Mathematical analysis and numerical simulation of a Reynolds-Koiter model for the elastohydrodynamic journal-bearing device

Iñigo Arregui; J. Jesús Cendán; Carlos Vázquez

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 2, page 325-343
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Arregui, Iñigo, Jesús Cendán, J., and Vázquez, Carlos. "Mathematical analysis and numerical simulation of a Reynolds-Koiter model for the elastohydrodynamic journal-bearing device." ESAIM: Mathematical Modelling and Numerical Analysis 36.2 (2010): 325-343. <http://eudml.org/doc/194107>.

@article{Arregui2010,
abstract = { 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. },
author = {Arregui, Iñigo, Jesús Cendán, J., Vázquez, Carlos},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Koiter model; Reynolds equation; free boundary problems; fixed point techniques.; fixed point techniques; fixed point},
language = {eng},
month = {3},
number = {2},
pages = {325-343},
publisher = {EDP Sciences},
title = {Mathematical analysis and numerical simulation of a Reynolds-Koiter model for the elastohydrodynamic journal-bearing device},
url = {http://eudml.org/doc/194107},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Arregui, Iñigo
AU - Jesús Cendán, J.
AU - Vázquez, Carlos
TI - Mathematical analysis and numerical simulation of a Reynolds-Koiter model for the elastohydrodynamic journal-bearing device
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 2
SP - 325
EP - 343
AB - 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.
LA - eng
KW - Koiter model; Reynolds equation; free boundary problems; fixed point techniques.; fixed point techniques; fixed point
UR - http://eudml.org/doc/194107
ER -

References

top
  1. S. Alvarez, Problemas de frontera libre en teoría de lubricación. Ph.D. thesis, Universidad Complutense de Madrid (1986).  
  2. I. Arregui and C. Vázquez, Finite element solution of a Reynolds-Koiter coupled problem for the elastic journal bearing. Comput. Methods Appl. Mech. Engrg.190 (2001) 2051-2062.  Zbl1013.74019
  3. G. Bayada and M. Chambat, The transition between the Stokes equation and the Reynolds equation: A mathematical proof. Appl. Math. Optim.14 (1986) 73-93.  Zbl0701.76039
  4. G. Bayada and M. Chambat, Sur quelques modélisations de la zone de cavitation en lubrification hydrodynamique. J. Theoret. Appl. Mech.5 (1986) 703-729.  Zbl0621.76030
  5. G. Bayada, M. Chambat and C. Vázquez, Characteristics method for the formulation and computation of a free boundary cavitation problem. J. Comput. Appl. Math.98 (1998) 191-212.  Zbl0948.76064
  6. G. Bayada, J. Durany and C. Vázquez, Existence of solution for a lubrication problem in elastic journal bearing devices with thin bearing. Math. Methods Appl. Sci.18 (1995) 255-266.  Zbl0820.35110
  7. M. Bernadou and P.G. Ciarlet, Sur l'ellipiticité du modèle linéaire de coques de W.T. Koiter. Lecture Notes in Appl. Sci. Engrg.34 (1976) 89-136.  Zbl0356.73066
  8. M. Bernadou, P.G. Ciarlet and B. Miara, Existence theorems for two-dimensional linear shell theories. J. Elasticity34 (1992) 645-667.  Zbl0808.73045
  9. H. Brézis, Analyse fonctionnelle. Masson, Paris (1983).  Zbl0511.46001
  10. A. Cameron, Basic lubrication theory. Ellis Horwood, West Sussex (1981).  
  11. Ph. Destuynder, Modélisation des coques minces élastiques. Masson, Paris (1990).  
  12. Ph. Destuynder and M. Salaün, 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.  Zbl0852.73059
  13. Ph. Destuynder and M. Salaün, 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.  Zbl0852.73059
  14. J. Durany, G. García and C. Vázquez, An elastohydrodynamic coupled problem between a piezoviscous Reynolds equation and a hinged plate model. RAIRO Modél. Math. Anal. Numér.31 (1997) 495-516.  Zbl0879.73044
  15. J. Durany, G. García and C. Vázquez, Simulation of a lubricated Hertzian contact problem under imposed load. Finite Elem. Anal. Des.38 (2002) 645-658.  Zbl1073.74048
  16. V. Girault and P.A. Raviart, Finite element aproximation of the Navier-Stokes equations. Lecture Notes in Math.749, Springer (1997).  
  17. T.G. Hughes, C.D. Elcoate and H.P. Evans, 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.  Zbl0949.76048
  18. D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications. SIAM, Philadelphia (2000).  Zbl0988.49003
  19. R. Verstappen, A simple numerical algorithm for elastohydrodynamic lubrication, based on a dynamic variation principle. J. Comput. Phys.97 (1991) 460-488.  Zbl0737.76014
  20. S.R. Wu, A penalty formulation and numerical approximation of the Reynolds-Hertz problem of elastohydrodynamic lubrication. Internat. J. Engrg. Sci.24 (1986) 1001-1013.  Zbl0583.73016
  21. S.R. Wu and J.T. Oden, A note on applications of adaptive finite elements to elastohydrodynamic lubrication problems. Comm. Appl. Numer. Methods3 (1987) 485-494.  Zbl0625.76041

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.