Skip to main content
Springer Nature Link
Log in

On Shape Optimization with Stochastic Loadings

  • Chapter
  • First Online:
Constrained Optimization and Optimal Control for Partial Differential Equations

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 160))

  • 2921 Accesses

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.

Mathematics Subject Classification (2000). 90C15, 74B05, 65N30, 65N38, 34E08, 49K45.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  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.

    Google Scholar 

  2. Albers, S. Online algorithms: a survey.Mathematical Programming 97 (2003), 3–26.

    MathSciNet  MATH  Google Scholar 

  3. Allaire, G. Shape Optimization by the Homogenization Method, vol. 146. Springer Applied Mathematical Sciences, 2002.

    Google Scholar 

  4. Allaire, G., Bonnetier, E., Francfort, G., and Jouve, F. Shape optimization by the homogenization method. Numerische Mathematik 76 (1997), 27–68.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  11. Atwal, P. Continuum limit of a double-chain model for multiload shape optimization. Journal of Convex Analysis 18, 1 (2011).

    MathSciNet  MATH  Google Scholar 

  12. Atwal, P. Hole-hole interaction in shape optimization via topology derivatives. PhD thesis, University of Bonn, in preparation.

    Google Scholar 

  13. Banichuk, N.V., and Neittaanmäki, P. On structural optimization with incomplete information. Mechanics Based Design of Structures and Machines 35 (2007), 75–95.

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  15. Ben-Tal, A., El-Ghaoui, L., and Nemirovski, A. Robust Optimization. Princeton University Press, Princeton and Oxford, 2009.

    MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  17. Bendsoe, M.P. Optimization of structural topology, shape, and material. Springer-Verlag, Berlin, 1995.

    MATH  Google Scholar 

  18. Burger, M., Hackl, B., and Ring, W. Incorporating topological derivatives into level set methods. J. Comp. Phys. 194 (2004), 344–362.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  20. Chang, F.R. Stochastic Optimization in Continuous Time. Cambridge University Press, Cambridge, 2004.

    MATH  Google Scholar 

  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.

    MATH  Google Scholar 

  22. Cherkaev, A., and Cherkaev, E. Principal compliance and robust optimal design. Journal of Elasticity 72 (2003), 71–98.

    MathSciNet  MATH  Google Scholar 

  23. Ciarlet, P.G. Mathematical Elasticity Volume I: Three-Dimensional Elasticity, vol. 20. Studies in Mathematics and its Applications, North-Holland, 1988.

    MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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. 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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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. Dias, G.P., Herskovits, J., and Rochinha, F.A. Simultaneous shape optimization and nonlinear analysis of elastic solids. Computational Mechanics (1998).

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  31. Fleming, W.H., and Rishel, R.W. Deterministic and Stochastic Optimal Control. Springer, New York, 1975.

    MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  34. Hackbusch, W. Integral Equations, vol. 120 of International Series of Numerical Mathematics. Birkhäuser Verlag, Basel, Boston, Berlin, 1995.

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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. 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.

    MathSciNet  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  40. Marsden, J., and Hughes, T. Mathematical foundations of elasticity. Dover Publications Inc., New York, 1993.

    MATH  Google Scholar 

  41. Marti, K. Stochastic Optimization Methods. Springer, Berlin, 2005.

    MATH  Google Scholar 

  42. Melchers, R. Optimality-criteria-based probabilistic structural design. Structural and Multidisciplinary Optimization 23, 1 (2001), 34–39.

    Google Scholar 

  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. Pennanen, T. Epi-convergent discretizations of multistage stochastic programs. Mathematics of Operations Research 30 (2005), 245–256.

    MathSciNet  MATH  Google Scholar 

  45. Pflug, G.C., and Römisch, W. Modeling, Measuring and Managing Risk. World Scientific, Singapore, 2007.

    MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  47. Ruszczyński, A., and Shapiro, A., eds. Handbooks in Operations Research and Management Sciences, 10: Stochastic Programming. Elsevier, Amsterdam, 2003.

    Google Scholar 

  48. Schultz, R. Stochastic programming with integer variables. Mathematical Programming 97 (2003), 285–309.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  50. Schulz, V., and Schillings, C. On the nature and treatment of uncertainties in aerodynamic design. AIAA Journal 47 (2009), 646–654.

    Google Scholar 

  51. Schumacher, A. Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpositionierungskriterien. PhD thesis, Universität – Gesamthochschule Siegen, 1996.

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  53. Shapiro, A., Dentcheva, D., and Ruszczyński, A. Lectures on Stochastic Programming. SIAM-MPS, Philadelphia, 2009.

    MATH  Google Scholar 

  54. Shewchuk, J. Triangle: Engineering a 2d quality mesh generator and Delaunay triangulator. Applied Computational Geometry: Towards Geometric Engineering 1148 (1996), 203–222.

    Google Scholar 

  55. Shewchuk, J. Delaunay refinement algorithms for triangular mesh generation. Computational Geometry: Theory and Applications 22 (2002), 21–74.

    MathSciNet  MATH  Google Scholar 

  56. Soko̷lowski, J., and Żchowski, A. On the topological derivative in shape optimization. SIAM J. Control Optim. 37, 4 (1999), 1251–1272.

    MathSciNet  Google Scholar 

  57. Soko̷lowski, J., and Żchowski, A. Topological derivatives of shape functionals for elasticity systems. Mech. Struct. & Mach. 29, 3 (2001), 331–349.

    Google Scholar 

  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.

    MathSciNet  Google Scholar 

  59. Soko̷lowski, J., and Zolésio, J.-P. Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, 1992.

    Google Scholar 

  60. Steinbach, M. Tree-sparse convex programs. Mathematical Methods of Operations Research 56 (2002), 347–376.

    MathSciNet  MATH  Google Scholar 

  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. 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.

    MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, D-53115, Bonn, Germany

    Pradeep Atwal & Sergio Conti

  2. Institut für Numerische Simulation, Universität Bonn, Endenicher Allee 60, D-53115, Bonn, Germany

    Benedict Geihe & Martin Rumpf

  3. Martin Pach and Rüdiger Schultz Fakultät für Mathematik, Universität Duisburg-Essen, Forsthausweg 2, D-47057, Duisburg, Germany

    Martin Pach & Rüdiger Schultz

Authors
  1. Pradeep Atwal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergio Conti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Benedict Geihe
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Martin Pach
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Martin Rumpf
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Rüdiger Schultz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. , Department Mathematik, Universität Erlangen-Nürnberg, Martensstr. 3, Erlangen, 91058, Germany

    Günter Leugering

  2. , Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Geschossbau 2, Dortmund, 44221, Germany

    Sebastian Engell

  3. , Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Berlin, 12489, Germany

    Andreas Griewank

  4. , Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, Hamburg, 20146, Germany

    Michael Hinze

  5. , Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 293, Heidelberg, 69120, Germany

    Rolf Rannacher

  6. , FB 4 - Department Mathematik, Universität Trier, Trier, 54286, Germany

    Volker Schulz

  7. Fakultät für Mathematik, M1, Lehrstuhl für Mathematische Optimierung, Technische Universität München, Boltzmannstr. 3, Garching bei München, 85748, Germany

    Michael Ulbrich

  8. , Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostrasse 15, Darmstadt, 64293, Germany

    Stefan Ulbrich

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Basel AG

About this chapter

Cite this chapter

Atwal, P., Conti, S., Geihe, B., Pach, M., Rumpf, M., Schultz, R. (2012). On Shape Optimization with Stochastic Loadings. In: Leugering, G., et al. Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics, vol 160. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0133-1_12

Download citation

Publish with us

Policies and ethics