{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T19:03:03Z","timestamp":1775674983775,"version":"3.50.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,2,12]],"date-time":"2025-02-12T00:00:00Z","timestamp":1739318400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,2,12]],"date-time":"2025-02-12T00:00:00Z","timestamp":1739318400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100004731","name":"Natural Science Foundation of Zhejiang Province","doi-asserted-by":"publisher","award":["LQ24A010002"],"award-info":[{"award-number":["LQ24A010002"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["K21LB24"],"award-info":[{"award-number":["K21LB24"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2025,4]]},"DOI":"10.1007\/s10915-025-02815-2","type":"journal-article","created":{"date-parts":[[2025,2,12]],"date-time":"2025-02-12T06:43:12Z","timestamp":1739342592000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Numerical Studies of a Class of Thermoviscoelastic Frictional Contact Problem Described by Fractional Differential Hemivariational Inequalities"],"prefix":"10.1007","volume":"103","author":[{"given":"Hailing","family":"Xuan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoliang","family":"Cheng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0810-6921","authenticated-orcid":false,"given":"Lele","family":"Yuan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,2,12]]},"reference":[{"key":"2815_CR1","volume-title":"Theortical Numerical Analysis: A Functional Analysis Framework","author":"KE Atkinson","year":"2009","unstructured":"Atkinson, K.E., Han, W.: Theortical Numerical Analysis: A Functional Analysis Framework, 3rd edn. Springer, New York (2009)","edition":"3"},{"key":"2815_CR2","doi-asserted-by":"crossref","first-page":"841","DOI":"10.1016\/j.jmaa.2014.09.078","volume":"423","author":"K Bartosz","year":"2015","unstructured":"Bartosz, K., Cheng, X., Kalita, P., et al.: Rothe method for parabolic variational\u2013hemivariational inequalities. J. Math. Anal. Appl. 423, 841\u2013862 (2015)","journal-title":"J. Math. Anal. Appl."},{"key":"2815_CR3","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-75934-0","volume-title":"The Mathematical Theory of Finite Elements Methods","author":"SC Brenner","year":"2008","unstructured":"Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Elements Methods, 3rd edn. Spring, New York (2008)","edition":"3"},{"key":"2815_CR4","doi-asserted-by":"crossref","first-page":"5424","DOI":"10.1007\/s40314-018-0642-2","volume":"37","author":"O Chau","year":"2018","unstructured":"Chau, O.: Numerical analysis of a thermal contact problem with adhesion. Comput. Appl. Math. 37, 5424\u20135455 (2018)","journal-title":"Comput. Appl. Math."},{"key":"2815_CR5","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1016\/j.wear.2010.10.033","volume":"271","author":"A Chudzikiewicz","year":"2011","unstructured":"Chudzikiewicz, A., My\u015bli\u0144ski, A.: Thermoelastic wheel-rail contact problem with elastic graded materials. Wear 271, 417\u2013425 (2011)","journal-title":"Wear"},{"key":"2815_CR6","volume-title":"The Finite Element Method for Elliptic Problems","author":"PG Ciarlet","year":"1978","unstructured":"Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)"},{"key":"2815_CR7","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1016\/j.nonrwa.2014.08.012","volume":"22","author":"M Cocou","year":"2015","unstructured":"Cocou, M.: A class of dynamic contact problems with Coulomb friction in viscoelasticity. Nonlinear Anal. Real World Appl. 22, 508\u2013519 (2015)","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"2815_CR8","doi-asserted-by":"crossref","first-page":"1794","DOI":"10.1002\/oca.2765","volume":"42","author":"E Essoufi","year":"2021","unstructured":"Essoufi, E., Zafrar, A.: Optimal control of friction coefficient in Signorini contact problems. Opt. Control Appl. Methods 42, 1794\u20131811 (2021)","journal-title":"Opt. Control Appl. Methods"},{"key":"2815_CR9","volume":"424","author":"F Guo","year":"2022","unstructured":"Guo, F., Wang, J., Han, J.: Dynamic viscoelastic unilateral constrained contact problems with thermal effects. Appl. Math. Comput. 424, 127034 (2022)","journal-title":"Appl. Math. Comput."},{"key":"2815_CR10","doi-asserted-by":"crossref","DOI":"10.1090\/amsip\/030","volume-title":"Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity","author":"W Han","year":"2002","unstructured":"Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, vol. 30. RI-International Press, Somerville (2002)"},{"key":"2815_CR11","doi-asserted-by":"crossref","first-page":"3891","DOI":"10.1137\/140963248","volume":"46","author":"W Han","year":"2014","unstructured":"Han, W., Mig\u00f3rski, S., Sofonea, M.: A class of variational\u2013hemivariational inequalities with applications to frictional contact problems. SIAM J. Math. Anal. 46, 3891\u20133912 (2014)","journal-title":"SIAM J. Math. Anal."