Second-Order Sensitivities of Inelastic Finite-Element Response by Direct Differentiation
Publication: Journal of Engineering Mechanics
Volume 134, Issue 10
Abstract
In this paper analytical equations are developed and implemented to obtain second-order derivatives of finite-element responses with respect to input parameters. The work extends previous work on first-order response sensitivity analysis. Of particular interest in this study is the computational feasibility of obtaining second-order response sensitivities. In the past, the straightforward finite difference approach has been available, but this approach suffers from serious efficiency and accuracy concerns. In this study it is demonstrated that analytical differentiation of the response algorithm and subsequent implementation on the computer provides second-order sensitivities at a significantly reduced cost. The sensitivity results are consistent with and have the same numerical precision as the ordinary response. The computational cost advantage of the direct differentiation approach increases as the problem size increases. Several novel implementation techniques are developed in this paper to optimize the computational efficiency. The derivations and implementations are demonstrated and verified with two finite-element analysis examples.
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Acknowledgments
The first writer gratefully acknowledges his University Graduate Fellowship from The Univ. of British Columbia, Vancouver. The study is also sponsored by the second writer’s Discovery Grant from the National Science and Engineering Research Council of Canada (NSERC), which is hereby gratefully acknowledged.
References
Bebamzadeh, A. (2008). “Numerical modeling of composite manufacturing: Reliability, sensitivity, and model errors.” Ph.D. thesis, Univ. of British Columbia, Vancouver B.C., Canada.
Chen, C. J., and Choi, K. K. (1994). “Continuum approach for second-order shape design sensitivity of three-dimensional elastic solids.” AIAA J., 32(10), 2099–2107.
Choi, K. K., and Santos, J. L. T. (1987). “Design sensitivity analysis of non-linear structural systems. I: Theory.” Int. J. Numer. Methods Eng., 24(11), 2039–2055.
Conte, J. P., Vijalapura, P. K., and Meghella, M. (2003). “Consistent finite-element response sensitivity analysis.” J. Eng. Mech., 129(12), 1380–1393.
Ditlevsen, O., and Madsen, H. O. (1996). Structural reliability methods, Wiley, Chichester, N.Y.
Haukaas, T., and Der Kiureghian, A. (2005). “Parameter sensitivity and importance measures in nonlinear finite element reliability analysis.” J. Eng. Mech., 131(10), 1013–1026.
Hwang, H. Y., Choi, K. K., and Chang, K. H. (1997). “Second-order shape design sensitivity using P-version finite element analysis.” Struct. Optim., 14(2–3), 91–99.
Kita, E., Kataoka, Y., and Kamiya, N. (1997). “Application of element-free trefftz method to second-order design sensitivity analysis of two-dimensional elastic problem.” JSME Int. J., Ser. A, 40(4), 375–381.
Kleiber, M., Antunez, H., Hien, T., and Kowalczyk, P. (1997). Parameter sensitivity in nonlinear mechanics, Wiley, West Sussex, U.K.
Lee, B. W., and Lim, O. K. (1998). “Application of stochastic finite element method to optimal design of structures.” Comput. Struct., 68(5), 491–497.
Liu, P. L., and Der Kiureghian, A. (1991). “Optimization algorithms for structural reliability.” Comput. Struct., 9(3), 161–178.
Luenberger, D. G. (1984). Linear and nonlinear programming, 2nd Ed., Addison-Wesley, Reading, Mass.
Ozaki, I., Kimura, F., and Berz, M. (1995). “Higher-order sensitivity analysis of finite element method by automatic differentiation.” Comput. Mech., 16(4), 223–234.
Park, S., Kapania, R. K., and Kim, S. J. (1999). “Nonlinear transient response and second-order sensitivity using time finite element method.” AIAA J., 37(5), 613–622.
Polak, E. (1997). Optimization: Algorithms and consistent approximations, Applied Mathematical Sciences Series, Vol. 124, Springer, New York.
Roth, C., and Grigoriu, M. (2001). “Sensitivity analysis of dynamic systems subjected to seismic loads.” Rep. No MCEER-01-0003, Multidisciplinary Center for Earthquake Engineering Research, State Univ. of New York, Buffalo, N.Y.
Scott, M. H., Franchin, P., Fenves, G. L., and Filippou, F. C. (2003). “Response sensitivity for nonlinear beam-column elements.” J. Struct. Eng., 130(9), 1281–1288.
Simo, J. C., and Hughes, T. J. R. (1998). “Computational inelasticity.” Interdisciplinary applied mathematics, Springer, New York.
Tsay, J. J., and Arora, J. S. (1990). “Nonlinear structural design sensitivity analysis for path dependent problems. I: General theory.” Comput. Methods Appl. Mech. Eng., 81(2), 183–208.
Zhang, Y., and Der Kiureghian, A. (1993). “Dynamic response sensitivity of inelastic structures.” Comput. Methods Appl. Mech. Eng., 108(1–2), 23–36.
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© 2008 ASCE.
History
Received: Mar 19, 2007
Accepted: Mar 31, 2008
Published online: Oct 1, 2008
Published in print: Oct 2008
Notes
Note. Associate Editor: Jiun-Shyan Chen
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