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The asymptotic concentration approach combined with isogeometric analysis for topology optimization of two-dimensional linear elasticity structures

  • Received: 26 February 2023 Revised: 01 May 2023 Accepted: 06 May 2023 Published: 12 May 2023
  • We propose an asymptotic concentration approach combined with isogeometric analysis (IGA) for the topology optimization of two-dimensional (2D) linear elasticity structures under the commonly-used framework of the solid isotropic materials and penalty (SIMP) model. Based on the SIMP framework, the B-splines are used as basis functions to describe geometric model in structural finite element analysis, which closely combines geometric modeling with structural analysis. Isogeometric analysis is utilized to define the geometric characteristics of the 2D linear elasticity structures, which can greatly improve the calculation accuracy. In addition, to eliminate the gray-scale intervals usually caused by the intermediate density in the SIMP approach, we utilize the asymptotic concentration method to concentrate the intermediate density values on either 0 or 1 gradually. Consequently, the intermediate densities representing gray-scale intervals in topology optimization results are sufficiently eliminated by virtue of the asymptotic concentration method. The effectiveness and applicability of the proposed method are illustrated by several typical examples.

    Citation: Mingtao Cui, Wang Li, Guang Li, Xiaobo Wang. The asymptotic concentration approach combined with isogeometric analysis for topology optimization of two-dimensional linear elasticity structures[J]. Electronic Research Archive, 2023, 31(7): 3848-3878. doi: 10.3934/era.2023196

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  • We propose an asymptotic concentration approach combined with isogeometric analysis (IGA) for the topology optimization of two-dimensional (2D) linear elasticity structures under the commonly-used framework of the solid isotropic materials and penalty (SIMP) model. Based on the SIMP framework, the B-splines are used as basis functions to describe geometric model in structural finite element analysis, which closely combines geometric modeling with structural analysis. Isogeometric analysis is utilized to define the geometric characteristics of the 2D linear elasticity structures, which can greatly improve the calculation accuracy. In addition, to eliminate the gray-scale intervals usually caused by the intermediate density in the SIMP approach, we utilize the asymptotic concentration method to concentrate the intermediate density values on either 0 or 1 gradually. Consequently, the intermediate densities representing gray-scale intervals in topology optimization results are sufficiently eliminated by virtue of the asymptotic concentration method. The effectiveness and applicability of the proposed method are illustrated by several typical examples.



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    [1] M. P. Bendsøe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Eng., 71 (1988), 197–224. https://doi.org/10.1016/0045-7825(88)90086-2 doi: 10.1016/0045-7825(88)90086-2
    [2] M. Zhou, G. I. N. Rozvany, The COC algorithm, Part Ⅱ: topological, geometrical and generalized shape optimization, Comput. Methods Appl. Mech. Eng., 89 (1991), 309–336. https://doi.org/10.1016/0045-7825(91)90046-9 doi: 10.1016/0045-7825(91)90046-9
    [3] H. P. Mlejnek, Some aspects of the genesis of structures, Struct. Optim., 5 (1992), 64–69. https://doi.org/10.1007/BF01744697 doi: 10.1007/BF01744697
