Skip to main content
SpringerLink
Log in
Book cover

Optimization of Large Structural Systems pp 235–256Cite as

Recent Advances in Approximation Concepts for Optimum Structural Design

  • Chapter

Part of the book series: NATO ASI Series ((NSSE,volume 231))

Abstract

This paper reviews the basic approximation concepts used in structural optimization. It also discusses some of the most recent developments in that area since the introduction of approximation concepts in the mid-seventies. The paper distinguishes between local, medium-range and global approximations; it covers functions approximations and problem approximations. It shows that, although the lack of comparative data established on reference test cases prevents an accurate assessment, there have been significant improvements. The largest number of developments have been in the areas of local function approximations and use of intermediate variable and response quantities. It appears also that some new methodologies emerge which could greatly benefit from the introduction of new computer architectures.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arora, J.S., (1976) “Survey of Structural Reanalysis Techniques,” J. Structural Division, ASCE, Vol 102, No. ST4, pp. 783–802

    Google Scholar 

  2. Barthelemy, B., Haftka, R.T., Madapur, U., and Sankaranarayanan, S., (1991), “Integrated Analysis and Design using 3D Finite Elements,” AIM J.,in press.

    Google Scholar 

  3. Barthelemy, J.-F.M., Chang, K.J., and Rogers, J.L.Jr., (1988a) “Shuttle Solid Rocket Booster Bolted Field Joint Shape Optimization,” J. Spacecraft and Rockets, Vol. 25, No. 2, Mar-Apr 1988, pp. 117–124.

    Google Scholar 

  4. Barthelemy, J.-F.M., Riley, M.F., (1988), “Improved Multilevel Optimization Approach to the Design of Complex Engineering Systems,” AIM J., Vol. 26, No. 3, pp. 353–360.

    Google Scholar 

  5. Belegundu, A.D., Rajan, S.D., and Rajgopal, J. (1990) “Exponential Approximations in Optimal Design,” work-in-progress paper at AIAA/ASME/ASCH/AHS/ASC 31st Structures, Structural Dynamics and Materials Conference, Apr. 2–4, Long Beach, CA, in NASA CP 3064, pp. 137–150.

    Google Scholar 

  6. Bennett, J.A. (1981), “Application of Linear Constraint Approximation to Frame Structures,” Proc. of International Symposium on Optimum Structural Design, Tucson, AZ, Oct. 19–22, pp. 7.9–7.15.

    Google Scholar 

  7. Box, G.E.P., and Draper, N.R., (1987), Empirical Model-Building and Response Surfaces, Wiley, New York.

    MATH  Google Scholar 

  8. Braibant, V., and Fleury, C., (1985), “An Approximation Concepts Approach to Shape Optimal Design,” Computer Methods in Applied Mechanics and Engineering, Vol. 53, pp. 119–148.

    Article  MathSciNet  MATH  Google Scholar 

  9. Brown, R.T., and Nachlas, J.A., (1985), “Structural Optimization of Laminated Conical Shells,” AIM J., Vol, 23, No., 5, pp. 781–787.

    Google Scholar 

  10. Canfield, R.A., (1990), “High-Quality Approximation of Eigenvalues in Structural Optimization,” AIM J., Vol. 28, No. 6, pp. 1116–1122.

    Google Scholar 

  11. Chan, A.S.L., and Turlea, E., (1978), “An Approximate Method for Structural Optimization,” Computers and Structures, Vol. 8, pp. 357–363.

    Article  MATH  Google Scholar 

  12. Chang, K.-J., Haftka, R.T., Giles, G.L., and Kao, P.-J., “Sensitivity Based Scaling for Correlating Structal Response from Different Analytical Models,” AIAA Paper 91–0925, Proc. of AIAA/ASME/ASCE/AHS/ASC 32nd Structures, Structural Dynamics and Materials Conference,Baltimore, MD, April 8–10.

    Google Scholar 

  13. Ding, Y., (1987), “Optimum Design of Sandwich Constructions,” Computers and Structures, Vol. 25, No. 1, pp. 51–68.

    Article  MATH  Google Scholar 

  14. Ding, Y., and Esping, B.J.D., (1986), “Optimum Design of Frames with Beams of Different Cross-Sectional Shapes,” Proc. of AIAA/ASME/ASCE/AHS 27th Structures, Structural Dynamics and Materials Conference, San Antonio, TX, May 19–21, Part I, pp. 262–275.

