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Progressive Quadratic Approximation Method for Effective Constructing the Second-Order Response Surface Models in the Large Scaled System Design

대형 설계 시스템의 효율적 반응표면 근사화를 위한 점진적 이차 근사화 기법

Hong, Gyeong-Jin;Kim, Min-Su;Choe, Dong-Hun
홍경진;김민수;최동훈

  • Published : 2000.12.01

Abstract

For effective construction of second-order response surface models, an efficient quad ratic approximation method is proposed in the context of trust region model management strategy. In the proposed method, although only the linear and quadratic terms are uniquely determined using 2n+1 design points, the two-factor interaction terms are mathematically updated by normalized quasi-Newton formula. In order to show the numerical performance of the proposed approximation method, a sequential approximate optimizer is developed and solves a typical unconstrained optimization problem having 2, 6, 10, 15, 30 and 50 design variables, a gear reducer system design problem and two dynamic response optimization problems with multiple objectives, five objectives for one and two objectives for the other. Finally, their optimization results are compared with those of the CCD or the 50% over-determined D-optimal design combined with the same trust region sequential approximate optimizer. These comparisons show that the proposed method gives more efficient than others.

Keywords

Progressive Quadratic Approximation;Response Surface;Sequential Approximate Optimization;Trust Region

References

  1. Vanderplaats, G. N., 1995, DOT-Design Optimization Tools Users Manual, Vanderplaats Research & Development, Inc.
  2. 김민수, 최동훈, 1998, '근사 확장 라그랑지 승수 기법을 적용한 동적 반응 최적화,' 대한기계학회 논문집 A 권, 제 22 권, 제 7 호, pp. 1135-1147
  3. Azam and Li, W. C., 1989, 'Multi-Level Design Optimization using Global Monotonicity Analysis,' ASME, Journal of Mechanism, Transmission and Automation in Design, Vol. 111, pp. 259-263
  4. Haug, E. J. and Arora, J. S., 1979, Applied optimal Design, Wiley-Interscience, New York, pp. 341-352
  5. Osyzka, A., 1984, Multicriterion optimization in Engineering with Fortran programs, Ellis Horwood: Chichester, pp. 31-39
  6. Box, M. J. and Draper, N. R., 1971, 'Factorial Designs, the IX'XI Criterion, and Some Related Matters,' Technometrics, Vol. 13(4), pp. 731-742 https://doi.org/10.2307/1266950
  7. Corana, 1987, 'Minimizing Multimodal Functions of Continuous Variables with the Simulated Annealing Algorithm,' ACM Transaction on Mathematical Software, Vol. 13(3), pp. 262-280 https://doi.org/10.1145/29380.29864
  8. Box, G. E. and Draper, N. R., 1987, Empirical Model Building and Response Surface, John Wiley & Sons, New York
  9. Haim, D., Giunta, A. A., Holzwarth, M. M., Mason, W. H., Waston L. T. and Haftka R. T., 1999, 'Comparison of optimization Software Packages for an Aircraft Multidisciplinary Design Optimization problem,' Design Optimization: International Journal for product & Process Improvement, Vol. 1(1), pp. 9-23
  10. Celis, M. R., Dennis, J. E. and Tapia, R. A., 1985, 'A Trust Region Strategy for Nonlinear Equality Constrained Optimization,' Numerical optimization (Boggs PT, Byrd RH and Schnabel RB Eds), SIAM, Philadelpia, pp. 71-88
  11. Nelson, Il 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
  12. 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
  13. 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
  14. Fletcher, R., 1972, 'An Algorithm for Solving Linearly Constrained Optimization Problems,' Math. Prog, Vol. 2, pp. 133-165 https://doi.org/10.1007/BF01584540
  15. Carpenter, W. C., 1993, 'Effect of Design Selection on Response Surface Performance,' Contractor Report 4520, NASA, June
  16. Fletcher, R., 1987, Practical Method of Optimization, John Wiley & Sons, Chichester
  17. Alexandrov, N., 1996, 'Robustness Properties of a Trust Region Frame Work for Managing Approximizations 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
  18. Barthelemy, J. F. and Haftka, R. T., 1991, 'Recent Advances in Approximation Concepts for Optimum Structural Design,' NASA, TM, 104032
  19. 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
  20. Bloebaum, C. L., Hong, W., and Peck, A., 1994, 'Improved Move Limit Strategy for Approximate Optimization,' Proceedings of the 5th AIAA/USAF/NASA/ISSMO Symposium, AIAA 94-4337-CP, Panama City, Florida, September 7-9, pp. 843-850
  21. Unal, R., Lepsch, R. A. and McMilin, M. L., 1988, 'Response Surface Model Building and Multidisciplinary Optimization Using D-Optimal Designs,' Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, Missouri, September 2-4, AIAA-98-4759, pp. 405-411
  22. 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 https://doi.org/10.1177/1063293X9600400405