Numerical Solutions of Direct and Inverse Problems for a Time Fractional Viscoelastoplastic Equation
Publication: Journal of Engineering Mechanics
Volume 143, Issue 7
Abstract
This paper is devoted to numerical solutions of direct and inverse problems for the nonlinear nonlocal time fractional equation , where . After solving the direct problem by the method of lines, a numerical method based on discretization of the minimization problem, steepest descent method, and least-squares approach is proposed for the solution of the inverse problem. Numerical examples with noise-free and noisy data illustrate applicability and accuracy of the proposed method to some extent.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This research has been supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) through the project Nr 113F373. The authors thank the referees for their very careful reading and for pointing out several mistakes as well as for the useful comments and suggestions. The authors also thank Prof. Roberto Ballarini, Editor-in-Chief, since he strongly encouraged submission of a revised version of the paper.
References
Adolfsson, K., and Enelund, M. (2003). “Fractional derivative viscoelasticity at large deformations.” Non-linear Dyn., 33(3), 301–321.
Adolfsson, K., Enelund, M., and Olsson, P. (2005). “On the fractional order model of viscoelasticity.” Mech. Time-Depend. Mater., 9(1), 15–34.
Bagley, R. L. (1989). “Power law and fractional calculus model of viscoelasticity.” AIAA J., 27(10), 1412–1417.
Bagley, R. L., and Torvik, P. J. (1983a). “A theoretical basis for the application of fractional calculus to viscoelasticity.” J. Rheol., 27(3), 201–210.
Bagley, R. L., and Torvik, P. J. (1983b). “Fractional calculus—A different approach to the analysis of viscoelastically damped structures.” AIAA J., 21(5), 741–748.
Bagley, R. L., and Torvik, P. J. (1986). “On the fractional calculus model of viscoelastic behavior.” J. Rheol., 30(1), 133–155.
Beda, T., and Chevalier, Y. (2004). “New methods for identifying rheological parameter for fractional derivative modeling of viscoelastic behavior.” Mech. Time-Depend. Mater., 8(2), 105–118.
Cheng, J., Nakagawa, J., Yamamoto, M., and Yamazaki, T. (2009). “Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation.” Inverse Prob., 25(11).
Coussot, C., Kalyanam, S., Yapp, R., and Insana, M. F. (2009). “Fractional derivative models for ultrasonic characterization of polymer and breast tissue viscoelasticity.” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 56(4), 715–726.
Craiem, D., and Magin, R. L. (2010). “Fractional order models of viscoelasticity as an alternative in the analysis of red blood cell (RBC) membrane mechanics.” Phys. Biol., 7(1), .
Diethelm, K., Ford, N. J., and Freed, A. D. (2004). “Detailed error analysis for a fractional Adams method.” Numer. Algorithms, 36(1), 31–52.
Diethelm, K., and Freed, A. D. (1999). “The FracPECE subroutine for the numerical solution of dierential equations of fractional order.” Conf. on Orschung und Wissenschaftliches Rechnen: Beitrge zum Heinz-Billing-Preis, Max Planck Society for the Advancement of Science e.V., Munich, Germany.
Garrappa, R. (2010). “On linear stability of predictor-corrector algorithms for fractional differential equations.” Int. J. Comput. Math., 87(10), 2281–2290.
Gerasimov, D. N., Kondratieva, V. A., and Sinkevich, O. A. (2010). “An anomalous non-self-similar infiltration and fractional diffusion equation.” Phys. D., 239(16), 1593–1597.
Gerolymatou, E., Vardoulakis, I., and Hilfer, R. (2006). “Modeling infiltration by means of a non-linear fractional diffusion model.” J. Phys. D: Appl. Phys., 39(18), 4104–4110.
Graney, L., and Richardson, A. A. (1981). “The numerical solution of non-linear partial differential equations by the method of lines.” J. Comput. Appl. Math., 7(4), 229–236.
