Minimal Surfaces and the Plateau Problem: Numerical Methods and Applications

Sapagovas, Mifodijus; Būda, Vytautas; Maskeliūnas, Saulius; Štikonienė, Olga; Štikonas, Artūras

doi:10.15388/24-INFOR552

Informatica

Minimal Surfaces and the Plateau Problem: Numerical Methods and Applications

Volume 35, Issue 2 (2024), pp. 401–420

Mifodijus Sapagovas

Vytautas Būda Saulius Maskeliūnas

Olga Štikonienė

Artūras Štikonas

https://doi.org/10.15388/24-INFOR552

Pub. online: 9 April 2024 Type: Research Article

Open Access

Received
1 February 2024

Accepted
1 April 2024

Published
9 April 2024

Abstract

This article offers a comprehensive overview of the results obtained through numerical methods in solving the minimal surface equation, along with exploring the applications of minimal surfaces in science, technology, and architecture. The content is enriched with practical examples highlighting the diverse applications of minimal surfaces.

References

Agalcev, A., Sapagovas, M. (1967). The solution of the equation of minimal surface by finite-difference method. Lithuanian Mathematical Journal, 7(3), 373–379 (in Russian). https://doi.org/10.15388/LMJ.1967.19976.

Ambrazevičius, A. (1981). Solvability of the problem in a conical capillary. Problems of mathematical analysis, 8, 3–23.

Ambrazevičius, A.P. (1991). Finding the form of the surface of a liquid in a conical container for a given volume of the liquid. I. Lithuanian Mathematical Journal, 22(1), 1–6. https://doi.org/10.1007/BF00967921.

Burneika, J. (2002). Forma, kompozicija, dizainas. VDA leidykla.

Concus, P. (1967). Numerical solution of the minimal surface equation. Mathematics of Computation, 21(99), 340–350. https://doi.org/10.2307/2003235.

Čiupaila, R., Sapagovas, M. (2002). Solution of the system of parametric equations of the sessile drop. Nonlinear Analysis: Modelling and Control, 7(2), 201–206. https://doi.org/10.3846/13926292.2002.9637192.

Čiupaila, R., Sapagovas, M., Štikonienė, O. (2013). Numerical solution of nonlinear elliptic equation with nonlocal condition. Nonlinear Analysis: Modelling and Control, 18(4), 412–426. https://doi.org/10.15388/NA.18.4.13970.

Douglas, J. (1927–1928). A method of numerical solution of the problem of Plateau. Annals of Mathematics, 29(1/4), 180–188. https://doi.org/10.2307/1967991.

Drew, P. (1976). Frei Otto: Form and Structure. Westview Press.

Fomenko, A.T. (1989). The Plateau Problem: Historical Survey. Gordon & Breach, Williston, VT.

Glotov, D., Hames, W.E., Meir, A.J., Ngoma, S. (2016). An integral constrained parabolic problem with applications in thermochronology. Computers & Mathematics with Applications, 71(11), 2301–2312. https://doi.org/10.1016/j.camwa.2016.01.017.

Greenspan, D. (1965). On approximating extremals of functionals. Part I. The method and examples for boundary value problems. International Computing Centre Bulletin, University of Roma, 4, 99–120.

Halbrecht, H. (2009). On the numerical solution of Plateau’s problem. Applied Numerical Mathematics, 59(11), 2785–2800. https://doi.org/10.1016/j.apnum.2008.12.028.

Hyde, S., Andersson, S., Larsson, K., Blum, Z., Landh, T., Lidin, S., Ninham, B.W. (1997). The Language of Shape. The Role of Curvature in Condensed Matter: Physics, Chemistry and Biology. Elsevier, Amsterdam.

Johnson, C., Thomée, V. (1975). Error estimation for a finite element approximation of a minimal surface. Mathematics of Computation, 29(130), 343–349. https://doi.org/10.2307/2005555.

Kasuba, A. (2019). Kasuba Works. https://www.kasubaworks.com.

Larsson, K. (2005). In: Lynch, M.L., Spicer, P.T. (Eds.) Bicontinuous Liquid Crystals. CRC Press Taylor & Francis Group, Boca Raton, USA, pp. 3–13. Chapter 1. Bicontinuous cubic liquid crystalline materials: a historical perspective and modern assessment.

Lipkovski, J., Lipkovski, A. (2015). Form-finding software and minimal surface equation: a comparative approach. Filomat, 29(10), 2447–2455. https://doi.org/10.2298/FIL1510447L.

Otto, F., Rasch, B. (1996). Finding Form: Towards an Architecture of the Minimal. Axel Menges.

Pan, Q., Xu, G. (2011). Construction of minimal subdivision surface with a given boundary. Computer-Aided Design, 43(4), 374–380. https://doi.org/10.1016/j.cad.2010.12.013.

Ragulskis, K., Sapagovas, M., Čiupaila, R., Jurkulnevičius, A. (1986). Numerical experiment in stationary problems of liquid metal contact. Vibrotechnika, 4(57), 105–111.

Razumas, V. (2005). In: Lynch, M.L., Spicer, P.T. (Eds.) Bicontinuous Liquid Crystals. CRC Press Taylor & Francis Group, Boca Raton, USA, pp. 169–211. Chapter 7. Bicontinuous cubic phases of lipids with entrapped proteins: structural features and bioanalytical applications.

