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An Accurate Method for First and Second Derivatives of Dynamic Responses

Published online by Cambridge University Press:  31 August 2011

Q. Liu ,
J. Zhang ,
L. Gu  and
L. Yan
Q. Liu*
Affiliation:
Department of Civil Engineering, Guangxi University of Technology, Liuzhou 545006, PR ChinaDepartment of Mechanical Engineering, University of Nebraska Lincoln, NE 68588, U.S.A.College of Civil and Architecture Engineering, Guangxi University, Nanning 530004, PR China
J. Zhang
Affiliation:
Department of Mechanical Engineering, Indiana University – Purdue UniversityIndianapolis (IUPUI), Indianapolis, IN 46202, U.S.A.
L. Gu
Affiliation:
Department of Mechanical Engineering, University of Nebraska Lincoln, NE 68588, U.S.A.
L. Yan
Affiliation:
College of Civil and Architecture Engineering, Guangxi University, Nanning 530004, PR China
*
*Associate Professor, corresponding author
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Abstract

This paper has developed an accurate method for calculating the first and second derivatives of dynamic responses with respect to the design variables of structures subjected to dynamic loads. An efficient algorithm to calculate the dynamic responses, their first and second derivatives with respect to the design variables is formulated based on the Newmark-β method. The algorithm is achieved by direct differentiation and only a single dynamics analysis is required. An example is demonstrated with the new method proposed in this paper and the analytical method. The comparative numerical results show the new method is highly accurate compared to the analytical method.

Keywords

First derivative of dynamic responseSensitivity analysisSecond derivative of dynamic responseHessian matrix
Type
Articles
Information
Journal of Mechanics , Volume 27 , Issue 3 , September 2011 , pp. 389 - 398
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2011

