Structural optimization

, Volume 5, Issue 3, pp 129–144 | Cite as

Approximation concepts for optimum structural design — a review

  • J. -F. M. Barthelemy
  • R. T. Haftka
Review Article

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 function 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.

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References

  1. Arora, J.S. 1976: Survey of structural reanalysis techniques.J. Struc. Div. ASCE 102, 783–802Google Scholar
  2. Barthelemy, B.; Haftka, R.T.; Madapur, U; Sankaranarayanan, S. 1991: Integrated analysis and design using 3D finite elements.AIAA J. 29, 791–797Google Scholar
  3. Barthelemy, J.-F.M.; Chang, K.J.; Rogers, J.L., Jr. 1988: Shuttle solid rocket booster bolted field joint shape optimization.J. Spacecraft and Rockets 25, 117–124Google Scholar
  4. Barthelemy, J.-F.M.; Riley, M.F. 1988: Improved multilevel optimization approach to the design of complex engineering systems.AIAA J. 26, 353–360Google Scholar
  5. Belegundu, A.D.; Rajan, S.D.; Rajgopal, J. 1990: Exponential approximations in optimal design.NASA CP 3064, 137–150Google Scholar
  6. Bennett, J.A. 1981: Application of linear constraint approximation to frame structures.Proc. Int. Symp. Optimum Structural Design (held in Tucson, AZ), pp. 7.9–7.15Google Scholar
  7. Box, G.E.P.; Draper, N.R. 1987:Empirical model-building and response surfaces. New York: John Wiley & SonsGoogle Scholar
  8. Braibant, V; Fleury, C. 1985: An approximation concepts approach to shape optimal design.Comp. Meth. Appl. Mech. Eng. 53, 119–148Google Scholar
  9. Brown, R.T.; Nachlas, J.A. 1985: Structural optimization of laminated conical shells.AIAA J. 23, 781–787Google Scholar
  10. Canfield, R.A. 1990: High-quality approximation of eigenvalue in structural optimization.AIAA J. 28, 1116–1122Google Scholar
  11. Carpenter, W.C.; Barthelemy, J.-F.M. 1992: Comparison of polynomial approximations and artificial neural nets for response surfaces in engineering optimization.Proc. AIAA/ASME/ASCE/AHS/ASC 33rd Structures, Structural Dynamics and Materials Conf. (held in Dallas, TX)Google Scholar
  12. Chan, A.S.L.; Turlea, E. 1978: An approximate method for structural optimization.Comp. & Struct. 8, 357–363Google Scholar
  13. Chang, K.J.; Haftka, R.T.; Giles, G.L.; Kao, P.-J. 1991: Sensitivity based scaling for correlating structural response from different analytical models.AIAA Paper 91-0925, Proc. AIAA/AME/ASCE/AHS/ASC 32nd Structures, Structural Dynamics and Materials Conf. (held in Baltimore, MD)Google Scholar
  14. Ding, Y. 1987: Optimum design of sandwich constructions.Comp. & Struct. 25, 51–68Google Scholar
  15. Ding, Y.; Esping, B.J.D. 1986: Optimum design of frames with beams of different cross-sectional shapes.Proc. AIAA/ASME/ASCE/AHS 27th Structures, Structural Dynamics and Materials Conf. (held in San Antonio, TX) Part I, pp. 262–275Google Scholar
  16. Duffin, R.J.; Peterson, E.L.; Zener, C.M. 1967:Geometric programming. New York: John Wiley & SonsGoogle Scholar
  17. Fadel, G.M.; Riley, M.F.; Barthelemy, J.-F.M. 1990: Two point exponential approximation method for structural optimization.Struc. Optim. 2, 117–124Google Scholar
  18. Federov, V.V. 1972:Theory of optimal experiments. New York: Academic PressGoogle Scholar
  19. Fleury, C. 1988: A convex linearization method using second order information.Proc. Fourth SAS-World Conf. 2, 374–383Google Scholar
  20. Fleury, C. 1989a: Efficient approximation concepts using second order information.Int. J. Num. Meth. Eng. 28, 2041–2058Google Scholar
  21. Fleury, C. 1989b: First and second order convex approximation strategies in structural optimization.Struc. Optim. 1, 3–10Google Scholar
  22. Fleury, C.; Braibant, V. 1986: Structural optimization. A new dual method using mixed variables.Int. J. Num. Meth. Eng. 23, 409–428Google Scholar
  23. Fleury, C.; Sander, G. 1983: Dual methods for optimizing finite element flexural systems.Comp. Meth. Appl. Mech. Eng. 37, 249–275Google Scholar
