{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,22]],"date-time":"2026-03-22T00:49:03Z","timestamp":1774140543455,"version":"3.50.1"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T00:00:00Z","timestamp":1753142400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T00:00:00Z","timestamp":1753142400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100007041","name":"Universidad de Zaragoza","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100007041","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Virtual Reality"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>We develop a method for realistic haptic rendering of generalized solids that employs graph neural networks. In order to give these neural networks the required realism, biases of different types are used. In the case of solids with (hyper)elastic, hence reversible, behaviour, we use Hamiltonian neural networks. These networks employ a cognitive bias that ensures energy conservation during the simulation. In the case of dissipative solids (particularly visco-hyperelastic solids) we extend the cognitive bias so that it takes into account entropy production. In this way, regardless of the type of solid considered, the developed neural networks are able to provide the haptic devices with the necessary information for a realistic and, above all, physically consistent rendering. The results are tested using a set of haptic gloves and virtual reality glasses, which have shown excellent realism, but not without certain limitations due to the state of the art in the development of hardware of this type.<\/jats:p>","DOI":"10.1007\/s10055-025-01199-w","type":"journal-article","created":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T16:10:35Z","timestamp":1753200635000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A neural network architecture for physically-consistent haptic rendering"],"prefix":"10.1007","volume":"29","author":[{"given":"Quercus","family":"Hern\u00e1ndez","sequence":"first","affiliation":[]},{"given":"Pedro","family":"Martins","sequence":"additional","affiliation":[]},{"given":"Lucas","family":"Tesan","sequence":"additional","affiliation":[]},{"given":"Ic\u00edar","family":"Alfaro","sequence":"additional","affiliation":[]},{"given":"David","family":"Gonz\u00e1lez","sequence":"additional","affiliation":[]},{"given":"Francisco","family":"Chinesta","sequence":"additional","affiliation":[]},{"given":"El\u00edas","family":"Cueto","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,22]]},"reference":[{"key":"1199_CR1","doi-asserted-by":"crossref","unstructured":"Aliabadi FM (2020) Boundary element methods. In: Encyclopedia of continuum mechanics, pp 182\u2013193. Springer","DOI":"10.1007\/978-3-662-55771-6_18"},{"key":"1199_CR2","unstructured":"Barbi\u010d J, James D (2007) Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduced deformable models. In: Proceedings of the 2007 ACM SIGGRAPH\/eurographics symposium on computer animation, pp 171\u2013180. Eurographics Association"},{"key":"1199_CR3","volume-title":"A first course in finite elements","author":"J Belytschko","year":"2007","unstructured":"Belytschko J (2007) A first course in finite elements. Wiley, New York"},{"key":"1199_CR4","doi-asserted-by":"crossref","unstructured":"Chen W, Liu G, Zhang Y, Shirinzadeh B (2016) An artificial neural network based haptic rendering of contact with deformable bodies. In: 2016 IEEE international conference on mechatronics and automation, pp 2332\u20132337. IEEE","DOI":"10.1109\/ICMA.2016.7558929"},{"issue":"2","key":"1199_CR5","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1016\/j.pbiomolbio.2010.09.016","volume":"103","author":"H Courtecuisse","year":"2010","unstructured":"Courtecuisse H, Jung H, Allard J, Duriez C, Lee DY, Cotin S (2010) Gpu-based real-time soft tissue deformation with cutting and haptic feedback. Prog Biophys Mol Biol 103(2):159\u2013168","journal-title":"Prog Biophys Mol Biol"},{"key":"1199_CR6","doi-asserted-by":"publisher","unstructured":"Deo D, De S (2009) PhyNeSS: a physics-driven neural networks-based surgery simulation system with force feedback. In: Proceedings of 3rd Joint EuroHaptics conference and symposium on haptic interfaces for virtual environment and teleoperator systems, world haptics 2009, pp 