Advances in Engineering Software 46 (2012) 4–18

Contents lists available at ScienceDirect

Advances in Engineering Software

journal homepage: www.elsevier .com/locate /advengsoft

Stabilized fully-coupled finite elements for elastohydrodynamiclubrication problems

W. Habchi a,⇑, D. Eyheramendy b, P. Vergne c, G. Morales-Espejel d

a Lebanese American University (LAU), Department of Industrial and Mechanical Engineering, Byblos, Lebanonb LMA, Ecole Centrale Marseille, CNRS UPR7051, 13451 Marseille Cedex 20, Francec Université de lyon, CNRS INSA-Lyon, LaMCoS, UMR 5259, F-69621 Villeurbanne, Franced SKF Engineering and Research Center, Nieuwegein, The Netherlands

a r t i c l e i n f o

Article history:Available online 13 November 2010

Keywords:Finite elementsElastohydrodynamic lubricationArtificial diffusionFull-system approachThermal effectsNon-Newtonian lubricants

0965-9978/$ - see front matter � 2010 Civil-Comp Ltdoi:10.1016/j.advengsoft.2010.09.010

⇑ Corresponding author.E-mail address: wassim.habchi@lau.edu.lb (W. Ha

a b s t r a c t

This work presents a model for elastohydrodynamic (EHD) lubrication problems. A finite element full-system approach is employed. The hydrodynamic and elastic problems are solved simultaneously whichleads to fast convergence rates. The free boundary problem at the contact’s exit is handled by a penaltymethod. For highly loaded contacts, the standard Galerkin solution of Reynolds equation exhibits anoscillatory behaviour. The use of artificial diffusion techniques is proposed to stabilize the solution. Thisapproach is then extended to account for non-Newtonian lubricant behaviour and thermal effects. Arti-ficial diffusion procedures are also introduced to stabilize the solution at high loads.

� 2010 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays, rolling element bearings are essential componentsin any mechanical system that includes rotating parts. This is be-cause the direct contact between the rotating parts may damagethem and lead to system failure. Moreover, direct contact leadsto high frictional losses and a significant increase in the energyconsumption of the system. Thus, bearings are introduced to pre-vent large surface metal-to-metal contact. In order to further im-prove the functioning of these components, they are lubricated.In other words, a lubricant (oil, grease, etc.) is introduced betweenthe contacting surfaces (rolling element and inner/outer raceway).It is therefore important for the good functioning of a bearing tohave a reliable tool to estimate film thicknesses and frictionallosses in the contact.

In this work, we are mainly interested in a particular lubricationregime known as elastohydrodynamic (EHD). In this regime, thecontacting bodies are separated by a full lubricant film. The pres-sure generated in the contact is high enough to induce a significantelastic deformation of the contacting bodies. Therefore, hydrody-namic and elastic effects are strongly coupled. When studyingthese contacts, it is not necessary to consider the often rather com-plex geometry of the contacting machine elements. Since the filmthickness and contact width are generally small compared to the

d and Elsevier Ltd. All rights reser

bchi).

local radius of curvature of the running surfaces, the surface geom-etry in the contact area can be accurately approximated by parab-oloids. This approximation allows a simplification of the contactgeometry. The latter can be reduced to the contact between aparaboloid and a flat surface. Two types of contacts may be distin-guished: line contacts (see Fig. 1) where the contacting elementsare assumed to be infinitely long in one of the principal directions(e.g.: cylindrical roller bearings) or point contacts (see Fig. 2)which occur between two parabolically shaped surfaces with localradii of curvature R1x and R2x in the x-direction and R1y and R2y inthe y-direction. A special case of a point contact is the circularcontact which occurs if the radii of curvature of the contacting ele-ments in both principal directions are equal. This paper is mainlyfocused on the circular contact configuration. However, whereneeded, some line contact results are shown for demonstrativepurposes.

In the past few decades, a large number of numerical solutions ofthe isothermal elastohydrodynamic lubrication (EHL) problememerged. Most of these studies used a weak coupling resolutionof the Reynolds (hydrodynamic) and elasticity equations. The twoequations were solved separately and an iterative procedure wasapplied between their corresponding solutions. A direct conse-quence of the weak coupling resolution is in general, a slowconvergence rate of the global system. This can be partially ex-plained by the use of underrelaxation techniques that are often nec-essary as a compensation to the loss of information that occursduring the iterative coupling process. Very few authors attemptedto solve the fully-coupled problem. Among them, we cite Oh and

ved.

u1

u2h(x)

Rz

x u1

u2h(x)

R1

R2

Fig. 1. EHL line contact and the equivalent reduced geometry.

u1

u2h(x,y)

Rxz

x u1

u2h(x,y)

R1x

R2x

R2y

R1y

Ry

Fig. 2. EHL point contact and the equivalent reduced geometry.

W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18 5

Rohde [1] who tried to solve the problem as one integro-differentialequation using Newton’s method, or also Holmes et al. [2] who usedthe differential deflection method to compute the elastic deforma-tion of the contacting bodies. Although these attempts succeeded infinding a solution of the EHD problem with outstanding conver-gence rates (a few iterations were enough to obtain a convergedsolution), they suffered from three major drawbacks. First, becauseof the simultaneous solution of all pressure updates, the implemen-tation of the cavitation condition at the outlet of the contact wasrather tedious and second, the elastic deformation calculationwas based on a half-space approach (Boussinesq solution). In thisapproach, the deformation at each point is related to pressure atall discretization points of the computational domain. This leadsto a full Jacobian matrix. Despite some attempts to reduce the sizeof this matrix by neglecting the smallest terms (e.g. see [2]), thecomputational effort and memory storage were still heavy. Finally,the Jacobian matrix was nearly singular for heavily loaded contactswhich made the resolution process under such conditions particu-larly difficult. This is probably why this approach did not have asmuch success as the weakly coupled one, mostly based on a finitedifference discretization of the corresponding equations and forwhich several efficient techniques were developed such as Multi-grid techniques that were introduced to lubrication problems byLubrecht [3] and further improved by Venner [4] who extendedthe solution to highly loaded contacts.

In an earlier work [5], the authors had already introduced a fi-nite element approach to isothermal Newtonian EHL problems.The hydrodynamic and elastic problems were fully-coupledwhereas an iterative procedure was established to ensure the loadbalance. The free boundary problem arising at the contact’s exitwas handled by a heavyside function. Later on, the authors ex-tended this approach to account for non-Newtonian lubricantbehaviour and thermal effects in lightly loaded contacts [6]. In thispaper, a finite element fully-coupled resolution procedure is intro-duced. The load balance equation is integrated directly into thematrix system, providing thus a full-coupling of the three EHLequations (hydrodynamic, elastic and load balance equations). Apenalty method is introduced to handle the free boundary problemin a straightforward manner. Appropriate stabilization techniquesare applied to the Reynolds equation in order to extend the solu-tion to highly loaded contacts. This approach has the advantageof providing a sparse Jacobian matrix, giving thus an efficient res-

olution of the fully-coupled problem with fast convergence rates.Then, this approach is extended to, first, non-Newtonian lubricantsand second, thermal effects under light and heavy loads. In bothcases, generalized forms of the Reynolds equation are considered.

2. Isothermal Newtonian approach

In this section, the lubricant is assumed to behave as a Newto-nian fluid and thermal effects are neglected in the contact (temper-ature throughout the lubricant film is considered constant andequal to the ambient temperature T0). The contacting surfacesare submitted to a prescribed external load F and have constantunidirectional surface velocities in the x-direction.

2.1. Mathematical model

This section presents the mathematical model used to describethe behaviour of EHD contacts under isothermal Newtonian re-gime. This model is the basis of the EHL solver presented in thiswork, from which, an extension to a more physical modelling canbe carried out.

