Skip to main content
SpringerLink
Log in

Modelling and Approximation Strategies in Optimization — Global and Mid-Range Approximations, Response Surface Methods, Genetic Programming, Low / High Fidelity Models

  • Conference paper
Emerging Methods for Multidisciplinary Optimization

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 425))

Abstract

In this chapter global and mid-range approximations of the objective and constraint functions are introduced, discussed and illustrated by examples of real-life applications. Particular attention is paid to the development of techniques applicable to difficult design optimization problems in which the objective and constraint functions are computationally expensive, can be affected by numerical noise and at some combinations of design variables could be impossible to evaluate Genetic programming methodology is introduced as a systematic way of selecting a structure of high quality global approximations. Mechanistic approximations and techniques based on the interaction of high and low fidelity numerical models are discussed and illustrated by examples.

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

Access this chapter

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Alvarez, L.F., Hughes, D.C., and Toropov, V.V. (2000). Optimization of the manufacturing process for Roman cement. In Sienz, J., ed., Engineering Design Optimization. Process and Product Improvement. Proceedings of 2nd ASMO UK/ISSMO Conference, 27–32.

    Google Scholar 

  • Ashill, P.R., Wood, R.F., and Weeks, D.J. (1987) An improved, semi-inverse version of the Viscous, Garabedian and Korn Method (VGK), RAE Technical Report 87002.

    Google Scholar 

  • Audze, P., and Eglais, V. (1977). New approach for planing out of experiments. Problems of Dynamics and Strengths 35: 104–107 (in Russian).

    Google Scholar 

  • Barthelemy, J.-F.M., and Haftka R.T. (1993). Approximation concepts for optimum structural design–a review. Structural Optimization 5: 129–144.

    Article  Google Scholar 

  • Booker, A.J. (1998). Design and analysis of computer experiments. AIAA-98–4757. Proceedings of 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, AIAA. Part 1: 118–128.

    Google Scholar 

  • Booker, A.J., Dennis, J.E., Jr., Frank, P.D., Serafini, D.B., and Torzson, V. (1999). A rigorous framework for optimization of expensive functions by surrogates. Structural Optimization 17: 1–13.

    Article  Google Scholar 

  • Box, G.E.P., and Draper, N.R. (1987). Empirical model-building and response surfaces. New York: John Wiley and Sons.

    MATH  Google Scholar 

  • Burgee, S., Giunta, A.A., Balabanov, V., Grossman, B., Mason, W.H., Narducci, R., Haftka, R.T., and Watson, L.T. (1996). A coarse-grained parallel variable-complexity multidisciplinary optimization paradigm. International Journal of Supercomputer Applications and High Performance Computing 4: 269–299.

    Article  Google Scholar 

  • Chung, H.-S. and Alonso, J.J. (2000). Comparison of approximation models with merit functions for design optimization. AIAA 2000–4754. Proceedings of 8th AIAA/USAF/NASA/ ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA: AIAA.

    Google Scholar 

  • Etman, L.F.P., Adriaens, J.M.T.A., van Slagmaat, M.T.P., and Schoofs, A.J.G. (1996). Crash worthiness design optimization using multipoint sequential linear programming. Structural Optimization 12: 222–228.

    Article  Google Scholar 

  • Free, J.W., Parkinson, A.R., Bryce, G.R., and Balling, R.J. (1987). Approximations of computationally expensive and noisy functions for constrained nonlinear optimization. Journal of Mechanisms, Transmissions, and Automation in Design 109: 528–532.

    Article  Google Scholar 

  • Giunta, A.A., Dudley, J.M., Narducci, R., Grossman, B., Haftka, R.T., Mason, W.H., and Watson, L.T. (1994). Noisy aerodynamic response and smooth approximations in HSCT design. AIAA-94–4376-CP. Proceedings of 5th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary and Structural Optimization, Panama City, Florida: AIAA. 1117–1128.

    Google Scholar 

  • Giunta, A.A., Balabanov, V., Haim, D., Grossman, B., Mason, W.H., Watson, L.T., and Haftka, R.T. (1997). Multidisciplinary optimization of a supersonic transport using design of experiments theory and response surface modelling. Aeronautical Journal 101: 347–356.

    Google Scholar 

  • Gray, G., Li, Y., Murray-Smith, D., and Sharman, K. (1996). Structural system identification using genetic programming and a block diagram oriented simulation tool. Electronics Letters 32: 1422–1424.

