Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Topology optimization with functionally graded multi-material for elastic buckling criteria

  • Minh-Ngoc Nguyen (Department of Architectural Engineering, Sejong University) ;
  • Dongkyu Lee (Department of Architectural Engineering, Sejong University) ;
  • Joowon Kang (School of Architecture, Yeungnam University) ;
  • Soomi Shin (School of Architecture, Yeungnam University)
  • Received : 2021.08.31
  • Accepted : 2022.12.12
  • Published : 2023.01.10
⟨ Previous Next ⟩

Abstract

This research presents a multi-material topology optimization for functionally graded material (FGM) and nonFGM with elastic buckling criteria. The elastic buckling based multi-material topology optimization of functionally graded steels (FGSs) uses a Jacobi scheme and a Method of Moving Asymptotes (MMA) as an expansion to revise the design variables shown first. Moreover, mathematical expressions for modified interpolation materials in the buckling framework are also described in detail. A Solid Isotropic Material with Penalization (SIMP) as well as a modified penalizing material model is utilized. Based on this investigation on the buckling constraint with homogenization material properties, this method for determining optimal shape is presented under buckling constraint parameters with non-homogenization material properties. For optimal problems, minimizing structural compliance like as an objective function is related to a given material volume and a buckling load factor. In this study, conflicts between structural stiffness and stability which cause an unfavorable effect on the performance of existing optimization procedures are reduced. A few structural design features illustrate the effectiveness and adjustability of an approach and provide some ideas for further expansions.

Keywords

Acknowledgement

This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2022R1A2C1003776).