},{"key":"2815_CR12","doi-asserted-by":"crossref","first-page":"2942","DOI":"10.1016\/j.camwa.2019.12.027","volume":"79","author":"D Han","year":"2020","unstructured":"Han, D., Han, W., Jureczka, M., Ochal, A.: Numerical analysis of a contact problem with wear. Comput. Math. Appl. 79, 2942\u20132951 (2020)","journal-title":"Comput. Math. Appl."},{"key":"2815_CR13","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.jmaa.2016.05.012","volume":"443","author":"J Han","year":"2016","unstructured":"Han, J., Mig\u00f3rski, S.: A quasistatic viscoelastic frictional contact problem with multivalued normal compliance, unilateral constraint and material damage. J. Math. Anal. Appl. 443, 57\u201380 (2016)","journal-title":"J. Math. Anal. Appl."},{"key":"2815_CR14","first-page":"1","volume":"303","author":"J Han","year":"2017","unstructured":"Han, J., Mig\u00f3rski, S., Zeng, H.: Weak solvability of a fractional viscoelastic frictionless contact problem. Appl. Math. Comput. 303, 1\u201318 (2017)","journal-title":"Appl. Math. Comput."},{"key":"2815_CR15","doi-asserted-by":"crossref","unstructured":"Hao, J., Wang, J., Han, J.: History-dependent fractional hemivariational inequality with time-delay system for a class of new frictionless quasistatic contact problems. Math. Mech. Solids (2021)","DOI":"10.1177\/10812865211054185"},{"key":"2815_CR16","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1080\/02331934.2022.2133544","volume":"73","author":"J Hao","year":"2024","unstructured":"Hao, J., Wang, J., Lu, L.: Coupled system of fractional hemivariational inequalities with applications. Optimization 73, 969\u2013994 (2024)","journal-title":"Optimization"},{"key":"2815_CR17","doi-asserted-by":"crossref","first-page":"3147","DOI":"10.1016\/j.cma.2004.08.006","volume":"194","author":"S H\u00fceber","year":"2005","unstructured":"H\u00fceber, S., Wohlmuth, B.I.: A primal\u2013dual active set strategy for non-linear multibody contact problems. Comput. Methods Appl. Mech. Eng. 194, 3147\u20133166 (2005)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2815_CR18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00033-019-1141-z","volume":"70","author":"P Kalita","year":"2019","unstructured":"Kalita, P., Szafraniec, P., Shillor, M.: A frictional contact problem with wear diffusion. Z. Angew. Math. Phys. 70, 1\u201317 (2019)","journal-title":"Z. Angew. Math. Phys."},{"key":"2815_CR19","volume-title":"Theory and Applications of Fractional Differential Equations","author":"A Kilbas","year":"2006","unstructured":"Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)"},{"key":"2815_CR20","doi-asserted-by":"crossref","first-page":"4729","DOI":"10.1016\/j.na.2012.03.023","volume":"75","author":"A Kulig","year":"2012","unstructured":"Kulig, A., Mig\u00f3rski, S.: Solvability and continuous dependence results for second order nonlinear evolution inclusions with a Volterra-type operator. Nonlinear Anal. 75, 4729\u20134746 (2012)","journal-title":"Nonlinear Anal."},{"key":"2815_CR21","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/j.nonrwa.2018.05.014","volume":"44","author":"A Kulig","year":"2018","unstructured":"Kulig, A.: Variational\u2013hemivariational approach to quasistatic viscoplastic contact problem with normal compliance, unilateral constraint, memory term, friction and damage. Nonlinear Anal. Real World Appl. 44, 401\u2013416 (2018)","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"2815_CR22","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1080\/01630563.2012.706673","volume":"34","author":"C Li","year":"2013","unstructured":"Li, C., Zeng, F.: The finite difference methods for fractional ordinary differential equations. Numer. Funct. Anal. Optim. 34, 149\u2013179 (2013)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"2815_CR23","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.camwa.2024.04.024","volume":"166","author":"M Ling","year":"2024","unstructured":"Ling, M., Xiao, W., Han, W.: Numerical analysis of a history-dependent mixed hemivariational\u2013variational inequality in contact problems. Comput. Math. Appl. 166, 65\u201376 (2024)","journal-title":"Comput. Math. Appl."},{"key":"2815_CR24","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-26455-2","volume-title":"Theory of Elasticity","author":"AI Lurie","year":"2005","unstructured":"Lurie, A.I., Belyaev, A.: Theory of Elasticity. Springer, Berlin (2005)"},{"key":"2815_CR25","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4614-4232-5","volume-title":"Nonlinear Inclusions and Hemivariational Inequalities","author":"S Mig\u00f3rski","year":"2013","unstructured":"Mig\u00f3rski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Springer, New York (2013)"},{"key":"2815_CR26","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1016\/j.nonrwa.2013.07.002","volume":"15","author":"S Mig\u00f3rski","year":"2014","unstructured":"Mig\u00f3rski, S., Szafraniec, P.: A class of dynamic frictional contact problems governed by a system of hemivariational inequalities in thermoviscoelasticity. Nonlinear Anal. Real World Appl. 15, 158\u2013171 (2014)","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"2815_CR27","volume":"125","author":"S Migorski","year":"2023","unstructured":"Migorski, S., Cai, D.: A general differential quasi variational-hemivariational inequality: well-posedness and application. Sci. Numer. Simul. 