Structural Engineering and Mechanics

  • 제76권4호
  • /
  • Pages.529-540
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  • 2020
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  • 1225-4568(pISSN)
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  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Robust concurrent topology optimization of multiscale structure under load position uncertainty

  • Cai, Jinhu (School of Mechanical Engineering and Automation, Beihang University) ;
  • Wang, Chunjie (School of Mechanical Engineering and Automation, Beihang University)
  • 투고 : 2020.06.09
  • 심사 : 2020.07.26
  • 발행 : 2020.11.25
⟨ 이전 논문 다음 논문 ⟩

초록

Concurrent topology optimization of macrostructure and microstructure has attracted significant interest due to its high structural performance. However, most of the existing works are carried out under deterministic conditions, the obtained design may be vulnerable or even cause catastrophic failure when the load position exists uncertainty. Therefore, it is necessary to take load position uncertainty into consideration in structural design. This paper presents a computational method for robust concurrent topology optimization with consideration of load position uncertainty. The weighted sum of the mean and standard deviation of the structural compliance is defined as the objective function with constraints are imposed to both macro- and micro-scale structure volume fractions. The Bivariate Dimension Reduction method and Gauss-type quadrature (BDRGQ) are used to quantify and propagate load uncertainty to calculate the objective function. The effective properties of microstructure are evaluated by the numerical homogenization method. To release the computation burden, the decoupled sensitivity analysis method is proposed for microscale design variables. The bi-directional evolutionary structural optimization (BESO) method is used to obtain the black-and-white designs. Several 2D and 3D examples are presented to validate the effectiveness of the proposed robust concurrent topology optimization method.

키워드

과제정보

This work was supported by the National Science Foundation of China (Grant No51635002).

