Analytical surfaces with a plane contour and with superellipses of the main frame

Analytical method of definition of surface considerably simplifies the following operation of design of curvilinear shell structures and shells in comparison with other methods of definition. Having used the universality of superellipses which form the family of close plane curves that are symmetrical relatively two coordinate axes, one can assume them as a three of curves of the main frame of a design surface. The three of different surfaces with identical main framework will be obtained as a result of plane-and-parallel translation of every of three superellipses along another director ellipse under condition of going of the mobile superellipse through symmetrical points of the third superellipse of the main framework. This method of formation of the surfaces gained wide distribution in many branches of building, technics, and science. In a paper, all known surfaces with main frames of three superellipses are described and illustrated by twenty-nine figures. More than nine tens of them were brought out. Some surfaces were taken as middle surfaces of thin building shells. Their stress-strain state was determined by FEM. The presented results and a list of references containing 32 names will help to find new directions in research of surfaces and shells of this type that have some advantages.

1. Weisstein E.W. Superellipse. From MathWorld - A Wolfram Web Resource. https://mathworld.wolfram.com/Superellipse.html

2. Hein P. “Piet Heine”, 2021 https://piethein.com/piet-hein/

3. Flanagan D.L., Hefner O.V. Surface molding- new tool for the engineer (Man-Computer Graphics/MCG/ allows operator control through oscilloscope via light sensitive pen). Astronautics and Aeronautics. 1967. 4. Pp. 58-62.

4. Gardner M. The superellipse: a curve that lies between the ellipse and the rectangle. Scientific American. 1965. 21. Pp. 222–234.

5. Barr A. Superquadrics and angle-preserving transformations. IEEE Computer Graphics Applications. 1981. 1. Pp. 11–23.

6. Bar M., Neta M. Humans prefer curved visual objects. Psychological science. 2006. 17(8). Pp. 645-648.

7. Talu S. D. L. Complex 3D shapes with superellipsoids, supertoroids and convex polyhedrons. Journal of Engineering Studies and Research. 2011. Vol. 17. no. 4. Pp. 96–100.

8. Xiaoming Zhang, Paul L. Rosin. Superellipse fitting to partial data. Pattern Recognition (Computer Science). 36. No 3. 2003. Pp. 743-752

9. Kadir can Erbaş. Surface Area of Superellipsoids and its Application to Physics Problems: Chapter 3. In book: New-Applications-In-Basic-Sciences. September 2022. Publisher: Iksad Publishing House. Baskent University, Ankara, Turkey.

10. Avdon’ev E.Ya. Analytical description of the ship hull surfaces. Prikladnaya Geometriya i Inzhenernaya Grafika. 1972. Iss. 15. Pp. 156-160 (In Russ.).

11. Strashnov S.V. Utilizing superellipses in computer modeling of architectural and engineering structures. Bulletin of the South Ural State University. Ser. Construction Engineering and Architecture. 2023. 23(4). Pp. 67–76. (in Russ.). DOI: 10.14529/build230408

12. Lamé G. Examen de differentes méthodes employées pour résoudre les problèmes de géometrie. M. V. Courcier imprimeur Libraire, 1818. (New Edition 2008 from Editions Gabay).

13. Krivoshapko S.N. Hydrodynamic surfaces. Sudostroenie. 2021. No 3. Pp. 64-67. (In Russ.).

14. Krivoshapko S.N. Tangential developable and hydrodynamic surfaces for early stage of ship shape design. Ships and Offshore Structures. 2023. 18:5. Pp. 660-668. DOI: 10.1080/17445302.2022.2062165

15. Karnevich V.V. Hydrodynamic surfaces with midship section in the form of the Lame curves. RUDN Journal of Engineering Researches. 2021. 22(4): 323-328 DOI: 10.22363/2312-8143-2021-22-4-323-328

16. Krivoshapko S.N. Algebraic ship hull surfaces with a main frame from three plane curves in coordinate planes. RUDN Journal of Engineering Research. 2022. Vol. 23. No. 3. Pp. 207-212. doi:10.22363/2312-8143-2022-23-3-207-212. (in Russ.)

17. Avdon’ev E.Ya., Protod’yakonov S.M. Geometrical investigation of some surfaces of the highest orders. Prikladnaya Geometriya i Inzhenernaya Grafika. 1975. Iss. 20. Pp. 138-142.