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References

REFERENCES

1. Keulen, F. van., Haftka, R. T. and Kim, N. H., “Review of Options for Structural Design Sensitivity Analysis. Part 1: Linear Systems,” Computer Methods in Applied Mechanics and Engineering, 194, pp. 3213–3243 (2005).CrossRefGoogle Scholar
2. Hsieh, C. C. and Arora, J. S., “Design Sensitivity Analysis and Optimization of Dynamic Response,” Computer Methods in Applied Mechanics and Engineering, 43, pp. 195–219 (1984).Google Scholar
3. Bogomolni, M., Kirsch, U. and Sheinman, I., “Efficient Design Sensitivities of Structures Subjected to Dynamic Loading,” International Journal of Solids and Structures, 43, pp. 5485–5500 (2006).Google Scholar
4. Kirsch, U., Bogomolni, M. and Keulen, F. van., “Efficient Finite-Difference Design-Sensitivities,” AIAA Journal, 43, pp. 399–405 (2005).CrossRefGoogle Scholar
5. Adelman, H. M. and Haftka, R. T., “Sensitivity Analysis of Discrete Structural Systems,” AIAA Journal, 24, pp. 823–832 (1986).CrossRefGoogle Scholar
6. Haftka, R. T. and Adelman, H. M., “Recent Developments in Structural Sensitivity Analysis,” Structure Multidiscipline Optimization, 1, pp. 137–151 (1989).CrossRefGoogle Scholar
7. Hsieh, C. C. and Arora, J. S., “An Efficient Method for Dynamic Response Optimization,” AIAA Journal, 23, pp. 1484–1486 (1985).CrossRefGoogle Scholar
8. Hsieh, C. C. and Arora, J. S., “A Hybrid Formulation for Treatment of Point-Wise State Variable Constraints in Dynamic Response Optimization,” Computer Methods in Applied Mechanics and Engineering, 48, pp. 171–189 (1985).Google Scholar
9. Hsieh, C. C. and Arora, J. S., “Structural Design Sensitivity Analysis with General Boundary Conditions-Dynamic Problem,” International Journal of Numerical Methods in Engineering, 21, pp. 267–283 (1985).CrossRefGoogle Scholar
10. Sousa, L. G., Cardoso, J. B. and Valido, A. J., “Optimal Cross-Section and Configuration Design of Elastic-Plastic Structures Subjected to Dynamic Cyclic Loading,” Structure Optimization, 13, pp. 112–118 (1997).CrossRefGoogle Scholar
11. Feng, T. T., Arora, J. S. and Haug, E. J., “Optimal Structural Design Under Dynamic Loads,” International Journal of Numerical Methods in Engineering, 11, pp. 39–52 (1977).CrossRefGoogle Scholar
12. Wang, S. and Choi, K. K., “Continuum Design Sensitivity of Transient Response Using Ritz and Mode Acceleration Method,” AIAA Journal, 30, pp. 1099–1109 (1992).Google Scholar
13. Dutta, A. and Ramakrishnan, C. V., “Accurate Computation of Design Derivatives for Plates and Shells Under Transient Dynamic Loads,” Engineering and Computers, 17, pp. 7–27 (2000).CrossRefGoogle Scholar
14. Yamakawa, H., “Optimum Structural Designs for Dynamic Response. In: E. Atrek et al. New Directions in Optimum Structural Design,” Wiley pp. 249–266, New York (1984).Google Scholar
15. Greene, W. H. and Haftka, R. T., “Computational Aspects of Sensitivity Calculations in Linear Transient Structural Analysis,” Structure Optimization, 3, pp. 176–201 (1991).Google Scholar
16. Paul, A. C., Dutta, A. and Ramakrishnan, C. V., “Accurate Computation of Design Sensitivities for Dynamically Loaded Structures with Displacement Constraints,” AIAA Journal, 34, pp. 1670–1677 (1996).Google Scholar
17. Arora, J. S. and Cardoso, J. B., “A design sensitivity analysis principle and its implementation into ADINA,” Computers and Structres, 32, pp. 691–705 (1989).Google Scholar
18. Arora, J. S. and Cardoso, J. B., “Variational Principle for Shape Design Sensitivity Analysis,” AIAA Journal, 30, pp. 538–547 (1992).Google Scholar
19. Arora, J. S. and Haug, E. J., “Methods of Design Sensitivity Analysis in Structural Optimization,” AIAA Journal, 17, pp. 970–974 (1979).CrossRefGoogle Scholar
20. Tortorelli, D. A., Lu, S. C. Y. and Haber, R. B., “Design Sensitivity Analysis for Elastodynamic Systems,” Mechanics of Structures and Machines, 18, pp. 77–106 (1990).CrossRefGoogle Scholar
21. Dutta, A. and Ramakrishnan, C. V., “Accurate Computation of Design Sensitivities for Structures Under Transient Dynamic Loads Using Time Marching Scheme,” International Journal of Numerical Methods in Engineering, 41, pp. 977–999 (1998).Google Scholar
22. Dutta, A. and Ramakrishnan, C. V., “Accurate Computation of Design Sensitivities for Structures Under Transient Dynamic Loads with Constraints on Stress,” Computers and Structures, 66, pp. 463–472 (1998).CrossRefGoogle Scholar
23. Hsieh, C. C. and Arora, J. S., “Algorithms for Point-Wise State Variable Constraints in Structural Optimization,” Computers and Structures, 22, pp. 225–238 (1986).CrossRefGoogle Scholar
24. Hsieh, C. C. and Arora, J. S., “Design Sensitivity Analysis and Optimization of Dynamic Response,” Computer Methods in Applied Mechanics and Engineering, 43, pp. 195–219 (1984).Google Scholar
25. Kocer, F. Y. and Arora, J. S., “Optimal Design of Latticed Towers Subjected to Earthquake Loading,” Journal of Structure Engineering, 128, pp. 197–204 (2002).CrossRefGoogle Scholar
26. Chahande, A. I. and Arora, J. S., “Development of a Multiplier Method for Dynamic Response Optimization Problem,” Structural Optimization, 6, pp. 69–78 (1993).CrossRefGoogle Scholar
27. Cassis, J. H. and Schmit, L. A., “Optimum Structural Design with Dynamic Constraints,” Proceedings, Journal of Structural Engineering, ASCE, 102, pp. 2053–2071 (1976).Google Scholar
28. Reklaitis, G., Ravindran, V. A. and Ragsdell, K. M., “Engineering Optimization Methods and Applications,” John Wiley & Sons, New York (1983).Google Scholar
29. Qimao, Liu, Jing, Zhang and Liubin, Yan, “An Accurate Numerical Method of Calculating First and Second Derivatives of Dynamic Response Based on the Precise Time Step Integration Method [J],” European Journal of Mechanics—A/Solids, 29, pp. 370–377 (2010).Google Scholar