2.1.1. EHL theory and equationsThree main equations define an EHL problem: the Reynolds

equation which describes the pressure distribution p in the contactarea, the linear elasticity equations which determine the elasticdeformation of the contacting elements and the load balance equa-tion which ensures that the correct load F is applied. All equationsare written in dimensionless form using the Hertzian dry contactparameters [7] (i.e. Hertzian contact pressure ph and Hertzian con-tact radius a). The Reynolds [8] equation describing the dimension-less pressure distribution P for a steady state circular contactproblem with unidirectional surface velocities u1 and u2 in the X-direction is given by:

r � ðerPÞ � @ð�qHÞ@X

¼ 0 ð1Þ

where e ¼ �qH3

�lk , k ¼ 12umlRR2

a3ph,um ¼ u1þu2

2 and R ¼ R1R2R1þR2

This equation stems from the Navier–Stokes equations to whichthe thin film simplifying assumptions are applied. H is the filmthickness. The dimensionless viscosity �l and density �q vary with

XZ

Y

Contact area

cΩ

bΩ∂ U=0

Fig. 3. Geometry of the EHL problem.

6 W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18

pressure throughout the contact domain Xc (see Fig. 3) making theproblem highly non-linear. The different laws of these transportproperties that are used in this work are provided in Appendix A.

Neglecting body loads, the linear elasticity equations consist infinding the displacement vector U = {u, v, w} over the computa-tional domain X such that:

divðrÞ ¼ 0 with r ¼ C esðUÞ ð2Þ

where r is the stress tensor, es the strain tensor and C the compli-ance matrix. The film thickness H contains three contributions:the separation of the solid bodies H0, the original undeformedgeometry and the elastic deflection of the components d:

HðX;YÞ ¼ H0 þX2 þ Y2

2þ dðX;YÞ with dðX; YÞ ¼ jwðX; YÞj ð3Þ

Finally, the load balance equation is written in dimensionless formas follows:Z

Xc

PðX;YÞdX ¼ 2p3

ð4Þ

where 2p/3 corresponds to the dimensionless external load. Thisequation ensures that the correct external load is applied. The latteris controlled by the value of the film thickness constant H0.

2.1.2. Boundary conditionsStarting with Reynolds equation, it is admitted that p equals the

ambient pressure at the boundary of the contact domain Xc. Inpractice, it is defined as zero and thus, the pressure solved for cor-responds to the pressure rise above the ambient level value:

p ¼ 0 on @Xc ð5Þ

Moreover, since the lubricant is assumed to be at liquid state in-side the film, pressures lower than the vapour pressure are physi-cally impossible. The fluid will cavitate and the pressure willremain constant and equal to the vapour pressure. Since in mostsituations the vapour pressure of the lubricant is of the same orderof magnitude as the ambient pressure which is very small com-pared to the contact pressure, the Reynolds’ [8] cavitation bound-ary condition requires that:

p P 0 on Xc and p ¼ rp �~nc ¼ 0 on the cavitation boundary

ð6Þ

where~nc is the outward normal vector to the outlet boundary of thecontact also called cavitation boundary. It is clear that determiningthe exact location of this boundary (specified by the appearance ofnegative pressures in the solution of Reynolds’ equation) is a freeboundary problem since the pressure distribution is not known ‘‘a

priori’’. It is important to note that the first part of Eq. (6) ensuresthat the cavitation of the lubricant and the film break-up at the out-let of the contact are taken into account, whereas the second partensures the mass conservation of the lubricant flow.

Finally, the boundary conditions of the elastic problem are de-fined as follows:

U ¼ 0 at the bottom boundary @Xb

rn ¼ r �~n ¼ �P at the contact area boundary Xc

rn ¼ 0 elsewhere

8><>: ð7Þ

The boundary conditions defined in this section, associated to thedifferent EHL equations provided in Section 2.1.1 completely definethe physical problem to be solved.

2.1.3. Geometrical and material considerationsA half-space configuration can be assumed because the size of

the contact is very small compared to the size of the solids. In prac-tice, the latter corresponds to a cube with large dimensions com-pared to the size of the contact. In a previous paper [5], theauthors showed that the dimensions of the cube should be at least60 times larger than the Hertzian contact radius so that it can beconsidered as a half-space. The contact domain Xc is located onthe centre of the upper face of the cube and extends fromX = �4.5 to 1.5 and Y = �3.0 to 3.0 (See Fig. 3). Note that Reynoldsequation is two-dimensional and is solved on Xc whereas the elas-ticity equations are three-dimensional and are solved on the entirecubic domain X.

Remark: In the line contact case, the half-space structure is asquare with large dimensions compared to the contact domainwhich this time is one-dimensional and is also located on thecentre of the upper side of the square. In addition, a plane strainanalysis is applied to the linear elasticity problem.

In order to simplify the computational model, an equivalentproblem is defined to replace the elastic deformation computationfor both contacting bodies under the same pressure distribution.The equivalent model is defined by applying Eq. (2) to a body thathas the following material properties (see Appendix B):

Eeq ¼E2

1E2ð1þ t2Þ2 þ E22E1ð1þ t1Þ2

½E1ð1þ t2Þ þ E2ð1þ t1Þ�2and

teq ¼E1t2ð1þ t2Þ þ E2t1ð1þ t1Þ

E1ð1þ t2Þ þ E2ð1þ t1Þ

Remark: For the particular case of the two contacting elementsbeing made out of the same material (E, t), the equivalent mate-rial properties are Eeq = E/2 and teq = t. Therefore, the total elas-tic deflection would be twice the elastic deflection of each body.

Moreover, by multiplying the equivalent Young’s Modulus by a/R the dimensionless displacement vector is obtained directly anddividing it by ph allows the use of the dimensionless pressure dis-tribution P as a pressure load in the contact area. Hence, the equiv-alent material properties become:

Eeq ¼E2

1E2ð1þ t2Þ2 þ E22E1ð1þ t1Þ2

½E1ð1þ t2Þ þ E2ð1þ t1Þ�2� a

Rphand

teq ¼E1t2ð1þ t2Þ þ E2t1ð1þ t1Þ

E1ð1þ t2Þ þ E2ð1þ t1Þð8Þ

The previous simplification is equivalent to considering that oneof the bodies is rigid while the other (that has the equivalent mate-rial properties defined in Eq. (8)) accommodates the total elastic

W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18 7

deflection of both surfaces. This avoids running a similar calculationtwice (once for every solid body).

2.2. Numerical model

This section describes the numerical model used for the model-ling of EHL contacts operating under an isothermal Newtonianregime.

2.2.1. The penalty methodAs pointed out previously, at the outlet of the contact a free

boundary problem arises. It is treated by means of a penalty meth-od. The latter was introduced to EHL problems by Wu [9]. This ap-proach introduces an additional penalty term to the Reynoldsequation (1) that becomes:

r � ðerPÞ � @ð�qHÞ@X

� n � P� ¼ 0 ð9Þ

where n is an arbitrary large positive number and P� = min (P, 0)corresponds to the negative part of the pressure distribution. Notethat the penalty term (�n � P�) has no effect where P P 0 and theconsistency of Reynolds’ equation is preserved. However, in the out-let region of the contact, where P < 0, the penalty term dominatesEq. (9), provided that the arbitrary constant n has a sufficiently largevalue. Hence, the negative pressures are forced towards zero by thepresence of the penalty term and the physical constraint that P P 0over the entire computational domain is automatically satisfied. Wualso showed that this approach satisfies the Reynolds’ boundaryconditions and thus the mass flow rate conservation throughoutthe contact. In addition, this method is very straightforward andeasy to implement.

2.2.2. Finite element formulationsThe full-system approach used in this work consists in solving

Reynolds equation, the linear elasticity equations and the load bal-ance equation simultaneously. In the Reynolds’ equation, thedimensionless film thickness H is replaced by its expression givenin Eq. (3)whereas the dimensionless viscosity �l and density �q arereplaced by one of the expressions provided in Appendix A. TheReynolds’ and linear elasticity equations are partial differentialequations whereas the load balance equation is an ordinary inte-gral equation that is added directly to the system along with theintroduction of an additional unknown H0. Hence, the unknownsof this model are the pressure distribution P, the elastic deforma-tion of the contacting elements U = {u, v, w} and the film thicknessconstant H0.

