{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T23:47:16Z","timestamp":1769039236763,"version":"3.49.0"},"reference-count":49,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,26]],"date-time":"2022-01-26T00:00:00Z","timestamp":1643155200000},"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":["391977956 - SFB 1357"],"award-info":[{"award-number":["391977956 - SFB 1357"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>The plane\u2013cube intersection problem has been discussed in the literature since 1984 and iterative solutions to it have been used as part of piecewise linear interface construction (PLIC) in computational fluid dynamics simulation codes ever since. In many cases, PLIC is the bottleneck of these simulations regarding computing time, so a faster analytic solution to the plane\u2013cube intersection would greatly reduce the computing time for such simulations. We derive an analytic solution for all intersection cases and compare it to the previous solution from Scardovelli and Zaleski (Scardovelli, R.; Zaleski, S. Analytical relations connecting linear interfaces and volume fractions in rectangular grids. J. Comput. Phys.2000, 164, 228\u2013237), which we further improve to include edge cases and micro-optimize to reduce arithmetic operations and branching. We then extend our comparison regarding computing time and accuracy to include two different iterative solutions as well. We find that the best choice depends on the employed hardware platform: on the CPU, Newton\u2013Raphson is fastest with compiler optimization enabled, while analytic solutions perform better than iterative solutions without. On the GPU, the fastest method is our optimized version of the analytic SZ solution. We finally provide details on one of the applications of PLIC\u2014curvature calculation for the Volume-of-Fluid model used for free surface fluid simulations in combination with the lattice Boltzmann method.<\/jats:p>","DOI":"10.3390\/computation10020021","type":"journal-article","created":{"date-parts":[[2022,1,26]],"date-time":"2022-01-26T11:02:53Z","timestamp":1643194973000},"page":"21","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Analytic Solution to the Piecewise Linear Interface Construction Problem and Its Application in Curvature Calculation for Volume-of-Fluid Simulation Codes"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4652-8383","authenticated-orcid":false,"given":"Moritz","family":"Lehmann","sequence":"first","affiliation":[{"name":"Biofluid Simulation and Modeling, Theoretische Physik VI, University of Bayreuth, 95448 Bayreuth, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5597-1160","authenticated-orcid":false,"given":"Stephan","family":"Gekle","sequence":"additional","affiliation":[{"name":"Biofluid Simulation and Modeling, Theoretische Physik VI, University of Bayreuth, 95448 Bayreuth, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,26]]},"reference":[{"key":"ref_1","unstructured":"Youngs, D.L. (1982). Time-dependent multi-material flow with large fluid distortion. Numerical Methods in Fluid Dynamics, Academic Press."},{"key":"ref_2","unstructured":"Youngs, D.L. (1984). An Interface Tracking Method for a 3D Eulerian Hydrodynamics Code, Atomic Weapons Research Establishment (AWRE). Technical Report."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.camwa.2012.05.012","article-title":"On enhanced non-linear free surface flow simulations with a hybrid LBM\u2013VOF model","volume":"65","author":"Grilli","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1006\/jcph.2000.6567","article-title":"Analytical relations connecting linear interfaces and volume fractions in rectangular grids","volume":"164","author":"Scardovelli","year":"2000","journal-title":"J. Comput. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.compfluid.2016.05.014","article-title":"A simple volume-of-fluid reconstruction method for three-dimensional two-phase flows","volume":"134","author":"Kawano","year":"2016","journal-title":"Comput. Fluids"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1186\/s43591-021-00018-8","article-title":"Ejection of marine microplastics by raindrops: A computational and experimental study","volume":"1","author":"Lehmann","year":"2021","journal-title":"Microplastics Nanoplastics"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1186\/s43591-021-00010-2","article-title":"Tracing the horizontal transport of microplastics on rough surfaces","volume":"1","author":"Laermanns","year":"2021","journal-title":"Microplastics Nanoplastics"},{"key":"ref_8","unstructured":"Lehmann, M. (2019). High Performance Free Surface LBM on GPUs. [Master\u2019s Thesis, University of Bayreuth]."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"043302","DOI":"10.1103\/PhysRevE.93.043302","article-title":"Curvature estimation from a volume-of-fluid indicator function for the simulation of surface tension and wetting with a free-surface lattice Boltzmann method","volume":"93","author":"Bogner","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s10955-005-8879-8","article-title":"Lattice Boltzmann model for free surface flow for modeling foaming","volume":"121","author":"Thies","year":"2005","journal-title":"J. Stat. Phys."