Abstract
Previous studies on topology optimization subject to stress constraints usually considered von Mises or Drucker–Prager criterion. In some engineering applications, e.g., the design of concrete structures, the maximum first principal stress (FPS) must be controlled in order to prevent concrete from cracking under tensile stress. This paper presents an effective approach to dealing with this issue. The approach is integrated with the bi-directional evolutionary structural optimization (BESO) technique. The p-norm function is adopted to relax the local stress constraint into a global one. Numerical examples of compliance minimization problems are used to demonstrate the effectiveness of the proposed algorithm. The results show that the optimized design obtained by the method has slightly higher compliance but significantly lower stress level than the solution without considering the FPS constraint. The present methodology will be useful for designing concrete structures.
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References
Amir O (2017) Stress-constrained continuum topology optimization: a new approach based on elasto-plasticity. Struct Multidiscip Optim 55:1797–1818. https://doi.org/10.1007/s00158-016-1618-8
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202. https://doi.org/10.1007/BF01650949
Bendsøe MP (1995) Optimization of structural topology, shape, and material. Springer, Berlin Heidelberg, Berlin, Heidelberg
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224. https://doi.org/10.1016/0045-7825(88)90086-2
Blachowski B, Tauzowski P, Lógó J (2020) Yield limited optimal topology design of elastoplastic structures. Struct Multidiscip Optim. https://doi.org/10.1007/s00158-019-02447-9
Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50:2143–2158. https://doi.org/10.1002/nme.116
Cai K (2011) A simple approach to find optimal topology of a continuum with tension-only or compression-only material. Struct Multidiscip Optim 43:827–835. https://doi.org/10.1007/s00158-010-0614-7
Cai K, Gao Z, Shi J (2014) Topology optimization of continuum structures with bi-modulus materials. Eng Optim 46:244–260. https://doi.org/10.1080/0305215X.2013.765001
Cai K, Cao J, Shi J et al (2016) Optimal layout of multiple bi-modulus materials. Struct Multidiscip Optim 53:801–811. https://doi.org/10.1007/s00158-015-1365-2
Cheng GD, Guo X (1997) ε-Relaxed approach in structural topology optimization. Struct Optim 13:258–266. https://doi.org/10.1007/BF01197454
Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43:1453–1478. https://doi.org/10.1002/(SICI)1097-0207(19981230)43:8<1453::AID-NME480>3.0.CO;2-2
Duysinx P, Sigmund O (1998) New developments in handling stress constraints in optimal material distribution. In: 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. American Institute of Aeronautics and Astronautics, Reston, Virigina
Duysinx P, Van Miegroet L, Lemaire E et al (2008) Topology and generalized shape optimization: why stress constraints are so important? Int J Simul Multidiscip Des Optim 2:253–258. https://doi.org/10.1051/ijsmdo/2008034
Fan Z, Xia L, Lai W et al (2019) Evolutionary topology optimization of continuum structures with stress constraints. Struct Multidiscip Optim 59:647–658. https://doi.org/10.1007/s00158-018-2090-4
He Y, Cai K, Zhao ZL, Xie YM (2020) Stochastic approaches to generating diverse and competitive structural designs in topology optimization. Finite Elem Anal Des 173:103399. https://doi.org/10.1016/j.finel.2020.103399
Herfelt MA, Poulsen PN, Hoang LC (2019) Strength-based topology optimisation of plastic isotropic von Mises materials. Struct Multidiscip Optim 59:893–906. https://doi.org/10.1007/s00158-018-2108-y
Holmberg E, Torstenfelt B, Klarbring A (2013) Stress constrained topology optimization. Struct Multidiscip Optim 48:33–47. https://doi.org/10.1007/s00158-012-0880-7
Huang X, Xie YM (2007a) Bidirectional evolutionary topology optimization for structures with geometrical and material nonlinearities. AIAA J 45:308–313. https://doi.org/10.2514/1.25046
Huang X, Xie YM (2007b) Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem Anal Des 43:1039–1049. https://doi.org/10.1016/j.finel.2007.06.006
Huang X, Xie YM (2008) Optimal design of periodic structures using evolutionary topology optimization. Struct Multidiscip Optim 36:597–606. https://doi.org/10.1007/s00158-007-0196-1
Huang X, Xie YM (2009) Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Comput Mech 43:393–401. https://doi.org/10.1007/s00466-008-0312-0
Huang X, Xie YM (2010) Evolutionary topology optimization of continuum structures: methods and applications. John Wiley & Sons, Ltd, Chichester
Huang X, Xie YM, Lu G (2007) Topology optimization of energy-absorbing structures. Int J Crashworthiness 12:663–675. https://doi.org/10.1080/13588260701497862
Kirsch U (1990) On singular topologies in optimum structural design. Struct Optim 2:133–142. https://doi.org/10.1007/BF01836562