2.2.2.1. Galerkin formulation. A standard Galerkin formulation is ap-plied to the Reynolds’ and linear elasticity equations. The latter isobtained by multiplying each equation by a given weighting func-tion, WP and WU respectively, and integrating it over the corre-sponding domain of application. Finally, integration by parts isapplied to the equations revealing thus the boundary terms. Forthe sake of simplicity, the zero boundary terms are omitted. Theload balance equation is weighted by WH0 . The resultant systemof equations becomes:

Find (P, U, H0) e SP � SU � R such that 8ðWP;WU ;WH0 Þ 2SP � SU � R, one has:R

Xc�erP � rWPdXþ

RXc

�qH @WP@X dX�

RXc

n � P�WPdX ¼ 0RX�CesðUÞ � esðWUÞdXþ

RXc�P �~nWUdX ¼ 0R

XcPWH0 dX� 2p

3 WH0 ¼ 0

8>><>>: ð10Þ

where SP ¼ fP 2 H1ðXcÞ=P ¼ 0 on @Xcg and SU ¼ fU 2 H1ðXÞ=U ¼0 on @Xbg

Note that Reynolds’ equation is solved on the two-dimensionalcontact domain Xc whereas the linear elasticity equations aresolved on the three-dimensional domain X.

2.2.2.2. Approximated formulation. Let us now write the discreteform of the previous system of equations. Consider Xh ¼fX1; . . . ;Xneg a finite element partition of X such that: �X ¼ [ne

e¼1�Xe,

�X ¼ X [ @X, �Xe ¼ Xe [ @Xe and Xe \Xe0 = u if e – e0. ne denotes

the total number of elements in the partition while oX and oXe de-note respectively the boundaries of the domain X and the elementXe. Let Xce be the set of elements defined by Xce ¼ fXe \Xc=Xce–;gand let nce be the total number of elements belonging to Xce. LetSh

P � SP and ShU � SU . The discrete functions Ph and Uh defining these

spaces have the same characteristics as their analytical equivalentsP and U defined in the previous section. Moreover, Ph 2 Ll andUh 2 Ll0 where Ll and Ll0 are the sets of piecewise polynomials ofdegrees equal to l and l

0respectively defined within each element

Xe. The approximate functions PhðeÞ and UhðeÞ (within an elemente) of P and U respectively, are given by:

PhðeÞ ¼XnP

i¼1

PðeÞi NPi and UhðeÞ ¼XnU

i¼1

UðeÞi NUi ð11Þ

where PðeÞi and UðeÞi are the nodal values of P and U respectively,associated to the interpolation functions NPi and NUi within the ele-ment e (nP and nU being their respective numbers). The weighting

functions WP and WU are approximated in a similar way by WhðeÞ

P

and WhðeÞ

U respectively:

WhðeÞ

P ¼XnP

i¼1

W ðeÞPi NPi and WhðeÞ

U ¼XnU

i¼1

W ðeÞUi NUi ð12Þ

where W ðeÞP;i and W ðeÞ

U;i are the nodal values of the weight functions WP

and WU within the element e respectively. The discrete form of thesystem of Eq. (10) is obtained by replacing the field variables P andU by their discrete equivalents Ph and Uh respectively and theweighting functions WP and WU by Wh

P and WhU respectively:

Find ðPh;Uh;H0Þ 2 Shp � Sh

U � R such that 8ðWhP ;W

hU ;WH0 Þ 2

ShP � Sh

U � R, one has:

RXh

c�erPh � rWh

PdXþR

Xhc

�qH @WhP

@X dX�R

Xhcn � Ph�Wh

PdX ¼ 0RXh �CesðUhÞ � esðWh

UÞdXþR

Xhc�Ph �~nWh

UdX ¼ 0RXh

cPhWH0 dX� 2p

3 WH0 ¼ 0

8>>><>>>:

ð13Þ

Again, for the sake of simplicity, the zero boundary integrals havebeen omitted. The unknowns of the discrete system of equations(13) are the nodal values of P and U and the value of the film thick-ness constant parameter H0.

Remark: The circular contact problem is symmetric with respectto the ZX-plane. This symmetry is taken into account in order toreduce its size. Hence, the linear elasticity problem requires asymmetry boundary condition on the symmetry plane ZX(U �~n ¼ 0() v ¼ 0 on @Xs). Similarly, Reynolds’ equationrequires an additional symmetry boundary condition(rP �~n ¼ 0() @P=@Y ¼ 0 on oXcs) and the computed dimen-sionless load should equal p/3 instead of 2p/3.

2.2.3. Stability issuesThe solution of Reynolds equation is known to be unstable in the

central contact area (high pressure region) for highly loaded con-tacts [10–14]. In this section, we provide a method to cure theseinstabilities that lead to an oscillatory behaviour of the solution.

8 W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18

The line contact case is also treated for demonstrative purposes. Infact, let us rewrite the Reynolds equation in a different way:

RðPÞ ¼ �r � ðerPÞ þ H@�q@P

@P@Xþ �q

@H@X¼ 0 ð14Þ

Let b = bX in the line contact case or b = {bX, bY} in the circular con-tact case with bX ¼ H @�q

@P and bY = 0. Finally, let Q ¼ ��q @H@X, then Eq.

(14) can be written:

RðPÞ ¼ �r � ðerPÞ|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}«Diffusion»

þb � rP|fflfflfflfflffl{zfflfflfflfflffl}«Convection»

�Q|{z}«Source»

ð15Þ

The penalty term is not mentioned for the sake of simplicity and be-cause it is nil in the region of interest of this section (high pressureregion). But keep in mind that this term should be added to Eq. (15)during the resolution process. Note that Eq. (15) is the Reynolds’equation in compact notation for both line and circular contacts.For the line contact case, the differential operators are unidirec-tional in the X-direction whereas in the circular contact case theyare bidirectional (in the X and Y-directions).

Eq. (15) has the form of the classical Diffusion–Convectionequation (applied to P) with a source term Q. We can clearly iden-tify the diffusion term (left) with a diffusion coefficient e and theconvection term (centre) with a convection operator b � r. Forhighly loaded contacts, e becomes very small. In fact, �q exhibits aslight increase while �l is increased by several orders of magnitudeand H becomes smaller. Therefore, the convection-like term inEq. (15) becomes dominant. It is well known that the standardGalerkin formulation, associated with the finite element methodworks well only when the diffusion term is dominant [15–18]. Infact, the central differencing property of the Galerkin method iswell suited only for elliptic problems (dominated by diffusion).When convection becomes dominant, the Galerkin formulation isno longer appropriate and gives rise to spurious oscillations inthe solution. One way to get rid of these oscillations is obtainedby using special stabilized formulations. A various number of thesetechniques can be found in the literature such as ‘‘artificial diffu-sion’’ [15,17,19] or ‘‘Discontinuous Galerkin’’ methods [20,21]. Inthe following, we propose the use of ‘‘artificial diffusion’’ tech-niques to cure the spurious oscillations of the solution for highlyloaded contacts.

2.2.3.1. Line contact. As mentioned earlier, in the case of a highlyloaded contact, Reynolds equation becomes convection-domi-nated. The solution exhibits an oscillatory behaviour (See Fig. 4,Left). In order to overcome such a problem Brooks and Hughes[15] introduced the so called Streamline Upwind Petrov Galerkin(SUPG) method. The discrete weak variational form for the SUPGmethod applied to Reynolds equation (15) is given by:

Fig. 4. Heavily loaded line contact problem and the effect of stabilization. Left: Standu1 = u2 = 1 m/s, R = 15 mm, A1 = 19.17 �C, A2 = 4.07 � 10�3 MPa�1, B1 = 0.23, B2 = 0.0249 M

Find Ph 2 ShP such that 8Wh

P 2 ShP , one has:Z

Xhc

erPh � rWhPdXþ

ZXh

c

ðb � rPh � QÞWhPdX

þXnce

e¼1

ZXce

RðPhÞsðb � rWhPÞdX ¼ 0 ð16Þ

The first two terms represent the standard Galerkin method appliedto Reynolds equation while the last term represents the additionalstabilizing term that is added to the interior Xce of each discretiza-tion element.

