{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:45:58Z","timestamp":1740123958676,"version":"3.37.3"},"reference-count":67,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T00:00:00Z","timestamp":1724025600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T00:00:00Z","timestamp":1724025600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001711","name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","doi-asserted-by":"publisher","award":["HOGAEMS n.200021_188589"],"award-info":[{"award-number":["HOGAEMS n.200021_188589"]}],"id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SCHO 1562\/6-1","SCHO 1562\/6-1","CRC - TRR361\/F90","CRC - TRR361\/F90"],"award-info":[{"award-number":["SCHO 1562\/6-1","SCHO 1562\/6-1","CRC - TRR361\/F90","CRC - TRR361\/F90"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2024,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a special, e.g., hierarchical, finite element basis construction. Using insights from de Rham complex approximation with splines, we show that additional conditions are here unnecessary. Spanning tree techniques can be adapted to operate on a hexahedral mesh resulting from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.<\/jats:p>","DOI":"10.1007\/s10444-024-10181-0","type":"journal-article","created":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T06:55:28Z","timestamp":1724050528000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems"],"prefix":"10.1007","volume":"50","author":[{"given":"Bernard","family":"Kapidani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2104-9167","authenticated-orcid":false,"given":"Melina","family":"Merkel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9150-0219","authenticated-orcid":false,"given":"Sebastian","family":"Sch\u00f6ps","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rafael","family":"V\u00e1zquez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,8,19]]},"reference":[{"issue":"7","key":"10181_CR1","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1049\/ip-a-1:19880072","volume":"135","author":"R Albanese","year":"1988","unstructured":"Albanese, R., Rubinacci, G.: Integral formulation for 3d eddy-current computation using edge elements. IEE Proc. Sci. Meas. Tech. 135(7), 457\u2013462 (1988). https:\/\/doi.org\/10.1049\/ip-a-1:19880072","journal-title":"IEE Proc. Sci. Meas. Tech."},{"issue":"1137\/1","key":"10181_CR2","first-page":"9781611975543","volume":"10","author":"DN Arnold","year":"2018","unstructured":"Arnold, D.N.: Finite Element Exterior Calculus. SIAM 10(1137\/1), 9781611975543 (2018)","journal-title":"Finite Element Exterior Calculus. SIAM"},{"issue":"2","key":"10181_CR3","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1090\/S0273-0979-10-01278-4","volume":"47","author":"DN Arnold","year":"2010","unstructured":"Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus from Hodge theory to numerical stability. Bull. Am. Math. Soc. 47(2), 281\u2013354 (2010). https:\/\/doi.org\/10.1090\/S0273-0979-10-01278-4","journal-title":"Bull. Am. Math. Soc."},{"key":"10181_CR4","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1017\/s096249291400004x","volume":"23","author":"LB Da Veiga","year":"2014","unstructured":"Da Veiga, L.B., Buffa, A., Sangalli, G., V\u00e1zquez, R.: Mathematical analysis of variational isogeometric methods. Acta. Num. 23, 157\u2013287 (2014). https:\/\/doi.org\/10.1017\/s096249291400004x","journal-title":"Acta. Num."