DOI QR코드

DOI QR Code

Multi-material polygonal topology optimization for functionally graded isotropic and incompressible linear elastic structures

  • Thanh T. Banh (Department of Architectural Engineering, Sejong University) ;
  • Joowon Kang (Department of Architecture, Yeungnam University) ;
  • Soomi Shin (Research Institute of Industrial Technology, Pusan National University) ;
  • Dongkyu Lee (Department of Architectural Engineering, Sejong University)
  • 투고 : 2023.12.12
  • 심사 : 2024.04.22
  • 발행 : 2024.05.10

초록

This paper proposes an effective method for optimizing the structure of functionally graded isotropic and incompressible linear elastic materials. The main emphasis is on utilizing a specialized polytopal composite finite element (PCE) technique capable of handling a broad range of materials, addressing common volumetric locking issues found in nearly incompressible substances. Additionally, it employs a continuum model for bi-directional functionally graded (BFG) material properties, amalgamating these aspects into a unified property function. This study thus provides an innovative approach that tackles diverse material challenges, accommodating various elemental shapes like triangles, quadrilaterals, and polygons across compressible and nearly incompressible material properties. The paper thoroughly details the mathematical formulations for optimizing the topology of BFG structures with various materials. Finally, it showcases the effectiveness and efficiency of the proposed method through numerous numerical examples.