    Google Scholar 

  15. Duffin, R.J., Peterson, E.L., and Zener, C.M.,(1967), Geometric Programming,John Wiley.

    Google Scholar 

  16. Fadel, G.M., Riley, M.F., and Barthelemy, J.-F.M., (1990), “Two Point Exponential Approximation Method for Structural Optimization,” Structural Optimization, Vol. 2, pp. 117–124.

    Article  Google Scholar 

  17. Fleury, C., (1988), “A Convex Linearization Method using Second Order Information,” Proc. of Fourth SAS-World Conference, Oct. 17–19, Vol. 2, pp. 374–383.

    Google Scholar 

  18. Fleury, C., (1989a), “Efficient Approximation Concepts using Second Order Information,” International Journal for Numerical Methods in Engineering, Vol. 28, pp. 2041–2058.

    Article  MathSciNet  MATH  Google Scholar 

  19. Fleury, C., (1989b), “First and Second Order Convex Approximation Strategies in Structural Optimization,” Structural Optimization, Vol. 1, pp. 3–10.

    Article  Google Scholar 

  20. Fleury, C., and Braibant, V., (1986), “Structural Optimization: a New Dual Method Using Mixed Variables,” Int. J. Num. Meth. Eng., Vol. 23, pp. 409–428.

    Article  MathSciNet  MATH  Google Scholar 

  21. Fleury, C., and Sander, G., (1983), “Dual Methods for Optimizing Finite Element Flexural Systems,” Computer Methods in Applied Mechanics and Engineering, Vol. 37, pp. 249–275.

    Article  MATH  Google Scholar 

  22. Fleury, C., and Smaoui, H., (1988), “Convex Approximation Strategies in Structural Optimization,” Discretization Methods and Structural Optimization Procedures and Applications, Eschenauer, H.A., and Thierauf, G., Eds., Springer-Verlag, Berlin, pp. 118–126.

    Google Scholar 

  23. Free, J.W., Parkinson, A.R., Bryce, and G.R., Balling, R.J., (1987), “Approximation of Computationally Expensive and Noisy Functions for Constrained Nonlinear Optimization,” Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 109, pp. 528–532.

    Google Scholar 

  24. Fuchs, M.B., (1980), “ Linearized Homogeneous Constraints in Structural Design,” Int. J. Mech. Sci., Vol. 22, pp. 33–40.

    Article  MATH  Google Scholar 

  25. Fuchs, M.B., and Haj Ali, R.M., (1990), “A Family of Homogeneous Analysis Models for the Design of Scalable Structures,” Structural Optimization, Vol. 2, pp. 143–152.

    Article  Google Scholar 

  26. Haftka, R.T., (1988), “First-and Second-Order Constraint Approximations in Structural Optimization,” Comp. Mech., Vol. 3, pp. 89–104.

    Article  MATH  Google Scholar 

  27. Haftka, R.T., (1991) “Combining Local and Global Approximations,” AIAA Journal Vol. 29, no. 9, pp. 1523–1525.

    Google Scholar 

  28. Haftka, R.T., Gurdal, Z., and Kamat, M.P. (1990) Elements of Structural Optimization,2nd Ed., Kluwer Academic Publishers Group, the Netherlands.

    Google Scholar 

  29. Haftka, R.T., Nachlas, J.A., Watson, L.T., Rizzo, T., and Desai, R., (1989), ‘Two-Point Constraint Approximation in Structural Optimization,“ Comp. Meth. Appl. Mech. Eng., Vol 60, pp. 289–301.

    Article  Google Scholar 

  30. Haftka, R.T., and Shore, C.P., (1979), “Approximation Method for Combined Thermal/Structural Design”, NASA TP-1428.

    Google Scholar 

  31. Haftka, R.T., and Starnes, J. (1976), “Applications of a Quadratic Extended Interior Penalty Function for Structural Optimization,” AIM J., Vol. 14, No. 6, pp. 718–724.

    MATH  Google Scholar 

  32. Hajela, P., (1982), “Further Developments in the Controlled Growth Approach for Optimal Structural Synthesis,” Paper 82-DET-62, Proc. of ASME 1982 Design Automation Conference, Arlington, VA.

    Google Scholar 

  33. Hajela, P., (1986), “Geometric Programming Strategies in Large-Scale Structural Synthesis,” AIM Journal, Vol. 24, No. 7, pp. 1173–1178.