  24. Fleury, C.; Smaoui, H. 1988: Convex approximation strategies in structural optimization. In: Eschenauer, H.A.; Thierauf, G. (eds.)Discretization methods and structural optimization procedures and applications, pp. 118–126. Berlin, Heidelberg, New York: SpringerGoogle Scholar
  25. Free, J.W.; Parkinson, A.R.; Bryce, G.R.; Balling, R.J. 1987: Approximation of computationally expensive and noisy functions for constrained nonlinear optimization.J. of Mech. Trans. Auto. Des. 109, 528–532Google Scholar
  26. Fuchs, M.B. 1980: Linearized homogeneous constraints in structural design.Int. J. Mech. Sci. 22, 33–40Google Scholar
  27. Fuchs, M.B.; Haj Ali, R.M. 1990: A family of homogeneous analysis models for the design of scalable structures.Struct. Optim. 2, 143–152Google Scholar
  28. Giles, G.L. 1986: Equivalent plate analysis of aircraft wing box structures with general planform geometry.J. Aircraft 23, 859–864Google Scholar
  29. Haftka, R.T. 1991: Combining local and global approximations.AIAA J. 29, 1523–1525Google Scholar
  30. Haftka, R.T. 1988: First- and second-order constraint approximations in structural optimization.Comp. Mech. 3, 89–104Google Scholar
  31. Haftka, R.T.; Gurdal, Z. 1992:Elements of structural optimization. Dordrecht: KluwerGoogle Scholar
  32. Haftka, R.T.; Nachlas, J.A.; Watson, L.T.; Desai, R. 1989: Twopoint constraint approximation in structural optimization.Comp. Meth. Appl. Mech. Eng. 60, 289–301Google Scholar
  33. Haftka, R.T.; Shore, C.P. 1979: Approximation method for combined thermal/structural design.NASA TP-1428 Google Scholar
  34. Haftka, R.T.; Starnes, J. 1976: Applications of a quadratic extended interior penalty function for structural optimization.AIAA J. 14, 718–724Google Scholar
  35. Hajela, P. 1982: Further developments in the controlled growth approach for optimal structural synthesis.Proc. ASME 1982 Design Automatic Conf. (held in Arlington, VA)Google Scholar
  36. Hajela, P. 1986: Geometric programming strategies in large-scale structural synthesis.AIAA J. 24, 1173–1178Google Scholar
  37. Hajela, P.; Berke, L. 1990: Neurobiological computational models in structural analysis and design.Proc. 31st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conf. (held in Long Beach, CA), Part I, pp. 345–355Google Scholar
  38. Hajela, P.; Sobieszczanski-Sobieski, J. 1981: The controlled growth method — a tool for structural optimization.Proc. AIAA/ASME/ASCE/AHS 22nd Structures, Structural Dynamics and Materials Conf. (held in Atlanta, GA) Part I, 206–215Google Scholar
  39. Hansen, S.K.; Vanderplaats, G.N. 1990: Approximation method for configuration optimization of trusses.AIAA J. 28, 161–168Google Scholar
  40. Haug, E.J.; Arora, J.S. 1979:Applied optimal design. New York: John Wiley & SonsGoogle Scholar
  41. Jawed, A.H.; Morris, A.J. 1984: Approximate higher-order sensitivities in structural design.Eng. Optim. 7, 121–142Google Scholar
  42. Jawed, A.H.; Morris, A.J. 1985: Higher-order updates for dynamics responses in structural optimization.Comp. Meth. Appl. Mech. Eng. 49, 175–201Google Scholar
  43. Kegl, M.; Butinar, B.; Oblak, M. 1991: Optimization: a convex approximation with variable conservativeness.Z. angew. Math. Mech. 71, T703-T704Google Scholar
  44. Kirsch, U. 1984: Approximate behavior models for optimum structural design. In: Atrek, E.; Gallagher, R.H.; Ragsdell, K.M.; Zienkiewicz, O.C. (eds.)New directions in optimal structural design, pp. 365–384. New York: John Wiley & SonsGoogle Scholar
  45. Kirsch, U. 1991: Reduced basis approximations of structural displacements for optimal design.AIAA J. 29, 1751–1758Google Scholar
  46. Kirsch, U.; Toledano, G. 1983: Approximate reanalysis for modification of structural geometry.Comp. & Struct. 16, 269–277Google Scholar
  47. Kodiyalam, S.; Vanderplaats, G.N. 1989: Shape optimization of 3D continuum structures via force approximation technique.AIAA J. 27, 1256–1263Google Scholar