},{"key":"10181_CR5","doi-asserted-by":"publisher","unstructured":"Bortot, L., Auchmann, B., CortesGarcia, I., De Gersem, H., Maciejewski, M., Mentink, M., Sch\u00f6ps, S., Van Nugteren, J., Verweij, A.: A coupled A-H formulation for magneto-thermal transients in high-temperature superconducting magnets. IEEE Trans. Appl. Super. 30(5), (2020). https:\/\/doi.org\/10.1109\/TASC.2020.2969476. Arxiv:1909.03312","DOI":"10.1109\/TASC.2020.2969476"},{"key":"10181_CR6","unstructured":"Bossavit, A.: On finite elements for the electricity equation, pp. 85\u201392, (1982). Academic Press"},{"issue":"2","key":"10181_CR7","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1108\/eb010005","volume":"4","author":"A Bossavit","year":"1985","unstructured":"Bossavit, A.: Two dual formulations of the 3-d eddy-currents problem. COMPEL 4(2), 103\u2013116 (1985). https:\/\/doi.org\/10.1108\/eb010005","journal-title":"COMPEL"},{"issue":"1","key":"10181_CR8","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1108\/eb010040","volume":"7","author":"A Bossavit","year":"1988","unstructured":"Bossavit, A.: Results for benchmark problem 5, the Bath-cube experiment: an aluminium block in an alternating field. COMPEL 7(1), 81\u201388 (1988). https:\/\/doi.org\/10.1108\/eb010040","journal-title":"COMPEL"},{"key":"10181_CR9","doi-asserted-by":"publisher","unstructured":"Bossavit, A.: Computational electromagnetism: variational formulations. Complementarity. Edge Elements (1998). https:\/\/doi.org\/10.1016\/B978-0-12-118710-1.X5000-4. Academic Press","DOI":"10.1016\/B978-0-12-118710-1.X5000-4"},{"issue":"3","key":"10181_CR10","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1108\/03321649910274892","volume":"18","author":"A Bossavit","year":"1999","unstructured":"Bossavit, A.: On the Lorenz gauge. COMPEL 18(3), 323\u2013336 (1999). https:\/\/doi.org\/10.1108\/03321649910274892","journal-title":"COMPEL"},{"key":"10181_CR11","doi-asserted-by":"publisher","unstructured":"Bossavit, A., Kettunen, L.: Yee-like schemes on a tetrahedral mesh, with diagonal lumping. Int. J. Numer. Model. Electron. Network. Dev. Field 12(1-2), 129\u2013142 (1999).https:\/\/doi.org\/10.1002\/(SICI)1099-1204(199901\/04)12:1\/2<129::AID-JNM327>3.0.CO;2-G","DOI":"10.1002\/(SICI)1099-1204(199901\/04)12:1\/2<129::AID-JNM327>3.0.CO;2-G"},{"key":"10181_CR12","doi-asserted-by":"publisher","unstructured":"Buffa, A., D\u0142olz, J., Kurz, S., Sch\u00f6ps, S., V\u00e1zquez, R., Wolf, F.: Multipatch approximation of the de Rham sequence and its traces in isogeometric analysis. Numer. Math. 144(1), 201\u2013236 (2019). https:\/\/doi.org\/10.1007\/s00211-019-01079-x. Arxiv:1806.01062","DOI":"10.1007\/s00211-019-01079-x"},{"issue":"2","key":"10181_CR13","doi-asserted-by":"publisher","first-page":"818","DOI":"10.1137\/100786708","volume":"49","author":"A Buffa","year":"2011","unstructured":"Buffa, A., Rivas, J., Sangalli, G., V\u00e1zquez, R.: Isogeometric discrete differential forms in three dimensions. SIAM J. Numer. Anal. 49(2), 818\u2013844 (2011). https:\/\/doi.org\/10.1137\/100786708","journal-title":"SIAM J. Numer. Anal."