Robust concurrent topology optimization of multiscale structure under single or multiple uncertain load cases. (English) Zbl 07843255
Summary: Concurrent topology optimization of macrostructure and material microstructure has attracted significant interest in recent years. However, most of the existing works assumed deterministic load conditions, thus the obtained design might have poor performance in practice when uncertainties exist. Therefore, it is necessary to take uncertainty into account in structural design. This article proposes an efficient method for robust concurrent topology optimization of multiscale structure under single or multiple load cases. The weighted sum of the mean and standard deviation of the structural compliance is minimized and constraints are imposed to both the volume fractions of macrostructure and microstructure. The effective properties of the microstructure are calculated via the homogenization method. An efficient sensitivity analysis method is proposed based on the superposition principle and orthogonal similarity transformation of real symmetric matrices. To further reduce the computational cost, an efficient decoupled sensitivity analysis method for microscale design variables is proposed. The bidirectional evolutionary structural optimization method is employed to obtain black and white designs for both macrostructure and microstructure. Several two-dimensional and three-dimensional numerical examples are presented to demonstrate the effectiveness of the proposed approach and the effects of load uncertainty on the optimal design of both macrostructure and microstructure.
{© 2019 John Wiley & Sons, Ltd.}
{© 2019 John Wiley & Sons, Ltd.}
MSC:
| 74Pxx | Optimization problems in solid mechanics |
| 74Sxx | Numerical and other methods in solid mechanics |
| 35Qxx | Partial differential equations of mathematical physics and other areas of application |
Keywords:
concurrent topology optimization; multiscale structure; numerical homogenization; sensitivity analysis; uncertain load casesReferences:
| [1] | EschenauerHA, OlhoffN. Topology optimization of continuum structures: a review. Appl Mech Rev. 2001;54(4):331‐390. https://doi.org/10.1115/1.1388075. · doi:10.1115/1.1388075 |
| [2] | BendsøeMP, SigmundO. Topology Optimization Theory, Method and Applications. Germany: Springer; 2003. |
| [3] | ZhuJH, ZhangWH, XiaL. Topology optimization in aircraft and aerospace structures design. Arch Comput Methods Eng. 2016;23(4):595‐622. https://doi.org/10.1007/s11831‐015‐9151‐2. · Zbl 1360.74128 · doi:10.1007/s11831‐015‐9151‐2 |
| [4] | BendsøeMP, KikuchiN. Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng. 1988;71(2):197‐224. https://doi.org/10.1016/0045‐7825(88)90086‐2. · Zbl 0671.73065 · doi:10.1016/0045‐7825(88)90086‐2 |
| [5] | BendsøeMP. Optimal shape design as a material distribution problem. Struct Optim. 1989;1(4):193‐202. https://doi.org/10.1007/BF01650949. · doi:10.1007/BF01650949 |
| [6] | ZhouM, RozvanyG. The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng. 1991;89(1):309‐336. https://doi.org/10.1016/0045‐7825(91)90046‐9. · doi:10.1016/0045‐7825(91)90046‐9 |
| [7] | XieY, StevenG. A simple evolutionary procedure for structural optimization. Comput Struct. 1993;49(5):885‐896. https://doi.org/10.1016/0045‐7949(93)90035‐C. · doi:10.1016/0045‐7949(93)90035‐C |
| [8] | HuangX, XieM. Evolutionary Topology Optimization of Continuum Structures: Methods and Applications. Chichester, West Sussex: John Wiley & Sons; 2010. · Zbl 1279.90001 |
| [9] | WangMY, WangX, GuoD. A level set method for structural topology optimization. Comput Methods Appl Mech Eng. 2003;192(1):227‐246. https://doi.org/10.1016/S0045‐7825(02)00559‐5. · Zbl 1083.74573 · doi:10.1016/S0045‐7825(02)00559‐5 |
| [10] | AllaireG, JouveF, ToaderAM. Structural optimization using sensitivity analysis and a level‐set method. J Comput Phys. 2004;194(1):363‐393. https://doi.org/10.1016/j.jcp.2003.09.032. · Zbl 1136.74368 · doi:10.1016/j.jcp.2003.09.032 |