},{"key":"ref_11","unstructured":"Th\u00fcrey, N., K\u00f6rner, C., and R\u00fcde, U. (2005). Interactive Free Surface Fluids with the Lattice Boltzmann Method, University of Erlangen-Nuremberg. Technical Report 05-4."},{"key":"ref_12","unstructured":"Pohl, T. (2008). High Performance Simulation of Free Surface Flows Using the Lattice Boltzmann Method, Verlag Dr. Hut."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"984","DOI":"10.1016\/j.procs.2011.04.104","article-title":"Free-surface lattice-Boltzmann simulation on many-core architectures","volume":"4","author":"Schreiber","year":"2011","journal-title":"Procedia Comput. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5838","DOI":"10.1016\/j.jcp.2009.04.042","article-title":"An accurate adaptive solver for surface-tension-driven interfacial flows","volume":"228","author":"Popinet","year":"2009","journal-title":"J. Comput. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1080\/10618560701440915","article-title":"An improved three-dimensional model for interface pressure calculations in free-surface flows","volume":"21","author":"Jafari","year":"2007","journal-title":"Int. J. Comput. Fluid Dyn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1002\/fld.1282","article-title":"Lattice Boltzmann-based single-phase method for free surface tracking of droplet motions","volume":"53","author":"Xing","year":"2007","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_17","unstructured":"Donath, S. (2011). Wetting Models for a Parallel High-Performance Free Surface Lattice Boltzmann Method: Benetzungsmodelle F\u00fcr Eine Parallele Lattice-Boltzmann-Methode Mit Freien Oberfl\u00e4chen, Verlag Dr. Hut."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.compfluid.2010.12.027","article-title":"Verification of surface tension in the parallel free surface lattice Boltzmann method in waLBerla","volume":"45","author":"Donath","year":"2011","journal-title":"Comput. Fluids"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1016\/j.camwa.2013.06.007","article-title":"Free surface lattice Boltzmann with enhanced bubble model","volume":"67","author":"Anderl","year":"2014","journal-title":"Comput. Math. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3628","DOI":"10.1016\/j.camwa.2010.01.054","article-title":"A new approach to the lattice Boltzmann method for graphics processing units","volume":"61","author":"Obrecht","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_21","unstructured":"Wittmann, M. (2016). Hardware-effiziente, hochparallele Implementierungen von Lattice-Boltzmann-Verfahren f\u00fcr komplexe Geometrien. [Ph.D. Thesis, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg (FAU)]."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1016\/j.camwa.2013.10.002","article-title":"Optimized implementation of the Lattice Boltzmann Method on a graphics processing unit towards real-time fluid simulation","volume":"67","author":"Delbosc","year":"2014","journal-title":"Comput. Math. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Herschlag, G., Lee, S., Vetter, J.S., and Randles, A. (2018, January 21\u201325). GPU data access on complex geometries for d3q19 lattice Boltzmann method. Proceedings of the 2018 IEEE International Parallel and Distributed Processing Symposium (IPDPS), Vancouver, BC, Canada.","DOI":"10.1109\/IPDPS.2018.00092"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2566","DOI":"10.1016\/j.cpc.2014.06.003","article-title":"Memory transfer optimization for a lattice Boltzmann solver on Kepler architecture nVidia GPUs","volume":"185","author":"Mawson","year":"2014","journal-title":"Comput. Phys. Commun."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"924","DOI":"10.1016\/j.camwa.2012.05.002","article-title":"Comparison of different propagation steps for lattice Boltzmann methods","volume":"65","author":"Wittmann","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2380","DOI":"10.1016\/j.camwa.2009.08.052","article-title":"LBM based flow simulation using GPU computing processor","volume":"59","author":"Kuznik","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Kr\u00fcger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., and Viggen, E.M. (2017). The Lattice Boltzmann Method, Springer International Publishing.","DOI":"10.1007\/978-3-319-44649-3"},{"key":"ref_28","unstructured":"Chapman, S., Cowling, T.G., and Burnett, D. (1990). The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"012035","DOI":"10.1088\/1742-6596\/877\/1\/012035","article-title":"Accuracy and Numerical Stabilty Analysis of Lattice Boltzmann Method with Multiple Relaxation Time for Incompressible Flows","volume":"877","author":"Purqon","year":"2017","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Wu, X., and