},{"key":"10181_CR14","doi-asserted-by":"publisher","first-page":"1143","DOI":"10.1016\/j.cma.2009.12.002","volume":"199","author":"A Buffa","year":"2010","unstructured":"Buffa, A., Sangalli, G., V\u00e1zquez, R.: Isogeometric analysis in electromagnetics: B-splines approximation. Comput. Meth. Appl. Mech. Eng. 199, 1143\u20131152 (2010). https:\/\/doi.org\/10.1016\/j.cma.2009.12.002","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"10181_CR15","doi-asserted-by":"publisher","first-page":"1291","DOI":"10.1016\/j.jcp.2013.08.015","volume":"257","author":"A Buffa","year":"2013","unstructured":"Buffa, A., Sangalli, G., V\u00e1zquez, R.: Isogeometric methods for computational electromagnetics: B-spline and T-spline discretizations. J. Comput. Phys. 257, 1291\u20131320 (2013). https:\/\/doi.org\/10.1016\/j.jcp.2013.08.015","journal-title":"J. Comput. Phys."},{"key":"10181_CR16","doi-asserted-by":"publisher","unstructured":"B\u00ed\u00adr\u00f3, O., Preis, K.: On the use of the magnetic vector potential in the finite-element analysis of three-dimensional eddy currents. IEEE Trans. Magn. 25(4), 3145\u20133159 (1989) https:\/\/doi.org\/10.1109\/20.34388","DOI":"10.1109\/20.34388"},{"issue":"3","key":"10181_CR17","doi-asserted-by":"publisher","first-page":"1307","DOI":"10.1109\/20.376269","volume":"31","author":"O B\u00edr\u00f3","year":"1995","unstructured":"B\u00edr\u00f3, O., Preis, K., Richter, K.R.: Various FEM formulations for the calculation of transient 3D eddy currents in nonlinear media. IEEE Trans. Magn. 31(3), 1307\u20131312 (1995). https:\/\/doi.org\/10.1109\/20.376269","journal-title":"IEEE Trans. Magn."},{"key":"10181_CR18","doi-asserted-by":"publisher","first-page":"65","DOI":"10.2528\/PIER00080103","volume":"32","author":"M Clemens","year":"2001","unstructured":"Clemens, M., Weiland, T.: Discrete electromagnetism with the finite integration technique. PIER 32, 65\u201387 (2001). https:\/\/doi.org\/10.2528\/PIER00080103","journal-title":"PIER"},{"issue":"2","key":"10181_CR19","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1109\/20.996149","volume":"38","author":"M Clemens","year":"2002","unstructured":"Clemens, M., Weiland, T.: Regularization of eddy-current formulations using discrete grad-div operators. IEEE Trans. Magn. 38(2), 569\u2013572 (2002). https:\/\/doi.org\/10.1109\/20.996149","journal-title":"IEEE Trans. Magn."},{"key":"10181_CR20","unstructured":"Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to algorithms. MIT Press (2001)"},{"key":"10181_CR21","unstructured":"de Boor, C.: A practical guide to splines, Applied Mathematical Sciences, vol.\u00a027, rev. edn. Springer (2001)"},{"key":"10181_CR22","doi-asserted-by":"publisher","unstructured":"De\u00a0Los\u00a0Santos, E., Alonso\u00a0Rodr\u00ed\u00adguez, A.M., Rapetti, F.: Construction of a spanning tree for high-order edge elements. Int. J. Numer. Model. Electron. Network. Dev. Field 36(3), e3080 (2023) https:\/\/doi.org\/10.1002\/jnm.3080","DOI":"10.1002\/jnm.3080"},{"issue":"2","key":"10181_CR23","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/BF01232924","volume":"79","author":"HK Dirks","year":"1996","unstructured":"Dirks, H.K.: Quasi-stationary fields for microelectronic applications. Electr. Eng. 79(2), 145\u2013155 (1996). https:\/\/doi.org\/10.1007\/BF01232924","journal-title":"Electr. Eng."},{"key":"10181_CR24","doi-asserted-by":"publisher","unstructured":"Dular, J., Harutyunyan, M., Bortot, L., Sch\u00f6ps, S., Vanderheyden, B., Geuzaine, C.: On the stability of mixed finite-element formulations for high-temperature superconductors. IEEE Trans. Appl. Super. 