DOI QR코드

DOI QR Code

Stress-based topology optimization under buckling constraint using functionally graded materials

  • Minh-Ngoc Nguyen (Department of Architectural Engineering, Sejong University) ;
  • Dongkyu Lee (Department of Architectural Engineering, Sejong University) ;
  • Soomi Shin (Research Institute of Industrial Technology, Pusan National University)
  • 투고 : 2021.03.08
  • 심사 : 2023.07.13
  • 발행 : 2024.04.25

초록

This study shows functionally graded material structural topology optimization under buckling constraints. The SIMP (Solid Isotropic Material with Penalization) material model is used and a method of moving asymptotes is also employed to update topology design variables. In this study, the quadrilateral element is applied to compute buckling load factors. Instead of artificial density properties, functionally graded materials are newly assigned to distribute optimal topology materials depending on the buckling load factors in a given design domain. Buckling load factor formulations are derived and confirmed by the resistance of functionally graded material properties. However, buckling constraints for functionally graded material topology optimization have not been dealt with in single material. Therefore, this study aims to find the minimum compliance topology optimization and the buckling load factor in designing the structures under buckling constraints and generate the functionally graded material distribution with asymmetric stiffness properties that minimize the compliance. Numerical examples verify the superiority and reliability of the present method.

키워드

과제정보

This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2022R1A2C1003776, 2021R1I1A1A01054901).

