Abstract
The majority of topology optimization methods for porous infill designs is based on the assumption of deterministic loads. However, in practice, quantities such as positions, weights, and directions of applied loads may change accidentally. Deterministic load-based designs might deliver poor structural performance under loading uncertainties. Such uncertain factors need to be taken into account in topological optimization to seek robust results. This paper presents a novel robust concurrent topology optimization method for the design of uniform/non-uniform porous infills under the accidental change of loads. A combination of moving morphable bars (MMBs) and loading uncertainties is proposed to directly model multiscale structures and seek robust designs. The macro- and microscopic structures can be simultaneously optimized through the minimization of the weighted sum of the expected compliance and standard deviation. The geometries of adaptive geometric components (AGCs) are straightforwardly optimized. The AGCs consist of two classes of geometric components: macroscopic bars describing the overall structure and microscopic bars describing the material microstructures. Automatic mesh-refinement is utilized to enhance computing efficiency. Numerical examples demonstrate that robust porous design can be obtained with only one global volume constraint while the material continuity of neighboring unit cells and the structural porosity can be maintained without additional constraints. The robust designs yield a more robust structural performance along with a smaller standard deviation compared with deterministic porous designs under loading uncertainties.
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References
Allaire G, Jouve F, Toader A-M (2002) A level-set method for shape optimization. Comptes Rendus Math 334:1125–1130. https://doi.org/10.1016/S1631-073X(02)02412-3
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. https://doi.org/10.1016/j.compstruc.2010.11.004
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202. https://doi.org/10.1007/BF01650949
Bendsoe 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
Bourdin B, Chambolle A (2003) Design-dependent loads in topology optimization. ESAIM Control Optim Calc Var 9:19–48. https://doi.org/10.1051/cocv
Cai J, Wang C, Fu Z (2020) Robust concurrent topology optimization of multiscale structure under single or multiple uncertain load cases. Int J Numer Methods Eng 121:1456–1483. https://doi.org/10.1002/nme.6275
Changizi N, Jalalpour M (2017) Robust topology optimization of frame structures under geometric or material properties uncertainties. Struct Multidiscip Optim 56:791–807. https://doi.org/10.1007/s00158-017-1686-4
Chen S, Chen W, Lee S (2010) Level set based robust shape and topology optimization under random field uncertainties. Struct Multidiscip Optim 41:507–524. https://doi.org/10.1007/s00158-009-0449-2
Cheng L, Liu J, To AC (2018) Concurrent lattice infill with feature evolution optimization for additive manufactured heat conduction design. Struct Multidiscip Optim 58:511–535. https://doi.org/10.1007/s00158-018-1905-7
Csébfalvi A (2018) A new compliance-function-shape-oriented robust approach for volume-constrained continuous topology optimization with uncertain loading directions. Period Polytech Civ Eng 62:219–225. https://doi.org/10.3311/PPci.11398
Csébfalvi A, Lógó J (2018) A critical analysis of expected-compliance model in volume-constrained robust topology optimization with normally distributed loading directions, using a minimax-compliance approach alternatively. Adv Eng Softw 120:107–115. https://doi.org/10.1016/j.advengsoft.2018.02.003
da Silva GA, Cardoso EL (2016) Topology optimization of continuum structures subjected to uncertainties in material properties. Int J Numer Methods Eng 106:192–212. https://doi.org/10.1002/nme.5126
da Silva GA, Cardoso EL (2017) Stress-based topology optimization of continuum structures under uncertainties. Comput Methods Appl Mech Eng 313:647–672. https://doi.org/10.1016/j.cma.2016.09.049
da Silva GA, Beck AT, Cardoso EL (2018) Topology optimization of continuum structures with stress constraints and uncertainties in loading. Int J Numer Methods Eng 113:153–178. https://doi.org/10.1002/nme.5607
Deng J, Chen W (2017) Concurrent topology optimization of multiscale structures with multiple porous materials under random field loading uncertainty. Struct Multidiscip Optim 56:1–19. https://doi.org/10.1007/s00158-017-1689-1
Deng J, Yan J, Cheng G (2013) Multi-objective concurrent topology optimization of thermoelastic structures composed of homogeneous porous material. Struct Multidiscip Optim 47:583–597. https://doi.org/10.1007/s00158-012-0849-6