31(6) (2021) https:\/\/doi.org\/10.1109\/TASC.2021.3098724Arxiv:2106.00313","DOI":"10.1109\/TASC.2021.3098724"},{"issue":"3","key":"10181_CR25","doi-asserted-by":"publisher","first-page":"1356","DOI":"10.1109\/20.376278","volume":"31","author":"P Dular","year":"1995","unstructured":"Dular, P., Nicolet, A., Genon, A., Legros, W.: A discrete sequence associated with mixed finite elements and its gauge condition for vector potentials. IEEE Trans. Magn. 31(3), 1356\u20131359 (1995). https:\/\/doi.org\/10.1109\/20.376278","journal-title":"IEEE Trans. Magn."},{"issue":"6","key":"10181_CR26","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/TMAG.2019.2906099","volume":"55","author":"P D\u0142otko","year":"2019","unstructured":"D\u0142otko, P., Kapidani, B., Pitassi, S., Specogna, R.: Fake conductivity or cohomology: which to use when solving eddy current problems with $$h$$-formulations? IEEE Trans. Magn. 55(6), 1\u20134 (2019). https:\/\/doi.org\/10.1109\/TMAG.2019.2906099","journal-title":"IEEE Trans. Magn."},{"issue":"6","key":"10181_CR27","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/tmag.2017.2661480","volume":"53","author":"P D\u0142otko","year":"2017","unstructured":"D\u0142otko, P., Kapidani, B., Specogna, R.: Topoprocessor: an efficient computational topology toolbox for h-oriented eddy current formulations. IEEE Trans. Magn. 53(6), 1\u20134 (2017). https:\/\/doi.org\/10.1109\/tmag.2017.2661480","journal-title":"IEEE Trans. Magn."},{"issue":"3","key":"10181_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/TMAG.2017.2749618","volume":"54","author":"P D\u0142otko","year":"2018","unstructured":"D\u0142otko, P., Kapidani, B., Specogna, R.: Lean cohomology computation for electromagnetic modeling. IEEE Trans. Magn. 54(3), 1\u20134 (2018). https:\/\/doi.org\/10.1109\/TMAG.2017.2749618","journal-title":"IEEE Trans. Magn."},{"issue":"4","key":"10181_CR29","doi-asserted-by":"publisher","first-page":"1601","DOI":"10.1137\/090766334","volume":"48","author":"P D\u0142otko","year":"2010","unstructured":"D\u0142otko, P., Specogna, R.: Critical analysis of the spanning tree techniques. SIAM J. Numer. Anal. 48(4), 1601\u20131624 (2010)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"10181_CR30","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1109\/TMAG.2013.2281076","volume":"50","author":"P D\u0142otko","year":"2014","unstructured":"D\u0142otko, P., Specogna, R.: Lazy cohomology generators: a breakthrough in (co)homology computations for CEM. IEEE Trans. Magn. 50(2), 577\u2013580 (2014). https:\/\/doi.org\/10.1109\/TMAG.2013.2281076","journal-title":"IEEE Trans. Magn."},{"key":"10181_CR31","doi-asserted-by":"publisher","unstructured":"D\u0142otko, P., Specogna, R., Trevisan, F.: Automatic generation of cuts on large-sized meshes for the $${T}-{\\Omega }$$ geometric eddy-current formulation. Comput. Methods Appl. Mech. Eng. 198(47), 3765\u20133781 (2009). https:\/\/doi.org\/10.1016\/j.cma.2009.08.007https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0045782509002539","DOI":"10.1016\/j.cma.2009.08.007"},{"key":"10181_CR32","doi-asserted-by":"publisher","unstructured":"Friedrich, L.A.J.: Isogeometric analysis of nonlinear eddy current problems. Dissertation, Technische Universiteit Eindhoven (2020) https:\/\/doi.org\/10.6100\/5074ed73-fcec-4a93-8a5c-699abcf6e896","DOI":"10.6100\/5074ed73-fcec-4a93-8a5c-699abcf6e896"},{"issue":"2","key":"10181_CR33","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/TMAG.2019.2953625","volume":"56","author":"LAJ Friedrich","year":"2020","unstructured":"Friedrich, L.A.J., Gysen, B.L.J., Jansen, J.W., Lomonova, E.A.: Analysis of motional eddy currents in the slitted stator core of an axial-flux permanent-magnet machine. IEEE Trans. Magn. 56(2), 1\u20134 (2020). https:\/\/doi.org\/10.1109\/TMAG.2019.2953625","journal-title":"IEEE Trans. Magn."