Meshless methods are an alternative procedure for solving partial differential equations in opposition to the numerical methods that require structured meshes. In this work, the meshless method with radial basis functions (MMRB) is compared to the finite difference method (FDM) for solving the Reynolds equation applied to lubricated finite bearing applications. The performance of these two methods is compared based on the precision of estimating the normal force applied to the sliding surface of the bearing. Different mesh families are tested for different bearing configurations. Results show that the MMRB is better than the FDM for nonrectangular geometries with coarser meshes. For rectangular domains without discontinuities, the FDM is still the best choice for solving the problem.

References

References
1.
Belytschko
,
T.
,
Krongauz
,
Y.
,
Organ
,
D.
,
Fleming
,
M.
, and
Krysl
,
P.
,
1996
, “
Meshless Methods: An Overview and Recent Developments
,”
Comp. Meth. Appl. Mech. Eng.
,
139
, pp.
3
47
.10.1016/S0045-7825(96)01078-X
2.
Viana
,
S. A.
,
Rodger
,
D.
, and
Lai
,
H. C.
,
2007
, “
Overview of Meshless Methods
,”
Int. Comp. Soc. News.
,
14
, pp.
3
6
. Available at: http://www.compumag.org/jsite/images/stories/newsletter/ICS-07-14-2-Rodger.pdf
3.
Li
,
J.
,
Cheng
,
A. H. D.
, and
Chen
,
C. S.
,
2003
, “
A Comparison of Efficiency and Error Convergence of Multiquadric Collocation Method and Finite Element Method
,”
Eng. Anal. Bound Cond.
,
27
, pp.
251
257
.10.1016/S0955-7997(02)00081-4
4.
Kansa
,
E. J.
,
1990
, “
Multiquadrics - A Scattered Data Approximation Scheme With Application to Computational Fluid-Dynamics
,”
Comput. Math. Appl.
,
19
(
8–9
), pp.
127
161
.10.1016/0898-1221(90)90270-T
5.
Wang
,
J. G.
, and
Liu
,
G. R.
,
2002
, “
A Point Interpolation Meshless Method Based on Radial Basis Functions
,”
Int. J. Num. Meth. Eng.
,
54
, pp.
1623
1648
.10.1002/nme.489
6.
Li
,
M.
,
Chen
,
C. S.
, and
Tsai
,
C. H.
,
2010
, “
Meshless Method Based on Radial Basis Functions for Solving Parabolic Partial Differential Equations With Variable Coefficients
,”
Numer. Heat Transf. B
,
57
(
5
), pp.
333
347
.10.1080/10407790.2010.481489
7.
Yao
,
G.
,
Sarler
,
B.
, and
Chen
,
C. S.
,
2011
, “
A Comparison of Three Explicit Local Meshless Methods Using Radial Basis Functions
,”
Eng. Anal. Bound. Element.
,
35
(
3
), pp.
600
609
.10.1016/j.enganabound.2010.06.022
8.
Madych
,
W. R.
,
1992
, “
Miscellaneous Error Bounds for Multiquadric and Related Interpolators
,”
Comput. Math. Appl.
,
24
(
12
), pp.
121
138
.10.1016/0898-1221(92)90175-H
9.
Schaback
,
R.
,
1995
, “
Error Estimates and Condition Numbers for Radial Basis Function Interpolation
,”
Adv. Comput. Math.
,
3
, pp.
251
264
.10.1007/BF02432002
10.
Pandey
,
A. K.
,
Pratap
,
R.
, and
Chau
,
F. S.
,
2007
, “
Analytical Solution of the Modified Reynolds Equation for Squeeze Film Damping in Perforated MEMS Structures
,”
Sens. Actuat. A
,
135
, pp.
839
848
.10.1016/j.sna.2006.09.006
11.
Chasalevris
,
A.
, and
Sfyris
,
D.
,
2013
, “
Evaluation of the Finite Journal Bearing Characteristics, Using the Exact Analytical Solution of the Reynolds Equation
,”
Tribol. Int.
,
57
, pp.
216
234
.10.1016/j.triboint.2012.08.011
12.
Smith
,
G. D.
,
1985
,
Numerical Solution of Partial Differential Equations
,
3rd ed.
,
Oxford University Press
,
New York
.
13.
Li
,
J.
,
Hon
,
Y. C.
, and
Chen
,
C. S.
,
2002
, “
Numerical Comparisons of Two Meshless Methods Using Radial Basis Functions
,”
Eng. Anal. Bound. Element.
,
26
, pp.
205
225
.10.1016/S0955-7997(01)00101-1
14.
Powel
,
H.
, and
Barraco
, V
.
,
2002
, “
A Comparison Analysis Between Unsymmetric and Symmetric Radial Basis Function Collocation Methods for the Numerical Solution of Partial Differential Equations
,”
Comput. Math. Appl.
,
43
, pp.
551
583
10.1016/S0898-1221(01)00305-4.
15.
Curzon
,
A. E.
,
1991
, “
A Set of Solutions of Reynolds Equation for Finite Slider Bearings
,”
Tribol. Int.
,
24
(
4
), pp.
207
212
.10.1016/0301-679X(91)90045-B
16.
George
,
P. L.
,
1991
,
Automatic Mesh Generation—Application to Finite Element Methods
,
Wiley
,
New York.
17.
Russo
,
F. H.
, and
Santos
, I
. F.
,
1998
, “
Tilting-Pad Journal Bearings With Electronic Radial Injection
,”
ASME J. Tribol.
,
120
(3), pp.
583
594
.10.1115/1.2834591
18.
Chen
,
Y.
,
Davis
,
T. A.
, and
Hager
,
W. W.
, and
Rajamanickam
,
S.
,
2008
, “
Algorithm 887: CHOLMOD, Supernodal Sparse Cholesky Factorization and Update/Downdate
,”
ACM Trans. Math. Softw.
,
35
(
3
), pp.
22
, 1–14.10.1145/1391989.1391995
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