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A Static Condensation Reduced Basis Element Approach for the Reynolds Lubrication Equation

Published online by Cambridge University Press:  05 December 2016

Eduard Bader*
Affiliation:
Aachen Institute for Advanced Study in Computational Engineering Science, RWTH Aachen University, Schinkelstr. 2, 52062 Aachen, Germany
Martin A. Grepl*
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany
Siegfried Müller*
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany
*
*Corresponding author. Email addresses:bader@aices.rwth-aachen.de (E. Bader), grepl@igpm.rwth-aachen.de (M. A. Grepl), mueller@igpm.rwth-aachen.de (S. Müller)
*Corresponding author. Email addresses:bader@aices.rwth-aachen.de (E. Bader), grepl@igpm.rwth-aachen.de (M. A. Grepl), mueller@igpm.rwth-aachen.de (S. Müller)
*Corresponding author. Email addresses:bader@aices.rwth-aachen.de (E. Bader), grepl@igpm.rwth-aachen.de (M. A. Grepl), mueller@igpm.rwth-aachen.de (S. Müller)
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Abstract

In this paper, we propose a Static Condensation Reduced Basis Element (SCRBE) approach for the Reynolds Lubrication Equation (RLE). The SCRBE method is a computational tool that allows to efficiently analyze parametrized structures which can be decomposed into a large number of similar components. Here, we extend the methodology to allow for a more general domain decomposition, a typical example being a checkerboard-pattern assembled from similar components. To this end, we extend the formulation and associated a posteriori error bound procedure. Our motivation comes from the analysis of the pressure distribution in plain journal bearings governed by the RLE. However, the SCRBE approach presented is not limited to bearings and the RLE, but directly extends to other component-based systems. We show numerical results for plain bearings to demonstrate the validity of the proposed approach.

Type
Research Article
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Bader, E.. A reduced basis element approach for the reynolds lubrication equation. Master's thesis, RWTH Aachen University, 2012.Google Scholar
[2] Bai, S., Peng, X., Li, Y., and Sheng, S.. A hydrodynamic laser surface-textured gas mechanical face seal. Tribology Letters, 38(2):187194, 2010.Google Scholar
[3] Bhushan, B.. Principle and Applications of Tribology, 2nd Edition. Tribology Series. Wiley, 2013.Google Scholar
[4] Chan, T. F. and Mathew, T. P.. Domain decomposition algorithms. Acta Numerica, 3:61143, 1 1994.Google Scholar
[5] Craig, R. R. Jr. and Bampton, M. C.. Coupling of substructures for dynami analyses. AIAA Journal, 6(7):13131319, 1968.Google Scholar
[6] Eftang, J. L., Huynh, D. B. P., Knezevic, D. J., Rønquist, E. M., and Patera, A. T.. Adaptive port reduction in static condensation. In Proceedings of 7th Vienna Conference on Mathematical Modelling (MATHMOD 2012), 2012.Google Scholar
[7] Eftang, J. L. and Patera, A. T.. Port reduction in parametrized component static condensation: approximation and a posteriori error estimation. Int. J. Numer. Methods Eng., 96(5):269302, 2013.Google Scholar
[8] Gadeschi, G. B., Backhaus, K., and Knoll, G.. Numerical analysis of laser-textured piston-rings in the hydrodynamic lubrication regime. Journal of Tribology, 134(4):8 pages, 2012.Google Scholar
[9] Hesthaven, J. S., Rozza, G., and Stamm, B.. Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics, 2015.Google Scholar
[10] Hurty, W. C.. Dynamic analysis of structural systems using component modes. AIAA Journal, 3(4):678685, 1965.Google Scholar
[11] Huynh, D. B. P.. A static condensation reduced basis element approximation: Application to three-dimensional acoustic muffler analysis. International Journal of Computational Methods, 11(03):1343010, 2014.Google Scholar
[12] Huynh, D. B. P., Knezevic, D. J., and Patera, A. T.. A static condensation reduced basis element method: approximation and a posteriori error estimation. ESAIM: Math. Model. Num., 47:213251, 1 2013.CrossRefGoogle Scholar
[13] Huynh, D. B. P., Knezevic, D. J., and Patera, A. T.. A static condensation reduced basis element method: complex problems. Comput. Methods Appl. Mech. Engrg., 259(0):197216, 2013.Google Scholar
[14] Huynh, D. B. P., Rozza, G., Sen, S., and Patera, A. T.. A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math., 345(8):473478, 2007.Google Scholar
[15] Kango, S., Singh, D., and Sharma, R.. Numerical investigation on the influence of surface texture on the performance of hydrodynamic journal bearing. Meccanica, 47(2):469482, 2012.CrossRefGoogle Scholar
[16] Kumar Gupta, K., Kumar, R., Kumar, H., and Sharma, M.. Study on effect of surface texture on the performance of hydrodynamic journal bearing. International Journal of Engineering and Advanced Technology, 3(1):4954, 2013.Google Scholar
[17] Maday, Y. and Rønquist, E. M.. A reduced-basis element method. J. Sci. Comput., 17:447459, 2002. 10.1023/A:1015197908587.Google Scholar
[18] Maday, Y. and Rønquist, E. M.. The reduced basis element method: Application to a thermal fin problem. SIAM J. Sci. Comput., 26(1):240258, 2004.Google Scholar
[19] Murty, K.. Note on a bard-type scheme for solving the complementarity problem. Opsearch, 11:123130, 1974.Google Scholar
[20] Pinkus, O. and Sternlicht, B.. Theory of hydrodynamic lubrication. McGraw-Hill, 1961.Google Scholar
[21] Prud’homme, C., Rovas, D. V., Veroy, K., Machiels, L., Maday, Y., Patera, A. T., and Turinici, G.. Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluid. Eng., 124(1):7080, 2002.Google Scholar
[22] Rozza, G., Huynh, D. B. P., and Patera, A. T.. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Archives of Computational Methods in Engineering, 15(3):229275, 2008.Google Scholar
[23] Vallaghé, S. and Patera, A.. The static condensation reduced basis element method for a mixed-mean conjugate heat exchanger model. SIAM J. Sci. Comput., 36(3):B294–B320, 2014.Google Scholar
[24] Veroy, K., Prud’homme, C., Rovas, D. V., and Patera, A. T.. A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, 2003. AIAA Paper 2003-3847.Google Scholar