},{"issue":"3","key":"10181_CR34","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1108\/eb010071","volume":"9","author":"K Fujiwara","year":"1990","unstructured":"Fujiwara, K., Nakata, T.: Results for benchmark problem 7 (asymmetrical conductor with a hole). COMPEL 9(3), 137\u2013154 (1990)","journal-title":"COMPEL"},{"issue":"1","key":"10181_CR35","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/TASC.2020.3013028","volume":"31","author":"F Grilli","year":"2021","unstructured":"Grilli, F., Pardo, E., Morandi, A., G\u00f6m\u00f6ry, F., Solovyov, M., Zerme\u00f1o, V.M.R., Brambilla, R., Benkel, T., Riva, N.: Electromagnetic modeling of superconductors with commercial software: possibilities with two vector potential-based formulations. IEEE Trans. Appl. Super. 31(1), 1\u20139 (2021). https:\/\/doi.org\/10.1109\/TASC.2020.3013028","journal-title":"IEEE Trans. Appl. Super."},{"key":"10181_CR36","volume-title":"Algebraic topology","author":"A Hatcher","year":"2001","unstructured":"Hatcher, A.: Algebraic topology. Cambridge University Press, New York (2001)"},{"issue":"6","key":"10181_CR37","doi-asserted-by":"publisher","first-page":"682","DOI":"10.1109\/tmag.2007.915991","volume":"44","author":"R Hiptmair","year":"2008","unstructured":"Hiptmair, R., Kramer, F., Ostrowski, J.: A robust Maxwell formulation for all frequencies. IEEE Trans. Magn. 44(6), 682\u2013685 (2008). https:\/\/doi.org\/10.1109\/tmag.2007.915991","journal-title":"IEEE Trans. Magn."},{"issue":"5","key":"10181_CR38","doi-asserted-by":"publisher","first-page":"1405","DOI":"10.1137\/S0097539701386526","volume":"31","author":"R Hiptmair","year":"2002","unstructured":"Hiptmair, R., Ostrowski, J.: Generators of $$h_1(\\gamma _{h},\\mathbb{Z} )$$ for triangulated surfaces: construction and classification. SIAM J. Comput. 31(5), 1405\u20131423 (2002)","journal-title":"SIAM J. Comput."},{"key":"10181_CR39","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110143","volume":"433","author":"F Holderied","year":"2021","unstructured":"Holderied, F., Possanner, S., Wang, X.: MHD-kinetic hybrid code based on structure-preserving finite elements with particles-in-cell. J. Comput. Phys. 433, 110143 (2021). https:\/\/doi.org\/10.1016\/j.jcp.2021.110143","journal-title":"J. Comput. Phys."},{"key":"10181_CR40","doi-asserted-by":"publisher","first-page":"4135","DOI":"10.1016\/j.cma.2004.10.008","volume":"194","author":"TJR Hughes","year":"2005","unstructured":"Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Meth. Appl. Mech. Eng. 194, 4135\u20134195 (2005). https:\/\/doi.org\/10.1016\/j.cma.2004.10.008","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"10181_CR41","doi-asserted-by":"publisher","unstructured":"Jackson, J.D.: Classical electrodynamics, 3rd edn. Wiley & Sons (1998) https:\/\/doi.org\/10.1017\/CBO9780511760396","DOI":"10.1017\/CBO9780511760396"},{"key":"10181_CR42","doi-asserted-by":"publisher","unstructured":"Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational homology. Springer New York (2004) https:\/\/doi.org\/10.1007\/b97315","DOI":"10.1007\/b97315"},{"issue":"5","key":"10181_CR43","doi-asserted-by":"publisher","first-page":"466","DOI":"10.1109\/20.106354","volume":"26","author":"A Kameari","year":"1990","unstructured":"Kameari, A.: Calculation of transient 3D eddy-current using edge elements. IEEE Trans. Magn. 26(5), 466\u2013469 (1990). https:\/\/doi.org\/10.1109\/20.106354","journal-title":"IEEE Trans. Magn."