Hairer, E., and Wanner, G. (1991). Solving ordinary differential equations. II: Stiff and differential-algebraic problems, Springer, Berlin.
Hasanov, A., and Tatar, S. (2009). “Solutions of linear and non-linear problems related to torsional rigidity of a beam.” Comput. Mater. Sci., 45(2), 494–498.
Hasanov, A., and Tatar, S. (2010a). “An inversion method for identification of elastoplastic properties of a beam from torsional experiment.” Int. J. Non-Linear Mech., 45(5), 562–571.
Hasanov, A., and Tatar, S. (2010b). “Semi-analytic inversion method for determination elastoplastic properties of power hardening materials from limited torsional experiment.” Inverse Prob. Sci. Eng., 18(2), 265–278.
Jin, B., and Rundell, W. (2012). “An inverse problem for a one-dimensional time-fractional diffusion problem.” Inverse Prob., 28(7), .
Kachanov, L. M. (1971). Fundamentals of the theory of plasticity: Applied mathematics and mechanics 65, North-Holland Publishing Company, Amsterdam, Netherlands.
Kilbas, A. A., Srivastava, H. M., and Trujillo, J. J. (2006). Theory and applications of fractional differential equations, Elsevier, Amsterdam, Netherlands.
Koeller, R. C. (1984). “Applications of fractional calculus to the theory of viscoelasticity.” J. Appl. Mech., 51(2), 299–307.
Langenbach, A. (1964). “Verallgemeinerte und exakte Losungen des Problems der elastischen-plastischen Torsion von Stabben.” Math. Nachr., 28(3–4), 219–234 (in German).
Lewandowski, R., and Chorazyczewski, B. (2010). “Identification of the parameters of the Kelvin-Voigt and the Maxwell fractional models, used to modeling of viscoelastic dampers.” Comput. Struct., 88(1), 1–17.
Li, G., Zhang, D., Jia, X., and Yamamoto, M. (2013). “Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time fractional diffusion equation.” Inverse Prob., 29(6), .
Liu, J. J., and Yamamoto, M. (2010). “A backward problem for the time-fractional diffusion equation.” Appl. Anal., 89(11), 1769–1788.
Mainardi, F., Luchko, Y., and Pagnini, G. (2001). “The fundamental solution of the space-time fractional diffusion equation.” Fract. Calculus Appl. Anal., 4(2), 153–192.
Mamedov, A. (1995). “An inverse problem related to the determination of elastoplastic properties of a cyclindrical bar.” Int. J. Non-Linear Mech., 30(1), 23–32.
Mamedov, A. (1998). “Determination of elasto-plastic properties of a bimetallic and hollow bar.” Int. J. Non-Linear Mech., 33(2), 385–392.
MATLAB 8.1 [Computer software]. MathWorks, Natick, MA.
Mendiguren, J., Cortés, F., and Galdos, L. (2012). “A generalised fractional derivative model to represent elastoplastic behaviour of metals.” Int. J. Mech. Sci., 65(1), 12–17.
Podlubny, I. (1999). Fractional differential equations, Academic Press, San Diego.
Podlubny, I., Chechkin, A. V., Skovranek, T., Chen, Y. Q., and Vinagre, B. (2009). “Matrix approach to discrete fractional calculus. II: Partial fractional differential equations.” J. Comput. Phys., 228(8), 3137–3153.
Pritz, T. (2003). “Five-parameter fractional derivative model for polymeric damping materials.” J. Sound Vib., 265(5), 935–952.
Sakamoto, K., and Yamamoto, M. (2011a). “Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems.” J. Math. Anal. Appl., 382(1), 426–447.
Sakamoto, K., and Yamamoto, M. (2011b). “Inverse source problem with a final overdetermination for a fractional diffusion equation.” Math. Controls Related Fields, 1(4), 509–518.
Sasso, M., Palmieri, G., and Amodi, D. (2011). “Application of fractional derivative models in linear viscoelastic problems.” Mech. Time-Depend. Mater., 15(4), 367–387.