| [11] | SigmundO, MauteK. Topology optimization approaches. Struct Multidiscip Optim. 2013;48(6):1031‐1055. https://doi.org/10.1007/s00158‐013‐0978‐6. · doi:10.1007/s00158‐013‐0978‐6 |
| [12] | DeatonJD, GrandhiRV. A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim. 2014;49(1):1‐38. https://doi.org/10.1007/s00158‐013‐0956‐z. · doi:10.1007/s00158‐013‐0956‐z |
| [13] | EdgarJ, TintS. Additive manufacturing technologies: 3D printing, rapid prototyping, and direct digital manufacturing, 2nd Edition. Johnson Matthey Technol Rev. 2015;59(3):193‐198. https://doi.org/10.1595/205651315X688406. · doi:10.1595/205651315X688406 |
| [14] | ThompsonMK, MoroniG, VanekerT, et al. Design for additive manufacturing: trends, opportunities, considerations, and constraints. CIRP Ann. 2016;65(2):737‐760. https://doi.org/10.1016/j.cirp.2016.05.004. · doi:10.1016/j.cirp.2016.05.004 |
| [15] | NgoTD, KashaniA, ImbalzanoG, NguyenKT, HuiD. Additive manufacturing (3D printing): a review of materials, methods, applications and challenges. Compos Part B. 2018;143:172‐196. https://doi.org/10.1016/j.compositesb.2018.02.012. · doi:10.1016/j.compositesb.2018.02.012 |
| [16] | RodriguesH, GuedesJ, BendsoeM. Hierarchical optimization of material and structure. Struct Multidiscip Optim. 2002;24(1):1‐10. https://doi.org/10.1007/s00158‐002‐0209‐z. · doi:10.1007/s00158‐002‐0209‐z |
| [17] | CoelhoPR, GuedesJM, RodriguesHC. A hierarchical model for concurrent material and topology optimisation of three‐dimensional structures. Struct Multidiscip Optim. 2008;35(2):107‐115. https://doi.org/10.1007/s00158‐007‐0141‐3. · doi:10.1007/s00158‐007‐0141‐3 |
| [18] | CoelhoPG, CardosoJB, FernandesPR, RodriguesHC. Parallel computing techniques applied to the simultaneous design of structure and material. Adv Eng Softw. 2011;42(5):219‐227. https://doi.org/10.1016/j.advengsoft.2010.10.003. · Zbl 1308.74126 · doi:10.1016/j.advengsoft.2010.10.003 |
| [19] | LiuL, YanJ, ChengG. Optimum structure with homogeneous optimum truss‐like material. Comput Struct. 2008;86(13):1417‐1425. https://doi.org/10.1016/j.compstruc.2007.04.030. · doi:10.1016/j.compstruc.2007.04.030 |
| [20] | NiuB, YanJ, ChengG. Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Struct Multidiscip Optim. 2008;39(2):115. https://doi.org/10.1007/s00158‐008‐0334‐4. · doi:10.1007/s00158‐008‐0334‐4 |
| [21] | YanJ, GuoX, ChengG. Multi‐scale concurrent material and structural design under mechanical and thermal loads. Comput Mech. 2016;57(3):437‐446. https://doi.org/10.1007/s00466‐015‐1255‐x. · Zbl 1382.74103 · doi:10.1007/s00466‐015‐1255‐x |
| [22] | XuB, HuangX, ZhouS, XieY. Concurrent topological design of composite thermoelastic macrostructure and microstructure with multi‐phase material for maximum stiffness. Compos Struct. 2016;150:84‐102. https://doi.org/10.1016/j.compstruct.2016.04.038. · doi:10.1016/j.compstruct.2016.04.038 |
| [23] | DuJ, YangR. Vibro‐acoustic design of plate using bi‐material microstructural topology optimization. J Mech Sci Technol. 2015;29(4):1413‐1419. https://doi.org/10.1007/s12206‐015‐0312‐x. · doi:10.1007/s12206‐015‐0312‐x |
| [24] | ZuoZH, HuangX, RongJH, XieYM. Multi‐scale design of composite materials and structures for maximum natural frequencies. Mater Des. 2013;51:1023‐1034. https://doi.org/10.1016/j.matdes.2013.05.014. · doi:10.1016/j.matdes.2013.05.014 |
| [25] | ZhaoJ, YoonH, YounBD. An efficient concurrent topology optimization approach for frequency response problems. Comput Methods Appl Mech Eng. 2019;347:700‐734. https://doi.org/10.1016/j.cma.2019.01.004. · Zbl 1440.74332 · doi:10.1016/j.cma.2019.01.004 |
| [26] | VicenteW, ZuoZ, PavanelloR, CalixtoT, PicelliR, XieY. Concurrent topology optimization for minimizing frequency responses of two‐level hierarchical structures. Comput Methods Appl Mech Eng. 2016;301:116‐136. https://doi.org/10.1016/j.cma.2015.12.012. · Zbl 1425.74380 · doi:10.1016/j.cma.2015.12.012 |
