Steel and Composite Structures

  • 제48권1호
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  • Pages.73-88
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  • 2023
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  • 1229-9367(pISSN)
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  • 1598-6233(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

A smooth boundary scheme-based topology optimization for functionally graded structures with discontinuities

  • Thanh T. Banh (Department of Architectural Engineering, Sejong University) ;
  • Luu G. Nam (Department of Architectural Engineering, Sejong University) ;
  • Dongkyu Lee (Department of Architectural Engineering, Sejong University)
  • 투고 : 2021.06.08
  • 심사 : 2023.06.27
  • 발행 : 2023.07.10
⟨ 이전 논문 다음 논문 ⟩

초록

This paper presents a novel implicit level set method for topology optimization of functionally graded (FG) structures with pre-existing discontinuities (pre-cracks) using radial basis functions (RBF). The mathematical formulation of the optimization problem is developed by incorporating RBF-based nodal densities as design variables and minimizing compliance as the objective function. To accurately capture crack-tip behavior, crack-tip enrichment functions are introduced, and an eXtended Finite Element Method (X-FEM) is employed for analyzing the mechanical response of FG structures with strong discontinuities. The enforcement of boundary conditions is achieved using the Hamilton-Jacobi method. The study provides detailed mathematical expressions for topology optimization of systems with defects using FG materials. Numerical examples are presented to demonstrate the efficiency and reliability of the proposed methodology.

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. Allaire, G., Gournay, F., Jouve, F. and Toader, A.M. (2005), "Structural optimization using topological and shape sensitivity via a level set method", Control Cybernetic., 34(1), 59-80. 
  2. Allaire, G., Jouve F. and Toader A.M. (2004), Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194(1), https://doi.org/10.1016/j.jcp.2003.09.032. 
  3. 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. 
  4. 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. 
  5. Banh, T.T. and Lee D. (2018), "Multi-material topology optimization design for continuum structures with crack patterns", Compos. Struct., 185, 193-209. http://dx.doi.org/10.12989/scs.2018.29.5.635. 
  6. Banh, T.T., Lieu X.Q., Kang, J., Ju Y., Shin ,S. and Lee, D. (2023a), "A robust dynamic unified multi-material topology optimization method for functionally graded structures", Struct. Multidiscipl. Optimiz., 66, 75. https://doi.org/10.1007/s00158-023-03501-3. 
  7. Banh, T.T., Lieu, X.Q., Kang, J., Ju, Y., Shin S. and Lee, D (2023b), "A novel robust stress-based multimaterial topology optimization model for structural stability framework using refined adaptive continuation method", Eng. Comput., https://doi.org/10.1007/s00366-023-01829-4. 
  8. Banh, T.T., Nguyen, Q.X., Herrmann, M., Filippou, F.C. and Lee, D. (2018), "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. 
  9. Belytschko, T. and Black, T. (1999), "Elastic crack growth in finite elements with minimal remeshing", Int. J. Numer. Meth. Eng., 45, 601-620. https://doi.org/10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S. 
  10. Bendsoe, M.P. (1989), "Optimal shape design as a material distribution problem", Struct. Optimiz., 1(4), 193-202. https://doi.org/10.1007/BF01650949. 
  11. Bendsoe, M.P. (1995), Optimization of Structures Topology, Shape, Material, New York: Springer, USA. 
  12. Buhmann, M.D. (2004), Radial Basis Functions: Theory and Implementations, volume 12 of Cambridge Monographs on Applied and Computational Mathematics, 89-98, Cambridge University Press, New York, NY, USA. 
  13. Cheng, A.D., Golberg, M.A., Kansa, E.J. and Zammito, G. (2003), "Exponential convergence and H-c multiquadric collocation method for partial differential equations", Numer. Meth. Partial Equat., 19, 571-594. https://doi.org/10.1002/num.10062. 
  14. Ciu, M.T., Luo, C.C., Li, G. and Pan, M. (2021), "The parameterized level set method for structural topology optimization with shape sensitivity constraint factor", Eng. Comput., 37, 855-872. https://doi.org/10.1007/s00366-019-00860-8. 