    MATH  Google Scholar 

  34. Hajela, P., and Berke, L. (1990), “Neurobiological Computational Models in Structural Analysis and Design,” Proc. of 31st A1AA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Apr. 2–4, Long Beach, CA, Part I, pp. 345–355.

    Google Scholar 

  35. Hajela, P., and Sobieszczanski-Sobieski, J., (1981) ‘The Controlled Growth Method–A Tool for Structural Optimization,“ Proc. of AIAA/ASME/ASCE/AHS 22nd Structures, Structural Dynamics and Materials Conference, Apr. 6–8, Atlanta, GA, Part I, pp. 206–215.

    Google Scholar 

  36. Hansen, S.K., and Vanderplaats, G.N., (1990), “Approximation Method for Configuration Optimization of Trusses,” AIM J., Vol. 28, No. 1, pp. 161–168.

    Google Scholar 

  37. Jawed, A.H., and Morris, A.J., (1984), “Approximate Higher-Order Sensitivities in Structural Design,” Engineering Optimization, Vol. 7, No. 2, pp. 121–142.

    Article  Google Scholar 

  38. Jawed, A.H., and Morris, A.J., (1985), “Higher-Order Updates for Dynamic Responses in Structural Optimization,” Computer Methods in Applied Mechanics and Engineering, Vol 49, pp. 175–201.

    Article  MATH  Google Scholar 

  39. Kirsch, U., (1984), “Approximate Behaviour Models for Optimum Structural Design,” New Directions in Optimal Structural Design, Atrek, E. et al., Eds., John Wiley and Sons, New York, pp. 365–384.

    Google Scholar 

  40. Kirsch, U., and Toledano, G. (1983), “Approximate Reanalysis for Modifications of Structural Geometry,” Computers and Structures, Vol 16, No. 1–4, pp. 269–277.

    MATH  Google Scholar 

  41. Kodiyalam, S., and Vanderplaats, G.N., (1989), “Shape Optimization of 3D Continuum Structures Via Force Approximation Technique,” AIM J., Vol. 27, No. 1, pp. 161–168.

    Google Scholar 

  42. Kreisselmeier, G., and Steinhauser, R., (1979), “Systematic Control Design by Optimizing a Vector Performance Index,” Proc. of IFAC Symposium on Computer Aided Design of Control Systems, Zurich, Switzerland, pp. 113–117.

    Google Scholar 

  43. Lawson, J.S., Batchelor, C., Parkinson, A.R., and Talbert, J., (1989) “Consideration of Variance and Bias in the Choice of a Saturated Second-Order Design for use in Engineering Optimization,” Report EDML 89–7, Engineering Design Methods Laboratory, Brigham Young University.

    Google Scholar 

  44. Lust, R.V., (1990), “Structural Optimization with Crashworthiness Constraints,” to appear in Proc. of III Air Force/NASA Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, San Francisco, CA, Sep. 24–26.

    Google Scholar 

  45. Lust, R.V., and Schmit, L.A., (1986), “Alternative Approximation Concepts for Space Frame Synthesis,” AIM J., Vol. 24, No. 10, pp. 1676–1684.

    MATH  Google Scholar 

  46. Manning, R.A., Lust, R.V., and Schmit, L.A., (1986), “Behavior Sensitivities for Control-Augmented Structures,” in Proc. of NASA-Va. Tech. Symposium on Sensitivity Analysis in Engineering, Hampton, VA, 25–26 Sep. 1986, H.M. Adelman and R.T. Haftka Compilers, NASA CP 2457, pp. 33–57.

    Google Scholar 

  47. McCullers, L.A., and Lynch, R.W., (1972) “Composite Wing Design for Aeroelastic Requirements,” in Proc. of Conference on Fibrous Composites in Flight Vehicle Design, AFFDL TR-72–130, pp. 951–972.

    Google Scholar 

  48. Mills-Curran, W.C., and Schmit, L.A.Jr., (1983), “Structural Optimization with Dynamic Behavior Constraints,” Proc. of AIAAIASMEIASCEIAHS 24th Structures, Structural Dynamics and Materials Conference, Lake Tahoe, NV, May 2–4, Part 1, pp. 369–382.

    Google Scholar 

  49. Mills-Curran, W.C., Lust, R.V., and Schmit, L.A., (1983), “Approximations Method for Space Frame Synthesis,” AIM J., Vol. 21, No. 11, pp. 1571–1580.