    Article  Google Scholar 

  • Gray, G., Murray-Smith, D., Sharman, K., and Li, Y. (1996). Nonlinear model structure identification using genetic programming. In Late-breaking papers of Genetic Programming ‘86, Stanford, CA.

    Google Scholar 

  • Haftka, R.T. (1997). Optimization and experiments — a survey. In Tatsumi, T., Watanabe, E., and Kambe, T., eds., Theoretical and Applied Mechanics 1996, Proceedings of XIX International Congress of Theoretical and Applied Mechanics, Elsevier. 303–321.

    Google Scholar 

  • Jin, R., Chen, W. and Simpson, T.W. (2000). Comparative studies of metamodeling techniques under multiple modeling criteria. AIAA 2000–4801. Proceedings of 8th AIAA/USAF/NASA/ ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA: AIAA.

    Google Scholar 

  • van Keulen, F., Toropov, V.V., and Markine, V.L. (1996). Recent refinements in the multi-point approximation method in conjunction with adaptive mesh refinement. In McCarthy, J.M., ed., Proceedings of ASME Design Engineering Technical Conferences and Computers in Engineering Conference, Irvine, CA: 96-DETC/DAC-1451. 1–12.

    Google Scholar 

  • van Keulen, F., Polynkine, A.A., and Toropov, V.V. (1997). Shape optimization with adaptive mesh refinement: Target error selection strategies. Engineering Optimization 28: 95–125.

    Article  Google Scholar 

  • van Keulen, F., and Toropov, V.V. (1997). New developments in structural optimization using adaptive mesh refinement and multi-point approximations. Engineering Optimization 29: 217–234.

    Article  Google Scholar 

  • van Keulen, F., and Toropov, V.V. (1998). Multipoint approximations for structural optimization problems with noisy response functions. Proceedings of 1st ISSMO/NASA Internet Conference on Approximations and Fast Reanalysis in Engineering Optimization. Published on a CD ROM by ISSMO/NASA/AIAA.

    Google Scholar 

  • van Keulen, F., and Toropov, V.V. (1999). The multipoint approximation method in a parallel computing environment. Zeitschrift fuer Angewandte Mathematik and Mechanik 79: 567–570.

    Google Scholar 

  • van Keulen, F., Haftka, R.T., and Qu, X.-Y. (2000). Noise and discontinuity issues in response surfaces based on functions and derivatives. AIAA-00–1363. Proceedings of 41 st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference. Kinnear, K.E. (1994). Advances in genetic programming, MIT Press.

    Google Scholar 

  • Koza, J.R. (1992). Genetic Programming: on the programming of computers by means of natural selection MIT Press.

    Google Scholar 

  • Lewis, R.M. (1998). Using sensitivity information in the construction of kriging models for design optimization. AIAA-98–4799. Proceedings of 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis: AIAA. Part 2: 730–737.

    Google Scholar 

  • Madsen, J.I., Shyy, W., and Haftka, R.T. (2000). Response surface techniques for diffuser shape optimization. AIAA Journal 38: 1512–1518.

    Article  ADS  Google Scholar 

  • Madsen, K., and Hegelund, P. (1991). Non-gradient subroutines for non-linear optimization. Institute for Numerical Analysis, Technical University of Denmark, Report NI-91–05.

    Google Scholar 

  • McKay, B., Willis, M.J., Hidden, H.G., Montague, G.A., and Barton, G.W. (1996). Identification of industrial processes using genetic programming. In Friswell, M.I., and Mottershead, J.E., eds., Proceedings of International Conference on Identification in Engineering Systems, Swansea, The Cromwell Press Ltd. 328–337.

    Google Scholar 

  • Markine, V.L., Meijers, P., Meijaard, J.P., and Toropov, V.V. (1996). Optimization of the dynamic response of linear mechanical systems using a multipoint approximation technique. In Bestie, D., and Schiehlen, W., eds., Proceedings of IUTAM Symposium on Optimization of Mechanical Systems, Stuttgart, Kluwer. 189–196.

    Chapter  Google Scholar 

  • Markine, V.L., Meijers, P., Meijaard, J.P., and Toropov, V.V. (1996). Multilevel optimization of dynamic behaviour of a linear mechanical system with multipoint approximation. Engineering Optimization 25: 295–307.

