Сomparative analysis of geometric methods for constructing polyhedral mesh models from archimedean solids

Аuthors

Zayats E. E.^*, Fevralskikh A. V.^**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: ee.zaiats@yandex.ru
**e-mail: a.fevralskih@gmail.com

Abstract

The capabilities of modern software packages and approaches to implementing the finite volume method for modeling continuum mechanics often run into conflicting requirements for the detailing of computer models and the reliability of the description of the processes being modeled. One of the essential stages of modeling is the construction of grid models. Modern software packages mainly develop methods for constructing grid models consisting of terahedrons and hexahedrons, and only in rare cases - from polyhedral elements. The use of grid models based on polyhedrons for finite-volume gas-dynamic calculations seems most preferable due to a number of reasons. Firstly, a polyhedral finite volume allows obtaining a larger amount of information at each integration step for approximating the solution. Secondly, the use of finite volumes of this type allows one to partially avoid the problem of cell deformation when constructing a grid with distributed dimensions. These circumstances lead to faster convergence of calculations based on polyhedral models. The aim of this work is to develop an efficient algorithm for constructing polyhedral grids based on the methods of densest filling of space with Archimedean solids and regular polyhedra. A block diagram of the algorithm and the results of its testing in the form of a comparative analysis of the obtained grid models by the number of cells, vertices and faces are given. The most efficient method is recognized to be based on cuboctahedrons and octahedrons, which provides the greatest information content of the grid with a smaller discretization step. The obtained results can be used in the development of efficient grid generators for continuous media dynamics problems.

Keywords:

polyhedral mesh, Archimedean solids, finite volume method

References

1. Baker T. J. Developments and trends in three-dimensional mesh generation // Applied Numerical Mathematics. 1989. Vol. 5, No. 4. P. 275–304. DOI: 10.1016/0168-9274(89)90012-3
2. Sosnowski M., Krzywanski J., Grabowska K., Gnatowska R. Polyhedral meshing in numerical analysis of conjugate heat transfer // EPJ Web Conf. 2018. Vol. 180. P. 6. DOI: 10.1051/epjconf/201818002096
3. Vershkov V. A., Voronich I. V., Vyshinskii V. V. Trudy MAI. 2015. No. 82.
4. Platonov I. M., Bykov L. V. Trudy MAI. 2016. No. 89.
5. Nikitchenko Yu. A., Berezko M. E., Krasavin E. E. Trudy MAI. 2023. No. 131.
6. Dmitriev V. G., Korovaitseva E. A., Popova A. R. Trudy MAI. 2024. No. 137.
7. Galanin M. P., Shcheglov I. A. Preprinty IPM im. M. V. Keldysha RAN. 2006. No. 9.
8. Yakobovskii M. V., Grigor'ev S. K. Preprinty IPM im. M. V. Keldysha RAN. 2018. No. 109. DOI: 10.20948/prepr-2018-109.
9. Shtabel' N. V. Matematika i ee prilozheniya: fundamental'nye problemy nauki i tekhniki: Sbornik trudov vserossiiskoi konferentsii. 2015. pp. 139–145.
10. Kutishcheva A. Yu., Markov S. I. Vysokoproizvoditel'nye vychislitel'nye sistemy i tekhnologii. 2023. Vol. 7, No. 1. pp. 70–77.
11. Fevral'skikh A. V., Gramuzov E. M., Kupchik V. S. Patent RU 2024688271, 30.10.2024.
12. Fevral'skikh A. V. Russkii inzhener. 2024. No. 4 (85). pp. 32–35.
13. Makhnev M. S., Fevral'skikh A. V. Trudy MAI. 2019. No. 109. DOI: 10.34759/trd-2019-109-23
14. Strelets D. Yu., Lavrishcheva L. S., Staroverov N. N., Novoselov V. N., Fevral'skikh A. V., Bashkirov I. G. Vestnik mashinostroeniya. 2024. Vol. 103, No. 8. pp. 672–678. DOI: 10.36652/0042-4633-2024-103-8-672-678
15. Popov E. V. Metod natyanutykh setok v zadachakh geometricheskogo modelirovaniya (The method of stretched grids in geometric modeling problems), Doctor’s thesis, Nizhny Novgorod, 2001, 248 p.
16. Popov E. V., Popova T. P. Minimal surface form finding and visualization using stretched grid method // Scientific Visualisation. 2021. Vol. 13. No. 1. P. 54–68. DOI: 10.26583/sv.13.1.05
17. Kopysov S. P., Novikov A. K., Ponomarev A. B., Rychkov V. N., Sagdeeva Yu. A. Programmnye produkty i sistemy. 2008. No. 2. URL: https://cyberleninka.ru/article/n/programmnaya-sreda-raschetnyh-setochnyh-modeley-dlya-parallelnyh-v...
18. Schoberl J. NETGEN An advancing front 2D/3D-mesh generator based on abstract rules // Comput Visual Sci. 1997. Vol. 1. P. 41–52. DOI: 10.1007/s007910050004
19. Garimella R. V., Kim J., Berndt M. Polyhedral Mesh Generation and Optimization for Non-manifold Domains // In: Sarrate, J., Staten, M. (eds) Proceedings of the 22nd International Meshing Roundtable. Springer, Cham. 2014. P. 313–330. DOI: 10.1007/978-3-319-02335-9_18
20. Scroggs M. W., Dokken J. S., Richardson C. N., Wells G. N. Construction of Arbitrary Order Finite Element Degree-of-Freedom Maps on Polygonal and Polyhedral Cell Meshes // ACM Trans. Math. Softw. 2022. Vol. 48, No. 2. P. 1–23. DOI: 10.1145/3524456
21. Lee S. Y. Polyhedral Mesh Generation and A Treatise on Concave Geometrical Edges // Procedia Engineering. 2015. Vol. 124. P. 174–186. DOI: 10.1016/j.proeng.2015.10.131
22. Vennindzher M. Modeli mnogogrannikov (Polyhedron models), Moscow, Mir, 1974, 243 p.
23. Vilenkin N. Ya. Kombinatorika (Combinatorics), Moscow, Nauka, 1969, 328 p.
24. Williams R. Polyhedra packing and space filling // The Geometrical Foundation of Natural Structure: A Source Book of Design. – New York: Dover Publications. 1979. P. 164–199.

Download