Another interesting technique was proposed by Hughes et al.[17] based on the fact that stabilization terms may be obtainedby minimizing the square of the equation’s residual. It is calledthe Galerkin Least Squares (GLS) method. The discrete weak varia-tional form for this method is given by:

Find Ph 2 ShP such that 8Wh

P 2 ShP , one has:Z

Xhc

erPh � rWhPdXþ

ZXh

c

ðb � rPh � QÞWhPdX

þXnce

e¼1

ZXce

RðPhÞsðb � rWhP �r � ðerWh

PÞÞdX ¼ 0 ð17Þ

Once again, we can identify the standard Galerkin formulation inthe first two terms and the additional stabilizing term in the lastpart. The most important feature of SUPG and GLS is that both areresidual based techniques. In other words, they do not affect thesolution of the original problem. Actually, the additional terms van-ish at the converged solution (R � 0) and therefore the consistencyof Reynolds equation is preserved.

The definition of the tuning parameter s remained ‘‘intuitive’’for a long time. An example of a theoretical formulation was intro-duced by Hughes [18] in the mid 90s. Since, several formulationshave been proposed by various authors. In this work we shall adoptthe definition given by Galeão et al. [16] who provided an exten-sion of this parameter to high order approximations:

s ¼ he

2jbjl nðPeÞ

with : Pe ¼jbjhe

2eland nðPeÞ ¼ cothðPeÞ �

1Pe

ð18Þ

where he and Pe are respectively the characteristic length and the lo-cal Peclet number of the element e. Pe defines the convection-to-dif-fusion ratio inside an element e. Whenever Pe > 1, convectionbecomes dominant and stability problems mentioned earlier arelikely to occur.

Fig. 4 shows a typical case of a heavily loaded steel–steel linecontact. Lagrange quintic elements are used for the hydrodynamic

ard Galerkin, Centre: SUPG, Right: GLS (ph = 3 GPa, lR = 0.012 Pa s, a� = 23 GPa�1,Pa�1, C1 = 16.04, C2 = 18.18 �C, Tg(0) = �73.86 �C, lg = 1012 Pa s and T = 40 �C).

W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18 9

problem and quadratic elements for the elastic part, with a meshdensity of 400 elements in the Hertzian contact region(�1 6 X 6 1). The total number of dofs is 23,000 where only2000 dofs are dedicated to the hydrodynamic part and the rest tothe elastic calculation. This is to be expected since the former isone-dimensional while the latter is two-dimensional. The Dowsonand Higginson [22] formula and the WLF [23] model (see AppendixA) are used as density–pressure and viscosity–pressure relation-ships respectively. The solution exhibits an oscillatory behaviourwhen a Standard Galerkin formulation is used. On the other hand,the use of SUPG or GLS formulations completely smoothes out thisspurious behaviour.

Remark: Using high order elements for the hydrodynamic prob-lem, as an alternative to refining the mesh, allows having a goodprecision for its solution without inducing any unnecessaryincrease in the number of dofs of the two-dimensional (linecontact case) or three-dimensional (circular contact case) elas-tic problem.

2.2.3.2. Circular contact. In the case of a heavily loaded circular con-tact problem, the same behaviour is observed for a standard Galer-kin formulation and the solution exhibits serious oscillations in thecentral area of the contact as can be seen in Fig. 5 (Left). The SUPGand GLS formulations are the same as those given in Eqs. (16) and(17) respectively. But in this case, both methods only succeed inreducing the amplitude of the oscillations without completelysmoothing them out as can be seen in Fig. 5 (Centre). In the two-dimensional case of the convection–diffusion equation, it is ofcommon use to add additional terms to the stabilized SUPG orGLS formulations such as ‘‘Isotropic Diffusion (ID)’’ terms [24].The latter succeed in smoothing out the remaining oscillationswithout a significant perturbation in the solution of the originalproblem. These terms are defined as:

ID ¼Xnce

e¼1

ZXce

qidhejbj

2lrPh � rWh

PdX ð19Þ

The coefficient qid represents the relative amount of ‘‘Isotropic Dif-fusion’’ with respect to the original method. The terms in Eq. (19)are added to the stabilized SUPG or GLS formulations given inEqs. (16) and (17) respectively.

Fig. 5 shows the case of a heavily loaded steel–steel point con-tact. The same lubricant is used as in the line contact case with thesame viscosity–pressure and density–pressure relationships (see

Fig. 5. Heavily loaded circular contact problem and the effect of stabilization. Left: Stana� = 23 GPa�1, u1 = u2 = 1 m/s, R = 12.7 mm, A1 = 19.17 �C, A2 = 4.07 � 10�3 MPa�1, B1 = 0.2T = 40 �C).

previous section). This case corresponds to a severe EHL contactthat can rarely be found in real life contacts. Therefore, it has noimportant physical relevance, but it is used here in the only pur-pose of demonstrating the stability of the current model. Lagrangequintic elements are used for the hydrodynamic problem and qua-dratic elements for the elastic part. The total number of dofs is75,000 where 35,000 dofs are dedicated to the hydrodynamic partand the rest to the elastic calculation. The solution exhibits anoscillatory behaviour when a Standard Galerkin formulation isused. The use of SUPG or GLS formulations reduces these oscilla-tions without completely smoothing them out. And finally, theaddition of ‘‘Isotropic Diffusion’’ terms eliminates the remainingoscillations in the solution (qid is set to 0.5).

It is important to note that the ID terms are non-residualbased and therefore the consistency of the Reynolds equation islost. But, fortunately, the addition of these terms to the stabilizedformulations does not significantly affect the solution of the ori-ginal problem as can be seen in Fig. 6. In order to evaluate theeffect of these terms on the solution of the circular contact prob-lem, a test case was carried out under the same conditions men-tioned earlier with the only difference that the Hertzian pressureis taken to be 0.68 GPa. This allows getting the standard Galerkinsolution (the contact is not heavily loaded) in order to compare itto the stabilized GLS formulation with/without addition of ‘‘IsotropicDiffusion’’.

Fig. 6 shows the dimensionless pressure profile along the cen-tral line in the X-direction for the case mentioned earlier. Globally,the additional ID terms do not significantly affect the solution(Left). By zooming on the pressure spike’s region (Right) we cannote that a slight difference can be observed in this region. Finally,note that the more ‘‘Isotropic Diffusion’’ is added, the more thepressure spike is affected. In fact, for qid = 0.5, the pressure spikedeviates from that of the standard Galerkin or GLS solutions morethan for qid = 0.25. The dimensionless central and minimum filmthicknesses Hc and Hm along with their corresponding relativedeviations for the different formulations with respect to the stan-dard Galerkin solution are reported in Table 1:

Table 1 clearly confirms that the effect of the additional IDterms on the film thickness results is also negligible and that,again, the more ID is added, the more the solution is affected. Fromhere on, unless stated otherwise, whenever ‘‘Isotropic Diffusion’’ isemployed, a value of qid = 0.5 is considered.

Remark: Note that, a single formulation can be used forboth lightly and heavily loaded contacts, since the stabilized

dard Galerkin, Centre: SUPG/GLS, Right: SUPG/GLS + ID (ph = 3 GPa, lR = 0.012 Pa s,3, B2 = 0.0249 MPa�1, C1 = 16.04, C2 = 18.18 �C, Tg(0) = �73.86 �C, lg = 1012 Pa s and

Fig. 6. Effect of «Isotropic Diffusion» on the dimensionless pressure solution of Reynolds’ equation in the case of a circular contact (ph = 0.68 GPa, lR = 0.012 Pa s,a� = 23 GPa�1, u1 = u2 = 1 m/s, R = 12.7 mm, A1 = 19.17 �C, A2 = 4.07 � 10�3 MPa�1, B1 = 0.23, B2 = 0.0249 MPa�1, C1 = 16.04, C2 = 18.18 �C, Tg(0) = �73.86 �C, lg = 1012 Pa s andT = 40 �C).