  48. Kreisselmeier, G.; Steinhauser, R. 1979: Systematic control design by optimizing a vector performance index.Proc. IFAC Symp. on Computer aided Design of Control Systems (held in Zürich, Switzerland), pp. 113–117Google Scholar
  49. Larsson, T.; Rönnqvist, M. 1993: A second-order approximation method for structural optimization.Struct. Optim. (submitted)Google Scholar
  50. Lawson, J.S.; Batchelor, C.; Parkinson, A.R.; 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 UniversityGoogle Scholar
  51. Lust, R.V. 1990: Structural optimization with crashworthiness constraints.Proc. III Air Force/NASA Symp. on Recent Advances in Multidisciplinary Analysis and Optimization (held in San Francisco, CA)Google Scholar
  52. Lust R.V.; Schmit, L.A. 1986: Alternative approximation concepts for space frame synthesis.AIAA J. 24, 1676–1684Google Scholar
  53. Manning, R.A.; Lust, R.V.; Schmit, L.A. 1986: Behavior sensitivities for control-augmented structures.Proc. NASA-Va. Tech. Symp. Sensitvities Analysis in Engineering (held in Hampton, VA).NASA CP 2457, 33–57Google Scholar
  54. McCullers, L.A.; Lynch, R.W. 1972: Composite wing design for aeroelastic requirements.Proc. Conf. on Fibrous Composite in Flight Vehicle Design. AFFDL TR-72-130, 951–972Google Scholar
  55. Mills-Curran, W.C.; Lust, R.V.; Schmit, L.A. 1983: Approximation methods for space frame synthesis.AIAA J. 21, 1571–1580Google Scholar
  56. Mills-Curran, W.C.; Schmit, L.A., Jr. 1983: Structural optimization with dynamic behavior constraints.Proc. AIAA/ASME/ASCE/AHS 24th Structures, Structural Dynamics and Materials Conf. (held in Lake Tahoe, NV), Part I, 161–170Google Scholar
  57. Miura, H.; Chargin, K.L. 1991: New approximation of frequency response for structural synthesis and parameter identification.Proc. Ninth Int. Modal Analysis Conf. and Exhibit (held in Florence, Italy)Google Scholar
  58. Miura, H.; Schmit, L.A. 1978: Second order approximation of natural frequency constraints in structural synthesis.Int. J. Num. Meth. Eng. 13, 337–351Google Scholar
  59. Mistree, F.; Hughes, O.F.; Phuoc, H.B. 1981: An optimization method for the design of large, highly constrained complex systems.Eng. Opt. 5, 179–197Google Scholar
  60. Moore, G.J.; Vanderplaats, G.N. 1990: Improved approximations for static stress constraints in shape optimal design of shell structures.Proc. AIAA/ASME/ASCE/AHS/ASC 31st Structures, Structural Dynamics and Materials Conf. (held in Long Beach, CA), Part I, 161–170Google Scholar
  61. Morris, A.J. 1972: Structural optimization by geometric programming.Int. J. Solids and Struct. 8, 847–874Google Scholar
  62. Morris, A.J. 1974: The optimization of statically indeterminate structures by means of approximate geometric programming.Second Symp. on Structural Optimization, AGARD-CP-123, 6.1–6.15Google Scholar
  63. Murthy, D.V.; Haftka, R.T. 1988: Approximations to eigenvalues of modified general matrices.Comp. & Struct. 29, 903–917Google Scholar
  64. Noor, A.K.; Lowder, H.E. 1975: Structural reanalysis via a mixed method.Comp. & Struct. 5, 9–12Google Scholar
  65. Pedersen, P. 1981: The integrated approach of FEM-SLP for solving problems of optimal design. In: Haug, E.J.; Cea, J. (eds.)Optimization of distributed parameters structures,1, 757–780. Amsterdam: Sijthoff and NoordhoffGoogle Scholar
  66. Pickett, R.M., Jr.; Rubinstein, M.F.; Nelson, R.B. 1973: Automated structural synthesis using a reduced number of design coordinates.AIAA J. 11, 489–494Google Scholar
  67. Prasad, B. 1983: Explicit constraint approximation forms in structural optimization. Part 1: analyses and projections.Comp. Meth. Appl. Mech. Eng. 40, 1–26Google Scholar
  68. Prasad, B. 1984a: Explicit constraint approximation forms in structural optimization. Part 2: numerical experiences.Comp. Meth. Appl. Mech. Eng. 46, 15–38Google Scholar
  69. Prasad, B. 1984b: Novel concepts for constraint treatments and approximations in efficient structural synthesis.AIAA J. 22, 957–966Google Scholar