참고문헌

  1. Allaire, G., Jouve, F. and Toader, A.M. (2004), "Structural optimization using sensitivity analysis and a level-set method", J. Comput. Phys., 194(1), 363-393. https://doi.org/10.1016/j.jcp.2003.09.032.
  2. Banh, T.T. and Lee, D. (2019), "Topology optimization of multidirectional variable thickness thin plate with multiple materials", Struct. Multidisciplinary Opt., 59(5), 1503-1520. https://doi.org/10.1007/s00158-018-2143-8.
  3. Banh, T.T., Nguyen, X.Q., Herrmann, M., Filippou, F.C. and Lee, D. (2020), "Multiphase material topology optimization of mindlin-reissner plate with nonlinear variable thickness and winkler foundation", Steel Compos. Struct., 35(1), 129-145. https://doi.org/10.12989/scs.2020.35.1.129.
  4. Bendsoe, M.P. (1989), "Optimal shape design as a material distribution problem", Struct. Optimization, 1(4), 193-202. https://doi.org/10.1007/BF01650949.
  5. Bendsoe, M.P. and Kikuchi, N. (1988), "Generating optimal topologies in structural design using a homogenization method", Comput Methods Appl. Mech. Eng., 71(2), 197-224. https://doi.org/10.1016/0045-7825(88)90086-2.
  6. Cai, J., Wang, C. and Fu, Z. (2019), "Robust concurrent topology optimization of multiscale structure under single or multiple uncertain load cases", J. Numeric Methods Eng., 121, 1456-1483. https://doi.org/10.1002/nme.6275.
  7. Coelho, P.G., Fernandes, P.R., Guedes, J.M. and Rodrigues, H.C. (2008), "A hierarchical model for concurrent material and topology optimisation of three dimensional structures", Struct. Multidisciplinary Opt., 35(2), 107-115. https://doi.org/10.1007/s00158-007-0141-3.
  8. Cui, C. Ohmori, H. and Sasaki, M. (2003), "Computational morphogenesis of 3d structures by extended eso method", J. Assoc. Shell Spatial Struct., 44(1), 51-61.
  9. Deaton, J.D. and Grandhi, R.V. (2014), "A survey of structural and multidisciplinary continuum topology optimization: post 2000", Struct. Multidisciplinary Opt., 49(1), 1-38. https://doi.org/10.1007/s00158-013-0956-z.
  10. Doan, Q.H., Lee, D., Lee, J. and Kang, J. (2019), "Design of buckling constrained multiphase material structures using continuum topology optimization", Meccanica, 54(8),1179-1201. https://doi.org/10.1007/s11012-019-01009-z.
  11. Doan, Q.H., Lee, D., Lee, J. and Kang, J. (2020), "Multi-material structural topology optimization with decision making of stiffness design criteria", Adv. Eng. Informatics, 45, 101098. https://doi.org/10.1016/j.aei.2020.101098.
  12. Du, J. and Yang, R. (2015), "Vibro-acoustic design of plate using bi-material microstructural topology optimization", J. Mech. Sci. Technol., 29(4), 1413-1419. https://doi.org/10.1007/s12206-015-0312-x.
  13. Dunning, P.D., Kim, H.A. and Mullineux, G. (2011), "Introducing loading uncertainty in topology optimization", AIAA J., 49(4), 760-768. https://doi.org/10.2514/1.J050670.
  14. Edgar, J. and Tint, S. (2015), "Additive manufacturing technologies: 3d printing, rapid prototyping, and direct digital manufacturing", Johnson Matthey Technol. Review, 59(3), 193-198. https://doi.org/10.1595/205651315X688406.
  15. Fan, Z., Yan, J., Wallin, M., Ristinmaa, M., Niu, B. and Zhao, G (2019), "Multiscale eigenfrequency optimization of multimaterial lattice structures based on the asymptotic homogenization method", Struct. Multidisciplinary Opt., https://doi.org/10.1007/s00158-019-02399-0.
  16. Fukushima, J., Suzuki, K. and Kikuchi, N. (1992), "Shape and topology optimization of a car body with multiple loading conditions", Technical Report, 101(6), 1025-1029. https://doi.org/10.4271/920777.
  17. Guest, J.K. and Igusa, T. (2008), "Structural optimization under uncertain loads and nodal locations", Comput Methods Appl. Mech. Eng., 198(1), 116-124. https://doi.org/10.1016/j.cma.2008.04.009.
  18. Guo, X. and Cheng, G. (2010), "Recent development in structural design and optimization", Acta Mechanica Sinica, 26(6), 807-823. https://doi.org/10.1007/s10409-010-0395-7.
  19. Guo, X., Zhao, X., Zhang, W., Yan, J. and Sun, G. (2015), "Multiscale robust design and optimization considering load uncertainties", Comput Methods Appl. Mech. Eng., 283, 994-1009. https://doi.org/10.1016/j.cma.2014.10.014.
  20. Jeong, S., Seong, H.K., Kim, C.W. and Yoo, J. (2001), "Structural design considering the uncertainty of load positions using the phase field design method", Finite Element Anal. Design, 54(4), 331-390. https://doi.org/10.1115/1.1388075.
