Abstract
The aim of this paper is to study the topology optimization for mechanical systems with hybrid material and geometric uncertainties. The random variations are modeled by a memory-less transformation of random fields which ensures their physical admissibility. The stochastic collocation method combined with the proposed material and geometry uncertainty models provides robust designs by utilizing already developed deterministic solvers. The computational cost is decreased by using of sparse grids and discretization refinement that are proposed and demonstrated as well. The method is utilized in the design of minimum compliance structure. The proposed algorithm provides a computationally cheap alternative to previously introduced stochastic optimization methods based on Monte Carlo sampling by using adaptive sparse grids method.
Similar content being viewed by others
References
Allaire G, Dapogny C (2014) A linearized approach to worst-case design in parametric and geometric shape optimization. Math Models Methods Appl Sci 24:2199–2257
Ben-Tal A, Nemirovski A (1997) Robust truss topology design via semidefinite programming. SIAM J Optim 7:991–1016
Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190:3443–3459
Chen S, Chen W (2011) A new level-set based approach to shape and topology optimization under geometric uncertainty. Struct Multidiscip Optim 44:1–18
Chen SK, Chen W, Lee SH (2010) Level set based robust shape and topology optimization under random field uncertainties. Struct Multidiscip Optim 41:507–524
Cherkaev E, Cherkaev A (2008) Minimax optimization problem of structural design. Comput Struct 86:1426–1435
Dunning PD, Kim HA, Mullineux G (2011) Introducing loading uncertainty in topology optimization. AIAA J 49:760–768
Guest J, Igusa T (2008) Structural optimization under uncertain loads and nodal locations. Comput Methods Appl Mech Eng 198:116–124
Kang Z, Wang YQ (2011) Structural topology optimization based on non-local Shepard interpolation of density field. Comput Methods Appl Mech Engrg 200:3515–3525
Keshavarzzadeha V, Fernandeza F, Tortorelli DA (2017) Topology optimization under uncertainty via non-intrusive polynomial chaos expansion. Comput Methods Appl Mech Eng 318:120–147
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
Lazarov BS, Schevenels M, Sigmund O (2012a) Topology optimization considering material and geometric uncertainties using stochastic collocation methods. Struct Multidiscip Optim 46:597–612
Lazarov BS, Schevenels M, Sigmund O (2012b) Topology optimization with geometric uncertainties by perturbation techniques. Int J Numer Methods Eng 90:1321–1336
Logo J (2007) New type of optimality criteria method in case of probabilistic loading conditions. Mech Based Des Struct Mach 35:147–162
Logo J, Ghaemi M, Rad MM (2009) Optimal topologies in case of probabilistic loading: the influence of load correlation. Mech. Based Des. Struct. Mach 37:327–348
MATLAB. Version 8.5 (R2015a) (2015) Natick, Massachusetts: The MathWorks Inc.
Matsui K, Terada K (2004) Continuous approximation of material distribution for topology optimization. Internat J Numer Methods Engrg 59:1925–1944
Schevenels M, Lazarov BS, Sigmund O (2011) Robust topology optimization accounting for spatially varying manufacturing errors. Comput Methods Appl Mech Eng 200:3613–3627
Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mech Sinica 25:227–239
Tootkaboni M, Asadpoure A, Guest JK (2012) Topology optimization of continuum structures under uncertainty – a polynomial chaos approach. Comput Methods Appl Mech Engrg 201–204:263–275
Witteveen JAS, Bijl H (2006) Modeling arbitrary uncertainties using gramschmidt polynomial chaos. Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, pp 1–17
Zhao J, Wang C (2014) Robust topology optimization of structures under loading uncertainty. AIAA J 52:398–407
Zhao Q, Chen X, Ma ZD, Lin Y (2015) Robust Topology Optimization Based on Stochastic Collocation Methods under Loading Uncertainties. Math Probl Eng, vol. 2015, 1–14
Zhou M, Lazarov BS, Sigmund O (2014) Topology optimization for optical projection lithography with manufacturing uncertainties. Appl Opt 53:2720–2729
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rostami, S.A.L., Ghoddosian, A. Topology optimization of continuum structures under hybrid uncertainties. Struct Multidisc Optim 57, 2399–2409 (2018). https://doi.org/10.1007/s00158-017-1868-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-017-1868-0