Finite-Element Sensitivity for Plasticity Using Complex Variable Methods
Publication: Journal of Engineering Mechanics
Volume 141, Issue 2
Abstract
The complex variable FEM (ZFEM) has been enhanced in this research to compute derivatives with respect to shape, material properties (elastic modulus, yield stress, plastic modulus, hardening parameters), and loads for a nonlinear solid mechanics model undergoing plastic deformation. This method presents a new and novel approach that uses complex variables to estimate derivatives within an incremental-iterative procedure for the solution of nonlinear finite-element equations. ZFEM offers significant advantages over real-valued finite-element analysis in that highly accurate derivative information may be obtained from a single analysis. The method has been implemented within a commercial finite-element software package using the user element and user material options. A strategy was developed to allow the software’s solver algorithm to handle complex variable operations needed by ZFEM to perform sensitivity analysis. Numerical results confirm the high accuracy of the method through the analysis of a thick-walled cylinder case using perfectly plastic, bilinear hardening, and Ramberg-Osgood plasticity models.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This work was supported by the Clarkson Aerospace Corp. through funding under prime contract FA8650-05-D-1912 from the Air Force Research Laboratory and subprime contract 12-S567-018-02-C1. Support by the program manager Pat Golden AFRL/RXCM is gratefully acknowledged.
References
ABAQUS 6.12 [Computer software]. Providence, RI, Dassault Systèmes Simulia Corp.
Bathe, K. (1996). Finite element procedures, Prentice Hall, Englewood Cliffs, NJ.
Chen, W. (1975). Limit analysis and soil plasticity, Elsevier Scientific Publishing, Amsterdam.
Chen, W. (1982). Plasticity in reinforced concrete, McGraw Hill, New York.
Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., and Knuth, D. E. (1996). “On the LambertW function.” Adv. Comput. Math., 5(1), 329–359.
Corless, R. M., and Jeffrey, D. J. (2002). Artificial intelligence, automated reasoning, and symbolic computation, Springer, Berlin.
Dassault Systèmes Simulia Corp. (2012a). ABAQUS standard users manual, Vol. II, Version 6.12, Providence, RI.
Dassault Systèmes Simulia Corp. (2012b). ABAQUS user subroutines reference manual, Version 6.12, Providence, RI.
de Souza Neto, E., Peric, D., and Owen, D. (2008). Computational methods for plasticity: Theory and applications, Wiley, Hoboken, NJ.
Desai, C., and Abel, J. (1972). Introduction to the finite element method, Van Nostrand-Reinhold, New York.
Hertelé, S., Waele, W. D., Denys, R., and Verstraete, M. (2012). “Full-range stress-strain behaviour of contemporary pipeline steels: Part I. Model description.” Int. J. Press. Vessels Piping, 92(Apr), 34–40.
Hill, R. (1950). Mathematical theory of plasticity, Oxford University Press, London.
Kim, S., Ryu, J., and Cho, M. (2011). “Numerically generated tangent stiffness matrices using the complex variable derivative method for nonlinear structural analysis.” Comput. Methods Appl. Mech. Eng., 92(1–4), 34–40.
Kojic, M., and Bathe, K. (2005). Inelastic analysis of solids and structures, Springer, Berlin.
Krieg, R. D., and Krieg, D. B. (1977). “Accuracies of numerical solution methods for the elastic-perfectly plastic model.” J. Pressure Vessel Technol., 99(4), 510–515.
Lazzarin, P., and Livieri, P. (1997). “Different solutions for stress and strain fields in autofrettaged thick-walled cylinders.” Int. J. Press. Vessels Piping, 71(3), 231–238.
Lyness, J. N., and Moler, C. B. (1967). “Numerical differentiation of analytic functions.” SIAM J. Numer. Anal., 4(2), 202–210.
MATLAB R2013b [Computer software]. MathWorks, Natick, MA.
Miehe, C. (1996). “Numerical computation of algorithmic (consistent) tangent moduli in large-strain computational inelasticity.” Comput. Methods Appl. Mech. Eng., 134(3–4), 223–240.
Millwater, H., Wagner, D., Baines, A., and Lovelady, K. (2013). “Improved WCTSE method for the generation of 2D weight functions through implementation into a commercial finite element code.” Eng. Fract. Mech., 109, 302–309.
Mises, R. (1928). “Mechanik der plastischen formänderung der kristallen.” J. Pressure Vessel Technol., 8(3), 161–185.
Peréz-Foguet, A., Rodríguez-Ferran, A., and Huerta, A. (2000). “Numerical differentiation for local and global tangent operators in computational plasticity.” Comput. Methods Appl. Mech. Eng., 189(1), 277–296.
Squire, W., and Trapp, G. (1998). “Using complex variables to estimate derivatives of real functions.” SIAM Rev., 40(1), 110–112.
Tsay, J. J., and Arora, J. S. (1990). “Nonlinear structural design sensitivity analysis for path dependent problems. Part 1: General theory.” Comput. Methods Appl. Mech. Eng., 81(2), 183–208.
Voorhees, A., Millwater, H., and Bagley, R. (2011). “Complex variable methods for shape sensitivity of finite element models.” Finite Elem. Anal. Des., 47(10), 1146–1156.
Wagner, D., and Millwater, H. (2012). “2D weight function development using a complex Taylor series expansion method.” Eng. Fract. Mech., 86(May), 23–37.
Zhang, Y., and Kiureghian, A. D. (1993). “Dynamic response sensitivity of inelastic structures.” Comput. Methods Appl. Mech. Eng., 108(1–2), 23–26.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 141 • Issue 2 • February 2015
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Jan 17, 2014
Accepted: Apr 25, 2014
Published online: Jul 29, 2014
Published in print: Feb 1, 2015
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.