Degenerate diffusion with Preisach hysteresis

Chiara Gavioli; Pavel Krejčí

doi:10.3934/dcdss.2023154

Article Contents

2023, Volume 16, Issue 12: 3677-3708. Doi: 10.3934/dcdss.2023154

This issue Previous Article On some doubly-nonlinear parabolic equations posed in $ \mathbb{R}^{{d}} $ Next Article A nonlocal Gray-Scott model: Well-posedness and diffusive limit

Degenerate diffusion with Preisach hysteresis

Chiara Gavioli^1, and
Pavel Krejčí^2, ,

1.
Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria
2.
Faculty of Civil Engineering, Czech Technical University, Thákurova 7, CZ-16629 Praha 6, Czech Republic

^*Corresponding author: Pavel Krejčí
^*Corresponding author: Pavel Krejčí

Received: March 2023

Revised: August 2023

Early access: August 2023

Published: December 2023

The first author is supported by the Austrian Science Fund (FWF) grants V662, Y1292, and F65, and by the OeAD WTZ grants CZ02/2022 and CZ09/2023; the second author is supported by the GAČR Grant No. 20-14736S, and by the European Regional Development Fund Project No. CZ.02.1.01/0.0/0.0/16_019/0000778

Abstract / Introduction Full Text(HTML) Figure(3) Related Papers Cited by

Abstract

Fluid diffusion in unsaturated porous media manifests strong hysteresis effects due to surface tension on the liquid-gas interface. We describe hysteresis in the pressure-saturation relation by means of the Preisach operator, which makes the resulting evolutionary PDE strongly degenerate. We prove the existence and uniqueness of a strong global solution in arbitrary space dimension using a special weak convexity concept.

Keywords:

Mathematics Subject Classification: 35K65, 47J40, 74N30.

Citation:

Full Text(HTML)

Figure 1. Typical experimental hysteresis dependence in porous media between the logarithm soil suction $ \psi $ and the volumetric water content $ \theta $

Download: Full-size image PowerPoint slide

Figure 2. Evolution of the memory curve and of the hysteresis diagram associated with the input sequence $ 0 {\to} u_1 {\to} u_2 {\to} u_3 {\to} u_4 \to u_5 $

Download: Full-size image PowerPoint slide

Figure 3. Backward time step in $ \Omega_+ $ (left) and in $ \Omega_- $ (right)

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	B. Albers, Modeling the hysteretic behavior of the capillary pressure in partially saturated porous media: A review, Acta Mech., 225 (2014), 2163-2189. doi: 10.1007/s00707-014-1122-4.
[2]	M. Al Saaideh, N. Alatawneh and M. Al Janaideh, A generalized Prandtl-Ishlinskii model for hysteresis modeling in electromagnetic devices, Proceedings of ECCE 2021 – IEEE Energy Conversion Congress and Exposition, Vancouver, Canada, (2021), 4513-4518.
[3]	M. Balato, S. Perna, C. Petrarca and C. Visone, A simple scheme for the inversion of a Preisach like hysteresis operator in saturation conditions, AIP Adv., 12 (2022), 035047.
[4]	G. Bertotti, Hysteresis in Magnetism: For Physicists, Materials Scientists, and Engineers, Academic Press, New York, 1998.
[5]	O. V. Besov, V. P. Il'in and S. M. Nikol'skii, Integral Representations of Functions and Imbedding Theorems, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, DCHalsted Press, New York-Toronto-London, 1979.
[6]	B. Detmann and P. Krejčí, A multicomponent flow model in deformable porous media, Math. Methods Appl. Sci., 42 (2019), 1894-1906. doi: 10.1002/mma.5482.
[7]	M. Eleuteri, J. Kopfová and P. Krejčí, Magnetohydrodynamic flow with hysteresis, SIAM J. Math. Anal., 41 (2009), 435-464. doi: 10.1137/080718383.
[8]	M. Eleuteri and P. Krejčí, Asymptotic behaviour of a Neumann parabolic problem with hysteresis, Z. Angew. Math. Mech., 87 (2007), 261-277. doi: 10.1002/zamm.200610299.
[9]	E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ric. Mat., 8 (1959), 24-51.
[10]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, Classics Appl. Math., 69. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011 doi: 10.1137/1.9781611972030.ch1.
[11]	M. Hilpert, On uniqueness for evolution problems with hysteresis, Mathematical Models for Phase Change Problems, Internat. Ser. Numer. Math., Birkhäuser Verlag, Basel, 88 (1989), 377-388.
[12]	P. Krejčí, Hysteresis and periodic solutions of semilinear and quasilinear wave equations, Math. Z., 193 (1986), 247-264. doi: 10.1007/BF01174335.
[13]	P. Krejčí, The Preisach hysteresis model: Error bounds for numerical identification and inversion, Discrete Contin. Dyn. Syst. - S., 6 (2013), 101-119. doi: 10.3934/dcdss.2013.6.101.
[14]	P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakuto Int. Ser. Math. Sci. Appl., Vol. 8, Gakkōtosho, Tokyo, 1996.
[15]	K. Kuhnen, Modeling, identification and compensation of complex hysteretic nonlinearities - A modified Prandtl-Ishlinskii approach, Eur. J. Control, 9 (2003), 407-418. doi: 10.3166/ejc.9.407-418.
[16]	L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, 13 (1959), 115-162.
[17]	H. Q. Pham, D. G. Fredlund and S. Lee Barbour, A study of hysteresis models for soil-water characteristic curves, Can. Geotech. J., 42 (2005), 1548-1568.
[18]	F. Preisach, Über die magnetische Nachwirkung, Z. Phys., 94 (1935), 277-302.
[19]	M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1967.
[20]	M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Texts Appl. Math., 13. Springer-Verlag, New York, 2004.
[21]	B. Schweizer, Hysteresis in porous media: Modelling and analysis, Interfaces Free Boundaries, 19 (2017), 417-447. doi: 10.4171/IFB/388.
[22]	A. Visintin, Differential Models of Hysteresis, Appl. Math. Sci., 111. Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.