Applications of Mathematics, Vol. 68, No. 6, pp. 795-828, 2023
On forward and inverse uncertainty quantification for a model for a magneto mechanical device involving a hysteresis operator
Olaf Klein
Received March 31, 2023. Published online October 12, 2023. OPEN ACCESS
Abstract: Modeling real world objects and processes one may have to deal with hysteresis effects but also with uncertainties. Following D. Davino, P. Krejčí, and C. Visone (2013), a model for a magnetostrictive material involving a generalized Prandtl-Ishlinskiĭ-operator is considered here. Using results of measurements, some parameters in the model are determined and inverse Uncertainty Quantification (UQ) is used to determine random densities to describe the remaining parameters and their uncertainties. Afterwards, the results are used to perform forward UQ and to compare the generated outputs with measured data. This extends some of the results from O. Klein, D. Davino, and C. Visone (2020).
References: [1] M. Al Janaideh, C. Visone, D. Davino, P. Krejčí: The generalized Prandtl-Ishlinskii model: Relation with the Preisach nonlinearity and inverse compensation error. American Control Conference (ACC). Portland, Oregon (2014), 4759-4764. DOI 10.1109/ACC.2014.6858952
[2] M. Brokate, J. Sprekels: Hysteresis and Phase Transitions. Applied Mathematical Sciences 121. Springer, New York (1996). DOI 10.1007/978-1-4612-4048-8 | MR 1411908 | Zbl 0951.74002
[3] D. Davino, P. Krejčí, C. Visone: Fully coupled modeling of magneto-mechanical hysteresis through 'thermodynamic' compatibility. Smart Mater. Struct. 22 (2013), Article ID 095009. DOI 10.1088/0964-1726/22/9/095009
[4] D. Davino, C. Visone: Rate-independent memory in magneto-elastic materials. Discrete Contin. Dyn. Syst., Ser. S 8 (2015), 649-691. DOI 10.3934/dcdss.2015.8.649 | MR 3356455 | Zbl 1302.93028
[5] G. Grimmett, D. Stirzaker: Probability and Random Processes. Oxford University Press, New York (2001). MR 2059709 | Zbl 1015.60002
[6] J. Kaipio, E. Somersalo: Statistical and Computational Inverse Problems. Applied Mathematical Sciences 160. Springer, New York (2005). DOI 10.1007/b138659 | MR 2102218 | Zbl 1068.65022
[7] O. Klein, D. Davino, C. Visone: On forward and inverse uncertainty quantification for models involving hysteresis operators. Math. Model. Nat. Phenom. 15 (2020), Article ID 53, 19 pages. DOI 10.1051/mmnp/2020009 | MR 4175380 | Zbl 07372583
[8] O. Klein, P. Krejčí: Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations. Nonlinear Anal., Real World Appl. 4 (2003), 755-785. DOI 10.1016/S1468-1218(03)00013-0 | MR 1978561 | Zbl 1043.47047
[9] M. A. Krasnosel'skij, A. V. Pokrovskij: Systems with Hysteresis. Springer, Berlin (1989). DOI 10.1007/978-3-642-61302-9 | MR 0987431 | Zbl 0665.47038
[10] P. Krejčí: Hysteresis, Convexity and Dissipation in Hyperbolic Equations. GAKUTO International Series. Mathematical Sciences and Applications 8. Gakkotosho, Tokyo (1996). MR 2466538 | Zbl 1187.35003
[11] P. Krejčí: Long-time behavior of solutions to hyperbolic equations with hysteresis. Evolutionary Equations. Vol. II. Handbook of Differential Equations. Elsevier, Amsterdam (2005), 303-370. DOI 10.1016/S1874-5717(06)80007-1 | MR 2182830 | Zbl 1098.35030
[12] P. M. Lee: Bayesian Statistics: An Introduction. John Wiley & Sons, Chichester (2012). MR 3237439 | Zbl 1258.62028
[13] S. Marelli, B. Sudret: UQLab: A framework for uncertainty quantification in Matlab. Vulnerability, Uncertainty, and Risk: Quantification, Mitigation, and Management. American Society of Civil Engineers, Reston (2014). DOI 10.1061/9780784413609.257
[14] I. D. Mayergoyz: Mathematical Models of Hysteresis. Springer, New York (1991). DOI 10.1007/978-1-4612-3028-1 | MR 1083150 | Zbl 0723.73003
[15] M. Moustapha, C. Lataniotis, P. Wiederkehr, P.-R. Wagner, D. Wicaksono, S. Marelli, B. Sudret: UQLib User Manual. ETH, Zürich (2022), Available at https://www.uqlab.com/uqlib-user-manual.
[16] Y. Pawitan: In All Likelihood: Statistical Modelling and Inference Using Likelihood. Oxford Science Publications. Oxford University Press, Oxford (2001). Zbl 1013.62001
[17] R. C. Smith: Uncertainty Quantification: Theory, Implementation, and Applications. Computational Science & Engineering 12. SIAM, Philadelphia (2014). DOI 10.1137/1.9781611973228 | MR 3155184 | Zbl 1284.65019
[18] T. J. Sullivan: Introduction to Uncertainty Quantification. Texts in Applied Mathematics 63. Springer, Cham (2015). DOI 10.1007/978-3-319-23395-6 | MR 3364576 | Zbl 1336.60002
[19] A. Visintin: Differential Models of Hysteresis. Applied Mathematical Sciences 111. Springer, Berlin (1994). DOI 10.1007/978-3-662-11557-2 | MR 1329094 | Zbl 0820.35004
[20] C. Visone, M. Sjöström: Exact invertible hysteresis models based on play operators. Phys. B, Condens. Matter 343 (2004), 148-152. DOI 10.1016/j.physb.2003.08.087
[21] P.-R. Wagner, J. Nagel, S. Marelli, B. Sudret: UQLab User Manual: Bayesian Inversion for Model Calibration and Validation. ETH, Zürich (2022), Available at https://www.uqlab.com/inversion-user-manual.
Affiliations: Olaf Klein, Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany, email: olaf.klein@wias-berlin.de
