A fast compound direct iterative algorithm for solving transient line contact elastohydrodynamic lubrication problems

Abstract

A fast compound direct iterative algorithm for solving transient line contact elastohydrodynamic lubrication (EHL) problems is presented. First, by introducing a special matrix splitting iteration method into the traditional compound direct iterative method, the full matrices for the linear systems of equations are transformed into sparse banded ones with any half-bandwidth; then, an extended Thomas method which can solve banded linear systems with any half-bandwidth is derived to accelerate the computing speed. Through the above two steps, the computational complexity of each iteration is reduced approximately from O\left(N^{\mathrm{3}}/\mathrm{3}\right) to O\left({\beta }^{\mathrm{2}}N\right), where N is the total number of nodes, and \beta is the half-bandwidth. Two kinds of numerical results of transient EHL line contact problems under sinusoidal excitation or pure normal approach process are obtained. The results demonstrate that the new algorithm increases computing speed several times more than the traditional compound direct iterative method with the same numerical precision. Also the results show that the new algorithm can get the best computing speed and robustness when the ratio, half-bandwidth to total number of nodes, is about 7.5%–10.0% in moderate load cases.

Key words： elastohydrodynamic lubrication; transient; line contact; matrix splitting iteration method; the Thomas method

Appendix

Tab.1

Notations
/\mathbf{A},/{\mathbf{A}}^{*}	iterative matrices
b	half width of contact zone (m)
C_i	loading coefficient
C_w	ratio of w_d to w_{\mathrm{0}}
D_{ki},D{X}_{ki}	influence coefficients
er{r}_p	pressure convergency criterion
er{r}_w	load convergency criterion
E^{\prime }	equivalent elastic modulus (Pa)
E_K^E,E_K^W,E_K^O	coefficients of differential equation
f	frequency of excitation (Hz)
G	dimensionless material parameter
h	film thickness (\mu m)
h_{min}	minimum film thickness (\mu m)
H	dimensionless film thickness
H_{\mathrm{00}}	dimensionless central offset film thickness
{\mathbf{L}}^{*}	lower triangular matrix
m	order of matrix
{\mathbf{M}}_{sub}	submatrix
N	total number of nodes
O(N)	proportional to N
p	pressure (Pa)
p_H	maximum Hertzian pressure (Pa)
P	dimensionless pressure
\mathbf{P}	pressure matrix
R	equivalent curvature radius (m)
t	time (s)
t_{old},t_{new}	calculation time (hour)
T	dimensionless time
T_s	average number of outer loop
u	entrainment velocity (m/s)
U	dimensionless velocity parameter
{\mathbf{U}}^{*}	upper triangular matrix
w	applied load per unit length (N/m)
w_{\mathrm{0}}	applied load per unit length under steady condition (N/m)
w_d	amplitude of dynamic load (N/m)
W	dimensionless load parameter
x	abscissa along rolling direction (m)
X	dimensionless abscissa
X_{inlet}	inlet point dimensionless abscissa
X_{outlet}	oulet point dimensionless abscissa
X^{\prime }	additional dimensionless abscissa
z	Roelands parameter
\alpha	Barus’s piezoviscous coefficient ({Pa}^{-\mathrm{1}})
\beta	half-bandwidth of iterative matrix
\gamma ,\sigma ,\tau	elements of L^{*} and U^{*}
ϵ	half-bandwidth of banded matrix
\eta	viscosity of lubricant (Pa\cdot s)
{\eta }_{\mathrm{0}}	inlet viscosity of lubricant (Pa\cdot s)
\overline{\eta }	dimensionless viscosity
\lambda	derivative of \overline{\rho } with respect to P
\rho	density of lubricant (kg/m^3)
{\rho }_{\mathrm{0}}	inlet density of lubricant (kg/m^3)
\overline{\rho }	dimensionless density
\xi	comprehensive parameter
{\omega }_P	low relaxation factor of /P
{\omega }_H	low relaxation factor of /H_{\mathrm{00}}
\mathrm{\Delta }T	step of dimensionless time
{\mathrm{\Omega }}_e	dimensionless excitation frequency
Subscripts
e,j,l	refers to the number of matrix elements
i	refers to the number of P_i
k	refers to the position of X_k
Superscripts
n	refers to discrete time T_n
s	refers to the number of iterations

