Abstract
In this article, we propose a novel method to obtain a near-optimal frame structure, based on the solution of a homogenization-based topology optimization model. The presented approach exploits the equivalence between Michell’s problem of least-weight trusses and a compliance minimization problem using optimal rank-2 laminates in the low volume fraction limit. In a fully automated procedure, a discrete structure is extracted from the homogenization-based continuum model. This near-optimal structure is post-optimized as a frame, where the bending stiffness is continuously decreased, to allow for a final design that resembles a truss structure. Numerical experiments show excellent behavior of the method, where the final designs are close to analytical optima, and obtained in less than 10 minutes, for various levels of detail, on a standard PC.
Similar content being viewed by others
References
Aage N, Nobel-jørgensen M, Andreasen CS, Sigmund O (2013) Interactive topology optimization on hand-held devices. Struct Multidiscip Optim 47(1):1–6. https://doi.org/10.1007/s00158-012-0827-z
Achtziger W (2007) On simultaneous optimization of truss geometry and topology. Struct Multidiscip Optim 33(4):285–304. https://doi.org/10.1007/s00158-006-0092-0
Bendsøe MP, Haber RB (1993) The michell layout problem as a low volume fraction limit of the perforated plate topology optimization problem: an asymptotic study. Structural optimization 6(4):263–267. https://doi.org/10.1007/BF01743385
Bendsøe MP, Ben-Tal A, Zowe J (1994) Optimization methods for truss geometry and topology design. Structural optimization 7(3):141–159. https://doi.org/10.1007/BF01742459
Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158. https://doi.org/10.1002/nme.116
Bourdin B, Kohn R (2008) Optimization of structural topology in the high-porosity regime. J Mech Phys Solids 56(3):1043–1064. https://doi.org/10.1016/j.jmps.2007.06.002
Bruns T (2006) Zero density lower bounds in topology optimization. Comput Methods Appl Mech Eng 196(1):566–578. https://doi.org/10.1016/j.cma.2006.06.007
Bruns T, Tortorelli D (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459. https://doi.org/10.1016/S0045-7825(00)00278-4
Dobbs MW, Felton LP (1969) Optimization of truss geometry. J Struct Div 95(10):2105–2118
Dorn WS, Gomory RE, Greenberg HJ (1964) Automatic design of optimal structures. Journal de Mecanique 3:25–52
Gao G, yu Liu Z, bin Li Y, feng Qiao Y (2017) A new method to generate the ground structure in truss topology optimization. Eng Optim 49(2):235–251. https://doi.org/10.1080/0305215X.2016.1169050
Gilbert M, Tyas A (2003) Layout optimization of large-scale pin-jointed frames. Eng Comput 20(8):1044–1064. https://doi.org/10.1108/02644400310503017
Graczykowski C, Lewiński T (2010) Michell cantilevers constructed within a half strip. tabulation of selected benchmark results. Struct Multidiscip Optim 42(6):869–877. https://doi.org/10.1007/s00158-010-0525-7
Groen JP, Sigmund O (2017) Homogenization-based topology optimization for high-resolution manufacturable micro-structures. Int J Numer Methods Eng :1–18. https://doi.org/10.1002/nme.5575
He L, Gilbert M (2015) Rationalization of trusses generated via layout optimization. Struct Multidiscip Optim 52(4):677–694. https://doi.org/10.1007/s00158-015-1260-x
Hemp WS (1973) Optimum structures. Clarendon Press, Oxford
Lewiński T, Rozvany GIN (2008) Exact analytical solutions for some popular benchmark problems in topology optimization iii: L-shaped domains. Struct Multidiscip Optim 35(2):165–174. https://doi.org/10.1007/s00158-007-0157-8
Lewiński T, Zhou M, Rozvany G (1994a) Extended exact least-weight truss layouts—part ii: Unsymmetric cantilevers. Int J Mech Sci 36(5):399–419. https://doi.org/10.1016/0020-7403(94)90044-2
Lewiński T, Zhou M, Rozvany G (1994b) Extended exact solutions for least-weight truss layouts—part i: Cantilever with a horizontal axis of symmetry. Int J Mech Sci 36(5):375–398. https://doi.org/10.1016/0020-7403(94)90043-4
Martínez P, Martí P, Querin OM (2007) Growth method for size, topology, and geometry optimization of truss structures. Struct Multidiscip Optim 33(1):13–26. https://doi.org/10.1007/s00158-006-0043-9
Michell A (1904) The limits of economy of material in frame-structures. Phil Mag 8(47):589–597. https://doi.org/10.1080/14786440409463229
Pantz O, Trabelsi K (2008) A post-treatment of the homogenization method for shape optimization. SIAM J Control Optim 47(3):1380–1398. https://doi.org/10.1137/070688900
Pantz O, Trabelsi K (2010) Construction of minimization sequences for shape optimization. In: 15th international conference on methods and models in automation and robotics (MMAR), pp 278–283. https://doi.org/10.1109/MMAR.2010.5587222
Pedersen P (1969) On the minimum mass layout of trusses. In: AGARD conference proceedings no 36, symposium on structural optimization, pp 36–70
Pedersen P (1989) On optimal orientation of orthotropic materials. Structural optimization 1(2):101–106. https://doi.org/10.1007/BF01637666
Pedersen P (1990) Bounds on elastic energy in solids of orthotropic materials. Structural optimization 2(1):55–63. https://doi.org/10.1007/BF01743521
Ramos JrAS, Paulino GH (2016) Filtering structures out of ground structures – a discrete filtering tool for structural design optimization. Struct Multidiscip Optim 54(1):95–116. https://doi.org/10.1007/s00158-015-1390-1
Rozvany GIN (1998) Exact analytical solutions for some popular benchmark problems in topology optimization. Structural optimization 15(1):42–48. https://doi.org/10.1007/BF01197436
Rule W K (1994) Automatic truss design by optimized growth. J Struct Eng 120(10):3063–3070
Sokół T (2011) A 99 line code for discretized michell truss optimization written in mathematica. Struct Multidiscip Optim 43(2):181–190. https://doi.org/10.1007/s00158-010-0557-z
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
Washizawa T, Asai A, Yoshikawa N (2004) A new approach for solving singular systems in topology optimization using krylov subspace methods. Struct Multidiscip Optim 28(5):330–339. https://doi.org/10.1007/s00158-004-0439-3
Zegard T, Paulino GH (2014) Grand - ground structure based topology optimization for arbitrary 2d domains using matlab. Struct Multidiscip Optim 50(5):861–882. https://doi.org/10.1007/s00158-014-1085-z
Zhou K, Li X (2008) Topology optimization for minimum compliance under multiple loads based on continuous distribution of members. Struct Multidiscip Optim 37(1):49–56. https://doi.org/10.1007/s00158-007-0214-3
Zhou K, Li X (2011) Topology optimization of truss-like continua with three families of members model under stress constraints. Struct Multidiscip Optim 43(4):487–493. https://doi.org/10.1007/s00158-010-0584-9
Acknowledgements
The authors acknowledge the support of the Villum Fonden through the Villum investigator project InnoTop. The authors would also like to thank Andreas Bærentzen and Niels Aage for valuable discussions during the preparation of the work. Finally, the authors wish to thank Krister Svanberg for providing the MATLAB MMA code.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Gregoire Allaire
Rights and permissions
About this article
Cite this article
Larsen, S.D., Sigmund, O. & Groen, J.P. Optimal truss and frame design from projected homogenization-based topology optimization. Struct Multidisc Optim 57, 1461–1474 (2018). https://doi.org/10.1007/s00158-018-1948-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-018-1948-9