  70. Pritchard, J.I.; Adelman, H.M. 1990: Differential equation based method for accurate approximations in optimization.Proc. AIAA/ASME/ASCE/AHS/ASC 31st Structures, Structural Dynamics and Materials Conf. (held in Long Beach, CA)Google Scholar
  71. Pritchard, J.I.; Adelman, H.M. 1991: Differential equation based method for accurate modal approximations.AIAA J. 29, 484–486Google Scholar
  72. Rajamaran, A.; Schmit, L.A., Jr. 1981: Basis reduction concepts in large scale structural synthesis.Eng. Optim. 5, 91–104Google Scholar
  73. Rasmussen, J. 1990: Accumulated approximations — a new method for structural optimization by iterative improvements.Preprint of IIrd Air Force/NASA Symp. on Recent Advances in Multidisciplinary Anaylsis and Optimzation (held in San Francisco, CA), pp. 253–258Google Scholar
  74. Rasmussen, J. 1991: Private communicationGoogle Scholar
  75. Reinschmidt, K.F.; Cornell, C.A.; Brotchie, J.F. 1966: Iterative design and structural optimization.92, 281–918Google Scholar
  76. Renwei, X.; Peng, L. 1987: Structural optimization based on second-order approximations of functions and dual theory.Comp. Meth. Appl. Mech. Eng. 65, 101–114Google Scholar
  77. Ricketts, R.H.; 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 Conf. (held in San Diego, CA)Google Scholar
  78. Rommel, B.A. 1983: The development of FAST-FLOW 8A program for flutter optimization to satisfy multiple flutter requirements.AGARD Conf. Proc. 345, Aeroelastic Considerations in the Preliminary Design of Aircraft, 8.1–8.17Google Scholar
  79. Sacks, J.; Welch, W.J.; Michell, T.J.; Wynn, H.P. 1989: Design and analysis of computer experiments.Statistical Sci. 4, 409–435Google Scholar
  80. Salajegheh, E.; Vanderplaats, G.N. 1986/1987: An efficient approximation method for structural synthesis with reference to space structures.Space Struct. J. 2, 165–175Google Scholar
  81. Salama, M.; Ramanathan, R.K.; Schmit, L.A., Jr.; Sarma, I.S. 1984: Influence of analysis and design models on minimum weight design.Proc. NASA Symp. on Recent Experiences in Multidiscipinary Analysis and Optim. (held in Hampton, VA),NASA CP 2327 Part 1, 329–342Google Scholar
  82. Schmit, L.A., Jr.; Farshi, B. 1974: Some approximation concepts for structural synthesis.AIAA J. 12, 692–699Google Scholar
  83. Schmit, L.A., Jr.; Miura, H. 1976: Approximation concepts for efficient structural synthesis.NASA CR-2552 Google Scholar
  84. Schoofs, A.J.G. 1987:Experimental design and structural optimization. Technical University of Eindhoven: Ph.D. DissertationGoogle Scholar
  85. Schoofs, A.J.G.; Klink, M.B.M.; van Campen, D.H. 1992: Approximation of structural optimization problems by means of designed numerical experiments.Struct. Optim. 4, 206–212Google Scholar
  86. Sepulveda, A.E.; Thomas, H.L.; Schmit, L.A., Jr. 1991: Improved transient response approximations for control augmented structural optimzation.Proc. PACAM II (presented in Valparaiso, Chile), 611–614Google Scholar
  87. Smaoui, H.; Fleury, C.; Schmit, L.A., Jr. 1988: Advances in dual algorithms and convex approximations methods.Proc. AIAA/ASME/ASCE/AHS 29th Structures, Structural Dynamics and Materials Conf. (held in Williamsburg, VA), Part 3, pp. 1339–1347Google Scholar
  88. Sobieszczcanski-Sobieski, J.; Loendorf, D. 1972: A mixed optimization method for automated design of fuselage structures.J. Aircraft. 9, 805–811Google Scholar
  89. Sobiesczcanski-Sobieski, J.; James, B.B.; Dovi, A.R. 1985: Structural optimzation by multilevel decomposition.AIAA J. 23, 1775–1782Google Scholar
  90. Starnes, J.H., Jr.; Haftka, R.T. 1979: Preliminary design of composite wings for buckling, stress and displacement constraints.J. Aircraft 16, 564–570Google Scholar
  91. Storaasli, O.O.; Sobieszczanski-Sobieski, J. 1974: On the accuracy of the Taylor approximation for structure resizing.AIAA J. 12, 231–233Google Scholar