},{"key":"10181_CR44","unstructured":"Kapidani, B.: Topoprocessor (2023). https:\/\/github.com\/bkapidani\/topoprocessor"},{"issue":"10","key":"10181_CR45","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/TMAG.2016.2582471","volume":"52","author":"B Kapidani","year":"2016","unstructured":"Kapidani, B., D\u0142otko, P., Alotto, P., Bettini, P., Specogna, R.: Computation of relative 1-cohomology generators from a 1-homology basis for eddy currents boundary integral formulations. IEEE Trans. Magn. 52(10), 1\u20136 (2016). https:\/\/doi.org\/10.1109\/TMAG.2016.2582471","journal-title":"IEEE Trans. Magn."},{"key":"10181_CR46","doi-asserted-by":"publisher","unstructured":"Kapidani, B., Merkel, M., Sch\u00f6ps, S., V\u00e1zquez, R.: Tree-cotree decomposition of isogeometric mortared spaces in H(curl) on multi-patch domains. Comput. Meth. Appl. Mech. Eng. 395, 114949 (2022) https:\/\/doi.org\/10.1016\/j.cma.2022.114949Arxiv:2110.15860","DOI":"10.1016\/j.cma.2022.114949"},{"key":"10181_CR47","doi-asserted-by":"publisher","unstructured":"Kettunen, L., Forsman, K., Bossavit, A.: Formulation of the eddy current problem in multiply connected regions in terms of h. Int. J. Numer. Meth. Eng. 41(5), 935\u2013954 (1998). https:\/\/doi.org\/10.1002\/(SICI)1097-0207(19980315)41:5<935::AID-NME321>3.0.CO;2-F","DOI":"10.1002\/(SICI)1097-0207(19980315)41:5<935::AID-NME321>3.0.CO;2-F"},{"issue":"3","key":"10181_CR48","doi-asserted-by":"publisher","first-page":"1466","DOI":"10.1109\/20.767243","volume":"35","author":"L Kettunen","year":"1999","unstructured":"Kettunen, L., Forsman, K., Bossavit, A.: Gauging in Whitney spaces. IEEE Trans. Magn. 35(3), 1466\u20131469 (1999). https:\/\/doi.org\/10.1109\/20.767243","journal-title":"IEEE Trans. Magn."},{"key":"10181_CR49","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1016\/j.cma.2012.08.007","volume":"247","author":"SK Kleiss","year":"2012","unstructured":"Kleiss, S.K., Pechstein, C., J\u00fcttler, B., Tomar, S.: IETI - isogeometric tearing and interconnecting. Comput. Meth. Appl. Mech. Eng. 247, 201\u2013215 (2012). https:\/\/doi.org\/10.1016\/j.cma.2012.08.007","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"issue":"8","key":"10181_CR50","doi-asserted-by":"publisher","first-page":"3916","DOI":"10.1063\/1.338583","volume":"61","author":"PR Kotiuga","year":"1987","unstructured":"Kotiuga, P.R.: On making cuts for magnetic scalar potentials in multiply connected regions. JAP 61(8), 3916\u20133918 (1987). https:\/\/doi.org\/10.1063\/1.338583","journal-title":"JAP"},{"issue":"3","key":"10181_CR51","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1119\/1.2397095","volume":"75","author":"J Larsson","year":"2007","unstructured":"Larsson, J.: Electromagnetics from a quasistatic perspective. Am. J. Phys. 75(3), 230\u2013239 (2007). https:\/\/doi.org\/10.1119\/1.2397095","journal-title":"Am. J. Phys."