Abstract
This paper reviews a novel method for integrated topology and packaging optimization in lightweight systems design. Presented as the component-existence model, this approach introduces a new class of packaging design variable coupled with traditional material pseudo-densities in a standard topology workflow. This method adopts a modified interpolation scheme compatible with standard finite element mesh discretizations, and through development of so-called design fields and variable mapping functions, applies gradient-based optimization and analytical sensitivity expressions for driving design updates. The resulting framework can consider multiple overlapping components, enhances effective component mobility in the design domain, and features numerical efficiency comparable to standard topology-only problems. This method is demonstrated in six numerical case studies and compared against solutions from full factorial analysis, with resulting integrated designs approaching or exceeding benchmark configurations within ±5%.
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References
Ambrozkiewicz O, Kriegesmann B (2021) Simultaneous topology and fastener layout optimization of assemblies considering joint failure. Int. J. Numer. Methods Eng. 122(1):294–319. https://doi.org/10.1002/nme.6538
Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in MATLAB using 88 lines of code. Struct. Multidiscip. Optim. 43(1):1–16. https://doi.org/10.1007/s00158-010-0594-7
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71(2):197–224. https://doi.org/10.1016/0045-7825(88)90086-2
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch. Appl. Mech. 69(9):635–654. https://doi.org/10.1007/s004190050248
Bendsøe MP, Sigmund O (2004) Topology optimization: theory, methods and applications. Springer-Verlag
Carrick C, Kim IY (2019) Packaging optimization using the dynamic vector fields method. Int. J. Numer. Methods Eng. 120(7):860–879. https://doi.org/10.1002/nme.6161
Chickermane H, Gea HC (1997) Design of multi-component structural systems for optimal layout topology and joint locations. Eng. Comput. 13(4):235–243. https://doi.org/10.1007/BF01200050
Crispo L, Bohrer R, Roper SWK, Kim IY (2021) Spatial gradient interface detection in topology optimization for an unstructured mesh. Struct. Multidiscip. Optim. 63(1):515–522. https://doi.org/10.1007/s00158-020-02688-z
Dandurand B, Guarneri P, Fadel GM, Wiecek MM (2014) Bilevel multiobjective packaging optimization for automotive design. Struct. Multidiscip. Optim. 50(4):663–682. https://doi.org/10.1007/s00158-014-1120-0
Dong H, Fadel GM, Blouin VY (2005) Packing optimization by enhanced rubber band analogy:681–689. https://doi.org/10.1115/DETC2005-85502
Dowsland KA, Dowsland WB (1995) Solution approaches to irregular nesting problems. Eur. J. Oper. Res. 84(3):506–521. https://doi.org/10.1016/0377-2217(95)00019-M
Drira A, Pierreval H, Hajri-Gabouj S (2007) Facility layout problems: a survey. Annu. Rev. Control. 31(2):255–267. https://doi.org/10.1016/j.arcontrol.2007.04.001
Florea V, Pamwar M, Sangha B, Kim IY (2020) Simultaneous single-loop multimaterial and multijoint topology optimization. Int. J. Numer. Methods Eng. 121(7):1558–1594. https://doi.org/10.1002/nme.6279
Gao H-H, Zhu J-H, Zhang W-H, Zhou Y (2015) An improved adaptive constraint aggregation for integrated layout and topology optimization. Comput. Methods Appl. Mech. Eng. 289:387–408. https://doi.org/10.1016/j.cma.2015.02.022
Gu X, Yang K, Wu M, Zhang Y, Zhu J, Zhang W (2021) Integrated optimization design of smart morphing wing for accurate shape control. Chin. J. Aeronaut. 34(1):135–147. https://doi.org/10.1016/j.cja.2020.08.048
Guest JK, Prévost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int. J. Numer. Methods Eng. 61(2):238–254. https://doi.org/10.1002/nme.1064
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). https://doi.org/10.1115/1.4027609
Jaranson J, Ahmed M (2015) MMLV: lightweight interior systems design, SAE 2015 World Congress & Exhibition. SAE International, pp 1–8. https://doi.org/10.4271/2015-01-1236
Kang Z, Wang Y (2013) Integrated topology optimization with embedded movable holes based on combined description by material density and level sets. Comput. Methods Appl. Mech. Eng. 255:1–13. https://doi.org/10.1016/j.cma.2012.11.006
Kim IY, Kwak BM (2002) Design space optimization using a numerical design continuation method. Int. J. Numer. Methods Eng. 53(8):1979–2002. https://doi.org/10.1002/nme.369
Kim SY, Kim IY, Mechefske CK (2012) A new efficient convergence criterion for reducing computational expense in topology optimization: reducible design variable method. Int. J. Numer. Methods Eng. 90(6):752–783. https://doi.org/10.1002/nme.3343
