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Sequential Semidefinite Program for Maximum Robustness Design of Structures under Load Uncertainty

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Abstract

A robust structural optimization scheme as well as an optimization algorithm are presented based on the robustness function. Under the uncertainties of the external forces based on the info-gap model, the maximization of the robustness function is formulated as an optimization problem with infinitely many constraints. By using the quadratic embedding technique of uncertainty and the S-procedure, we reformulate the problem into a nonlinear semidefinite programming problem. A sequential semidefinite programming method is proposed which has a global convergent property. It is shown through numerical examples that optimum designs of various linear elastic structures can be found without difficulty.

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References

  1. TSOMPANAKIS, Y., and PAPADRAKAKIS, M., Large-Scale Reliability-Based Structural Optimization, Structural and Multidisciplinary Optimization, Vol. 26, pp. 429–440, 2004.

    Article  Google Scholar 

  2. KHARMANDA, G., OLHOFF, N., MOHAMED, A., and LEMAIRE, M., Reliability-Based Topology Optimization, Structural and Multidisciplinary Optimization, Vol. 26, pp. 295–307, 2004.

    Article  Google Scholar 

  3. CHOI, K. K., TU, J., and PARK, Y. H., Extensions of Design Potential Concept for Reliability-Based Design Optimization to Nonsmooth and Extreme Cases, Structural and Multidisciplinary Optimization, Vol. 22, pp. 335–350, 2001.

    Article  Google Scholar 

  4. JUNG, D. H., and LEE, B. C., Development of a Simple and Efficient Method for Robust Optimization, International Journal for Numerical Methods in Engineering, Vol. 53, pp. 2201–2215, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  5. DOLTSINIS, I., and KANG, Z., Robust Design of Structures Using Optimization Methods, Computer Methods in Applied Mechanics and Engineering, Vol. 193, pp. 2221–2237, 2004.

    Article  MATH  Google Scholar 

  6. BEN-HAIM, Y., and ELISHAKOFF, I., Convex Models of Uncertainty in Applied Mechanics, Elsevier, New York, NY, 1990.

    MATH  Google Scholar 

  7. PANTELIDES, C. P., and GANZERLI, S., Design of Trusses under Uncertain Loads using Convex Models, ASCE Journal of Structural Engineering, Vol. 124, pp. 318–329, 1998.

    Article  Google Scholar 

  8. BEN-TAL, A., and NEMIROVSKI, A., Robust Optimization: Methodology and Applications, Mathematical Programming, Vol. 92B, pp. 453–480, 2002.

    Article  MathSciNet  Google Scholar 

  9. BEN-TAL, A., and NEMIROVSKI, A., Robust Truss Topology Optimization via Semidefinite Programming, SIAM Journal on Optimization, Vol. 7, pp. 991–1016, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  10. KOČVARA, M., ZOWE, J., and NEMIROVSKI, A., Cascading: An Approach to Robust Material Optimization, Computers and Structures, Vol. 76, pp. 431–442, 2000.

    Article  Google Scholar 

  11. HAN, J. S., and KWAK, B. M., Robust Optimization using a Gradient Index: MEMS Applications, Structural and Multidisciplinary Optimization, Vol. 27, pp. 469–478, 2004.

    Article  Google Scholar 

  12. CALAFIORE, G., and EL GHAOUI, L., Ellipsoidal Bounds for Uncertain Linear Equations and Dynamical Systems, Automatica, Vol. 40, pp. 773–787, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  13. BEN-HAIM, Y., Information-Gap Decision Theory, Academic Press, London, UK, 2001.

    MATH  Google Scholar 

  14. KANNO, Y., and TAKEWAKI, I., Robustness Analysis of Trusses with Separable Load and Structural Uncertainties, International Journal of Solids and Structures, Vol. 43, pp. 2646–2669, 2006.

  15. WOLKOWICZ, H., SAIGAL, R., and VANDENBERGHE, L., Editors, Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, Kluwer Academic Publishers, Dordrecht, Netherlands, 2000.

    Google Scholar 

  16. JARRE, F., Some Aspects of Nonlinear Semidefinite Programming, System Modeling and Optimization: 20th IFIP Conference on System Modeling and Optimization, Edited by E. W. Sachs and R. Tichatschke, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 55–69, 2003.

  17. FUKUSHIMA, M., TAKAZAWA, K., OHSAKI, S., and IBARAKI, T., Successive Linearization Methods for Large-Scale Nonlinear Programming Problems, Japan Journal of Industrial and Applied Mathematics, Vol. 9, pp. 117–132, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  18. KANZOW, C., NAGEL, C., KATO, H., and FUKUSHIMA, M., Successive Linearization Methods for Nonlinear Semidefinite Programs, Computational Optimization and Applications, Vol. 31, pp. 251–273, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  19. KOJIMA, M., and TUNÇEL, L., Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets, SIAM Journal on Optimization, Vol. 10, pp. 750–778, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  20. SIMO, J. C., and HUGHES, T. J. R., Computational Inelasticity, Springer-Verlag, New York, NY, 1998.

    MATH  Google Scholar 

  21. BEN-TAL, A., and NEMIROVSKI, A., Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, SIAM, Philadelphia, Pennysylvania, 2001.

  22. OHSAKI, M., FUJISAWA, K., KATOH, N., and KANNO, Y., Semidefinite Programming for Topology Optimization of Truss under Multiple Eigenvalue Constraints, Computer Methods in Applied Mechanics and Engineering, Vol. 180, pp. 203–217, 1999.

    Article  MATH  Google Scholar 

  23. KANNO, Y., OHSAKI, M., and KATOH, N., Sequential Semidefinite Programming for Optimization of Framed Structures under Multimodal Buckling Constraints, International Journal of Structural Stability and Dynamics, Vol. 1, pp. 585–602, 2001.

    Article  Google Scholar 

  24. STURM, J. F., Using SeDuMi 1.02, a MATLAB Toolbox for Optimization over Symmetric Cones, Optimization Methods and Software, Vol. 11/12, pp. 625–653, 1999.

  25. THE MATH WORKS, Using MATLAB, The Math Works, Natick, Massachusetts, 2002.

  26. KANNO, Y., and TAKEWAKI, I., Sequential Semidefinite Program for Maximum Robustness Design of Structures under Load Uncertainties, Kyoto University, BGE Research Report 04–05, 2004 (revised 2005); available at http://www.archi.kyoto-u.ac.jp/∼bge/RR/.

  27. FUJISAWA, K., KOJIMA, M., and NAKATA, K., Exploiting Sparsity in Primal-Dual Interior-Point Methods for Semidefinite Programming, Mathematical Programming, Vol. 79, pp. 235–253, 1997.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Informatics, The University of Tokyo, Tokyo, Japan

    Y. Kanno (Assistant Professor)

  2. Department of Urban and Environmental Engineering, Kyoto University, Kyoto, Japan

    I. Takewaki (Professor)

Authors
  1. Y. Kanno
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  2. I. Takewaki
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Additional information

Communicated by K. K. Choi

The authors are grateful to the Associate Editor and two anonymous referees for handling the paper efficiently as well as for helpful comments and suggestions.

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Kanno, Y., Takewaki, I. Sequential Semidefinite Program for Maximum Robustness Design of Structures under Load Uncertainty. J Optim Theory Appl 130, 265–287 (2006). https://doi.org/10.1007/s10957-006-9102-z

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  • DOI: https://doi.org/10.1007/s10957-006-9102-z

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