Sakakibara, K., Shimizu, Y. (2022). Numerical analysis for the Plateau problem by the method of fundamental solutions. Numerical Analysis, 1–21. https://doi.org/10.48550/arXiv.2212.06508.

Sapagovas, M. (1982). The difference method for the solution of the problem of the equilibrium of a drop of liquid. Difference Equations and Their Applications, 31, 63–72 (in Russian).

Sapagovas, M. (1983). Numerical methods for the solution of the equation of a surface with prescribed mean curvature. Lithuanian Mathematical Journal, 23(3), 321–326. https://doi.org/10.1007/BF00966474.

Sapagovas, M. (1984). Difference Methods for Solution of Nonlinear Elliptic Equations. Doctor thesis, M.V. Keldysh Institute of Applied Mathematics, Moscow.

Sapagovas, M. (2023). Plateau problem and minimal surfaces: numerical methods and applications. Lietuvos matematikos rinkinys. LMD darbai, ser. B, 64, 1–15 (in Lithuanian). https://doi.org/10.15388/LMR.2023.33611.

Schumacher, H., Wardetzky, M. (2019). Variational convergence of discrete minimal surfaces. Numerische Mathematik, 141, 173–213. https://doi.org/10.1007/s00211-018-0993-z.

Tråsdahl, Ø., Rønquist, E.M. (2011). High order numerical approximation of minimal surfaces. Journal of Computational Physics, 230(11), 4795–4810.

Uraltseva, N.N. (1973). Solution of the capillarity problem. Vestnik, Leningrad University. Mathematics, (19), 54–64.

Valldeperas, M., Salis, A., Barauskas, J., Tiberg, F., Arnebrant, T., Razumas, V., Monduzzi, M., Nylander, T. (2019). Enzyme encapsulation in nanostructured self-assembled structures: Toward biofunctional supramolecular assemblies. Current Opinion in Colloid & Interface Science, 44, 130–142. https://doi.org/10.1016/j.cocis.2019.09.007.

Vogel, T.I. (1987). Stability of a Liquid Drop Trapped Between Two Parallel Planes. SIAM Journal on Applied Mathematics, 47(3), 516–525. http://www.jstor.org/stable/2101796.

Zareckas, V.-S.S., Ragulskienė, V.L. (1971). Mercury Switching Elements for Automation Devices. Automation Library. Energy.

Biographies

Sapagovas Mifodijus

https://orcid.org/0000-0002-7139-3468

mifodijus.sapagovas@mif.vu.lt

M. Sapagovas graduated from the Vilnius University in 1961. He received the doctoral degree in mathematics (PhD) at the Institute of Mathematics in Kiev in 1965. In 1986, he received the doctor habilitatus degree from the M. Keldysh Institute of Applied Mathematics in Moscow. M. Sapagovas is a professor (1989), a member of the Lithuanian Academy of Sciences (1987), professor-emeritus of the Statistics and Probability Group at the Institute of Data Science and Digital Technologies. The main field of scientific interest is the numerical methods for nonlinear PDE as well as mathematical modelling.

Būda Vytautas

vytautas.buda@ism.lt

V. Būda graduated from the Vilnius University in 1978. He received a Doctor’s degree in mathematics (PhD) at the Institute of Mathematics and Informatics in 1988. He worked as an associate professor at the Vilnius Gediminas Technical University and the University of Management and Economics (ISM, Vilnius) between 1988–2018. After retirement, he takes a position as a part-time lecturer at the ISM.

Maskeliūnas Saulius

https://orcid.org/0000-0002-3587-9655

saulius.maskeliunas@mif.vu.lt

S. Maskeliūnas received a doctoral degree in informatics from the Institute of Mathematics and Informatics in 1996. He is a researcher at the Cyber-Social Systems Engineering Group and deputy director at the Institute of Data Science and Digital Technologies. His general interests include: information systems and knowledge-based systems, in particular, (from most important now to past ones): semantic web, scientometrics, web services, ontological engineering, knowledge management and organisational memories, workflow automation, qualitative reasoning, expert systems.

Štikonienė Olga

https://orcid.org/0000-0002-0302-3449

olga.stikoniene@mif.vu.lt

O. Štikonienė graduated from the Lomonosov Moscow State University. She obtained her PhD in Mathematics at the Institute of Mathematics and Informatics in 1997. Currently, she is a professor at the Institute of Applied Mathematics at Vilnius University. Her research interests include numerical methods for nonlinear partial differential equations, nonlocal differential problems, and mathematical modelling in physics and medicine.

Štikonas Artūras

https://orcid.org/0000-0002-5872-5501

arturas.stikonas@mif.vu.lt

A. Štikonas studied at Vilnius University in 1980–1982 and Lomonosov Moscow State University in 1982–1986. PhD (mathematics and physics), Department of Numerical Mathematics, Academy of Science, USSR in 1990. In 2008, he defended habilitation thesis. Since 2017, he works as a professor and research professor at the Institute of Applied Mathematics, Vilnius University. His research focuses on numerical methods and problems with nonlocal conditions.

Full article Related articles

Open access article under the CC BY license.

Keywords

Plateau problem minimal surface equation numerical methods applications of minimal surfaces

Metrics

since January 2020

222 Article info views	103 Full article views
155 PDF downloads	39 XML downloads

RSS

Authors

Abstract

References

Biographies

Export citation

Copy and paste formatted citation

Download citation in file