The aim of this work is to deduce the existence of solution of a coupled problem arising in elastohydrodynamic lubrication. The lubricant pressure and concentration are modelled by Reynolds equation, jointly with the free-boundary Elrod-Adams model in order to take into account cavitation phenomena. The bearing deformation is solution of Koiter model for thin shells. The existence of solution to the variational problem presents some difficulties: the coupled character of the equations, the nonlinear multivalued operator associated to cavitation and the fact of writing the elastic and hydrodynamic equations on two different domains. In a first step, we regularize the Heaviside operator. Additional difficulty related to the different domains is circumvented by means of prolongation and restriction operators, arriving to a regularized coupled problem. This one is decoupled into elastic and hydrodynamic parts, and we prove the existence of a fixed point for the global operator. Estimations obtained for the regularized problem allow us to prove the existence of solution to the original one. Finally, a numerical method is proposed in order to simulate a real journal-bearing device and illustrate the qualitative and quantitative properties of the solution.
Mots clés : Koiter model, Reynolds equation, free boundary problems, fixed point techniques
@article{M2AN_2002__36_2_325_0,
author = {Arregui, I\~nigo and Cend\'an, J. Jes\'us and V\'azquez, Carlos},
title = {Mathematical analysis and numerical simulation of a {Reynolds-Koiter} model for the elastohydrodynamic journal-bearing device},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {325--343},
publisher = {EDP-Sciences},
volume = {36},
number = {2},
year = {2002},
doi = {10.1051/m2an:2002015},
mrnumber = {1906821},
zbl = {1045.35092},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2002015/}
}
TY - JOUR AU - Arregui, Iñigo AU - Cendán, J. Jesús AU - Vázquez, Carlos TI - Mathematical analysis and numerical simulation of a Reynolds-Koiter model for the elastohydrodynamic journal-bearing device JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2002 SP - 325 EP - 343 VL - 36 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2002015/ DO - 10.1051/m2an:2002015 LA - en ID - M2AN_2002__36_2_325_0 ER -
%0 Journal Article %A Arregui, Iñigo %A Cendán, J. Jesús %A Vázquez, Carlos %T Mathematical analysis and numerical simulation of a Reynolds-Koiter model for the elastohydrodynamic journal-bearing device %J ESAIM: Modélisation mathématique et analyse numérique %D 2002 %P 325-343 %V 36 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2002015/ %R 10.1051/m2an:2002015 %G en %F M2AN_2002__36_2_325_0
Arregui, Iñigo; Cendán, J. Jesús; Vázquez, Carlos. Mathematical analysis and numerical simulation of a Reynolds-Koiter model for the elastohydrodynamic journal-bearing device. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 2, pp. 325-343. doi : 10.1051/m2an:2002015. http://www.numdam.org/articles/10.1051/m2an:2002015/
[1] , Problemas de frontera libre en teoría de lubricación. Ph.D. thesis, Universidad Complutense de Madrid (1986).
[2] and, Finite element solution of a Reynolds-Koiter coupled problem for the elastic journal bearing. Comput. Methods Appl. Mech. Engrg. 190 (2001) 2051-2062. | Zbl
[3] and, The transition between the Stokes equation and the Reynolds equation: A mathematical proof. Appl. Math. Optim. 14 (1986) 73-93. | Zbl
[4] and, Sur quelques modélisations de la zone de cavitation en lubrification hydrodynamique. J. Theoret. Appl. Mech. 5 (1986) 703-729. | Zbl
[5] , and, Characteristics method for the formulation and computation of a free boundary cavitation problem. J. Comput. Appl. Math. 98 (1998) 191-212. | Zbl
[6] , and, Existence of solution for a lubrication problem in elastic journal bearing devices with thin bearing. Math. Methods Appl. Sci. 18 (1995) 255-266. | Zbl
[7] and, Sur l'ellipiticité du modèle linéaire de coques de W.T. Koiter. Lecture Notes in Appl. Sci. Engrg. 34 (1976) 89-136. | Zbl
[8] , and, Existence theorems for two-dimensional linear shell theories. J. Elasticity 34 (1992) 645-667. | Zbl
[9] , Analyse fonctionnelle. Masson, Paris (1983). | MR | Zbl
[10] , Basic lubrication theory. Ellis Horwood, West Sussex (1981).
[11] , Modélisation des coques minces élastiques. Masson, Paris (1990). | MR | Zbl
[12] and, A mixed finite element for shell model with free edge boundary conditions. Part I: The mixed variational formulation. Comput. Methods Appl. Mech. Engrg. 120 (1995) 195-217. | Zbl
[13] and, A mixed finite element for shell model with free edge boundary conditions. Part II: The numerical scheme. Comput. Methods Appl. Mech. Engrg. 120 (1995) 219-242. | Zbl
[14] , and, An elastohydrodynamic coupled problem between a piezoviscous Reynolds equation and a hinged plate model. RAIRO Modél. Math. Anal. Numér. 31 (1997) 495-516. | EuDML | Numdam | Zbl
[15] , and, Simulation of a lubricated Hertzian contact problem under imposed load. Finite Elem. Anal. Des. 38 (2002) 645-658. | Zbl
[16] and, Finite element aproximation of the Navier-Stokes equations. Lecture Notes in Math. 749, Springer (1997). | Zbl
[17] , and, A novel method for integrating first- and second-order differential equations in elastohydrodynamic lubrication for the solution of smooth isotermal, line contact problems. Internat. J. Numer. Methods Engrg. 44 (1999) 1099-1113. | Zbl
[18] and, An introduction to variational inequalities and their applications. SIAM, Philadelphia (2000). | MR | Zbl
[19] , A simple numerical algorithm for elastohydrodynamic lubrication, based on a dynamic variation principle. J. Comput. Phys. 97 (1991) 460-488. | Zbl
[20] , A penalty formulation and numerical approximation of the Reynolds-Hertz problem of elastohydrodynamic lubrication. Internat. J. Engrg. Sci. 24 (1986) 1001-1013. | Zbl
[21] and, A note on applications of adaptive finite elements to elastohydrodynamic lubrication problems. Comm. Appl. Numer. Methods 3 (1987) 485-494. | Zbl
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