Table 1Effect of «Isotropic Diffusion» on the dimensionless film thickness results in the caseof a circular contact (ph = 0.68 GPa, lR = 0.012 Pa s, a� = 23 GPa�1, u1 = u2 = 1 m/s,R = 12.7 mm, A1 = 19.17 �C, A2 = 4.07 � 10�3 MPa�1, B1 = 0.23, B2 = 0.0249 MPa�1,C1 = 16.04, C2 = 18.18 �C, Tg(0) = -73.86 �C, lg = 1012 Pa s and T = 40 �C).

Hc Hm (DH/H)c (DH/H)m

Standard Galerkin 0.137462 0.074807 – –GLS 0.136897 0.074519 0.41% 0.38%GLS + ID (qid = 0.25) 0.136752 0.074469 0.52% 0.45%GLS + ID (qid = 0.5) 0.136608 0.074419 0.62% 0.52%

10 W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18

formulations introduced earlier do not affect the solution of theformer.

2.2.4. Newton–Raphson procedureA Newton (Newton–Raphson) procedure is applied to the non-

linear system of equations formed by the stabilized Reynolds equa-tion, the linear elasticity equations and the load balance equation.This system can be rewritten under the following matrix form:

RstabðP;U;H0Þ ¼ 0J21P þ J22U ¼ 0J31P ¼ F3

8><>: ð20Þ

where (P, U, H0) is the vector of the nodal values of P and U, and theconstant film thickness parameter H0. Rstab(P, U, H0) denotes the dis-crete form of the non-linear stabilized Reynolds equation. It isapproximated by its linear part LP;U;H0 RstabðdP; dU; dH0Þ at (P, U, H0)obtained by a first order Taylor expansion:

LP;U;H0 RstabðdP; dU; dH0Þ

¼ RstabðP;U;H0Þ þ@Rstab

@P

����P;U;H0

dP þ @Rstab

@U

����P;U;H0

dU þ @Rstab

@H0

����P;U;H0

dH0

ð21Þ

Let J11 ¼@Rstab@P

���P;U;H0

, J12 ¼@Rstab@U

���P;U;H0

and J13 ¼@Rstab@H0

���P;U;H0

.

The elasticity and load balance equations are already linear andtherefore they are strictly equivalent to their linear part at(P, U, H0). Starting with an initial guess ðP0;U0;H0

0Þ of the solution,the linearized system of equations to solve at the ith iteration ofthe Newton procedure is defined by:

Find ðdPi;dUi;dHi0Þ such that :

LPi�1 ;Ui�1 ;Hi�10

RstabðdPi;dUi;dHi0Þ¼0

J21 � ðPi�1þdPiÞþ J22 � ðUi�1þdUiÞ¼0

J31 � ðPi�1þdPiÞ¼F3

8>><>>:

ð22Þ

where ðdPi; dUi; dHi0Þ is an increment vector. The Hertzian pressure

and elastic deformation profiles or a previously stored solutionare a good estimate for (P0, U0). Replacing LPi�1 ;Ui�1 ;Hi�1

0Rstab

ðdPi; dUi; dHi0Þ by its expression provided in Eq. (21), the system of

equations (22) becomes:

J11 J12 J13

J21 J22 ;J31 ; ;

264

375

i�1dP

dUdH0

8><>:

9>=>;

i

¼�RstabðP;U;H0Þ�J21P � J22U

F3 � J31P

8><>:

9>=>;

i�1

ð23Þ

The matrix on the left hand side is the Jacobian matrix wherewe can identify the coupling terms J12, J13, J21 and J31. We remindthe reader that in the present approach the Jacobian matrix issparse (more than 99% of the terms are zeros), and therefore thecomputational effort or memory usage required for inverting itare far less important than for a full matrix. The Newton procedureconsists in solving the linearized system of Eqs. (23) at every iter-ation i and adding the result ðdPi; dUi; dHi

0Þ to the vectorðPi�1;Ui�1;Hi�1

0 Þ obtained at the previous iteration:

P

U

H0

8><>:

9>=>;

i

¼P

U

H0

8><>:

9>=>;

i�1

þdP

dU

dH0

8><>:

9>=>;

i

ð24Þ

This system is solved using a direct solver and the procedure is re-peated until the convergence of the solution is reached i.e. until theEuclidian norm of the relative residual error vector falls below 10�6.

2.3. Quantitative analysis

In this section, a thorough quantitative analysis of the currentmodel is described. First, the effect of the penalty term on the pres-sure and film thickness solutions is analyzed then, a comparativestudy with a finite difference based model (using multigrid tech-niques) is carried out for a typical circular contact case. The lattervalidates the current approach and proves its efficiency.

W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18 11

2.3.1. Penalty term analysisIn Section 2.2.1, it was pointed out that the free boundary prob-

lem that arises at the outlet of the contact is treated by applying apenalty method. In this section, the effect of the penalty term onthe pressure and film thickness solutions is analyzed. For this pur-pose a typical steel–steel ball on plane contact is considered. Thelubricant is assumed to be compressible. Its density varies withpressure according to the Dowson and Higginson formula (seeAppendix A). The Roelands [25] model is used for viscosity-pres-sure dependence. The parameter qid is taken to be 0.5. Lagrangequintic elements are used for the hydrodynamic problem and qua-dratic elements for the elastic part. The mesh size is approximatelyequal to 0.5 in the inlet and outlet regions of the contact and 0.05in the central area. In practice, the penalty term’s parameter n inEq. (9) is taken as:

n ¼ n0 � h2e ð25Þ

where n0 is an arbitrary large positive number. The values of thecentral and minimum film thicknesses, the outlet absicca of thecontact (location of the free boundary Xout) on the central line inthe X-direction and the minimum pressure are listed in Table 2as a function of the value of n0.It is clear that a minimum valueof n0 is required in order to get converged pressure and film thick-ness solutions. In fact, note that beyond n0 = 106, any increase ofthe value of this parameter is useless and does not lead to any sig-

Table 2Effect of the penalty term on the pressure and film thickness solutions (F = 100 N,ph = 1 GPa, u1 = u2 = 0.8 m/s, R = 16 mm, lR = 0.04 Pa s, a = 22 GPa�1 andT = T0 = TR = 20 �C).

n0 Hc Hm Xout (Y = 0) Min(P)

102 0.082198 0.038382 0.9870 �3.0092 � 10�2

104 0.082214 0.039267 1.0403 �2.5630 � 10�3

106 0.082215 0.039298 1.0589 �9.5199 � 10�5

108 0.082215 0.039298 1.0602 �6.5941 � 10�5

Table 3Comparison of the current model with the Venner & Lubrecht [26] model for a typical cia = 22 GPa�1 and T = T0 = TR = 20 �C).

Venner & Lubrecht [26]

N dofs Hc Hm Niter

16,770 (128 � 128 mesh) 0.07887 0.03712 15366,306 (256 � 256 mesh) 0.08093 0.03848 107263,682 (512 � 512 mesh) 0.08144 0.03876 80

-4.5

cΩcΩ∂

cΩ∂

csΩ∂

Fig. 7. Meshing of the

nificant changes in the solution. This is why this value has beenemployed throughout this work. The condition number of the lin-ear system is of course influenced by the latter. As a direct linearsystem solver is used, it does not affect the solution. Finally, notethat the larger this parameter is, the closer the negative pressuresget to zero.

2.3.2. Validation and comparison with FD multigrid based modelA comparative study between the present finite element model

and a semi-system finite difference (with multigrid techniques)model is established. For this purpose, a representative test casecorresponding to a fairly loaded circular contact problem is takenfrom [26] in order to compare the results and efficiency of bothmodels. The lubricant properties and operating conditions are thesame as described in the previous section. The test is carried outfor several mesh sizes and the results are reported in Table 3. Infact, for the finite difference based model, the mesh diameter heis constant throughout the contact domain and equal to0.046875, 0.0234375 and 0.01171875 for the three cases men-tioned in Table 3 respectively whereas for the current model, heis variable (see Fig. 7), and approximately equal to 0.5 in the inletand outlet regions of the contact in the three test cases whereas inthe central part of the contact it is approximately equal to 0.15,0.075 and 0.05 respectively.

