Multi-objective concurrent isogeometric topology optimization of multiscale structures

Abstract

Multiscale structures require excellent multiphysical properties to withstand the loads in various complex engineering fields. In this study, a concurrent isogeometric topology optimization method is proposed to design multiscale structures with high thermal conductivity and low mechanical compliance. First, the mathematical description model of multi-objective topology optimization for multiscale structures is constructed, and a single-objective concurrent isogeometric topology optimization formulation for mechanical and thermal compliance is proposed. Then, by combining the isogeometric analysis method, the material interpolation model and decoupled sensitivity analysis scheme of the objective function are established on macro and micro scales. The solid isotropic material with penalization method is employed to update iteratively the macro and microstructure topologies simultaneously. Finally, the feasibility and advantages of the proposed approach are illustrated by several 2D and 3D numerical examples with different volume fractions, while the effects of volume fraction and different boundary conditions on the final configuration and multi-objective performance of the multiscale structure are explored. Results show that the isogeometric concurrent design of multiscale structures through multi-objective optimization can produce better multi-objective performance compared with a single-scale one.

Graphical abstract

Keywords

isogeometric topology optimization / multiscale structure / multi-objective optimization / thermal conductivity / mechanical compliance

References

[1] ZhengX Y, Lee H, WeisgraberT H, ShusteffM, DeOtteJ, DuossE B, Kuntz J D, BienerM M, GeQ, Jackson J A, KucheyevS O, FangN X, Spadaccini C M. Ultralight, ultrastiff mechanical metamaterials. Science, 2014, 344(6190): 1373–1377

[2] ShaikeeaA J D, CuiH C, O’Masta M, ZhengX Y R, DeshpandeV S. The toughness of mechanical metamaterials. Nature Materials, 2022, 21(3): 297–304

[3] PejmanR, Najafi A R. Multiphysics topology optimization of a multifunctional structural battery composite. Structural and Multidisciplinary Optimization, 2023, 66(3): 46

[4] WangY H, Sha W, XiaoM, QiuC W, GaoL. Deep-learning-enabled intelligent design of thermal metamaterials. Advanced Materials, 2023, 35(33): 2302387

[5] ZhengY F, Fu Z J, WangY J, LuX, QuJ P, ZhangC Z. Hierarchical design of material microstructures with thermal insulation properties. International Journal of Heat and Mass Transfer, 2022, 186: 122514

[6] SuX N, Chen W J, LiuS T. Multi-scale topology optimization for minimizing structural compliance of cellular composites with connectable graded microstructures. Structural and Multidisciplinary Optimization, 2021, 64(4): 2609–2625

[7] Giraldo-LondoñoO, MirabellaL, Dalloro L, PaulinoG H. Multi-material thermomechanical topology optimization with applications to additive manufacturing: design of main composite part and its support structure. Computer Methods in Applied Mechanics and Engineering, 2020, 363: 112812

[8] ChenW J, Zheng Y F, WangY J. Multi-objective topology optimization filled with multiple microstructures. Composite Structures, 2023, 304: 116322

[9] BendsøeM P, KikuchiN. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 1988, 71(2): 197–224

[10] SigmundO. A 99 line topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 2001, 21(2): 120–127

[11] AndreassenE, Clausen A, SchevenelsM, LazarovB S, Sigmund O. Efficient topology optimization in MATLAB using 88 lines of code. Structural and Multidisciplinary Optimization, 2011, 43(1): 1–16

[12] HuangX, Xie Y M. Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Computational Mechanics, 2009, 43(3): 393–401

[13] WangM Y, Wang X M, GuoD M. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003, 192(1–2): 227–246

[14] GuoX, ZhangW S, ZhongW L. Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. Journal of Applied Mechanics, 2014, 81(8): 081009

[15] KiranR, Nguyen-Thanh N, HuangJ Z, ZhouK. Buckling analysis of cracked orthotropic 3D plates and shells via an isogeometric-reproducing kernel particle method. Theoretical and Applied Fracture Mechanics, 2021, 114: 102993

[16] ZhangJ P, Luo T, ZhangD B, YinS H, HeH L, PengJ P. Multi-objective topology optimization of thermal-mechanical coupling anisotropic structures using the isogeometric analysis approach. Applied Mathematical Modelling, 2023, 117: 267–285

[17] HughesT J R, Cottrell J A, BazilevsY. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39–41): 4135–4195

[18] KatoJ, Yachi D, TeradaK, KyoyaT. Topology optimization of micro-structure for composites applying a decoupling multi-scale analysis. Structural and Multidisciplinary Optimization, 2014, 49(4): 595–608

[19] YanX, HuangX, ZhaY, XieY M. Concurrent topology optimization of structures and their composite microstructures. Computers & Structures, 2014, 133: 103–110

