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Decomposition and Multilevel Optimization

  • Raphael T. Haftka
  • Zafer Gürdal
Part of the Solid Mechanics And Its Applications book series (SMIA, volume 11)

Abstract

The resources required for the solution of an optimization problem typically increase with the dimensionality of the problem at a rate which is more than linear. That is, if we double the number of design variables in a problem, the cost of solution will typically more than double. Large problems may also require excessive computer memory allocations. For these reasons we often seek ways of breaking a large optimization problem into a series of smaller problems.

Keywords

Design Variable Collapse Load Member Force Portal Frame Multilevel Optimization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Giles, G.L. “Procedure for Automating Aircraft Wing Structural Design,” J. of the Structural Division, ASCE, 97 (ST1), pp. 99–113, 1971.Google Scholar
  2. [2.
    Sobieszczanski, J. and Loendorf, D., “A Mixed Optimization Method for Automated Design of Fuselage Structures”, J. of Aircraft, 9(12), pp. 805–811, 1972.CrossRefGoogle Scholar
  3. [3]
    Barthelemy, J.-F.,M., “Engineering Design Applications of Multilevel Optimization Methods,” in Computer-Aided Optimum Design of Structures: Applications (eds. C.A. Brcbbia and S. Hernandez), Springer-Verlag, pp. 113–122, 1989.Google Scholar
  4. [4]
    Sobieszczanski-Sobieski, J., James, B.B., and Dovi, A.R., “Structural Optimization by Multilevel Decomposition”, AIAA J., 23, 11, pp. 1775–1782, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Thareja, R. R., and Haftka, R. T., “Efficient Single-Level Solution of Hierarchical Problems in Structural Optimization”, AIAA J., 28, 3, pp. 506–514, 1990.zbMATHCrossRefGoogle Scholar
  6. [6]
    Thareja, R., and Haftka, R.T., “Numerical Difficulties Associated with using Equality Constraints to Achieve Multilevel Decomposition in Structural Optimization,” AIAA Paper No. 86-0854CP, Proceedings of the AIAA/ASME/ASCE/ AHS 27th Structures, Structural Dynamics and Materials Conference, San Antonio, Texas, May 1986, pp. 21–28.Google Scholar
  7. [7]
    Schmit L.A., and Mehrinfar, M., “Multilevel Optimum Design of Structures with Fiber-Composite Stiffened Panel Components”, AIAA J., 20, 1, pp. 138–147, 1982.zbMATHCrossRefGoogle Scholar
  8. [8]
    Kirsch, U., “Multilevel Optimal Design of Reinforced Concrete Structures”, Engineering Optimization, 6, pp. 207–212, 1983.CrossRefGoogle Scholar
  9. [9]
    Dantzig, G.B., and Wolfe, P., “The Decomposition Algorithm for Linear Program,” Econometrica, 29, No. 4, pp. 767–778, 1961.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Dantzig, G.B., “A Decomposition Principle for Linear Programs,” in Linear Programming and Extensions, Princeton Press, 1963.Google Scholar
  11. [11]
    Rosen, J.B., “Primal Partition Programming for Block Diagonal Matrices”, Numerische Mathematik, 6, pp. 250–260, 1964.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Geoffrion, A.M., “Elements of Large-Scale Mathematical Programming”, in Perspectives on Optimization (A.M. Geoffrion, editor) Addison Wesley, pp. 25–64, 1972.Google Scholar
  13. [13]
    Kirsch, U., “An Improved Multilevel Structural Synthesis Method”, J. Structural Mechanics, 13(2), pp. 123–144, 1985.CrossRefGoogle Scholar
  14. [14]
    Barthelemy, J.-F.M., and Sobieszczanski-Sobieski, J., “Extrapolation of Optimum Designs based on Sensitivity Derivatives,” AIAA J., 21, pp. 797–799, 1983.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Haftka, R.T., “An Improved Computational Approach for Multilevel Optimum Design”, J. of Structural Mechanics, 12, 2, pp. 245–261, 1984.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Sobieszczanski-Sobieski, J., James, B. B., and Riley, M. F., “Structural Sizing by Generalized, Multilevel Optimization”, AIAA J., 25, 1, pp. 139–145, 1987.CrossRefGoogle Scholar
  17. [17]
    Fox, R. L., and Schmit, L. A., “Advances in the Integrated Approach to Structural Synthesis”, J. of Spacecraft and Rockets, 3(6), pp.858–866, 1966.CrossRefGoogle Scholar
  18. [18]
    Haftka, R.T., “Simultaneous Analysis and Design”, AIAA J., 23, 7, pp. 1099–1103, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Smaoui, H., and Schmit. L.A., “An Integrated Approach to the Synthesis of Geometrically Non-linear Structures,” International Journal for Numerical Methods in Engineering, 26, pp. 555–570, 1988.zbMATHCrossRefGoogle Scholar
  20. [20]
    Ringertz, U.T., “Optimization of Structures with Nonlinear Response,” Engineering Optimization, 14, pp. 179–188, 1989.CrossRefGoogle Scholar
  21. [21]
    Haftka, R. T., and Kamat, M. P., “Simultaneous Nonlinear Structural Analysis and Design”, Computational Mechanics, 4, 6, pp. 409–416, 1989.zbMATHCrossRefGoogle Scholar
  22. [22]
    Chibani, L., Optimum Design of Structures, Springer-Verlag, Berlin, Heidelberg, 1989.zbMATHCrossRefGoogle Scholar
  23. [23]
    Bendsøc, M.P., Ben-Tal, A., and Haftka, R.T., “New Displacement-Based Methods for Optimal Truss Topology Design,” AIAA Paper 91-1215, Proceedings, AIAA/ASME/ASCE/AHS/ASC 32nd Structures, Structural Dynamics and Materials Conference, Baltimore, MD, April 8-10, 1991, Part 1, pp. 684–696.Google Scholar
  24. [24]
    Shin, Y., Haftka, R. T., and Plaut, R. H., “Simultaneous Analysis and Design for Eigenvalue Maximization”, AIAA J., 26, 6, pp. 738–744, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Pedersen, P., “On the Minimum Mass Layout of Trusses”, AGARD Conference Proceedings, No. 36 on Symposium on Structural Optimization, Turkey, October, 1969, pp. 11.1-11.17, 1970.Google Scholar
  26. [26]
    Vanderplaats, G.N., and Moses, F., “Automated Design of Trusses for Optimum Geometry”, J. of the Structural Division, ASCE, 98, ST3, pp. 671–690, 1972.Google Scholar
  27. [27]
    Spillers, W.R., “Iterative Design for Optimal Geometry”, J. of the Structural Division, ASCE, 101, ST7, pp.1435–1442, 1975.Google Scholar
  28. [28]
    Kirsch, U., “Synthesis of Structural Geometry using Approximation Concepts”, Computers and Structures, 15, 3, pp. 305–314, 1982.zbMATHCrossRefGoogle Scholar
  29. [29]
    Ginsburg, S., and Kirsch, U., “Design of Protective Structures against Blast”, J. of the Structural Division, ASCE, 109(6), pp. 1490–1506, 1983.CrossRefGoogle Scholar
  30. [30]
    Kirsch, U., “Multilevel Synthesis of Standard Building Structures,” Engineering Optimization, 7, pp. 105–120, 1984.CrossRefGoogle Scholar
  31. [31]
    Kirsch, U., “A Bounding Procedure for Synthesis of Prestressed Systems,” Computers and Structures, 20(5), pp. 885–895, 1985.zbMATHCrossRefGoogle Scholar
  32. [32]
    Sobieszczanski-Sobieski, J., “Sensitivity of Complex, Internally Coupled Systems,” AIAA Journal, 28(1), pp. 153–160, 1990.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1992

Authors and Affiliations

  • Raphael T. Haftka
    • 1
  • Zafer Gürdal
    • 2
  1. 1.Department of Aerospace and Ocean EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  2. 2.Department of Engineering Science and MechanicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA

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