{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T08:41:34Z","timestamp":1769848894609,"version":"3.49.0"},"reference-count":46,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["TRR361\/F90"],"award-info":[{"award-number":["TRR361\/F90"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["P 32911"],"award-info":[{"award-number":["P 32911"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper we formulate and analyze a space-time finite element\nmethod for the numerical simulation of rotating electric machines where\nthe finite element mesh is fixed in a space-time domain.\nBased on the Babu\u0161ka\u2013Ne\u010das theory we prove unique solvability\nboth for the continuous variational formulation and for a standard Galerkin\nfinite element discretization in the space-time domain. This approach\nallows for an adaptive resolution of the solution both in space and time,\nbut it requires the solution of the overall system of algebraic equations.\nWhile the use of parallel solution algorithms seems to be mandatory,\nthis also allows for a parallelization simultaneously in space and time.\nThis approach is used for the eddy current approximation of the Maxwell\nequations which results in an elliptic-parabolic interface problem.\nNumerical results for linear and nonlinear constitutive material relations\nconfirm the applicability and accuracy of the proposed approach.<\/jats:p>","DOI":"10.1515\/cmam-2024-0033","type":"journal-article","created":{"date-parts":[[2024,9,2]],"date-time":"2024-09-02T14:33:30Z","timestamp":1725287610000},"page":"441-457","source":"Crossref","is-referenced-by-count":5,"title":["A Space-Time Finite Element Method for the Eddy Current Approximation of Rotating Electric Machines"],"prefix":"10.1515","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8906-821X","authenticated-orcid":false,"given":"Peter","family":"Gangl","sequence":"first","affiliation":[{"name":"231591 Johann Radon Institute for Computational and Applied Mathematics , Altenberger Stra\u00dfe 69, 4040 Linz , Austria"}]},{"given":"Mario","family":"Gobrial","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Angewandte Mathematik , TU Graz , Steyrergasse 30, 8010 Graz , Austria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2552-3022","authenticated-orcid":false,"given":"Olaf","family":"Steinbach","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Angewandte Mathematik , TU Graz , Steyrergasse 30, 8010 Graz , Austria"}]}],"member":"374","published-online":{"date-parts":[[2024,9,3]]},"reference":[{"key":"2025032910202794305_j_cmam-2024-0033_ref_001","doi-asserted-by":"crossref","unstructured":"A.  Alonso Rodr\u00edguez and A.  Valli,\nEddy Current Approximation of Maxwell Equations,\nMS&A. Model. Simul. Appl. 4,\nSpringer, Milan, 2010.","DOI":"10.1007\/978-88-470-1506-7"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_002","doi-asserted-by":"crossref","unstructured":"R.  Andreev,\nStability of sparse space-time finite element discretizations of linear parabolic evolution equations,\nIMA J. Numer. Anal. 33 (2013), no. 1, 242\u2013260.","DOI":"10.1093\/imanum\/drs014"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_003","unstructured":"A.  Arkkio,\nAnalysis of induction motors based on the numerical solution of the magnetic field and circuit equations,\nDissertation, Acta polytechnica Scandinavica, 1987."},{"key":"2025032910202794305_j_cmam-2024-0033_ref_004","doi-asserted-by":"crossref","unstructured":"L.  Armijo,\nMinimization of functions having Lipschitz continuous first partial derivatives,\nPacific J. Math. 16 (1966), 1\u20133.","DOI":"10.2140\/pjm.1966.16.1"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_005","unstructured":"I.  Babu\u0161ka and A. K.  Aziz,\nThe Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations,\nAcademic Press, New York, 1972."},{"key":"2025032910202794305_j_cmam-2024-0033_ref_006","doi-asserted-by":"crossref","unstructured":"F.  Bachinger, U.  Langer and J.  Sch\u00f6berl,\nNumerical analysis of nonlinear multiharmonic eddy current problems,\nNumer. Math. 100 (2005), no. 4, 593\u2013616.","DOI":"10.1007\/s00211-005-0597-2"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_007","doi-asserted-by":"crossref","unstructured":"J. P. A.  Bastos and N.  Sadowski,\nElectromagnetic Modeling by Finite Element Methods,\nElectrical Comput. Eng.,\nCRC Press, Boca Raton, 2003.","DOI":"10.1201\/9780203911174"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_008","doi-asserted-by":"crossref","unstructured":"M.  