{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T07:21:51Z","timestamp":1772090511515,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T00:00:00Z","timestamp":1661299200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T00:00:00Z","timestamp":1661299200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["431968427"],"award-info":[{"award-number":["431968427"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100020618","name":"Universit\u00e4t Bayreuth","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100020618","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The phase-field method provides a powerful framework for microstructure evolution modeling in complex systems, as often required within the framework of integrated computational materials engineering. However, spurious grid friction, pinning and grid anisotropy seriously limit the resolution efficiency and accuracy of these models. The energetic resolution limit is determined by the maximum dimensionless driving force at which reasonable model operation is still ensured. This limit turns out to be on the order of 1 for conventional phase-field models. In 1D, grid friction and pinning can be eliminated by a global restoration of Translational Invariance (TI) in the discretized phase-field equation. This is called the sharp phase-field method, which allows to choose substantially coarser numerical resolutions of the diffuse interface without the appearance of pinning. In 3D, global TI restricts the beneficial properties to a few specific interface orientations. We propose an accurate scheme to restore TI locally in the local interface normal direction. The new sharp phase-field model overcomes grid friction and pinning in three-dimensional simulations, and can accurately operate at dimensionless driving forces up to the order of <jats:inline-formula><jats:alternatives><jats:tex-math>$$10^{4}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mn>10<\/mml:mn>\n                    <mml:mn>4<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. At one-grid-point interface resolutions, exceptional degrees of isotropy can be achieved, if further the largely inhomogeneous latent heat release at the advancing solid-liquid interface is mitigated. Imposing a newly proposed source term regularization, the new model captures the formation of isotropic seaweed structures without spurious dendritic selection by grid anisotropy, even at one-grid-point interface resolutions.<\/jats:p>","DOI":"10.1007\/s00366-022-01729-z","type":"journal-article","created":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T20:02:30Z","timestamp":1661371350000},"page":"1699-1709","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Sharp phase-field modeling of isotropic solidification with a super efficient spatial resolution"],"prefix":"10.1007","volume":"39","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2799-9075","authenticated-orcid":false,"given":"Michael","family":"Fleck","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1189-9288","authenticated-orcid":false,"given":"Felix","family":"Schleifer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,8,24]]},"reference":[{"key":"1729_CR1","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1080\/09506608.2020.1757894","volume":"66","author":"W Kurz","year":"2021","unstructured":"Kurz W, Rappaz M, Trivedi R (2021) Progress in modelling solidification microstructures in metals and alloys. part ii: dendrites from 2001 to 2018. Int Mater Rev 66:30\u201376","journal-title":"Int Mater Rev"},{"key":"1729_CR2","doi-asserted-by":"publisher","unstructured":"Tourret D, Liu H, LLorca J (2021) Phase-field modeling of microstructure evolution: Recent applications, perspectives and challenges. Prog Mater Sci 100810 . https:\/\/doi.org\/10.1016\/j.pmatsci.2021.100810","DOI":"10.1016\/j.pmatsci.2021.100810"},{"key":"1729_CR3","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1146\/annurev-matsci-070218-010151","volume":"49","author":"MR Tonks","year":"2019","unstructured":"Tonks MR, Aagesen LK (2019) The phase field method: Mesoscale simulation aiding material discovery. Annu Rev Mater Res 49:79\u2013102. https:\/\/doi.org\/10.1146\/annurev-matsci-070218-010151","journal-title":"Annu Rev Mater Res"},{"key":"1729_CR4","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1016\/j.commatsci.2018.03.015","volume":"149","author":"AM Jokisaari","year":"2018","unstructured":"Jokisaari AM, Voorhees PW, Guyer JE, Warren JA, Heinonen O (2018) Phase field benchmark problems for dendritic growth and linear elasticity. Comp. Mater. Sci. 149:336\u2013347. https:\/\/doi.org\/10.1016\/j.commatsci.2018.03.015","journal-title":"Comp. Mater. Sci."