# American Institute of Mathematical Sciences

July  2019, 24(7): 3051-3066. doi: 10.3934/dcdsb.2018299

## Inverse parameter-dependent Preisach operator in thermo-piezoelectricity modeling

 Institute of Mathematics, Czech Academy of Sciences, Žitná 25,115 67 Praha 1, Czech Republic

* Corresponding author: Giselle A. Monteiro

Received  March 2018 Revised  June 2018 Published  October 2018

Hysteresis is an important issue in modeling piezoelectric materials, for example, in applications to energy harvesting, where hysteresis losses may influence the efficiency of the process.The main problem in numerical simulations is the inversion of the underlying hysteresis operator.Moreover, hysteresis dissipation is accompanied with heat production, which in turn increases thetemperature of the device and may change its physical characteristics. More accurate models thereforehave to take the temperature dependence into account for a correct energy balance.We prove here that the classical Preisach operator with a fairly general parameter-dependenceadmits a Lipschitz continuous inverse in the space of right-continuous regulated functions, propose a time-discrete and memory-discrete inversion algorithm, and show that higher regularity of the inputs leads to a higher regularity of the output of the inverse.

Citation: Pavel Krejčí, Giselle A. Monteiro. Inverse parameter-dependent Preisach operator in thermo-piezoelectricity modeling. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3051-3066. doi: 10.3934/dcdsb.2018299
##### References:

show all references

##### References:
 [1] Pavel Krejčí. The Preisach hysteresis model: Error bounds for numerical identification and inversion. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 101-119. doi: 10.3934/dcdss.2013.6.101 [2] Timoteo Carletti. The lagrange inversion formula on non--Archimedean fields, non--analytical form of differential and finite difference equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 835-858. doi: 10.3934/dcds.2003.9.835 [3] Alexander Pimenov, Dmitrii I. Rachinskii. Linear stability analysis of systems with Preisach memory. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 997-1018. doi: 10.3934/dcdsb.2009.11.997 [4] Rod Cross, Hugh McNamara, Leonid Kalachev, Alexei Pokrovskii. Hysteresis and post Walrasian economics. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 377-401. doi: 10.3934/dcdsb.2013.18.377 [5] Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773 [6] J. Samuel Jiang, Hans G. Kaper, Gary K Leaf. Hysteresis in layered spring magnets. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 219-232. doi: 10.3934/dcdsb.2001.1.219 [7] Stanislaw Migórski. Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1339-1356. doi: 10.3934/dcdsb.2006.6.1339 [8] Zhenhai Liu, Stanislaw Migórski. Noncoercive damping in dynamic hemivariational inequality with application to problem of piezoelectricity. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 129-143. doi: 10.3934/dcdsb.2008.9.129 [9] Linh V. Nguyen. A family of inversion formulas in thermoacoustic tomography. Inverse Problems & Imaging, 2009, 3 (4) : 649-675. doi: 10.3934/ipi.2009.3.649 [10] Jana Kopfová. Nonlinear semigroup methods in problems with hysteresis. Conference Publications, 2007, 2007 (Special) : 580-589. doi: 10.3934/proc.2007.2007.580 [11] Martin Brokate, Pavel Krejčí. Weak differentiability of scalar hysteresis operators. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2405-2421. doi: 10.3934/dcds.2015.35.2405 [12] Antonio DeSimone, Natalie Grunewald, Felix Otto. A new model for contact angle hysteresis. Networks & Heterogeneous Media, 2007, 2 (2) : 211-225. doi: 10.3934/nhm.2007.2.211 [13] Peter Seibt. A period formula for torus automorphisms. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1029-1048. doi: 10.3934/dcds.2003.9.1029 [14] Hans F. Weinberger, Xiao-Qiang Zhao. An extension of the formula for spreading speeds. Mathematical Biosciences & Engineering, 2010, 7 (1) : 187-194. doi: 10.3934/mbe.2010.7.187 [15] Gamaliel Blé, Carlos Cabrera. A generalization of Douady's formula. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6183-6188. doi: 10.3934/dcds.2017267 [16] Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232 [17] Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17 [18] Frank Natterer. Photo-acoustic inversion in convex domains. Inverse Problems & Imaging, 2012, 6 (2) : 315-320. doi: 10.3934/ipi.2012.6.315 [19] Piotr Fijałkowski. A global inversion theorem for functions with singular points. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 173-180. doi: 10.3934/dcdsb.2018011 [20] Liying Wang, Weiguo Zhao, Dan Zhang, Linming Zhao. A geometric inversion algorithm for parameters calculation in Francis turbine. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1373-1384. doi: 10.3934/dcdss.2015.8.1373

2018 Impact Factor: 1.008