  15. Dolbow, J.E. and Gosz, M. (2002), "On the computation of mixed-mode stress intesity factors in functionally graded materials", Int. J. Solid Struct., 39, 2557-2574. https://doi.org/10.1016/S0020-7683(02)00114-2. 
  16. Erdogan, F. and Wu, B.H. (1997), "The surface crack problem for a plate with functionally graded properties", J. Appl. Mech., 93, 130-142. https://doi.org/10.1016/j.jmbbm.2019.02.012. 
  17. Geiss, M.J., Barrera, J.L., Boddeti, and N., Maute, K. (2019), "A regularization scheme for explicit level-set XFEM topology optimization", J. Front. Mech. Eng., 14, 153-170. https://doi.org/10.1007/s11465-019-0533-2. 
  18. Ghasemi, T., Park, H.S. and Rabczuk, T. (2017), "A level-set based IGA formulation for topology optimization of flexoelectric materials", Comput. Meth. Appl. Mech. Eng., 313, 239-258. https://doi.org/10.1016/j.cma.2016.09.029. 
  19. Gu, P. and Asaro, R.J. (1997), "Crack deflection in functionally graded materials", Int. J. Solids Struct., 34, 3085-3098. https://doi.org/10.1016/S0020-7683(96)00175-8. 
  20. Guo X., Zhang W. and Zhong W. (2014), "Doing topology optimization explicitly and geometrically-A new moving morphable components based framework", J. Appl. Mech., 81, 081009. https://doi.org/10.1115/1.4027609. 
  21. Guo, X., Zhang, W.S., Wang, M.Y. and Wei, P. (2011), "Stress-related topology optimization via level set approach", Comput. Meth. Appl. Mech. Engrg., 200, 3439-3452. https://doi.org/10.1016/j.cma.2011.08.016. 
  22. Huynh, H.D., Zhuang, X. and Nguyen, X.H. (2020), "A polytree-based adaptive scheme for modeling linear fracture mechanics using a coupled XFEM-SBFEM approach", Eng. Anal. Bound. Element., 115, 72-85. https://doi.org/10.1016/j.enganabound.2019.11.001. 
  23. 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. 
  24. Islam, S.U., Khan, W., Ullah, B. and Ullah, Z. (2020), "The localized radial basis functions for parameterized level set based structural optimization", Eng. Anal. Bound. Element., 113, 296-305. https://doi.org/10.1016/j.enganabound.2020.01.008. 
  25. Jahangiry, H.A. and Tavakkoli, S.M. (2017), "An isogeometrical approach to structural level set topology optimization", Comput. Methods Appl. Mech. Engrg., 319, 240-257. https://doi.org/10.1016/j.cma.2017.02.005. 
  26. Jiang, Y.T. and Zhao, M. (2020), "Topology optimization under design-dependent loads with the parameterized level-set method based on radial-basis functions", Comput. Meth. Appl. Mech. Engrg., 369, 113235. https://doi.org/10.1016/j.cma.2020.113235. 
  27. Kansa, E.J., Power, H, Fasshauer, G.E. and Ling. L. (2004), "A volumetric integral radial basis function method for time-dependent partial differential equations I. Formulation", Eng. Anal. Bound. Element., 28, 1191-1206. https://doi.org/10.1016/j.enganabound.2004.01.004. 
  28. Kefal, A., Sohouli, A., Erkan, O., Yildiz, M. and Suleman, A. (2019), "Topology optimization of cracked structures using peridynamics", Continuum Mech. Thermodyn, 31, 1645-1672. https://doi.org/10.1007/s00161-019-00830-x. 
  29. 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. 
  30. 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. 
  31. 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. 
  32. Kumar, R., Lal, A., Singh B.N. and Singh, J. (2019), "Meshfree approach on buckling and free vibration analysis of porous FGM plate with proposed IHHSDT resting on the foundation", Curved Layered Struct., 6(1), 192-211. 10.1515/cls-2019-0017. 