18. Strashnov S.V. Computer simulation of new forms of shell structures. Geometry & Graphics. 2022. No. 4. Pp. 26-34. DOI: https://doi.org/10.12737/2308-4898-2022-10-4-26-34

19. Gokhan Budak and Serdar Beji. Computational resistance analyses of a generic submarine hull form and its geometric variants. The Journal of Ocean Technology. 2016. 11(2). Pp. 77-86.

20. Moonesun Mohammad, Yuri M. Korol, Hosein Dalayeli. CFD analysis on the bare hull form of submarines for minimizing the resistance. International Journal of Maritime Technology (IJMT). 2015. Vol. 3 / Winter 2015: 1-16 [Available online at: http://ijmt.ir/browse.php?a_code=A-10-450-1&sid=1&slc_lang=en].

21. Krivoshapko S.N. Surfaces with a main framework of three given curves which include one circle. Structural Mechanics of Engineering Constructions and Buildings. 2023. 19(2). Pp. 210–219. http://doi.org/10.22363/1815-5235-2023-19-2-210-219

22. Mamieva I.A., Karnevich V.V. Geometry and static analysis of thin shells with ruled middle surfaces of three superellipses as main frame. Building and Reconstruction. 2023. No. 1(105). Pp. 16-27. DOI 10.33979/2073-7416-2023-105-1-16-27. – EDN LSIOLJ.

23. Gbaguidi Aisse G.L., Aleshina O.O., Mamieva I.А. Investigation of three thin shells with ruled middle surfaces with the same main frame. Structural Mechanics of Engineering Constructions and Buildings. 2024. Vol. 20. № 2. Pp. 146–158. http://doi.org/10.22363/1815-5235-2024-20-2-146-158

24. Iraida A. Mamieva. Ruled algebraic surfaces with a main frame from three superellipses. Structural Mechanics of Engineering Constructions and Buildings. 2022. 18(4). Pp. 387-395 DOI 10.22363/1815-5235-2022-18-4- 387-395

25. Krivoshapko S.N. Surfaces of diagonal translation of the velaroidal type on a rhomb plan. Building and Reconstruction. 2023. № 2 (106). Pp. 59-69. DOI: 10.33979/2073-7416-2023-106-2-59-69.

26. Aleshina O.O. Geometry and static analysis of thin shells in the form of a surface of diagonal translation of velaroidal type. Structural Mechanics of Engineering Constructions and Buildings. 2023. Vol. 19. No 2. Pp. 84 – 93. DOI 10.22363/1815-5235-2023-19-1-84-93

27. Krivoshapko S.N., Aleshina O.O., Ivanov V.N. Static analysis of shells with middle surfaces containing the main frame from three given superellipses. Structural Mechanics and Analysis of Constructions. 2022. No. 6. Pp. 18-27. doi:10.37538/0039-2383.2022.6.18.27 (in Russ.)

28. Tupikova E.M. Shells in the form of algebraic ruled surfaces on a rhombic base. Structural Mechanics of Engineering Constructions and Buildings. 2023; 19(5): 510–519 (In Russ.). http://doi.org/10.22363/1815-5235-2023-19-5-510-519

29. Volkov G.F. A translation shell of negative curvature. Armotsementnie Konstruktzii v Stroitelstve. Leningrad: GSI, 1963. Pp. 48 – 58 (In Russ.).

30. Zakerdoost H., Ghassemi H., Ghiasi M. Ship hull form optimization by evolutionary algorithm in order to diminish the drag. J. Marine Sci. Appl. 2013. 12: 170-179. DOI: 10.1007/s11804-013-1182-1

31. Andrun M., Blagojević B., Bašić J. The influence of numerical parameters in the finite-volume method on the Wigley hull resistance. Proceedings of the Institution of Mechanical Engineers. Part M. Journal of Engineering for the Maritime Environment. 2018. Vol 233 (4), pp. 1123-1132. 10.1177/1475090218812956

32. Strashnov S.V., Rynkovskaya M.I. To the question of the classification for analytical surfaces. Geometry & Graphics. 2022. Vol. 10. No. 1. Pp. 36-43 DOI: 10.12737/2308-4898-2022-10-1-36-43.