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Theoretical and experimental validation of the variable-thickness topology optimization approach for the rib-stiffened panels

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Abstract

In this paper, we consider compliance minimization problems within the variable-thickness approach for the rib-stiffened plates subjected to a transverse loading. It is known, that such optimization problems are usually not well posed and their solutions become strongly mesh-dependent. To overcome this issue, we introduce additional regularization constraint on the thickness gradient and evaluate the convergence and efficiency of considered method. Variable thickness is defined based on topology optimization approach introducing additional design variables in the nodes of the shell-type elements. Numerical solutions are provided by using finite element simulations within Mindlin–Reissner theory and method of moving asymptotes. Possibility for the well-converged optimal solutions for the benchmark problems with rib-stiffened panels loaded by the systems of concentrated forces is shown. Parametric studies are provided to analyse the effects of the shape functions order, values of penalty factors and initial conditions for the plate thickness. Recommendations for the optimal settings of the considered method are established. Theoretical and experimental assessments on the advantages and accuracy of the variable-thickness approach are given based on comparison of the obtained solutions to the standard design for the plates with regular stiffening.

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References

  1. Huybrechts, S.M., Meink, T.E., Wegner, P.M., Ganley, J.M.: Manufacturing theory for advanced grid stiffened structures. Compos. A Appl. Sci. Manuf. 33(2), 155–161 (2002). https://doi.org/10.1016/S1359-835X(01)00113-0

    Article  Google Scholar 

  2. Lozano-Galant, J.A., Payá-Zaforteza, I.: Structural analysis of Eduardo Torroja’s Frontón de Recoletos’ roof. Eng. Struct. 33(3), 843–854 (2011). https://doi.org/10.1016/j.engstruct.2010.12.006

    Article  Google Scholar 

  3. Challagulla, K.S., Georgiades, A., Kalamkarov, A.: Asymptotic homogenization modeling of smart composite generally orthotropic grid-reinforced shells: Part I-theory. Eur. J. Mech.-A/Solids. 29(4), 530–540 (2010). https://doi.org/10.1016/j.euromechsol.2010.03.007

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Hadjiloizi, D., Kalamkarov, A.L., Georgiades, A.: Plane stress analysis of magnetoelectric composite and reinforced plates: applications to wafer-and rib-reinforced plates and three-layered honeycomb shells. ZAMM-J. Appl. Math. Mech/./Zeitschrift für Angewandte Math. und Mech. 97(7), 786–814 (2017). https://doi.org/10.1002/zamm.201500228

    Article  ADS  MathSciNet  Google Scholar 

  5. Bedair, O.: Analysis and limit state design of stiffened plates and shells: a world view. Appl. Mech. Rev. 62(2), 020801 (2009). https://doi.org/10.1115/1.3077137

    Article  ADS  Google Scholar 

  6. Bedair, O.: Recent developments in modeling and design procedures of stiffened plates and shells. Recent Patents Eng. 7(3), 196–208 (2013). https://doi.org/10.2174/1872212107999131120161751

    Article  Google Scholar 

  7. Sigmund, O., Maute, K.: Topology optimization approaches. Struct. Multidiscip. Optim. 48(6), 1031–1055 (2013). https://doi.org/10.1007/s00158-013-0978-6

    Article  MathSciNet  Google Scholar 

  8. Wu, J., Sigmund, O., Groen, J.P.: Topology optimization of multi-scale structures: a review. Struct. Multidiscip. Optim. 63(3), 1455–1480 (2021). https://doi.org/10.1007/s00158-021-02881-8

    Article  MathSciNet  Google Scholar 

  9. Lin, H., Xu, A., Misra, A., Zhao, R.: An ANSYS APDL code for topology optimization of structures with multi-constraints using the BESO method with dynamic evolution rate (DER-BESO). Struct. Multidiscip. Optim. 62(4), 2229–2254 (2020). https://doi.org/10.1007/s00158-020-02588-2

    Article  MathSciNet  Google Scholar 

  10. Giorgio, I.: Lattice shells composed of two families of curved Kirchhoff rods: an archetypal example, topology optimization of a cycloidal metamaterial. Continuum Mech. Thermodyn. 33(4), 1063–1082 (2021). https://doi.org/10.1007/s00161-020-00955-4

    Article  ADS  MathSciNet  Google Scholar 

  11. Aage, N., Andreassen, E., Lazarov, B.S., Sigmund, O.: Giga-voxel computational morphogenesis for structural design. Nature 550(7674), 84–86 (2017). https://doi.org/10.1038/nature23911

