Skip to main content
SpringerLink
Log in

A polynomial dimensional decomposition-based method for robust topology optimization

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

Abstract

This paper implements a novel integration of the polynomial dimensional decomposition (PDD), topology derivative, and level-set method for robust topology optimization subject to a large number of random inputs. With this method, the influence of a large number of random inputs can be easily captured in an accurate manner. In addition, the stochastic moments and their sensitivities can be obtained from analytical expressions based on the PDD approximation of response functions and the deterministic topology derivative. Only a single stochastic analysis is required for evaluating the moments and their sensitivities in each iteration. The topology is described by the level-set function and its evolution is driven by solving the reaction-diffusion equation of the level-set function. An augmented Lagrange penalty formulation dovetails the stochastic topology derivatives of objective and constraints into the reaction term in the reaction-diffusion equation, which generates a new topology during the iteration process. The practical examples illustrate that the proposed method can render meaningful optimal designs for structures subject to several or a large number of random inputs.

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.

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
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  • Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393

    Article  MathSciNet  MATH  Google Scholar 

  • Allaire G, de Gournay F, Jouve F, Toader AM (2005) Structural optimization using topological and shape sensitivity via a level set method. Control Cybern 34(1):59–80

    MathSciNet  MATH  Google Scholar 

  • Amstutz S (2011) Connections between topological sensitivity analysis and material interpolation schemes in topology optimization. Struct Multidisc Optim 43(6):755–765

    Article  MathSciNet  MATH  Google Scholar 

  • Amstutz S, Novotny AA, Neto EAD (2012) Topological derivative-based topology optimization of structures subject to drucker-prager stress constraints. Comput Methods Appl Mech Eng 233:123–136

    Article  MathSciNet  MATH  Google Scholar 

  • Bojczuk D, Mroz Z (2009) Topological sensitivity derivative and finite topology modifications: application to optimization of plates in bending. Struct Multidisc Optim 39(1):1–15

    Article  MathSciNet  MATH  Google Scholar 

  • Burger M, Hackl B, Ring W (2004) Incorporating topological derivatives into level set methods. J Comput Phys 194(1):344–362

    Article  MathSciNet  MATH  Google Scholar 

  • Busbridge I (1948) Some integrals involving hermite polynomials. J Lond Math Soc 23:135–141

    Article  MathSciNet  MATH  Google Scholar 

  • Chatterjee T, Chakraborty S, Chowdhury R (2019) A critical review of surrogate assisted robust design optimization. Arch Comput Methods Eng 26(1):245–274

    Article  MathSciNet  Google Scholar 

  • Chen S, Chen W, Lee S (2010) Level set based robust shape and topology optimization under random field uncertainties. Struct Multidisc Optim 41(4):507–524

    Article  MathSciNet  MATH  Google Scholar 

  • Cheng G, Guo X (1997) \(\varepsilon \)-relaxed approach in structural topology optimization. Struct Optim 13(4):258–266

    Article  Google Scholar 

  • Cheng G, Jiang Z (1992) Study on topology optimization with stress constraints. Eng Optim 20(2):129–148

    Article  Google Scholar 

  • Datta G, Bhattacharjya S, Chakraborty S (2020) Efficient reliability-based robust design optimization of structures under extreme wind in dual response surface framework. Struct Multidisc Optim 62(5):2711–2730

    Article  MathSciNet  Google Scholar 

  • Du XP, Chen W (2000) Towards a better understanding of modeling feasibility robustness in engineering design. J Mech Des 122(4):385–394

    Article  Google Scholar 

  • Evans DH (1967) An application of numerical integration techniclues to statistical toleraucing. Technometrics 9(3):441–456

    MathSciNet  Google Scholar 

  • Garreau S, Guillaume P, Masmoudi M (2001) The topological asymptotic for PDE systems: the elasticity case. SIAM J Control Optim 39(6):1756–1778

    Article  MathSciNet  MATH  Google Scholar 

  • Grigoriu M (1991) Statistically equivalent solutions of stochastic mechanics problems. J Eng Mech 117(8):1906–1918

    Google Scholar 

  • Grigoriu M (2002) Stochastic calculus: applications in science and engineering. Springer, Berlin

    Book  MATH  Google Scholar 

  • Hong HP (1998) An efficient point estimate method for probabilistic analysis. Reliab Eng Syst Saf 59(3):261–267

    Article  Google Scholar 

  • Huang B, Du X (2007) Analytical robustness assessment for robust design. Struct Multidisc Optim 34(2):123–137

    Article  Google Scholar 

  • Kleiber M, Hien TD (1992) The stochastic finite element method. Wiley, New York

    MATH  Google Scholar 

  • Lee I, Choi K, Du L, Gorsich D (2008) Dimension reduction method for reliability-based robust design optimization. Comput Struct 86(13–14):1550–1562

    Article  MATH  Google Scholar 

  • Lee S, Chen W, Kwak B (2009) Robust design with arbitrary distributions using gauss-type quadrature formula. Struct Multidisc Optim 39(3):227–243

    Article  MathSciNet  MATH  Google Scholar 

  • Lopes CG, Santos RBd, Novotny AA (2015) Topological derivative-based topology optimization of structures subject to multiple load-cases. Latin Am J Solids Struct 12(5):834–860

    Article  Google Scholar 

  • Mazurek A (2012) Geometrical aspects of optimum truss like structures for three-force problem. Struct Multidisc Optim 45(1):21–32

    Article  MathSciNet  MATH  Google Scholar 

  • Mazurek A, Baker WF, Tort C (2011) Geometrical aspects of optimum truss like structures. Struct Multidisc Optim 43(2):231–242

    Article  Google Scholar 

  • Norato JA, Bendsoe MP, Haber RB, Tortorelli DA (2007) A topological derivative method for topology optimization. Struct Multidisc Optim 33(4–5):375–386

    Article  MathSciNet  MATH  Google Scholar 

  • Osher S, Fedkiw R, Piechor K (2004) Level set methods and dynamic implicit surfaces. Appl Mech Rev 57(3):B15–B15

    Article  Google Scholar 

  • Otomori M, Yamada T, Izui K, Nishiwaki S (2015) Matlab code for a level set-based topology optimization method using a reaction diffusion equation. Struct Multidisc Optim 51(5):1159–1172

    Article  MathSciNet  Google Scholar 

  • Rahman S (2008) A polynomial dimensional decomposition for stochastic computing. Int J Numer Meth Eng 76(13):2091–2116

    Article  MathSciNet  MATH  Google Scholar 

  • Rahman S (2009) Extended polynomial dimensional decomposition for arbitrary probability distributions. J Eng Mech-Asce 135(12):1439–1451

    Article  Google Scholar 

  • Rahman S (2010) Statistical moments of polynomial dimensional decomposition. J Eng Mech 136(7):923–927

    Google Scholar 

  • Rahman S (2014) Approximation errors in truncated dimensional decompositions. Math Comput 83(290):2799–2819

    Article  MathSciNet  MATH  Google Scholar 

  • Rahman S, Rao BN (2001) A perturbation method for stochastic meshless analysis in elastostatics. Int J Numer Meth Eng 50(8):1969–1991

    Article  MATH  Google Scholar 

  • Rahman S, Ren X (2014) Novel computational methods for high-dimensional stochastic sensitivity analysis. Int J Numer Meth Eng 98(12):881–916

    Article  MathSciNet  MATH  Google Scholar 

  • Rahman S, Xu H (2004) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probab Eng Mech 19(4):393–408

    Article  Google Scholar 

  • Ren X (2020) A polynomial dimensional decomposition framework based on topology derivatives for stochastic topology sensitivity analysis of high-dimensional complex systems and a type of benchmark problems. Probab Eng Mech 62:103104

    Article  Google Scholar 

  • Ren X, Rahman S (2013a) Robust design optimization by polynomial dimensional decomposition. Struct Multidisc Optim 48(1):127–148

    Article  MathSciNet  MATH  Google Scholar 

  • Ren X, Rahman S (2013b) Robust design optimization by polynomial dimensional decomposition. Struct Multidisc Optim 48(1):127–148

    Article  MathSciNet  MATH  Google Scholar 

  • Ren X, Yadav V, Rahman S (2015) Reliability-based design optimization by adaptive-sparse polynomial dimensional decomposition. Struct Multidisc Optim pp 1–28

  • Rosenblueth E (1981) Two-point estimates in probabilities. Appl Math Model 5(5):329–335

    Article  MathSciNet  MATH  Google Scholar 

  • Rozvany GI, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Optim 4(3–4):250–252

    Article  Google Scholar 

  • Sato Y, Izui K, Yamada T, Nishiwaki S (2020) Robust topology optimization of optical cloaks under uncertainties in wave number and angle of incident wave. Int J Numer Meth Eng 121(17):3926–3954

    Article  MathSciNet  Google Scholar 

  • Sethian JA (1999) Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, vol 3. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Shumacher A (1995) Topologieoptimierung von bauteilstrukturen unter verwendung von lochpositionierungkriterien. Universitat-Gesamthochschule-Siegen, Siegen, These de doctorat

