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

Jakub Kůdela; Pavel Popela

Kybernetika (2018)

  • Volume: 54, Issue: 6, page 1201-1217
  • ISSN: 0023-5954

Abstract

top
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).

How to cite

top

Kůdela, Jakub, and Popela, Pavel. "Chance constrained optimal beam design: Convex reformulation and probabilistic robust design." Kybernetika 54.6 (2018): 1201-1217. <http://eudml.org/doc/294449>.

@article{Kůdela2018,
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).},
author = {Kůdela, Jakub, Popela, Pavel},
journal = {Kybernetika},
keywords = {optimal design; stochastic programming; chance constrained optimization; probabilistic robust design; geometric programming},
language = {eng},
number = {6},
pages = {1201-1217},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Chance constrained optimal beam design: Convex reformulation and probabilistic robust design},
url = {http://eudml.org/doc/294449},
volume = {54},
year = {2018},
}

TY - JOUR
AU - Kůdela, Jakub
AU - Popela, Pavel
TI - Chance constrained optimal beam design: Convex reformulation and probabilistic robust design
JO - Kybernetika
PY - 2018
PB - Institute of Information Theory and Automation AS CR
VL - 54
IS - 6
SP - 1201
EP - 1217
AB - 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).
LA - eng
KW - optimal design; stochastic programming; chance constrained optimization; probabilistic robust design; geometric programming
UR - http://eudml.org/doc/294449
ER -

References

top
  1. Adam, L., Branda, M., 10.1007/s10957-016-0943-9, J. Optim. Theory Appl. 170 (2016), 2, 419-436. MR3527703DOI10.1007/s10957-016-0943-9
  2. Beck, A. T., Gomes, W. J. S., Lopez, R. H., Miguel, L. F. F., 10.1007/s00158-015-1253-9, Struct. Multidisciplin. Optim. 52 (2015), 3, 479-492. MR3399194DOI10.1007/s00158-015-1253-9
  3. Ben-Tal, A., Ghaoui, L. El, Nemirovski, A., 10.1515/9781400831050, Princeton University Press, 2009. MR2546839DOI10.1515/9781400831050
  4. Boyd, S. P., Vandenberghe, L., 10.1017/cbo9780511804441, Cambridge University Press, New York 2004. Zbl1058.90049MR2061575DOI10.1017/cbo9780511804441
  5. Calafiore, G. C., Campi, M. C., 10.1109/tac.2006.875041, IEEE Trans. Automat. Control 51 (2006), 5, 742-753. MR2232597DOI10.1109/tac.2006.875041
  6. Campi, M. C., Garatti, S., 10.1007/s10957-010-9754-6, J. Optim. Theory Appl. 148 (2011), 257-280. MR2780563DOI10.1007/s10957-010-9754-6
  7. Carè, A., Garatti, S., Campi, M. C., 10.1137/130928546, SIAM J. Optim. 25 (2015), 4, 2061-2080. MR3413595DOI10.1137/130928546
  8. Dupačová, J., Stochastic geometric programming with an application., Kybernetika 46 (2010), 3, 374-386. MR2676074
  9. Gandomi, A. H., Yang, X.-S., Alavi, A. H., 10.1007/s00366-011-0241-y, Engrg. Comput. 29 (2013), 1, 17-35. DOI10.1007/s00366-011-0241-y
  10. Grant, M., Boyd, S., 10.1007/978-1-84800-155-8_7, In: Recent Advances in Learning and Control (V. Blondel, S. Boyd and H. Kimura, eds.), Springer-Verlag Limited, Berlin 2008, pp. 95-110. MR2409077DOI10.1007/978-1-84800-155-8_7
  11. Haslinger, J., Mäkinen, R. A. E., 10.1137/1.9780898718690, SIAM, 2003. MR1969772DOI10.1137/1.9780898718690
  12. Laníková, I., Štěpánek, P., Šimůnek, P., 10.1260/1369-4332.17.4.495, Advances Structural Engrg. 17 (2014), 4, 495-511. DOI10.1260/1369-4332.17.4.495
  13. Lepš, M., Šejnoha, M., 10.1016/s0045-7949(03)00215-3, Computers Structures 81 (2003), 1, 1957-1966. DOI10.1016/s0045-7949(03)00215-3
  14. Luedtke, J., Ahmed, S., Nemhauser, G. L., 10.1007/s10107-008-0247-4, Math. Programm. Ser. A 122 (2010), 247-272. MR2546332DOI10.1007/s10107-008-0247-4
  15. Marek, P., Brozzetti, J., Gustar, M., 10.1115/1.1451167, TeReCo, Praha 2001. DOI10.1115/1.1451167
  16. Nemirovski, A., 10.1016/j.ejor.2011.11.006, Europ. J. Oper. Res. 219 (2012), 707-718. MR2898951DOI10.1016/j.ejor.2011.11.006
  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., 10.1007/s10957-009-9523-6, J. Optim. Theory Appl. 142 (2009), 399-416. MR2525799DOI10.1007/s10957-009-9523-6
  19. Rozvany, G. I. N., (eds.), T. Lewiński, CISM Courses and Lectures: Topology Optimization in Structural and Continuum Mechanics., Springer-Verlag, Wien 2014. MR3183768
  20. Ruszczynski, A., (eds.), A. Shapiro, Handbooks in Operations Research and Management Science: Stochastic Programming., Elsevier, Amsterdam 2003. MR2051792
  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. MR0934925
  23. Young, W. C., Budynas, R. G., Sadegh, A. M., Roark's Formulas for Stress and Strain. Seventh edition., McGraw-Hill Education, 2002. MR0112352
  24. Žampachová, E., Popela, P., Mrázek, M., Optimum beam design via stochastic programming., Kybernetika 46 (2010), 3, 571-582. MR2676092
  25. Zhuang, X., Pan, R., 10.1115/1.4005597, J. Mechan. Design 134 (2012), 2, Article number 021002. DOI10.1115/1.4005597

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.