Krog L, Tucker A, Kemp M, Boyd R (2004) Topology optimisation of aircraft wing box ribs, 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. Am Ins Aeronaut Astronautics. https://doi.org/10.2514/6.2004-4481
Li C, Kim IY (2014) Topology, size and shape optimization of an automotive cross car beam. Proceed Inst Mech Eng, Part D: J Automobile Eng 229(10):1361–1378. https://doi.org/10.1177/0954407014561279
Li D, Kim IY (2018) Multi-material topology optimization for practical lightweight design. Struct. Multidiscip. Optim. 58(3):1081–1094. https://doi.org/10.1007/s00158-018-1953-z
Li C, Kim IY, Jeswiet J (2014) Conceptual and detailed design of an automotive engine cradle by using topology, shape, and size optimization. Struct. Multidiscip. Optim. 51(2):547–564. https://doi.org/10.1007/s00158-014-1151-6
Liu P, Kang Z (2018) Integrated topology optimization of multi-component structures considering connecting interface behavior. Comput. Methods Appl. Mech. Eng. 341:851–887. https://doi.org/10.1016/j.cma.2018.07.001
Liu K, Tovar A (2014) An efficient 3D topology optimization code written in Matlab. Struct. Multidiscip. Optim. 50(6):1175–1196. https://doi.org/10.1007/s00158-014-1107-x
Liu T, Zhu J-H, Zhang W-H, Zhao H, Kong J, Gao T (2019) Integrated layout and topology optimization design of multi-component systems under harmonic base acceleration excitations. Struct. Multidiscip. Optim. 59(4):1053–1073. https://doi.org/10.1007/s00158-019-02200-2
Ma Z-D, Kikuchi N, Pierre C, Raju B (2005) Multidomain topology optimization for structural and material designs. J. Appl. Mech. 73(4):565–573. https://doi.org/10.1115/1.2164511
Miao Y, Blouin VY, Fadel GM (2003) Multi-objective configuration optimization with vehicle dynamics applied to midsize truck design:319–327. https://doi.org/10.1115/DETC2003/DAC-48735
Qian Z, Ananthasuresh GK (2004) Optimal embedding of rigid objects in the topology design of structures. Mech Based Design Struct Mach 32(2):165–193. https://doi.org/10.1081/SME-120030555
Qiao Z, Weihong Z, Jihong Z, Tong G (2012) Layout optimization of multi-component structures under static loads and random excitations. Eng. Struct. 43:120–128. https://doi.org/10.1016/j.engstruct.2012.05.013
Roper S, Vierhout G, Li D, Sangha B, Pamwar M, Kim IY (2019) Multi-material topology optimization and multi-material selection in design. SAE Int. https://doi.org/10.4271/2019-01-0843
Rozvany GIN (2008) A critical review of established methods of structural topology optimization. Struct. Multidiscip. Optim. 37(3):217–237. https://doi.org/10.1007/s00158-007-0217-0
Shah V, Pamwar M, Sangha B, Kim IY (2021) Material interface control in multi-material topology optimization using pseudo-cost domain method. Int. J. Numer. Methods Eng. 122(2):455–482. https://doi.org/10.1002/nme.6545
Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct. Multidiscip. Optim. 21(2):120–127. https://doi.org/10.1007/s001580050176
Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct. Multidiscip. Optim. 33(4-5):401–424. https://doi.org/10.1007/s00158-006-0087-x
Sigmund O, Maute K (2013) Topology optimization approaches. Struct. Multidiscip. Optim. 48(6):1031–1055. https://doi.org/10.1007/s00158-013-0978-6
Skszek, T. W., Zaluzec, M., Conklin, J., and Wagner, D. MMLV: project overview, SAE 2015 World Congress & Exhibition. SAE International, Detroit, USA 2015, pp. 1-8. https://doi.org/10.4271/2015-01-0407
Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int. J. Numer. Methods Eng. 24(2):359–373. https://doi.org/10.1002/nme.1620240207
Svanberg K (2002) A Class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J. Optim. 12(2):555–573. https://doi.org/10.1137/S1052623499362822
Thomas H, Zhou M, Schramm U (2002) Issues of commercial optimization software development. Struct. Multidiscip. Optim. 23(2):97–110. https://doi.org/10.1007/s00158-002-0170-x
van Dijk NP, Maute K, Langelaar M, van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct. Multidiscip. Optim. 48(3):437–472. https://doi.org/10.1007/s00158-013-0912-y
Vanderplaats GN (2007) Multidiscipline design optimization. Vanderplaats Research & Development, California
Vatanabe SL, Lippi TN, Lima CR (2016) d., Paulino, G. H., and Silva, E. C. N. Topology optimization with manufacturing constraints: a unified projection-based approach. Adv. Eng. Softw. 100:97–112. https://doi.org/10.1016/j.advengsoft.2016.07.002
Wang Y, Luo Z, Zhang X, Kang Z (2014) Topological design of compliant smart structures with embedded movable actuators. Smart Mater. Struct. 23(4):045024. https://doi.org/10.1088/0964-1726/23/4/045024
Wu, S., Kay, M., King, R., Vila-Parrish, A. R., Fitts, E. P., and Warsing, D. P. (2014) Multi-objective optimization of 3D packing problem in additive manufacturing.