References

1 KannanR. A high order spectral volume method for elastohydrodynamic lubrication problems: formulation and application of an implicit p-multigrid algorithm for line contact problems. Computers & Fluids, 2011, 48(1): 44-53

2 HabchiW, EyheramendyD, VergneP, Morales-EspejelG. Stabilized fully-coupled finite elements for elastohydrodynamic lubrication problems. Advances in Engineering Software, 2012, 46(1): 4-18

3 NoutaryM P, VennerC H, LubrechtA A. Grid generation in hydrodynamic and elastohydrodynamic lubrication using algebraic multigrid method. Proceedings of the Institution of Mechanical Engineers. Part J, Journal of Engineering Tribology, 2012, 226(5): 343-349

4 TanX, GoodyerC E, JimackP K, TaylorR I, WalkleyM A. Computational approaches for modelling elastohydrodynamic lubrication using multiphysics software. Proceedings of the Institution of Mechanical Engineers. Part J, Journal of Engineering Tribology, 2012, 226(6): 463-480

5 HamrockB J, DowsonD. Isothermal elastohydrodynamic lubrication of point contacts: part 1—theoretical formulation. Journal of Lubrication Technology, 1976, 98(2): 223

6 AiX, YuH. A full numerical solution for general transient elastohydrodynamic line contacts and its application. Wear, 1988, 121(2): 143-159

7 YangP, WenS. A forward iterative numerical method for steady-state elastohydrodynamically lubricated line contacts. Tribology International, 1990, 23(1): 17-22

8 DowsonD, HigginsonG R. A numerical solution to the elasto-hydrodynamic problem. Proceedings of the Institution of Mechanical Engineers Part C-Journal of Mechanical Engineering Science, 1959, 1(1): 6-15

9 DowsonD, HigginsonG R. Elasto-hydrodynamic lubrication. Oxford: Pergamon Press, 1977

10 EvansH P, SnidleR W. Inverse solution of Reynolds’ equation of lubrication under point-contact elastohydrodynamic conditions. Journal of Lubrication Technology, 1981, 103(4): 539-546

11 EvansHP, SnidleRW. The elastohydrodynamic lubrication of point contacts at heavy loads. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1982; 382(1782): 183-199

12 RohdeSM, OhKP. A unified treatment of thick and thin film elastohydrodynamic problems by using higher order element methods. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1975; 343(1634): 315-331

13 OhK P, RohdeS M. Numerical solution of the point contact problem using the finite element method. International Journal for Numerical Methods in Engineering, 1977, 11(10): 1507-1518

14 LubrechtA A, Ten NapelW E, BosmaR. Multigrid, an alternative method for calculating film thickness and pressure profiles in elastohydrodynamically lubricated line contacts. ASME Journal of Tribology, 1986, 108(4): 551-556

15 VennerC H. Multilevel solution of the EHL line and point contact problems. PhD Thesis, University of Twente, Endschende, the Netherlands, 1991

16 VennerC H. Higher-order multilevel solvers for the EHL line and point contact problem. ASME Journal of Tribology, 1994, 116(4): 741-750

17 NurgatE, ScalesL, BerzinsM. Solving EHL problems using iterative, multigrid, and homotopy methods. ASME Journal of Tribology, 1999, 121(1): 28-33

18 TaylorC, O'CallaghanJ F. A numerical solution of the elastohydrodynamic lubrication problem using finite elements. Proceedings of the Institution of Mechanical Engineers Part C-Journal of Mechanical Engineering Science, 1972, 14(4): 229-237

19 RoelandsC J A. Correlation aspects of the viscosity-temperature-pressure relationship of lubricating oils. PhD Thesis, Technische Hogeschool, Delft, the Netherlands, 1966

20 DowsonD, HigginsonG R. Elasto-hydrodynamic lubrication: the fundamentals of roller and gear lubrication. Oxford: Pergamon Press, 1966

21 ThomasL H. Elliptic problems in linear differential equations over a network. Watson Sci. Comput. Lab Report., Columbia University, New York, 1949