Abstract
The robust topology optimization of structures with truss-like lattice material under unknown but bounded uncertainties is studied in this paper. To consider the uncertainty of structures composed of truss-like lattice material during the production and service, a formulation of robust topology optimization is constructed, which takes into account the unknown but bounded uncertainties of both the magnitude and direction of load and the diameter of truss-like lattice material. Besides, the absolute robustness index and the relative robustness index are established to measure the robustness of the structure. By dividing the intervals of uncertain parameters, a method called the subinterval dimension-wise method is proposed to solve the difficulty of determining the response interval of structures caused by large uncertainty. Finally, two examples are given to illustrate the effectiveness of the proposed method and its applicability in complex structures, especially the structure with large uncertainty.
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References
Andreassen E, Andreasen CS (2014) How to determine composite material properties using numerical homogenization. Comput Mater Sci 83:488–495
Asadpoure A, Tootkaboni M, Guest JK (2011) Robust topology optimization of structures with uncertainties in stiffness – application to truss structures. Comput Struct 89:1131–1141
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654
Ben-Haim Y (1994) A non-probabilistic concept of reliability. Struct Saf 14:227–245
Deshpande VS, Fleck NA, Ashby MF (2001) Effective properties of the octet-truss lattice material. J Mech Phys Solids 49:1747–1769
Doltsinis I, Zhan K (2004) Robust design of structures using optimization methods. Comput Methods Appl Mech Eng 193:2221–2237
Dunning PD, Kim HA (2013) Robust topology optimization: minimization of expected and variance of compliance. AIAA J 51:2656–2664
Dunning PD, Kim HA, Mullineux G (2011) Introducing loading uncertainty in topology optimization. AIAA J 49:760–768
Elishakoff I (1998) Three versions of the finite element method based on concepts of either stochasticity, fuzziness, or anti-optimization. Appl Mech Rev 51:209–218
Guo X, Zhang W, Zhang L (2013) Robust structural topology optimization considering boundary uncertainties. Comput Methods Appl Mech Eng 253:356–368
Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. J Appl Mech 81(8)
Guo X, Zhao X, Zhang W, Yan J, Sun G (2015) Multi-scale robust design and optimization considering load uncertainties. Comput Methods Appl Mech Eng 283:994–1009
Huang X, Zhou SW, Xie YM, Li Q (2013) Topology optimization of microstructures of cellular materials and composites for macrostructures. Comput Mater Sci 67:397–407
Jiang C, Bi RG, Lu GY, Han X (2013) Structural reliability analysis using non-probabilistic convex model. Comput Methods Appl Mech Eng 254:83–98
Kang Z, Luo Y (2009) Non-probabilistic reliability-based topology optimization of geometrically nonlinear structures using convex models. Comput Methods Appl Mech Eng 198:3228–3238
Kharmanda G, Olhoff N, Mohamed A, Lemaire M (2004) Reliability-based topology optimization. Struct Multidiscip Optim 26:295–307
Kogiso N, Ahn W, Nishiwaki S, Izui K, Yoshimura M (2008) Robust topology optimization for compliant mechanisms considering uncertainty of applied loads. J Adv Mech Des Syst Manuf 2:96–107
Liu J, Gea HC (2018) Robust topology optimization under multiple independent unknown-but-bounded loads. Comput Methods Appl Mech Eng 329:464–479
Luo Y, Kang Z, Yue Z (2012) Maximal stiffness design of two-material structures by topology optimization with nonprobabilistic reliability. AIAA J 50:1993–2003
Park SI, Rosen DW, Choi SK, Duty CE (2014) Effective mechanical properties of lattice material fabricated by material extrusion additive manufacturing ☆. Addit Manuf 1-4:12–23
Peng X, Li J, Jiang S, Liu Z (2018) Robust topology optimization of continuum structures with loading uncertainty using a perturbation method. Eng Optim 50:584–598
Qiu Z (2003) Comparison of static response of structures using convex models and interval analysis method. Int J Numer Methods Eng 56:1735–1753
Qiu Z, Elishakoff I (1998) Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis. Comput Methods Appl Mech Eng 152:361–372
Qiu ZP, Wang L (2016) The need for introduction of non-probabilistic interval conceptions into structural analysis and design. Sci China Phys Mech Astron 59:114632
Qiu Z, Liu D, Wang L, Xia H (2019) Scale-span stress-constrained topology optimization for continuum structures integrating truss-like microstructures and solid material. Comput Methods Appl Mech Eng 355:900–925
Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Structural Optimization 4:250–252
