Journal of Mechanical Science and Technology

The Korean Society of Mechanical Engineers (대한기계학회)
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Efficient Approximation Method for Constructing Quadratic Response Surface Model

  • Park, Dong-Hoon (Director, Center of Innovative Design Optimization Technology, Hanyang University) ;
  • Hong, Kyung-Jin (Graduate Research Assistant, Department of Mechanical Design and Production Engineering, Hanyang University) ;
  • Kim, Min-Soo (Contract Professor of the BK21 Division for Research and Education in Mechanical Engineering, Hanyang University)
  • Published : 2001.07.01
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Abstract

For a large scaled optimization based on response surface methods, an efficient quadratic approximation method is presented in the context of the trust region model management strategy. If the number of design variables is η, the proposed method requires only 2η+1 design points for one approximation, which are a center point and tow additional axial points within a systematically adjusted trust region. These design points are used to uniquely determine the main effect terms such as the linear and quadratic regression coefficients. A quasi-Newton formula then uses these linear and quadratic coefficients to progressively update the two-factor interaction effect terms as the sequential approximate optimization progresses. In order to show the numerical performance of the proposed method, a typical unconstrained optimization problem and two dynamic response optimization problems with multiple objective are solved. Finally, their optimization results compared with those of the central composite designs (CCD) or the over-determined D-optimality criterion show that the proposed method gives more efficient results than others.

Keywords

References

  1. Alexandrov, N., 1996, 'Robustness Properties of a Trust Region Frame Work for Managing Approximations in Engineering Optimization,' Proceedings of the 6th AIAA/'NASA/USAF Multidisciplinary Analysis & Optimization Symposium, AIAA 96-4102-CP, Bellevue, Washington, September 7-9, pp. 1056-1059
  2. Bloebaum, C. L., Hong, W. and Peck, A., 1994, 'Improved Move Limit Strategy for Approximate Optimization,' Proceedings of the 5th AlAAj USAF/NASA/ISSMO Symposium, AIAA 94 -4337-CP, Panama City, Florida, September 7-9, pp. 843-850
  3. Box, G. E. and Draper, N. R., 1987, Empirical Model Building and Response Surfaces, John Wiley, New York
  4. Box, M. J. and Draper, N. R.. 1971, 'Factorial Designs, the |X'X| Criterion, and Some Related Matters,' Technometrics, Vol. 13, No. 4, pp. 731-742 https://doi.org/10.2307/1266950
  5. Carpenter, W. C, 1993, 'Effect of Design Selection on Response Surface Performance,' Contractor Report 4520, NASA, June
  6. Celis, M. R.. Dennis, J. E. and Tapia, R. A., 1985, 'A Trust Region Strategy for Nonlinear Equality Constrained Optimization,' Numerical optimization 1984 (Boggs PT, Byrd RH and Schnabel RB. Eds), SIAM, Philadelpia, pp. 71-88
  7. Corana, et al., 1987, 'Minimizing Multimodal Functions of Continuous Variables with the Simulated Annealing Algorithm,' ACM Transaction on Mathematical Software, Vol. 13, No. 3, pp. 262-280 https://doi.org/10.1145/29380.29864
  8. Dennis, J. E. and Torczon. T., 1996, 'Approximation Model Management for Optimization,' Proceedings of the 6th AIAA/NASA/USAF Multidisciplinary Analysis & Optimization Symposium, AIAA 96-4046, Bellevue, Washington, September 7-9, pp. 1044-1046
  9. Fletcher, R., 1972, 'An Algorithm for Solving Linearly Constrained Optimization Problems,' Math. Prog., Vol. 2, pp. 133-165 https://doi.org/10.1007/BF01584540
  10. Fletcher, R., 1987, Practical Method of Optimization, John Wiley & Sons: Chichester
  11. Grandhi, R. V., Haftka, R. T. and Watson, L. T., 1986, 'Design-Oriented identification of Critical Times in Transient Response,' AIAA Journal, Vol. 24, No. 4, pp. 649-656
  12. Haim, D., Giunta, A. A., Holzwarth, M. M., Mason, W. H., Watson, L. T. and Haftka, R. T., 1999, 'Comparison of optimization Software Packages for an Aircraft MultidiscipHnary Design Optimization problem,' Design Optimization: international Journal for product & Process Improvement, Vol. 1, No. 1, pp. 9-23
  13. Haug, E. J. and Arora, J. S., 1979, Applied optimal Design, Wiley-Interscience, New York, pp. 341-352
  14. Nelson, II. S. A. and Papalambros, P. Y., 1999, 'The Use of Trust Region Algorithms to Exploit Discrepancies in Function Computation Time Within Optimization Models,' ASME Journal of Mechanical Design, Vol. 121, pp. 552-556
  15. Osyzka, A., 1984, Multicriterion optimization in Engineering with Fortran programs. Ellis Horwood: Chichester, pp. 31-39
  16. Powell, M. J. D., 1975, 'Convergence Properties of a Class of Minimization Algorithms,' Nonlinear Programming 2 (Mangasarian OL, Meyer RR and Robinson SM Eds), Academic Press, New York
  17. Powell, M. J. D., 1978, 'A Fast Algorithm for Nonlinearly Constrained Optimization Calculations,' Numerical Analysis Proceedings Dundee 1977 (Watson GA. Eds), Springer-Verlag: Berlin, pp. 144-157
  18. Rodriguez, J. F., Renaud, J. E. and Watson, L. T., 1988, 'Trust Region Augmented Lagrangian methods for Sequential Response Surface Approximation and Optimization,' ASME Journal of Mechanical Design, Vol. 120, pp. 58-66
  19. Unal, R., Lepsch, R. A. and McMilin, M. L., 1988, 'Response Surface Model Building and MultidiscipHnary Optimization Using D-Optimal Designs,' Proceedings of the 7th AIAA/USAF/ NASA/ISSMO Symposium on MultidiscipHnary Analysis and Optimization, AIAA-98-4759, St. Louis, Missouri, September 2-4, pp. 405-411
  20. Wujek, B, Renaud, J. E, Batill, S. M. and Brockman, J. B., 1996, 'Concurrent Subspace Optimization Using Design Variable Sharing in a Distributed Computing Environment,' Concurrent Engineering: Research and Applications (CERA), Technomic Publishing Company Inc: December