},{"key":"1729_CR5","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.121.025501","volume":"121","author":"A Finel","year":"2018","unstructured":"Finel A, Le Bouar Y, Dabas B, Appolaire B, Yamada Y, Mohri T (2018) Sharp phase field method. Phys Rev Lett 121:025501. https:\/\/doi.org\/10.1103\/PhysRevLett.121.025501","journal-title":"Phys Rev Lett"},{"key":"1729_CR6","doi-asserted-by":"publisher","first-page":"695","DOI":"10.1006\/jcph.2001.6933","volume":"174","author":"K Glasner","year":"2001","unstructured":"Glasner K (2001) Nonlinear preconditioning for diffuse interfaces. J. Comp. Phys. 174:695\u2013711. https:\/\/doi.org\/10.1006\/jcph.2001.6933","journal-title":"J. Comp. Phys."},{"key":"1729_CR7","doi-asserted-by":"publisher","first-page":"1858","DOI":"10.1016\/j.apnum.2009.01.010","volume":"59","author":"M Weiser","year":"2009","unstructured":"Weiser M (2009) Pointwise nonlinear scaling for reaction-diffusion equations. Appl. Num. Math. 59:1858\u20131869. https:\/\/doi.org\/10.1016\/j.apnum.2009.01.010","journal-title":"Appl. Num. Math."},{"key":"1729_CR8","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.94.013001","volume":"94","author":"J-M Debierre","year":"2016","unstructured":"Debierre J-M, Gu\u00e9rin R, Kassner K (2016) Phase-field study of crystal growth in three-dimensional capillaries: Effects of crystalline anisotropy. Phys Rev E 94:013001. https:\/\/doi.org\/10.1103\/PhysRevE.94.013001","journal-title":"Phys Rev E"},{"key":"1729_CR9","doi-asserted-by":"publisher","first-page":"338","DOI":"10.1016\/j.commatsci.2018.02.003","volume":"147","author":"TZ Gong","year":"2018","unstructured":"Gong TZ, Chen Y, Cao YF, Kang XH, Li DZ (2018) Fast simulations of a large number of crystals growth in centimeter-scale during alloy solidification via nonlinearly preconditioned quantitative phase-field formula. Comp. Mater. Sci. 147:338\u2013352. https:\/\/doi.org\/10.1016\/j.commatsci.2018.02.003","journal-title":"Comp. Mater. Sci."},{"key":"1729_CR10","doi-asserted-by":"publisher","first-page":"474","DOI":"10.1137\/17M1150153","volume":"61","author":"J Shen","year":"2019","unstructured":"Shen J, Xu J, Yang J (2019) A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev 61:474\u2013506. https:\/\/doi.org\/10.1137\/17M1150153","journal-title":"SIAM Rev"},{"key":"1729_CR11","doi-asserted-by":"publisher","unstructured":"Ji, K., Molavi Tabrizi, A., Karma, A.: Isotropic finite-difference approximations for phase-field simulations of polycrystalline alloy solidification. J Comput Phys 111069 (2022). https:\/\/doi.org\/10.1016\/j.jcp.2022.111069","DOI":"10.1016\/j.jcp.2022.111069"},{"key":"1729_CR12","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1186\/s41313-021-00033-5","volume":"6","author":"S Sakane","year":"2022","unstructured":"Sakane S, Takaki T, Aoki T (2022) Parallel-gpu-accelerated adaptive mesh refinement for three-dimensional phase-field simulation of dendritic growth during solidification of binary alloy. Mater Theory 6:3. https:\/\/doi.org\/10.1186\/s41313-021-00033-5","journal-title":"Mater Theory"},{"key":"1729_CR13","doi-asserted-by":"publisher","first-page":"444","DOI":"10.1063\/1.1713333","volume":"35","author":"WW Mullins","year":"1964","unstructured":"Mullins WW, Sekerka RF (1964) Stability of a planar interface during solidification of a dilute binary alloy. J Appl Phys 35:444\u2013451. https:\/\/doi.org\/10.1063\/1.1713333","journal-title":"J Appl Phys"},{"key":"1729_CR14","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1016\/S0378-4371(97)00433-0","volume":"249","author":"EA Brener","year":"1998","unstructured":"Brener EA, M\u00fcller-Krumbhaar H, Temkin DE, Abel T (1998) Morphology diagram of possible structures in diffusional growth. Phys A 249:73\u201381. https:\/\/doi.org\/10.1016\/S0378-4371(97)00433-0","journal-title":"Phys A"},{"issue":"24","key":"1729_CR15","doi-asserted-by":"publisher","first-page":"4605","DOI":"10.1557\/jmr.2017.393","volume":"32","author":"M Fleck","year":"2017","unstructured":"Fleck M, Querfurth F, Glatzel U (2017) Phase field modeling of solidification in multi-component alloys with a case study on the Inconel 718 alloy. J Mater Res 32(24):4605\u20134615. https:\/\/doi.org\/10.1557\/jmr.2017.393","journal-title":"J Mater