CIRM - Videos & books Library - Stochastic variational inequalities for random mechanics

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Auteurs : Mertz, Laurent (Auteur de la conférence)
CIRM (Editeur )

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Résumé : The mathematical framework of variational inequalities is a powerful tool to model problems arising in mechanics such as elasto-plasticity where the physical laws change when some state variables reach a certain threshold [1]. Somehow, it is not surprising that the models used in the literature for the hysteresis effect of non-linear elasto-plastic oscillators submitted to random vibrations [2] are equivalent to (finite dimensional) stochastic variational inequalities (SVIs) [3]. This presentation concerns (a) cycle properties of a SVI modeling an elasto-perfectly-plastic oscillator excited by a white noise together with an application to the risk of failure [4,5]. (b) a set of Backward Kolmogorov equations for computing means, moments and correlation [6]. (c) free boundary value problems and HJB equations for the control of SVIs. For engineering applications, it is related to the problem of critical excitation [7]. This point concerns what we are doing during the CEMRACS research project. (d) (if time permits) on-going research on the modeling of a moving plate on turbulent convection [8]. This is a mixture of joint works and / or discussions with, amongst others, A. Bensoussan, L. Borsoi, C. Feau, M. Huang, M. Laurière, G. Stadler, J. Wylie, J. Zhang and J.Q. Zhong.

Codes MSC :
35R60 - PDEs with randomness, stochastic PDE
60H10 - Stochastic ordinary differential equations
60H30 - Applications of stochastic analysis (to PDE, etc.)
74H50 - Random vibrations (dynamical problems in solid mechanics)
74C05 - Small-strain, rate-independent theories

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2017-08-16_Mertz.mp4

Informations sur la Rencontre

Nom de la Rencontre : CEMRACS: Numerical methods for stochastic models: control, uncertainty quantification, mean-field / CEMRACS : Méthodes numériques pour équations stochastiques : contrôle, incertitude, champ moyen
Organisateurs de la Rencontre : Bouchard, Bruno ; Chassagneux, Jean-François ; Delarue, François ; Gobet, Emmanuel ; Lelong, Jérôme
Dates : 17/07/17 - 25/08/17
Année de la rencontre : 2017
URL de la Rencontre : http://conferences.cirm-math.fr/1556.html

Données de citation

DOI : 10.24350/CIRM.V.19217703
Citer cette vidéo: Mertz, Laurent (2017). Stochastic variational inequalities for random mechanics. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19217703
URI : http://dx.doi.org/10.24350/CIRM.V.19217703

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Bibliographie

[1] Duvaut, G., & Lions, J.L. (1976). Inequalities in mechanics and physics. Berlin: Springer-Verlag - http://dx.doi.org/10.1007/978-3-642-66165-5

[2] Karnopp, D., & Scharton, T.D. (1966). Plastic deformation in random vibration. The journal of the Acoustical society of America, 39(6), 1154-1161 - http://dx.doi.org/10.1121/1.1910005

[3] Bensoussan, A., & Turi, J. (2008). Degenerate Dirichlet problems related to the invariant measure of elasto-plastic oscillators. Applied Mathematics and Optimization, 58(1), 1-27 - https://doi.org/10.1007/s00245-007-9027-4

[4] Bensoussan, A., Mertz, L., & Yam, S.C.P. (2012). Long cycle behavior of the plastic deformation of an elasto-perfectly-plastic oscillator with noise. Comptes Rendus. Mathématique. Académie des Sciences, Paris, 350(17-18), 853-859 - http://dx.doi.org/10.1016/j.crma.2012.09.020

[5] Feau, C., Lauriere, M., & Mertz, L. (2017). Asymptotic formulae for the risk of failure related to an elasto- plastic problem with noise. To appear in Asymptotic Analysis -

[6] Mertz, L., Stadler, G., & Wylie, J. (2017). A backward Kolmogorov equation approach to compute means, moments and correlations of path-dependent stochastic dynamical systems. - https://arxiv.org/abs/1704.02170v1

[7] Takewaki, I., Moustafa, A., & Kohei, F. (2013). Improving the Earthquake Resilience of Buildings. Berlin: Springer-Verlag - https://doi.org/10.1007/978-1-4471-4144-0

[8] Zhong, J.Q, & Zhang, J. (2007). Modeling the dynamics of a free boundary on turbulent thermal convection. Physical Review E, 76, 016307 - http://dx.doi.org/10.1103/physreve.76.016307

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