On Shape Optimization with Stochastic Loadings

  • Pradeep Atwal
  • Sergio Conti
  • Benedict Geihe
  • Martin Pach
  • Martin Rumpf
  • Rüdiger Schultz
Chapter
Part of the International Series of Numerical Mathematics book series (ISNM, volume 160)

Abstract

This article is concerned with different approaches to elastic shape optimization under stochastic loading. The underlying stochastic optimization strategy builds upon the methodology of two-stage stochastic programming. In fact, in the case of linear elasticity and quadratic objective functionals our strategy leads to a computational cost which scales linearly in the number of linearly independent applied forces, even for a large set of realizations of the random loading. We consider, besides minimization of the expectation value of suitable objective functionals, also two different risk averse approaches, namely the expected excess and the excess probability. Numerical computations are performed using either a level set approach representing implicit shapes of general topology in combination with composite finite elements to resolve elasticity in two and three dimensions, or a collocation boundary element approach, where polygonal shapes represent geometric details attached to a lattice and describing a perforated elastic domain. Topology optimization is performed using the concept of topological derivatives. We generalize this concept, and derive an analytical expression which takes into account the interaction between neighboring holes. This is expected to allow efficient and reliable optimization strategies of elastic objects with a large number of geometric details on a fine scale.

Keywords

Shape optimization in elasticity two-stage stochastic programming risk averse optimization level set method boundary element method topological derivative 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adali, S., J.C. Bruch, J., Sadek, I., and Sloss, J. Robust shape control of beams with load uncertainties by optimally placed piezo actuators. Structural and Multidisciplinary Optimization 19, 4 (2000), 274–281.CrossRefGoogle Scholar
  2. 2.
    Albers, S. Online algorithms: a survey.Mathematical Programming 97 (2003), 3–26.MATHMathSciNetGoogle Scholar
  3. 3.
    Allaire, G. Shape Optimization by the Homogenization Method, vol. 146. Springer Applied Mathematical Sciences, 2002.Google Scholar
  4. 4.
    Allaire, G., Bonnetier, E., Francfort, G., and Jouve, F. Shape optimization by the homogenization method. Numerische Mathematik 76 (1997), 27–68.CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Allaire, G., de Gournay, F., Jouve, F., and Toader, A.-M. Structural optimization using topological and shape sensitivity via a level set method. Control and Cybernetics 34 (2005), 59–80.MATHMathSciNetGoogle Scholar
  6. 6.
    Allaire, G., and Jouve, F. A level-set method for vibration and multiple loads structural optimization. Comput. Methods Appl. Mech. Engrg. 194, 30-33 (2005), 3269–3290.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Allaire, G., Jouve, F., and de Gournay, F. Shape and topology optimization of the robust compliance via the level set method. ESAIM Control Optim. Calc. Var. 14 (2008), 43–70.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Allaire, G., Jouve, F., and Toader, A.-M. Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics 194, 1 (2004), 363–393.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Alvarez, F., and Carrasco, M. Minimization of the expected compliance as an alternative approach to multiload truss optimization. Struct. Multidiscip. Optim. 29 (2005), 470–476.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Amstutz, S., and Andrä, H. A new algorithm for topology optimization using a level-set method. Journal of Computational Physics 216 (2006), 573–588.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Atwal, P. Continuum limit of a double-chain model for multiload shape optimization. Journal of Convex Analysis 18, 1 (2011).MathSciNetGoogle Scholar
  12. 12.
    Atwal, P. Hole-hole interaction in shape optimization via topology derivatives. PhD thesis, University of Bonn, in preparation.Google Scholar
  13. 13.
    Banichuk, N.V., and Neittaanmäki, P. On structural optimization with incomplete information. Mechanics Based Design of Structures and Machines 35 (2007), 75–95.CrossRefGoogle Scholar
  14. 14.
    Bastin, F., Cirillo, C., and Toint, P. Convergence theory for nonconvex stochastic programming with an application to mixed logit. Math. Program. 108 (2006),207–234.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Ben-Tal, A., El-Ghaoui, L., and Nemirovski, A. Robust Optimization. Princeton University Press, Princeton and Oxford, 2009.MATHGoogle Scholar
  16. 16.