  21. Jeong, S., Seong, H.K., Kim, C.W. and Yoo, J. (2019), "Structural design considering the uncertainty of load positions using the phase field design method", Finite Element Anal. Design, 161, 1-15. https://doi.org/10.1016/j.finel.2019.04.002.
  22. Jun, Y. Guo, X. and Cheng, G. (2016), "Multiscale concurrent material and structural design under mechanical and thermal loads", Comput. Mech., 57(3), 437-446. https://doi.org/10.1007/s00466-015-1255-x.
  23. Kumar, S., Tejani, G.G., Pholdee, N. and Bureerat, S. (2020), "Multi-objective modified heat transfer search for truss optimization", Eng. Comput., https://doi.org/10.1007/s00366-020-01010-1.
  24. Lee, S. H., Chen, W. and Kwak, B.M. (2009), "Robust design with arbitrary distributions using gauss-type quadrature formula", Struct. Multidisciplinary Opt., 39(3), 227-243. https://doi.org/10.1007/s00158-008-0328-2.
  25. Li, Q., Xu, R., Liu, J., Liu, S. and Zhang, S. (2019), "Topology optimization design of multiscale structures with alterable microstructural length-width ratios", Compos. Struct., 230, 111454. https://doi.org/10.1016/j.compstruct.2019.111454.
  26. Liang, X. and Du, J. (2019), "Concurrent multiscale and multi-material topological optimization of vibro-acoustic structures", Comput Methods Appl. Mech. Eng., 349: 117-148. https://doi.org/10.1016/j.cma.2019.02.010.
  27. Liu, J., Wen, G., Qing, Q. and Xie, Y.M. (2017), "An efficient method for topology optimization of continuum structures in the presence of uncertainty in loading direction", J. Comput. Methods, 14(05), 1750054. https://doi.org/10.1142/S0219876217500542.
  28. Liu, J., Wen, G., Qing, Q., Li, F. and Xie, Y.M. (2018), "Robust topology optimization for continuum structures with random loads", Eng. Comput., 35(2), 710-732. https://doi.org/10.1108/EC-10-2016-0369.
  29. Liu, L., Yan, J. and Cheng, G. (2008), "Optimum structure with homogeneous optimum truss-like material", Comput. Struct., 86(13), 1417-1425. https://doi.org/10.1016/j.compstruc.2007.04.030.
  30. Logo, J. (2012), "Simp type topology optimization procedure considering uncertain load position", Periodica Polytechnica Civil Engineering, 56(2), 213-217. https://doi.org/10.3311/pp.ci.2012-2.07.
  31. M. Zhou and G.I.N. Rozvany (1991), The coc algorithm, part ii: Topological, geometrical and generalized shape optimization. Comput Methods Appl. Mech. Eng., 89(1), 309-336. https://doi.org/10.1016/0045-7825(91)90046-9.
  32. Madsen, H., Krenk, S. and Lind, N.C. (2006), Methods Structural Safety, Courier Corporation, Massachusetts, USA.
  33. Niu, B., Yan, J. and Cheng, G. (2008), "Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency", Struct. Multidisciplinary Opt., 39(2), 115. https://doi.org/10.1007/s00158-008-0334-4.
  34. Rodrigues, H., Guedes, J.M. and Bendsoe, M.P. (2002), "Hierarchical optimization of material and structure", Struct. Multidisciplinary Opt., 24(1), 1-10. https://doi.org/10.1007/s00158-002-0209-z.
  35. Sigmund, O. and Maute, K. (2013), "Topology optimization approaches", Struct. Multidisciplinary Opt., 48(6), 1031-1055. https://doi.org/10.1007/s00158-013-0978-6.
  36. Sigmund, O. and Petersson, J. (1998), "Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima", Struct. Opt., 16(1), 68-75. https://doi.org/10.1007/BF01214002.
  37. Stein, M. (1987), "Large sample properties of simulations using latin hypercube sampling", Technometrics, 29(2), 143-151. https://doi.org/10.1080/00401706.1987.10488205.
  38. Tejani, G.G., Kumar, S. and Gandomi, A.H. (2019b), "Multi-objective heat transfer search algorithm for truss optimization", Eng. Comput., 1-22. https://doi.org/10.1007/s00366-019-00846-6.
  39. Tejani, G.G., Pholdee, N., Bureerat, S. and Prayogo, D. (2018b), "Multiobjective adaptive symbiotic organisms search for truss optimization problems", Knowledge Based Syst., 161, 398-414. https://doi.org/10.1016/j.knosys.2018.08.005.
  40. Tejani, G.G., Pholdee, N., Bureerat, S., Prayogo, D. and Gandomi, A.H. (2019a), "Structural optimization using multi-objective modified adaptive symbiotic organisms search", Expert Syst. Appl., 125, 425-441. https://doi.org/10.1016/j.eswa.2019.01.068.
  41. Tejani, G.G., Savsani, V.J. and Patel, V.K. (2016), "Adaptive symbiotic organisms search (sos) algorithm for structural design optimization", J. Comput. Design Eng., 3(3), 226-249. https://doi.org/10.1016/j.jcde.2016.02.003.