  • Sokolowski J, Zochowski A (1999a) On the topological derivative in shape optimization. SIAM J Control Optim 37(4):1251–1272

    Article  MathSciNet  MATH  Google Scholar 

  • Sokolowski J, Zochowski A (1999b) Topological derivatives for elliptic problems. Inverse Prob 15(1):123

    Article  MathSciNet  MATH  Google Scholar 

  • Sokolowski J, Zochowski A (2001) Topological derivatives of shape functionals for elasticity systems. Mech Struct Mach 29(3):331–349

    Article  MathSciNet  MATH  Google Scholar 

  • Sokolowski J, Zochowski A (2009) Topological derivative in shape optimization. Encycl Optim pp 3908–3918

  • Sui Y, Yang D (1998) A new method for structural topological optimization based on the concept of independent continuous variables and smooth model. Acta Mech Sin 14(2):179–185

    Article  Google Scholar 

  • Volpi S, Diez M, Gaul NJ, Song H, Iemma U, Choi K, Campana EF, Stern F (2015) Development and validation of a dynamic metamodel based on stochastic radial basis functions and uncertainty quantification. Struct Multidisc Optim 51(2):347–368

    Article  Google Scholar 

  • Wang H, Kim NH (2006) Robust design using stochastic response surface and sensitivities. In: 11th AIAA/ISSMO Multidiscip Anal Optim Conf

  • Wang MY, Wang XM, Guo DM (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246

    Article  MathSciNet  MATH  Google Scholar 

  • Wang X, Geng X, Wang L, Wang R, Shi Q (2018) Motion error based robust topology optimization for compliant mechanisms under material dispersion and uncertain forces. Struct Multidisc Optim 57(6):2161–2175

    Article  MathSciNet  Google Scholar 

  • Xu H, Rahman S (2004) A generalized dimension-reduction method for multidimensional integration in stochastic mechanics. Int J Numer Meth Eng 61(12):1992–2019

    Article  MATH  Google Scholar 

  • Xu H, Rahman S (2005) Decomposition methods for structural reliability analysis. Probab Eng Mech 20(3):239–250

    Article  Google Scholar 

  • Xu Y, Gao Y, Wu C, Fang J, Li Q (2019) Robust topology optimization for multiple fiber-reinforced plastic (frp) composites under loading uncertainties. Struct Multidisc Optim 59(3):695–711

    Article  Google Scholar 

  • Yamada T, Izui K, Nishiwaki S, Takezawa A (2010) A topology optimization method based on the level set method incorporating a fictitious interface energy. Comput Methods Appl Mech Eng 199(45–48):2876–2891

    Article  MathSciNet  MATH  Google Scholar 

  • Yamazaki F, Shinozuka M, Dasgupta G (1988) Neumann expansion for stochastic finite element analysis. J Eng Mech 114(8):1335–1354

    Google Scholar 

  • Yi G, Sui Y (2016) Timp method for topology optimization of plate structures with displacement constraints under multiple loading cases. Struct Multidisc Optim 53(6):1185–1196

    Article  MathSciNet  Google Scholar 

  • Zhao J, Wang C (2014) Robust topology optimization under loading uncertainty based on linear elastic theory and orthogonal diagonalization of symmetric matrices. Comput Methods Appl Mech Eng 273:204–218

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao J, Youn BD, Yoon H, Fu Z, Wang C (2018) On the orthogonal similarity transformation (OST)-based sensitivity analysis method for robust topology optimization under loading uncertainty: a mathematical proof and its extension. Struct Multidisc Optim 58(1):51–60

    Article  MathSciNet  Google Scholar 

  • Zhou M, Rozvany G (1991) The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336

    Article  Google Scholar 

Download references

Acknowledgements

The authors acknowledge financial support from the US National Science Foundation under Grant No. CMMI-1635167 and the startup funding of Georgia Southern University.

Author information

Authors and Affiliations

  1. Mechanical Engineering Department, Georgia Southern University, Statesboro, GA, USA

    Xuchun Ren

  2. School of Civil Engineering, Chongqing Jiaotong University, Chongqing, China

    Xiaodong Zhang

Authors
  1. Xuchun Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaodong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuchun Ren.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Replication of results

The detailed data that support the findings of this study are available from the corresponding author upon reasonable request. The distribution of the associated software/code is planned to be initiated after several coming articles in this project are published.

Additional information

Responsible Editor: Junji Kato

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ren, X., Zhang, X. A polynomial dimensional decomposition-based method for robust topology optimization. Struct Multidisc Optim 64, 3527–3548 (2021). https://doi.org/10.1007/s00158-021-03036-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00158-021-03036-5

Keywords

Search

Navigation