Two-Stage Stochastic Optimization Meets Two-Scale Simulation

  • Sergio Conti
  • Benedict GeiheEmail author
  • Martin Rumpf
  • Rüdiger Schultz
Part of the International Series of Numerical Mathematics book series (ISNM, volume 165)


Risk averse stochastic optimization is investigated in the context of elastic shape optimization, allowing for microstructures in the admissible shapes. In particular, a two-stage model for shape optimization under stochastic loading with risk averse cost functionals is combined with a two-scale approach for the simulation of microstructured materials. The microstructure is composed of an elastic material with geometrically simple perforations located on a regular periodic lattice. Different types of microscopic geometries are investigated and compared to each other. In addition they are compared to optimal nested laminates, known to realize the optimal lower bound of compliance cost functionals. We combine this two-scale approach to elastic shapes with a two-stage stochastic programming approach to risk averse shape optimization, dealing with risk neutral and risk averse cost functionals in the presence of stochastic loadings.


Two-stage stochastic programming Risk averse optimization Two-scale elastic shape optimization Microstructure optimization Finite element method Boundary element method 

Mathematics Subject Classification (2010)

90C15 74B05 74P05 74Q05 74S05 74S15 49M29 



The authors would like to thank Martin Lenz for help with the boundary element method applied to microscopic cell problems. This work was supported by the Deutsche Forschungsgemeinschaft through the Schwerpunktprogramm 1253 Optimization with Partial Differential Equations.