  92. Svanberg, K. 1987: The method of moving asymptotes — a new method for structural optimization.Int. J. Num. Meth. Eng. 24, 359–373Google Scholar
  93. Svanberg, K. 1992a: The method of moving asymptotes (MMA), with some extensions. In: Rozvany, G.I.N. (ed)Optimization of large structural systems. (Proc. NATO ASI, Berchtesgarden, Germany, 1991), pp. 555–566. Dordrecht: Kluwer (to appear)Google Scholar
  94. Svanberg, K. 1992b: Some second order methods for structural optimization. In: Rozvany, G.I.N. (ed.)Optimization of large structural systems (Proc. NATO ASI, Berchtesgarden, Germany, 1991), pp. 567–578. Dordrecht: Kluwer (to appear)Google Scholar
  95. Templeman, A.B.; Winterbottom, S.K. 1974: Structural design application of geometric programming.Second Structural Optimization Symp., AGARD-CP-123, 5.1–5.16Google Scholar
  96. Thomas, H.L.; Sepulveda, A.E.; Schmit, L.A., Jr. 1991: Improved approximations for control augmented structural synthesis.AIAA J. (to appear)Google Scholar
  97. Thomas, H.L.; Sepulveda, A.E.; Schmit, L.A., Jr. 1990: Improved approximations for dynamic displacements using intermediate response quantities.Preprints of IIIrd Air Force/NASA Symp. on Recent Advances in Multidisciplinary Analysis and Optimization (held in San Francisco, CA)Google Scholar
  98. Thomas, H.L.; Vanderplaats, G.N. 1991: An improved approximation for stress constraints in plate structures.Proc. Opti91 (held in Boston, MA)Google Scholar
  99. Toropov, V.V. 1989: Simulation approach to structural optimization.Struc. Optim. 1, 37–46Google Scholar
  100. Vanderplaats, G.N. 1979: Efficient algorithm for numerical airfoil optimization.J. Aircraft 16, 842–847Google Scholar
  101. Vanderplaats, G.N.; Han, S.H. 1990: Arch shape optimization using force approximation methods.Struct. Optim. 2, 193–201Google Scholar
  102. Vanderplaats, G.N.; Kodiyalam, S. 1990: Two-level approximation method for stress constraints in structural optimization.AIAA J. 28, 948–951Google Scholar
  103. Vanderplaats, G.N.; Salajegheh, E. 1988: An efficient approximation technique for frequency constraints in frame optimization.Int. J. Num. Meth. Eng. 26, 1057–1069Google Scholar
  104. Vanderplaats, G.N.; Salajegheh, E. 1989: A new approximation method for stress constraints in structural synthesis.AIAA J. 27, 352–358Google Scholar
  105. White, K.P., Jr.; Gabler, H.C.III; Pilkey, W.D. 1986: Approximating dynamic response in small arrays using polynomial parameterizations and response surface methodolgy.The Shock and Vibration Buletin 55, 167–173Google Scholar
  106. 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, 3.798–3.811Google Scholar
  107. Woo, T.H. 1987: Space frame optimization subject to frequency constraints.AIAA J. 25, 1396–1404Google Scholar
  108. Wrenn, G.A.; Dovi, A.R. 1988: Multilevel decomposition approach to the preliminary design of a transport aircraft wing.J. Aircraft 25, 632–638Google Scholar
  109. Yoshida, N.; Vanderplaats, G.N. 1988: Structural optimization using beam elements.AIAA J. 26, 454–462Google Scholar
  110. Zhou, M. 1989: Geometrical optimization of trusses by a two-level approximation concept.Struct. Optim. 1, 235–240Google Scholar
  111. Zhou, M.; Xhia, R.W. 1990a: Two-level approximation concept in structural synthesis.Int. J. Num. Meth. Eng. 29, 1681–1699Google Scholar
  112. Zhou, M.; Xhia, R.W. 1990b: An efficient method of truss design for optimum geometry.Comp. & Struct. 35, 115–119Google Scholar
  113. Zienkiewicz, O.C.; Campbell, J.S. 1973: Shape optimization and sequential linear programming. In: Gallagher, R.H.; Zienkiewicz, O.C. (eds.)Optimum structural design. New York: John Wiley & SonsGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • J. -F. M. Barthelemy
    • 1
  • R. T. Haftka
    • 2
  1. 1.NASA Langley Research Center, MS246HamptonUSA
  2. 2.Department of Aerospace and Ocean EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgUSA

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