30\u201334. https:\/\/doi.org\/10.1109\/WHC.2009.4810896","DOI":"10.1109\/WHC.2009.4810896"},{"key":"1199_CR7","doi-asserted-by":"crossref","unstructured":"Desai S, Mattheakis M, Sondak D, Protopapas P, Roberts S (2021) Port-hamiltonian neural networks for learning explicit time-dependent dynamical systems. arXiv preprint arXiv:2107.08024","DOI":"10.1103\/PhysRevE.104.034312"},{"key":"1199_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10916-016-0459-8","volume":"40","author":"D Escobar-Castillejos","year":"2016","unstructured":"Escobar-Castillejos D, Noguez J, Neri L, Magana A, Benes B (2016) A review of simulators with haptic devices for medical training. J Med Syst 40:1\u201322","journal-title":"J Med Syst"},{"key":"1199_CR9","unstructured":"ESI Group: Press release: enabling a world premiere: 100% of the development of a vehicle done in virtual reality. Accessed on 2024-02-09 (2021). https:\/\/www.esi-group.com\/news\/enabling-a-world-premiere-100-of-the-development-of-a-vehicle-done-in-virtual-reality"},{"key":"1199_CR10","unstructured":"Fey M, Lenssen JE (2019) Fast graph representation learning with pytorch geometric. arXiv preprint arXiv:1903.02428"},{"key":"1199_CR11","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1016\/j.cma.2014.09.029","volume":"283","author":"D Gonz\u00e1lez","year":"2015","unstructured":"Gonz\u00e1lez D, Alfaro I, Quesada C, Cueto E, Chinesta F (2015) Computational vademecums for the real-time simulation of haptic collision between nonlinear solids. Comput Methods Appl Mech Eng 283:210\u2013223","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"2266","key":"1199_CR12","doi-asserted-by":"publisher","first-page":"20210068","DOI":"10.1098\/rspa.2021.0068","volume":"478","author":"A Goyal","year":"2022","unstructured":"Goyal A, Bengio Y (2022) Inductive biases for deep learning of higher-level cognition. Proc R Soc A 478(2266):20210068","journal-title":"Proc R Soc A"},{"key":"1199_CR13","unstructured":"Greydanus SJ, Dzumba M, Yosinski J (2019) Hamiltonian neural networks"},{"issue":"6","key":"1199_CR14","doi-asserted-by":"publisher","first-page":"6620","DOI":"10.1103\/PhysRevE.56.6620","volume":"56","author":"M Grmela","year":"1997","unstructured":"Grmela M, \u00d6ttinger HC (1997) Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys Rev E 56(6):6620","journal-title":"Phys Rev E"},{"issue":"4","key":"1199_CR15","doi-asserted-by":"publisher","first-page":"705","DOI":"10.1109\/TOH.2024.3382258","volume":"17","author":"N Heravi","year":"2024","unstructured":"Heravi N, Culbertson H, Okamura AM, Bohg J (2024) Development and evaluation of a learning-based model for real-time haptic texture rendering. IEEE Trans Haptics 17(4):705\u2013716","journal-title":"IEEE Trans Haptics"},{"issue":"9","key":"1199_CR16","doi-asserted-by":"publisher","first-page":"720","DOI":"10.1016\/j.robot.2007.05.003","volume":"55","author":"M Johnsson","year":"2007","unstructured":"Johnsson M, Balkenius C (2007) Neural network models of haptic shape perception. Robot Auton Syst 55(9):720\u2013727","journal-title":"Robot Auton Syst"},{"issue":"1","key":"1199_CR17","doi-asserted-by":"publisher","first-page":"62","DOI":"10.1109\/TOH.2021.3137936","volume":"15","author":"JB Joolee","year":"2021","unstructured":"Joolee JB, Jeon S (2021) Data-driven haptic texture modeling and rendering based on deep spatio-temporal networks. IEEE Trans Haptics 15(1):62\u201367","journal-title":"IEEE Trans Haptics"},{"key":"1199_CR18","first-page":"7092","volume":"32","author":"N Keriven","year":"2019","unstructured":"Keriven N, Peyr\u00e9 G (2019) Universal invariant and equivariant graph neural networks. Adv Neural Inf Process Syst 32:7092\u20137101","journal-title":"Adv Neural Inf Process Syst"},{"issue":"3","key":"1199_CR19","doi-asserted-by":"publisher","first-page":"380","DOI":"10.1109\/TVCG.2010.52","volume":"17","author":"J Lang","year":"2010","unstructured":"Lang J, Andrews S (2010) Measurement-based modeling of contact forces and textures for haptic rendering. IEEE Trans Vis Comput Graph 17(3):380\u2013391","journal-title":"IEEE Trans Vis Comput Graph"},{"key":"1199_CR20","doi-asserted-by":"crossref","unstructured":"Laycock SD, Day A (2003) Recent developments and applications of haptic devices. In: Computer graphics forum, vol. 