    Ben-Tal, A., Kočvara, M., Nemirovski, A., and Zowe, J. Free material design via semidefinite programming: the multiload case with contact conditions. SIAM J. Optim. 9 (1999), 813–832.CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Bendsoe, M.P. Optimization of structural topology, shape, and material. Springer-Verlag, Berlin, 1995.Google Scholar
  18. 18.
    Burger, M., Hackl, B., and Ring, W. Incorporating topological derivatives into level set methods. J. Comp. Phys. 194 (2004), 344–362.CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Burger, M., and Osher, S.J. A survey on level set methods for inverse problems and optimal design. European Journal of Applied Mathematics 16, 2 (2005), 263–301.CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Chang, F.R. Stochastic Optimization in Continuous Time. Cambridge University Press, Cambridge, 2004.CrossRefMATHGoogle Scholar
  21. 21.
    Cherkaev, A., and Cherkaev, E. Stable optimal design for uncertain loading conditions. In Homogenization, V. B. et al, ed., vol. 50 of Series on Advances in Mathematics for Applied Sciences. World Scientific, Singapore, 1999, pp. 193–213.Google Scholar
  22. 22.
    Cherkaev, A., and Cherkaev, E. Principal compliance and robust optimal design. Journal of Elasticity 72 (2003), 71–98.CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Ciarlet, P.G. Mathematical Elasticity Volume I: Three-Dimensional Elasticity, vol. 20. Studies in Mathematics and its Applications, North-Holland, 1988.MATHGoogle Scholar
  24. 24.
    Clements, D., and Rizzo, F. A Method for the Numerical Solution of Boundary Value Problems Governed by Second-order Elliptic Systems. IMA Journal of Applied Mathematics 22 (1978), 197–202.CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Conti, S., Held, H., Pach, M., Rumpf, M., and Schultz, R. Risk averse shape optimization. Siam Journal on Control and Optimization (2009). Submitted.Google Scholar
  26. 26.
    Conti, S., Held, H., Pach, M., Rumpf, M., and Schultz, R. Shape optimization under uncertainty – a stochastic programming perspective. SIAM J. Optim. 19 (2009), 1610–1632.CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    de Gournay, F., Allaire, G., and Jouve, F. Shape and topology optimization of the robust compliance via the level set method. ESAIM: Control, Optimisation and Calculus of Variations 14 (2007), 43–70.CrossRefGoogle Scholar
  28. 28.
    Delfour, M.C., and Zolésio, J. Geometries and Shapes: Analysis, Differential Calculus and Optimization. Adv. Des. Control 4. SIAM, Philadelphia, 2001.Google Scholar
  29. 29.
    Dias, G.P., Herskovits, J., and Rochinha, F.A. Simultaneous shape optimization and nonlinear analysis of elastic solids. Computational Mechanics (1998).Google Scholar
  30. 30.
    Du, Q., and Wang, D. Tetrahedral mesh generation and optimization based on centroidal Voronoi tesselations. International Journal for Numerical Methods in Engineering 56 (2003), 1355–1373.CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Fleming, W.H., and Rishel, R.W. Deterministic and Stochastic Optimal Control. Springer, New York, 1975.MATHGoogle Scholar
  32. 32.
    Garreau, S., Guillaume, P., and Masmoudi, M. The topological asymptotic for PDE systems: The elasticity case. SIAM J. Control Optim. 39, 6 (2001), 1756–1778.CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Guedes, J.M., Rodrigues, H.C., and Bendsoe, M.P. A material optimization model to approximate energy bounds for cellular materials under multiload conditions. Struct. Multidiscip. Optim. 25 (2003), 446–452.CrossRefMathSciNetGoogle Scholar
  34. 34.
    Hackbusch, W. Integral Equations, vol. 120 of International Series of Numerical Mathematics. Birkhäuser Verlag, Basel, Boston, Berlin, 1995.Google Scholar
  35. 35.
    Hackbusch, W., and Sauter, S. Composite finite elements for the approximation of PDEs on domains with complicated micro-structures. Numerische Mathematik 75 (1997), 447–472.CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    He, L., Kao, C.-Y., and Osher, S. Incorporating topological derivatives into shape derivatives based level set methods. Journal of Computational Physics 225, 1 (2007), 891–909.CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Huyse, L. Free-form airfoil shape optimization under uncertainty using maximum expected value and second-order second-moment strategies. ICASE report; no. 2001-18. ICASE, NASA Langley Research Center Available from NASA Center for Aerospace Information, Hampton, VA, 2001.Google Scholar
  38. 38.
    Liehr, F., Preusser, T., Rumpf, M., Sauter, S., and Schwen, L.O. Composite finite elements for 3D image based computing. Computing and Visualization in Science 12 (2009), 171–188.CrossRefMathSciNetGoogle Scholar
  39. 39.
    Liu, Z., Korvink, J.G., and Huang, R. Structure topology optimization: Fully coupled level set method via femlab. Structural and Multidisciplinary Optimization 29 (June 2005), 407–417.Google Scholar