  33. Li, L., Wang, M.Y. and Wei, P. (2011), "XFEM schemes for level set based structural optimization", Front. Mech. Eng., 7(4), 335-356. https://doi.org/10.1007/s11465-012-0351-2. 
  34. Liu, H., Zhong, H.M., Tian, Y., Ma, Q.P. and Wang, M.Y. (2019), "A novel subdomain level set method for structural topology optimization and its application in graded cellular structure design", Struct. Multidiscipl. Optimiz., 60, 2221-2247. https://doi.org/10.1007/s00158-019-02318-3. 
  35. Liu, T., Li, B., Wang, S.Y. and Gao, L. (2014), "Eigenvalue topology optimization of structures using a parameterized level set method", Struct. Multidiscipl. Optimiz., 50, 573-591. https://doi.org/10.1007/s00158-014-1069-z. 
  36. Luo, J.Z., Luo, Z., Chen, S.K., Tong, L.Y. and Wang, M. (2008a), "A new level set method for systematic design of hinge-free compliant mechanisms", Comput. Methods Appl. Mech. Engrg., 198, 318-331. https://doi.org/10.1016/j.cma.2008.08.003. 
  37. Luo, Z., Tong, L.Y., Wang, M.Y. and Wang, S.Y. (2007), "Shape and topology optimization of compliant mechanisms using a parameterization level set method", Comput. Modeling Eng. Sci., 227, 680-705. https://doi.org/10.1016/j.jcp.2007.08.011. 
  38. Luo, Z., Tong, L.Y., Wang, M.Y. and Wang, S.Y. (2008b), "A level set-based parameterization method for structural shape and topology optimization", Int J Numer Meth Engng, 76, 1-26. https://doi.org/10.1002/nme.2092. 
  39. Madych, W.R. and Nelson, S.A. (1992), "Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation", J. Approxim., Theory, 70, 97-114. https://doi.org/10.1016/0021-9045(92)90058-V. 
  40. Micchelli, C.A. (1986), "Interpolation of scattered data: distance matrices and conditionally positive definite", Construct. Approxim., 2, 11-22. https://doi.org/10.1007/BF01893414. 
  41. Moes, N. and Dolbow, J. and Belytschko, T. (1999), "A finite element method for crack growth without remeshing", Int. J. Numer. Meth. Eng., 46, 131-150. https://doi.org/10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J. 
  42. Morse, B.S., Yoo, T.S., Chen, D.T., Rheingans, P. and Subramanian, K.R. (2001), "Interpolating implicit surfaces from scattered surface data using compactly supported radial basic functions", Int Conf Shape Model Appl, 15, 89-98. 10.1109/SMA.2001.923379. 
  43. Nanthakumar, S.S., Lahmer, T., Zhuang, X., Zi, G. and Rabczuk, T. (2016), "Detection of material interfaces using a regularized level set method in piezoelectric structures", Inverse Prob. Sci. Eng., 1, 153-176. https://doi.org/10.1080/17415977.2015.1017485. 
  44. Nguyen, N.T., Thai, H.C., Luu, A.T., Nguyen, X.H. and Lee, J. (2019), "NURBS-based postbuckling analysis of functionally graded carbon nanotube-reinforced composite shells", Comput. Meth. Appl. Mech. Eng., 347, 983-1003. https://doi.org/10.1016/j.cma.2019.01.011. 
  45. Nguyen, P.A., Banh T.T., Lee, D., Lee, J., Kang, J. and Shin, S. (2018), "Design of multiphase carbon fiber reinforcement of crack existing concrete structures using topology optimization", Steel Compos. Struct., 29, 635-645. https://doi.org/10.1016/j.compstruct.2017.11.088. 
  46. Nguyen, X.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. 
  47. Osher, S. and Santosa, F. (2001), "Level set methods for optimization problems involving geometry and constraints: I. frequencies of a two-density inhomogeneous drum", J. Comput. Phys., 171(1), 272-288. https://doi.org/10.1006/jcph.2001.6789. 
