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Application of the sequential parametric convex approximation method to the design of robust trusses

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Abstract

We study an algorithm recently proposed, which is called sequential parametric approximation method, that finds the solution of a differentiable nonconvex optimization problem by solving a sequence of differentiable convex approximations from the original one. We show as well the global convergence of this method under weaker assumptions than those made in the literature. The optimization method is applied to the design of robust truss structures. The optimal structure of the model considered minimizes the total amount of material under mechanical equilibrium, displacements and stress constraints. Finally, Robust designs are found by considering load perturbations.

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References

  1. Achtziger, W.: Truss topology optimization including bar properties different for tension and compression. Struct. Optim. 12(1), 63–74 (1996)

    Article  Google Scholar 

  2. Achtziger, W.: Topology optimization of discrete structures: an introduction in view of computational and nonsmooth aspects. Topology optimization in structural mechanics. CISM Courses and Lectures, vol. 374, pp. 57–100. Springer, Vienna (1997)

  3. Achtziger, W., Bendsøe, M., Ben-Tal, A., Zowe, J.: Equivalent displacement based formulations for maximum strength truss topology design. Impact Comput. Sci. Eng. 4(4), 315–345 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Mathematical Programming. A Publication of the Mathematical Programming Society 95(1, Ser. B), 3–51: ISMP 2000, Part 3. Atlanta, GA (2003)

  5. Alvarez, F., Carrasco, M.: Minimization of the expected compliance as an alternative approach to multiload truss optimization. Struct. Multidiscip. Optim. 29(6), 470–476 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bazaraa, M., Sherali, H., Shetty, C.M.: Nonlinear Programming, 3rd edn. Wiley, Hoboken (2006)

    Book  MATH  Google Scholar 

  7. Beck, A., Ben-Tal, A., Tetruashvili, L.: A sequential parametric convex approximation method with applications to nonconvex truss topology design problems. J. Glob. Optim. 47(1), 29–51 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ben-Tal, A., Nemirovski, A.: Robust truss topology design via semidefinite programming. SIAM J. Optim. 7(4), 991–1016 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization. MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001)

  10. Bendsøe, M.: Optimization of Structural Topology, Shape, and Material. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  11. Bertsekas, D.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  12. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  13. Bruggi, M.: On an alternative approach to stress constraints relaxation in topology optimization. Struct. Multidiscip. Optim. 36(2), 125–141 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cheng, G., Guo, X.: \(\varepsilon \)-relaxed approach in structural topology optimization. Struct. Optim. 13(4), 258–266 (1997)

    Article  Google Scholar 

  15. Clarke, F.H.: Optimization and nonsmooth analysis. In: Classics in Applied Mathematics, vol. 5, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1990)

  16. Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, New York (2001)

    MATH  Google Scholar 

  17. Herskovits, J.: Feasible direction interior-point technique for nonlinear optimization. J. Optim. Theory Appl. 99(1), 121–146 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Herskovits, J., Mappa, P., Goulart, E., Mota Soares, C.M.: Mathematical programming models and algorithms for engineering design optimization. Comput. Methods Appl. Mech. Eng. 194(30–33), 3244–3268 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kang, Z., Bai, S.: On robust design optimization of truss structures with bounded uncertainties. Struct. Multidiscip. Optim. 47(5), 699–714 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Landau, L., Lifshitz, E.M.: Theory of Elasticity. Butterworth Heinemann, Oxford (1986)

    MATH  Google Scholar 

  21. Le, C., Norato, J., Bruns, T., Ha, C., Tortorelli, D.: Stress-based topology optimization for continua. Struct. Multidiscip. Optim. 41(4), 605–620 (2010)

    Article  Google Scholar 

  22. Luenberger, D.G.: Linear and Nonlinear Programming, 2nd edn. Kluwer Academic Publishers, Boston (2003)

    MATH  Google Scholar 

  23. Nocedal, J., Wright, S.: Numerical optimization. In: Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer, New York (2006)

  24. París, J., Navarrina, F., Colominas, I., Casteleiro, M.: Topology optimization of continuum structures with local and global stress constraints. Struct. Multidiscip. Optim. 39(4), 419–437 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Popov, E., Balan, T.: Engineering Mechanics of Solids. Prentice Hall, Upper Saddle River (1998)

    Google Scholar 

  26. Rozvany, G.: On design-dependent constraints and singular topologies. Struct. Multidiscip. Optim. 21(2), 164–172 (2001)

    Article  MathSciNet  Google Scholar 

  27. Stolpe, M.: Global optimization of minimum weight truss topology problems with stress, displacement, and local buckling constraints using branch-and-bound. Int. J. Numer. Methods Eng. 61(8), 1270–1309 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Stolpe, M., Svanberg, K.: On the trajectories of the epsilon-relaxation approach for stress-constrained truss topology optimization. Struct. Multidiscip. Optim. 21(2), 140–151 (2001)

    Article  Google Scholar 

  29. Stolpe, M., Svanberg, K.: A note on stress-constrained truss topology optimization. Struct. Multidiscip. Optim. 25(1), 62–64 (2003)

    Article  MATH  Google Scholar 

  30. Sturm, J.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(12), 625–653 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wright, S.J., Tenny, M.J.: A feasible trust-region sequential quadratic programming algorithm. SIAM J. Optim. 14(4), 1074–1105 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhu, Z., Jian, J.: An efficient feasible SQP algorithm for inequality constrained optimization. Nonlinear Anal. Real World Appl. 10(2), 1220–1228 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Instituto de Estructuras y Transporte, Facultad de Ingeniería, Universidad de la República, Julio Herrera y Reissig 565, Montevideo, Uruguay

    Alfredo Canelas

  2. Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Mons. Álvaro del Portillo 12455, Las Condes, Santiago, Chile

    Miguel Carrasco

  3. Facultad de Ingeniería, Universidad Diego Portales, Ejército 441, Santiago, Chile

    Julio López

Authors
  1. Alfredo Canelas
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  2. Miguel Carrasco
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  3. Julio López
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Corresponding author

Correspondence to Alfredo Canelas.

Additional information

The first author was supported by the Uruguayan Councils ANII and CSIC. The second and third authors were supported by CONICYT-Chile, via FONDECYT projects 1130905 and 1160894, respectively.

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Canelas, A., Carrasco, M. & López, J. Application of the sequential parametric convex approximation method to the design of robust trusses. J Glob Optim 68, 169–187 (2017). https://doi.org/10.1007/s10898-016-0460-2

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  • DOI: https://doi.org/10.1007/s10898-016-0460-2

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