Res"},{"key":"1729_CR16","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/PL00011060","volume":"16","author":"T Ihle","year":"2000","unstructured":"Ihle T (2000) Competition between kinetic and surface tension anisotropy in dendritic growth. Euro Phys J B 16:337\u2013344. https:\/\/doi.org\/10.1007\/PL00011060","journal-title":"Euro Phys J B"},{"key":"1729_CR17","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1023\/A:1015815928191","volume":"10","author":"J Bragard","year":"2002","unstructured":"Bragard J, Karma A, Lee YH, Plapp M (2002) Linking Phase-Field and Atomistic Simulations to Model Dendritic Solidification in Highly Undercooled Melts. Interf Sci 10:121. https:\/\/doi.org\/10.1023\/A:1015815928191","journal-title":"Interf Sci"},{"key":"1729_CR18","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/j.commatsci.2014.07.037","volume":"95","author":"K Reuther","year":"2014","unstructured":"Reuther K, Rettenmayr M (2014) Perspectives for cellular automata for the simulation of dendritic solidification: a review. Comput Mater Sci 95:213\u2013220","journal-title":"Comput Mater Sci"},{"key":"1729_CR19","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.84.031601","volume":"84","author":"M Plapp","year":"2011","unstructured":"Plapp M (2011) Unified derivation of phase-field models for alloy solidification from a grand-potential functional. Phys Rev E 84:031601. https:\/\/doi.org\/10.1103\/PhysRevE.84.031601","journal-title":"Phys Rev E"},{"key":"1729_CR20","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.96.033311","volume":"96","author":"M Ohno","year":"2017","unstructured":"Ohno M, Takaki T, Shibuta Y (2017) Variational formulation of a quantitative phase-field model for nonisothermal solidification in a multicomponent alloy. Phys Rev E 96:033311. https:\/\/doi.org\/10.1103\/PhysRevE.96.033311","journal-title":"Phys Rev E"},{"key":"1729_CR21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.98.023309","volume":"98","author":"LK Aagesen","year":"2018","unstructured":"Aagesen LK, Gao Y, Schwen D, Ahmed K (2018) Grand-potential-based phase-field model for multiple phases, grains, and chemical components. Phys Rev E 98:023309. https:\/\/doi.org\/10.1103\/PhysRevE.98.023309","journal-title":"Phys Rev E"},{"key":"1729_CR22","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1016\/j.commatsci.2017.09.029","volume":"142","author":"M Greenwood","year":"2018","unstructured":"Greenwood M, Shampur KN, Ofori-Opoku N, Pinomaa T, Wang L, Gurevich S, Provatas N (2018) Quantitative 3d phase field modelling of solidification using next-generation adaptive mesh refinement. Comp Mater Sci 142:153. https:\/\/doi.org\/10.1016\/j.commatsci.2017.09.029","journal-title":"Comp Mater Sci"},{"key":"1729_CR23","doi-asserted-by":"publisher","DOI":"10.1016\/j.pmatsci.2019.05.002","volume":"106","author":"L Gr\u00e1n\u00e1sy","year":"2019","unstructured":"Gr\u00e1n\u00e1sy L, T\u00f3th GI, Warren JA, Podmaniczky F, Tegze G, R\u00e1tkai L, Pusztai T (2019) Phase-field modeling of crystal nucleation in undercooled liquids: a review. Prog Mater Sci 106:100569. https:\/\/doi.org\/10.1016\/j.pmatsci.2019.05.002","journal-title":"Prog Mater Sci"},{"key":"1729_CR24","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.101.022802","volume":"101","author":"K Kim","year":"2020","unstructured":"Kim K, Sherman QC, Aagesen LK, Voorhees PW (2020) Phase-field model of oxidation: Kinetics. Phys Rev E 101:022802. https:\/\/doi.org\/10.1103\/PhysRevE.101.022802","journal-title":"Phys Rev E"},{"key":"1729_CR25","doi-asserted-by":"publisher","first-page":"288","DOI":"10.1016\/j.commatsci.2018.06.049","volume":"153","author":"M Fleck","year":"2018","unstructured":"Fleck M, Federmann H, Pogorelov E (2018) Phase-field modeling of li-insertion kinetics in single LiFePO4-nano-particles for rechargeable li-ion battery application. Comp Mater Sci 153:288\u2013296. https:\/\/doi.org\/10.1016\/j.commatsci.2018.06.049","journal-title":"Comp Mater Sci"},{"key":"1729_CR26","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1016\/j.actamat.2020.09.073","volume":"201","author":"A Dimokrati","year":"2020","unstructured":"Dimokrati A, Le Bouar Y, Benyoucef M, Finel A (2020) S-pfm model for ideal grain growth. Acta Mater 201:147\u2013157. https:\/\/doi.org\/10.1016\/j.actamat.2020.09.073","journal-title":"Acta Mater"},{"key":"1729_CR27","doi-asserted-by":"publisher","DOI":"10.1016\/j.intermet.2020.106745","volume":"120","author":"F Schleifer","year":"2020","unstructured":"Schleifer