참고문헌

  1. Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B.S. and Sigmund, O. (2011), "Efficient topology optimization in MATLAB using 88 lines of code", Struct. Multidiscipl. Optimiz., 43, 1-16. https://doi.org/10.1007/s00158-010-0594-7.
  2. Bandyopadhyay, A. and Bryan, H. (2018), "Additive manufacturing of multi-material structures", Mater. Sci. Eng.: R: Reports, 129, 1-16. https://doi.org/10.1016/j.mser.2018.04.001.
  3. Banh, T. and Dongkyu, L. (2018), "Multi-material topology optimization design for continuum structures with crack patterns", Compos. Struct., 186. https://doi.org/10.1016/j.compstruct.2017.11.088.
  4. Banh, T.T. (2018), "Multi-material topology optimization of Reissner-Mindlin plates using MITC4", Steel Compos. Struct., 27(1), 27-33. https://doi.org/10.12989/SCS.2018.27.1.027.
  5. Banh, T.T., Luu, N.G. and Lee, D. (2021), "A non-homogeneous multi-material topology optimization approach for functionally graded structures with cracks", Compos. Struct., 273, 114230.
  6. Bendsoe, M.P. and Sigmund, O. (2013), Topology Optimization: Theory, Methods, and Applications, Springer Science & Business Media.
  7. Blasques, J.P. (2014), "Multi-material topology optimization of laminated composite beams with eigenfrequency constraints", Compos. Struct., 111, 45-55. https://doi.org/10.1016/j.compstruct.2013.12.021.
  8. Blasques, J.P. and Mathias, S. (2012), "Multi-material topology optimization of laminated composite beam cross sections", Compos. Struct., 94, 3278-89. https://doi.org/10.1016/j.compstruct.2012.05.002.
  9. De Borst, R., Crisfield, M.A., Remmers, J.J. and Verhoosel, C.V. (2012), Nonlinear Finite Element Analysis of Solids and Structures. John Wiley & Sons.
  10. Doan, Q.H. and Dongkyu, L. (2016), "Multi- Material structural topology optimization method with elastic buckling constraints", Master Thesis, Sejong University, Seoul, Korea.
  11. Doan, Quoc Hoan, Dongkyu Lee, Jaehong Lee, and Joowon Kang (2019), "Design of buckling constrained multiphase material structures using continuum topology optimization", Meccanica, 54, 1179-201.10.1007/s11012-019-01009-z
  12. 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 eigensolver", Int. J. Numer. Meth. Eng., 107(12), 1029-1053. https://doi.org/10.1002/nme.5203
  13. Frazier, W.E. (2014), "Metal additive manufacturing: A review", J. Mater. Eng. Perform., 23, 1917-1928. https://doi.org/10.1007/s11665-014-0958-z
  14. Gao, X. and Haitao, M. (2015), "Topology optimization of continuum structures under buckling constraints", Comput. Struct., 157, 142-152. https://doi.org/10.1016/j.compstruc.2015.05.020.
  15. Gouker, R.M., Gupta, S.K., Bruck, H.A. and Holzschuh, T. (2006), "Manufacturing of multi-material compliant mechanisms using multi-material molding", Int. J. Adv. Manufact. Technol., 30, 1049-1075. https://doi.org/10.1007/s00170-005-0152-4.
  16. Huang, S.H., Liu, P., Mokasdar, A. and Hou, L. (2013), "Additive manufacturing and its societal impact: a literature review", Int. J. Adv. Manufact. Technol., 67, 1191-1203. https://doi.org/10.1007/s00170-012-4558-5.
  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. Konda, N. and Erdogan, F. (1994), "The mixed mode crack problem in a nonhomogeneous elastic medium", Eng. Fract. Mech., 47(4), 533-545. https://doi.org/10.1016/0013-7944(94)90253-4.
  19. Lieu, X.Q. and Lee, J. (2017), "Multiresolution topology optimization using isogeometric analysis", Int. J. Numer. Meth. Eng., 112, 2025-2047. https://doi.org/10.1002/nme.5593.
  20. Lieu, X.Q., Lee, J. (2019), "An isogeometric multimesh design approach for size and shape optimization of multidirectional functionally graded plates", Comput. Meth. Appl. Mech. Eng., 343, 407-437. https://doi.org/10.1016/j.cma.2018.08.017.
  21. Lieu, X.Q., Lee, J.H. (2017), "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. Lindgaard, E. and Dahl, J. (2013), "On compliance and buckling objective functions in topology optimization of snap-through problems", Struct. Multidiscipl. Optimiz., 47, 409-421. https://doi.org/10.1007/s00158-012-0832-2.
  23. Luong-Van, H., Nguyen-Thoi, T., Liu, G.R. and Phung-Van, P. (2014), "A cell-based smoothed finite element method using three-node shear-locking free Mindlin plate element (CS-FEM-MIN3) for dynamic response of laminated composite plates on viscoelastic foundation", Eng. Anal. Bound. Elements, 42, 8-19. https://doi.org/10.1016/j.enganabound.2013.11.008.
  24. Manickarajah, D. (1998), "Optimum design of structures with stability constraints using the evolutionary optimisation method", Victoria University of Technology.
  25. Neves, M.M., Sigmund, O. and Bendsoe, M.P. (2002), "Topology optimization of periodic microstructures with a penalization of highly localized buckling modes", Int. J. Numer. Meth. Eng., 54(6), 809-834. https://doi.org/10.1002/nme.449.
  26. Nguyen-Thoi, T., Luong-Van, H., Phung-Van, P., Rabczuk, T. and Tran-Trung, D. (2013), "Dynamic responses of composite plates on the Pasternak foundation subjected to a moving mass by a cell-based smoothed discrete shear gap (CS-FEM-DSG3) method", Int. J. Compos. Mater., 3(6), 19-27. https://doi.org/10.5923/s.cmaterials.201309.03.