Fig. 7 shows a typical meshing of the contact area Xc, delimitedby its boundaries oXc and oXcs, for a circular contact problem. Notethat the mesh is coarse in the inlet and outlet regions of the contactwhere the solution shows very small variations. On the other hand,a fine mesh is used in the central (Hertzian) contact area where thepressure gradients are more important. Finally, a finer mesh is usedat the outlet of the central contact area where the pressure spikeand film thickness constriction are generally observed. Thus, agood capture of the severe pressure gradients that occur in thisarea is obtained.

Table 3 gives the dimensionless central and minimum filmthicknesses obtained by both the Venner and Lubrecht and the cur-rent models for the test case described earlier using different mesh

rcular contact case (F = 100 N, ph = 1 GPa, u1 = u2 = 0.8 m/s, R = 16 mm, lR = 0.04 Pa s,

Current model

N dofs Hc Hm Niter

18,313 0.080950 0.038818 1139,836 0.081845 0.039165 1376,249 0.082215 0.039298 14

Y

X

0.1- 1.0 1.5

-3.0

cΩ∂

contact area Xc.

Fig. 8. 3D dimensionless pressure profile (Left), dimensionless pressure and film thickness profiles along the central line in the X-direction (Right) (F = 100 N, ph = 1 GPa,u1 = u2 = 0.8 m/s, R = 16 mm, lR = 0.04 Pa s, a = 22 GPa�1 and T = T0 = TR = 20 �C).

1st Newtonianplateau

2nd Newtonianplateau

Transition1st Newtonianplateau

2nd Newtonianplateau

Transition

µ1

µ2

Fig. 9. Typical behaviour of a non-Newtonian lubricant modelled by the Carreaumodel.

12 W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18

densities. The number of iterations Niter required by each model toreach the converged solution is also reported. For the multigridbased model, Niter corresponds to the equivalent number of itera-tions that would be carried out over the finest mesh level given inthe left column. The total number of dofs for the Venner and Lubr-echt model is divided by two with respect to the number given in[26] because this model does not take into account the symmetryof the problem (e.g.: 66,306 = (256 + 1) � (256/2 + 1) � 2). Notethat even for a total number of 18,313 dofs, the solution given bythe current model can be considered sufficiently accurate com-pared to the finest mesh case considered here. Also note, comparedto the Venner and Lubrecht model, the much smaller number ofiterations that is required to get a converged solution. This revealsthe outstanding convergence rate mentioned earlier.

The 3D dimensionless pressure profile for this case is given inFig. 8 (Left) and the corresponding plot of the dimensionless pres-sure and film thickness profiles along the central line in the X-direction are shown in Fig. 8 (Right).

This test allows the validation of the current model and showsthat the size of the system of equations to solve can be reducedcompared to finite difference based models. This is mainly due tothe use of the finite element method which enables non-regularnon-structured meshing. Therefore, fine meshing is used onlywhere needed as can be seen in Fig. 7. Moreover, in a previouswork [5], the authors showed that both models have the samecomplexity and since much faster convergence rates are obtained,then memory storage and computational time are considerably re-duced. In fact, a typical circular contact resolution takes, roughlyspeaking, between 1 and 3 min on a PC with a 2 GHz processorwhereas a line contact solution is obtained in less than 10 s.

3. Extension to a more physical modelling

In this section, the previously described model is extended to amore physical modelling of the contact’s behaviour. First, non-Newtonian lubricants are considered and then thermal effects aretaken into account. Both extensions introduce additional physicsto the problem, making it more complex, both on the numericaland the physical level.

3.1. Non-Newtonian effects

The classical isothermal EHL theory presented in Section 2 as-sumes that the lubricant behaves as a Newtonian fluid. However,when a non-Newtonian lubricant is used, its viscosity varies inthe thickness of the film due to shear stress variations. In general,

non-Newtonian lubricants have a shear-thinning behaviour. Inother words, their viscosity decreases with the increase of shearstress. In the current work, this behaviour is modelled using themodified version of the Carreau equation provided by Bair [27](see Appendix A). The latter is a very powerful shear-thinningmodel that takes into account the second Newtonian plateau thatcan occur at very high shear rates (see Fig. 9).

In order to account for the viscosity variations in the film thick-ness, Najji [28] introduced a generalized Reynolds equation thatcan be written in the following dimensionless form:

r � ðe0rPÞ �@ �qH u2 � �ge

�g0eðu2 � u1Þ

� �h i@X

¼ 0 ð26Þ

where e0 ¼ �qH3

k01�g00e� �ge

�g02e

� �, k0 ¼ R2lR

pha3 , 1�ge¼R 1

0dZ�g , 1

�g0e¼R 1

0Z�g dZ and

1�g00e¼R 1

0Z2

�g dZ.Note that if the generalized Newtonian viscosity g is replaced

by the Newtonian one l Eq. (26) becomes the classical Reynoldsequation. Finally, Eq. (26) can be written in the following form:

R0ðPÞ ¼ �r � ðe0rPÞ þ H u2 ��ge

�g0eðu2 � u1Þ

� �@�q@P

@P@X

þ �q@ H u2 � �ge

�g0eðu2 � u1Þ

� �h i@X

¼ 0 ð27Þ

Let b0X ¼ H u2 � �ge�g0eðu2 � u1Þ

� �@�q@P, b0Y ¼ 0 and Q 0 ¼

��q@ H u2�

�ge�g0eðu2�u1Þ

� �h i@X .

Then Eq. (27) becomes:

R0ðPÞ ¼ �r � ðe0rPÞ þ b0 � rP � Q 0 ¼ 0 ð28Þ

Table 4Lubricant properties for the non-Newtonian test case.

Lubricant properties

l1,R = 0.0705 Pa s av = 7.52 � 10�4 �C�1

l2,R = 0.0157 Pa s K 00R = 11.29Gc = 0.01 MPa b0K = 0 K�1

nc = 0.8 K0,R = 8.375 GPaB = 4.2 bK = 0.006765 K�1

R0 = 0.658 ec = �9.599 � 10�4 �C�1

Initial value for H0, P, U and

Solve the fully coupled non-linear problem using Newton’s method to get pressure and film

thickness distributions

Compute integral terms

Compute new and

Converged P

Yes

No

Get solution for postprocessing

Fig. 11. Flow diagram for the numerical modelling of an isothermal EHL circularcontact lubricated with a non-Newtonian lubricant.

W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18 13

Hence, the generalized Reynolds’ equation also has the form of aclassical Diffusion–Convection equation (applied to P) with a sourceterm Q

0. Again, thermal effects are neglected and temperature is

considered constant throughout the lubricant film and equal tothe ambient temperature T0.

3.1.1. Finite element procedureThe system of EHL equations formed by the generalized Rey-

nolds’ equation, the linear elasticity and the load balance equationis solved using a similar Newton Raphson approach as describedearlier. The finite element formulations for this system are not re-minded here. Those are similar to the ones provided in Section 2since the generalized Reynolds equation also has the form of a con-vection/diffusion equation. The free boundary problem is againtreated by applying a penalty method and for heavy loads, theproblem exhibits similar instability features as described in Sec-tion 2.2.3. This can be seen in Fig. 10 where a test case is carriedout with a typical shear-thinning lubricant for a steel-on-glass cir-cular contact. Note that, in practice, this case can never be realizedon an experimental apparatus because glass would not withstandsuch a load. The Tait–Doolittle [29,30] free volume model (seeAppendix A) is used to express the density and viscosity depen-dence on pressure. The lubricant properties can be found inTable 4.

The stabilized formulations will not be reminded in this section,these are the same as described in Section 2. Again, it is clear that astandard Galerkin formulation leads to an oscillatory behaviour ofthe solution. Applying GLS or SUPG formulations reduces theamplitude of these oscillations without completely smoothingthem out. And finally, adding the ID terms to the GLS or SUPG for-mulations completely smoothes out the remaining oscillations.