Wu, E. (2009, January 19\u201321). Bubble creation and multi-fluids interaction. Proceedings of the 2009 11th IEEE International Conference on Computer-Aided Design and Computer Graphics, Huangshan, China.","DOI":"10.1109\/CADCG.2009.5246926"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1646","DOI":"10.1016\/j.ijheatmasstransfer.2007.07.037","article-title":"Numerical simulation of film boiling on a sphere with a volume of fluid interface tracking method","volume":"51","author":"Yuan","year":"2008","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_32","first-page":"HW-008","article-title":"A VOF-based method for the simulation of thermocapillary flow","volume":"Volume 63","author":"Ma","year":"2010","journal-title":"APS Division of Fluid Dynamics Meeting Abstracts"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"110041","DOI":"10.1016\/j.oceaneng.2021.110041","article-title":"Modeling of solitary wave interaction with emerged porous breakwater using PLIC-VOF method","volume":"241","author":"Booshi","year":"2021","journal-title":"Ocean Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"90","DOI":"10.9753\/icce.v36.papers.90","article-title":"A lattice Boltzmann approach for three-dimensional tsunami simulation based on the PLIC-VOF method","volume":"36","author":"Sato","year":"2018","journal-title":"Coast. Eng. Proc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1016\/j.ijmecsci.2011.07.009","article-title":"Ejecting performance simulation of an innovative piezoelectric actuated lubrication generator for space mechanisms","volume":"53","author":"Sheng","year":"2011","journal-title":"Int. J. Mech. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1241","DOI":"10.1111\/j.1467-8659.2009.01671.x","article-title":"Visualization and Analysis-Oriented Reconstruction of Material Interfaces","volume":"Volume 29","author":"Meredith","year":"2010","journal-title":"Computer Graphics Forum"},{"key":"ref_37","unstructured":"NVIDIA (2021, July 21). Parallel Thread Execution ISA Version 6.4. Available online: https:\/\/docs.nvidia.com\/cuda\/parallel-thread-execution\/."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"104950","DOI":"10.1016\/j.compfluid.2021.104950","article-title":"NPLIC: A machine learning approach to piecewise linear interface construction","volume":"223","author":"Ataei","year":"2021","journal-title":"Comput. Fluids"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1129","DOI":"10.1175\/JAS-D-12-0100.1","article-title":"Effects of altitude on maximum raindrop size and fall velocity as limited by collisional breakup","volume":"70","author":"Prodi","year":"2013","journal-title":"J. Atmos. Sci."},{"key":"ref_40","unstructured":"Bourke, P. (2021, July 21). Polygonising a Scalar Field. Available online: http:\/\/paulbourke.net\/geometry\/polygonise\/."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1145\/37402.37422","article-title":"Marching cubes: A high resolution 3D surface construction algorithm","volume":"21","author":"Lorensen","year":"1987","journal-title":"ACM Siggraph Comput. Graph."},{"key":"ref_42","unstructured":"Vega, D., Abache, J., and Coll, D. (2019). A Fast and Memory-Saving Marching Cubes 33 Implementation with the Correct Interior Test. J. Comput. Graph. Tech. Vol., 3, Available online: https:\/\/jcgt.org\/published\/0008\/03\/01\/paper.pdf."},{"key":"ref_43","unstructured":"Parker, B., and Youngs, D. (1992). Two and Three Dimensional Eulerian Simulation of Fluid Flow with Material Interfaces, Atomic Weapons Establishment."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Pressley, A.N. (2010). Elementary Differential Geometry, Springer Science & Business Media.","DOI":"10.1007\/978-1-84882-891-9"},{"key":"ref_45","unstructured":"Abbena, E., Salamon, S., and Gray, A. (2017). Modern Differential Geometry of Curves and Surfaces with Mathematica, Chapman and Hall\/CRC."},{"key":"ref_46","first-page":"23","article-title":"Focal surfaces of discrete geometry","volume":"Volume 257","author":"Yu","year":"2007","journal-title":"ACM International Conference Proceeding Series"},{"key":"ref_47","unstructured":"Har\u2019el, Z. (1995). Curvature of Curves and Surfaces\u2014A Parabolic Approach, Department of Mathematics, Technion\u2013Israel Institute of Technology."},{"key":"ref_48","unstructured":"Jia, Y.B. (2018). Gaussian and Mean Curvatures, Iowa State University. Com S 477\/577 Notes."},{"key":"ref_49","unstructured":"Eberly, D. (2000). Least Squares Fitting of Data, Magic Software."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/10\/2\/21\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:08:29Z","timestamp":1760134109000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/10\/2\/21"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,26]]},"references-count":49,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["computation10020021"],"URL":"https:\/\/doi.org\/10.3390\/computation10020021","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,26]]}}}