키워드

과제정보

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

참고문헌

  1. Antonietti, P., Bruggi, M., Scacchi, S. and Verani M. (2017), "On the virtual element method for topology optimization on polygonal meshes: A numerical study", Comput. Mathem. Appl., 74, 1091-1109.
  2. Arslan, K. and Gunes, R. (2018), "Experimental damage evaluation of honeycomb sandwich structures with Al/B4c fgm face plates under high velocity impact loads", Compos. Struct., 202, 304-312. https://doi.org/10.1016/j.compstruct.2018.01.087.
  3. Banh, T.T., Lieu, Q.X., Kang, J., Ju, Y., Shin, S. and Lee, D. (2023c), "A novel robust stress-based multimaterial topology optimization model for structural stability framework using refined adaptive continuation method", Eng. Comput., 1-37.
  4. Banh, T.T., Lieu, X.Q., Lee, J., Kang, J. and Lee, D. (2023b), "A robust dynamic unified multi-material topology optimization method for functionally graded structures", Struct. Multidiscipl. Optimiz., 66. https://doi.org/10.1007/s00158-023-03501-3.
  5. Banh, T.T., Luu, G.N. and Lee, D. (2021b), "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.
  6. Banh, T.T., Luu, G.N. and Lee, D. (2023a), "A smooth boundary scheme-based topology optimization for functionally graded structures with discontinuities", Steel Compos. Struct., 48, 73-88. https://doi.org/10.12989/scs.2023.48.1.073.
  7. Banh, T.T., Luu, G.N., Lieu, X.Q., Lee, J., Kang, J. and Lee, D. (2021a), "Multiple bi-directional FGMs topology optimization approach with a preconditioned conjugate gradient multigrid", Steel Compos. Struct., 41, 385-402. https://doi.org/10.12989/scs.2021.41.3.385.
  8. Banh, T.T., Shin, S., Kang, J. and Lee, D. (2024a), "Frequencyconstrained topology optimization in incompressible multimaterial systems under design-dependent loads", Thin-Wall. Struct., 196, 111467. https://doi.org/10.1016/j.tws.2023.111467.
  9. Banh, T.T., Shin, S., Kang, J. and Lee, D. (2024b), "Comprehensive multi-material topology optimization for stress-driven design with refined volume constraint subjected to harmonic force excitation", Eng. Comput., https://doi.org/10.1007/s00366-023-01939-z.
  10. Bendsoe, M.P. and Kikuchi, N. (1988), "Generating optimal topologies in structural design using homogenization", Comput. Meth. Appl. Mech. Eng., 71, 197-224. https://doi.org/10.1016/0045-7825(88)90086-2.
  11. Bendsoe, M.P. and Sigmund, O. (1999), "Material interpolation schemes in topology optimization", Archive Appl. Mech., 69. 635-654. https://doi.org/10.1007/s004190050248
  12. Cai, K., Cao, J., Shi, J., Liu, L. and Qin, Q.H. (2016), "Optimal layout of multiple bi-modulus materials", Struct. Multidiscipl. Optimiz., 53, 801-811. https://doi.org/10.1007/s00158-015-1365-2
  13. Chau, P.K.N., Chau, K.N., Ngo, T., Hackl, K. and Nguyen, X.H. (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
  14. Hoang, V.N., Pham, T., Ho, D. and Nguyen, X.H. (2022), "Robust multiscale design of incompressible multi-materials under loading uncertainties", Eng. Comput., 38, 875-890. https://doi.org/10.1007/s00158-023-03501-3.
  15. Ilschner, B. (1996), "Processing-microstructure-property relationships in graded materials", J. Mech. Phys. Solids, 44, 647-656. https://doi.org/10.1016/0022-5096(96)00023-3.
  16. Kim, J.H. and H. Paulino, G.H. (2003), "An accurate scheme for mixed-mode fracture analysis of functionally graded materials using the interaction integral and micromechanics models", Int. J. Numer. Meth. Eng., 58, 1457-1497. https://doi.org/10.1002/nme.819.
  17. Kim, J.H. and Paulino, G.H. (2002), "Isoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials", J. Appl. Mech., 69(4), 502-514. https://doi.org/10.1115/1.1467094.
  18. Kim, J.H. and Paulino, G.H. (2004), "Consistent formulations of the interaction integral method for fracture of functionally graded materials", J. Appl. Mech., 72, 351-364. https://doi.org/10.1115/1.1876395.
  19. Li D. and Kim I.Y. (2018), "Multi-material topology optimization for practical lightweight design", Struct. Multidiscipl. Optimiz., 58, 1081-1094. https://doi.org/10.1007/s00158-018-1953-z
  20. Lieu, X.Q. and Lee J. (2017a), "Multiresolution topology optimization using isogeometric analysis", Int. J. Numer. Meth. Eng., 112, 2025-2047. https://doi.org/10.1002/nme.5593.
  21. Lieu, X.Q. and Lee J. (2017b), "A multi-resolution approach for multi-material topology optimization based on isogeometric analysis", Comput. Meth. Appl. Mech. Eng., 323, 272-302. https://doi.org/10.1016/j.cma.2017.05.009.
  22. Liu, P., Luo, Y. and Kang, Z. (2016), "Multi-material topology optimization considering interface behavior via XFEM and level set method", Comput. Meth. Appl. Mech. Eng., 308, 113-133. https://doi.org/10.1016/j.cma.2016.05.016
  23. Nguyen, M.N., Lee, D. and Kang, J., (2023), "Topology optimization with functionally graded multi-material for elastic buckling criteria", Steel Compos. Struct., 46, 33-51. https://doi.org/10.12989/scs.2023.46.1.033.
  24. Nguyen, X.H., Chau, K.N. and Chau K.N. (2019), "Polytopal composite finite elements", Comput. Methods Appl. Mech. Eng., 355, 405-437. https://doi.org/10.1016/j.cma.2019.06.030
  25. Paulino, G.H. and Silva, E.C.N. (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
  26. Pedersen, N.L. (2000), "Maximization of eigenvalue using topology optimization", Struct. Multidiscipl. Optimiz., 20, 2-11. https://doi.org/10.1007/s001580050130.
  27. Radhika, N., Kamireddy, T., Kanithi, R. 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., 102, 1705-1716. https://doi.org/10.1007/s12633-017-9657-3.
  28. 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. 1037-1067. https://doi.org/10.1016/S0022-5096(96)00114-7
  29. Smith, J.A., Mele, E., Rimington, R.P., Capel, A.J., Lewis, M.P., Sil- berschmidt, V.V. and Li, S. (2019), "Polydimethylsiloxane and poly(ether) ether ketone functionally graded composites for biomedical applications", J. Mech. Behavior Biomedic. Mater., 93, 130-142. https://doi.org/10.1016/j.jmbbm.2019.02.012.
  30. Stolpe, M. and Svanberg, K. (2001), "An alternative interpolation scheme for minimum compliance topology optimization", Struct. Multidiscipl. Optimiz., 22, 116-124. https://doi.org/10.1007/s001580100129.
  31. Svanberg, K. (1987), "The method of moving asymptotes - A new method for structural optimization", Int. J. Numer. Meth. Eng., 24, 359-373. https://doi.org/10.1002/nme.1620240207.
  32. Taheri, A.H. and Hassani, B. (2014), "Simultaneous isogeometrical shape and material design of functionally graded structures for optimal eigenfrequencies", Comput. Meth. Appl. Mech. Eng., 277, 46-80. https://doi.org/10.1016/j.cma.2014.04.014.
  33. Talischi, C., Paulino, G.H. and Pereira A. (2012), "PolyTop: A Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes", Struct. Multidiscipl. Optimiz., 45, 329-357. https://doi.org/10.1007/s00158-011-0696-x.
  34. Talischi, C., Paulino, G.H., Pereira, A. and Menezes I.F.M. (2009), "Polygonal finite elements for topology optimization: A unifying paradigm", Int. J. Numer. Meth. Eng., 82, 671-698.
  35. Tavakoli, R. and Mohseni, M. (2014), "Alternating active-phase algorithm for multimaterial topology optimization problems: A 115-line MATLAB implementation", Struct. Multidiscipl. Optimiz., 49, 621-642. http://dx.doi.org/10.1007/S00158- 013-0999- 1.
  36. Thomsen, J. (1992), "Topology optimization of structures composed of one or two materials", Struct. Optimiza., 66. 108-115. https://doi.org/10.1007/BF01744703
  37. Wang, Y., Luo, Z., Kang, Z. and Zhang, N. (2015), "A multimaterial level set-based topology and shape optimization method", Comput. Meth. Appl. Mech. Eng., 283, 1570-1586. https://doi.org/10.1016/j.cma.2014.11.002
  38. Wei, P. and Paulino, G.H. (2020), "A parameterized level set method combined with polygonal finite elements in topology optimization", Struct. Multidiscipl. Optimiz., 61, 1913-1928. https://doi.org/10.1007/s00158-019-02444-y
  39. Zhang, W., Feng, Z. and Cao, D. (2012), "Nonlinear dynamics analysis of aero engine blades", J. Dyn. Control, 10, 213-221. https://doi.org/10.1109/UKSIM.2011.48.
  40. Zhou, S.W. and Wang, M.Y. (2007), "Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multiphase transitiond", Struct. Multidiscipl. Optimiz., 33, 89-111. https://doi.org/10.1007/s00158-006-0035-9