  40. 40.
    Marsden, J., and Hughes, T. Mathematical foundations of elasticity. Dover Publications Inc., New York, 1993.Google Scholar
  41. 41.
    Marti, K. Stochastic Optimization Methods. Springer, Berlin, 2005.MATHGoogle Scholar
  42. 42.
    Melchers, R. Optimality-criteria-based probabilistic structural design. Structural and Multidisciplinary Optimization 23, 1 (2001), 34–39.CrossRefGoogle Scholar
  43. 43.
    Owen, S.J. A survey of unstructured mesh generation technology. In Proceedings of the 7th International Meshing Roundtable (Dearborn, Michigan, 1998), Sandia National Laboratories, pp. 239–267.Google Scholar
  44. 44.
    Pennanen, T. Epi-convergent discretizations of multistage stochastic programs. Mathematics of Operations Research 30 (2005), 245–256.CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Pflug, G.C., and Römisch, W. Modeling, Measuring and Managing Risk. World Scientific, Singapore, 2007.CrossRefMATHGoogle Scholar
  46. 46.
    Rumigny, N., Papadopoulos, P., and Polak, E. On the use of consistent approximations in boundary element-based shape optimization in the presence of uncertainty. Comput. Methods Appl. Mech. Engrg. 196 (2007), 3999–4010.CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Ruszczyński, A., and Shapiro, A., eds. Handbooks in Operations Research and Management Sciences, 10: Stochastic Programming. Elsevier, Amsterdam, 2003.Google Scholar
  48. 48.
    Schultz, R. Stochastic programming with integer variables. Mathematical Programming 97 (2003), 285–309.MATHMathSciNetGoogle Scholar
  49. 49.
    Schultz, R., and Tiedemann, S. Risk aversion via excess probabilities in stochastic programs with mixed-integer recourse. SIAM J. on Optimization 14, 1 (2003), 115–138.CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Schulz, V., and Schillings, C. On the nature and treatment of uncertainties in aerodynamic design. AIAA Journal 47 (2009), 646–654.CrossRefGoogle Scholar
  51. 51.
    Schumacher, A. Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpositionierungskriterien. PhD thesis, Universität – Gesamthochschule Siegen, 1996.Google Scholar
  52. 52.
    Sethian, J.A., and Wiegmann, A. Structural boundary design via level set and immersed interface methods. Journal of Computational Physics 163, 2 (2000), 489–528.CrossRefMATHMathSciNetGoogle Scholar
  53. 53.
    Shapiro, A., Dentcheva, D., and Ruszczyński, A. Lectures on Stochastic Programming. SIAM-MPS, Philadelphia, 2009.CrossRefMATHGoogle Scholar
  54. 54.
    Shewchuk, J. Triangle: Engineering a 2d quality mesh generator and Delaunay triangulator. Applied Computational Geometry: Towards Geometric Engineering 1148 (1996), 203–222.CrossRefGoogle Scholar
  55. 55.
    Shewchuk, J. Delaunay refinement algorithms for triangular mesh generation. Computational Geometry: Theory and Applications 22 (2002), 21–74.CrossRefMATHMathSciNetGoogle Scholar
  56. 56.
    Soko̷lowski, J., and Żchowski, A. On the topological derivative in shape optimization. SIAM J. Control Optim. 37, 4 (1999), 1251–1272.CrossRefMathSciNetGoogle Scholar
  57. 57.
    Soko̷lowski, J., and Żchowski, A. Topological derivatives of shape functionals for elasticity systems. Mech. Struct. & Mach. 29, 3 (2001), 331–349.CrossRefGoogle Scholar
  58. 58.
    Soko̷lowski, J., and Żchowski, A. Optimality conditions for simultaneous topology and shape optimization. SIAM Journal on Control and Optimization 42, 4 (2003), 1198–1221.CrossRefMathSciNetGoogle Scholar
  59. 59.
    Soko̷lowski, J., and Zolésio, J.-P. Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, 1992.Google Scholar
  60. 60.
    Steinbach, M. Tree-sparse convex programs. Mathematical Methods of Operations Research 56 (2002), 347–376.MATHMathSciNetGoogle Scholar
  61. 61.
    Wächter, A. An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications in Process Engineering. Phd thesis, Carnegie Mellon University, 2002.Google Scholar
  62. 62.
    Wächter, A., and Biegler, L. On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Mathematical Programming 106, 1 (2006), 25–57.CrossRefMATHMathSciNetGoogle Scholar
  63. 63.
    Zhuang, C., Xiong, Z., and Ding, H. A level set method for topology optimization of heat conduction problem under multiple load cases. Comput. Methods Appl. Mech. Engrg. 196 (2007), 1074–1084.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Pradeep Atwal
    • 1
  • Sergio Conti
    • 1
  • Benedict Geihe
    • 2
  • Martin Pach
    • 3
  • Martin Rumpf
    • 2
  • Rüdiger Schultz
    • 3
  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany
  2. 2.Institut für Numerische SimulationUniversität BonnBonnGermany
  3. 3.Martin Pach and Rüdiger Schultz Fakultät für MathematikUniversität Duisburg-EssenDuisburgGermany

Personalised recommendations