Tolerance range selection of topologically optimized structures with the effects of uncertainties of manufacturing process

Abstract

Robust design methods for topology optimization have received significant attention from researchers in recent years. There are various attempts in past to handle the manufacturing uncertainty of the topologically optimized components. In the present work, same issue is dealt by introducing a method of probabilistic distribution of material. Here, help from Karhunen–Loeve expansion of stochastic process, coupled with Monte Carlo method is taken. The proposed method retains the uncertainty characteristic of the specific manufacturing processes such as etching, e-beam lithography, laser micro machining, and milling. Hence, this method offers larger flexibility to the designer. The simulation for manufacturing uncertainty is utilized to determine the optimal tolerance range of the factors for robust and targeted performance, which is almost nonexistent in the literature. The methodology for tolerance range selection is capable to incorporate uncertainty of multiple factors simultaneously. In this method cross array design of experiment approach is used to analyze the effect of tolerance of each factor. The overall process for manufacturing uncertainty and tolerance range selection is illustrated using four benchmark problems. The chosen factors for considered structural problems are force, elasticity, volume fraction, and aspect ratio. Various combinations of tolerance ranges are used to simulate the performance of the optimized structure, which is expressed in terms of robustness and targeted values of compliance, and maximum deflection. Based on the simulated results of signal-to-noise ratio and mean values of performance, the combinations of tolerance range are suggested that gives a high level of robustness or targeted performance accuracy. To indicate the uniqueness of proposed approach, the obtained response for performances is compared with already available response for performance in literature for generalized approach. Current work is advantageous compared to usual robust design, and provides the performance for a specific scenario at each possible combination of tolerance ranges.

Keywords

Topology optimization manufacturing uncertainty geometric imperfections tolerance design cross array design of experiments robust design targeted performance solid isotropic material with penalization

Get full access to this article

View all access options for this article.

References

Sigmund

. On the design of compliant mechanisms using topology optimization. Mech Struct Mach 1997; 25: 493–524.

Kogiso

Ahn

Nishiwaki

. Robust topology optimization for compliant mechanisms considering uncertainty of applied loads. J Adv Mech Des Syst Manuf 2008; 2: 96–107.

Gournay

Allaire

Jouve

. Shape and topology optimization of the robust compliance via the level set method. ESAIM Control Optim Calc Var 2008; 14: 43–70.

Dunning

Kim

Mullineux

. Introducing loading uncertainty in topology optimization. AIAA J 2011; 49: 760–768.

Chen

Lee

. Level set based robust shape and topology optimization under random field uncertainties. Struct Multidiscip Optim 2010; 41: 507–524.

Sigmund

. Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 2007; 33: 401–424.

Sigmund

. Manufacturing tolerant topology optimization. Acta Mech Sin 2009; 25: 227–239.

Schevenels

Lazarov

Sigmund

. Robust topology optimization accounting for spatially varying manufacturing errors. Comput Methods Appl Mech Eng 2011; 200: 3613–3627.

Chen

. A new level-set based approach to shape and topology optimization under geometric uncertainty. Struct Multidiscip Optim 2011; 44: 1–18.

10.

Amir

Sigmund

Lazarov

. Efficient reanalysis techniques for robust topology optimization. Comput Methods Appl Mech Eng 2012; 245–246: 217–231.

11.

Lazarov

Schevenels

Sigmund

. Topology optimization considering material and geometric uncertainties using stochastic collocation methods. Struct Multidiscip Optim 2012; 46: 597–612.

12.

Lazarov

Schevenels

Sigmund

. Topology optimization with geometric uncertainties by perturbation techniques. Int J Numer Meth Eng 2012; 90: 1321–1336.

13.

Jansen

Lombaert

Diehl

. Robust topology optimization accounting for misplacement of material. Struct Multidiscip Optim 2013; 47: 317–333.

14.

Javed

Rout

. Investigation on parametric sensitivity of topologically optimized structures. Proc IMechE, Part C: J Mechanical Engineering Science 2012; 226: 2791–2804.

15.

Javed

Rout

. Tolerance range section of topologically optimized structure using combined array design of experiments approach. Proc IMechE, Part C: J Mechanical Engineering Science. 2013; 227(9): 2013–2038.

16.

David

HVS

. CRC standard curves and surfaces with mathematica, 2nd ed. Boca Raton, FL: Chapman & Hall/CRC Applied Mathematics & Nonlinear Science, 2006.

17.

Roger

Pol

. Stochastic finite elements: A spectral approach, New York: Springer-Verlag, 1991.

18.

Dongbin

. Numerical methods for stochastic computations: A spectral method approach, Princeton, NJ: Princeton University Press, 2010.

19.

Kharmanda

Olhohoff

Mohamed

. Reliability-based topology optimization. Struct Multidiscip Optim 2004; 26: 295–307.

20.

Jung

Cho

. Reliability-based topology optimization of geometrically nonlinear structures with loading and material uncertainties. Finite Elem Anal Des 2004; 43: 311–331.

21.

Maute

Frangopol

. Reliability-based design of MEMS mechanisms by topology optimization. Comput Struct 2003; 81: 813–824.

22.

Bendsøe

. Optimization of structural topology: Shape and material, Berlin: Springer, 1995.

23.

Bendsøe

Sigmund

. Topology optimization: Theory, methods and applications, 2nd ed. Berlin: Springer, 2003.

24.

Sung

. Robust design and analysis for quality engineering, London: Chapman & Hall Publication, 1988.

25.

Montgomery

. Design and analysis of experiments, 5th ed. New Delhi: Wiley, 2007.

26.

Bruns

. A reevaluation of the SIMP method with filtering and an alternative formulation for solid-void topology optimization. Struct Multidiscip Optim 2005; 30: 428–436.

27.

Svanberg

Werme

. A hierarchical neighborhood search method for topology optimization. Struct Multidiscip Optim 2005; 29: 325–340.

28.

Albert

Etman

LFP

. A simple heuristic for gray-scale suppression in optimality criterion-based topology optimization. Struct Multidiscip Optim 2009; 39: 217–225.

29.

Zhou

Rozvany

. The COC method. Part II. Topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 1991; 89: 309–336.