Steel and Composite Structures

  • 제41권3호
  • /
  • Pages.385-402
  • /
  • 2021
  • /
  • 1229-9367(pISSN)
  • /
  • 1598-6233(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Multiple bi-directional FGMs topology optimization approach with a preconditioned conjugate gradient multigrid

  • Banh, Thanh T. (Department of Architectural Engineering, Sejong University) ;
  • Luu, Nam G. (Department of Architectural Engineering, Sejong University) ;
  • Lieu, Qui X. (Faculty of Civil Engineering, Ho Chi Minh City University of Technology) ;
  • Lee, Jaehong (Department of Architectural Engineering, Sejong University) ;
  • Kang, Joowon (Department of Architecture, Yeungnam University) ;
  • Lee, Dongkyu (Department of Architectural Engineering, Sejong University)
  • 투고 : 2020.09.04
  • 심사 : 2021.09.30
  • 발행 : 2021.11.10
⟨ 이전 논문 다음 논문 ⟩

초록

This article aims to introduce a multigrid preconditioned conjugate gradient (MPCG)-oriented topology optimization (TO) methodology using multiple bi-directional functionally graded (BFG) models for the first time. For that purpose, the MPCG paradigm is integrated into the TO procedure to more effectively and quickly resolve linear algebraic systems arising from the differential equations' discretization. In addition, a refined BFG material interpolation in Solid Isotropic Material with Penalization (SIMP), which is based on an alternating active-phase algorithm, is produced. In which continuously altered macroscopic material properties are represented by an explicit exponential function. This paper describes the well-founded mathematical formulations for multi-material topology optimization of BFG structures in great detail. Finally, several numerical examples are tested to demonstrate the proposed approach's capability and efficiency.

키워드

과제정보

This research was supported by a grant (NRF-2020R1A4A2002855) from NRF (National Research Foundation of Korea) funded by MEST (Ministry of Education and Science Technology) of Korean government.

