Modelling and Approximation Strategies in Optimization — Global and Mid-Range Approximations, Response Surface Methods, Genetic Programming, Low / High Fidelity Models
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.
Keywords
Design Variable Approximation Strategy Tuning Parameter Constraint Function Pitching MomentPreview
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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.CrossRefGoogle 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.CrossRefGoogle Scholar
- Box, G.E.P., and Draper, N.R. (1987). Empirical model-building and response surfaces. New York: John Wiley and Sons.zbMATHGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefADSGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefzbMATHGoogle 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.CrossRefGoogle 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.CrossRefzbMATHADSGoogle 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.CrossRefGoogle Scholar
- Sobieszanski-Sobieski, J., and Haftka, R.T. (1997). Multidisciplinary aerospace design optimization: Survey of recent developments. Structural Optimization 14: 1–23.CrossRefGoogle Scholar
- Toropov V.V. (1989). Simulation approach to structural optimization. Structural Optimization 1: 37–46.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefADSGoogle Scholar