  48. Osher, S. and Sethian, J.A. (1988), "Fronts propagating with curvature-dependent speed: algorithms based on Hamilton Jacobi formulations", J. Comput. Phys., 79(1), 12-49. https://doi.org/10.1016/0021-9991(88)90002-2. 
  49. Paulino, G.H. and Silva, E.C.N. (2005), "Design of functionally graded structures using topology optimization", Mat. Sci. Forum, 492-493, 435-440. https://doi.org/10.4028/www.scientific.net/MSF.492-493.435. 
  50. Peng, D., Merriman B, Osher S, Zhao H. and Kang M. (1999), "A PDE-based fast local level set method", J. Comput. Phys., 155, 410-438. https://doi.org/10.1006/jcph.1999.6345. 
  51. Querin, O.M., Steven, G.P. and Xie, Y.M. (1998), "Evolutionary structural optimisation (ESO) using a bidirectional algorithm", Eng. Comput., 15(8), 1031-1048. https://doi.org/10.1108/02644409810244129. 
  52. 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. 
  53. Roodsarabi, M., Khatibinia, M. and Sarafrazi, S.R. (2016), "Hybrid of topological derivative-based level set method and isogeometric analysis for structural topology optimization", Steel Compos. Struct., 21(6), 1389-1410. http://dx.doi.org/10.12989/scs.2016.21.6.1389. 
  54. Rostami S.A.L., Ghoddosian A., Kolahdooz A. and Zhang, J. (2022), "Topology optimization of continuum structures under geometric uncertainty using a new extended finite element method", Eng. Optimiz., 54, 1692-1708. https://doi.org/10.1080/0305215X.2021.1957860. 
  55. Rostami, S.A.L., Kolahdooz, A., Chung, H., Shi, M. and Zhang, J. (2023), "Robust topology optimization of continuum structures with smooth boundaries using moving morphable components", Struct. Multidiscipl. Optimiz., 66, 121. https://doi.org/10.1007/s00158-023-03580-2. 
  56. Rostami, S.A.L., Kolahdooz, A. and Zhang, J. (2021), "Robust topology optimization under material and loading uncertainties using an evolutionary structural extended finite element method", Eng. Anal. Bound. Element., 133, 61-70. https://doi.org/10.1016/j.enganabound.2021.08.023. 
  57. Sethian, J.A. and Wiegmann, A. (2000), "Structural boundary design via level set and immersed interface methods", J. Comput. Phys., 163(2), 489-528. https://doi.org/10.1006/jcph.2000.6581. 
  58. Sha, W., Xiao, M., Gao, L. and Zhang, Y. (2021). "A new level set based multi-material topology optimization method using alternating active-phase algorithm", Comput. Methods Appl. Mech. Engrg, 377, 113674. https://doi.org/10.1016/j.cma.2021.113674. 
  59. Shakour, E. and Amir, O. (2021), "Topology optimization with precise evolving boundaries based on IGA and untrimming techniques", Comput. Methods Appl. Mech. Engrg., 374, 13564. https://doi.org/10.1016/j.cma.2020.113564. 
  60. Shobeiri, V. (2015), "The topology optimization design for cracked structures", Eng. Anal. Bound. Element., 58, 26-38. https://doi.org/10.1016/j.enganabound.2015.03.002. 
  61. Shu, L., Wang, M.Y., Fang, Z.D., Ma, Z.D. and Wei, P. (2011), "Level set based structural topology optimization for minimizing frequency response", J. Sound Vib. 330, 5820-5834. https://doi.org/10.1016/j.jsv.2011.07.026. 
  62. Sigmund, O. (2001), "A 99 line topology optimization code written in Matlab", Struct. Multidiscipl. Optimiz., 21, 120-127. https://doi.org/10.1007/s001580050176. 
  63. 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. 
  64. Sohouli, A., Kefal, A., Abdelhamid, A., Yildiz, M. and Suleman, A. (2020), "Continuous densitybased topology optimization of cracked structures using peridynamics", Struct. Multidiscipl. Optimiz., 62, 2375-2389. https://doi.org/10.1007/s00158-020-02608-1. 