    Article  ADS  Google Scholar 

  12. Cheng, K.T., Olhoff, N.: An investigation concerning optimal design of solid elastic plates. Int. J. Solids Struct. 17(3), 305–323 (1981). https://doi.org/10.1016/0020-7683(81)90065-2

    Article  MathSciNet  MATH  Google Scholar 

  13. Munoz, J., Pedregal, P.: A review of an optimal design problem for a plate of variable thickness. SIAM J. Control. Optim. 46(1), 1–13 (2007). https://doi.org/10.1137/050639569

    Article  MathSciNet  MATH  Google Scholar 

  14. Litvinov, V.: Optimal control of the natural frequency of a plate of variable thickness. USSR Comput. Math. Math. Phys. 19(4), 70–86 (1979). https://doi.org/10.1016/0041-5553(79)90157-5

    Article  MathSciNet  MATH  Google Scholar 

  15. Czarnecki, S., Lewiński, T.: On minimum compliance problems of thin elastic plates of varying thickness. Struct. Multidiscip. Optim. 48(1), 17–31 (2013). https://doi.org/10.1007/s00158-013-0893-x

    Article  MathSciNet  MATH  Google Scholar 

  16. Keng-Tuno, C.: On non-smoothness in optimal design of solid, elastic plates. Int. J. Solids Struct. 17(8), 795–810 (1981). https://doi.org/10.1016/0020-7683(81)90089-5

    Article  MATH  Google Scholar 

  17. Niordson, F.: Optimal design of elastic plates with a constraint on the slope of the thickness function. Int. J. Solids Struct. 19(2), 141–151 (1983). https://doi.org/10.1016/0020-7683(83)90005-7

    Article  MATH  Google Scholar 

  18. Bonnetier, E., Conca, C.: Approximation of Young measures by functions and application to a problem of optimal design for plates with variable thickness. Proc. R. Soci. Edinburgh Sect. A: Math. 124(3), 399–422 (1994). https://doi.org/10.1017/S0308210500028717

    Article  MathSciNet  MATH  Google Scholar 

  19. Antonić, N., Balenović, N.: Optimal design for plates and relaxation. Math. Commun. 4(1), 111–119 (1999)

    MathSciNet  MATH  Google Scholar 

  20. Bouchitté, G., Fragalà, I., Seppecher, P.: Structural optimization of thin elastic plates: the three dimensional approach. Arch. Ration. Mech. Anal. 202(3), 829–874 (2011). https://doi.org/10.1007/s00205-011-0435-x

    Article  MathSciNet  MATH  Google Scholar 

  21. Lam, Y., Santhikumar, S.: Automated rib location and optimization for plate structures. Struct. Multidiscip. Optim. 25(1), 35–45 (2003). https://doi.org/10.1007/s00158-002-0270-7

    Article  Google Scholar 

  22. Dugré, A., Vadean, A., et al.: Challenges of using topology optimization for the design of pressurized stiffened panels. Struct. Multidiscip. Optim. 53(2), 303–320 (2016). https://doi.org/10.1007/s00158-015-1321-1

    Article  Google Scholar 

  23. Träff, E.A., Sigmund, O., Aage, N.: Topology optimization of ultra high resolution shell structures. Thin-Walled Struct. 160, 107349 (2021). https://doi.org/10.1016/j.tws.2020.107349

    Article  Google Scholar 

  24. Banh, T.T., Lee, D.: Topology optimization of multi-directional variable thickness thin plate with multiple materials. Struct. Multidiscip. Optim. 59(5), 1503–1520 (2019). https://doi.org/10.1007/s00158-018-2143-8

    Article  Google Scholar 

  25. Wang, J., Chang, S., Liu, G., Liu, L., Wu, L.: Optimal rib layout design for noise reduction based on topology optimization and acoustic contribution analysis. Struct. Multidiscip. Optim. 56(5), 1093–1108 (2017). https://doi.org/10.1007/s00158-017-1705-5

    Article  MathSciNet  Google Scholar 

  26. Jiang, X., Liu, C., Du, Z., Huo, W., Zhang, X., Liu, F., et al.: A unified framework for explicit layout/topology optimization of thin-walled structures based on Moving Morphable Components (MMC) method and adaptive ground structure approach. Comput. Methods Appl. Mech. Eng. 396, 115047 (2022). https://doi.org/10.1016/j.cma.2022.115047

