Structural and Multidisciplinary Optimization

, Volume 37, Issue 2, pp 185–194 | Cite as

Optimization of shell buckling incorporating Karhunen-Loève-based geometrical imperfections

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Abstract

The optimization of shell buckling is performed considering peak normal force and absorbed internal energy in the presence of geometrical imperfections implemented through Karhunen-Loève expansions. Initially, the mass of a shell is minimized in the presence of random initial imperfections by allowing cutouts in the material, subject to constraints on the average peak force and average internal energy. Then, robustness is considered by minimizing the coefficient of variation of the normal peak force while constraining the average peak force and average internal energy. LS-OPT® is used both to generate an experimental design and to perform a Monte Carlo simulation (96 runs) using LS-DYNA® at each of the experimental design points. The effect of imperfections when minimizing the mass is not large, but when considering robustness, however, the optimal design has a substantially increased hole size and increased shell thickness, resulting in a heavier design with maximal robustness within the constraints.

Keywords

Karhunen-Loève expansions Geometrical imperfections Buckling optimization 

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References

  1. Arbocz J (2000) The effect of imperfect boundary conditions on the collapse behavior of anisotropic shells. Int J Solids Struct 37:6891–6915MATHCrossRefMathSciNetGoogle Scholar
  2. Arbocz J, Abramovich H (1979) The initial imperfection databank at the Delft University of Technology. Part 1. Technical Report LR-290, Delft University of Technology, Department of Aerospace EngineeringGoogle Scholar
  3. Arbocz J, Starnes JH (2002) Future directions and challenges in shell stability analysis. Thin Wall Struct 40:729–754CrossRefGoogle Scholar
  4. Bushnell D (1985) Computerized Buckling Analysis of Shells. Marthinus Nijhoff, DordrechtGoogle Scholar
  5. Elishakoff I (2000) Uncertain buckling: its past, present and future. Int J Solids Struct 37:6869–6889MATHCrossRefMathSciNetGoogle Scholar
  6. Ghanem RG, Spanos PD (1991) Stochastic finite elements—a spectral approach. Revised Edition, Dover, 2003. Springer, New YorkGoogle Scholar
  7. Le Mâıtre OP, Reagan MT, Najm HN, Ghanem RG, Knio OM (2002) A stochastic projection method for fluid flow. II. Random process. J Comp Phys 181:9–44MATHCrossRefGoogle Scholar
  8. Livermore Software Technology Corporation (LSTC) (2004) LS-DYNA manual version 970. Livermore, CAGoogle Scholar
  9. Livermore Software Technology Corporation (LSTC) (2006) LS-DYNA version 971. Livermore, CAGoogle Scholar
  10. Phoon KK, Huang SP, Quek ST (2002a) Simulation of second-order processes using Karhunen–Loève expansion. Comput Struct 80(12):1049–1060CrossRefMathSciNetGoogle Scholar
  11. Phoon KK, Huang SP, Quek ST (2002b) Implementation of Karhunen–Loève expansion for simulation using a wavelet-Galerkin scheme. Probabilist Eng Mech 17(3):293–303CrossRefGoogle Scholar
  12. Phoon KK, Huang HW, Quek ST (2004) Comparison between Karhunen–Loève and wavelet expansions for simulation of Gaussian processes. Comput Struct 82(13–14):985–991CrossRefGoogle Scholar
  13. Phoon KK, Huang SP, Quek ST (2005) Simulation of strongly non-Gaussian processes using Karhunen–Loève expansion. Probabilist Eng Mech 20(2):188–198CrossRefGoogle Scholar
  14. Rahman S, Xu H (2005) A meshless method for computational stochastic mechanics. Int J Comput Meth Eng Science Mech 6:41–58CrossRefGoogle Scholar
  15. Roux WJ, Craig KJ (2006) Validation of structural simulations considering stochastic process variation. Paper 06M-67, SAE Congress, Detroit, USAGoogle Scholar
  16. Roux WJ, Stander N, Günther F, Müllerschön H (2006) Stochastic analysis of highly nonlinear structures. Int J Numer Meth Eng 65:1221–1242MATHCrossRefGoogle Scholar
  17. Schenk CA, Schuëller GI (2003) Buckling analysis of cylindrical shells with random geometrical imperfections. Int J Nonlinear Mech 38:1119–1132MATHCrossRefGoogle Scholar
  18. Schenk CA, Schuëller GI (2005) Uncertainty assessment of large finite element systems. Lecture Notes in Applied and Computational Mechanics, Vol.24. Springer, Berlin, HeidelbergGoogle Scholar
  19. Stander N, Craig KJ (2002) On the robustness of a simple domain reduction scheme for simulation-based optimization. Eng Comput 19(4):431–450MATHCrossRefGoogle Scholar
  20. Stander N, Roux WJ, Eggleston TA, Craig KJ (2006) LS-OPT v3.1 User’s manual, Livermore Software Technology Corporation. Livermore, USAGoogle Scholar
  21. TrueGrid Manual (2005) XYZ Scientific Applications. Livermore, CAGoogle Scholar
  22. Turk M, Pentland A (1991) Eigenfaces for recognition. J Cognit Neurosci 3(1):71–86CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.University of PretoriaPretoriaSouth Africa
  2. 2.Livermore Software Technology CorporationLivermoreUSA

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