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

Inverse parameter-dependent Preisach operator in thermo-piezoelectricity modeling

Pavel Krejčí and  Giselle A. Monteiro ,

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  July 2019 Early access  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.

Keywords: Hysteresis, Preisach operator, inversion formula, piezoelectricity.
Mathematics Subject Classification: Primary: 34C55, 47J40; Secondary: 58C07, 49J40.
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:
[1]

M. Al Janaideh, F. Deasy and P. Krejčí, Inversion of hysteresis and creep operators, Physica B: Condensed Matter, 407 (2012), 1354-1356.   Google Scholar

[2]

M. Brokate and P. Krejčí, Weak differentiability of scalar hysteresis operators, Discrete Cont. Dyn. Systems A, 35 (2015), 2405-2421.  doi: 10.3934/dcds.2015.35.2405.  Google Scholar

[3]

M. Brokate and A. Visintin, Properties of the Preisach model for hysteresis, J. Reine Angew. Math., 402 (1989), 1-40.  doi: 10.1515/crll.1989.402.1.  Google Scholar

[4]

R. Cross, A. M. Krasnosel'skii and A. V. Pokrovskii, A time-dependent Preisach model, Physica B, 306 (2001), 206-210.   Google Scholar

[5]

D. Davino, A. Giustiniani and C. Visone, Magnetoelastic energy harvesting: Modeling and experiments, in Smart Actuation and Sensing Systems - Recent Advances and Future Challenges (G. Berselli, R. Vertechy and G. Vassura, eds.), InTech, (2012), 487-512. Google Scholar

[6]

D. Davino, P. Krejčí, A. Pimenov, D. Rachinskii and C. Visone, Analysis of an operator-differential model for magnetostrictive energy harvesting, Communications in Nonlinear Science and Numerical Simulation, 39 (2016), 504-519.  doi: 10.1016/j.cnsns.2016.04.004.  Google Scholar

[7]

D. Davino, P. Krejčí and C. Visone, Fully coupled modeling of magnetomechanical hysteresis through 'thermodynamic' compatibility, Smart Materials and Structures, 22 (2013), 095009. Google Scholar

[8]

B. Kaltenbacher and P. Krejčí, A thermodynamically consistent phenomenological model for ferroeletric and ferroelastic hysteresis, ZAMM - Z. Angew. Math. Mech., 96 (2016), 874-891.  doi: 10.1002/zamm.201400292.  Google Scholar

[9]

B. Kaltenbacher and P. Krejčí, Analysis of an optimization problem for a piezoelectric energy harvester to appear in Arch. Appl. Mech. doi: 10.1007/s00419-018-1459-6.  Google Scholar

[10]

M. Kamlah, Ferroelectric and ferroelastic piezoceramics modeling of electromechanical hysteresis phenomena, Continuum Mechanics and Thermodynamics, 13 (2001), 219-268.   Google Scholar

[11]

P. Krejčí, Hysteresis and periodic solutions of semilinear and quasilinear wave equations, Math. Z., 193 (1986), 247-264.  doi: 10.1007/BF01174335.  Google Scholar

[12]

P. Krejčí, On Maxwell equations with the Preisach hysteresis operator: the one-dimensional time-periodic case, Apl. Mat., 34 (1989), 364-374.   Google Scholar

[13]

P. Krejčí, The Kurzweil integral and hysteresis, Journal of Physics: Conference Series, 55 (2006), 144-154.   Google Scholar

[14]

P. Krejčí, The Preisach hysteresis model: Error bounds for numerical identification and inversion, Discrete Contin. Dyn. Syst Ser. S, 6 (2013), 101-119.  doi: 10.3934/dcdss.2013.6.101.  Google Scholar

[15]

P. Krejčí, H. Lamba, G. A. Monteiro and D. Rachinskii, The Kurzweil integral in financial market modeling, Math. Bohem., 141 (2016), 261-286.  doi: 10.21136/MB.2016.18.  Google Scholar

[16]

P. Krejčí, H. Lamba and D. Rachinskii, Global stability of a piecewise linear macroeconomic model with a continuum of equilibrium states and sticky expectation, arXiv: 1711.07563. Google Scholar

[17]

P. Krejčí and Ph. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.   Google Scholar

[18]

P. Krejčí and G. A. Monteiro, Oscillations of a temperature-dependent piezoelectric rod, arXiv: 1803.10000 Google Scholar

[19]

K. Kuhnen, Modeling, identification and compensation of complex hysteretic nonlinearities - A modified Prandtl-Ishlinskii approach, Eur. J. Control, 9 (2003), 407-418.   Google Scholar

[20]

F. Preisach, Über die magnetische Nachwirkung, Z. Phys., 94 (1935), 277-302.   Google Scholar

[21]

