Abstract
The coupling problem between the structural field and temperature field is widely encountered in engineering applications and holds significant research importance. In the context of thermo-mechanical coupling topology optimization, the thermal stress resulting from the temperature field can cause structural deformation, consequently impacting the structural performance. Therefore, it is crucial to conduct rational optimization designs to ensure favorable mechanical behavior and heat dissipation. This study utilizes thermo-mechanical coupling theory to perform multi-objective topology optimization, aiming to minimize compliance and heat dissipation weakness concurrently, thereby obtaining a more comprehensive design scheme with enhanced overall performance. Initially, a topological optimization model for the coupled thermo-mechanical problem is established. Subsequently, the objective functions of structural compliance and heat dissipation weakness are normalized, and their sensitivities are derived. Next, a multi-load case and multi-objective optimization algorithm based on Floating Projection Topological Optimization (FPTO) is proposed to minimize both structural compliance and heat dissipation weakness. By comparing the topological configuration and objective function values obtained using the Solid Isotropic Material with Penalization (SIMP) method, it is evident that the FPTO method achieves clear and smooth boundaries in the topological configuration, while yielding smaller objective function values. Additionally, under appropriate trade-off factors, the FPTO method achieves a more balanced topological structure and optimizes material distribution without increasing the structural volume, thus enabling lightweight structures, providing novel ideas and methods for engineering applications.
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The data in this study are available from the corresponding author on reasonable request.
References
Bode B, Herrmann K, Reusch J, Plappert S, Ehlers T, Gembarski PC, Lachmayer R (2023) Thermo-elastic topology optimization for high temperatures gradients using load separation. Procedia CIRP 119:576–581. https://doi.org/10.1016/j.procir.2023.03.113
Chen J, Zhao Q, Zhang L et al (2023) Topology optimization of transient thermo-elastic structure considering regional temperature control. Acta Mech Solida Sin 36(2):262–273. https://doi.org/10.1007/s10338-022-00377-6
Denk M, Rother K, Paetzold K (2020) Multi-objective topology optimization of heat conduction and linear elastostatic using weighted global criteria method. In: DS 106: proceedings of the 31st symposium design for X (DFX2020), pp 91–100. https://doi.org/10.35199/dfx2020.10
Denk M, Rother K, Gadzo E, Paetzold K (2022) Multi-objective topology optimization of frame structures using the weighted sum method. In: Proceedings of the Munich symposium on lightweight design 2021: tagungsband zum Münchner leichtbauseminar 2021. Springer, Berlin Heidelberg, pp 83–92. https://doi.org/10.1007/978-3-662-65216-9_8
Eschenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):331–390. https://doi.org/10.1115/1.1388075
Fang L, Wang X, Zhou H (2022) Topology optimization of thermoelastic structures using MMV method. Appl Math Model 103:604–618. https://doi.org/10.1016/j.apm.2021.11.008
Fawaz A, Hua Y, Le Corre S, Fan Y, Luo L (2022) Topology optimization of heat exchangers: a review. Energy 252:124053. https://doi.org/10.1016/j.energy.2022.124053
Francisco P, Faria L, Simões R (2020) Multi-objective and multi-load topology optimization and experimental validation of homogenized coupled fluid flow and heat transfer and structural stiffness. Struct Multidiscip Optim 62:2571–2598. https://doi.org/10.1007/s00158-020-02625-0
Gao T, Zhang W (2010) Topology optimization involving thermo-elastic stress loads. Struct Multidiscip Optim 42:725–738. https://doi.org/10.1007/s00158-010-0527-5
Guanghui SHI, Chengqi GUAN, Dongliang QUAN, Dongtao WU, Lei TANG, Tong GAO (2020) An aerospace bracket designed by thermo-elastic topology optimization and manufactured by additive manufacturing. Chin J Aeronaut 33(4):1252–1259. https://doi.org/10.1016/j.cja.2019.09.006
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 (2020) Smooth topological design of structures using the floating projection. Eng Struct 208:110330. https://doi.org/10.1016/j.engstruct.2020.110330
Huang X (2021) On smooth or 0/1 designs of the fixed-mesh element-based topology optimization. Adv Eng Softw 151:102942
Huang X, Li W (2022) Three-field floating projection topology optimization of continuum structures. Comput Methods Appl Mech Eng 399:115444. https://doi.org/10.1016/j.cma.2022.115444
Jihong ZHU, Han ZHOU, Chuang WANG, Lu ZHOU, Shangqin YUAN, Zhang W (2021) A review of topology optimization for additive manufacturing: status and challenges. Chin J Aeronaut 34(1):91–110. https://doi.org/10.1016/j.cja.2020.09.020
