Theoretical and experimental validation of the variable-thickness topology optimization approach for the rib-stiffened panels
Tóm tắt
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.
Tài liệu tham khảo
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
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
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
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
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
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
Sigmund, O., Maute, K.: Topology optimization approaches. Struct. Multidiscip. Optim. 48(6), 1031–1055 (2013). https://doi.org/10.1007/s00158-013-0978-6
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
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
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
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
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
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
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
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
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
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
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
Antonić, N., Balenović, N.: Optimal design for plates and relaxation. Math. Commun. 4(1), 111–119 (1999)
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
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
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
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
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
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
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
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
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
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
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)
Desmorat, B., Spagnuolo, M., Turco, E.: Stiffness optimization in nonlinear pantographic structures. Math. Mech. Solids 25(12), 2252–2262 (2020)
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)
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)
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)
Vasiliev, VV., Morozov, EV.: Advanced mechanics of composite materials and structures (2018)
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
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
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