{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,23]],"date-time":"2025-10-23T11:17:43Z","timestamp":1761218263556,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,11]],"date-time":"2020-07-11T00:00:00Z","timestamp":1594425600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004543","name":"China Scholarship Council","doi-asserted-by":"publisher","award":["201808260026"],"award-info":[{"award-number":["201808260026"]}],"id":[{"id":"10.13039\/501100004543","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Ministry of Education","award":["2019R1A6A3A13094308"],"award-info":[{"award-number":["2019R1A6A3A13094308"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we present several numerical simulation results of dendritic pattern formation using an isotropic crystal growth model, which is based on phase-field modeling, on curved surfaces. An explicit time-stepping method is used and the direct computing method to the Laplace\u2013Beltrami operator, which employs the point centered triangulation approximating Laplacian over the discretized surface with a triangular mesh, is adopted. Numerical simulations are performed not only on simple but also on complex surfaces with various curvatures, and the proposed method can simulate dendritic growth on complex surfaces. In particular, ice crystal growth simulation results on aircraft fuselage or metal bell-shaped curved surfaces are provided in order to demonstrate the practical relevance to our dendrite growth model. Furthermore, we perform several numerical parameter tests to obtain a best fitted set of parameters on simple surfaces. Finally, we apply this set of parameters to numerical simulation on complex surfaces.<\/jats:p>","DOI":"10.3390\/sym12071155","type":"journal-article","created":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T05:10:30Z","timestamp":1595394630000},"page":"1155","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Numerical Simulation of Dendritic Pattern Formation in an Isotropic Crystal Growth Model on Curved Surfaces"],"prefix":"10.3390","volume":"12","author":[{"given":"Sungha","family":"Yoon","sequence":"first","affiliation":[{"name":"Department of Mathematics, Korea University, Seoul 02841, Korea"}]},{"given":"Jintae","family":"Park","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Korea University, Seoul 02841, Korea"}]},{"given":"Jian","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1120-2455","authenticated-orcid":false,"given":"Chaeyoung","family":"Lee","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Korea University, Seoul 02841, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0484-9189","authenticated-orcid":false,"given":"Junseok","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Korea University, Seoul 02841, Korea"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"708","DOI":"10.1006\/jcph.2001.6726","article-title":"A front-tracking method for the computations of multiphase flow","volume":"169","author":"Tryggvason","year":"2001","journal-title":"J. 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