Schiesselt, H., Metzlert, R., Blument, A., and Nonnemacher, T. F. (1995). “Generalized viscoelastic models: Their fractional equations with solutions.” J. Wys. A: Math. Gen., 28(23), 6567.
Shen, S., Liu, F., Liu, Q., and Anh, V. (2015). “Numerical simulation of anomalous infiltration in porous media.” Numer. Algorithm, 68(3), 443–454.
Sumelka, J. W. (2014a). “Application of fractional continuum mechanics to rate independent plasticity.” Acta Mech., 225(11), 3247–3264.
Sumelka, J. W. (2014b). “Fractional viscoplasticity.” Mech. Res. Commun., 56, 31–36.
Suzuki, J. L., Zayernouri, M., Bittencourt, M. L., and Karniadakis, G. E. (2016). “Fractional-order uniaxial visco-elasto-plastic models for structural analysis.” Comput. Methods Appl. Mech. Eng., 308, 443–467.
Tatar, S. (2013a). “Monotonocity of input-output mapping related to inverse elastoplastic torsional problem.” Appl. Math. Model., 37(23), 9552–9561.
Tatar, S. (2013b). “Numerical solution of the non-linear direct problem related to inverse elastoplastic problem.” Inverse Prob. Sci. Eng., 21(1), 52–62.
Tatar, S., and Muradoglu, Z. (2014a). “A modification of the semi-analytic inversion method: Determination of the yield stress and a comparison with the parametrization algorithm.” Inverse Prob. Sci. Eng., 22(4), 543–556.
Tatar, S., and Muradoglu, Z. (2014b). “Numerical solution of the non-linear evolutional inverse problem related to elatoplastic torsional problem.” Appl. Anal., 93(6), 1187–1200.
Tatar, S., Tinaztepe, R., and Ulusoy, S. (2015). “Determination of an unknown source term in a space-time fractional diffusion equation.” J. Frac. Calc. Appl., 6(2), 94–101.
Tatar, S., Tinaztepe, R., and Ulusoy, S. (2016). “Simultaneous inversion for the exponents of the fractional time and space derivatives in the space-time fractional diffusion equation.” Appl. Anal., 95(1), 1–23.
Tatar, S., and Ulusoy, S. (2013). “A uniqueness result in an inverse problem for a space-time fractional diffusion equation.” Electron. J. Differ. Equ., 2015(258), 1–9.
Tatar, S., and Ulusoy, S. (2015) “An inverse source problem for a one-dimensional space-time fractional diffusion equation.” Appl. Anal., 94(11), 2233–2244.
Tatar, S., and Ulusoy, S. (2017). “Analysis of direct and inverse problems for a fractional elastoplasticity model.” Filomat, in press.
Timoshenko, S., and Goodier, S. N. (1951). Theory of elasticity, McGraw-Hill, New York.
Timoshenko, S., and Woinowsky-Krieger, S. (1959). Theory of plates and shells, 2nd Ed., McGraw-Hill, New York.
Welch, S. W. J., Rorre, R. A. L., and Duren, R. G. (1999). “Application of time-based fractional calculus methods to viscoelastic creep and stress relaxation of materials.” Mech. Time Depend. Mater., 3(3), 279–303.
Xu, X., Cheng, J., and Yamamoto, M. (2011). “Carleman estimate for a fractional diffusion equation with half order and application.” Appl. Anal., 90(9), 1355–1371.
Yamamoto, M., and Zhang, Y. (2012). “Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate.” Inverse Prob., 28(10), .
Zhang, Y., and Xu, X. (2011). “Inverse source problem for a fractional diffusion equation.” Inverse Prob., 27(3), .
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 143 • Issue 7 • July 2017
Copyright
©2017 American Society of Civil Engineers.
History
Received: Jul 24, 2016
Accepted: Nov 23, 2016
Published ahead of print: Feb 22, 2017
Published online: Feb 23, 2017
Published in print: Jul 1, 2017
Discussion open until: Jul 23, 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.