| [27] | XuB, JiangJS, XieYM. Concurrent design of composite macrostructure and multi‐phase material microstructure for minimum dynamic compliance. Compos Struct. 2015;128:221‐233. https://doi.org/10.1016/j.compstruct.2015.03.057. · doi:10.1016/j.compstruct.2015.03.057 |
| [28] | XuB, XieYM. Concurrent design of composite macrostructure and cellular microstructure under random excitations. Compos Struct. 2015;123:65‐77. https://doi.org/10.1016/j.compstruct.2014.10.037. · doi:10.1016/j.compstruct.2014.10.037 |
| [29] | YanX, HuangX, SunG, XieYM. Two‐scale optimal design of structures with thermal insulation materials. Compos Struct. 2015;120:358‐365. https://doi.org/10.1016/j.compstruct.2014.10.013. · doi:10.1016/j.compstruct.2014.10.013 |
| [30] | FerreiraRT, RodriguesHC, GuedesJM, HernandesJA. Hierarchical optimization of laminated fiber reinforced composites. Compos Struct. 2014;107:246‐259. https://doi.org/10.1016/j.compstruct.2013.07.051. · doi:10.1016/j.compstruct.2013.07.051 |
| [31] | WangY, WangMY, ChenF. Structure‐material integrated design by level sets. Struct Multidiscip Optim. 2016;54(5):1145‐1156. https://doi.org/10.1007/s00158‐016‐1430‐5. · doi:10.1007/s00158‐016‐1430‐5 |
| [32] | LiH, LuoZ, GaoL, QinQ. Topology optimization for concurrent design of structures with multi‐patch microstructures by level sets. Comput Methods Appl Mech Eng. 2018;331:536‐561. https://doi.org/10.1016/j.cma.2017.11.033. · Zbl 1439.74284 · doi:10.1016/j.cma.2017.11.033 |
| [33] | SivapuramR, DunningPD, KimHA. Simultaneous material and structural optimization by multiscale topology optimization. Struct Multidiscip Optim. 2016;54(5):1267‐1281. https://doi.org/10.1007/s00158‐016‐1519‐x. · doi:10.1007/s00158‐016‐1519‐x |
| [34] | XiaL, BreitkopfP. Recent advances on topology optimization of multiscale nonlinear structures. Arch Comput Methods Eng. 2017;24(2):227‐249. https://doi.org/10.1007/s11831‐016‐9170‐7. · Zbl 1364.74086 · doi:10.1007/s11831‐016‐9170‐7 |
| [35] | XiaL, XiaQ, HuangX, XieYM. Bi‐directional evolutionary structural optimization on advanced structures and materials: a comprehensive review. Arch Comput Methods Eng. 2018;25(2):437‐478. https://doi.org/10.1007/s11831‐016‐9203‐2. · Zbl 1392.74074 · doi:10.1007/s11831‐016‐9203‐2 |
| [36] | GuoX, ZhaoX, ZhangW, YanJ, SunG. Multi‐scale robust design and optimization considering load uncertainties. Comput Methods Appl Mech Eng. 2015;283:994‐1009. https://doi.org/10.1016/j.cma.2014.10.014. · Zbl 1423.74747 · doi:10.1016/j.cma.2014.10.014 |
| [37] | DunningPD, KimHA, MullineuxG. Introducing loading uncertainty in topology optimization. AIAA J. 2011;49(4):760‐768. https://doi.org/10.2514/1.J050670. · doi:10.2514/1.J050670 |
| [38] | ToriiAJ. Robust compliance‐based topology optimization: a discussion on physical consistency. Comput Methods Appl Mech Eng. 2019;352:110‐136. https://doi.org/10.1016/j.cma.2019.04.022. · Zbl 1441.74170 · doi:10.1016/j.cma.2019.04.022 |
| [39] | XuY, GaoY, WuC, FangJ, LiQ. Robust topology optimization for multiple fiber‐reinforced plastic (FRP) composites under loading uncertainties. Struct Multidiscip Optim. 2019;59(3):695‐711. https://doi.org/10.1007/s00158‐018‐2175‐0. · doi:10.1007/s00158‐018‐2175‐0 |
| [40] | DengJ, ChenW. Concurrent topology optimization of multiscale structures with multiple porous materials under random field loading uncertainty. Struct Multidiscip Optim. 2017;56(1):1‐19. https://doi.org/10.1007/s00158‐017‐1689‐1. · doi:10.1007/s00158‐017‐1689‐1 |
| [41] | ZhengJ, LuoZ, LiH, JiangC. Robust topology optimization for cellular composites with hybrid uncertainties. Int J Numer Methods Eng. 2018;115(6):695‐713. https://doi.org/10.1002/nme.5821. · Zbl 07878399 · doi:10.1002/nme.5821 |
| [42] | ZhengJ, LuoZ, JiangC, GaoJ. Robust topology optimization for concurrent design of dynamic structures under hybrid uncertainties. Mech Syst Signal Process. 2019;120:540‐559. https://doi.org/10.1016/j.ymssp.2018.10.026. · doi:10.1016/j.ymssp.2018.10.026 |