Xia L, Zhu J, Zhang W (2012a) Sensitivity analysis with the modified Heaviside function for the optimal layout design of multi-component systems. Comput. Methods Appl. Mech. Eng. 241-244:142–154. https://doi.org/10.1016/j.cma.2012.06.005
Xia L, Zhu J, Zhang W (2012b) A superelement formulation for the efficient layout design of complex multi-component system. Struct. Multidiscip. Optim. 45(5):643–655. https://doi.org/10.1007/s00158-011-0720-1
Xia L, Zhu J, Zhang W, Breitkopf P (2013) An implicit model for the integrated optimization of component layout and structure topology. Comput. Methods Appl. Mech. Eng. 257:87–102. https://doi.org/10.1016/j.cma.2013.01.008
Yetis FA, Saitou K (2002) Decomposition-based assembly synthesis based on structural considerations. J. Mech. Des. 124(4):593–601. https://doi.org/10.1115/1.1519276
Zhang W, Zhang Q (2009) Finite-circle method for component approximation and packing design optimization. Eng. Optim. 41(10):971–987. https://doi.org/10.1080/03052150902890056
Zhang W, Xia L, Zhu J, Zhang Q (2011) Some recent advances in the integrated layout design of multicomponent systems. J Mech Design 133(10). https://doi.org/10.1115/1.4005083
Zhang J, Zhang WH, Zhu JH, Xia L (2012) Integrated layout design of multi-component systems using XFEM and analytical sensitivity analysis. Comput. Methods Appl. Mech. Eng. 245-246:75–89. https://doi.org/10.1016/j.cma.2012.06.022
Zhang W, Zhong W, Guo X (2015) Explicit layout control in optimal design of structural systems with multiple embedding components. Comput. Methods Appl. Mech. Eng. 290:290–313. https://doi.org/10.1016/j.cma.2015.03.007
Zhang, W., Zhu, J., and Gao, T. Integrated layout and topology optimization, Topology Optimization in Engineering Structure Design. Elsevier, 2016, pp. 159-215.
Zhang Z, Zhang R, Zhu J, Gao T, Chen F, Zhang W (2021) Integrated batteries layout and structural topology optimization for a solar-powered drone. Chin. J. Aeronaut. https://doi.org/10.1016/j.cja.2020.10.020
Zhou M, Fleury R, Shyy Y-K, Thomas H, Brennan J (2002) Progress in topology optimization with manufacturing constraints, 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. Am Inst Aeronaut Astronaut. https://doi.org/10.2514/6.2002-5614
Zhou Y, Zhang W, Zhu J, Xu Z (2016) Feature-driven topology optimization method with signed distance function. Comput. Methods Appl. Mech. Eng. 310:1–32. https://doi.org/10.1016/j.cma.2016.06.027
Zhu JH, Zhang WH (2010) Integrated layout design of supports and structures. Comput. Methods Appl. Mech. Eng. 199(9):557–569. https://doi.org/10.1016/j.cma.2009.10.011
Zhu J, Zhang W, Beckers P, Chen Y, Guo Z (2008) Simultaneous design of components layout and supporting structures using coupled shape and topology optimization technique. Struct. Multidiscip. Optim. 36(1):29–41. https://doi.org/10.1007/s00158-007-0155-x
Zhu J, Zhang W, Beckers P (2009) Integrated layout design of multi-component system. Int. J. Numer. Methods Eng. 78(6):631–651. https://doi.org/10.1002/nme.2499
Zhu JH, Beckers P, Zhang WH (2010) On the multi-component layout design with inertial force. J. Comput. Appl. Math. 234(7):2222–2230. https://doi.org/10.1016/j.cam.2009.08.073
Zhu J, Zhang W, Xia L, Zhang Q, Bassir D (2012) Optimal packing configuration design with finite-circle method. J. Intell. Robot. Syst. 67(3):185–199. https://doi.org/10.1007/s10846-011-9645-6
Zhu J-H, Gao H-H, Zhang W-H, Zhou Y (2015) A Multi-point constraints based integrated layout and topology optimization design of multi-component systems. Struct. Multidiscip. Optim. 51(2):397–407. https://doi.org/10.1007/s00158-014-1134-7
Zhu J-H, Zhang W-H, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Archives of Computational Methods in Engineering 23(4):595–622. https://doi.org/10.1007/s11831-015-9151-2
Zhu J-H, Guo W-J, Zhang W-H, Liu T (2017) Integrated layout and topology optimization design of multi-frame and multi-component fuselage structure systems. Struct. Multidiscip. Optim. 56(1):21–45. https://doi.org/10.1007/s00158-016-1645-5
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This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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Sufficient information for replicating results is provided throughout the manuscript. Of importance are the inclusion of methodology details such as design field descriptions, priority schemes, and mapping functions. Also included are details for the numerical implementation and case studies, including phase-based settings and case study setup. Pseudocode is also provided in the Appendix for further clarification of the iTOPO workflow. Appropriate references are specified for aspects not immediately within the scope of this paper (e.g., standard checkerboard filtering).
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Roper, S.W.K., Kim, I.Y. Integrated topology and packaging optimization using coupled material and component pseudo-densities. Struct Multidisc Optim 64, 3345–3380 (2021). https://doi.org/10.1007/s00158-021-02992-2
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DOI: https://doi.org/10.1007/s00158-021-02992-2