},{"key":"10181_CR52","doi-asserted-by":"publisher","unstructured":"Ledger, P.D., Zaglmayr, S.: hp-finite element simulation of three-dimensional eddy current problems on multiply connected domains. Comput. Methods Appl. Mech. Eng. 199(49), 3386\u20133401 (2010). https:\/\/doi.org\/10.1016\/j.cma.2010.07.010https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0045782510002227","DOI":"10.1016\/j.cma.2010.07.010"},{"issue":"8","key":"10181_CR53","doi-asserted-by":"publisher","first-page":"1897","DOI":"10.1109\/tmtt.2003.815263","volume":"51","author":"SC Lee","year":"2003","unstructured":"Lee, S.C., Lee, J.F., Lee, R.: Hierarchical vector finite elements for analyzing waveguiding structures. IEEE Trans. Microw. Theor. Tech. 51(8), 1897\u20131905 (2003). https:\/\/doi.org\/10.1109\/tmtt.2003.815263","journal-title":"IEEE Trans. Microw. Theor. Tech."},{"key":"10181_CR54","doi-asserted-by":"publisher","unstructured":"Manges, J.B., Cendes, Z.J.: Tree-cotree decompositions for first-order complete tangential vector finite elements. Int. J. Numer. Meth. Eng. 40(9), 1667\u20131685 (1997). https:\/\/doi.org\/10.1002\/(SICI)1097-0207(19970515)40:9<1667::AID-NME133>3.0.CO;2-9","DOI":"10.1002\/(SICI)1097-0207(19970515)40:9<1667::AID-NME133>3.0.CO;2-9"},{"key":"10181_CR55","unstructured":"Merkel, M., Kapidani, B., Sch\u00f6ps, S., V\u00e1zquez, R.: Isogeometric mortaring for the 3D simulation of electric machines. In: 23rd Conference on the Computation of Electromagnetic Fields (COMPUMAG 2021). International Compumag Society (2022). http:\/\/www.compumag2021.com"},{"issue":"3","key":"10181_CR56","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1007\/BF01396415","volume":"35","author":"JC N\u00e9d\u00e9lec","year":"1980","unstructured":"N\u00e9d\u00e9lec, J.C.: Mixed finite elements in $${R}^3$$. Numer. Math. 35(3), 315\u2013341 (1980). https:\/\/doi.org\/10.1007\/BF01396415","journal-title":"Numer. Math."},{"issue":"5","key":"10181_CR57","doi-asserted-by":"publisher","first-page":"B1195","DOI":"10.1137\/130906556","volume":"35","author":"M Pellikka","year":"2013","unstructured":"Pellikka, M., Suuriniemi, S., Kettunen, L., Geuzaine, C.: Homology and cohomology computation in finite element modeling. SIAM J. Sci. Comput. 35(5), B1195\u2013B1214 (2013). https:\/\/doi.org\/10.1137\/130906556","journal-title":"SIAM J. Sci. Comput."},{"key":"10181_CR58","doi-asserted-by":"publisher","unstructured":"Pitassi, S., Ghiloni, R., Specogna, R.: Inverting the discrete curl operator: a novel graph algorithm to find a vector potential of a given vector field. J. Comput. Phys., 466, 111404 (2022). https:\/\/doi.org\/10.1016\/j.jcp.2022.111404https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0021999122004661","DOI":"10.1016\/j.jcp.2022.111404"},{"key":"10181_CR59","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1007\/s10915-011-9500-8","volume":"51","author":"A Ratnani","year":"2012","unstructured":"Ratnani, A., Sonnendr\u00fccker, E.: An arbitrary high-order spline finite element solver for the time domain Maxwell equations. J. Sci. Comput. 51, 87\u2013106 (2012)","journal-title":"J. Sci. Comput."},{"issue":"2","key":"10181_CR60","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1109\/20.996146","volume":"38","author":"Z Ren","year":"2002","unstructured":"Ren, Z.: T-$${\\Omega }$$ formulation for eddy-current problems in multiply connected regions. IEEE Trans. Magn. 38(2), 557\u2013560 (2002). https:\/\/doi.org\/10.1109\/20.996146","journal-title":"IEEE Trans. Magn."