  27. Nguyen-Thoi, T., Phung-Van, P., Luong-Van, H., Nguyen-Van, H. and Nguyen-Xuan, H. (2013), "A cell-based smoothed three-node Mindlin plate element (CS-MIN3) for static and free vibration analyses of plates", Comput. Mech., 51, 65-81. https://doi.org/10.1016/j.commatsci.2014.04.043.
  28. Nguyen-Xuan, H. (2017), "A polytree-based adaptive polygonal finite element method for topology optimization", Int. J. Numer. Meth. Eng., 110, 972-1000. https://doi.org/10.1002/nme.5448.
  29. Nguyen-Xuan, H., 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.
  30. Nguyen-Xuan, H., Nguyen-Hoang, S., Rabczuk, T. and Hackl, K. (2017), "A polytree-based adaptive approach to limit analysis of cracked structures", Comput. Meth. Appl. Mech. Eng., 313, 1006-1039. https://doi.org/10.1016/j.cma.2016.09.016.
  31. Obielodan, J.O., Ceylan, A., Murr, L.E. and Stucker, B.E. (2010), "Multi-material bonding in ultrasonic consolidation", Rapid Prototyping J., 16(3), 180-188. https://doi.org/10.1108/13552541011034843.
  32. Olhoff, N. and Rasmussen, S.H. (1977), "On single and bimodal optimum buckling loads of clamped columns", Int. J. Solids Struct., 13(7), 605-614. https://doi.org/10.1016/0020-7683(77)90043-9.
  33. Paulino, G.H. and Silva, E.C.N. (2005), "Design of functionally graded structures using topology optimization", Mater. Sci. Forum, 492, 435-440. www.scientific.net/MSF.492-493.435. https://doi.org/10.4028/www.scientific.net/MSF.492-493.435
  34. Paulino, G.H., Sutradhar, A. and Gruy, L.J. (2003), "Boundary element methods for functionally graded materials", WIT Transactions Modelling and Simulation, 34.
  35. Phung-Van, P., Abdel-Wahab, M., Liew, K.M., Bordas, S.P.A. and Nguyen-Xuan, H. (2015), "Isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates using higher-order shear deformation theory", Compos. Struct., 123, 137-149. https://doi.org/10.1016/j.compstruct.2014.12.021.
  36. Phung-Van, P., De Lorenzis, L., Thai, C.H., Abdel-Wahab, M. and Nguyen-Xuan, H. (2015), "Analysis of laminated composite plates integrated with piezoelectric sensors and actuators using higher-order shear deformation theory and isogeometric finite elements", Comput. Mater. Sci., 96, 495-505. https://doi.org/10.1016/j.commatsci.2014.04.068.
  37. Phung-Van, P., Ferreira, A.J.M., Nguyen-Xuan, H. and Wahab, M.A. (2017), "An isogeometric approach for size-dependent geometrically nonlinear transient analysis of functionally graded nanoplates", Compos. Part B: Eng., 118, 125-134. https://doi.org/10.1016/j.compositesb.2017.03.012.
  38. Phung-Van, P., Lieu, Q.X., Nguyen-Xuan, H. and Wahab, M.A. (2017), "Size-dependent isogeometric analysis of functionally graded carbon nanotube-reinforced composite nanoplates", Compos. Struct., 166, 120-135. https://doi.org/10.1016/j.compstruct.2017.01.049.
  39. Phung-Van, P., Nguyen, L.B., Tran, L.V., Dinh, T.D., Thai, C.H., Bordas, S.P.A. and Nguyen-Xuan, H. (2015), "An efficient computational approach for control of nonlinear transient responses of smart piezoelectric composite plates", Int. J. Non-Linear Mech., 76, 190-202. https://doi.org/10.1016/j.ijnonlinmec.2015.06.003.
  40. Phung-Van, P., Tran, L.V., Ferreira, A.J.M., Nguyen-Xuan, H. and Abdel-Wahab, M.J.N.D. (2017), "Nonlinear transient isogeometric analysis of smart piezoelectric functionally graded material plates based on generalized shear deformation theory under thermo-electro-mechanical loads", Nonlinear Dyn., 87, 879-894. https://doi.org/10.1007/s11071-016-3085-6.
  41. Phung-Van, P., Luong-Van, H., Nguyen-Thoi, T. and Nguyen-Xuan, H. (2014), "A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) based on the C0-type higher-order shear deformation theory for dynamic responses of Mindlin plates on viscoelastic foundations subjected to a moving sprung vehicle", Int. J. Numer. Meth. Eng., 98(13), 988-1014. https://doi.org/10.1002/nme.4662.
  42. 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.
  43. 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.
  44. 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.
  45. Vaezi, M., Chianrabutra, S., Mellor, B. and Yang, S. (2013), "Multiple material additive manufacturing-Part 1: a review: this review paper covers a decade of research on multiple material additive manufacturing technologies which can produce complex geometry parts with different materials", Virtual Phys. Prototyping, 8(1), 19-50. https://doi.org/10.1080/17452759.2013.778175.
  46. Van den Boom, S.J. (2014), "Topology optimisation including buckling analysis", MS Dissertation, Delft University of Technology.
  47. Xingjun, G. and Haitao, M. (2014), "A new method for dealing with pseudo modes in topology optimization of continua for free vibration", Chinese J. Theoretic. Appl. Mech., 46(5), 739-746.
  48. Yang, Q., Zhang, P., Cheng, L., Min, Z., Chyu, M. and To, A.C. (2016), "Finite element modeling and validation of thermomechanical behavior of Ti-6Al-4V in directed energy deposition additive manufacturing", Additive Manufact., 12, 169-177. https://doi.org/10.1016/j.addma.2016.06.012.
  49. Yi, B., Zhou, Y., Yoon, G.H. and Saitou, K. (2019), "Topology optimization of functionally-graded lattice structures with buckling constraints", Comput. Meth. Appl. Mech. Eng., 354, 593-619. https://doi.org/10.1016/j.cma.2019.05.055.
  50. Zhou, M. (2004), "Topology optimization for shell structures with linear buckling responses", In WCCM VI. Beijing, China.