Kybernetika 54 no. 6, 1201-1217, 2018

Chance constrained optimal beam design: Convex reformulation and probabilistic robust design

Jakub Kůdela and Pavel PopelaDOI: 10.14736/kyb-2018-6-1201

Abstract:

In this paper, we are concerned with a civil engineering application of optimization, namely the optimal design of a loaded beam. The developed optimization model includes ODE-type constraints and chance constraints. We use the finite element method (FEM) for the approximation of the ODE constraints. We derive a convex reformulation that transforms the problem into a linear one and find its analytic solution. Afterwards, we impose chance constraints on the stress and the deflection of the beam. These chance constraints are handled by a sampling method (Probabilistic Robust Design).

Keywords:

stochastic programming, optimal design, chance constrained optimization, probabilistic robust design, geometric programming

Classification:

90C15, 90C30, 65C05, 49M25

paper.pdf

References:

  1. L. Adam and M. Branda: Nonlinear chance constrained problems: Optimality conditions, regularization and solvers. J. Optim. Theory Appl. 170 (2016), 2, 419-436.   DOI:10.1007/s10957-016-0943-9
  2. A. T. Beck, W. J. S. Gomes, R. H. Lopez and L. F. F. Miguel: A comparison between robust and risk-based optimization under uncertainty. Struct. Multidisciplin. Optim. 52 (2015), 3, 479-492.   DOI:10.1007/s00158-015-1253-9
  3. A. Ben-Tal, L. El Ghaoui and A. Nemirovski: Robust Optimization. Princeton University Press, 2009.   DOI:10.1515/9781400831050
  4. S. P. Boyd and L. Vandenberghe: Convex Optimization. Cambridge University Press, New York 2004.   DOI:10.1017/cbo9780511804441
  5. G. C. Calafiore and M. C. Campi: The Scenario approach to robust control design. IEEE Trans. Automat. Control 51 (2006), 5, 742-753.   DOI:10.1109/tac.2006.875041
  6. M. C. Campi and S. Garatti: A Sampling-and-discarding approach to chance-constrained optimization: feasibility and optimality. J. Optim. Theory Appl. 148 (2011), 257-280.   DOI:10.1007/s10957-010-9754-6
  7. A. Carè, S. Garatti and M. C. Campi: Scenario min-max optimization and the risk of empirical costs. SIAM J. Optim. 25 (2015), 4, 2061-2080.   DOI:10.1137/130928546
  8. J. Dupačová: Stochastic geometric programming with an application. Kybernetika 46 (2010), 3, 374-386.   CrossRef
  9. A. H. Gandomi, X.-S. Yang and A. H. Alavi: Cuckoo search algorithm: A metaheuristic approach to solve structural optimization problems. Engrg. Comput. 29 (2013), 1, 17-35.   DOI:10.1007/s00366-011-0241-y
  10. M. Grant and S. Boyd: Graph implementations for nonsmooth convex programs. In: Recent Advances in Learning and Control (V. Blondel, S. Boyd and H. Kimura, eds.), Springer-Verlag Limited, Berlin 2008, pp. 95-110.   DOI:10.1007/978-1-84800-155-8\_7
  11. J. Haslinger and R. A. E. Mäkinen: Introduction to Shape Optimization: Theory, Approximation, and Computation (Advances in Design and Control). SIAM, 2003.   DOI:10.1137/1.9780898718690
  12. I. Laníková, P. Štěpánek and P. Šimůnek: Optimized Design of concrete structures considering environmental aspects. Advances Structural Engrg. 17 (2014), 4, 495-511.   DOI:10.1260/1369-4332.17.4.495
  13. M. Lepš and M. Šejnoha: New approach to optimization of reinforced concrete beams. Computers Structures 81 (2003), 1, 1957-1966.   DOI:10.1016/s0045-7949(03)00215-3
  14. J. Luedtke, S. Ahmed and G. L. Nemhauser: An integer programming approach for linear programs with probabilistic constraints. Math. Programm. Ser. A 122 (2010), 247-272.   DOI:10.1007/s10107-008-0247-4
  15. P. Marek, J. Brozzetti and M. Gustar: Probabilistic Assessment of Structures using Monte Carlo Simulation. TeReCo, Praha 2001.   DOI:10.1115/1.1451167
  16. A. Nemirovski: On safe tractable approximations of chance constraints. Europ. J. Oper. Res. 219 (2012), 707-718.   DOI:10.1016/j.ejor.2011.11.006
  17. E. Oberg, F. D. Jones and H. H. Ryffel: Machinery's Handbook Guide. 29th edition. Industrial Press, 2012.   CrossRef
  18. B. K. Pagnoncelli, S. Ahmed and A. Shapiro: Sample average approximation method for chance constrained programming: Theory and applications. J. Optim. Theory Appl. 142 (2009), 399-416.   DOI:10.1007/s10957-009-9523-6
  19. G. I. N. Rozvany and T. Lewiński (eds.): CISM Courses and Lectures: Topology Optimization in Structural and Continuum Mechanics. Springer-Verlag, Wien 2014.   CrossRef
  20. A. Ruszczynski and A. Shapiro (eds.): Handbooks in Operations Research and Management Science: Stochastic Programming. Elsevier, Amsterdam 2003.   CrossRef
  21. Z. Šabartová and P. Popela: Beam design optimization model with FEM based constraints. Mendel J. Ser. 1 (2012), 422-427.   CrossRef
  22. I. M. Smith and D. V. Griffiths: Programming the Finite Element Method. Fourth edition. John Wiley and Sons, New York 2004.   CrossRef
  23. W. C. Young, R. G. Budynas and A. M. Sadegh: Roark's Formulas for Stress and Strain. Seventh edition. McGraw-Hill Education, 2002.   CrossRef
  24. E. Žampachová, P. Popela and M. Mrázek: Optimum beam design via stochastic programming. Kybernetika 46 (2010), 3, 571-582.   CrossRef
  25. X. Zhuang and R. Pan: A sequential sampling strategy to improve reliability-based design optimization with implicit constraint functions. J. Mechan. Design 134 (2012), 2, Article number 021002.   DOI:10.1115/1.4005597