| [43] | HuangX, XieY. Convergent and mesh‐independent solutions for the bi‐directional evolutionary structural optimization method. Finite Elem Anal Des. 2007;43(14):1039‐1049. https://doi.org/10.1016/j.finel.2007.06.006. · doi:10.1016/j.finel.2007.06.006 |
| [44] | HuangX, XieYM. A further review of ESO type methods for topology optimization. Struct Multidiscip Optim. 2010;41(5):671‐683. https://doi.org/10.1007/s00158‐010‐0487‐9. · doi:10.1007/s00158‐010‐0487‐9 |
| [45] | ZhaoJ, WangC. Robust structural topology optimization under random field loading uncertainty. Struct Multidiscip Optim. 2014;50(3):517‐522. https://doi.org/10.1007/s00158‐014‐1119‐6. · doi:10.1007/s00158‐014‐1119‐6 |
| [46] | ZhaoJ, WangC. Robust topology optimization under loading uncertainty based on linear elastic theory and orthogonal diagonalization of symmetric matrices. Comput Methods Appl Mech Eng. 2014;273:204‐218. https://doi.org/10.1016/j.cma.2014.01.018. · Zbl 1296.74085 · doi:10.1016/j.cma.2014.01.018 |
| [47] | ChenS, ChenW, LeeS. Level set based robust shape and topology optimization under random field uncertainties. Struct Multidiscip Optim. 2010;41(4):507‐524. https://doi.org/10.1007/s00158‐009‐0449‐2. · Zbl 1274.74323 · doi:10.1007/s00158‐009‐0449‐2 |
| [48] | ZhaoJ, YounBD, YoonH, FuZ, WangC. On the orthogonal similarity transformation (OST)‐based sensitivity analysis method for robust topology optimization under loading uncertainty: a mathematical proof and its extension. Struct Multidiscip Optim. 2018;58(1):51‐60. https://doi.org/10.1007/s00158‐018‐2013‐4. · doi:10.1007/s00158‐018‐2013‐4 |
| [49] | XiaL, BreitkopfP. Design of materials using topology optimization and energy‐based homogenization approach in Matlab. Struct Multidiscip Optim. 2015;52(6):1229‐1241. https://doi.org/10.1007/s00158‐015‐1294‐0. · doi:10.1007/s00158‐015‐1294‐0 |
| [50] | CarrascoM, IvorraB, RamosAM. Stochastic topology design optimization for continuous elastic materials. Comput Methods Appl Mech Eng. 2015;289:131‐154. https://doi.org/10.1016/j.cma.2015.02.003. · Zbl 1423.74727 · doi:10.1016/j.cma.2015.02.003 |
| [51] | ZhaoJ, YoonH, YounBD. An efficient decoupled sensitivity analysis method for multiscale concurrent topology optimization problems. Struct Multidiscip Optim. 2018;58(2):445‐457. https://doi.org/10.1007/s00158‐018‐2044‐x. · doi:10.1007/s00158‐018‐2044‐x |
| [52] | SigmundO, PeterssonJ. Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh‐dependencies and local minima. Structural optimization. 1998;16(1):68‐75. https://doi.org/10.1007/BF01214002. · doi:10.1007/BF01214002 |
| [53] | MarlerR, AroraJ. Survey of multi‐objective optimization methods for engineering. Struct Multidiscip Optim. 2004;26(6):369‐395. https://doi.org/10.1007/s00158‐003‐0368‐6. · Zbl 1243.90199 · doi:10.1007/s00158‐003‐0368‐6 |
| [54] | KimIY, deWeckO. Adaptive weighted sum method for bi‐objective optimization: Pareto front generation. Struct Multidiscip Optim. 2005;29:149‐158. |
| [55] | MarlerRT, AroraJS. The weighted sum method for multi‐objective optimization: new insights. Struct Multidiscip Optim. 2010;41(6):853‐862. https://doi.org/10.1007/s00158‐009‐0460‐7. · Zbl 1274.90359 · doi:10.1007/s00158‐009‐0460‐7 |
| [56] | CaoMJ, MaHT, WeiP. A novel robust design method for improving stability of optimized structures. Acta Mech Sinica. 2015;31(1):104‐111. https://doi.org/10.1007/s10409‐015‐0007‐7. · doi:10.1007/s10409‐015‐0007‐7 |
| [57] | YanX, HuangX, ZhaY, XieY. Concurrent topology optimization of structures and their composite microstructures. Comput Struct. 2014;133:103‐110. https://doi.org/10.1016/j.compstruc.2013.12.001. · doi:10.1016/j.compstruc.2013.12.001 |
| [58] | DunningPD, KimHA. Robust topology optimization: minimization of expected and variance of compliance. AIAA J. 2013;51(11):2656‐2664. https://doi.org/10.2514/1.J052183. · doi:10.2514/1.J052183 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