Bolten, S.  Friedhoff, J.  Hahne and S.  Sch\u00f6ps,\nParallel-in-time simulation of an electrical machine using MGRIT,\nComput. Vis. Sci. 23 (2020), no. 1\u20134, Paper No. 14.","DOI":"10.1007\/s00791-020-00333-2"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_009","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and L. R.  Scott,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_010","unstructured":"A.  Cesarano, C.  Dapogny and P.  Gangl,\nSpace-time shape optimization of rotating electric machines,\npreprint (2024), https:\/\/arxiv.org\/abs\/2402.07017."},{"key":"2025032910202794305_j_cmam-2024-0033_ref_011","doi-asserted-by":"crossref","unstructured":"L. D.  Dalcin, R. R.  Paz, P. A.  Kler and A.  Cosimo,\nParallel distributed computing using Python,\nAdv. Water Resour. 34 (2011), 1124\u20131139.","DOI":"10.1016\/j.advwatres.2011.04.013"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_012","doi-asserted-by":"crossref","unstructured":"A.  Ern and J.-L.  Guermond,\nTheory and Practice of Finite Elements,\nAppl. Math. Sci. 159,\nSpringer, New York, 2004.","DOI":"10.1007\/978-1-4757-4355-5"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_013","doi-asserted-by":"crossref","unstructured":"R. D.  Falgout, J. E.  Jones and U. M.  Yang,\nThe design and implementation of hypre, a library of parallel high performance preconditioners,\nNumerical Solution of Partial Differential Equations on Parallel Computers,\nLect. Notes Comput. Sci. Eng. 51,\nSpringer, Berlin (2006), 267\u2013294.","DOI":"10.1007\/3-540-31619-1_8"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_014","unstructured":"E.  Frank,\nFree-form optimization of electric machines based on shape derivatives,\nMaster\u2019s thesis, Johannes Kepler University Linz, 2010."},{"key":"2025032910202794305_j_cmam-2024-0033_ref_015","doi-asserted-by":"crossref","unstructured":"S.  Friedhoff, J.  Hahne, I.  Kulchytska-Ruchka and S.  Sch\u00f6ps,\nExploring parallel-in-time approaches for Eddy current problems,\nProgress in Industrial Mathematics at ECMI 2018,\nMath. Ind. 30,\nSpringer, Cham (2019), 373\u2013379.","DOI":"10.1007\/978-3-030-27550-1_47"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_016","doi-asserted-by":"crossref","unstructured":"M. J.  Gander,\n50 years of time parallel time integration,\nMultiple Shooting and Time Domain Decomposition Methods,\nContrib. Math. Comput. Sci. 9,\nSpringer, Cham (2015), 69\u2013113.","DOI":"10.1007\/978-3-319-23321-5_3"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_017","doi-asserted-by":"crossref","unstructured":"M. J.  Gander, I.  Kulchytska-Ruchka, I.  Niyonzima and S.  Sch\u00f6ps,\nA new parareal algorithm for problems with discontinuous sources,\nSIAM J. Sci. Comput. 41 (2019), no. 2, B375\u2013B395.","DOI":"10.1137\/18M1175653"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_018","doi-asserted-by":"crossref","unstructured":"M. J.  Gander and M.  Neum\u00fcller,\nAnalysis of a new space-time parallel multigrid algorithm for parabolic problems,\nSIAM J. Sci. Comput. 38 (2016), no. 4, A2173\u2013A2208.","DOI":"10.1137\/15M1046605"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_019","doi-asserted-by":"crossref","unstructured":"P.  Gangl, M.  Gobrial and O.  Steinbach,\nA parallel space-time finite element method for the simulation of an electric motor,\nDomain Decomposition Methods in Science and Engineering XXVII,\nLect. Notes Comput. Sci. Eng. 149,\nSpringer, Cham (2024), 255\u2013262.","DOI":"10.1007\/978-3-031-50769-4_30"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_020","doi-asserted-by":"crossref","unstructured":"C.  Geuzaine and J.-F.  Remacle,\nGmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities,\nInternat. J. Numer. Methods Engrg. 79 (2009), no. 11, 1309\u20131331.","DOI":"10.1002\/nme.2579"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_021","doi-asserted-by":"crossref","unstructured":"J.  Gyselinck, L.  Vandevelde, P.  Dular, C.  Geuzaine and W.  Legros,\nA general method for the frequency domain FE modeling of rotating electromagnetic devices,\nIEEE Trans. Magnet. 39 (2003), no. 3, 1147\u20131150.","DOI":"10.1109\/TMAG.2003.810381"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_022","doi-asserted-by":"crossref","unstructured":"N.  Ida and J. P. A.  