Complex Perturbation Method for Sensitivity Analysis of Nonlinear Trusses
This article has a reply.
VIEW THE REPLYThis article has a reply.
VIEW THE REPLYPublication: Journal of Structural Engineering
Volume 143, Issue 1
Abstract
A numerical method based on complex analysis is presented for obtaining response sensitivities of three-dimensional (3D) trusses with geometrical and material nonlinearities. To this end, a complex perturbation method for sensitivity analysis is proposed that can be used in conjunction with the finite-element method. Performance of the complex perturbation method is demonstrated by evaluating response sensitivities of linear elastic, hyperelastic, and hyperelastic-plastic 3D trusses. To validate the method, the sensitivity data obtained from the complex perturbation method are compared with analytical sensitivity data and the central difference method. Advantages and limitations of complex perturbation method are discussed, and it is shown that the complex perturbation method can be used to obtain sensitivity data of analytical quality.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The presented work is supported in part by the U.S. National Science Foundation through Grant CMS-1055314. Any opinions, findings, conclusions, and recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the sponsors.
References
Beck, T. (1994). “Automatic differentiation of iterative processes.” J. Comput. Appl. Math., 50(1–3), 109–118.
Bonet, J., and Wood, R. D. (2008). Nonlinear continuum mechanics for finite element analysis, 2nd Ed., Cambridge University Press, Cambridge, U.K.
Cerviño, L. I., and Bewley, T. R. (2003). “On the extension of the complex-step derivative technique to pseudospectral algorithms.” J. Comput. Phys., 187(2), 544–549.
Conte, J., Vijalapura, P., and Meghella, M. (2003). “Consistent finite-element response sensitivity analysis.” J. Eng. Mech., 1380–1393.
CPSSL-FEA [Computer software]. Univ. of Notre Dame, Notre Dame, IN.
Griewank, A., and Walther, A. (2008). Evaluating derivatives, Society for Industrial and Applied Mathematics, Philadelphia.
Gu, Q., Conte, J. P., Elgamal, A., and Yang, Z. (2009). “Finite element response sensitivity analysis of multi-yield-surface J2 plasticity model by direct differentiation method.” Comput. Methods Appl. Mech. Eng., 198(30–32), 2272–2285.
Hoffman, J. D., and Frankel, S. (2001). Numerical methods for engineers and scientists, CRC Press, Boca Raton, FL.
Jeffrey, F., and Juan, A. (2011). “The development of hyper-dual numbers for exact second-derivative calculations.” 49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, American Institute of Aeronautics and Astronautics, Reston, VA.
Kiran, R., and Khandelwal, K. (2014a). “Complex step derivative approximation for numerical evaluation of tangent moduli.” Comput. Struct., 140(0), 1–13.
Kiran, R., and Khandelwal, K. (2014b). “Numerically approximated Cauchy integral (NACI) for implementation of constitutive models.” Finite Elem. Anal. Des., 89(0), 33–51.
Kiran, R., and Khandelwal, K. (2015). “Automatic implementation of finite strain anisotropic hyperelastic models using hyper-dual numbers.” Comput. Mech., 55(1), 229–248.
Kleiber, M., Antúnez, H., Hien, T. D., and Kowalczyk, P. (1997). Parameter sensitivity in nonlinear mechanics: Theory and finite element computations, Wiley, New York.
Kok-Lam, L., John, C., Yang, C., and Jongrae, K. (2005). “New complex-step derivative approximations with application to second-order Kalman filtering.” AIAA Guidance, Navigation, and Control Conf. and Exhibit, American Institute of Aeronautics and Astronautics, Reston, VA.
Korelc, J. (2008). “Automation of finite element method.” Nonlinear finite element methods, P. Wriggers, ed., Springer, Heidelberg, Germany, 483–508.
Korelc, J. (2009). “Automation of primal and sensitivity analysis of transient coupled problems.” Comput. Mech., 44(5), 631–649.
Lantoine, G., Russell, R. P., and Dargent, T. (2012). “Using multicomplex variables for automatic computation of high-order derivatives.” ACM Trans. Math. Softwares, 38(3), 1–21.
Lyness, J., and Moler, C. (1967). “Numerical differentiation of analytic functions.” SIAM J. Numer. Anal., 4(2), 202–210.
Martins, J. R. R. A., Sturdza, P., and Alonso, J. J. (2003). “The complex-step derivative approximation.” ACM Trans. Math. Software, 29(3), 245–262.
MATLAB version R2016a [Computer software]. MathWorks, Natick, MA.
Michaleris, P., Tortorelli, D. A., and Vidal, C. A. (1994). “Tangent operators and design sensitivity formulations for transient non-linear coupled problems with applications to elastoplasticity.” Int. J. Numer. Methods Eng., 37(14), 2471–2499.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (2007). Numerical recipes: The art of scientific computing, 3rd Ed., Cambridge University Press, New York.
Rall, L. B. (1986). “The arithmetic of differentiation.” Math. Mag., 59(5), 275–282.
Scott, M. (2007). “Analytical sensitivity of plastic rotations in beam-column elements.” J. Struct. Eng., 1341–1345.
Scott, M. (2012). “Response sensitivity of geometrically nonlinear force-based frame elements.” J. Struct. Eng., 1963–1972.
Tanaka, M., Fujikawa, M., Balzani, D., and Schröder, J. (2014). “Robust numerical calculation of tangent moduli at finite strains based on complex-step derivative approximation and its application to localization analysis.” Comput. Methods Appl. Mech. Eng., 269(0), 454–470.
van Keulen, F., Haftka, R. T., and Kim, N. H. (2005). “Review of options for structural design sensitivity analysis. Part 1: Linear systems.” Comput. Methods Appl. Mech. Eng., 194(30–33), 3213–3243.
Zhang, Y., and Der Kiureghian, A. (1993). “Dynamic response sensitivity of inelastic structures.” Comput. Methods Appl. Mech. Eng., 108(1), 23–36.
Information & Authors
Information
Published In
Journal of Structural Engineering
Volume 143 • Issue 1 • January 2017
Copyright
© 2016 American Society of Civil Engineers.
History
Received: May 21, 2015
Accepted: Jun 1, 2016
Published online: Aug 22, 2016
Published in print: Jan 1, 2017
Discussion open until: Jan 22, 2017
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.