Evaluating the effectiveness of the method of topological optimization of reinforced panels based on analytical solutions to benchmark problems

DOI: 10.34759/trd-2023-129-07

Аuthors

Kyaw Y. K.^*, Rabinsky L. N.^**

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

*e-mail: kyawyeko23@gmail.com
**e-mail: rabinskiy@mail.ru

Abstract

This article regards a variant of the problem of a flat freely supported plate loaded with a concentrated force applied at a displacement relative to the center of the plate and acting along the normal to its surface. For such a problem, the topological optimization technique formulated earlier by the authors for the models of plates of variable thickness offers a simple solution for selecting the optimal orientation of stiffening ribs that ensures minimum deflections of the plate under load. In the topological optimization problem, the resulting stiffening ribs are arranged symmetrically relative to the plate central plane, and the angle between them depends on the magnitude of the load application point displacement. To check the efficiency of the applied optimization technique the authors suggested considering a similar problem for a plane-space frame, in which the element structure repeats the arrangement of the stiffness ribs that appear in the solution of the topological optimization problem for the plate. The solution for the frame deformation problem can be easily constructed in analytical closed form. It is possible to determine the optimum angle of opening of the frame elements for a given location of the load application point based on this solution. The article demonstrates that there is a qualitative consistency between the optimal geometry of the frame found from the analytical solution and the optimal geometry of the corresponding reinforced plate found from the solution of the topological optimization problem. Particularly, the same characteristic dependence of the optimal angle of stiffening ribs opening on the magnitude of load application point displacement relative to the center of the plate is established.

Keywords:

topological optimization, reinforced panels, plane-space frame, analytical solution

References

1. Bendsøe M., Sigmund O. Tbopology optimization: theory, methods, and applications, Springer, 2003.
2. Sigmund, K. Maute. Topology optimization approaches, Structural and Multidisciplinary Optimization, 2013, vol. 48 (6), pp. 1031–1055. DOI:10.1007/S00158-013-0978-6
3. Wang C. et al. Structural topology optimization considering both performance and manufacturability: strength, stiffness, and connectivity, Structural and Multidisciplinary Optimization, 2021, vol. 63, pp. 1427-1453. DOI:10.1007/s00158-020-02769-z
4. Ferrari F., Sigmund O. Revisiting topology optimization with buckling constraints, Structural and Multidisciplinary Optimization, 2019, vol. 59, no. 5, pp. 1401-1415. DOI:10.1007/s00158-019-02253-3
5. Picelli R., Vicente W.M., Pavanello R., Xie Y.M. Evolutionary topology optimization for natural frequency maximization problems considering acoustic—structure interaction, Finite Elements in Analysis and Design, 2015, vol. 106, pp. 56–64. URL: https://doi.org/10.1016/j.finel.2015.07.010
6. Svanberg. The method of moving asymptotes — a new method for structural optimization, International Journal for numerical methods in engineering, 1987, vol. 24 (2), pp. 359–373. DOI:10.1002/NME.1620240207
7. Lam Y., Santhikumar S. Automated rib location and optimization for plate structures, Struct Multidisc Optim, 2003, vol. 25, pp. 35–45. URL: https://doi.org/10.1007/s00158-002-0270-7
8. Linyuan Li, Chang Liu, Weisheng Zhang, Zongliang Du, Xu Guo. Combined model-based topology optimization of stiffened plate structures via MMC approach, International Journal of Mechanical Sciences, 2021, vol. 208, pp. 106682. URL: https://doi.org/10.1016/j.ijmecsci.2021.106682
9. Keng-Tung Cheng, Niels Olhoff. An investigation concerning optimal design of solid elastic plates, International Journal of Solids and Structures, 1981, vol. 17(3), pp. 305–323. DOI:10.1016/0020-7683(81)90065-2
10. Kyaw Ye Ko, Solyaev Yu.O. Topological optimization of reinforced panels loaded with concentrated forces // Trudy MAI, 2021, no. 120. URL: https://trudymai.ru/eng/published.php?ID=161420. DOI: 10.34759/trd-2021-120-07
11. Lur’e K.A., Cherkaev A.V. Izvestiya AN SSSR. Mekhanika tverdogo tela, 1976, no. 6, pp. 157-159.
12. Valery V. Vasiliev, Evgeny V. Morozov. Advanced mechanics of composite materials and structures, Elsevier, 2018.
13. Julio Munoz and Pablo Pedregal. A review of an optimal design problem for a plate of variable thickness, SIAM Journal on control and optimization, 2007, vol. 46(1), pp. 1-13. DOI:10.1137/050639569
14. E.A. Träff, O. Sigmund, N. Aage. Topology optimization of ultra high resolution shell structures, Thin-Walled Structures, 2021, vol. 160, pp. 107349. DOI:10.1016/j.tws.2020.107349
15. Bouchitt´e, I. Fragal`a, P. Seppecher. Structural optimization of thin elastic plates: the three dimensional approach Archive for rational mechanics and analysis, 2011, vol. 202 (3), pp. 829–874. DOI:10.1007/s00205-011-0435-x
16. Dugr ́e, A. Vadean, et al. Challenges of using topology optimization for the design of pressurized stiffened panels, Structural and Multidisciplinary Optimization, 2016, vol. 53(2), pp. 303–320. DOI:10.1007/s00158-015-1321-1
17. Rais-Rohani M., Lokits J. Reinforcement layout and sizing optimization of composite submarine sail structures, Struct Multidisc Optim, 2007, vol. 34, pp. 75-90. URL: https://doi.org/10.1007/s00158-006-0066-2
18. Martirosov M.I., Khomchenko A.V. Trudy MAI, 2022, no. 126. URL: https://trudymai.ru/eng/published.php?ID=168990. DOI: 10.34759/trd-2022-126-04
19. Dudchenko A.A., Le Kim Kyong, Lur’e S.A. Trudy MAI, 2012, no. 50. URL: https://trudymai.ru/eng/published.php?ID=28792
20. Sigmund. On benchmarking and good scientific practise in topology optimization, Structural and Multidisciplinary Optimization, 2022, vol. 65 (11), pp. 1–10. DOI:10.1007/s00158-022-03427-2

Download