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Research Papers: Hydrodynamic Lubrication

Comparison of Iterative Methods for the Solution of Compressible-Fluid Reynolds Equation

[+] Author and Article Information
Nenzi Wang

Department of Mechanical Engineering, Chang Gung University, 259 Wen-Hwa 1st Road, Tao-Yuan 333, Taiwannenzi@mail.cgu.edu.tw

Shih-Hung Chang

Department of Mechanical Engineering, Chang Gung University, 259 Wen-Hwa 1st Road, Tao-Yuan 333, Taiwand9622004@stmail.cgu.edu.tw

Hua-Chih Huang

Mechanical and Systems Research Laboratories, Industrial Technology Research Institute, No. 191, Gung Ye 38 Road, Taichung Industrial Area, Tai-Chung 407, Taiwanhuachih.huang@itri.org.tw

J. Tribol 133(2), 021702 (Mar 18, 2011) (7 pages) doi:10.1115/1.4003149 History: Received August 30, 2010; Revised November 20, 2010; Published March 18, 2011; Online March 18, 2011

This study presents an efficacy comparison of iterative solution methods for solving the compressible-fluid Reynolds equation in modeling air- or gas-lubricated bearings. A direct fixed-point iterative (DFI) method and Newton’s method are employed to transform the Reynolds equation in a form that can be solved iteratively. The iterative solution methods examined are the Gauss–Seidel method, the successive over-relaxation (SOR) method, the preconditioned conjugate gradient (PCG) method, and the multigrid method. The overall solution time is affected by both the transformation method and the iterative method applied. In this study, Newton’s method shows its effectiveness over the straightforward DFI method when the same iterative method is used. It is demonstrated that the use of an optimal relaxation factor is of vital importance for the efficiency of the SOR method. The multigrid method is an order faster than the PCG and optimal SOR methods. Also, the multigrid and PCG methods involve an extended coding work and are less flexible in dealing with gridwork and boundary conditions. Consequently, a compromise has to be made in terms of ease of use as well as programming effort for the solution of the compressible-fluid Reynolds equation.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

The convergence test for the air bearing running at 30,000 rpm and ε=0.9

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Figure 2

The pressure distributions along the centerline of the journal bearing in the direction of sliding

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Figure 3

The effect of convergence tolerance on the center-point pressure and CPU time. The air bearing model is linearized by Newton’s method and solved by the GS method.

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Figure 4

The CPU times for solving the air bearing model under various relaxation factors. Both cases are solved by the SOR method.

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Figure 5

The effect of outer iteration number on the center-point pressure of the bearing and CPU time. The model is linearized by Newton’s method and solved by the GS method.

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Figure 6

The effect of outer iteration number on the center-point pressure of the bearing and CPU time. The model is linearized by Newton’s method and solved by the SOR method.

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Figure 7

The variation of ‖u‖2 in the beginning of computation using the PCG method (ω=1.95)

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Figure 8

The CPU time for converged solution when inner iteration number is fixed. The bearing model is linearized by Newton’s method and solved by the PCG method (ω=1.95).

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Figure 9

The effect of outer iteration number on the center-point pressure of the bearing and CPU time. The model is linearized by Newton’s method and solved by the multigrid method.

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