    MATH  Google Scholar 

  50. Miura, H., and Chargin, K.L. (1991), “New Approximation of Frequency Response for Structural Synthesis and Parameter Identification,” Proc. of Ninth International Modal Analysis Confernce and Exhibit, Florence, Italy, 15–18 Apr. 1991.

    Google Scholar 

  51. Miura, J., and Schmitt, L.A., (1978) “Second Order Approximation of Natural Frequency Constraints in Structural Synthesis,” International Journal of Numerical Methods in Engineering, Vol. 13, No. 2, pp. 337–351.

    Article  MATH  Google Scholar 

  52. Mistree, F., Hughes, O.F., Phuoc, H.B., (1981), “An Optimization Method for the Design of Large, Highly Constrained Complex Systems,” Eng. Opt., Vol. 5, pp. 179–197.

    Article  Google Scholar 

  53. Moore, G.J., and Vanderplaats, G.N., (1990), “Improved Approximations for Static Stress Constraints in Shape Optimal Design of Shell Structures,” Proc. of AIAAIASME/ASCE/AHS/ ASC 31st Structures, Structural Dynamics and Materials Conference, Apr. 2–4, Long Beach, CA., Part I, pp. 161–170.

    Google Scholar 

  54. Morris, A.J., (1972), “Structural Optimization by Geometric Programming,” Int. J. Solids and Structures, Vol. 8, pp. 847–874.

    Article  MATH  Google Scholar 

  55. Moms, Al, (1974), ‘The Optimization of Statically Indeterminated Structures by Means of Approximate Geometric Programming,“ Second Symposium on Structural Optimization, AGARD-CP-123, pp. 6.1–6.15.

    Google Scholar 

  56. Murthy, D.V., and Haftka, R.T. (1988) “Approximations to Eigenvalues of Modified General Matrices,” Computers and Structures, Vol. 29, No. 5, pp. 903–917.

    Article  MathSciNet  MATH  Google Scholar 

  57. Noor, A.K., and Lowder, H.E., (1975a), “Structural Reanalysis via a Mixed Method,” Computers and Structures, Vol. 5, pp. 9–12.

    Article  Google Scholar 

  58. Pedersen, P., (1981), ‘The Integrated Approach of FEM-SLP for Solving Problems of Optimal Design,“ Optimization of Distributed Parameters Structures, Vol. 1, pp. 757–780 Haug, E.J., and Cea, J., Eds., Stijthoff and Noordhoff, Amsterdam.

    Google Scholar 

  59. Pickett, R.M.Jr., Rubinstein, M.F., and Nelson, R.B. (1973), “Automated Structural Synthesis using a Reduced Number of Design Coordinates,” AIM J., Vol. 11, No. 4, pp. 489–494.

    Google Scholar 

  60. Prasad, B., (1983), “Explicit Constraint Approximation Forms in Structural Optimization, Part I: Analyses and Projections,” Comp. Meth. Appl. Mech. Eng., Vol. 40, pp. 1–26.

    Article  MATH  Google Scholar 

  61. Prasad, B., (1984a), “Explicit Constraint Approximation Forms In Structural Optimization. Part 2: Numerical Experiences,” Computer Methods in Applied Mechanics and Engineering, Vol. 46, pp. 15–38.

    Article  MATH  Google Scholar 

  62. Prasad, B., (1984b), “Novel Concepts for Constraint Treatments and Approximations in Efficient Structural Synthesis,” AIAA J., Vol. 22, No. 7, pp. 957–966.

    Article  MATH  Google Scholar 

  63. Pritchard, J.I., and Adelman, H.M., (1990), “Differential Equation Based Method for Accurate Approximations in Optimization,” Proc. ofAIAA/ASME/ASCEIAHS/ASC 31st Structures, Structural Dynamics and Materials Conference, Apr. 2–4, Long Beach, CA, to appear AIAA J. see also NASA TM-102639.

    Google Scholar 

  64. Rajamaran, A., and Schmit, L.A.Jr., (1981), “Basis Reduction Concepts in Large Scale Structural Synthesis,” Engineering Optimization, Vol. 5, pp. 91–104.

    Article  Google Scholar 

  65. Rasmussen, J., (1990), “Accumulated Approximations–A New Method for Structural Optimization by Iterative Improvements,” To appear in: Proc. of 111rd Air Force/NASA Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, Sep. 24–26, San Francisco, CA.