22, pp 117\u2013132. Wiley Online Library","DOI":"10.1111\/1467-8659.00654"},{"key":"1199_CR21","doi-asserted-by":"crossref","unstructured":"Laycock SD, Day A (2007) A survey of haptic rendering techniques. In: Computer graphics forum, vol. 26, pp 50\u201365. Wiley Online Library","DOI":"10.1111\/j.1467-8659.2007.00945.x"},{"issue":"3","key":"1199_CR22","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1002\/rcs.266","volume":"5","author":"A Maciel","year":"2009","unstructured":"Maciel A, Halic T, Lu Z, Nedel LP, De S (2009) Using the physx engine for physics-based virtual surgery with force feedback. Int J Med Robot Comput Assisted Surg 5(3):341\u2013353","journal-title":"Int J Med Robot Comput Assisted Surg"},{"key":"1199_CR23","unstructured":"Mart\u00ednez Mart\u00ednez F (2014) Determining the biomechanical behavior of the liver using medical image analysis and evolutionary computation. PhD thesis, Universitat Polit\u00e8cnica de Val\u00e8ncia, Spain"},{"issue":"2","key":"1199_CR24","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1002\/cnm.887","volume":"23","author":"K Miller","year":"2007","unstructured":"Miller K, Joldes G, Lance D, Wittek A (2007) Total lagrangian explicit dynamics finite element algorithm for computing soft tissue deformation. Commun Numer Methods Eng 23(2):121\u2013134","journal-title":"Commun Numer Methods Eng"},{"issue":"3","key":"1199_CR25","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1016\/j.cmpb.2008.04.008","volume":"91","author":"S Niroomandi","year":"2008","unstructured":"Niroomandi S, Alfaro I, Cueto E, Chinesta F (2008) Real-time deformable models of non-linear tissues by model reduction techniques. Comput Methods Programs Biomed 91(3):223\u2013231","journal-title":"Comput Methods Programs Biomed"},{"issue":"5","key":"1199_CR26","doi-asserted-by":"publisher","first-page":"574","DOI":"10.1002\/cnm.1491","volume":"28","author":"S Niroomandi","year":"2011","unstructured":"Niroomandi S, Alfaro I, Gonzalez D, Cueto E, Chinesta F (2011) Real time simulation of surgery by reduced order modelling and x-fem techniques. Int J Numer Methods Biomed Eng 28(5):574\u2013588","journal-title":"Int J Numer Methods Biomed Eng"},{"issue":"5","key":"1199_CR27","doi-asserted-by":"publisher","first-page":"586","DOI":"10.1002\/cnm.2544","volume":"29","author":"S Niroomandi","year":"2013","unstructured":"Niroomandi S, Gonz\u00e1lez D, Alfaro I, Bordeu F, Leygue A, Cueto E, Chinesta F (2013a) Real-time simulation of biological soft tissues: a PGD approach. Int J Numer Methods Biomed Eng 29(5):586\u2013600","journal-title":"Int J Numer Methods Biomed Eng"},{"issue":"3","key":"1199_CR28","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1002\/nme.4531","volume":"96","author":"S Niroomandi","year":"2013","unstructured":"Niroomandi S, Alfaro I, Gonz\u00e1lez D, Cueto E, Chinesta F (2013b) Model order reduction in hyperelasticity: a proper generalized decomposition approach. Int J Numer Methods Eng 96(3):129\u2013149","journal-title":"Int J Numer Methods Eng"},{"key":"1199_CR29","doi-asserted-by":"crossref","unstructured":"Otaduy MA, Okamura A, Subramanian S (2016) Haptic technologies for direct touch in virtual reality. In: ACM SIGGRAPH 2016 Courses, pp 1\u2013123","DOI":"10.1145\/2897826.2927307"},{"key":"1199_CR30","unstructured":"Otaduy Trist\u00e1n MA (2004) 6-dof haptic rendering using contact levels of detail and haptic textures. PhD thesis, The University of North Carolina at Chapel Hill, North Carolina"},{"key":"1199_CR31","volume-title":"High fidelity haptic rendering","author":"MA Otaduy","year":"2022","unstructured":"Otaduy MA, Lin MC (2022) High fidelity haptic rendering. Springer, Germany"},{"issue":"6","key":"1199_CR32","doi-asserted-by":"publisher","first-page":"6633","DOI":"10.1103\/PhysRevE.56.6633","volume":"56","author":"HC \u00d6ttinger","year":"1997","unstructured":"\u00d6ttinger HC, Grmela M (1997) Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys Rev E 56(6):6633","journal-title":"Phys Rev E"},{"issue":"2","key":"1199_CR33","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1016\/j.jss.2009.04.018","volume":"156","author":"L Panait","year":"2009","unstructured":"Panait L, Akkary E, Bell RL, Roberts KE, Dudrick SJ, Duffy AJ (2009) The role of haptic feedback in laparoscopic simulation