3.1.2. Global numerical procedureThe same numerical procedure as described in Section 2 is em-

ployed with only a few slight differences. At every Newton resolu-tion, the integral terms in the generalized Reynolds equation arecomputed using the solution of the previous resolution. Thus aniterative linearization process is established as shown in the flowdiagram of Fig. 11. This iterative process is repeated until the con-vergence of the solution is obtained i.e. in this case, until the max-imum absolute difference between the pressure solutions at twoconsecutive resolutions falls below 10�3. The initial values forthe pressure profile P and the elastic deflection profile U corre-spond to those of a dry/Hertzian contact or a suitable previouslystored solution. As for the initial value of se, it is obtained by

Fig. 10. Stabilization effects on circular EHD contacts lubricated with a non-NewtonianT = T0 = TR = 40 �C).

assuming the lubricant is Newtonian and using the initial valuesof P, U and H0.

It is clear that the global numerical procedure requires morecomputational efforts than the one for the Newtonian approachand consequently, more cpu time is required. This is due to, firstthe calculation of the integral terms that is required at everyiteration of the global procedure and second, the inconsistent lin-earization of the formulation for these terms (the contribution ofthe integral terms to the Jacobian matrix is omitted). In fact, thisinconsistent linearization process leads to an additional loop inthe general procedure which requires in its turn 10–20 iterations.Roughly speaking, for a typical case, the calculations may takebetween 10 and 20 min on a personal computer with a 2 GHzprocessor.

lubricant (F = 1000 N, ph = 1.66 GPa, a� = 18.53 GPa�1, u1 = u2 = 1 m/s, R = 12.7 mm,

14 W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18

3.2. Thermal effects

In this section, the temperature throughout the lubricant film isnot considered constant anymore. In fact, two heat sources lead tothe global increase of temperature: the compressibility of thelubricant and frictional shear. Hence, not only the viscosity of thelubricant varies in the film thickness, but also its density. Thus anew generalized Reynolds equation that takes into account thesevariations is introduced by Yang and Wen [31]. The latter can bewritten in the following dimensionless form:

r � ðe00rPÞ � @ð�q�HÞ@X

¼ 0 ð29Þ

where

e00 ¼�q�g

� �e

H3

k00�q�g

� �e¼

�ge �q0e�g0e� �q00e k00 ¼ umR2lR

a3ph

�q� ¼ ½�q0e �geðu2 � u1Þ þ �qeu1�

um�qe ¼

Z 1

0�qdZ

�q0e ¼Z 1

0�qZ Z

0

dZ0

�gdZ�q00e ¼

Z 1

0qZ Z

0

Z0dZ0

�gdZ

1�ge¼Z 1

0

dZ�g

1�g0e¼Z 1

0

ZdZ�g

Eq. (29) is the most general form of Reynolds equation. It is va-lid for both Newtonian and non-Newtonian lubricants (for a New-tonian lubricant the generalized Newtonian viscosity g is replacedby the Newtonian one l). It takes into account the variations ofboth viscosity and density across the film thickness. In fact, thechanges in density are due to temperature variations across the lu-bricant film whereas the changes in viscosity stem from both tem-perature and (when a generalized Newtonian lubricant isconsidered) shear rate variations across the film. Moreover, bothdensity and viscosity are allowed to vary with pressure and tem-perature throughout the contact domain according to the relation-ships presented in Appendix A. Note that if the temperature in thelubricant film is assumed to be constant and equal to the ambienttemperature (T = T0 = cst) Eq. (29) reduces to the generalized Rey-nolds equation (26), and furthermore, if the generalized Newtonianviscosity g is replaced by the Newtonian one l, this equation re-duces to the classical Reynolds equation (1). Finally, letq0 ¼ �qðP; T ¼ T0Þ be the two dimensional function defined on thecontact domain Xc and describing the density variations over thelatter with respect to pressure considering a constant temperatureT = T0. Eq. (29) can be written in the following form:

R00ðPÞ ¼ �r � ðe00rPÞ þ@ q0

�q�q0

� �H

h i@X

¼ 0

¼ �r � ðe00rPÞ þ�q�

q0

� �H@q0

@P@P@Xþ q0

@�q�q0

� �H

h i@X

¼ 0 ð30Þ

Let b00X ¼�q�q0

� �H @q0

@P , b00Y ¼ 0 and Q 00 ¼ �q0

@�q�q0

� �H

h i@X .

Then Eq. (30) becomes:

R00ðPÞ ¼ �r � ðe00rPÞ þ b00 � rP � Q 00 ¼ 0: ð31Þ

Hence, the generalized Reynolds equation with thermal effectsalso has the form of a classical Diffusion–Convection equation (ap-plied to P) with a source term Q 00. The thermal model for temper-ature variations in the lubricant film and the contacting elementswill not be described here since this is not the main topic of this

paper but all the details can be found in a previous work of theauthors [6]. The latter is based on the resolution of the 3D energyequation in the lubricant film and the solid bodies.

3.2.1. Finite element procedureThe system of EHL equations formed by the generalized Rey-

nolds equation with thermal effects, the linear elasticity equationsand the load balance equation is solved using a similar Newton–Raphson approach as described in Section 2. The finite element for-mulations for this system are not reminded here. Those are similarto the ones provided in Section 2 since the generalized Reynoldsequation with thermal effects also has the form of a convection/dif-fusion equation. The free boundary problem is treated by applyingthe penalty method and for heavy loads, the solution of the gener-alized Reynolds equation with thermal effects exhibits similarinstability features as described previously. The same stabilizationtechniques are used to avoid them (based on a convection/diffu-sion form of Reynolds equation). This can be seen in Fig. 12 wherea test case with the same lubricant and operating conditions as inthe previous section is carried out. Only this time, temperature isno longer considered as a constant throughout the lubricant filmand solid bodies.

The stabilized formulations will not be reminded in this section,these are the same as provided in Section 2. Again, it is clear that astandard Galerkin formulation leads to an oscillatory behaviour ofthe solution. Applying GLS or SUPG formulations reduces theamplitude of these oscillations without completely smoothingthem out. And finally, adding the ID terms to the GLS or SUPG for-mulations completely smoothes out the remaining oscillations.

3.2.2. Global numerical procedureThe global numerical procedure for thermal EHL (TEHL) model-

ling is much more complex than for the isothermal case. The start-ing point consists in defining initial values for P, U, H0, T and se. Theinitial values for the pressure profile P and the elastic deflectionprofile U correspond to those of a dry/Hertzian contact or a suitablepreviously stored solution. As for the initial value of se, it is ob-tained by assuming the lubricant is Newtonian and using the initialvalues of P, U, T and H0. The initial temperature field is taken to beconstant and equal to the ambient temperature T0 throughout thecontacting solids and the lubricant film.

After the initial values of the different variables are defined, thesystem formed by the generalized Reynolds equation with thermaleffects, the linear elasticity and the load balance equations issolved using a Newton–Raphson procedure. At every Newton res-olution, the integral terms in the generalized Reynolds’ equationwith thermal effects are computed using the solution of the previ-ous resolution. Thus an iterative procedure is introduced. It is re-peated until the convergence of the solution is obtained i.e. inthis case, until the maximum absolute difference between thepressure solutions at two consecutive resolutions falls below 10�3.

Then the system formed by the energy equations of the contact-ing elements and the lubricant film is solved for a given pressureprofile P and lubricant’s velocity field U

!f . The latter is computed

for a given viscosity distribution obtained using the last pressureand temperature solutions. The new temperature distribution isused to compute the new value of the lubricant’s velocity field.The latter is again injected in the energy equations for a new res-olution. This is repeated until the convergence of the temperaturesolution is obtained i.e. in this case, until the maximum relativedifference between the temperature solutions at two consecutiveiterations falls below 10�3.

