In order to obtain a fast solution scheme, the trajectory piecewise linear (TPWL) method is applied to the transient elastohydrodynamic (EHD) line contact problem for the first time. TPWL approximates the nonlinearity of a dynamical system by a weighted superposition of reduced linearized systems along specified trajectories. The method is compared to another reduced order model (ROM), based on Galerkin projection, Newton–Raphson scheme and an approximation of the nonlinear reduced system functions. The TPWL model provides further speed-up compared to the Newton–Raphson based method at a high accuracy.
Issue Section:
Elastohydrodynamic Lubrication
References
1.
Christensen
, H.
, 1962
, “The Oil Film in a Closing Gap
,” Proc. R. Soc. London A
, 266
(1326
), pp. 312
–328
.2.
Lubrecht
, A. A.
, 1987
, “The Numerical Solution of the Elastohydrodynamically Lubricated Line and Point Contact Problem, Using Multigrid Techniques
,” Ph.D. thesis, University of Twente, Enschede, The Netherlands.3.
Brandt
, A.
, and Lubrecht
, A. A.
, 1990
, “Multilevel Matrix Multiplication and Fast Solution of Integral Equations
,” J. Comput. Phys.
, 90
(2
), pp. 348
–370
.4.
Ren
, N.
, Zhu
, D.
, Chen
, W. W.
, Liu
, Y.
, and Wang
, Q. J.
, 2008
, “A Three-Dimensional Deterministic Model for Rough Surface Line-Contact EHL Problems
,” ASME J. Tribol.
, 131
(1
), p. 011501
.5.
Hu
, Y.-Z.
, and Zhu
, D.
, 1999
, “A Full Numerical Solution to the Mixed Lubrication in Point Contacts
,” ASME J. Tribol.
, 122
(1
), pp. 1
–9
.6.
Chen
, W. W.
, Liu
, S.
, and Wang
, Q. J.
, 2008
, “Fast Fourier Transform Based Numerical Methods for Elasto-Plastic Contacts of Nominally Flat Surfaces
,” ASME J. Appl. Mech.
, 75
(1
), p. 011022
.7.
Habchi
, W.
, and Issa
, J.
, 2013
, “Fast and Reduced Full-System Finite Element Solution of Elastohydrodynamic Lubrication Problems: Line Contacts
,” Adv. Eng. Software
, 56
, pp. 51
–62
.8.
Habchi
, W.
, 2014
, “Reduced Order Finite Element Model for Elastohydrodynamic Lubrication: Circular Contacts
,” Tribol. Int.
, 71
, pp. 98
–108
.9.
Maier
, D.
, Hager
, C.
, Hetzler
, H.
, Fillot
, N.
, Vergne
, P.
, Dureisseix
, D.
, and Seemann
, W.
, 2015
, “A Nonlinear Model Order Reduction Approach to the Elastohydrodynamic Problem
,” Tribol. Int.
, 82
, pp. 484
–492
.10.
Carlberg
, K.
, Bou-Mosleh
, C.
, and Farhat
, C.
, 2011
, “Efficient Nonlinear Model Reduction Via a Least-Squares Petrov–Galerkin Projection and Compressive Tensor Approximations
,” Int. J. Numer. Methods Eng.
, 86
(2
), pp. 155
–181
.11.
Carlberg
, K.
, Farhat
, C.
, Cortial
, J.
, and Amsallem
, D.
, 2013
, “The {GNAT} Method for Nonlinear Model Reduction: Effective Implementation and Application to Computational Fluid Dynamics and Turbulent Flows
,” J. Comput. Phys.
, 242
, pp. 623
–647
.12.
Rewienski
, M.
, and White
, J.
, 2003
, “A Trajectory Piecewise-Linear Approach to Model Order Reduction and Fast Simulation of Nonlinear Circuits and Micromachined Devices
,” IEEE Trans. Comput.-Aided Des. Integr. Circ. Syst.
, 22
(2
), pp. 155
–170
.13.
Rewienski
, M. J.
, 2003
, “A Trajectory Piecewise-Linear Approach to Model Order Reduction of Nonlinear Dynamical Systems
,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.14.
Albunni
, M. N.
, 2010
, “Model Order Reduction of Moving Nonlinear Electromagnetic Devices
,” Ph.D. thesis, Technische Universität München, Munich, Germany.15.
Nahvi
, S. A.
, un Nabi
, M.
, and Janardhanan
, S.
, 2013
, “Trajectory Piece-Wise Quasi-Linear Approximation of Large Non-Linear Dynamic Systems
,” Int. J. Model. Identif. Control
, 19
(4
), pp. 369
–377
.16.
Reynolds
, O.
, 1886
, “On the Theory of Lubrication and Its Application to Mr. Beauchamp Tower's Experiments, Including an Experimental Determination of the Viscosity of Olive Oil
,” Philos. Trans. R Soc. London
, 177
, pp. 157
–234
.17.
Wu
, S.
, 1986
, “A Penalty Formulation and Numerical Approximation of the Reynolds–Hertz Problem of Elastohydrodynamic Lubrication
,” Int. J. Eng. Sci.
, 24
(6
), pp. 1001
–1013
.18.
Flamant
, A.
, 1892
, “Sur la Répartition des Pressions dans un Solide Rectangulaire Chargé Transversalement
,” C.R. Acad. Sci.
, 114
, pp. 1465
–1468
.19.
Hertz
, H.
, 1881
, “Über die Berührung Fester Elastischer Körper
,” J. Reine Angew. Math.
, 92
, pp. 156
–171
.20.
Venner
, C.
, 1991
, “Multilevel Solution of the EHL Line and Point Contact Problems
,” Ph.D. thesis, University of Twente, Enschede, The Netherlands.21.
Volkwein
, S.
, 1999
, Proper Orthogonal Decomposition and Singular Value Decomposition
, Karl-Franzens-Universität Graz & Technische Universität Graz
, Graz, Austria
.22.
Goodyer
, C. E.
, 2001
, “Adaptive Numerical Methods for Elastohydrodynamic Lubrication
,” Ph.D. thesis, University of Leeds, Leeds, UK.23.
Dowson
, D.
, and Higginson
, G.
, 1966
, Elastohydrodynamic Lubrication: The Fundamentals of Roller and Gear Lubrication
, Pergamon Press
, Oxford, UK
.24.
Roelands
, C.
, 1966
, “Correlational Aspects of the Viscosity–Temperature–Pressure Relationship of Lubricating Oils
,” Ph.D. thesis, Technical University Delft, Delft, The Netherlands.Copyright © 2016 by ASME
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