References

  1. Rahmatalla, S. and Swan, C.C. (2003), "Continuum topology optimization of buckling-sensitive structures", AIAA J., 41(6), 1180-1189. https://doi.org/10.2514/2.2062.
  2. Gao, X. and Ma, H. (2014), "A new method for dealing with pseudo modes in topology optimization of continua for free vibration", Lixue Xuebao/Chinese J. Theoretical Appl. Mech., 46(6), 739-746. https://doi.org/10.6052/0459-1879-13-406.
  3. Gao, X. and Ma, H. (2015), "Topology optimization of continuum structures under buckling constraints", Comput. Struct., 157, 142-152. https://doi.org/10.1016/j.compstruc.2015.05.020.
  4. Dunning, P.D, Ovtchinnikov, E., Scott, J. and Kim, H.A. (2016), "Level-set topology optimization with many linear buckling constraints using an efficient and robust eigen solver", Int. J. Numer. Meth. Engng., 107(12), 1029-1053. https://doi.org/10.1002/nme.5203.
  5. Sigmund, O. and Torquato, S. (1997), "Design of materials with extreme thermal expansion using a three-phase topology optimization method", J. Mech. Phys. Solids, 45(6), 1037-1067. https://doi.org/10.1016/S0022-5096(96)00114-7.
  6. Doan, Q.H and Lee, D.K. (2017), "Optimum topology design of multi-material structures with non-spurious buckling constraints", Adv. Eng. Softw., 114, 110-120. https://doi.org/10.1016/j.advengsoft.2017.06.002.
  7. Doan, Q.H., Lee, D.K., Lee, J.H. and Kang, J.W. (2019), "Design of buckling constrained multiphase material structures using continuum topology optimization", Meccanica, 54, 1179-1201. https://doi.org/10.1007/s11012-019-01009-z.
  8. Olhoff, N. and Rasmussen, S.H (1977), "On single and bimodal optimum buckling loads of clamped columns", Int. J. Solids Struct., 13(7), 605-164. https://doi.org/10.1016/0020-7683(77)90043-9.
  9. Federico, F., Lazarov, B.S. and Sigmund, O. (2018), "Eigenvalue topology optimization via efficient multilevel solution of the frequency response", 115(7), 872-892. https://doi.org/10.1002/nme.5829.
  10. Federico, F. and Sigmund, O. (2019), "Revisiting topology optimization with buckling constraints", Struct Multidisc Optim, 59, 1401-1415. https://doi.org/10.1007/s00158-019-02253-3.
  11. Andreassen, E., Federico, F. and Sigmund, O. and Diaz, A.R. (2017), "Frequency response as a surrogate eigenvalue problem in topology optimization", Int. J. Numer. Meth. Engng, 113(8), 1214-1229. https://doi.org/10.1002/nme.5563.
  12. Federico, F. and Sigmund, O. and Guest, J.K. (2021), "Topology optimization with linearized buckling criteria in 250 lines of Matlab", Struct Multidisc Optim, 63, 3045-3066. https://doi.org/10.1007/s00158-021-02854-x.
  13. Behrou, R., Lotfi, R., Carstensen, J.V., Ferrari, F. and James K.G. (2021), "Revisiting element removal for density-based structural topology optimization with reintroduction by Heaviside projection", Comput. Meth. Appl. Mech. Eng., 380, 113799. https://doi.org/10.1016/j.cma.2021.113799.
  14. Townsend, S. and Kim, H.A. (2019), "A level set topology optimization method for the buckling of shell structures", Struct Multidisc. Optim., 60, 1783-1800. https://doi.org/10.1007/s00158-019-02374-9.
  15. Wu, C., Fang, F. and Li, Q. (2021), "Multi-material topology optimization for thermal buckling criteria", Comput Methods Appl. Mech. Eng., 346, 1136-1155. https://doi.org/10.1016/j.cma.2018.08.015.
  16. Banh, T.T and Lee, D.K (2019), "Topology optimization of multidirectional variable thickness thin plate with multiple materials", Struct. Multidisc. Optim., 59, 1503-152. https://doi.org/10.1007/s00158-018-2143-8.
  17. Banh, T.T and Lee, D.K (2020), "Multiphase material topology optimization of Mindlin-Reissner plate with nonlinear variable thickness and Winkler foundation", Steel Compos. Struct., 35(1), 129-145. http://dx.doi.org/10.12989/scs.2020.35.1.129.
  18. Banh, T.T, Shin, S.M. and Lee, D.K. (2018), "Topology optimization for thin plate on elastic foundations by using multi-mateirial", Steel Compos. Struct., 27(2), 27-33. https://doi.org/10.12989/scs.2018.27.2.177.
  19. Banh, T.T and Lee, D.K (2018), "Multi-material topology optimization of Reissner-Mindlin plate using MICT4", Steel Compos. Struct., 27(1), 27-33. https://doi.org/10.12989/scs.2018.27.1.027.
  20. Picelli, R., Townsend, S., Brampton, C., Norato, J., Kim H.A (2018), "Stress-based shape and topology optimization with the level set method", Comput. Meth. Appl. Mech. Eng., 329, 1-23. https://doi.org/10.1016/j.cma.2017.09.001.
  21. Radhika, N.,Teja, K., Rahul, K. and Shivashankar, A. (2018), "Fabrication of Cu-Sn-Ni /SiC FGM for automotive applications: investigation of its mechanical and tribological properties", Environ. Sci. Pollut. Res., 10, 1705-1716. https://doi.org/10.1007/s12633-017-9657-3.
  22. Smith, J.A., Mele, E., Rimington, R.P., Capel, A.J., Lewis, M.P., Silberschmidt, V.V. and Li, S. (2019), "Polydimethylsiloxane and poly(ether) ether ketone functionally graded composites for biomedical applications", J. Mech. Behavior Biomed. Mater., 93, 130-142. https://doi.org/10.1016/j.jmbbm.2019.02.012.
