Skip to main content
SpringerLink
Log in

Topology optimization considering material and geometric uncertainties using stochastic collocation methods

  • Research Paper
  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Abstract

The aim of this paper is to introduce the stochastic collocation methods in topology optimization for mechanical systems with material and geometric uncertainties. The random variations are modeled by a memory-less transformation of spatially varying Gaussian 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 discussed in details and solutions to decrease it, like sparse grids and discretization refinement are proposed and demonstrated as well. The method is utilized in the design of compliant mechanisms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Notes

  1. Figures 12, 20 and 21 in Wang et al. (2011) show the normalized length scale as a function of the projection threshold η, In these figures the vertical axis is erroneously labeled as b/2R. The correct label is b/R, where b is the length scale and R is the filter radius.

References

  • Asadpoure A, Tootkaboni M, Guest JK (2011) Robust topology optimization of structures with uncertainties in stiffness—application to truss structures. Comput Struct 89(11–12):1131–1141

    Article  Google Scholar 

  • Ben-Tal A, Nemirovski A (1997) Robust truss topology design via semidefinite programming. SIAM J Optim 7(4):991–1016

    Article  MathSciNet  MATH  Google Scholar 

  • Bendsøe MP, Sigmund O (2004) Topology optimization—theory, methods and applications. Springer, Berlin

    MATH  Google Scholar 

  • Beyer HG, Sendhoff B (2007) Robust optimization—a comprehensive survey. Comput Methods Appl Mech Eng 196(33–34):3190–3218

    Article  MathSciNet  MATH  Google Scholar 

  • Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50:2143–2158

    Article  MathSciNet  MATH  Google Scholar 

  • Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190:3443–3459

    Article  MATH  Google Scholar 

  • Chen S, Chen W (2011) A new level-set based approach to shape and topology optimization under geometric uncertainty. Struct Multidisc Optim 44:1–18

    Article  Google Scholar 

  • Chen S, Chen W, Lee S (2010) Level set based robust shape and topology optimization under random field uncertainties. Structural and Multidisciplinary Optimization 41:507–524

    Article  MathSciNet  Google Scholar 

  • Ditlevsen O (1996) Dimension reduction and discretization in stochastic problems by regression method. In: Casciati F, Roberts B (eds) Mathematical models for structural reliability analysis. CRS Press, Boca Raton

    Google Scholar 

  • Elesin Y, Lazarov B, Jensen J, Sigmund O (2012) Design of robust and efficient photonic switches using topology optimization. Photonics Nanostruct Fundam Appl 10(1):153–165

    Article  Google Scholar 

  • Field Jr R, Grigoriu M (2009) Model selection for a class of stochastic processes or random fields with bounded range. Probab Eng Mech 24(3):331–342

    Article  Google Scholar 

  • Ghanem R, Spanos P (2003) Stochastic finite elements—a spectral approach. Dover, New York

    Google Scholar 

  • Grigoriu M (1998) Simulation of stationary non-Gaussian translation processes. J Eng Mech - ASCE 124(2):121–126

    Article  Google Scholar 

  • Guest J, Igusa T (2008) Structural optimization under uncertain loads and nodal locations. Comput Methods Appl Mech Eng 198(1):116–124

    Article  MathSciNet  MATH  Google Scholar 

  • Guest J, Prevost J, 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

    Article  MathSciNet  MATH  Google Scholar 

  • Karhunen K (1947) Über lineare methoden in der wahrscheinlichkeitsrechnung. Amer Acad Sci, Fennicade, Ser A 37:3–79

    Google Scholar 

  • Kawamoto A, Matsumori T, Yamasaki S, Nomura T, Kondoh T, Nishiwaki S (2011) Heaviside projection based topology optimization by a pde-filtered scalar function. Struct Multidisc Optim 44(1):19–24

    Article  Google Scholar 

  • 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(1):96–107

    Google Scholar 

  • Lazarov BS, Sigmund O (2011) Filters in topology optimizat ion based on Helmholtz type differential equations. Int J Numer Methods Eng 86(6):765–781

    Article  MathSciNet  MATH  Google Scholar 

  • Lazarov BS, Schevenels M, Sigmund O (2011) Robust design of large-displacement compliant mechanisms. Mech Sci 2(2):175–182

    Article  Google Scholar 

  • Lazarov BS, Schevenels M, Sigmund O (2012) Topology optimization with geometric uncertainties by perturbation techniques. Int J Numer Methods Eng (in print). doi:10.1002/nme.3361