training. J Surg Res 156(2):312\u2013316","journal-title":"J Surg Res"},{"key":"1199_CR34","unstructured":"Pfaff T, Fortunato M, Sanchez-Gonzalez A, Battaglia PW (2020) Learning mesh-based simulation with graph networks. arXiv preprint arXiv:2010.03409"},{"issue":"10","key":"1199_CR35","doi-asserted-by":"publisher","first-page":"1230","DOI":"10.1002\/nme.5252","volume":"108","author":"C Quesada","year":"2016","unstructured":"Quesada C, Gonz\u00e1lez D, Alfaro I, Cueto E, Chinesta F (2016) Computational vademecums for real-time simulation of surgical cutting in haptic environments. Int J Numer Methods Eng 108(10):1230\u20131247","journal-title":"Int J Numer Methods Eng"},{"issue":"3","key":"1199_CR36","doi-asserted-by":"publisher","DOI":"10.1002\/cnm.2926","volume":"34","author":"C Quesada","year":"2017","unstructured":"Quesada C, Alfaro I, Gonz\u00e1lez D, Chinesta F, Cueto E (2017) Haptic simulation of tissue tearing during surgery. Int J Numer Methods Biomed Eng 34(3):e2926","journal-title":"Int J Numer Methods Biomed Eng"},{"key":"1199_CR37","doi-asserted-by":"publisher","unstructured":"Quesada C, Bad\u00edas A, Gonz\u00e1lez D, Alfaro I, Chinesta F, Cueto E (2023) Chapter 22 - surgery simulators based on model-order reduction. In: Chinesta F, Cueto E, Payan Y, Ohayon J (eds) Reduced order models for the biomechanics of living organs. biomechanics of living organs, pp 435\u2013452. Academic Press. https:\/\/doi.org\/10.1016\/B978-0-32-389967-3.00029-9. https:\/\/www.sciencedirect.com\/science\/article\/pii\/B9780323899673000299","DOI":"10.1016\/B978-0-32-389967-3.00029-9"},{"key":"1199_CR38","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","volume":"378","author":"M Raissi","year":"2019","unstructured":"Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686\u2013707","journal-title":"J Comput Phys"},{"issue":"2","key":"1199_CR39","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1016\/j.jsurg.2019.09.006","volume":"77","author":"K Rangarajan","year":"2020","unstructured":"Rangarajan K, Davis H, Pucher PH (2020) Systematic review of virtual haptics in surgical simulation: A valid educational tool? J Surg Educ 77(2):337\u2013347","journal-title":"J Surg Educ"},{"issue":"2","key":"1199_CR40","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1109\/MCG.2004.1274058","volume":"24","author":"K Salisbury","year":"2004","unstructured":"Salisbury K, Conti F, Barbagli F (2004) Haptic rendering: introductory concepts. IEEE Comput Graphics Appl 24(2):24\u201332","journal-title":"IEEE Comput Graphics Appl"},{"key":"1199_CR41","unstructured":"Sanchez-Gonzalez A, Godwin J, Pfaff T, Ying R, Leskovec J, Battaglia P (2020) Learning to simulate complex physics with graph networks. In: International conference on machine learning, pp 8459\u20138468. PMLR"},{"key":"1199_CR42","doi-asserted-by":"crossref","unstructured":"Tes\u00e1n L, Gonz\u00e1lez D, Martins P, Cueto E (2024) Thermodynamics-informed graph neural networks for real-time simulation of digital human twins. arXiv preprint arXiv:2412.12034","DOI":"10.26754\/jjii3a.202410903"},{"key":"1199_CR43","doi-asserted-by":"crossref","unstructured":"Vlachos C, Moustakas K (2024) High\u2013fidelity haptic rendering through implicit neural force representation. In: International conference on human haptic sensing and touch enabled computer applications, pp 493\u2013506. Springer","DOI":"10.1007\/978-3-031-70058-3_40"}],"container-title":["Virtual Reality"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10055-025-01199-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10055-025-01199-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10055-025-01199-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T07:56:56Z","timestamp":1761119816000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10055-025-01199-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,22]]},"references-count":43,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["1199"],"URL":"https:\/\/doi.org\/10.1007\/s10055-025-01199-w","relation":{},"ISSN":["1434-9957"],"issn-type":[{"value":"1434-9957","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,22]]},"assertion":[{"value":"21 March 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 July 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"114"}}