    [4] G. I. N. Rozvany, M. P. Bendsøe, U. Kirsch, Layout optimization of structures, Appl. Mech. Rev., 48 (1995), 41–119. https://doi.org/10.1115/1.3005097 doi: 10.1115/1.3005097
    [5] A. Rietz, Sufficiency of a finite exponent in SIMP (power law) method, Struct. Multidiscip. Optim., 21 (2001), 159–163. https://doi.org/10.1007/s001580050180 doi: 10.1007/s001580050180
    [6] M. Cui, P. Li, J. Wang, T. Gao, M. Pan, An improved optimality criterion combined with density filtering method for structural topology optimization, Eng. Optim., 55 (2023), 416–433. https://doi.org/10.1080/0305215X.2021.2010728 doi: 10.1080/0305215X.2021.2010728
    [7] Y. M. Xie, G. P. Steven, A simple evolutionary procedure for structural optimization, Comput. Struct., 49 (1993), 885–896. https://doi.org/10.1016/0045-7949(93)90035-C doi: 10.1016/0045-7949(93)90035-C
    [8] O. M. Querin, G. P. Steven, Y. M. Xie, Evolutionary structural optimisation using an additive algorithm, Finite Elem. Anal. Des., 34 (2000), 291–308. https://doi.org/10.1016/S0168-874X(99)00044-X doi: 10.1016/S0168-874X(99)00044-X
    [9] O. M. Querin, G. P. Steven, Y. M. Xie, Evolutionary structural optimization (ESO) using a bidirectional algorithm, Eng. Comput., 15 (1998), 1031–1048. https://doi.org/10.1108/02644409810244129 doi: 10.1108/02644409810244129
    [10] S. Osher, J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12–49. https://doi.org/10.1016/0021-9991(88)90002-2 doi: 10.1016/0021-9991(88)90002-2
    [11] G. Allaire, F. Jouve, A. M. Toader, A level-set method for shape optimization, C.R. Math., 334 (2002), 1125–1130. https://doi.org/10.1016/S1631-073X(02)02412-3 doi: 10.1016/S1631-073X(02)02412-3
    [12] G. Allaire, F. Jouve, A. M. Toader, Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194 (2004), 363–393. https://doi.org/10.1016/j.jcp.2003.09.032 doi: 10.1016/j.jcp.2003.09.032
    [13] G. Allaire, F. Gournay, F. Jouve, A.M. Toader, Structural optimization using topological and shape sensitivity via a level set method, Control Cybern., 34 (2005), 59–80. Available from: file:///C:/Users/97380/Downloads/Structural_optimization_using_topol.pdf.
    [14] M. Y. Wang, X. Wang, D. Guo, A level set method for structural topology optimization, Comput. Methods Appl. Mech. Eng., 192 (2003), 227–246. https://doi.org/10.1016/S0045-7825(02)00559-5 doi: 10.1016/S0045-7825(02)00559-5
    [15] M. Cui, H. Chen, J. Zhou, A level-set based multi-material topology optimization method using a reaction diffusion equation, Comput.-Aided Des., 73 (2016), 41–52. https://doi.org/10.1016/j.cad.2015.12.002 doi: 10.1016/j.cad.2015.12.002
    [16] S. S. Nanthakumar, T. Lahmer, X. Zhuang, G. Zi, T. Rabczuk, Detection of material interfaces using a regularized level set method in piezoelectric structures, Inverse Probl. Sci. Eng., 24 (2016), 153–176. https://doi.org/10.1080/17415977.2015.1017485 doi: 10.1080/17415977.2015.1017485
    [17] M. Cui, M. Pan, J. Wang, P. Li, A parameterized level set method for structural topology optimization based on reaction diffusion equation and fuzzy PID control algorithm, Electron. Res. Arch., 30 (2022), 2568–2599. https://doi.org/10.3934/era.2022132 doi: 10.3934/era.2022132
    [18] M. Cui, C. Luo, G. Li, M. Pan, The parameterized level set method for structural topology optimization with shape sensitivity constraint factor, Eng. Comput., 37 (2021), 855–872. https://doi.org/10.1007/s00366-019-00860-8 doi: 10.1007/s00366-019-00860-8