  42. Tejani, G.G., Savsani, V.J., Bureerat, S. and Patel, V.K. (2018a), "Topology and size optimization of trusses with static and dynamic bounds by modified symbiotic organisms search", J. Comput. Civil Eng., 32(2), https://doi.org/10.1061/(asce)cp.1943-5487.0000741.
  43. Wang, D. and Gao, W. (2019), "Robust topology optimization under load position uncertainty", J. Numeric Methods Eng., 120(11): 1249-1272. https://doi.org/10.1002/nme.6180.
  44. Wang, M.Y., Wang, X. and Guo, D. (2003), "A level set method for structural topology optimization", Comput Methods Appl. Mech. Eng., 192(1), 227-246. https://doi.org/10.1016/S0045-7825(02)00559-5.
  45. Wu, J., Gao, J., Luo, Z. and Brown, T. (2016), "Robust topology optimization for structures under interval uncertainty", Adv. Eng. Software, 99, 36-48. https://doi.org/10.1016/j.advengsoft.2016.05.002.
  46. Xia, L. and Breitkopf, P. (2015), "Design of materials using topology optimization and energybased homogenization approach in matlab", Struct. Multidisciplinary Opt., 52(6), 1229-1241. https://doi.org/10.1007/s00158-015-1294-0.
  47. Xiaodong, H. and Xie, Y.M. (2007), "Convergent and mesh-independent solutions for the bi-directional evolutionary Struct. Opt. method", Finite Element Anal. Design, 43(14), 1039-1049. https://doi.org/10.1016/j.finel.2007.06.006.
  48. Xiaodong, H. and Xie, Y.M. (2009), "Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials", Comput. Mech., 43(3),393-401. https://doi.org/10.1007/s00466-008-0312-0.
  49. Xiaodong, H. and Xie, Y.M. (2010), Evolutionary topology optimization of continuum structures: methods and applications, John Wiley & Sons, New Jersey, USA.
  50. Xiaodong, H. and Xie, Y.M. (2010b), "A further review of eso type methods for topology optimization", Struct. Multidisciplinary Opt., 41(5), 671-683. https://doi.org/10.1007/s00158-010-0487-9.
  51. Xie, Y.M. and Steven, G.P. (1993), "A simple evolutionary procedure for structural optimization", Comput. Struct., 49(5),885 - 896. https://doi.org/10.1016/0045-7949(93)90035-C.
  52. Xu, B. and Xie, Y.M. (2015), "Concurrent design of composite macrostructure and cellular microstructure under random excitations", Compos. Struct., 123, 65-77. https://doi.org/10.1016/j.compstruct.2014.10.037.
  53. Xu, B., Huang, X., Zhou, S.W. and Xie, Y.M. (2016), "Concurrent topological design of composite thermoelastic macrostructure and microstructure with multi-phase material for maximum stiffness", Compos. Struct., 150, 84-102. https://doi.org/10.1016/j.compstruct.2016.04.038.
  54. Xu, H. and Rahman, S. (2004), "A generalized dimension-reduction method for multidimensional integration in stochastic mechanics", J. Numeric Methods Eng., 61(12), 1992-2019. https://doi.org/10.1002/nme.1135.
  55. Yan, X. Xiaodong, H., Sun, G. and Xie, Y.M. (2015), "Two-scale optimal design of structures with thermal insulation materials", Compos. Struct., 120, 358-365. https://doi.org/10.1016/j.compstruct.2014.10.013.
  56. Yang, R.J. and Chahande, A.I. (1995), "Automotive applications of topology optimization", Struct. Optimization, 9(3), 245-249. https://doi.org/10.1007/BF01743977.
  57. Zhao, J. and Wang, C. (2014), "Robust topology optimization under loading uncertainty based on linear elastic theory and orthogonal diagonalization of symmetric matrices", Comput Methods Appl. Mech. Eng., 273, 204-218. https://doi.org/10.1016/j.cma.2014.01.018.
  58. Zhao, J., Yoon, H. and Youn, B.D. (2018), "An efficient decoupled sensitivity analysis method for multiscale concurrent topology optimization problems", Struct. Multidisciplinary Opt., 58(2), 445-457. https://doi.org/10.1007/s00158-018-2044-x.
  59. Zhao, J., Yoon, H. and Youn, B.D. (2019), "An efficient concurrent topology optimization approach for frequency response problems", Comput Methods Appl. Mech. Eng., 347, 700-734. https://doi.org/10.1016/j.cma.2019.01.004.
  60. Zheng, Y., Da, D., Li, H., Xiao, M. and Gao, L. (2020), "Robust topology optimization for multi-material structures under interval uncertainty", Appl. Math. Modelling, 78:627-647. https://doi.org/10.1016/j.apm.2019.10.019.
  61. Zheng, Y., Gao, L., Xiao, M., Li, H. and Luo, Z. (2018), "Robust topology optimization considering load uncertainty based on a semianalytical method", J. Adv. Manufacturing Technol., 94(9), 3537-3551. https://doi.org/10.1007/s00170-017-1002-x.
  62. Zuo, Z.H., Huang, X., Rong, J.H. and Xie, Y.M. (2013), "Multi-scale design of composite materials and structures for maximum natural frequencies", Mater. Design, 51: 1023-1034. https://doi.org/10.1016/j.matdes.2013.05.014.