Article
Full entry |
PDF
(1.3 MB)
Feedback
PDF
(1.3 MB)
Feedback
Keywords:
optimal design; stochastic programming; chance constrained optimization; probabilistic robust design; geometric programming
optimal design; stochastic programming; chance constrained optimization; probabilistic robust design; geometric programming
Summary:
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).
References:
[1] Adam, L., Branda, M.: 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 | MR 3527703
[2] Beck, A. T., Gomes, W. J. S., Lopez, R. H., Miguel, L. F. F.: 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 | MR 3399194
[3] Ben-Tal, A., Ghaoui, L. El, Nemirovski, A.: Robust Optimization. Princeton University Press, 2009. DOI 10.1515/9781400831050 | MR 2546839
[4] Boyd, S. P., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York 2004. DOI 10.1017/cbo9780511804441 | MR 2061575 | Zbl 1058.90049
[5] Calafiore, G. C., Campi, M. C.: The Scenario approach to robust control design. IEEE Trans. Automat. Control 51 (2006), 5, 742-753. DOI 10.1109/tac.2006.875041 | MR 2232597
[6] Campi, M. C., Garatti, S.: 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 | MR 2780563
[7] Carè, A., Garatti, S., Campi, M. C.: Scenario min-max optimization and the risk of empirical costs. SIAM J. Optim. 25 (2015), 4, 2061-2080. DOI 10.1137/130928546 | MR 3413595
[8] Dupačová, J.: Stochastic geometric programming with an application. Kybernetika 46 (2010), 3, 374-386. MR 2676074
[9] Gandomi, A. H., Yang, X.-S., Alavi, A. H.: 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] Grant, M., Boyd, S.: 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 | MR 2409077
[11] Haslinger, J., Mäkinen, R. A. E.: Introduction to Shape Optimization: Theory, Approximation, and Computation (Advances in Design and Control). SIAM, 2003. DOI 10.1137/1.9780898718690 | MR 1969772
[12] Laníková, I., Štěpánek, P., Šimůnek, P.: 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] Lepš, M., Šejnoha, M.: New approach to optimization of reinforced concrete beams. Computers Structures 81 (2003), 1, 1957-1966. DOI 10.1016/s0045-7949(03)00215-3
[14] Luedtke, J., Ahmed, S., Nemhauser, G. L.: 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 | MR 2546332
[15] Marek, P., Brozzetti, J., Gustar, M.: Probabilistic Assessment of Structures using Monte Carlo Simulation. TeReCo, Praha 2001. DOI 10.1115/1.1451167
[16] Nemirovski, A.: On safe tractable approximations of chance constraints. Europ. J. Oper. Res. 219 (2012), 707-718. DOI 10.1016/j.ejor.2011.11.006 | MR 2898951
[17] Oberg, E., Jones, F. D., Ryffel, H. H.: Machinery's Handbook Guide. 29th edition. Industrial Press, 2012.
[18] Pagnoncelli, B. K., Ahmed, S., Shapiro, A.: 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 | MR 2525799
[19] Rozvany, G. I. N., (eds.), T. Lewiński: CISM Courses and Lectures: Topology Optimization in Structural and Continuum Mechanics. Springer-Verlag, Wien 2014. MR 3183768
[20] Ruszczynski, A., (eds.), A. Shapiro: Handbooks in Operations Research and Management Science: Stochastic Programming. Elsevier, Amsterdam 2003. MR 2051792
[21] Šabartová, Z., Popela, P.: Beam design optimization model with FEM based constraints. Mendel J. Ser. 1 (2012), 422-427.
[22] Smith, I. M., Griffiths, D. V.: Programming the Finite Element Method. Fourth edition. John Wiley and Sons, New York 2004. MR 0934925
[23] Young, W. C., Budynas, R. G., Sadegh, A. M.: Roark's Formulas for Stress and Strain. Seventh edition. McGraw-Hill Education, 2002. MR 0112352
[24] Žampachová, E., Popela, P., Mrázek, M.: Optimum beam design via stochastic programming. Kybernetika 46 (2010), 3, 571-582. MR 2676092
[25] Zhuang, X., Pan, R.: 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
Search
Partner of