    [19] M. Cui, H. Chen, J. Zhou, F. Wang, A meshless method for multi-material topology optimization based on the alternating active-phase algorithm, Eng. Comput., 33 (2017), 871–884. https://doi.org/10.1007/s00366-017-0503-4 doi: 10.1007/s00366-017-0503-4
    [20] B. Bourdin, A. Chambolle, Design-dependent loads in topology optimization, ESAIM. Control. Optim. Calc. Var., 9 (2003), 19–48. https://doi.org/10.1051/cocv:2002070 doi: 10.1051/cocv:2002070
    [21] M. Cui, J. Wang, P. Li, M. Pan, Topology optimization of plates with constrained Layer damping treatments using a modified guide-weight method, J. Vib. Eng. Technol., 10 (2022), 19–36. https://doi.org/10.1007/s42417-021-00361-3 doi: 10.1007/s42417-021-00361-3
    [22] T. Hughes, J. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Eng., 194 (2005), 4135–4195. https://doi.org/10.1016/j.cma.2004.10.008 doi: 10.1016/j.cma.2004.10.008
    [23] J. Cottrell, T. Hughes, Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, Chichester, 2009. https://doi.org/10.1002/9780470749081
    [24] Y. Bazilevs, D. L. B. Veiga, J. Cottrell, T. Hughes, G. Sangalli, Isogeometric analysis: approximation, stability and error estimates for h-refined meshes, Math. Models Methods Appl. Sci., 16 (2006), 1031–1090. https://doi.org/10.1142/S0218202506001455 doi: 10.1142/S0218202506001455
    [25] B. Hassani, M. Khanzadi, S. M. Tavakkoli, An isogeometrical approach to structural topology optimization by optimality criteria, Struct. Multidiscip. Optim., 45 (2012), 223–233. https://doi.org/10.1007/s00158-011-0680-5 doi: 10.1007/s00158-011-0680-5
    [26] A. V. Kumar, A. Parthasarathy, Topology optimization using B-spline finite elements, Struct. Multidiscip. Optim., 44 (2011), 471–481. https://doi.org/10.1007/s00158-011-0650-y doi: 10.1007/s00158-011-0650-y
    [27] X. P. Qian, Topology optimization in B-spline space, Comput. Methods Appl. Mech. Eng., 265 (2013), 15–35. https://doi.org/10.1016/j.cma.2013.06.001 doi: 10.1016/j.cma.2013.06.001
    [28] Q. X. Lieu, J. Lee, Multiresolution topology optimization using isogeometric analysis, Int. J. Numer. Methods Eng., 112 (2017), 2025–2047. https://doi.org/10.1002/nme.5593 doi: 10.1002/nme.5593
    [29] Q. X. Lieu, J. Lee, A multi-resolution approach for multi-material topology optimization based on isogeometric analysis, Comput. Methods Appl. Mech. Eng., 323 (2017), 272–302. https://doi.org/10.1016/j.cma.2017.05.009 doi: 10.1016/j.cma.2017.05.009
    [30] A. H. Taheri, K. Suresh, An isogeometric approach to topology optimization of multi-material and functionally graded structures, Int. J. Numer. Methods Eng., 109 (2017), 668–696. https://doi.org/10.1002/nme.5303 doi: 10.1002/nme.5303
    [31] M. Montemurro, On the structural stiffness maximisation of anisotropic continua under inhomogeneous Neumann-Dirichlet boundary conditions, Compos. Struct., 287 (2022), 115289. https://doi.org/10.1016/j.compstruct.2022.115289 doi: 10.1016/j.compstruct.2022.115289
    [32] M. Montemurro, K. Refai, A. Catapano, Thermal design of graded architected cellular materials through a CAD-compatible topology optimisation method, Compos. Struct., 280 (2022), 114862. https://doi.org/10.1016/j.compstruct.2021.114862 doi: 10.1016/j.compstruct.2021.114862
    [33] G. Costa, M. Montemurro, Eigen-frequencies and harmonic responses in topology optimisation: a CAD-compatible algorithm, Eng. Struct., 214 (2020), 110602. https://doi.org/10.1016/j.engstruct.2020.110602 doi: 10.1016/j.engstruct.2020.110602