참고문헌

  1. Banh T.T. and Lee D. (2018), "Multi-material topology optimization design for continuum structures with crack patterns", Compos. Struct., 186, 193-209. https://doi.org/10.1016/j.compstruct.2017.11.088.
  2. 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.
  3. Banh T.T., Nguyen Q.X., Herrmann M., Filippou F.C. and Lee D. (2020), "Multi-phase material topology optimization of Mindlin-Reissner plate with nonlinear variable thickness and Winkler foundation", Steel Compos. Struct., 35, 129-145. http://doi.org/10.12989/scs.2020.35.1.129.
  4. Bendsoe, M. and Kikuchi, N. (1988), "Generating optimal topologies in structural design using a homogenization method", Comput. Method. Appl. M., 17, 197-224. https://doi.org/10.1016/0045-7825(88)90086-2.
  5. Bendsoe M. and Sigmund O. (2004), Topology optimization: Theory, Methods, and Applications, Springer-Verlag Berlin Heidelberg
  6. Bouguenina O., Belakhdar K., Tounsi A. and Bedia E.A.A. (2015), "Numerical analysis of FGM plates with variable thickness subjected to thermal buckling", Steel Compos. Struct., 19, 679-695. http://doi.org/10.12989/scs.2015.19.3.679
  7. Braess, D. and Hackbusch, W. (1983), "A new convergence proof for the multigrid method including the V-cycle", SIAM J. Numer. Anal., 20(5), 967-975. https://doi.org/10.1137/0720066.
  8. Brandt, A. (1973), "Multi-level adaptive technique (MLAT) for fast numerical solution to boundary value problems", Proceedings of the 3rd International Conference on Numerical Methods in Fluid Mechanics, 18, 82-89. https://doi.org/10.1007/BFb0118663.
  9. Brandt, A. (1977), "Multi-level adaptive solutions to boundary-value problems", Math. Comput., 31(138), 333-390. https://doi.org/10.2307/2006422.
  10. Delale, F. and Erdogan, F. (1983), "The crack problem for a nonhomogeneous plane", J. Appl. Mech., 50(3), 609-614. https://doi.org/10.1115/1.3167098.
  11. Erdogan, F. and Wu, B.H. (1997), "The surface crack problem for a plate with functionally graded properties", J. Appl. Mech., 64(3), 449-456. https://doi.org/10.1115/1.2788914.
  12. Fedorenko, R. (1962), "A relaxation method for solving elliptic difference equations", Comput. Math. Math. Phys., 1(4), 1092-1096. https://doi.org/10.1016/0041-5553(62)90031-9.
  13. Fedorenko, R. (1964), "The speed of convergence of one iterative process", Comput. Math. Math. Phys., 4(3), 227-235. https://doi.org/10.1016/0041-5553(64)90253-8.
  14. Ghabraie, K. (2012), "Applications of topology optimization techniques in seismic design of structure", Structural Seismic Design Optimization and Earthquake Engineering: Formulations and Applications, 10, 232-268. 10.4018/978-1-4666-1640-0.ch010.
  15. Gu, P. and Asaro, R.J. (1997), "Crack deflection in functionally graded materials", Int. J. Solid.Struct., 34(24), 3085-3098. https://doi.org/10.1016/S0020-7683(96)00175-8.
  16. Hackbusch, W. (1985), Multi-grid methods and applications, Springer Series in Computational Mathematics, Springer-Verlag Berlin Heidelberg.
  17. Hackbusch, W. (1976), "Ein iteratives verfahren zur schnellen auflosung elliptischer randwertprobleme", Tech. Rep., 76(12).
  18. Hackbusch, W. (1981), "On the convergence of multi-grid iterations", Beitrage zur Numer. Math., 9, 213-239. https://doi.org/10.1007/978-3-662-02427-0_7
  19. Hestenes, M.R. and Stiefel, E. (1952), "Methods of conjugate gradients for solving linear systems", J. Res. National Bureau Standards, 49, 409-435. https://doi.org/10.6028/jres.049.044.
  20. Kim, J. and Paulino, G.H. (2002), "Isoparametric graded finite elements for non-homogeneous isotropic and orthotropic materials", J. Appl. Mech., 69, 502-514. https://doi.org/10.1115/1.1467094.
  21. Konda, N. and Erdogan, F. (1994), "The mixed mode crack problem in a non-homogeneous elastic medium", Eng. Fract. Mech., 47(4), 533-545. https://doi.org/10.1016/0013-7944(94)90253-4.
  22. Li, H., Xiao, M., Zhang, Y. and Gao, L. (2020), "Robust topology optimization of thermoelastic metamaterials considering hybrid uncertainties of material property", Compos. Struct., 248, 112477. https://doi.org/10.1016/j.compstruct.2020.112477.
  23. Mirjavadi, S.S., Afshari, B.M., Shafiei, N., Hamouda, A.M.S. and Kazemi, M. (2017), "Thermal vibration of two-dimensional functionally graded (2D-FG) porous Timoshenko nanobeams", Steel Compos. Struct., 25, 415-426. http://dx.doi.org/10.12989/scs.2017.25.4.415.
  24. Paulino, G., Sutradhar, A. and Gray, L. (2002), "Boundary element methods for functionally graded materials", Bound. Elem., 34, 10. https://doi.org/10.2495/BT030141.
  25. Paulino, G.H. and Silva, E.C.N. (2005), "Design of functionally graded structures using topology optimization", Mater. Sci. Forum, 435-440.
  26. Sigmund, O. (1997), "On the design of compliant mechanisms using topology optimization", Mech. Struct. Machines, 25, 493-524. https://doi.org/10.1080/08905459708945415.
  27. Silva, E.C.N. and Paulino, G.H. (2004), "Topology optimization applied to the design of functionally graded material (FGM) structures", Proceedings of 21st international congress of theoretical and applied mechanics (ICTAM), 15-21 August 2004, Warsaw.
  28. Silva G.A., Beack A.T. and Sigmund Q. (2020), "Topology optimization of compliant mechanisms considering stress constraints, manufacturing uncertainty and geometric nonlinearity", Comput. Method. Appl. M., 365, 112872. https://doi.org/10.1016/j.cma.2020.112972.
  29. Tavakoli, R. and Mohseni, S.M. (2014), "Alternating active-phase algorithm for multimaterial topology optimization problems: A 115-line MATLAB implementation", Struct. Multidiscip. O., 49, 621-642. https://doi.org/10.1007/s00158-013-0999-1.
  30. Zhou, S. and Wang, M.Y. (2006), "Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multi-phase transition", Tructural Multidiscip. O., 33, 89-111. https://doi.org/10.1007/s00158-006-0035-9.