Deng J, Pedersen CBW, Chen W (2019) Connected morphable components-based multiscale topology optimization. Front Mech Eng 14:129–140. https://doi.org/10.1002/nme.1620240207
Duan Z, Jung Y, Yan J, Lee I (2020) Reliability-based multi-scale design optimization of composite frames considering structural compliance and manufacturing constraints. Struct Multidiscip Optim 61:2401–2421. https://doi.org/10.1007/s00158-020-02517-3
Fu J, Li H, Gao L, Xiao M (2019) Design of shell-infill structures by a multiscale level set topology optimization method. Comput Struct 212:162–172. https://doi.org/10.1016/j.compstruc.2018.10.006
Groen JP, Sigmund O (2018) Homogenization-based topology optimization for high-resolution manufacturable microstructures. Int J Numer Methods Eng 113:1148–1163. https://doi.org/10.1002/nme.5575
Groen JP, Wu J, Sigmund O (2019) Homogenization-based stiffness optimization and projection of 2D coated structures with orthotropic infill. Comput Methods Appl Mech Eng 349:722–742. https://doi.org/10.1016/j.cma.2019.02.031
Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. J Appl Mech 81:081009. https://doi.org/10.1115/1.4027609
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. https://doi.org/10.1016/j.cma.2014.10.014
Hoang VN, Jang GW (2017) Topology optimization using moving morphable bars for versatile thickness control. Comput Methods Appl Mech Eng 317:153–173. https://doi.org/10.1016/j.cma.2016.12.004
Hoang VN (2020) An explicit topology optimization method using moving polygonal morphable voids (MPMVs). Sci Tech Dev J 23:536–540. https://doi.org/10.32508/stdj.v23i2.2067
Hoang VN, Nguyen-Xuan H (2020) Extruded-geometric-component-based 3D topology optimization (accepted). Comput Methods Appl Mech Eng 113293. https://doi.org/10.1016/j.cma.2020.113293
Hoang VN, Nguyen NL, Nguyen-Xuan H (2020a) Topology optimization of coated structure using moving morphable sandwich bars. Struct Multidiscip Optim 61:491–506. https://doi.org/10.1007/s00158-019-02370-z
Hoang VN, Nguyen NL, Tran P, Qian M, Nguyen-Xuan H (2020b) Adaptive concurrent topology optimization of cellular composites for additive manufacturing. JOM. 72:2378–2390. https://doi.org/10.1007/s11837-020-04158-9
Jansen M, Lombaert G, Diehl M, Lazarov BS, Sigmund O, Schevenels M (2013) Robust topology optimization accounting for misplacement of material. Struct Multidiscip Optim 47:317–333. https://doi.org/10.1007/s00158-012-0835-z
Jansen M, Lombaert G, Schevenels M (2015) Robust topology optimization of structures with imperfect geometry based on geometric nonlinear analysis. Comput Methods Appl Mech Eng 285:452–467. https://doi.org/10.1016/j.cma.2014.11.028
Kazemi H, Vaziri A, Norato JA (2018) Topology optimization of structures made of discrete geometric components with different materials. J Mech Des 140:111401. https://doi.org/10.1115/1.4040624
Lazarov BS, Schevenels M, Sigmund O (2012a) Topology optimization considering material and geometric uncertainties using stochastic collocation methods. Struct Multidiscip Optim 46:597–612. https://doi.org/10.1007/s00158-012-0791-7
Lazarov BS, Schevenels M, Sigmund O (2012b) Topology optimization with geometric uncertainties by perturbation techniques. Int J Numer Methods Eng 90:1321–1336. https://doi.org/10.1002/nme.3361
Li H, Luo Z, Gao L, Qin Q (2018) Topology optimization for concurrent design of structures with multi-patch microstructures by level sets. Comput Methods Appl Mech Eng 331:536–561. https://doi.org/10.1016/j.cma.2017.11.033
Nguyen H, Hoang V, Jang G (2020) Moving morphable patches for three-dimensional topology optimization with thickness control. Comput Methods Appl Mech Eng 368:113186. https://doi.org/10.1016/j.cma.2020.113186
Norato JA, Bell BK, Tortorelli DA (2015) A geometry projection method for continuum-based topology optimization with discrete elements. Comput Methods Appl Mech Eng 293:306–327. https://doi.org/10.1016/j.cma.2015.05.005
Radman A, Huang X, Xie YM (2013) Topology optimization of functionally graded cellular materials. J Mater Sci 48:1503–1510. https://doi.org/10.1007/s10853-012-6905-1
Richardson JN, Filomeno Coelho R, Adriaenssens S (2015) Robust topology optimization of truss structures with random loading and material properties: a multiobjective perspective. Comput Struct 154:41–47. https://doi.org/10.1016/j.compstruc.2015.03.011
Saxena A (2011) Topology design with negative masks using gradient search. Struct Multidiscip Optim 44:629–649. https://doi.org/10.1007/s00158-011-0649-4
Schevenels M, Lazarov BS, Sigmund O (2011) Robust topology optimization accounting for spatially varying manufacturing errors. Comput Methods Appl Mech Eng 200:3613–3627. https://doi.org/10.1016/j.cma.2011.08.006
