Mathematical Proof for Structural Optimization with Equivalent Static Loads Transformed from Dynamic Loads

동하중에서 변환된 등가정하중에 의한 최적화 방법의 수학적 고찰

  • 박경진 (한양대학교 기계설계학과) ;
  • 강병수 (한양대학교 기계설계학과 대학원)
  • Published : 2003.02.01


Generally, structural optimization is carried out based on external static loads. All forces have dynamic characteristics in the real world. Mathematical optimization with dynamic loads is extremely difficult in a large-scale problem due to the behaviors in the time domain. The dynamic loads are often transformed into static loads by dynamic factors, design codes, and etc. Therefore, the optimization results can give inaccurate solutions. Recently, a systematic transformation has been proposed as an engineering algorithm. Equivalent static loads are made to generate the same displacement field as the one from dynamic loads at each time step of dynamic analysis. Thus, many load cases are used as the multiple leading conditions which are not costly to include in modern structural optimization. In this research, it is mathematically proved that the solution of the algorithm satisfies the Karush-Kuhn-Tucker necessary condition. At first, the solution of the new algorithm is mathematically obtained. Using the termination criteria, it is proved that the solution satisfies the Karush-Kuhn-Tucker necessary condition of the original dynamic response optimization problem. The application of the algorithm is discussed.


Dynamic Response Optimization;Equivalent Static Load;Karush-Kuhn-Tucker Necessary Condition;Structural Optimization


  1. Taylor, J.E., and Bendsoe, M.P., 1984, 'An Interpretation for Min-Max Structural Design Problems Including a Method for Relaxing Constraints,' International Journal of Solids and Structures, Vol. 30, No. 4, pp. 301-314
  2. Kegel, M.S. Butinar, B.J., and Oblak, M.N., 1992, 'Optimization of Mechanical Systems: On Strategy of Nonlinear First Order Approximation,' International Journal for Numerical Methods in Engineering, Vol. 33, pp. 223-234
  3. Choi, W.S., and Park, G.J., 2002, 'Quasi-Static Structural Optimization Using Equivalent Static Loads Transformed Dynamic Loads at All the Time Intervals,' Computer Methods in Applied Mechanics and Engineering, Vol. 191, pp. 2105-2122
  4. Mokhtar S. Bazaraa, Hanif D. Sherall, and Shetty, C. M., 1993, Nonlinear Programming: Theory and Algorithm, John Wiley&Sons, Inc
  5. Choi, W.S., Kang, B.S., Park, G.J., and Ryu, J.B., 1998, 'Optimization of Self-propelled Howitzer for Weight Reduction under Equivalent Static Load,' KSME Fall Conference, Vol. A, pp. 402-407
  6. Kang, B.S., Choi, W.S., and Park, G.J., 2000, 'Structural Optimization under Equivalent Static Loads Transformed from Dynamic Loads Based on Displacement,' Transactions of the KSME, A, Vol. 24, No. 8, pp. 1949-1957
  7. Choi, W.S., and Park, G.J., 1999, 'Transformation of Dynamic Load into Equivalent Static Loads Based on Modal Analysis,' International Journal for Numerical Methods in Engineering, Vol. 46, pp. 29-43<29::AID-NME661>3.0.CO;2-D
  8. Choi, W.S., 1999, 'Transformation of dynamic loads into equivalent static loads and structural optimization,' Ph.D. Dissertation, Hanyang University, Seoul, Korea
  9. Choi, W.S., Park, G.J., Shin, M.J., and Kim, D.S., 1995, 'Transformation of a Dynamic Load into an Equivalent Static Load and Shape Optimization of the Road Arm,' KSME Fall Conference, Vol. 1, pp. 609-614
  10. Choi, W.S., Kang, S.C., Shin, M.J., and Park, G.J., 1996, 'Transformation of a Dynamic Load into an Equyivalent Static Load and Shape Optimization of the Road Arm in Self-Propelled Howitzer,' Transactions of the KSME. A, Vol. 20, No. 12, pp. 3767-3781
  11. Sienkiewicz, Z., and Wilczynski, B., 1996, 'Shape Optimization of a Dynamically Loaded Machine Foundation Coupled to a Semi-Infinite Inelastic Medium,' Structural Optimization, Vol. 12, pp. 29-34
  12. Kang, B.S., Choi, W.S., and Park, G.J., 2001, 'Structural Optimization Under Equivalent Static Loads Transformed from Dynamic Loads Based on Displacement,' Computers & Structures, Vol. 79, pp. 145-154
  13. Mills-Curran, W.C., and Schmit, L.A., 1985, 'Structural Optimization with Dynamic Behavior Constraints,' AIAA Journal, Vol. 23, No. 1, pp. 132-138
  14. Chahande, A.I., and Arora, J.S., 1993, 'Development of a Multiplier for Dynamic Response Optimization Problems,' Structural Optimization, Vol. 6, pp. 69-78
  15. Kim, M.S., and Choi, D.H., 1997, 'Multibody Dynamic Response Optimization with ALM and Approximate Line Search,' Multibody System Dynamics, Kluwer Academic Publishers, Vol. 1, No. 1, pp. 47-64
  16. Greene, W.H., and Haftka, R.T., 1989, 'Computational Aspects of Sensitivity Calculations in Transient Structural Analysis,' Computers & Structures, Vol. 32, No. 2, pp. 433-443
  17. 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
  18. C.C. Hsieh and J.S. Arora, 1984, 'Design Sensitivity Analysis and Optimization of Dynamic Response, Computer Methods in Applied Mechanics and Engineering,' Vol. 43, pp. 195-219
  19. Cassis, J.H., and Schmit, L.A., 1976, 'Optimum Structural Design with Dynamic Constraints,' ASCE, Journal of Structural Division, ST10, pp. 2053-2071
  20. Rangachargulu, M.A.V., and Done, G.T.S., 1979, 'A Survey of Structural Design under Dynamic Constraints,' Shock and Vibration Digest, Vol. 11 No. 12, pp. 15-25
  21. Feng, T.T., Arora, J.S., and Haug, E.J., 1977, 'Optimal Structural Design under Dynamic Loads,' International Journal for Numerical Methods in Engineering, Vol. 11, pp. 39-52
  22. Schmit, L.A., 1981, 'Structural Synthesis-Its Genesis and Development,' AIAA Journal, Vol. 19, No. 10, pp. 1249-1263
  23. Hansen, S.R., and Vanderplaats, G.N., 1990, 'Approximation Method for Configuration Optimization of Trusses,' AIAA Journal, Vol. 28, No. 1, pp. 161-168
  24. Vanderplaats, G.N., 1982, 'Structural Optimization-Past, Present, and Future,' AIAA Journal, Vol. 20, No. 7, pp. 992-100
  25. Ashley, H., 1982, 'On Making Things the Best-Aeronautical Uses of Optimization,' Journal of Aircraft, Vol. 19, No. 21, pp. 5-28
  26. Haftka, R.T., and Gurdal, Z., 1993, Elements of Structural Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands
  27. Haug, E.J., and Arora, J.S., 1979, Applied Optimal Design, John Wiley and Sons, New York, New York,

Cited by

  1. Statistical Space-Time Metamodels Based on Multiple Responses Approach for Time-Variant Dynamic Response of Structures vol.34, pp.8, 2010,
  2. Optimization of the Television Packing System Using Equivalent Static Loads vol.39, pp.3, 2015,