{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T01:18:54Z","timestamp":1778548734467,"version":"3.51.4"},"reference-count":27,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,12,5]],"date-time":"2020-12-05T00:00:00Z","timestamp":1607126400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100011019","name":"Nemzeti Kutat\u00e1si Fejleszt\u00e9si \u00e9s Innov\u00e1ci\u00f3s Hivatal","doi-asserted-by":"publisher","award":["KH 130378"],"award-info":[{"award-number":["KH 130378"]}],"id":[{"id":"10.13039\/501100011019","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011019","name":"Nemzeti Kutat\u00e1si Fejleszt\u00e9si \u00e9s Innov\u00e1ci\u00f3s Hivatal","doi-asserted-by":"publisher","award":["K124366(124508)"],"award-info":[{"award-number":["K124366(124508)"]}],"id":[{"id":"10.13039\/501100011019","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012550","name":"Nemzeti Kutat\u00e1si, Fejleszt\u00e9si \u00e9s Innovaci\u00f3s Alap","doi-asserted-by":"publisher","award":["BME-NC"],"award-info":[{"award-number":["BME-NC"]}],"id":[{"id":"10.13039\/501100012550","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The recent results attained from a thermodynamically conceived numerical scheme applied on wave propagation in viscoelastic\/rheological solids are generalized here, both in the sense that the scheme is extended to four spacetime dimensions and in the aspect of the virtues of a thermodynamical approach. Regarding the scheme, the arrangement of which quantity is represented where in discretized spacetime, including the question of appropriately realizing the boundary conditions, is nontrivial. In parallel, placing the problem in the thermodynamical framework proves to be beneficial in regards to monitoring and controlling numerical artefacts\u2014instability, dissipation error, and dispersion error. This, in addition to the observed preciseness, speed, and resource-friendliness, makes the thermodynamically extended symplectic approach that is presented here advantageous above commercial finite element software solutions.<\/jats:p>","DOI":"10.3390\/e22121376","type":"journal-article","created":{"date-parts":[[2020,12,6]],"date-time":"2020-12-06T22:27:12Z","timestamp":1607293632000},"page":"1376","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Four Spacetime Dimensional Simulation of Rheological Waves in Solids and the Merits of Thermodynamics"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1227-0572","authenticated-orcid":false,"given":"\u00c1ron","family":"Pozs\u00e1r","sequence":"first","affiliation":[{"name":"Department of Energy Engineering, Faculty of Mechanical Engineering, BME, 1521 Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2492-0392","authenticated-orcid":false,"given":"M\u00e1ty\u00e1s","family":"Sz\u00fccs","sequence":"additional","affiliation":[{"name":"Department of Energy Engineering, Faculty of Mechanical Engineering, BME, 1521 Budapest, Hungary"},{"name":"Montavid Thermodynamic Research Group, 1112 Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5822-6035","authenticated-orcid":false,"given":"R\u00f3bert","family":"Kov\u00e1cs","sequence":"additional","affiliation":[{"name":"Department of Energy Engineering, Faculty of Mechanical Engineering, BME, 1521 Budapest, Hungary"},{"name":"Montavid Thermodynamic Research Group, 1112 Budapest, Hungary"},{"name":"Wigner Research Centre for Physics, Department of Theoretical Physics, Institute for Particle and Nuclear Physics, 1525 Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2708-7065","authenticated-orcid":false,"given":"Tam\u00e1s","family":"F\u00fcl\u00f6p","sequence":"additional","affiliation":[{"name":"Department of Energy Engineering, Faculty of Mechanical Engineering, BME, 1521 Budapest, Hungary"},{"name":"Montavid Thermodynamic Research Group, 1112 Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"114001","DOI":"10.1088\/1361-6382\/aa69e3","article-title":"First report of long term measurements of the MGGL laboratory in the M\u00e1tra mountain range","volume":"34","author":"Bulik","year":"2017","journal-title":"Class. Quantum Gravity"},{"key":"ref_2","first-page":"1693","article-title":"Long term measurements from the M\u00e1tra Gravitational and Geophysical Laboratory","volume":"228","author":"Bulik","year":"2019","journal-title":"Eur. Phys. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/s40948-020-00155-z","article-title":"Investigation of relationship between dynamic and static deformation moduli of rocks","volume":"6","author":"Davarpanah","year":"2020","journal-title":"Geomech. Geophys. Geo-Energy Geo-Resour."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"F\u00fcl\u00f6p, T. (2020). Wave propagation in rocks\u2014Investigating the effect of rheology. Period. Polytech. Civ. Eng., to appear.","DOI":"10.3311\/PPci.16096"},{"key":"ref_5","unstructured":"F\u00fcl\u00f6p, T., and Sz\u00fccs, M. (2020, January 12\u201314). A solution method for determining rheological time dependence around tunnels. Proceedings of the EUROCK2020, Trondheim, Norway."},{"key":"ref_6","unstructured":"F\u00fcl\u00f6p, T., and Sz\u00fccs, M. (2018). Analytical solution method for rheological problems of solids. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1177","DOI":"10.1016\/j.ijheatmasstransfer.2018.06.067","article-title":"Implicit numerical schemes for generalized heat conduction equations","volume":"126","author":"Rieth","year":"2018","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"F\u00fcl\u00f6p, T., Kov\u00e1cs, R., Sz\u00fccs, M., and Fawaier, M. (2020). Thermodynamical extension of a symplectic numerical scheme with half space and time shifts demonstrated on rheological waves in solids. Entropy, 22.","DOI":"10.3390\/e22020155"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1007\/s00161-014-0392-3","article-title":"Distinguished rheological models for solids in the framework of a thermodynamical internal variable theory","volume":"27","author":"Asszonyi","year":"2015","journal-title":"Contin. Mech. Thermodyn."},{"key":"ref_10","unstructured":"Hairer, E., Lubich, C., and Wanner, G. (2006). Geometric Numerical Integration, Springer. [2nd ed.]."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1515\/jnet-2018-0038","article-title":"Numerical stability with help from entropy: Solving a set of 13 moment equations for shock tube problem","volume":"44","author":"Zinner","year":"2019","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_12","unstructured":"Shang, X., and \u00d6ttinger, H.C. (2018). Structure-preserving integrators for dissipative systems based on reversible-irreversible splitting. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"776","DOI":"10.1002\/nme.5532","article-title":"Energy-Entropy-Momentum integration schemes for general discrete non-smooth dissipative problems in thermomechanics","volume":"112","author":"Portillo","year":"2017","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"445206","DOI":"10.1088\/1751-8121\/ab4767","article-title":"Contact variational integrators","volume":"52","author":"Vermeeren","year":"2019","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1673","DOI":"10.1088\/1361-6544\/aaa10e","article-title":"Variational discretization of the nonequilibrium thermodynamics of simple systems","volume":"31","author":"Yoshimura","year":"2018","journal-title":"Nonlinearity"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Cou\u00e9raud, B., and Gay-Balmaz, F. (2020). Variational discretization of thermodynamical simple systems on Lie groups. Discret. Cont. Dyn. Syst. S, 13.","DOI":"10.3934\/dcdss.2020064"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1841","DOI":"10.1016\/j.cma.2010.02.014","article-title":"Algorithms for coupled problems that preserve symmetries and the laws of thermodynamics: Part I: Monolithic integrators and their application to finite strain thermoelasticity","volume":"199","author":"Romero","year":"2010","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2235","DOI":"10.1016\/j.cma.2010.03.016","article-title":"Algorithms for coupled problems that preserve symmetries and the laws of thermodynamics: Part II: Fractional step methods","volume":"199","author":"Romero","year":"2010","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Berezovski, A., and V\u00e1n, P. (2017). Internal Variables in Thermoelasticity, Springer.","DOI":"10.1007\/978-3-319-56934-5"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1007\/s00707-019-2372-y","article-title":"Numerical scheme for simulation of transient flows of non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient and the Cauchy stress tensor","volume":"230","author":"Tierra","year":"2019","journal-title":"Acta Mech."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1825","DOI":"10.1002\/mma.2558","article-title":"Kinematic quantities of finite elastic and plastic deformation","volume":"35","year":"2012","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1007\/s00161-016-0545-7","article-title":"Galilean relativistic fluid mechanics","volume":"29","year":"2017","journal-title":"Contin. Mech. Thermodyn."},{"key":"ref_23","unstructured":"Matolcsi, T. (2004). Ordinary Thermodynamics: Nonequilibrium Homogeneous Processes, Akad\u00e9miai Kiad\u00f3 (Publishing House of the Hungarian Academy of Sciences)."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"591","DOI":"10.3311\/PPci.8628","article-title":"Elastic, thermal expansion, plastic and rheological processes\u2014Theory and experiment","volume":"60","author":"Asszonyi","year":"2016","journal-title":"Period. Civ. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1515\/jnet-2018-0074","article-title":"Kluitenberg\u2013Verh\u00e1s rheology of solids in the GENERIC framework","volume":"44","year":"2019","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Malhotra, V.M., and Carino, N.J. (2003). Handbook on Nondestructive Testing of Concrete, CRC Press. [2nd ed.].","DOI":"10.1201\/9781420040050"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"165","DOI":"10.3311\/PPme.15559","article-title":"Characterization of impacts of elastic-plastic spheres","volume":"64","author":"Kossa","year":"2020","journal-title":"Period. Polytech. Mech. Eng."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/12\/1376\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:41:28Z","timestamp":1760179288000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/12\/1376"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,5]]},"references-count":27,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["e22121376"],"URL":"https:\/\/doi.org\/10.3390\/e22121376","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,5]]}}}