Kambampati S, Gray JS, Kim HA (2020) Level set topology optimization of structures under stress and temperature constraints. Comput Struct 235:106265. https://doi.org/10.1016/j.compstruc.2020.106265
Li Q, Chen W, Liu S, Fan H (2018) Topology optimization design of cast parts based on virtual temperature method. Comput Aided Des 94:28–40. https://doi.org/10.1016/j.cad.2017.08.002
Li D, Zhang X, Guan Y, Zhan J (2010) Topology optimization of thermo-mechanical continuum structure. In: 2010 IEEE/ASME international conference on advanced intelligent mechatronics, IEEE, pp 403–408. https://doi.org/10.1109/AIM.2010.5695845
Liu K, Tovar A (2014) An efficient 3D topology optimization code written in Matlab. Struct Multidiscip Optim 50:1175–1196. https://doi.org/10.1007/s00158-014-1107-x
Long K, Wang X, Gu X (2018) Multi-material topology optimization for the transient heat conduction problem using a sequential quadratic programming algorithm. Eng Optim 50(12):2091–2107. https://doi.org/10.1080/0305215X.2017.1417401
Luo Y, Zhou M, Wang MY, Deng Z (2014) Reliability based topology optimization for continuum structures with local failure constraints. Comput Struct 143:73–84. https://doi.org/10.1016/j.compstruc.2014.07.009
Maute K (2014) Topology optimization of coupled multi-physics problems. Topology optimization in structural and continuum mechanics. Springer, Vienna, pp 421–437
Meng Q, Xu B, Huang C, Duan Z, Han P (2022) Topology optimization of thermo-elastic structures considering stiffness, strength, and temperature constraints over a wide range of temperatures. Int J Numer Meth Eng 123(7):1627–1653. https://doi.org/10.1002/nme.6909
Ooms T, Vantyghem G, Thienpont T, Van Coile R, De Corte W (2023) Compliance-based topology optimization of structural components subjected to thermo-mechanical loading. Struct Multidiscip Optim 66(6):126. https://doi.org/10.1007/s00158-023-03563-3
Pedersen P, Pedersen NL (2010) Strength optimized designs of thermoelastic structures. Struct Multidiscip Optim 42:681–691. https://doi.org/10.1007/s00158-010-0535-5
Vantyghem G, De Corte W, Steeman M, Boel V (2019) Density-based topology optimization for 3D-printable building structures. Struct Multidiscip Optim 60:2391–2403. https://doi.org/10.1007/s00158-019-02330-7
Yan XL, **e L, Chen JW, Hua HY, Huang XD (2020) A density-constrained topology optimization method. Mech Sci Technol Aerosp Eng 35:56–62. https://doi.org/10.13433/j.cnki.1003-8728.20200077
Yan X, Chen J, Hua H, Zhang Y, Huang X (2021) Smooth topological design of structures with minimum length scale and chamfer/round controls. Comput Methods Appl Mech Eng 383:113939. https://doi.org/10.1016/j.cma.2021.113939
Yang Z, Guo F, Weng J, Du F (2022) Cage structural topology optimization considering thermo-mechanical coupling. Adv Mech Eng 14(12):16878132221139968. https://doi.org/10.1177/16878132221139969
Yao QY, Zhao CY, Zhao Y, Wang H, Li W (2021) Topology optimization for heat transfer enhancement in latent heat storage. Int J Therm Sci 159:106578. https://doi.org/10.1016/j.ijthermalsci.2020.10
Yuan B, Ye H, Li J, Wei N, Sui Y (2023) Topology optimization of geometrically nonlinear structures under thermal–mechanical coupling. Acta Mech Solida Sin 36(1):22–33. https://doi.org/10.1007/s10338-022-00342-3
Zhang L, Zhao Q, Chen J (2022) Reliability-based topology optimization of thermo-elastic structures with stress constraint. Mathematics 10(7):1091. https://doi.org/10.3390/math10071091
Zheng Y, Liu B, Chen W, **a Z, Zhang C (2022) Topology optimization of hierarchical structures based on floating projection. Int J Mech Sci 231:107595. https://doi.org/10.1016/j.ijmecsci.2022.107595
Zhu X, Zhao C, Wang X, Zhou Y, Hu P, Ma ZD (2019) Temperature-constrained topology optimization of thermo-mechanical coupled problems. Eng Optim. https://doi.org/10.1080/0305215X.2018.1554065
Acknowledgements
The authors gratefully acknowledge that this work was financially supported by the Natural Science Foundation of Fujian Province, Guided Project of Fujian Province and Research Startup Foundation of Fujian University of Technology.
Funding
This work was financially supported by the Natural Science Foundation of Fujian Province (grant no. 2022J01921), Guided Project of Fujian Province (grant no. 2020H0020) and Research Startup Foundation of Fujian University of Technology (grant no. GY-Z17004).
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Conceptualization, D.F.H. and X.L.Y.; methodology, D.F.H. and X.L.Y. investigation, D.F.H; writing original draft preparation, S.S.Z.; simulation analysis, S.S.Z. and D.F.H; writing review and editing, S.S.Z. and D.F.H. All authors read and approved the final manuscript.
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Huang, D., Zhou, S. & Yan, X. Multi-objective topology optimization design of thermal-mechanical coupling structure based on FPTO method. Optim Eng (2024). https://doi.org/10.1007/s11081-024-09890-8
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DOI: https://doi.org/10.1007/s11081-024-09890-8