Sivapuram R, Dunning PD, Kim HA (2016) Simultaneous material and structural optimization by multiscale topology optimization. Struct Multidiscip Optim 54:1267–1281. https://doi.org/10.1007/s00158-016-1519-x
Sohouli A, Yildiz M, Suleman A (2018) Efficient strategies for reliability-based design optimization of variable stiffness composite structures. Struct Multidiscip Optim 57:689–704. https://doi.org/10.1007/s00158-017-1771-8
Svanberg K (1987) The method of moving asymptotes - a new method for structural optimization. Int J Numer Methods Eng 24:359–373. https://doi.org/10.1002/nme.1620240207
Taguchi G (1986) Introduction to quality engineering: designing quality into products and processes. Qual Res. https://doi.org/10.1002/qre.4680040216
Tootkaboni M, Asadpoure A, Guest JK (2012) Topology optimization of continuum structures under uncertainty - a polynomial chaos approach. Comput Methods Appl Mech Eng 201–204:263–275. https://doi.org/10.1016/j.cma.2011.09.009
Vicente WM, Zuo ZH, Pavanello R, Calixto TKL, Picelli R, Xie YM (2016) Concurrent topology optimization for minimizing frequency responses of two-level hierarchical structures. Comput Methods Appl Mech Eng 301:116–136. https://doi.org/10.1016/j.cma.2015.12.012
Wadbro E, Niu B (2019) Multiscale design for additive manufactured structures with solid coating and periodic infill pattern. Comput Methods Appl Mech Eng 357:112605. https://doi.org/10.1016/j.cma.2019.112605
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
Wang X, Long K, Hoang VN, Hu P (2018) An explicit optimization model for integrated layout design of planar multi-component systems using moving morphable bars. Comput Methods Appl Mech Eng 342:46–70. https://doi.org/10.1016/j.cma.2018.07.032
Wu Y, Li E, He ZC, Lin XY, Jiang HX (2020) Robust concurrent topology optimization of structure and its composite material considering uncertainty with imprecise probability. Comput Methods Appl Mech Eng 364:1–43. https://doi.org/10.1016/j.cma.2020.112927
Xia L, Breitkopf P (2017) Recent advances on topology optimization of multiscale nonlinear structures. Arch Comput Methods Eng 24:227–249. https://doi.org/10.1007/s11831-016-9170-7
Xie YM, Steven GP (1993) A simple approach to structural optimization. Compurers Struct 49:885–896. https://doi.org/10.1016/0045-7949(93)90035-C
Yan J, Guo X, Cheng G (2016) Multi-scale concurrent material and structural design under mechanical and thermal loads. Comput Mech 57:437–446. https://doi.org/10.1007/s00466-015-1255-x
Yan J, Duan Z, Lund E, Wang J (2017) Concurrent multi-scale design optimization of composite frames with manufacturing constraints. Struct Multidiscip Optim 56:519–533. https://doi.org/10.1007/s00158-017-1750-0
Yang H, Huang J (2020) An explicit structural topology optimization method based on the descriptions of areas. Struct Multidiscip Optim 61:1123–1156. https://doi.org/10.1007/s00158-019-02414-4
Zhang W, Chen J, Zhu X, Zhou J, Xue D, Lei X, Guo X (2017) Explicit three dimensional topology optimization via moving morphable void (MMV) approach. Comput Methods Appl Mech Eng 322:590–614. https://doi.org/10.1016/j.cma.2017.05.002
Zhang W, Song J, Zhou J, Du Z, Zhu Y, Sun Z, Guo X (2018) Topology optimization with multiple materials via moving morphable component (MMC) method. Int J Numer Methods Eng 113:1653–1675. https://doi.org/10.1002/nme.5714
Zhang W, Li D, Kang P, Guo X, Youn S (2020) Explicit topology optimization using IGA-based moving morphable void ( MMV ) approach. Comput Methods Appl Mech Eng 360:112685. https://doi.org/10.1016/j.cma.2019.112685
Zhao Q, Chen X, Ma ZD, Lin Y (2015) Robust topology optimization based on stochastic collocation methods under loading uncertainties. Math Probl Eng 2015. https://doi.org/10.1155/2015/580980
Zheng J, Luo Z, Li H, Jiang C (2018) Robust topology optimization for cellular composites with hybrid uncertainties. Int J Numer Methods Eng 115:695–713. https://doi.org/10.1002/nme.5821
Zheng J, Luo Z, Jiang C, Gao J (2019) Robust topology optimization for concurrent design of dynamic structures under hybrid uncertainties. Mech Syst Signal Process 120:540–559. https://doi.org/10.1016/j.ymssp.2018.10.026
Zhou M, Rozvany GIN (1991) The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89:309–336. https://doi.org/10.1016/0045-7825(91)90046-9
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Research is supported by Vingroup Innovation Foundation (VINIF) in project code VINIF.2019.DA04.
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Hoang, VN., Pham, T., Tangaramvong, S. et al. Robust adaptive topology optimization of porous infills under loading uncertainties. Struct Multidisc Optim 63, 2253–2266 (2021). https://doi.org/10.1007/s00158-020-02800-3
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DOI: https://doi.org/10.1007/s00158-020-02800-3