    Article  Google Scholar 

  • Markine, V.L., Meijers, P., Meijaard, J.P., and Toropov, V.V. (1998). Optimization of mechanisms using direct differentiation and a multipoint approximation method. Engineering Optimization 31: 141–160.

    Article  Google Scholar 

  • Markine, V.L., Esveld, C., Kok, A.W.M., de Man, A.P., and Toropov, V.V. (1998). Optimization of a high-speed railway track using multipoint approximation method. AIAA-98–4980. Proceedings of 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis: AIAA. Part 2: 1328–1331.

    Google Scholar 

  • Markine, V.L., Meijers, P., and Toropov, V.V. (1998). Optimization of workspace of a manipulator for a flight simulator using multipoint approximation method. AIAA-98–4955.

    Google Scholar 

  • Proceedings of 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis: AIAA. Part 2: 1922–1932.

    Google Scholar 

  • Markine V.L. (1999). Optimization of the dynamic behaviour of mechanical systems. PhD thesis, Delft University of Technology, The Netherlands. Shaker Publishing B. V.

    Google Scholar 

  • Markine, V.L., Toropov, V.V., and Esveld, C. (2000). Optimization of ballastless railway track using multipoint approximations: issue of domain-dependent calculability. AIAA-00–4714. Proceedings of 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA. Published on a CD ROM by AIAA.

    Google Scholar 

  • Matherton, G. (1963). Principles of geostatistics. Economic Geology 58: 1246–1266.

    Article  Google Scholar 

  • Narducci, R., Grossman, B., Valorani, M., Dadone, A., and Haftka, R.T. (1995). Optimization methods for non-smooth or noisy objective functions in fluid design problems. AIAA 95–1648 CP. Proceedings of 12th AIAA Computational Fluid Dynamics Conference.

    Google Scholar 

  • Papalambros, P.Y. and Wilde, D.J. (2000). Principles of optimal design. Modeling and computation. 2“ ed. Cambridge University Press.

    Google Scholar 

  • Polynkine, A.A., van Keulen, F., and Toropov, V.V. (1995). Optimization of geometrically nonlinear thin-walled structures using the multipoint approximation method. Structural Optimization 9: 105–116.

    Article  Google Scholar 

  • Polynkine, A.A., van Keulen, F., and Toropov, V.V. (1996). Optimization of geometrically non-linear structures based on a multi-point approximation method and adaptivity. Engineering Computations 13: 76–97.

    Article  MATH  Google Scholar 

  • Polynkine, A.A., van Keulen, F., de Boer, H., Bergsma, O.K., and Beukers, A. (1996). Shape optimization of thermoformed continuous fibre reinforced thermoplastic products. Structural Optimization 11: 228–234.

    Article  Google Scholar 

  • Rasmussen, J. (1990). Accumulated approximations–a new method for structural optimization by iterative improvements. Proceedings of 3rd USAF/NASA Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, San Francisco, CA. 253–258.

    Google Scholar 

  • Rikards, R. (1993). Elaboration of optimal design models for objects from data of experiments“. In Pedersen, P., ed., Optimal design with advanced materials, The Frithiof Niordson volume. Proceedings of the IUTAM Symposium, Lyngby, Denmark, Elsevier. 149–162.

    Google Scholar 

  • Roux, W.J., Stander, N., and Haftka, R.T. (1998). Response surface approximations for structural optimization, International Journal for Numerical Methods in Engineering 42: 517–534.

    Article  MATH  ADS  Google Scholar 

  • Schoofs, A.J.G. (1987). Experimental design and structural optimization: Ph.D. thesis, Eindhoven, The Netherlands.

    Google Scholar 

  • Seyant, N.E., Bloor, M.I.G., and Wilson, M.J. (2000). Aerodynamic design of a flying wing using response surface methodology. Journal of Aircraft 37: 562–569.

    Article  Google Scholar 

  • Sobieszanski-Sobieski, J., and Haftka, R.T. (1997). Multidisciplinary aerospace design optimization: Survey of recent developments. Structural Optimization 14: 1–23.

    Article  Google Scholar 

  • Toropov V.V. (1989). Simulation approach to structural optimization. Structural Optimization 1: 37–46.