  1. 1.
    G. Allaire, Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, vol. 146 (Springer, New York, 2002)Google Scholar
  2. 2.
    G. Allaire, E. Bonnetier, G. Francfort, F. Jouve, Shape optimization by the homogenization method. Numer. Math. 76, 27–68 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    G. Allaire, F. Jouve, A level-set method for vibration and multiple loads structural optimization. Comput. Methods Appl. Mech. Eng. 194(30–33), 3269–3290 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    G. Allaire, F. Jouve, F. de Gournay, Shape and topology optimization of the robust compliance via the level set method. ESAIM Control Optim. Calc. Var. 14, 43–70 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    R. Astley, J. Harrington, K. Stol, Mechanical modelling of wood microstructure, an engineering approach. Ipenz Trans. 24(1/EMCh), 21–29 (1997)Google Scholar
  6. 6.
    P. Atwal, S. Conti, B. Geihe, M. Pach, M. Rumpf, R. Schultz, On shape optimization with stochastic loadings, in Constrained Optimization and Optimal Control for Partial Differential Equations, ed. by G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich, S. Ulbrich. International Series of Numerical Mathematics, vol. 160, ch. 2 (Springer, Basel, 2012), pp. 215–243Google Scholar
  7. 7.
    M. Avellaneda, Optimal bounds and microgeometries for elastic two-phase composites. SIAM J. Appl. Math. 47(6), 1216–1228 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    N.V. Banichuk, P. Neittaanmäki, On structural optimization with incomplete information. Mech. Based Des. Struct. Mach. 35, 75–95 (2007)CrossRefGoogle Scholar
  9. 9.
    C. Barbarosie, Shape optimization of periodic structures. Comput. Mech. 30(3), 235–246 (2003)CrossRefzbMATHGoogle Scholar
  10. 10.
    C. Barbarosie, A.-M. Toader, Shape and topology optimization for periodic problems. I. The shape and the topological derivative. Struct. Multidiscip. Optim. 40(1–6), 381–391 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    C. Barbarosie, A.-M. Toader, Shape and topology optimization for periodic problems. II. Optimization algorithm and numerical examples. Struct. Multidiscip. Optim. 40(1–6), 393–408 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    C. Barbarosie, A.-M. Toader, Optimization of bodies with locally periodic microstructure. Mech. Adv. Mater. Struct. 19(4), 290–301 (2012)CrossRefGoogle Scholar
  13. 13.
    M. P. Bendsøe, Optimization of Structural Topology, Shape, and Material (Springer, Berlin, 1995)CrossRefGoogle Scholar
  14. 14.
    M. P. Bendsøe, A. Díaz, N. Kikuchi, Topology and generalized layout optimization of elastic structures, in Topology Design of Structures (Sesimbra, 1992). NATO Advanced Science Institutes Series E: Applied Sciences, vol. 227 (Kluwer Academic, Dordrecht, 1993), pp. 159–205Google Scholar
  15. 15.
    A. Ben-Tal, L. El-Ghaoui, A. Nemirovski, Robust Optimization (Princeton University Press, Princeton/Oxford, 2009)CrossRefzbMATHGoogle Scholar
  16. 16.
    A. Ben-Tal, M. Kočvara, A. Nemirovski, J. Zowe, Free material design via semidefinite programming: the multiload case with contact conditions. SIAM J. Optim. 9, 813–832 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    J.R. Birge, F. Louveaux, Introduction to Stochastic Programming. Springer Series in Operations Research (Springer, New York, 1997)Google Scholar
  18. 18.
    E. Bonnetier, F. Jouve, Checkerboard instabilities in topological shape optimization algorithms, in Proceedings of the Conference on Inverse Problems, Control and Shape Optimization (PICOF’98), Carthage, 1998Google Scholar
  19. 19.
    B. Bourdin, R.V. Kohn, Optimization of structural topology in the high-porosity regime. J. Mech. Phys. Solids 56(3), 1043–1064 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    A. Braides, A. Defranceschi, Homogeneization of Multiple Integrals (Claredon Press, Oxford, 1998)Google Scholar
  21. 21.
    G. Buttazzo, G. Dal Maso, Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim. 23, 17–49 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    A. Cherkaev, E. Cherkaev, Principal compliance and robust optimal design. J. Elast. 72, 71–98 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    D. Cioranescu, P. Donato, An Introduction to Homogenization (Oxford University Press, Oxford, 1999)zbMATHGoogle Scholar
  24. 24.
    D. Cioranescu, J.S.J. Paulin, Homogenization of Reticulated Structures (Springer, New York, 1999)CrossRefzbMATHGoogle Scholar
  25. 25.
    S. Conti, H. Held, M. Pach, M. Rumpf, R. Schultz, Shape optimization under uncertainty – a stochastic programming perspective. SIAM J. Optim. 19, 1610–1632 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    S. Conti, H. Held, M. Pach, M. Rumpf, R. Schultz, Risk averse shape optimization. SIAM J. Control Optim. 49, 927–947 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    D. Dentcheva, A. Ruszczyński, Optimization with stochastic dominance constraints. SIAM J. Optim. 14, 548–566 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    B. Geihe, M. Lenz, M. Rumpf, R. Schultz, Risk averse elastic shape optimization with parametrized fine scale geometry. Math. Program. 141(1–2), 383–403 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Y. Grabovsky, R.V. Kohn, Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. I. The confocal ellipse construction. J. Mech. Phys. Solids 43(6), 933–947 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    J.M. Guedes, H.C. Rodrigues, M.P. Bendsøe, A material optimization model to approximate energy bounds for cellular materials under multiload conditions. Struct. Multidiscip. Optim. 25, 446–452 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Z. Hashin, The elastic moduli of heterogeneous materials. Trans. ASME Ser. E. J. Appl. Mech. 29, 143–150 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11, 127–140 (1963)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    J. Haslinger, M. Kočvara, G. Leugering, M. Stingl, Multidisciplinary free material optimization. SIAM J. Appl. Math. 70(7), 2709–2728 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    P. Henning, M. Ohlberger, The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains. Numer. Math. 113(4), 601–629 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    S.J. Hollister, N. Kikuchi, Homogenization theory and digital imaging: a basis for studying the mechanics and design principles of bone tissue. Biotechnol. Bioeng. 43, 586–596 (1994)CrossRefGoogle Scholar
  36. 36.
    V. Jikov, V. Zhikov, S. Kozlov, O. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer, Berlin/New York, 1994)CrossRefGoogle Scholar
  37. 37.
    D.P. Kouri, M. Heinkenschloss, D. Ridzal, B.G. van Bloemen Waanders, A trust-region algorithm with adaptive stochastic collocation for PDE optimization under uncertainty. SIAM J. Sci. Comput. 35(4), A1847–A1879 (2013)CrossRefzbMATHGoogle Scholar
  38. 38.
    R. Melchers, Optimality-criteria-based probabilistic structural design. Struct. Multidiscip. Optim. 23(1), 34–39 (2001)CrossRefGoogle Scholar
  39. 39.
    N. Miller, A. Ruszczyński, Two-stage stochastic linear programming: Modeling and decomposition. Oper. Res. 59, 125–132 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    F. Murat, L. Tartar, Calcul des variations et homogénéisation. In Homogenization Methods: Theory and Applications in Physics (Bréau-sans-Nappe, 1983). Collect. Dir. Études Rech. Élec. France, vol. 57 (Eyrolles, Paris, 1985), pp. 319–369Google Scholar
  41. 41.
    G.C. Pflug, W. Römisch, Modeling, Measuring and Managing Risk (World Scientific, Singapore, 2007)CrossRefzbMATHGoogle Scholar
  42. 42.
    A. Ruszczyński, A. Shapiro (Eds.), Handbooks in Operations Research and Management Sciences, 10: Stochastic Programming (Elsevier, Amsterdam, 2003)Google Scholar
  43. 43.
    V. Schulz, C. Schillings, On the nature and treatment of uncertainties in aerodynamic design. AIAA J. 47, 646–654 (2009)CrossRefGoogle Scholar
  44. 44.
    A. Shapiro, Minimax and risk averse multistage stochastic programming. Eur. J. Oper. Res. 219, 719–726 (2012)CrossRefzbMATHGoogle Scholar
  45. 45.
    O. Sigmund, On the optimality of bone microstructure, in IUTAM Symposium on Synthesis in Bio Solid Mechanics, Copenhagen (Springer, 2002), pp. 221–234Google Scholar
  46. 46.
    L. Tartar, Estimations fines des coefficients homogénéisés, in Ennio De Giorgi Colloquium (Paris, 1983). Research notes in mathematics, vol. 125 (Pitman, Boston, 1985), pp. 168–187Google Scholar
  47. 47.
    A. Wächter, An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications in Process Engineering. Phd thesis, Carnegie Mellon University, 2002Google Scholar
  48. 48.
    A. Wächter, L. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  49. 49.
    W. E, B. Engquist, Z. Huang, Heterogeneous multiscale method: a general methodology for multiscale modeling. Phys. Rev. B 67(9), 092101–1–092101–4 (2003)Google Scholar
  50. 50.
    W. E, P. Ming, P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Am. Math. Soc. 18(1), 121–156 (2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Sergio Conti
    • 1
  • Benedict Geihe
    • 2
    Email author
  • Martin Rumpf
    • 2
  • Rüdiger Schultz
    • 3
  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany
  2. 2.Institut für Numerische SimulationUniversität BonnBonnGermany
  3. 3.Fakultät für MathematikUniversität Duisburg-EssenEssenGermany

Personalised recommendations