F, Holzinger M, Lin Y-Y, Glatzel U, Fleck M (2020) Phase-field modeling of a $$\\gamma$$\/$$\\gamma ^{\\prime \\prime }$$ microstructure in nickel-base superalloys with high $$\\gamma ^{\\prime \\prime }$$ volume fraction. Intermetallics 120:106745","journal-title":"Intermetallics"},{"key":"1729_CR28","doi-asserted-by":"publisher","unstructured":"Schleifer F, Fleck M, Holzinger M, Lin Y-Y, Glatzel U (2020) Phase-field modeling of $$\\gamma ^{\\prime }$$ and $$\\gamma ^{\\prime \\prime }$$ precipitate size evolution during heat treatment of Ni-base superalloys. Superalloys 2020, pp. 500\u2013508. Springer, Cham. Chap. 49. https:\/\/doi.org\/10.1007\/978-3-030-51834-9_49","DOI":"10.1007\/978-3-030-51834-9_49"},{"key":"1729_CR29","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.82.021606","volume":"82","author":"K Kassner","year":"2010","unstructured":"Kassner K, Gu\u00e9rin R, Ducousso T, Debierre J-M (2010) Phase-field study of solidification in three-dimensional channels. Phys Rev E 82:021606. https:\/\/doi.org\/10.1103\/PhysRevE.82.021606","journal-title":"Phys Rev E"},{"key":"1729_CR30","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1080\/14786430903193241","volume":"90","author":"M Fleck","year":"2010","unstructured":"Fleck M, H\u00fcter C, Pilipenko D, Spatschek R, Brener EA (2010) Pattern formation during diffusion limited transformations in solids. Phil Mag 90:265. https:\/\/doi.org\/10.1080\/14786430903193241","journal-title":"Phil Mag"},{"key":"1729_CR31","doi-asserted-by":"publisher","unstructured":"Eiken J (2012) Numerical solution of the phase-field equation with minimized discretization error. IOP Conf Ser Mater Sci Eng 33:012105. https:\/\/doi.org\/10.1088\/1757-899X\/33\/1\/012105","DOI":"10.1088\/1757-899X\/33\/1\/012105"},{"key":"1729_CR32","doi-asserted-by":"publisher","first-page":"5887","DOI":"10.1103\/PhysRevA.39.5887","volume":"39","author":"G Caginalp","year":"1989","unstructured":"Caginalp G (1989) Stefan and hele-shaw type models as asymptotic limits of the phase-field equations. Phys Rev A 39:5887\u20135896. https:\/\/doi.org\/10.1103\/PhysRevA.39.5887","journal-title":"Phys Rev A"},{"key":"1729_CR33","doi-asserted-by":"publisher","first-page":"4323","DOI":"10.1103\/PhysRevE.57.4323","volume":"57","author":"A Karma","year":"1998","unstructured":"Karma A, Rappel W-J (1998) Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys Rev E 57:4323\u20134349. https:\/\/doi.org\/10.1103\/PhysRevE.57.4323","journal-title":"Phys Rev E"},{"key":"1729_CR34","doi-asserted-by":"publisher","first-page":"462","DOI":"10.3139\/146.110295","volume":"4","author":"M Fleck","year":"2010","unstructured":"Fleck M, Brener EA, Spatschek R, Eidel B (2010) Elastic and plastic effects on solid-state transformations: a phase field study. Int J Mater Res 4:462. https:\/\/doi.org\/10.3139\/146.110295","journal-title":"Int J Mater Res"},{"key":"1729_CR35","doi-asserted-by":"publisher","first-page":"2972","DOI":"10.1103\/PhysRevE.49.2972","volume":"49","author":"T Ihle","year":"1994","unstructured":"Ihle T, M\u00fcller-Krumbhaar H (1994) Fractal and compact growth morphologies in phase transitions with diffusion transport. Phys Rev E 49:2972\u20132991. https:\/\/doi.org\/10.1103\/PhysRevE.49.2972","journal-title":"Phys Rev E"},{"key":"1729_CR36","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.72.011601","volume":"72","author":"B Utter","year":"2005","unstructured":"Utter B, Bodenschatz E (2005) Double dendrite growth in solidification. Phys Rev E 72:011601. https:\/\/doi.org\/10.1103\/PhysRevE.72.011601","journal-title":"Phys Rev E"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-022-01729-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-022-01729-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-022-01729-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T19:17:09Z","timestamp":1685560629000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-022-01729-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,24]]},"references-count":36,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["1729"],"URL":"https:\/\/doi.org\/10.1007\/s00366-022-01729-z","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,8,24]]},"assertion":[{"value":"7 March 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 August 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 August 2022","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 have no conflict of interest to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}