    Google Scholar 

  • Li CC, Kiureghian AD (1993) Optimal discretization of random fields. J Eng Mech - ASCE 119:1136–1154

    Article  Google Scholar 

  • Loève M (1955) Probability theory. Van Nostrand, Princeton

    MATH  Google Scholar 

  • Logo J, Ghaemi M, Rad M (2009) Optimal topologies in case of probabilistic loading: the influence of load correlation. Mech Based Des Struct Mach 37(3):327–348

    Article  Google Scholar 

  • Maitre OPL, Knio OM (2010) Spectral methods for uncertainty quantification: with applications to computational fluid dynamics. Springer, Berlin

    Book  MATH  Google Scholar 

  • Morokoff WJ, Caflisch RE (1995) Quasi-monte carlo integration. J Comput Phys 122(2):218–230

    Article  MathSciNet  MATH  Google Scholar 

  • Schevenels M, Lazarov B, Sigmund O (2011) Robust topology optimization accounting for spatially varying manufacturing errors. Comput Methods Appl Mech Eng 200(49–52):3613–3627

    Article  MATH  Google Scholar 

  • Schueller GI, Jensen HA (2008) Computational methods im optimization considering uncertainties—an overview. Comput Methods Appl Mech Eng 198:2–13

    Article  MathSciNet  MATH  Google Scholar 

  • Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25:493–524

    Article  Google Scholar 

  • Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33:401–424

    Article  Google Scholar 

  • Smolyak S (1963) Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl Akad Nauk USSR 4:240–243

    Google Scholar 

  • Sobol IM (1998) On quasi-monte carlo integrations. Math Comput Simul 47(2–5):103–112

    Article  MathSciNet  Google Scholar 

  • Stefanou G (2009) The stochastic finite element method: past, present and future. Comput Methods Appl Mech Eng 198(9–12):1031–1051

    Article  MATH  Google Scholar 

  • Sudret B, Der Kiureghian A (2000) Stochastic finite elements and reliability—a state-of-the-art report report n ucb/semm-2000/08. Tech. rep., University of California, Berkeley

  • Tsompanakis Y, Lagaros ND, Papadrakakis M (eds) (2008) Structural design optimization considering uncertainties, structures & infrastructures series, vol 1. Taylor & Francis, New York

    Google Scholar 

  • Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidisc Optim 43(6):767–784

    Article  Google Scholar 

  • Wiener N (1938) The homogeneous chaos. Am J Math 60(4):897–936

    Article  MathSciNet  Google Scholar 

  • Xiu D (2007) Efficient collocational approach for parametric uncertainty analysis. Commun Comput Phys 2:293–309

    MathSciNet  MATH  Google Scholar 

  • Xiu D (2010) Numerical methods for stochastic computations: a spectral method approach. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Xiu D, Hesthaven JS (2005) High-order collocation methods for differential equations with random inputs. SIAM J Sci Comput 27(3):1118–1139

    Article  MathSciNet  MATH  Google Scholar 

  • Xu S, Cai Y, Cheng G (2010) Volume preserving nonlinear density filter based on heaviside functions. Struct Multidisc Optim 41:1615–1488

    MathSciNet  Google Scholar 

  • Zhang J, Ellingwood B (1994) Orthogonal series expansions of random fields in first-order reliability analysis. J Eng Mech - ASCE 120(12):2660–2677

    Article  Google Scholar 

Download references

Acknowledgements

This work was financially supported by Villum Fonden and by a grant from the Danish Center of Scientific Computing (DCSC). The second author is a member of K.U.Leuven-BOF PFV/10/002 OPTEC-Optimization in Engineering Center.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, Nils Koppels Alle, B. 404, 2800, Lyngby, Denmark

    Boyan S. Lazarov & Ole Sigmund

  2. Department of Architecture, Urbanism and Planning, K.U.Leuven, Kasteelpark Arenberg 1, 3001, Leuven, Belgium

    Mattias Schevenels

Authors
  1. Boyan S. Lazarov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mattias Schevenels
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ole Sigmund
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boyan S. Lazarov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lazarov, B.S., Schevenels, M. & Sigmund, O. Topology optimization considering material and geometric uncertainties using stochastic collocation methods. Struct Multidisc Optim 46, 597–612 (2012). https://doi.org/10.1007/s00158-012-0791-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-012-0791-7

Keywords

Search

Navigation