  23. Almeida, S.R.M., Paulino, G.H. and Silva, E.C.N. (2010), "Layout and material gradation in topology optimization of functionally graded structures: A global-local approach", 42, 855-868. https://doi.org/10.1007/s00158-010-0514-x.
  24. Luo, Y., Li, Q. and Liu, S. (2019), "A projection-based method for topology optimization of structures with graded surfaces", Int. J. Numer. Meth. Engng., 118, 654-677.https://doi.org/10.1002/nme.6031.
  25. Banh, T.T, Luu G.N. and Lee, D.K. (2021), "A non-homogeneous multi-material topology optimization approach for functionally graded structures with cracks", Compos. Struct., 273, 114230. https://doi.org/10.1016/j.compstruct.2021.114230.
  26. Deng, J. and Chen, W. (2017), "Concurrent topology optimization of multiscale structures with multiple porous materials under random field loading uncertainty", Struct. Multidisc. Optim., 56, 1-19. https://doi.org 10.1007/s00158-017-1689-1.
  27. Hoang, V.N., Tran, P., Vu, V.T, Nguyen, X.H. (2020), "Design of lattice structures with direct multiscale topology optimization", Compos. Struct., 252, 112718. https://doi.org/10.1016/j.compstruct.2020.112718.
  28. Sivanpuram, R., Dunning, P.D. and Kim, H.A. (2016), "Simultaneous material and structural optimization by multiscale topology optimization", Struct Multidisc. Optim., 54, 1267-1281. https://doi.org/10.1007/s00158-016-1519-x. Xia, L. and Breitkopf, P. (2014), "Concurrent topology
  29. Xia, L. and Breitkopf, P. (2016), "Recent advances on topology optimization of multiscale nonlinear structures", Arc. Comput. Meth. Eng., 24, 227-249. https://doi.org/10.1007/s11831-016-9170-7.
  30. Xia, L. and Breitkopf, P. (2016), “Recent advances on topology optimization of multiscale nonlinear structures”, Arc. Comput. Meth. Eng., 24, 227-249. https://doi.org/10.1007/s11831-016-9170-7.
  31. Singh, R., Kumar, R., Farina, I., Colangelo, F., Feo, L. and Fraternali, F. (2019), "Multi-material additive manufacturing of sustainable innovative materials and structures", Polymers, 11, 62. https://doi.org/10.3390/polym11010062.
  32. Svanberg, K. (1987), "The method of moving asymptotes-a new method for structural optimization", Int. J. Numer. Meth. Engng., 24(2), 359-373. https://doi.org/10.1002/nme.1620240207.
  33. Bendsoe, M.P. and Sigmund, O. (2004), Topology Optimization. Theory, Methods, and Applications. 2nd ed., corrected printing. 
  34. Paulino, G.H., Sutradhar, A. and Gray, J.L. (2002), “Boundary element methods for functionally graded materials”, Bound. Element., 4.
  35. Silva, N.C.A. and Paulino, G.H. (2004), "Topology optimization applied to the design of functionally graded material (FGM) structures", In: Proceedings of 21st international congress of theoretical and applied mechanics (ICTAM), 15-21.
  36. Paulino, G.H. and Silva, N.C.E. (2005), "Design of functionally graded structures using topology optimization", Mater. Sci. Forum, 492-493, 435-440. https://doi.org/10.4028/www.scientific.net/MSF.492-493.435.
  37. Amio, R.C.R., Vatanabe S.L. and Silva E.C.N. (2013), "Design, manufacturing and characterization of functionally graded flextensional piezoelectric actuators", J. Physics: Conference Series, 419, 012003. https://doi.org/10.1088/1742-6596/419/1/012003.
  38. Zhou, M. (2004), "Topology optimization for shell structures with linear buckling responses", In WCCM VI. Beijing, China. Hoang, V.N, and Xuan, H.N. (2020), "Extruded-geometriccomponent-based 3D topology optimization", Comput. Meth. Appl. Mech. Eng., 371, 113293. https://doi.org/10.1016/j.cma.2020.113293.
  39. Hoang, V.N, and Xuan, H.N. (2020), “Extruded-geometriccomponent-based 3D topology optimization”, Comput. Meth. Appl. Mech. Eng., 371, 113293. https://doi.org/10.1016/j.cma.2020.113293.
  40. Hoang, V.N, Pham T, Tangaramvong, S, Bordas, S.P.A and Xuan, H.N. (2021)," Robust adaptive topology optimization of porous infills under loading uncertainties", Struct. Multidiscip. Optimiz., 63, 2253-2266. https://doi.org/10.1007/s00158-020-02800-3.
  41. Hoang, V.N, Pham T, Ho, D, and Xuan, H.N. (2022), "Robust multiscale design of incompressible multi-materials under loading uncertainties", Eng. Comput., 38, 875-890. https://doi.org/10.1007/s00366-021-01372-0.
  42. Xuan, H.N, Liu G.R., Bordas, S., Natarajan, S. and Rabczuk, T, (2013), "An adaptive singular ES-FEM for mechanics problems with singular field of arbitrary order", Comput. Meth. Appl. Mech. Eng., 253, 252-273. https://doi.org/10.1016/j.cma.2012.07.017.
  43. Chau, K.N., Chau, K.N., Ngo, T., Hackl, K. and Xuan, H.N., (2018), "A polytree-based adaptive polygonal finite element method for multi-material topology optimization", Comput. Meth. Appl. Mech. Eng., 332, 712-739. https://doi.org/10.1016/j.cma.2017.07.035.