Thermalization of rate-independent processes by entropic regularization

T. J. Sullivan; M. Koslowski; F. Theil; Michael Ortiz

doi:10.3934/dcdss.2013.6.215

Article Contents

2013, Volume 6, Issue 1: 215-233. Doi: 10.3934/dcdss.2013.6.215

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Thermalization of rate-independent processes by entropic regularization

1.
Applied & Computational Mathematics and Graduate Aerospace Laboratories, California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125-9400
2.
School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907-2088
3.
Mathematics Institute, University of Warwick, Coventry, CV4 7AL
4.
Graduate Aerospace Laboratories, California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125

Received: May 2011

Revised: July 2011

Published: February 2013

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Abstract

We consider the effective behaviour of a rate-independent process when it is placed in contact with a heat bath. The method used to ``thermalize'' the process is an interior-point entropic regularization of the Moreau--Yosida incremental formulation of the unperturbed process. It is shown that the heat bath destroys the rate independence in a controlled and deterministic way, and that the effective dynamics are those of a non-linear gradient descent in the original energetic potential with respect to a different and non-trivial effective dissipation potential.

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Mathematics Subject Classification: Primary: 47J35, 82C35.

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References

[1]	L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, second ed., 2008.
[2]	S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004.
[3]	R. Jordan and D. Kinderlehrer, An extended variational principle, Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., Dekker, New York, 177 (1996), 187-200.
[4]	R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
[5]	M. Koslowski, "A Phase-Field Model of Dislocations in Ductile Single Crystals," Ph. D. thesis, California Institute of Technology, Pasadena, California, USA, 2003.
[6]	A. Mielke, Evolution of rate-independent systems, Evolutionary Equations. Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559.
[7]	\bysame, Modeling and analysis of rate-independent processes, January 2007, Lipschitz Lecture held in Bonn: http://www.wias-berlin.de/people/mielke/papers/Lipschitz07Mielke.pdf.
[8]	A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25 (2009), 585-615.
[9]	A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.
[10]	J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299.
[11]	J. P. Penot and M. Théra, Semicontinuous mappings in general topology, Arch. Math. (Basel), 38 (1982), 158-166.
[12]	D. Preiss, Geometry of measures in $\mathbbR^n$distribution, rectifiability, and densities: , Ann. of Math., 125 (1987), 537-643.
[13]	T. J. Sullivan, "Analysis of Gradient Descents in Random Energies and Heat Baths," Ph. D. thesis, Mathematitics Institute, University of Warwick, Coventry, UK, 2009.
[14]	T. J. Sullivan, M. Koslowski, F. Theil and M. Ortiz, On the behavior of dissipative systems in contact with a heat bath: application to Andrade creep, J. Mech. Phys. Solids, 57 (2009), 1058-1077.
[15]	K. Yosida, "Functional Analysis," Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York, 1965.