Schevenels M, Lazarov BS, Sigmund O (2015) Robust topology optimization accounting for spatially varying manufacturing errors. Comput Methods Appl Mech Eng 200:3613–3627
Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48:1031–1055
Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373
Taguchi G (1993) Robust technology development. Mech Eng-CIME 115:60–63
Tootkaboni M, Asadpoure A, Guest JK (2012) Topology optimization of continuum structures under uncertainty–a polynomial chaos approach. Comput Methods Appl Mech Eng 201:263–275
Wang L, Liu Y (2020) A novel method of distributed dynamic load identification for aircraft structure considering multi-source uncertainties. Struct Multidiscip Optim 61(5):1929–1952
Wang C, Qiu Z (2016) Subinterval perturbation methods for uncertain temperature field prediction with large fuzzy parameters. Int J Therm Sci 100:381–390
Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Springer-Verlag New York, Inc., New York
Wang X, Wang L, Qiu Z (2014) A feasible implementation procedure for interval analysis method from measurement data. Appl Math Model 38:2377–2397
Wang L, Liu D, Yang Y, Wang X, Qiu Z (2017a) A novel method of non-probabilistic reliability-based topology optimization corresponding to continuum structures with unknown but bounded uncertainties. Comput Methods Appl Mech Eng 326:573–595
Wang X, Geng X, Wang L, Wang R, Shi Q (2017b) Motion error based robust topology optimization for compliant mechanisms under material dispersion and uncertain forces. Struct Multidiscip Optim 57(6):2161–2175
Wang L, Xia H, Yang Y, Cai Y, Qiu Z (2018a) A novel approach of reliability-based topology optimization for continuum structures under interval uncertainties. Rapid Prototyp J 25(9):1455–1474
Wang L, Cai Y, Liu D (2018b) Multiscale reliability-based topology optimization methodology for truss-like microstructures with unknown-but-bounded uncertainties. Comput Methods Appl Mech Eng 339:358–388
Wang L, Xiong C, Wang X, Xu M, Li Y (2018c) A dimension-wise method and its improvement for multidisciplinary interval uncertainty analysis. Appl Math Model 59:680–695
Wang X, Ren Q, Chen W, Liu Y, Wang L, Ding X (2019a) Structural design optimization based on the moving baseline strategy. Acta Mech Solida Sin 33(3):307–326
Wang L, Wang X, Li Y, Hu J (2019b) A non-probabilistic time-variant reliable control method for structural vibration suppression problems with interval uncertainties. Mech Syst Signal Process 115:301–322
Wang L, Liu Y, Liu Y (2019c) An inverse method for distributed dynamic load identification of structures with interval uncertainties. Adv Eng Softw 131:77–89
Wang X, Luo Z, Geng X (2020) Experimental verification of robust topology optimization for compliant mechanism. Rapid Prototyp J 26(9):1485–1502
Wu J, Luo Z, Zhang N, Zhang Y (2015) A new uncertain analysis method and its application in vehicle dynamics. Mech Syst Signal Process 50:659–675
Xia B, Yu D (2012) Modified sub-interval perturbation finite element method for 2D acoustic field prediction with large uncertain-but-bounded parameters. J Sound Vib 331:3774–3790
Xia B, Yu D, Liu J (2013) Interval and subinterval perturbation methods for a structural-acoustic system with interval parameters. J Fluids Struct 38:146–163
Xiong C, Wang L, Liu G, Shi Q (2019) An iterative dimension-by-dimension method for structural interval response prediction with multidimensional uncertain variables. Aerosp Sci Technol 86:572–581
Xu M, Qiu Z (2014) A dimension-wise method for the static analysis of structures with interval parameters. Sci China Phys Mech Astron 57:1934–1945
Zhang X, He J, Takezawa A, Kang Z (2018) Robust topology optimization of phononic crystals with random field uncertainty. Int J Numer Methods Eng 115:1154–1173
Zhao J, Wang C (2014a) Robust structural topology optimization under random field loading uncertainty. Struct Multidiscip Optim 50:517–522
Zhao J, Wang C (2014b) Robust topology optimization under loading uncertainty based on linear elastic theory and orthogonal diagonalization of symmetric matrices. Comput Methods Appl Mech Eng 273:204–218
Acknowledgements
This study is funded by the National Nature Science Foundation of China (No.12072007, No.11772026), the Defense Industrial Technology Development Program (No. JCKY2017208B001, No. JCKY2018601B001, No. JCKY2019209C004), and the Aeronautical Science Foundation of China (20182951014).
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Liu, D., Qiu, Z. A subinterval dimension-wise method for robust topology optimization of structures with truss-like lattice material under unknown but bounded uncertainties. Struct Multidisc Optim 64, 1241–1258 (2021). https://doi.org/10.1007/s00158-021-02911-5
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DOI: https://doi.org/10.1007/s00158-021-02911-5