Stochastic Sensitivity Analysis for Robust Topology Optimization

Conference paper


Topology optimization under uncertainty poses extreme difficulty to the already challenging topology optimization problem. This paper presents a new computational method for calculating topological sensitivities of statistical moments of high-dimensional complex systems subject to random inputs. The proposed method, capable of evaluating stochastic sensitivities for large-scale, robust topology optimization (RTO) problems, integrates a polynomial dimensional decomposition (PDD) of multivariate stochastic response functions and deterministic topology derivatives. In addition, the statistical moments and their topology sensitivities are both determined concurrently from a single stochastic analysis. When applied in collaboration with the gradient based optimization algorithm, the proposed method affords the ability of solving industrial-scale RTO design problems. Numerical examples indicate that the new method developed provides computationally efficient solutions.


Stochastic sensitivity analysis Polynomial dimensional decomposition Robust design topology optimization Topological derivatives 



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


  1. 1.
    Bendsøe, M.P., Sigmund, O.: Material interpolation schemes in topology optimization. Arch. Appl. Mech. 69(9–10), 635–654 (1999)zbMATHGoogle Scholar
  2. 2.
    Bendsøe, M.P., Sigmund, O.: Topology optimization: Theory, methods and applications (2003)Google Scholar
  3. 3.
    Busbridge, I.W.: Some integrals involving hermite polynomials. J. Lond. Math. Soc. 23, 135–141 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Garreau, S., Guillaume, P., Masmoudi, M.: The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim. 39(6), 1756–1778 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Grigoriu, M.: Statistically equivalent solutions of stochastic mechanics problems. J. Eng. Mech. 117(8), 1906–1918 (1991)CrossRefGoogle Scholar
  6. 6.
    Grigoriu, M.: Stochastic Calculus: Applications in Science and Engineering. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  7. 7.
    Guzina, B.B., Bonnet, M.: Topological derivative for the inverse scattering of elastic waves. Q. J. Mech. Appl. Math. 57, 161–179 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Huang, B., Du, X.: Analytical robustness assessment for robust design. Struct. Multi. Optim. 34(2), 123–137 (2007)CrossRefGoogle Scholar
  9. 9.
    Lee, S.H., Chen, W., Kwak, B.M.: Robust design with arbitrary distributions using gauss-type quadrature formula. Struct. Multi. Optim. 39(3), 227–243 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Liu, P., Luo, Y.J., Kang, Z.: Multi-material topology optimization considering interface behavior via xfem and level set method. Comput. Methods Appl. Mech. Eng. 308, 113–133 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lurie, A.I., Belyaev, A.: Theory of Elasticity. Foundations of Engineering Mechanics. Springer, Heidelberg (2010)Google Scholar
  12. 12.
    Rahman, S.: A polynomial dimensional decomposition for stochastic computing. Int. J. Numer. Methods Eng. 76(13), 2091–2116 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rahman, S.: Extended polynomial dimensional decomposition for arbitrary probability distributions. J. Eng. Mech. ASCE 135(12), 1439–1451 (2009)CrossRefGoogle Scholar
  14. 14.
    Rahman, S.: Statistical moments of polynomial dimensional decomposition. J. Eng. Mech. 136(7), 923–927 (2010)CrossRefGoogle Scholar
  15. 15.
    Rahman, S., Ren, X.C.: Novel computational methods for high-dimensional stochastic sensitivity analysis. Int. J. Numer. Methods Eng. 98(12), 881–916 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ren, X.C., Rahman, S.: Robust design optimization by polynomial dimensional decomposition. Struct. Multi. Optim. 48(1), 127–148 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sigmund, O., Petersson, J.: Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct. Multi. Optim. 16(1), 68–75 (1998)CrossRefGoogle Scholar
  18. 18.
    Sui, Y.K., Yang, D.Q.: A new method for structural topological optimization based on the concept of independent continuous variables and smooth model. Acta Mech. Sinica 14(2), 179–185 (1998)CrossRefGoogle Scholar
  19. 19.
    Wang, H., Kim, N.H.: Robust design using stochastic response surface and sensitivities. In: 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference (2006)Google Scholar
  20. 20.
    Wang, M.Y., Wang, X.M.: Color level sets: a multi-phase method for structural topology optimization with multiple materials. Comput. Methods Appl. Mech. Eng. 193(6), 469–496 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wang, M.Y., Wang, X.M., Guo, D.M.: A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192(1), 227–246 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Xu, H., Rahman, S.: A generalized dimension-reduction method for multidimensional integration in stochastic mechanics. Int. J. Numer. Methods Eng. 61(12), 1992–2019 (2004)CrossRefzbMATHGoogle Scholar
  23. 23.
    Xu, H., Rahman, S.: Decomposition methods for structural reliability analysis. Probab. Eng. Mech. 20(3), 239–250 (2005)CrossRefGoogle Scholar
  24. 24.
    Yamazaki, F., Shinozuka, M., Dasgupta, G.: Neumann expansion for stochastic finite element analysis. J. Eng. Mech. 114(8), 1335–1354 (1988)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringGeorgia Southern UniversityStatesboroUSA
  2. 2.State Key Laboratory of Bridge Engineering Structural DynamicsChina Merchant Chongqing Communications Research and Design Institute Co., Ltd.ChoingqingChina

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