Le C, Norato J, Bruns T et al (2010) Stress-based topology optimization for continua. Struct Multidiscip Optim 41:605–620. https://doi.org/10.1007/s00158-009-0440-y
Le C, Bruns T, Tortorelli D (2011) A gradient-based, parameter-free approach to shape optimization. Comput Methods Appl Mech Eng 200:985–996. https://doi.org/10.1016/j.cma.2010.10.004
Liang QQ, Xie YM, Steven GP (2000) Optimal topology selection of continuum structures with displacement constraints. Comput Struct 77:635–644. https://doi.org/10.1016/S0045-7949(00)00018-3
Liu B, Guo D, Jiang C et al (2019) Stress optimization of smooth continuum structures based on the distortion strain energy density. Comput Methods Appl Mech Eng 343:276–296. https://doi.org/10.1016/j.cma.2018.08.031
Luo Y, Bao J (2019) A material-field series-expansion method for topology optimization of continuum structures. Comput Struct 225:106122. https://doi.org/10.1016/j.compstruc.2019.106122
Luo Y, Wang MY, Deng Z (2013a) Stress-based topology optimization of concrete structures with prestressing reinforcements. Eng Optim 45:1349–1364. https://doi.org/10.1080/0305215X.2012.734816
Luo Y, Wang MY, Kang Z (2013b) An enhanced aggregation method for topology optimization with local stress constraints. Comput Methods Appl Mech Eng 254:31–41. https://doi.org/10.1016/j.cma.2012.10.019
Picelli R, Townsend S, Brampton C et al (2018) Stress-based shape and topology optimization with the level set method. Comput Methods Appl Mech Eng 329:1–23. https://doi.org/10.1016/j.cma.2017.09.001
Rozvany GIN (2001) On design-dependent constraints and singular topologies. Struct Multidiscip Optim 21:164–172. https://doi.org/10.1007/s001580050181
Schmit LA (1960) Structural design by systematic synthesis. In: Proc. of the second ASCE conference on electronic computation. ASCE, Pittsburgh, pp 105–122
Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16:68–75. https://doi.org/10.1007/BF01214002
Takezawa A, Yoon GH, Jeong SH et al (2014) Structural topology optimization with strength and heat conduction constraints. Comput Methods Appl Mech Eng 276:341–361. https://doi.org/10.1016/j.cma.2014.04.003
Tortorelli DA, Haber RB (1989) First-order design sensitivities for transient conduction problems by an adjoint method. Int J Numer Methods Eng 28:733–752. https://doi.org/10.1002/nme.1620280402
Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246. https://doi.org/10.1016/S0045-7825(02)00559-5
Xia L, Da D, Yvonnet J (2018a) Topology optimization for maximizing the fracture resistance of quasi-brittle composites. Comput Methods Appl Mech Eng 332:234–254. https://doi.org/10.1016/j.cma.2017.12.021
Xia L, Zhang L, Xia Q, Shi T (2018b) Stress-based topology optimization using bi-directional evolutionary structural optimization method. Comput Methods Appl Mech Eng 333:356–370. https://doi.org/10.1016/j.cma.2018.01.035
Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885–896. https://doi.org/10.1016/0045-7949(93)90035-C
Xie YM, Steven GP (1994) A simple approach to structural frequency optimization. Comput Struct 53:1487–1491. https://doi.org/10.1016/0045-7949(94)90414-6
Xie YM, Steven GP (1997) Evolutionary structural optimization. Springer, London
Xie YM, Yang K, He Y, Zhao ZL, Cai K (2019) How to obtain diverse and efficient structural designs through topology optimization. In: Proc. of the IASS Annual Symposium 2019. IASS, Barcelona, Vol. 2019, No. 17, pp 1–8
Xiong Y, Yao S, Zhao ZL, Xie YM (2020) A new approach to eliminating enclosed voids in topology optimization for additive manufacturing. Addit Manuf 32:101006. https://doi.org/10.1016/j.addma.2019.101006
Yang RJ, Chen CJ (1996) Stress-based topology optimization. Struct Optim 12:98–105. https://doi.org/10.1007/BF01196941
Yang K, Zhao ZL, He Y et al (2019) Simple and effective strategies for achieving diverse and competitive structural designs. Extrem Mech Lett 30:100481. https://doi.org/10.1016/j.eml.2019.100481
Zhao ZL, Zhou S, Feng XQ, Xie YM (2018) On the internal architecture of emergent plants. J Mech Phys Solids 119:224–239. https://doi.org/10.1016/j.jmps.2018.06.014
Zhao ZL, Zhou S, Cai K, Xie YM (2020a) A direct approach to controlling the topology in structural optimization. Comput Struct 227:106141. https://doi.org/10.1016/j.compstruc.2019.106141
Zhao ZL, Zhou S, Feng XQ, Xie YM (2020b) Morphological optimization of scorpion telson. J Mech Phys Solids 135:103773. https://doi.org/10.1016/j.jmps.2019.103773
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The authors received financial support from the National Natural Science Foundation of China (51778283 and 51678082) and the Australian Research Council (FL190100014, DE200100887).
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Chen, A., Cai, K., Zhao, ZL. et al. Controlling the maximum first principal stress in topology optimization. Struct Multidisc Optim 63, 327–339 (2021). https://doi.org/10.1007/s00158-020-02701-5
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DOI: https://doi.org/10.1007/s00158-020-02701-5