    Google Scholar 

  66. Rasmussen, J. (1991), private communication.

    Google Scholar 

  67. Renwei, X., and Peng, L., (1987), “Structural Optimization based on Second-Order Approximations of Functions and Dual Theory,” Computer Methods in Applied Mechanics and Engineering, Vol. 65, pp. 101–114.

    Article  MATH  Google Scholar 

  68. Ricketts, R.H., and Sobieszczanski-Sobieski, J., (1977) “Simplified and Refined Structural Modeling for Economical Flutter Analysis and Design,” AIAA Paper 77–421, Presented at AIAA/ASME/SAE 18th Structures, Structural Dynamics and Materials Conference, San Diego, CA.

    Google Scholar 

  69. Rommel, B.A., (1983) “The Developement of FAST-FLOW (A Program for Flutter Optimization to Satisfy Multiple Flutter Requirements),” in AGARD Conference Proceedings 354, Aeroelastic Considerations in the Preliminary Design of Aircraft, pp. 8. 1–8. 17.

    Google Scholar 

  70. Salajegheh, E., and Vanderplaats, G.N., (1986/1987), “An Efficient Approximation Method for Structural Synthesis with Reference to Space Structures,” Space Struct. J., Vol 2, pp. 165–175.

    Google Scholar 

  71. Salama, M., Ramanathan, R.K., Schmit, L.A.Jr., and Sarma, I.S. “Influence of Analysis and Design Models on Minimum Weight Design,” in Proc. of NASA Symposium on Recent Experiences in Multidisciplinary Analysis and Optimization, Apr. 24–26, 1984, Hampton, VA, NASA CP 2327, Part 1, pp. 329–342.

    Google Scholar 

  72. Schmit, L.A.Jr., and Farshi, B., (1974), “Some Approximation Concepts for Structural Synthesis,”, AIM J., Vol. 12, No. 5, pp. 692–699.

    Google Scholar 

  73. Schmit, L.A. Jr., and Miura, H., (1976), “Approximation Concepts for efficient Structural Synthesis,” NASA CR-2552.

    Google Scholar 

  74. Schoofs, A.J.G. (1987), “Experimental Design and Structural Optimization,” PhD Dissertation, Technical University of Eindhoven, 1987.

    Google Scholar 

  75. Sepulveda, A.E., Thomas, H.L., and Schmit, L.A.Jr., (1991) “Improved Transient Response Approximations for Control Augmented Structural Optimization,” Proc. of PACAM 11, Valparaiso, Chile, Jan. 2–4, 1991, pp. 611–614.

    Google Scholar 

  76. Smaoui, H., Fleury, C., and Schmit, L.A.Jr., (1988), “Advances in Dual Algorithms and Convex Approximations Methods,” Proc. of AIAA/ASME/ASCE/AHS 29th Structures, Structural Dynamics and Materials Conference, Williamsburg, VA, Apr. 18–20, Part 3, pp. 1339–1347.

    Google Scholar 

  77. Sobiesczcanski, J., and Loendorf, D., (1972) “A Mixed Optimization Method for Automated Design of Fuselage Structures,” J. Aircraft, Vol. 9, No. 12, pp. 805–811.

    Article  Google Scholar 

  78. Sobieszczanski-Sobieski, J., James, B.B., and Dovi, A.R., (1985), “Structural Optimization by Multilevel Decomposition,” AIAA J., Vol. 23, No. 11, pp. 1775–1782.

    Article  MathSciNet  MATH  Google Scholar 

  79. Starnes, J.H. Jr, and Haftka, R.T., (1979) ‘Preliminary Design of Composite Wings for Buckling, Stress, and Displacement Constraints,“ J. Aircraft, Vol. 16, pp. 564–570.

    Article  Google Scholar 

  80. - Storaasli, O.O., and Sobieszczanski-Sobieski, J. (1974), “On the Accuracy of the Taylor Approximation for Structure Resizing,” AIAA J., Vol. 12, pp. 231–233.

    Article  Google Scholar 

  81. Templeman, A.B., Winterbottom, S.K., (1974), “Structural Design Applications of Geometric Programming,” Second Structural Optimization Symposium, AGARD-CP-123, pp. 5. 1–5. 16.

    Google Scholar 

  82. Thomas, H.L., Sepulveda, A.E., and Schmit, L.A. Jr., (1990), “Improved Approximations for Dynamic Displacements using Intermediate Response Quantities,” To appear in: Proc. of lIlyd Air Force/NASA Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, Sep. 24–26, San Francisco, CA.

    Google Scholar 

  83. Thomas, H.L, Sepulveda, A.E., and Schmit, L.A. Jr., (1991), “Improved Approximations for Control Augmented Structural Synthesis,” AIM J., To appear.

    Google Scholar 

  84. Thomas, H.L., and Vanderplaats, G.N., (1991), “An Improved Approximation for Stresses Constraints in Plate Structures,” Proc. of Opti91, Boston, MA, 25–27 June 1991.

    Google Scholar 

  85. Toropov, V.V., (1989), “Simulation Approach to Structural Optimization,” Structural Optimization, Vol. 1, pp. 37–46.

    Article  Google Scholar 

  86. Vanderplaats, G.N., (1979), “Efficient Algorithm for Numerical Airfoil Optimization,” J. Aircraft, Vol. 16, No. 2, pp. 842–847.

    Article  Google Scholar 

  87. Vanderplaats G.N., and Han, S.H., (1990), “Arch Shape Optimization using Force Approximation Methods,” Structural Optimization, Vol. 2, pp. 193–201.

    Article  Google Scholar 

  88. Vanderplaats, G.N., and Kodiyalam, S., (1990), “Two-Level Approximation Method for Stress Constraints in Structural Optimization,” AIAA J., Vol. 28, No. 5, pp. 948–951.

    Google Scholar 

  89. Vanderplaats, G.N., and Salajegheh, E., (1988), “An Efficient Approximation Technique for Frequency Constraints in Frame Optimization,” International Journal for Numerical Methods in Engineering, Vol. 26, pp. 1057–1069.

    Article  MATH  Google Scholar 

  90. Vanderplaats, G.N., and Salajegheh, E. (1989), “A New Approximation Method for Stress Constraints in Structural Synthesis,” AIM J., Vol. 27, No. 3., pp. 352–358.

    Google Scholar 

  91. White, K.P.Jr., Hollowell, W.T., Gabler, H.C.III, Pilkey, W.D., (1985), “Simulation Optimization of the Crashworthiness of a Passenger Vehicle in Frontal Collision using Response Surface Methodology,” SAE Transactions, Sec. 3, pp. 3.798–3.811.

    Google Scholar 

  92. White, K.P.Jr., Gabler, H.C.III, and Pilkey, W.D., (1986), “Approximating Dynamic Response in Small Arrays using Polynomial Parameterization and Response Surface Methodology,” The Shock and Vibration Bulletin, Vol. 55, Part 3, pp. 167–173.

    Google Scholar 

  93. Woo, T.H. (1987), “Space Frame Optimization Subject to Frequency Constraints,” AIM J., Vol. 25, No. 10, pp. 1396–1404.

    Google Scholar 

  94. Yoshida, N., and Vanderplaats, G.N., (1988), “Structural Optimization using Beam Elements,” AIM J., Vol. 26, No. 4, pp. 454–462.

    MATH  Google Scholar 

  95. Thou, M., and Xhia, R.W. (1990), ‘Two-Level Approximation Concept in Structural Synthesis,“ Int. J. Num. Meth. Engr., Vol 29, pp. 1681–1699.

    Article  Google Scholar 

  96. Zienkiewicz, O.C., and Campbell, J.S., (1973), “Shape Optimization and Sequential Linear Programming,” in Optimum Structural Design, Gallagher, R.H. and Zienkiewicz, O.C., Eds., Wiley, New York.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. NASA/Langley Research Center, Hampton, VA, 23665-5225, USA

    J.-F. M. Barthelemy (Senior Aerospace Engineer, Interdisciplinary Research Office, Structural Dynamics Division)

  2. Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

    R. T. Haftka (Christopher Kraft Professor of Aerospace and Ocean Engineering)

Authors
  1. J.-F. M. Barthelemy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. T. Haftka
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Essen University, Germany

    G. I. N. Rozvany (Professor of Structural Design) (Professor of Structural Design)

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Barthelemy, JF.M., Haftka, R.T. (1993). Recent Advances in Approximation Concepts for Optimum Structural Design. In: Rozvany, G.I.N. (eds) Optimization of Large Structural Systems. NATO ASI Series, vol 231. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-9577-8_10

Download citation

  • DOI: https://doi.org/10.1007/978-94-010-9577-8_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-9579-2

  • Online ISBN: 978-94-010-9577-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation