 9, 12 pages. DOI:10.1111\/cgf.14104"},{"key":"e_1_3_4_46_1","unstructured":"Maplesoft. 2023. Maple. (2023). Retrieved from https:\/\/www.maplesoft.com\/"},{"key":"e_1_3_4_47_1","doi-asserted-by":"publisher","DOI":"10.1145\/3453986"},{"key":"e_1_3_4_48_1","doi-asserted-by":"publisher","unstructured":"Marcin Ma\u017adziarz. 2010. Unified Isoparametric 3D Lagrangefinite elements. Computer Modeling in Engineering & Sciences 66 1 (2010) 124. DOI:10.3970\/cmes.2010.066.001","DOI":"10.3970\/cmes.2010.066.001"},{"key":"e_1_3_4_49_1","doi-asserted-by":"publisher","DOI":"10.1145\/2010324.1964932"},{"key":"e_1_3_4_50_1","doi-asserted-by":"crossref","unstructured":"Aaron Meurer Christopher P. Smith Mateusz Paprocki Ondr\u0306ej C\u0306ert\u0131k Sergey B. Kirpichev Matthew Rocklin Amit Kumar Sergiu Ivanov Jason K. Moore Sartaj Singh and others. 2017. SymPy: Symbolic computing in Python. PeerJ Computer Science 3 (2017) e103.","DOI":"10.7717\/peerj-cs.103"},{"key":"e_1_3_4_51_1","first-page":"14","volume-title":"Proceedings of the 34th International Conference on Neural Information Processing Systems.","author":"Moses William S.","year":"2020","unstructured":"William S. Moses and Valentin Churavy. 2020. Instead of rewriting foreign code for machine learning, automatically synthesize fast gradients. In Proceedings of the 34th International Conference on Neural Information Processing Systems.Curran Associates Inc., Red Hook, NY, USA, 14 pages."},{"key":"e_1_3_4_52_1","first-page":"239","volume-title":"Proceedings of the Graphics Interface 2004.","author":"M\u00fcller Matthias","year":"2004","unstructured":"Matthias M\u00fcller and Markus Gross. 2004. Interactive virtual materials. In Proceedings of the Graphics Interface 2004.Canadian Human-Computer Communications Society, Waterloo, CAN, 239\u2013246."},{"key":"e_1_3_4_53_1","first-page":"1","volume-title":"Proceedings of the ACM SIGGRAPH\/Eurographics Symposium on Computer Animation","author":"Narain Rahul","year":"2016","unstructured":"Rahul Narain, Matthew Overby, and George E. Brown. 2016. ADMM \\(\\supseteq\\) projective dynamics: Fast simulation of general constitutive models. In Proceedings of the ACM SIGGRAPH\/Eurographics Symposium on Computer Animation. 1\u20138."},{"key":"e_1_3_4_54_1","volume-title":"Numerical Optimization (2nd. ed.)","author":"Nocedal Jorge","year":"2006","unstructured":"Jorge Nocedal and Stephen J. Wright. 2006. Numerical Optimization (2nd. ed.). Springer, New York, NY, USA."},{"key":"e_1_3_4_55_1","volume-title":"Analytical Differentiation on a Digital Computer","author":"Nolan J. F.","year":"1953","unstructured":"J. F. Nolan. 1953. Analytical Differentiation on a Digital Computer. Master\u2019s thesis. Massachusetts Institute of Technology."},{"key":"e_1_3_4_56_1","unstructured":"NVIDIA Corporation. cuSPARSE Library. (n. d.). DOI:https:\/\/docs.nvidia.com\/cuda\/cusparse\/"},{"key":"e_1_3_4_57_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(98)00219-9"},{"key":"e_1_3_4_58_1","unstructured":"Adam Paszke Sam Gross Francisco Massa Adam Lerer James Bradbury Gregory Chanan Trevor Killeen Zeming Lin Natalia Gimelshein Luca Antiga Alban Desmaison Andreas K\u00f6pf Edward Yang Zach DeVito Martin Raison Alykhan Tejani Sasank Chilamkurthy Benoit Steiner Lu Fang Junjie Bai and Soumith Chintala. 2019. PyTorch: An imperative style high-performance deep learning library. In Proceedings of the 33rd International Conference on Neural Information Processing Systems. Curran Associates Inc. Red Hook NY USA."