Bastos,\nElectromagnetics and Calculation of Fields,\nSpringer, New York, 1997.","DOI":"10.1007\/978-1-4612-0661-3"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_023","doi-asserted-by":"crossref","unstructured":"I.  Kulchytska-Ruchka and S.  Sch\u00f6ps,\nEfficient parallel-in-time solution of time-periodic problems using a multiharmonic coarse grid correction,\nSIAM J. Sci. Comput. 43 (2021), no. 1, C61\u2013C88.","DOI":"10.1137\/20M1314756"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_024","doi-asserted-by":"crossref","unstructured":"U.  Langer, S. E.  Moore and M.  Neum\u00fcller,\nSpace-time isogeometric analysis of parabolic evolution problems,\nComput. Methods Appl. Mech. Engrg. 306 (2016), 342\u2013363.","DOI":"10.1016\/j.cma.2016.03.042"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_025","doi-asserted-by":"crossref","unstructured":"U.  Langer, D.  Pauly and S.  Repin,\nMaxwell\u2019s Equations\u2014Analysis and Numerics,\nRadon Ser. Comput. Appl. Math. 24,\nDe Gruyter, Berlin, 2019.","DOI":"10.1515\/9783110543612"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_026","doi-asserted-by":"crossref","unstructured":"U.  Langer and A.  Schafelner,\nAdaptive space-time finite element methods for non-autonomous parabolic problems with distributional sources,\nComput. Methods Appl. Math. 20 (2020), no. 4, 677\u2013693.","DOI":"10.1515\/cmam-2020-0042"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_027","doi-asserted-by":"crossref","unstructured":"U.  Langer, O.  Steinbach, F.  Tr\u00f6ltzsch and H.  Yang,\nUnstructured space-time finite element methods for optimal control of parabolic equations,\nSIAM J. Sci. Comput. 43 (2021), no. 2, A744\u2013A771.","DOI":"10.1137\/20M1330452"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_028","doi-asserted-by":"crossref","unstructured":"C.  Mellak, J.  Deuringer and A.  Muetze,\nImpact of aspect ratios of solid rotor, large air gap induction motors on run-up time and energy input,\nIEEE Trans. Indust. Appl. 58 (2022), no. 5, 6045\u20136056.","DOI":"10.1109\/TIA.2022.3180030"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_029","unstructured":"J.  Ne\u010das,\nSur une m\u00e9thode pour r\u00e9soudre les \u00e9quations aux d\u00e9riv\u00e9es partielles du type elliptique, voisine de la variationnelle,\nAnn. Sc. Norm. Super. Pisa Cl. Sci. (3) 16 (1962), 305\u2013326."},{"key":"2025032910202794305_j_cmam-2024-0033_ref_030","doi-asserted-by":"crossref","unstructured":"M.  Neum\u00fcller and E.  Karabelas,\nGenerating admissible space-time meshes for moving domains in \n                  \n                     \n                        \n                           (\n                           \n                              d\n                              +\n                              1\n                           \n                           )\n                        \n                     \n                     \n                     (d+1)\n                  \n                dimensions,\nSpace-Time Methods\u2014Applications to Partial Differential Equations,\nRadon Ser. Comput. Appl. Math. 25,\nDe Gruyter, Berlin (2019), 185\u2013206.","DOI":"10.1515\/9783110548488-006"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_031","doi-asserted-by":"crossref","unstructured":"D. R. Q.  Pacheco and O.  Steinbach,\nSpace-time finite element tearing and interconnecting domain decomposition methods,\nDomain Decomposition Methods in Science and Engineering XXVI,\nLect. Notes Comput. Sci. Eng. 145,\nSpringer, Cham (2022), 479\u2013486.","DOI":"10.1007\/978-3-030-95025-5_51"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_032","doi-asserted-by":"crossref","unstructured":"C.  Pechstein and B.  J\u00fcttler,\nMonotonicity-preserving interproximation of B-H-curves,\nJ. Comput. Appl. Math. 196 (2006), no. 1, 45\u201357.","DOI":"10.1016\/j.cam.2005.08.021"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_033","doi-asserted-by":"crossref","unstructured":"P.  Putek,\nNonlinear magnetoquasistatic interface problem in a permanent-magnet machine with stochastic partial differential equation constraints,\nEng. Optim. 51 (2019), no. 12, 2169\u20132192.","DOI":"10.1080\/0305215X.2019.1577403"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_034","doi-asserted-by":"crossref","unstructured":"N.  Sadowski, Y.  Lefevre, M.  Lajoie-Mazenc and J.  