},{"issue":"3","key":"10181_CR61","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1108\/eb010070","volume":"9","author":"D Rodger","year":"1990","unstructured":"Rodger, D.: Benchmark problem 5 - the Bath cube (revised). COMPEL 9(3), 125\u2013135 (1990). https:\/\/doi.org\/10.1108\/eb010070","journal-title":"COMPEL"},{"key":"10181_CR62","doi-asserted-by":"publisher","unstructured":"Rodr\u00ed\u00adguez, A.A., Bertolazzi, E., Ghiloni, R., Valli, A.: Construction of a finite element basis of the first de Rham cohomology group and numerical solution of 3d magnetostatic problems. SIAM J. Numer. Anal. 51(4), 2380\u20132402 (2013) https:\/\/doi.org\/10.1137\/120890648","DOI":"10.1137\/120890648"},{"issue":"2","key":"10181_CR63","doi-asserted-by":"publisher","first-page":"374","DOI":"10.1108\/03321640510586015","volume":"24","author":"J Sch\u00f6berl","year":"2005","unstructured":"Sch\u00f6berl, J., Zaglmayr, S.: High order N\u00e9d\u00e9lec elements with local complete sequence properties. COMPEL 24(2), 374\u2013384 (2005). https:\/\/doi.org\/10.1108\/03321640510586015","journal-title":"COMPEL"},{"key":"10181_CR64","unstructured":"Spink, M., Claxton, D., de\u00a0Falco, C., V\u00e1zquez, R.: NURBS package (2021). https:\/\/gnu-octave.github.io\/packages\/nurbs\/. Accessed 14 Oct 2023"},{"key":"10181_CR65","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2528\/PIER00080101","volume":"32","author":"E Tonti","year":"2001","unstructured":"Tonti, E.: Finite formulation of the electromagnetic field. PIER 32, 1\u201344 (2001). https:\/\/doi.org\/10.2528\/PIER00080101","journal-title":"PIER"},{"key":"10181_CR66","doi-asserted-by":"publisher","unstructured":"V\u00e1zquez, R.: A new design for the implementation of isogeometric analysis in Octave and Matlab: GeoPDEs 3.0. Comput. Math. Appl. 72(3), 523\u2013554 (2016). https:\/\/doi.org\/10.1016\/j.camwa.2016.05.010","DOI":"10.1016\/j.camwa.2016.05.010"},{"key":"10181_CR67","doi-asserted-by":"publisher","unstructured":"Webb, J.P., Forghani, B.: The low-frequency performance of $$H-\\phi $$ and $$T-\\Omega $$ methods using edge elements for 3d eddy current problems. IEEE Trans. Magn. 29(6), 2461\u20132463 (1993). https:\/\/doi.org\/10.1109\/20.280983","DOI":"10.1109\/20.280983"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-024-10181-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-024-10181-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-024-10181-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,21]],"date-time":"2024-10-21T11:14:10Z","timestamp":1729509250000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-024-10181-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,19]]},"references-count":67,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2024,10]]}},"alternative-id":["10181"],"URL":"https:\/\/doi.org\/10.1007\/s10444-024-10181-0","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"type":"print","value":"1019-7168"},{"type":"electronic","value":"1572-9044"}],"subject":[],"published":{"date-parts":[[2024,8,19]]},"assertion":[{"value":"16 October 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 July 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 August 2024","order":3,"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 no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}}],"article-number":"92"}}