[20] ChenW J, Tong L Y, LiuS T. Concurrent topology design of structure and material using a two-scale topology optimization. Computers & Structures, 2017, 178: 119–128

[21] GroenJ P, Thomsen C R, SigmundO. Multi-scale topology optimization for stiffness and de-homogenization using implicit geometry modeling. Structural and Multidisciplinary Optimization, 2021, 63(6): 2919–2934

[22] YuC, WangQ F, XiaZ H, Wang Y J, MeiC, LiuY H. Multiscale topology optimization for graded cellular structures based on level set surface cutting. Structural and Multidisciplinary Optimization, 2022, 65(1): 32

[23] CoelhoP G, Cardoso J B, FernandesP R, RodriguesH C. Parallel computing techniques applied to the simultaneous design of structure and material. Advances in Engineering Software, 2011, 42(5): 219–227

[24] GaoJ, CaoX F, XiaoM, Yang Z Q, ZhouX Q, LiY, GaoL, YanW T, Rabczuk T, MaiY W. Rational designs of mechanical metamaterials: formulations, architectures, tessellations and prospects. Materials Science and Engineering R, 2023, 156: 100755

[25] LiS S, Zhu Y C, GuoX. Optimisation of spatially varying orthotropic porous structures based on conformal mapping. Computer Methods in Applied Mechanics and Engineering, 2022, 391: 114589

[26] LiuH, ZongH M, ShiT L, Xia Q. M-VCUT level set method for optimizing cellular structures. Computer Methods in Applied Mechanics and Engineering, 2020, 367: 113154

[27] GroenJ P, Wu J, SigmundO. Homogenization-based stiffness optimization and projection of 2D coated structures with orthotropic infill. Computer Methods in Applied Mechanics and Engineering, 2019, 349: 722–742

[28] LiuP, KangZ, LuoY J. Two-scale concurrent topology optimization of lattice structures with connectable microstructures. Additive Manufacturing, 2020, 36: 101427

[29] LiQ H, Xu R, WuQ B, LiuS T. Topology optimization design of quasi-periodic cellular structures based on erode–dilate operators. Computer Methods in Applied Mechanics and Engineering, 2021, 377: 113720

[30] Hu Y, LuoY F, LiuS T. Two-scale concurrent topology optimization method of hierarchical structures with self-connected multiple lattice-material domains. Composite Structures, 2021, 272: 114224

[31] AlAli M, Shimoda M. Investigation of concurrent multiscale topology optimization for designing lightweight macrostructure with high thermal conductivity. International Journal of Thermal Sciences, 2022, 179: 107653

[32] HuangX, Xie Y M, JiaB, LiQ, ZhouS W. Evolutionary topology optimization of periodic composites for extremal magnetic permeability and electrical permittivity. Structural and Multidisciplinary Optimization, 2012, 46(3): 385–398

[33] BayatM, Zinovieva O, FerrariF, AyasC, Langelaar M, SpangenbergJ, SalajegheR, Poulios K, MohantyS, SigmundO, HattelJ. Holistic computational design within additive manufacturing through topology optimization combined with multiphysics multi-scale materials and process modelling. Progress in Materials Science, 2023, 138: 101129

[34] AlAli M, Shimoda M. Toward multiphysics multiscale concurrent topology optimization for lightweight structures with high heat conductivity and high stiffness using MATLAB. Structural and Multidisciplinary Optimization, 2022, 65(7): 207

[35] TakezawaA, Yoon G H, JeongS H, KobashiM, Kitamura M. Structural topology optimization with strength and heat conduction constraints. Computer Methods in Applied Mechanics and Engineering, 2014, 276: 341–361

[36] YanX L, Huang X D, SunG Y, XieY M. Two-scale optimal design of structures with thermal insulation materials. Composite Structures, 2015, 120: 358–365

[37] XuB, HuangX, ZhouS W, Xie Y M. Concurrent topological design of composite thermoelastic macrostructure and microstructure with multi-phase material for maximum stiffness. Composite Structures, 2016, 150: 84–102

[38] WangY, Gao J, LuoZ, BrownT, ZhangN. Level-set topology optimization for multimaterial and multifunctional mechanical metamaterials. Engineering Optimization, 2017, 49(1): 22–42

[39] AlAli M, Shimoda M. On concurrent multiscale topology optimization for porous structures under hygro-thermo-elastic multiphysics with considering evaporation. International Journal for Numerical Methods in Engineering, 2023, 124(14): 3219–3249

[40] BaoY H, Wei Z S, JiaZ Y, WangD Z, ZhangX P, KangZ. Mechanical metamaterial design with the customized low-frequency bandgap and negative Poisson’s ratio via topology optimization. Extreme Mechanics Letters, 2024, 67: 102124