Cros,\nFinite element torque calculation in electrical machines while considering the mouvement,\nIEEE Trans. Magnet. 28 (1992), 1410\u20131413.","DOI":"10.1109\/20.123957"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_035","unstructured":"J.  Sch\u00f6berl,\nNetgen\/NGSolve (version 6.2.2302), 2019."},{"key":"2025032910202794305_j_cmam-2024-0033_ref_036","doi-asserted-by":"crossref","unstructured":"C.  Schwab and R.  Stevenson,\nSpace-time adaptive wavelet methods for parabolic evolution problems,\nMath. Comp. 78 (2009), no. 267, 1293\u20131318.","DOI":"10.1090\/S0025-5718-08-02205-9"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_037","doi-asserted-by":"crossref","unstructured":"O.  Steinbach,\nNumerical Approximation Methods for Elliptic Boundary Value Problems,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-68805-3"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_038","doi-asserted-by":"crossref","unstructured":"O.  Steinbach,\nSpace-time finite element methods for parabolic problems,\nComput. Methods Appl. Math. 15 (2015), no. 4, 551\u2013566.","DOI":"10.1515\/cmam-2015-0026"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_039","doi-asserted-by":"crossref","unstructured":"O.  Steinbach and P.  Gaulhofer,\nOn space-time finite element domain decomposition methods for the heat equation,\nDomain Decomposition Methods in Science and Engineering XXVI,\nLect. Notes Comput. Sci. Eng. 145,\nSpringer, Cham (2022), 547\u2013554.","DOI":"10.1007\/978-3-030-95025-5_59"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_040","doi-asserted-by":"crossref","unstructured":"O.  Steinbach and H.  Yang,\nAn algebraic multigrid method for an adaptive space-time finite element discretization,\nLarge-Scale Scientific Computing,\nLecture Notes in Comput. Sci. 10665,\nSpringer, Cham (2018), 66\u201373.","DOI":"10.1007\/978-3-319-73441-5_6"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_041","doi-asserted-by":"crossref","unstructured":"O.  Steinbach and H.  Yang,\nSpace-time finite element methods for parabolic evolution equations: Discretization, a posteriori error estimation, adaptivity and solution,\nSpace-Time Methods\u2014Applications to Partial Differential Equations,\nRadon Ser. Comput. Appl. Math. 25,\nDe Gruyter, Berlin (2019), 207\u2013248.","DOI":"10.1515\/9783110548488-007"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_042","doi-asserted-by":"crossref","unstructured":"O.  Steinbach and M.  Zank,\nCoercive space-time finite element methods for initial boundary value problems,\nElectron. Trans. Numer. Anal. 52 (2020), 154\u2013194.","DOI":"10.1553\/etna_vol52s154"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_043","doi-asserted-by":"crossref","unstructured":"R.  Stevenson and J.  Westerdiep,\nStability of Galerkin discretizations of a mixed space-time variational formulation of parabolic evolution equations,\nIMA J. Numer. Anal. 41 (2021), no. 1, 28\u201347.","DOI":"10.1093\/imanum\/drz069"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_044","unstructured":"V.  Thom\u00e9e,\nGalerkin Finite Element Methods for Parabolic Problems,\nSpringer Ser. Comput. Math. 25,\nSpringer, Berlin, 2006."},{"key":"2025032910202794305_j_cmam-2024-0033_ref_045","doi-asserted-by":"crossref","unstructured":"K.  Urban and A. T.  Patera,\nAn improved error bound for reduced basis approximation of linear parabolic problems,\nMath. Comp. 83 (2014), no. 288, 1599\u20131615.","DOI":"10.1090\/S0025-5718-2013-02782-2"},{"key":"2025032910202794305_j_cmam-2024-0033_ref_046","doi-asserted-by":"crossref","unstructured":"M.  Wolfmayr,\nA posteriori error estimation for time-periodic eddy current problems,\nComput. Methods Appl. Math. 24 (2024), no. 2, 511\u2013528.","DOI":"10.1515\/cmam-2023-0119"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2024-0033\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2024-0033\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,29]],"date-time":"2025-03-29T10:24:32Z","timestamp":1743243872000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2024-0033\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,3]]},"references-count":46,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,11,14]]},"published-print":{"date-parts":[[2025,4,1]]}},"alternative-id":["10.1515\/cmam-2024-0033"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2024-0033","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,3]]}}}