125, 107379 (2023)","journal-title":"Sci. Numer. Simul."},{"key":"2815_CR28","doi-asserted-by":"crossref","DOI":"10.1002\/zamm.201800172","volume":"99","author":"Z Peng","year":"2019","unstructured":"Peng, Z.: Existence of a class of variational inequalities modelling quasi-static viscoelastic contact problems. Z. Angew. Math. Mech. 99, e201800172 (2019)","journal-title":"Z. Angew. Math. Mech."},{"key":"2815_CR29","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2021.114234","volume":"388","author":"M Poluektov","year":"2022","unstructured":"Poluektov, M., Figiel, L.: A cut finite-element method for fracture and contact problems in large-deformation solid mechanics. Comput. Methods Appl. Mech. Eng. 388, 114234 (2022)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2815_CR30","volume-title":"Variational\u2013Hemivariational Inequalities with Applications","author":"M Sofonea","year":"2018","unstructured":"Sofonea, M., Mig\u00f3rski, S.: Variational\u2013Hemivariational Inequalities with Applications. Chapman and Hall\/CRC Press, New York (2018)"},{"key":"2815_CR31","volume":"137","author":"X Sun","year":"2024","unstructured":"Sun, X., Cheng, X., Xuan, H.: Error estimates and numerical simulations of a thermoviscoelastic contact problem with damage and long memory. Commun. Nonlinear Sci. Numer. Simul. 137, 108165 (2024)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"2815_CR32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-020-03162-2","volume":"2021","author":"J Webb","year":"2021","unstructured":"Webb, J.: A fractional Gronwall inequality and the asymptptic behaviour of global solutions of caputo fractional problems. Electron. J. Differ. Equ. 2021, 1\u201322 (2021)","journal-title":"Electron. J. Differ. Equ."},{"key":"2815_CR33","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.camwa.2021.07.003","volume":"98","author":"Y Weng","year":"2021","unstructured":"Weng, Y., Chen, T., Li, X., Huang, N.: Rothe method and numerical analysis for a new class of fractional differential hemivariational inequality with an application. Comput. Math. Appl. 98, 118\u2013138 (2021)","journal-title":"Comput. Math. Appl."},{"key":"2815_CR34","doi-asserted-by":"crossref","first-page":"1224","DOI":"10.1080\/00036811.2023.2237048","volume":"103","author":"H Xuan","year":"2024","unstructured":"Xuan, H., Cheng, X., Xiao, Q.: Analysis of a doubly-history dependent variational-hemivariational inequality arising in adhesive contact problem. Appl. Anal. 103, 1224\u20131240 (2024)","journal-title":"Appl. Anal."},{"key":"2815_CR35","doi-asserted-by":"crossref","first-page":"569","DOI":"10.4208\/nmtma.OA-2019-0130","volume":"13","author":"H Xuan","year":"2020","unstructured":"Xuan, H., Cheng, X., Han, W., Xiao, Q.: Numerical analysis of a dynamic contact problem with history-dependent operators. Numer. Math. Theory Methods Appl. 13, 569\u2013594 (2020)","journal-title":"Numer. Math. Theory Methods Appl."},{"key":"2815_CR36","first-page":"2781","volume":"26","author":"H Xuan","year":"2021","unstructured":"Xuan, H., Cheng, X.: Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete Continu. Dyn. Syst. Ser. B 26, 2781\u20132804 (2021)","journal-title":"Discrete Continu. Dyn. Syst. Ser. B"},{"key":"2815_CR37","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.cnsns.2017.07.016","volume":"56","author":"S Zeng","year":"2018","unstructured":"Zeng, S., Mig\u00f3rski, S.: A class of time-fractional hemivariational inequalities with application to frictional contact problem. Commun. Nonlinear Sci. Numer. Simul. 56, 34\u201348 (2018)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02815-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-025-02815-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02815-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,30]],"date-time":"2025-03-30T00:43:25Z","timestamp":1743295405000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-025-02815-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,12]]},"references-count":37,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,4]]}},"alternative-id":["2815"],"URL":"https:\/\/doi.org\/10.1007\/s10915-025-02815-2","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,12]]},"assertion":[{"value":"27 July 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 January 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 January 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 February 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"4"}}