},{"issue":"3","key":"e_1_3_4_59_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2998441","article-title":"Firedrake: Automating the finite element method by composing abstractions","volume":"43","author":"Rathgeber Florian","year":"2016","unstructured":"Florian Rathgeber, David A Ham, Lawrence Mitchell, Michael Lange, Fabio Luporini, Andrew TT McRae, Gheorghe-Teodor Bercea, Graham R Markall, and Paul HJ Kelly. 2016. Firedrake: Automating the finite element method by composing abstractions. ACM Transactions on Mathematical Software 43, 3 (2016), 1\u201327.","journal-title":"ACM Transactions on Mathematical Software"},{"key":"e_1_3_4_60_1","doi-asserted-by":"publisher","DOI":"10.7907\/DF7X-F354"},{"key":"e_1_3_4_61_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.14607"},{"key":"e_1_3_4_62_1","first-page":"22","volume-title":"Proceedings of the ACM SIGGRAPH 2019 Courses.","author":"Schroeder Craig","year":"2019","unstructured":"Craig Schroeder. 2019. Practical course on computing derivatives in code. In Proceedings of the ACM SIGGRAPH 2019 Courses.Association for Computing Machinery, New York, NY, USA, 22 pages. DOI:10.1145\/3305366.3328073"},{"key":"e_1_3_4_63_1","first-page":"1","volume-title":"Proceedings of the ACM SIGGRAPH Courses","author":"Sifakis Eftychios","year":"2012","unstructured":"Eftychios Sifakis and Jernej Barbic. 2012. FEM simulation of 3D deformable solids. In Proceedings of the ACM SIGGRAPH Courses. 1\u201350. DOI:10.1145\/2343483.2343501"},{"key":"e_1_3_4_64_1","doi-asserted-by":"publisher","DOI":"10.1145\/3180491"},{"key":"e_1_3_4_65_1","doi-asserted-by":"publisher","DOI":"10.1145\/3241041"},{"key":"e_1_3_4_66_1","doi-asserted-by":"publisher","DOI":"10.2312\/SGP\/SGP07\/109-116"},{"key":"e_1_3_4_67_1","first-page":"25","volume-title":"Proceedings of the ACM SIGGRAPH\/Eurographics Symposium on Computer Animation.","author":"Stomakhin Alexey","year":"2012","unstructured":"Alexey Stomakhin, Russell Howes, Craig Schroeder, and Joseph M. Teran. 2012. Energetically consistent invertible elasticity. In Proceedings of the ACM SIGGRAPH\/Eurographics Symposium on Computer Animation.Eurographics Association, Goslar, DEU, 25\u201332."},{"key":"e_1_3_4_68_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073368.1073394"},{"key":"e_1_3_4_69_1","doi-asserted-by":"publisher","unstructured":"L. Westhofen J. A. Fern\u00e1ndez-Fern\u00e1ndez S. R. Jeske and J. Bender. 2024. Strongly coupled simulation of magnetic rigid bodies. Computer Graphics Forum 43 8 (2024) e15185. DOI:10.1111\/cgf.15185","DOI":"10.1111\/cgf.15185"},{"key":"e_1_3_4_70_1","unstructured":"Inc. Wolfram Research. 2023. Mathematica Version 13.2. (2023). Retrieved from https:\/\/www.wolfram.com\/mathematicaChampaign IL 2022."},{"key":"e_1_3_4_71_1","volume-title":"Nonlinear Finite Element Methods","author":"Wriggers Peter","year":"2008","unstructured":"Peter Wriggers. 2008. Nonlinear Finite Element Methods. Springer Science and Business Media."},{"key":"e_1_3_4_72_1","doi-asserted-by":"publisher","DOI":"10.1145\/3550454.3555430"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3764928","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T10:06:50Z","timestamp":1759831610000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3764928"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,7]]},"references-count":71,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,2,28]]}},"alternative-id":["10.1145\/3764928"],"URL":"https:\/\/doi.org\/10.1145\/3764928","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,7]]},"assertion":[{"value":"2023-11-13","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-08-05","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-10-07","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}