C. Visone and M. Sjöström, Exact invertible hysteresis models based on play operators, Physica B: Condensed Matter, 343 (2004), 148-152.   Google Scholar

show all references

References:
[1]

M. Al Janaideh, F. Deasy and P. Krejčí, Inversion of hysteresis and creep operators, Physica B: Condensed Matter, 407 (2012), 1354-1356.   Google Scholar

[2]

M. Brokate and P. Krejčí, Weak differentiability of scalar hysteresis operators, Discrete Cont. Dyn. Systems A, 35 (2015), 2405-2421.  doi: 10.3934/dcds.2015.35.2405.  Google Scholar

[3]

M. Brokate and A. Visintin, Properties of the Preisach model for hysteresis, J. Reine Angew. Math., 402 (1989), 1-40.  doi: 10.1515/crll.1989.402.1.  Google Scholar

[4]

R. Cross, A. M. Krasnosel'skii and A. V. Pokrovskii, A time-dependent Preisach model, Physica B, 306 (2001), 206-210.   Google Scholar

[5]

D. Davino, A. Giustiniani and C. Visone, Magnetoelastic energy harvesting: Modeling and experiments, in Smart Actuation and Sensing Systems - Recent Advances and Future Challenges (G. Berselli, R. Vertechy and G. Vassura, eds.), InTech, (2012), 487-512. Google Scholar

[6]

D. Davino, P. Krejčí, A. Pimenov, D. Rachinskii and C. Visone, Analysis of an operator-differential model for magnetostrictive energy harvesting, Communications in Nonlinear Science and Numerical Simulation, 39 (2016), 504-519.  doi: 10.1016/j.cnsns.2016.04.004.  Google Scholar

[7]

D. Davino, P. Krejčí and C. Visone, Fully coupled modeling of magnetomechanical hysteresis through 'thermodynamic' compatibility, Smart Materials and Structures, 22 (2013), 095009. Google Scholar

[8]

B. Kaltenbacher and P. Krejčí, A thermodynamically consistent phenomenological model for ferroeletric and ferroelastic hysteresis, ZAMM - Z. Angew. Math. Mech., 96 (2016), 874-891.  doi: 10.1002/zamm.201400292.  Google Scholar

[9]

B. Kaltenbacher and P. Krejčí, Analysis of an optimization problem for a piezoelectric energy harvester to appear in Arch. Appl. Mech. doi: 10.1007/s00419-018-1459-6.  Google Scholar

[10]

M. Kamlah, Ferroelectric and ferroelastic piezoceramics modeling of electromechanical hysteresis phenomena, Continuum Mechanics and Thermodynamics, 13 (2001), 219-268.   Google Scholar

[11]

P. Krejčí, Hysteresis and periodic solutions of semilinear and quasilinear wave equations, Math. Z., 193 (1986), 247-264.  doi: 10.1007/BF01174335.  Google Scholar

[12]

P. Krejčí, On Maxwell equations with the Preisach hysteresis operator: the one-dimensional time-periodic case, Apl. Mat., 34 (1989), 364-374.   Google Scholar

[13]

P. Krejčí, The Kurzweil integral and hysteresis, Journal of Physics: Conference Series, 55 (2006), 144-154.   Google Scholar

[14]

P. Krejčí, The Preisach hysteresis model: Error bounds for numerical identification and inversion, Discrete Contin. Dyn. Syst Ser. S, 6 (2013), 101-119.  doi: 10.3934/dcdss.2013.6.101.  Google Scholar

[15]

P. Krejčí, H. Lamba, G. A. Monteiro and D. Rachinskii, The Kurzweil integral in financial market modeling, Math. Bohem., 141 (2016), 261-286.  doi: 10.21136/MB.2016.18.  Google Scholar

[16]

P. Krejčí, H. Lamba and D. Rachinskii, Global stability of a piecewise linear macroeconomic model with a continuum of equilibrium states and sticky expectation, arXiv: 1711.07563. Google Scholar

[17]

P. Krejčí and Ph. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.   Google Scholar

[18]

P. Krejčí and G. A. Monteiro, Oscillations of a temperature-dependent piezoelectric rod, arXiv: 1803.10000 Google Scholar

[19]

K. Kuhnen, Modeling, identification and compensation of complex hysteretic nonlinearities - A modified Prandtl-Ishlinskii approach, Eur. J. Control, 9 (2003), 407-418.   Google Scholar

[20]

F. Preisach, Über die magnetische Nachwirkung, Z. Phys., 94 (1935), 277-302.   Google Scholar

[21]

C. Visone and M. Sjöström, Exact invertible hysteresis models based on play operators, Physica B: Condensed Matter, 343 (2004), 148-152.   Google Scholar

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