  65. 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. 
  66. Wang, M.Y. and Wang, X.M. (2004b), "PDE-driven level sets, shape sensitivity and curvature flow for structural topology optimization", Comput. Modeling Eng. Sci., 6(4), 373-396. https://doi.org/10.3970/cmes.2004.006.373. 
  67. Wang, M.Y,.and Wang, X.M. (2004d), "Color level sets: A multi-phase method for structural topology optimization with multiple materials", Comput. Meth. Appl. Mech. Engrg., 193, 469-496. https://doi.org/10.1016/j.cma.2003.10.008. 
  68. Wang, M.Y. and Zhou, S.W. (2004c), "Synthesis of shape and topology of multi-material structures with a phase-field method", J. Comput.-Aided Mater. Des., 11, 117-138. https://doi.org/10.1007/s10820-005-3169-y. 
  69. Wang, M.Y, Wang, X.M. and Guo, D.M. (2004a), "Structural shape and topology optimization in a level-set-based framework of region representation", Struct. Multidiscipl. Optimiz., 27, 1-19. https://doi.org/10.1007/s00158-003-0363-y. 
  70. Wang, M.Y., Wang, X. and Guo D. (2003), "A level set method for structural topology optimization", Comput. Meth. Appl. Mech. Engrg., 192, 227-246. https://doi.org/10.1016/S0045-7825(02)00559-5. 
  71. Wang, S.Y. and Wang, M.Y. (2006a), "Radial basis functions and level set method for structural topology optimization", Int J. Numer. Meth. Engng., 65, 2060-2090. https://doi.org/10.1002/nme.1536. 
  72. Wang, S.Y. and Wang, M.Y. (2006b), "Structural shape and topology optimization using an implicit free boundary parametrization method", Comput. Modeling Eng. Sci., 13(2), 119-147. https://doi.org/10.3970/cmes.2006.013.119. 
  73. Wei, P., Li, Z., Li, Z.P. and Wang, M.Y. (2018), "An 88-line MATLAB code for the parameterized level set method-based topology optimization using radial basis functions", Struct. Multidiscipl. Optimiz., 58, 831-849. https://doi.org/10.1007/s00158-018-1904-8. 
  74. 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. 
  75. Wei, P. and Wang, M.Y. (2009), "Piecewise constant level set method for structural topology optimization", Int J Numer Meth Engng, 78, 379-402. https://doi.org/10.1002/nme.2478. 
  76. Xia, Q., Shi, T.L., Liu, S.Y. and Wang, M.Y. (2012), "A level set solution to the stress-based structural shape and topology optimization", Comput. Struct., 90-91, 55-64. https://doi.org/10.1016/j.compstruc.2011.10.009. 
  77. Xia, Q., Shi, T.L. and Wang, M.Y. (2011), "A level set based shape and topology optimization method for maximizing the simple or repeated first eigenvalue of structure vibration", Struct. Multidiscipl. Optimiz., 43, 473-485.  https://doi.org/10.1007/s00158-010-0595-6
  78. Xia, Q., Wang, M.Y. and Shi, T.L. (2014), "A level set method for shape and topology optimization of both structure and support of continuum structures", Comput. Meth. Appl. Mech. Engrg., 272, 340-353  https://doi.org/10.1016/j.cma.2014.01.014
  79. Xia, Q., Wang, M.Y. and Shi, T.L. (2015), "Topology optimization with pressure load through a level set method", Comput. Meth. Appl. Mech. Engrg., 283, 177-195. https://doi.org/10.1016/j.cma.2014.01.014. 
  80. 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. 
  81. 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. 
  82. Zhang, W.S., Gou, X., Wang, M.Y. and Wei, P. (2013), "Optimal topology design of continuum structures with stress concentration alleviation via level set method", Int. J. Numer. Meth. Engng., 93, 942-959. https://doi.org/10.1002/nme.4416.