| Title:
|
Chance constrained optimal beam design: Convex reformulation and probabilistic robust design (English) |
| Author:
|
Kůdela, Jakub |
| Author:
|
Popela, Pavel |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
54 |
| Issue:
|
6 |
| Year:
|
2018 |
| Pages:
|
1201-1217 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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). (English) |
| Keyword:
|
optimal design |
| Keyword:
|
stochastic programming |
| Keyword:
|
chance constrained optimization |
| Keyword:
|
probabilistic robust design |
| Keyword:
|
geometric programming |
| MSC:
|
49M25 |
| MSC:
|
65C05 |
| MSC:
|
90C15 |
| MSC:
|
90C30 |
| idZBL:
|
Zbl 07031769 |
| idMR:
|
MR3902629 |
| DOI:
|
10.14736/kyb-2018-6-1201 |
| . |
| Date available:
|
2019-02-18T14:49:59Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147605 |
| . |
| Reference:
|
[1] Adam, L., Branda, M.: Nonlinear chance constrained problems: Optimality conditions, regularization and solvers..J. Optim. Theory Appl. 170 (2016), 2, 419-436. MR 3527703, 10.1007/s10957-016-0943-9 |
| Reference:
|
[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. MR 3399194, 10.1007/s00158-015-1253-9 |
| Reference:
|
[3] Ben-Tal, A., Ghaoui, L. El, Nemirovski, A.: Robust Optimization..Princeton University Press, 2009. MR 2546839, 10.1515/9781400831050 |
| Reference:
|
[4] Boyd, S. P., Vandenberghe, L.: Convex Optimization..Cambridge University Press, New York 2004. Zbl 1058.90049, MR 2061575, 10.1017/cbo9780511804441 |
| Reference:
|
[5] Calafiore, G. C., Campi, M. C.: The Scenario approach to robust control design..IEEE Trans. Automat. Control 51 (2006), 5, 742-753. MR 2232597, 10.1109/tac.2006.875041 |
| Reference:
|
[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. MR 2780563, 10.1007/s10957-010-9754-6 |
| Reference:
|
[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. MR 3413595, 10.1137/130928546 |
| Reference:
|
[8] Dupačová, J.: Stochastic geometric programming with an application..Kybernetika 46 (2010), 3, 374-386. MR 2676074 |
| Reference:
|
[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. 10.1007/s00366-011-0241-y |
| Reference:
|
[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. MR 2409077, 10.1007/978-1-84800-155-8_7 |
| Reference:
|
[11] Haslinger, J., Mäkinen, R. A. E.: Introduction to Shape Optimization: Theory, Approximation, and Computation (Advances in Design and Control)..SIAM, 2003. MR 1969772, 10.1137/1.9780898718690 |
| Reference:
|
[12] Laníková, I., Štěpánek, P., Šimůnek, P.: Optimized Design of concrete structures considering environmental aspects..Advances Structural Engrg. 17 (2014), 4, 495-511. 10.1260/1369-4332.17.4.495 |
| Reference:
|
[13] Lepš, M., Šejnoha, M.: New approach to optimization of reinforced concrete beams..Computers Structures 81 (2003), 1, 1957-1966. 10.1016/s0045-7949(03)00215-3 |
| Reference:
|
[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. MR 2546332, 10.1007/s10107-008-0247-4 |
| Reference:
|
[15] Marek, P., Brozzetti, J., Gustar, M.: Probabilistic Assessment of Structures using Monte Carlo Simulation..TeReCo, Praha 2001. 10.1115/1.1451167 |
| Reference:
|
[16] Nemirovski, A.: On safe tractable approximations of chance constraints..Europ. J. Oper. Res. 219 (2012), 707-718. MR 2898951, 10.1016/j.ejor.2011.11.006 |
| Reference:
|
[17] Oberg, E., Jones, F. D., Ryffel, H. H.: Machinery's Handbook Guide. 29th edition..Industrial Press, 2012. |
| Reference:
|
[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. MR 2525799, 10.1007/s10957-009-9523-6 |
| Reference:
|
[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 |
| Reference:
|
[20] Ruszczynski, A., (eds.), A. Shapiro: Handbooks in Operations Research and Management Science: Stochastic Programming..Elsevier, Amsterdam 2003. MR 2051792 |
| Reference:
|
[21] Šabartová, Z., Popela, P.: Beam design optimization model with FEM based constraints..Mendel J. Ser. 1 (2012), 422-427. |
| Reference:
|
[22] Smith, I. M., Griffiths, D. V.: Programming the Finite Element Method. Fourth edition..John Wiley and Sons, New York 2004. MR 0934925 |
| Reference:
|
[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 |
| Reference:
|
[24] Žampachová, E., Popela, P., Mrázek, M.: Optimum beam design via stochastic programming..Kybernetika 46 (2010), 3, 571-582. MR 2676092 |
| Reference:
|
[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. 10.1115/1.4005597 |
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