    [34] T. Rodriguez, M. Montemurro, P. L. Texier, J. Pailhès, Structural displacement requirement in a topology optimization algorithm based on isogeometric entities, J. Optim. Theory Appl., 184 (2020), 250–276. https://doi.org/10.1007/s10957-019-01622-8 doi: 10.1007/s10957-019-01622-8
    [35] T. Roiné, M. Montemurro, J. Pailhès, Stress-based topology optimisation through non-uniform rational basis spline hyper-surfaces, Mech. Adv. Mater. Struct., 29 (2022), 3387–3407. https://doi.org/10.1080/15376494.2021.1896822 doi: 10.1080/15376494.2021.1896822
    [36] H. Ghasemi, H. S. Park, T. Rabczuk, A level-set based IGA formulation for topology optimization of flexoelectric materials, Comput. Methods Appl. Mech. Eng., 313 (2017), 239–258. https://doi.org/10.1016/j.cma.2016.09.029 doi: 10.1016/j.cma.2016.09.029
    [37] H. Ghasemi, H. S. Park, T. Rabczuk, A multi-material level set-based topology optimization of flexoelectric composites, Comput. Methods Appl. Mech. Eng., 332 (2018), 47–62. https://doi.org/10.1016/j.cma.2017.12.005 doi: 10.1016/j.cma.2017.12.005
    [38] Y. J. Wang, H. Xu, D. Pasini, Multiscale isogeometric topology optimization for lattice materials, Comput. Methods Appl. Mech. Eng., 316 (2017), 568–585. https://doi.org/10.1016/j.cma.2016.08.015 doi: 10.1016/j.cma.2016.08.015
    [39] Y. Gai, X. Zhu, Y. J. Zhang, W. Hou, P. Hu, Explicit isogeometric topology optimization based on moving morphable voids with closed B-spline boundary curves, Struct. Multidiscip. Optim., 61 (2020), 963–982. https://doi.org/10.1007/s00158-019-02398-1 doi: 10.1007/s00158-019-02398-1
    [40] M. P. Bendsøe, O. Sigmund, Topology Optimization, Berlin, Heidelberg: Springer, 2004. https://doi.org/10.1007/978-3-662-05086-6
    [41] O. Sigmund, Morphology-based black and white filters for topology optimization, Struct. Multidiscip. Optim., 33 (2007), 401–424. https://doi.org/10.1007/s00158-006-0087-x doi: 10.1007/s00158-006-0087-x
    [42] M. Cui, X. Yang, Y. Zhang, C. Luo, G. Li, An asymptotically concentrated method for structural topology optimization based on the SIMLF interpolation, Int. J. Numer. Methods Eng., 115 (2018), 1175–1216. https://doi.org/10.1002/nme.5840 doi: 10.1002/nme.5840
    [43] G. Costa, M. Montemurro, J. Pailhès, A 2D topology optimisation algorithm in NURBS framework with geometric constraints, Int. J. Mech. Mater. Des., 14 (2018), 669–696. https://doi.org/10.1007/s10999-017-9396-z doi: 10.1007/s10999-017-9396-z
    [44] H. J. Kim, Y. D. Seo, S. K. Youn, Isogeometric analysis for trimmed CAD surfaces, Comput. Methods Appl. Mech. Eng., 198 (2009), 2982–2995. https://doi.org/10.1016/j.cma.2009.05.004 doi: 10.1016/j.cma.2009.05.004
    [45] G. Costa, M. Montemurro, J. Pailhès, Minimum length scale control in a NURBS-based SIMP method, Comput. Methods Appl. Mech. Eng., 354 (2019), 963–989. https://doi.org/10.1016/j.cma.2019.05.026 doi: 10.1016/j.cma.2019.05.026
    [46] K. Svanberg, The method of moving asymptotes – a new method for structural optimization, Int. J. Numer. Methods Eng., 24 (1987), 359–373. https://doi.org/10.1002/nme.1620240207 doi: 10.1002/nme.1620240207
    [47] K. Svanberg, M. Werme, Topology optimization by sequential integer linear programming, in IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, Springer Netherlands, (2006), 425–436. https://doi.org/10.1007/1-4020-4752-5_42
    [48] M. Werme, Using the sequential linear integer programming method as a post-processor for stress-constrained topology optimization problems, Int. J. Numer. Methods Eng., 76 (2008), 1544–1567. https://doi.org/10.1002/nme.2378 doi: 10.1002/nme.2378
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