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Li, L., Liu, C., Zhang, W., Du, Z., Guo, X.: Combined model-based topology optimization of stiffened plate structures via MMC approach. Int. J. Mech. Sci. 208, 106682 (2021). https://doi.org/10.1016/j.ijmecsci.2021.106682

    Article  Google Scholar 

  28. Ji, J., Ding, X., Xiong, M.: Optimal stiffener layout of plate/shell structures by bionic growth method. Comput. Struct. 135, 88–99 (2014). https://doi.org/10.1016/j.compstruc.2014.01.022

    Article  Google Scholar 

  29. Liu, D., Hao, P., Zhang, K., Tian, K., Wang, B., Li, G., et al.: On the integrated design of curvilinearly grid-stiffened panel with non-uniform distribution and variable stiffener profile. Mater. Design. 190, 108556 (2020). https://doi.org/10.1016/j.matdes.2020.108556

    Article  Google Scholar 

  30. Giorgio, I., Ciallella, A., Scerrato, D.: A study about the impact of the topological arrangement of fibers on fiber-reinforced composites: some guidelines aiming at the development of new ultra-stiff and ultra-soft metamaterials. Int. J. Solids Struct. 203, 73–83 (2020)

    Article  Google Scholar 

  31. Desmorat, B., Spagnuolo, M., Turco, E.: Stiffness optimization in nonlinear pantographic structures. Math. Mech. Solids 25(12), 2252–2262 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Shekarchizadeh, N., Abali, B.E., Barchiesi, E., Bersani, A.M.: Inverse analysis of metamaterials and parameter determination by means of an automatized optimization problem. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Math. und Mech. 101(8), e202000277 (2021)

    MathSciNet  Google Scholar 

  33. Seppecher, P., Alibert, J.J., Lekszycki, T., Grygoruk, R., Pawlikowski, M., Steigmann, D., et al.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Continuum Mech. Thermodyn. 31(4), 851–884 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  34. Giorgio, I., Andreaus, U., Alzahrani, F., Hayat, T., Lekszycki, T., et al.: On mechanically driven biological stimulus for bone remodeling as a diffusive phenomenon. Biomech. Model. Mechanobiol. 18(6), 1639–1663 (2019)

    Article  Google Scholar 

  35. Vasiliev, VV., Morozov, EV.: Advanced mechanics of composite materials and structures (2018)

  36. Svanberg, K.: The method of moving asymptotes - a new method for structural optimization. Int. J. Numer. Meth. Eng. 24(2), 359–373 (1987). https://doi.org/10.1002/nme.1620240207

    Article  MathSciNet  MATH  Google Scholar 

  37. Lyngdoh, G., Doner, S., Yuan, R., Chelidze, D., et al.: Experimental monitoring and modeling of fatigue damage for 3D-printed polymeric beams under irregular loading. Int. J. Mech. Sci. 233, 107626 (2022). https://doi.org/10.1016/j.ijmecsci.2022.107626

    Article  Google Scholar 

  38. Sigmund, O.: On benchmarking and good scientific practise in topology optimization. Struct. Multidiscip. Optim. 65(11), 1–10 (2022). https://doi.org/10.1007/s00158-022-03427-2

    Article  Google Scholar 

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Acknowledgements

This work was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant agreement 075-15-2022-1023).

Funding

This work was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant agreement 075-15-2022-1023).

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Author notes

  1. Kyaw Ye Ko, Sergey Lurie, Arseniy Babaytsev, Lev Rabinskiy and Ivan Kondakov have contributed equally to this work.

Authors and Affiliations

  1. Moscow Aviation Institute, Volokolamskoe av., 4, Moscow, Russia, 125993

    Kyaw Ye Ko, Yury Solyaev, Sergey Lurie, Arseniy Babaytsev & Lev Rabinskiy

  2. Central Aerohydrodynamic Institute, Zhukovskiy str. 1, Zhukovskiy, Russia, 140180

    Ivan Kondakov

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  1. Kyaw Ye Ko
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Correspondence to Yury Solyaev.

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Author contributions

Conceptualization IK, YS, SL; Methodology YS, KYK; Formal analysis and investigation KYK, YS, AB; Writing— original draft preparation YS; Writing—review and editing SL; Funding acquisition LR, IK; Supervision LR.

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Ko, K.Y., Solyaev, Y., Lurie, S. et al. Theoretical and experimental validation of the variable-thickness topology optimization approach for the rib-stiffened panels. Continuum Mech. Thermodyn. 35, 1787–1806 (2023). https://doi.org/10.1007/s00161-023-01224-w

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