    Article  Google Scholar 

  • Toropov, V.V., Filatov, A.A., and Polynkin, A.A. (1993). Multiparameter structural optimiza- tion using FEM and multipoint explicit approximations. Structural Optimization 6: 7–14.

    Article  Google Scholar 

  • Toropov, V.V., and van der Giessen, E. (1993). Parameter identification for nonlinear constitutive models: Finite element simulation–optimization–nontrivial experiments. In Pedersen, P., ed., Optimal design with advanced materials, The Frithiof Niordson volume. Proceedings of the IUTAMSymposium, Lyngby, Denmark, Elsevier. 113–130.

    Google Scholar 

  • Toropov, V.V., and Carlsen, H. (1994). Optimization of Stirling engine performance based on multipoint approximation technique. In Gilmore, B.J. et al., eds., Advances in Design Automation 1994, Robust Design Applications, Decomposition and Design Optimization, Optimization Tools and Applications. 20th Design Automation Conference, Minneapolis. Vol. 2: 531–536.

    Google Scholar 

  • Toropov, V.V., Van Keulen, F., and Markine, V.L. (1995). Structural optimization in the presence of numerical noise. 1st World Congress of Structural and Multidisciplinary Optimization, Extended Abstracts–Posters, Goslar. 24–25.

    Google Scholar 

  • Toropov, V.V., van Keulen, F., Markine, V.L., and de Boer, H. (1996). Refinements in the multi-point approximation method to reduce the effects of noisy responses. Proceedings of 6th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue, WA: AIAA. 941–951.

    Google Scholar 

  • Toropov, V.V., and Markine, V.L. (1996). The use of simplified numerical models as midrange approximations. Proceedings of 6th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue, WA: AIAA. 952–958.

    Google Scholar 

  • Toropov, V.V., Markine, V.L., Meijers, P., and Meijaard, J.P. (1997). Optimization of a dynamic system using multipoint approximations and simplified numerical model. In Gutkowski, W., and Mroz., Z., eds., Proceedings of 2nd World Congress of Structural and Multidisciplinary Optimization, 613–618.

    Google Scholar 

  • Toropov, V.V., and Alvarez, L.F. (1998). Approximation model building for design optimization using genetic programming methodology. AIAA-98–4769. Proceedings of 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis: AIAA. Part 1: 490–498.

    Google Scholar 

  • Toropov, V.V., and Markine, V.L. (1998). Use of simplified numerical models as approximations: Application to a dynamic optimal design problem. Proceedings of 1st ISSMO/NASA Internet Conference on Approximations and Fast Reanalysis in Engineering Optimization. Published on a CD ROM by ISSMO/NASA/AIAA.

    Google Scholar 

  • Toropov, V.V., Markine, V.L., and Holden, C.M.E. (1999). Use of mid-range approximations for optimization problems with functions of domain-dependent calculability, Proceedings of 3rd World Congress of Structural and Multidisciplinary Optimization, Buffalo, NY.

    Google Scholar 

  • Valorani, M., and Dadone, A. (1995). Sensitivity derivatives for non-smooth or noisy objective functions in fluid design problems. Proceedings of ICFD Conference on Numerical Methods for Fluid Dynamics, Oxford University.

    Google Scholar 

  • Venkataraman, S., Haftka, R., and Johnson, T. (1998). Design of shell structures for buckling using correction response surface approximations. AIAA-98–4855. Proceedings of 7th AIAA/USAF/ NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis: AIAA. 1131–1144.

    Google Scholar 

  • Venter, G., Haftka, R.T., and Starnes, J.H. Jr. (1998). Construction of response surface approximations for design optimization. AIAA Journal 36: 2242–2249.

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Engineering, University of Bradford, UK

    V. V. Toropov

Authors
  1. V. V. Toropov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. The University of Liverpool, UK

    Jan Blachut

  2. University of Siegen, Germany

    Hans A. Eschenauer

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Wien

About this paper

Cite this paper

Toropov, V.V. (2001). Modelling and Approximation Strategies in Optimization — Global and Mid-Range Approximations, Response Surface Methods, Genetic Programming, Low / High Fidelity Models. In: Blachut, J., Eschenauer, H.A. (eds) Emerging Methods for Multidisciplinary Optimization. International Centre for Mechanical Sciences, vol 425. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2756-8_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-2756-8_5

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-83335-3

  • Online ISBN: 978-3-7091-2756-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation