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Adaptive finite element methods for hydrodynamic lubrication with cavitation


INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2007; 72:1584–1604 Published online 2 April 2007 in Wiley InterScience (www.interscience.wiley.com). DOI:

10.1002/nme.2051

Adaptive nite element methods for hydrodynamic lubrication with cavitation
Bertil Nilsson1, , and Peter Hansbo2
Surfaces Research Group, School of Business and Engineering, Halmstad University, Halmstad, Sweden 2 Department of Applied Mechanics, Chalmers University of Technology, Goteborg S-412 96, Sweden
1 Functional

SUMMARY We present an adaptive nite element method for a cavitation model based on Reynolds’ equation. A posteriori error estimates and adaptive algorithms are discussed, and numerical examples illustrating the theory are supplied. Copyright q 2007 John Wiley & Sons, Ltd.
Received 31 May 2006; Revised 11 January 2007; Accepted 19 February 2007 KEY WORDS:

Reynolds’ equation; cavitation; nite element; penalty

1. INTRODUCTION The motivation for this work is the need for accurate computations of the hydrostatic pressure in a lubricant entrapped between the tool and workpiece in a metal-forming process or in a sliding bearing. The ultimate goal is to be able to optimize the surface structure so as to take advantage of cavitation effects in the lubricant. Often, the computations performed in order to assess the effects of surface pit geometries are based on highly simplied assumptions, see, e.g. Etsion et al. [1, 2], Wang et al. [3]. We propose to instead solve the full model numerically as part of an optimization loop. To this end, we here initiate a study of adaptive nite element modelling of hydrodynamic lubrication including cavitation effects. The cavitation will introduce steep pressure gradients that cannot be resolved on a coarse computational mesh. Consequently, an adaptive algorithm to automatically rene the mesh locally, based on error estimation, is crucial for accurate results. To our knowledge, the only
Correspondence

to: Bertil Nilsson, Functional Surfaces Research Group, School of Business and Engineering, Halmstad University, Halmstad, Sweden. E-mail: bertil.nilsson@set.hh.se Contract/grant sponsor: The Swedish Foundation for Strategic Research through the ProViking Graduate School

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2007 John Wiley & Sons, Ltd.

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paper dealing with adaptivity for this problem proper is Wu and Oden [4], where an a priori error estimate was used as error indicator. Using such an approach, the problem of unknown constants precludes accurate estimation of the error and allows only error indication, i.e. information about where the error is large (in a relative sense). We follow instead the a posteriori approach laid out in Johnson and Hansbo [5], and in particular Johnson [6], which deals with a problem closely related to ours (however, without numerical examples). In this paper, we will focus on control of the error in energy-like norms (root-mean-square control of pressure gradients) and goal-oriented adaptive control for functionals of the error (e.g. of the pressure resultant). The cavitation problem that we consider can be written as a variational inequality. We will use a penalty approach to reformulate the problem as a variational equality, and our adaptive algorithm will be closely tied to the penalty formulation. For a more general method for error control of variational inequalities, applicable also to the problem at hand, we refer to Suttmeier [7]. We emphasize that the basic theory for a posteriori error estimates for the problem at hand is not new but was given by Johnson [6] for energy norm and by French et al. [8] for pointwise errors. Our contribution lies in the more general error estimates, in the implementation details, and in the numerical examples.

2. THE CONTINUOUS PROBLEM Consider a thin lubricant with viscosity enclosed between two surfaces 1 and motion. We assume that 1 (identied with the x y-plane) is stationary and that velocity v = (V, 0, 0). The Reynolds’ equation can then be written as (H 3 P) = 6 V *H *x
2 2

in relative moves with

where H (x, y) is the local thickness of the lubricant lm, and P is the pressure. For the physical reasoning behind this model, see, e.g. Capriz and Cimatti [9]. We assume that P is zero at the boundaries of the domain of interest (zero taken as the atmospheric pressure). The lubricant cannot support subatmospheric pressure, so an additional condition is P 0 in . In order to incorporate this condition into the model, it is often written as a variational inequality as follows. Let c be a typical thickness of the lm and set p := Pc2 , 6 V d := H , c and f := *d *x

Then the cavitation model can be formulated as follows: let
1 K = {v ∈ H0 ( ) : v 0}

and seek p ∈ K such that d 3 p (v p) d see [9].
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

f (v p) d

v ∈ K

(1)

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In order to solve the cavitation problem numerically, we rst introduce the following regularized 1 version of (1): given a small penalty parameter ∈ R+ , we seek p ∈ H0 ( ) such that d 3 p v d + where (s) = 0, s/ , s 0 s<0 d 3 ( p )v d = fvd
1 v ∈ H0 ( )

(2)

This formulation was studied by Scholz [10] in the context of obstacle problems, and was used as a starting point for formulating a posteriori error estimates by Johnson [6] (see also Wu [11] for application to the problem at hand). From [10], we know that the solution of (2) converges to the solution of (1) in the sense that d 3 | p p |2 d C f2d

The idea used in [6, 10] was to tie to the mesh size h in a nite element method for solving (2). In order to make dimensional sense (which is important for the conditioning of the discrete system of equations), it is clear that h 2 , in which case the error in the penalty formulation is of the same order as the discretization error of a linear nite element method. Thus, this approach is best suited for low-order nite element methods (linear and bilinear). For higher-order nite elements, we will either have a penalty error dominating the discretization error or, alternatively, with h q , q>2, obtain a badly conditioned system of equations. 3. FINITE ELEMENT APPROXIMATION 3.1. Formulation Let T = {T } be a locally quasiuniform triangulation of into simplexes T of local mesh size h (in the following regarded as a piecewise constant function such that h(x) = h|T for x ∈ T ) and let
1 Vh = {v ∈ H0 ( ): v|T ∈ P 1 (T ), T ∈ T}

i.e. we will use constant-strain triangles. Furthermore, we will tie the penalty parameter local mesh size, following [6, 10], according to = 1 h 2 , where is a constant. We seek ph ∈ Vh such that d 3 ph v d + or, explicitly, d 3 ph v d + where we used the notation w := min(w, 0)
Copyright q 2007 John Wiley & Sons, Ltd.
d 3 h 2 ph v d =

to the

d 3 ( ph )v d =

fvd

v ∈ Vh

(3)

fvd

v ∈ Vh

(4)

Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

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This non-linear problem we solved iteratively using xed-point iterations. We lumped the mass matrix resulting from the penalty term using nodal quadrature, and the condition ph 0 was checked nodewise. In all nodes where ph <0, penalty was applied. 3.2. A posteriori error control in the natural norm We consider error control in the natural norm induced by the nite element formulation, following Johnson [6], denoted by e = p ph . We have the following a posteriori error representation d 3/2 e where
= h f + (d 3 ph ) d 3 h 2 ph L 2 (T ) L2( )

+ d 3/2

1/2 1

h

e

L2( )

C1

T ∈T

T

+ C2

T ∈T

*T

(5)

T

*T

= 1 h 1/2 d 3 [n ph ] 2

L 2 (*T )

with [n ph ] denoting the jump in normal derivative across element sides *T , and [v] = v+ v v
+

on *Tint , on *T* ,

v ± = lim v(x nT )
↓0

This is a consequence of the following argument, given in [6]. We rst note the orthogonality relation d 3 e v d + Then, with of v → v ,
h

d 3 h 2 e v d = 0

v ∈ Vh

(6)

a suitable interpolant (e.g. that of Cl ment [12]) onto Vh , and using the monotonicity e (v w )(v w) (v w )2

we have d 3/2 e
2 L2( )

+ d 3/2

1/2 1 2 h e L2( )

=

d 3 |e|2 d + d 3 |e|2 d +

d 3 h 2 |e |2 d d 3 h 2 e e d d 3 h 2 p e d
d 3 h 2 ph e d d 3 h 2 ph e d

= =
Copyright q 2007 John Wiley & Sons, Ltd.

d 3 p e d + d 3 ph e d fed

d 3 ph e d

Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

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=

f (e

h e) d



d 3 ph (e
h e) d

h e) d

d 3 h 2 ph (e

Using integration by parts followed by Cauchy’s inequality and interpolation estimates yielding the constants C1 = sup
1 v∈H0 (

h 1 v h v L 2 (T ) , d 3/2 v L 2 (T ) )

C2 =

sup
1 v∈H0 ( )

h 1/2 v d 3/2 v

h v L 2 (*T ) L 2 (T )

the error representation formula (5) follows. It is clear that the constants C1 and C2 cannot be computed exactly, but they may be estimated as approximate solutions to the eigenvalue problem of nding u ∈ H 1 (K ) and ∈ R such that d 3 u v dx =
T T

(u

h u)(v



h v) dx

v ∈ H 1 (K )

Then C1 is given by C1 = max / h; C 2 is computed analogously. (For this computation, it is easier to let h denote the nodal interpolant, which, however, requires more smoothness than that assumed in the interpolation estimate in order to make sense, see [12]. This is a technical point of no practical importance in the current context.) For example, on an equilateral triangular element √ with H 1 (K ) replaced by P 2 (K ), and assuming d constant and h := 2 meas(T ), we nd C1 ≈ 0.501d 3/2 , 3.3. Goal-oriented a posteriori error control We next consider error control for functionals of the error, or ‘quantities of interest’, following Becker and Rannacher [13]. The total error in this approach is found as the product of two terms: 1. The residual, obtained by plugging the nite element solution into the differential equation. This quantity measures the inability of the nite element solution to solve the equation in a pointwise sense and is completely local. 2. The solution z of a linearized continuous dual problem; a generalized Green’s function (a.k.a. inuence function) which gives information about the effect of the local error upon the quantity of interest. The important questions to address in this context are the linearizarion of the dual problem and the computation of the dual solution z. In the following, we will discuss different ways to deal with these questions. We begin with a brief description of the goal-oriented approach in the present setting. Denote by e = p ph and e = ( p ) ( ph ). Dening the residual R ∈ H 1 ( ) by R, v :=
Copyright q



C2 ≈ 0.635d 3/2

fvd

d 3 ph v d

d 3 ( ph )v d

1 v ∈ H0 ( )

(7)

2007 John Wiley & Sons, Ltd.

Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

ADAPTIVE FINITE ELEMENT METHODS FOR HYDRODYNAMIC LUBRICATION

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where , denotes duality pairing, we have the following error equation: d 3 e v d + and the orthogonality property d 3 e v d + d 3e v d = 0 v ∈ Vh (9) d 3 e v d = R, v
1 v ∈ H0 ( )

(8)

In order to estimate functionals of the error, we follow [5, 13] and argue by duality as follows. For briefness of notation, assume that we have a general non-linear variational problem: nd p ∈ V such that A( p)v d = and a FEM counterpart: nd ph ∈ Vh such that A( ph )v d = fvd v ∈ Vh fvd v ∈ V

We then have the following Galerkin orthogonality property: (A( p) A( ph ))v d = 0 or B ( p ph )v d = 0 where B :=
0 1

v ∈ Vh

v ∈ Vh

A ( p + (1 ) ph ) d

(10)

Note that B constitutes an exact linearization, since
1 0

d A( p + (1 ) ph ) d = [A( p + (1 ) ph )]1 = A( p) A( ph ) 0 d
1 0

and
1 0

d A( p + (1 ) ph ) d = d

A ()

d ( p + (1 ) ph ) d = B ( p ph ) d

Next, we must dene a linearized dual continuous problem as follows: nd z such that BT z = g where g can be chosen freely, and where B T is the adjoint of B, dened by B T pv d =
Copyright q 2007 John Wiley & Sons, Ltd.

pB v d
Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

1590 for all p ∈ V and v ∈ V . Then

B. NILSSON AND P. HANSBO

( p ph )g d = = = =

( p ph )B T z d B ( p ph )z d (A( p) A( ph ))z d ( f A( ph ))(z
h z) d

(11)

To obtain an error estimate involving a quantity of interest, a suitable g must be chosen. In our case we are interested in the error in the pressure resultant, in which case g = 1 is the proper choice. We note also that in a scalar case B can be computed directly by B= A( p) A( ph ) p ph if p ph = 0, B =0 otherwise (12)

However, computing B exactly still requires knowledge of the exact solution and two possible practical strategies are: Use the rectangle rule for computing the integral in (10) to obtain B( p, ph ) ≈ A ( ph ). This is usually an inexpensive method since A is represented by the Newton matrix which is normally computed anyway in a non-linear iteration scheme, but it does introduce a rather severe linearization error. Compute an improved approximation ph of p and use B( p, ph ) ≈ B( ph , ph ). This requires is computed to a higher accuracy than a substantial additional computational effort since ph ph . Of course, ph can then be used as the solution used for design purposes, but technically the error is computed for ph , not ph . If we want to be economical, we then need some information as to how much the error decreases going from ph to ph which may not be so easy to come by. In this paper we shall consider and compare these two strategies applied to the lubrication problem. 1 Approximate linearization: Here we introduce the adjoint problem of nding z ∈ H0 ( ) such that d 3 z v d + for a given g, where we note that ( ph ) =
Copyright q

( ph )zv d =

gv d

1 v ∈ H0 ( )

(13)

0,
1

ph 0 , ph <0
Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

2007 John Wiley & Sons, Ltd.

ADAPTIVE FINITE ELEMENT METHODS FOR HYDRODYNAMIC LUBRICATION
1 ‘Exact’ linearization: Here we use (12) and search z ∈ H0 ( ) such that

1591

d 3 z v d + where b(x) =

bzv d =

gv d

1 v ∈ H0 ( )

(14)

d 3 e (x)/e(x) if e(x) = 0 0 if e(x) = 0

and we note that 0 b d 3 / . This idea was introduced by French et al. [8] to obtain maximum norm control of the error for a closely related problem. Having solved the dual problem numerically, we make the specic choice of v = e and deduce, following (11), that ge d ≈ f (z
h z) d



d 3 ph (z

h z) d



d 3 ( ph )(z

h z) d

(15)

This relation is then used for both the approximate and exact linearization cases. The strategy for numerical evaluation of the error is now as follows. In order to solve (14) approximately, we need to approximate b. This we do by solving the problem (3) on two meshes, the one used to compute ph and one where all elements have been divided once more. On the ner mesh, we also solve the dual problem (14) or, in case of the approximate linearization, (13). This is done in order to evaluate approximatively z h z. For indication of which elements that are to be rened, we note that the integral over can be written as a sum of element integrals. The size of the element integrals is then used as an indicator; we rene the 30% of the elements with the highest indicator in each adaptive step. We shall in particular be concerned with the choice g = 1, since this gives us the error in the pressure resultant. The pressure resultant is the quantity of interest in the lubrication problem; the ultimate goal is to optimize the pit geometries in such a way as to maximize this quantity.

4. NUMERICAL EXAMPLES In order to investigate the performance of the methods proposed, a few numerical examples will be presented. Unfortunately, experimental results demonstrating in detail the local behaviour of the pressure image is, to our knowledge, not published. Though some integrated experimental results, such as lift, has been given by Etsion et al. [1, 2] and Wang et al. [3]. The only parameter involved in the algorithm and used in the numerical calculations is , the global constant which ties the penalty parameter to the local mesh size, according to = 1 h 2 . We remark that, since h → 0 as the mesh is rened, the strength of the penalty increases automatically in areas where the mesh is rened. These are also the areas in which the error contribution is estimated to be large. In all examples that follow we set = 1000. An important strength of the method is that is just a potentiometer for speed of convergence and does not inuence the nal solution. All integrals involved are integrated using two-point Gauss quadrature. The characteristic channel height c is in all examples dened to be equal to the nominal channel height and all other unimportant scaling parameters is chosen so that the product 6 V := 1.
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

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B. NILSSON AND P. HANSBO

We consider a simple one-dimensional chain-tooth according to Figure 1. The geometry of the tooth is simply a half square and the channel height is twice the peak height of the tooth. The boundary conditions are p = 0 at the inlet and outlet of the domain. A comparison between a pure Reynolds solution and the adaptive nite element cavitation method with a posteriori error control in the natural norm is given in Figure 2. We can see that we essentially reconstruct the behaviour of the classical solution when the continuity boundary conditions are applied. The important note is of course that we do not need to a priori dene the boundary location between the uid and cavitation phases. As can be seen in Figure 2, the classical approach of rst computing p R from a pure Reynolds solution, followed by approximating p ≈ max(0, p R ) (known as the half-Sommerfeld condition, cf. [14]) is not very accurate.

Figure 1. One-dimensional tooth model.

Figure 2. One-dimensional tooth model. Comparison of pure Reynolds solution with proposed cavitation model.
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

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Figure 3. One-dimensional tooth model. Adaptivity renement progress.

Figure 4. One-dimensional tooth model. Convergence of a posteriori error representation according to relation (5).
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

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Figure 5. Hemi-spherical oil-pocket model.

Figure 6. Hemi-spherical oil-pocket model. Pure Reynolds solution on a coarse mesh.

In Figure 3 we visualize the adaptivity renement progress inserting new nodes. In each step, the elements that give the largest third of the element contributions to the total a posteriori error according to relation (5) are subdivided into two new ones. The decrease of the total error according to renement, measured by the number of nodes, is presented in Figure 4.
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

ADAPTIVE FINITE ELEMENT METHODS FOR HYDRODYNAMIC LUBRICATION

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Figure 7. Hemi-spherical oil-pocket model. Pure Reynolds solution on a semi-ne mesh.

Figure 8. Cavitation model. Initial mesh.

Copyright q

2007 John Wiley & Sons, Ltd.

Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

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Figure 9. Hemi-spherical oil-pocket model. Adaptivity progress.

Figure 10. Hemi-spherical oil-pocket model. Pressure iso levels.

4.2. Two-dimensional oil-pocket We consider a two-dimensional oil-pocket in the shape of a hemi-sphere, Figure 5. The geometry is furnished as follows. The channel height is twice the impact depth and half of the impact radius. Boundary condition is p = 0 on external boundary of the domain.
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

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Figure 11. Hemi-spherical oil-pocket model. Cavitation model, solution on a ne mesh.

Figure 12. Hemi-spherical oil-pocket model. Cavitation model, convergence of pressure in natural norm.
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

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Figure 13. Hemi-spherical oil-pocket model. Cavitation model, convergence of lift.

Figure 14. Innite array of pockets.

First a pure Reynolds solution on a coarse mesh, Figure 6, followed by solution on a semi-ne one, Figure 7. The following four Figures 8–11 illustrate error control in the natural norm starting with the initial mesh and after adaptive renement step 20. We can clearly see the focus towards regions with high gradients in the solution as a result of the error-oriented adaptive renement process.
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

ADAPTIVE FINITE ELEMENT METHODS FOR HYDRODYNAMIC LUBRICATION

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Figure 15. Innite array of pockets. Cavitation model, pressure distribution in one cell.

Figure 16. Innite array of pockets. Cavitation model, pressure distribution in one cell.
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

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B. NILSSON AND P. HANSBO

Figure 17. Innite array of pockets. Cavitation model, lift as function of area ratio d/w.

Figure 18. Innite array of pockets. Cavitation model, contour plot of lift according to area ratio d/w and depth/diameter ratio h/d.
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

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Figure 19. One-dimensional ‘hemi-spherical’ oil-pocket model. Bundle of pressure curves according to impact depth.

Figure 20. One-dimensional ‘hemi-spherical’ oil-pocket model. Lift according to impact depth.

Copyright q

2007 John Wiley & Sons, Ltd.

Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

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Figure 21. Hemi-spherical oil-pocket model. Convergence of residual error representation according to relation (15). Each dot represents an adaptive step.

The convergence of the pressure solution in the natural norm is shown in Figure 12, and the convergence of lift in Figure 13. 4.3. Optimization of two-dimensional oil-pocket area ratio We consider the same two-dimensional single oil-pocket layout as in the previous example, but now assembled as an innite array of quadrilateral cells, containing one centred pocket each, Figure 14. Our aim is to investigate how the lift is inuenced by the size and depth of the oil-pocket. One cell is considered and cyclic boundary conditions are imposed using multipoint constraints in order to simulate the innite array. The analysis is summarized in Figures 15–18. In the last gure, d/w denotes the oil-pocket impact diameter over cell width and h/d is the oil-pocket impact depth over diameter. Inspecting Figure 17 we can identify an optimal impact diameter but hardly do the same for the impact depth, which, however, is clearly identiable in a one-dimensional setting as evident in the last two gures: in Figure 19 the pressure curves for increasing impact depth clearly indicate a certain depth better than the others; the total lift as function of centre impact depth over nominal clearance between the surfaces accompanies the scenario, Figure 20. The worrisome question as to why the two-dimensional case is less predictive than the one-dimensional has to be left as an open question for the moment (cf. Section 5).
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

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Figure 22. Hemi-spherical oil-pocket model. Convergence in lift. Each dot represents an adaptivity step.

4.4. Goal-oriented a posteriori error control and comparison We consider again the two-dimensional oil-pocket with the shape of a hemi-sphere, Figure 5. In Figure 21 we present the decrease of the total residual error 1, e representation given in relation (15) relative to the number of nodes, for the two different strategies to compute the tangent matrix B. Finally, in Figure 22 we compare, relative to the convergence in lift, the formulation of error control in the natural norm with the two goal-oriented strategies.

5. DISCUSSION As expected, the goal-oriented nite element method is more effective in predicting the lift even for a rather coarse mesh. On the other hand, more work is needed at each renement step for the goal-oriented method, so it is not obvious to judge if one method is in favour over the other. Error estimation in the natural norm cannot give us information about the error in lift, but if the change in computational lift is monitored separately, the simpler natural norm adaptivity must, for our purposes, be considered sufciently good for adapting the mesh. In order to optimize and fully investigate the shape of the oil-pockets, our results indicate that the two-dimensional model does not have a good predictive quality. This we conjecture depends on the shortcomings of the Reynolds thin lm model. We believe that the simulations must rely on more accurate modelling, using incompressible Stokes or Navier–Stokes ow, at least in regions with rapidly varying height. Thus, future work will focus on the coupling of narrow regions with relative non-narrow regions (e.g. oil-pockets) using different physical models.
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme

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REFERENCES 1. Etsion I, Burstein L. A model for mechanical seals with regular microsurface structure. Tribology Transactions 1996; 39:677–683. 2. Etsion I, Kligerman Y, Halperin G. Analytical and experimental investigation of laser-textured mechanical seal faces. Tribology Transactions 1999; 42:511–516. 3. Wang X, Kato K, Adachi A, Aizawa K. Load carrying capacity map for the surface texture design of SiC thrust bearing sliding in water. Tribology International 2003; 36:189–197. 4. Wu SR, Oden JT. A note on applications of adaptive nite elements to elastohydrodynamic lubrication problems. Communications in Applied Numerical Methods 1987; 3:485–494. 5. Johnson C, Hansbo P. Adaptive nite element methods in computational mechanics. Computer Methods in Applied Mechanics and Engineering 1992; 101:143–181. 6. Johnson C. Adaptive nite element methods for the obstacle problem. Mathematical Models and Methods in Applied Sciences 1992; 2:483–487. 7. Suttmeier F-T. General approach for a posteriori error estimates for nite element solutions of variational inequalities. Computational Mechanics 2001; 27:317–323. 8. French DA, Larsson S, Nochetto RH. Pointwise a posteriori error analysis for an adaptive penalty nite element method for the obstacle problem. Computational Methods in Applied Mathematics 2001; 1:18–38. 9. Capriz G, Cimatti G. Free boundary problems in the theory of hydrodynamic lubrication: a survey. In Free Boundary Problems: Theory and Applications, vol. II, Fasano A, Primicerio M (eds). Pitman: Boston, 1983; 613–635. 10. Scholz R. Numerical solution of the obstacle problem by the penalty method. Computing 1984; 32:297–306. 11. Wu SR. A penalty formulation and numerical approximation of the Reynolds–Hertz problem of elastohydrodynamic lubrication. International Journal of Engineering Science 1984; 24:1001–1013. 12. Cl ment P. Approximation by nite element functions using local regularization. RAIRO Mod lisation e e Math matique et Analyse Num rique 1975; 9 R-2:77–84. e e 13. Becker R, Rannacher R. A feed-back approach to error control in nite element methods: basic analysis and examples. East-West Journal of Numerical Mathematics 1996; 4:237–264. 14. Fr ne J, Nicolas D, Degueurce B, Berthe D, Godet M. Hydrodynamic Lubrication. Elsevier: Amsterdam, 1997. e

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Int. J. Numer. Meth. Engng 2007; 72:1584–1604 DOI: 10.1002/nme


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