[41] GaoJ, LuoZ, XiaoM, Gao L, LiP G. A NURBS-based Multi-Material Interpolation (N-MMI) for isogeometric topology optimization of structures. Applied Mathematical Modelling, 2020, 81: 818–843

[42] KiyonoC Y, Vatanabe S L, SilvaE C N, ReddyJ N. A new multi-p-norm formulation approach for stress-based topology optimization design. Composite Structures, 2016, 156: 10–19

[43] WangY J, Xiao M, XiaZ H, LiP G, GaoL. From Computer-Aided Design (CAD) toward Human-Aided Design (HAD): an isogeometric topology optimization approach. Engineering, 2023, 22: 94–105

[44] SeoY D, Kim H J, YounS K. Isogeometric topology optimization using trimmed spline surfaces. Computer Methods in Applied Mechanics and Engineering, 2010, 199(49–52): 3270–3296

[45] WangY J, Benson D J. Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed elements. Frontiers of Mechanical Engineering, 2016, 11(4): 328–343

[46] GaiY D, Xing J, HuP. Efficient MATLAB implementation of NURBS-based IGA and material design using isogeometric topology optimization. Optimization and Engineering, 2023, 24(3): 1773–1808

[47] XiaL, Breitkopf P. Design of materials using topology optimization and energy-based homogenization approach in Matlab. Structural and Multidisciplinary Optimization, 2015, 52(6): 1229–1241

[48] GaoJ, LuoZ, XiaL, GaoL. Concurrent topology optimization of multiscale composite structures in Matlab. Structural and Multidisciplinary Optimization, 2019, 60(6): 2621–2651

[49] AndreassenE, Andreasen C S. How to determine composite material properties using numerical homogenization. Computational Materials Science, 2014, 83: 488–495

[50] WangY J, Liao Z Y, YeM, ZhangY, LiW H, XiaZ H. An efficient isogeometric topology optimization using multilevel mesh, MGCG and local-update strategy. Advances in Engineering Software, 2020, 139: 102733

[51] SigmundO. Materials with prescribed constitutive parameters: an inverse homogenization problem. International Journal of Solids and Structures, 1994, 31(17): 2313–2329

[52] WangD D, Xuan J C. An improved NURBS-based isogeometric analysis with enhanced treatment of essential boundary conditions. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37–40): 2425–2436

[53] GaoJ, LiH, GaoL, XiaoM. Topological shape optimization of 3D micro-structured materials using energy-based homogenization method. Advances in Engineering Software, 2018, 116: 89–102

[54] KatoJ, Yachi D, KyoyaT, TeradaK. Micro-macro concurrent topology optimization for nonlinear solids with a decoupling multiscale analysis. International Journal for Numerical Methods in Engineering, 2018, 113(8): 1189–1213

[55] ZhaoJ P, Yoon H, YounB D. An efficient decoupled sensitivity analysis method for multiscale concurrent topology optimization problems. Structural and Multidisciplinary Optimization, 2018, 58(2): 445–457

[56] BendsøeM P, SigmundO. Topology Optimization: Theory, Methods, and Applications. 2nd ed. Berlin: Springer, 2003

[57] DenkM, Rother K, PaetzoldK. Multi-objective topology optimization of heat conduction and linear elastostatic using weighted global criteria method. In: Proceedings of the 31st Symposium Design for X (DFX2020). The Design Society, 2020, 91–100

[58] YuC, WangQ F, MeiC, XiaZ H. Multiscale isogeometric topology optimization with unified structural skeleton. Computer Modeling in Engineering & Sciences, 2020, 122(3): 779–803

[59] AminzadehM, Tavakkoli S M. Multiscale topology optimization of structures by using isogeometrical level set approach. Finite Elements in Analysis and Design, 2024, 235: 104167

[60] MarlerR T, Arora J S. The weighted sum method for multi-objective optimization: new insights. Structural and Multidisciplinary Optimization, 2010, 41(6): 853–862

[61] CarpinelliG, Caramia P, MottolaF, ProtoD. Exponential weighted method and a compromise programming method for multi-objective operation of plug-in vehicle aggregators in microgrids. International Journal of Electrical Power & Energy Systems, 2014, 56: 374–384

[62] WuF H, Wang Z H, HanJ X, PeiG B. Research on multiobjective topology optimization of diesel engine cylinder block based on analytic hierarchy process. Mathematical Problems in Engineering, 2019, 2019(1): 6194634

Acknowledgements

This work was supported by the National Key R&D Program of China (Grant No. 2022YFB3302900) and the National Natural Science Foundation of China (Grant No. 52475261).

Conflict of Interest

The authors declare no conflict of interest.

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