Finally, since the temperature solution is obtained for a givenpressure distribution and vice versa, a final test is realized to checkthat the effects of the variations of any of the two solutions on theother one has become negligible i.e. to ensure the convergence of

Fig. 12. Stabilization effects on circular thermal EHD contacts (F = 1000 N, ph = 1.66 GPa, a� = 18.53 GPa�1, u1 = u2 = 1 m/s, R = 12.7 mm).

W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18 15

the global algorithm with respect to the coupling procedure. Thesame convergence criteria as listed above are employed.

The global numerical procedure is described in the flow chart ofFig. 13. More computational efforts and consequently cpu time arerequired than for an isothermal approach. In fact, in addition to thecomplementary operations that are required in the isothermalnon-Newtonian case, the solution of the thermal part comes asan additional computational task. Hence, for a typical case, the cal-culations may take ‘‘roughly’’ between 30 and 60 min on a personalcomputer with a 2 GHz processor.

N.B.: The additional computational tasks required for the non-Newtonian and thermal cases are common features that arerequired in any EHL solver regardless of the employedapproach. Hence, these cannot be considered as a drawback ofthe current model. In addition, since the core solver (i.e. New-ton–Raphson part) of the current model has been proven tobe more efficient than the state of the art existing ones (See Sec-

Initial value for H0, P, U, T and

Solve the fully coupled non-linear EHD problem to get pressure and film thickness distributions for a

given temperature profile T using Newton’s method

Compute integral terms

Compute new and

Converged PNo

Get solution for postprocessing

Solve fully coupled thermal problem to get temperature distributions in contacting solids and lubricant film for a

given pressure P and lubricant’s velocity field Uf

Compute new Uf

Converged T No

Converged P and T NoYes

Yes

Fig. 13. Flow diagram of the thermal EHL (TEHL) model.

tion 2), the current EHL solver remains more efficient even inthe non-Newtonian and thermal cases.

4. Conclusion

In this work, the authors introduced stabilized finite elementformulations for elastohydrodynamic lubrication problems consid-ering a Newtonian or non-Newtonian lubricant under isothermalor thermal regimes. In all cases, these formulations are based on‘‘artificial diffusion’’ techniques such as SUPG, GLS or ID whichare applied to the different Reynolds equations manipulated tohave the form of a classical Diffusion–Convection equation with asource term. It is shown that in the one-dimensional case of Rey-nolds equation (i.e.: line contacts) the SUPG or GLS formulationsare sufficient to remove the spurious behaviour of the solutionwhich occurs at heavy loads when convection becomes dominant.However, for the two-dimensional case (i.e.: circular contacts),additional ID terms are required to completely smooth out theoscillations. Fortunately, although these additional terms are notresidual based, they are shown to have a negligible effect on thesolution.

Acknowledgments

The authors thank Prof. E. Ioannides (SKF Group Product Tech-nical Director) for his kind permission to publish this work. Theyalso wish to express their gratitude to the French Ministry of Na-tional Education and Scientific Research for partially financing thisstudy.

Appendix A. Density and viscosity variations with pressure,temperature and shear stress

A.1. Density variations with pressure and temperature

In this section the models used in this work for describing thedensity dependence on both pressure and temperature are de-scribed. The R index stands for the value of the correspondingparameter at a reference state (pR, TR):

A.1.1. Dowson and HigginsonThe mathematical expression of the Dowson and Higginson [22]

model for density variations with pressure and temperature is gi-ven by:

qðp; TÞ ¼ qR 1þ 0:6� 10�9p

1þ 1:7� 10�9p� bDHðT � TRÞ

" #

16 W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18

A.1.2. Tait-equation of stateThe Tait [30] equation of state is written for the volume V rela-

tive to the volume at ambient pressure V0. The density data is ob-tained by simply inverting it:

VV0¼ 1� 1

1þ K 00ln 1þ p

K0ð1þ K 00Þ

� The initial bulk modulus K0 and the initial pressure rate of change ofbulk modulus K 00 are assumed to vary with temperature accordingto:

K0 ¼ K0R expð�bK TÞ and K 00 ¼ K 00R expðb0K TÞ

The volume at ambient pressure V0 relative to the ambient pressurevolume VR at the reference temperature TR is assumed to depend ontemperature according to:

V0

VR¼ 1þ aV ðT � TRÞ

The density relationship to pressure and temperature can be de-duced from the previous equations by a simple manipulation:

qðp; TÞ ¼ qRVR

V

� �¼ qR

VR

V0� V0

V

� �¼ qR

1V0=VR

� 1V=V0

� �

A.2. Viscosity variations

In this section the models used in this work for describing theviscosity dependence on pressure and temperature are described.Moreover, if the lubricant is non-Newtonian, the shear stressdependence is also provided:

A.2.1. Pressure and temperature dependenceWhen the considered lubricant has a Newtonian behaviour (i.e.:

the generalized Newtonian viscosity is equal to the Newtonian oneg = l), its viscosity only depends on pressure and temperature.

A.2.1.1. WLF. The mathematical expression of the WLF [23] modeldescribing the dependence of viscosity on both pressure and tem-perature is given by:

lðp; TÞ ¼ lg � 10�C1 �ðT�Tg ðpÞÞ�FðpÞC2þðT�Tg ðpÞÞ�FðpÞ

with :TgðpÞ ¼ Tgð0Þ þ A1 lnð1þ A2pÞFðpÞ ¼ 1� B1 lnð1þ B2pÞ

A1, A2, B1, B2, C1 and C2 are constants characterizing each fluid andlg is the viscosity at the glass transition temperature Tg. The func-tion Tg(p) represents the variation of the glass transition tempera-ture with respect to pressure based on experimental data whereasF(p) represents the variation of the thermal expansion coefficientwith pressure.

A.2.1.2. Roelands. The mathematical expression of the Roelands [25]model describing the dependence of viscosity on both pressure andtemperature is given by:

lðp; TÞ ¼ lR exp ðlnðlRÞ þ 9:67Þ"�1

(

þ 1þ 5:1� 10�9pÞZ0T � 138TR � 138

� ��S0 #)

whereZ0 ¼a

½5:1� 10�9ðlnðlRÞ þ 9:67Þ�

S0 ¼bRoeðTR � 138ÞlnðlRÞ þ 9:67

A.2.1.3. Doolittle free volume model. The Doolittle [29] model is basedon the free volume principle and is defined by the followingrelationship:

lðp; TÞ ¼ lR exp BR0

V1V1R

VVR� R0

V1V1R

� 11� R0

!" #

The relative occupied volume with respect to the reference state isgiven by the following relationship:

V1V1R

¼ 1þ ecðT � TRÞ

B and R0 are constants characterizing a given lubricant, whereas V/VR is defined by a given equation of state. This model is often asso-ciated to the Tait equation of state. Together they are known asthe Tait–Doolittle free volume density and viscosity model.

A.2.2. Shear stress dependenceFinally, when the lubricant is non-Newtonian, its viscosity

exhibits variations with respect to shear stress. In this work, themodified version of the Carreau model provided by Bair [27] isused to describe these variations. The latter is given by the follow-ing mathematical expression:

g ¼ l2 þl1 � l2

1þ seGc

� �bc� 1

nc�1

bc

where bc ¼ exp½0:657� 0:585 lnðncÞ�

where Gc is the liquid critical shear stress. This equation is a goodapproximation of the classical Carreau law for values of nc rangingfrom 0.3 to 0.8, which is the range of interest in EHL applications.

Appendix B. Equivalent elastic problem

In this appendix, the theory behind the equivalent elastic prob-lem is described. This problem is introduced in order to avoid solv-ing the elastic deflection problem twice, under the same loadingconditions, on the same geometry, using the respective materialproperties of solids 1 and 2. The equivalent material propertiesare obtained using the half-space theory. The latter states thatthe displacement d(x, y, z) of a point (x, y, z) produced by a concen-trated point force F acting normally to the surface z = 0 at the origin(see Fig. B1) is given, according to Love [32], by:

dðx; y; zÞ ¼ F4pl

z2

r3 þðkþ 2lÞF

4plðkþ lÞ1r

where k and l are the Lamé constants and r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ z2

p. The

Lamé constants are related to Young’s modulus E and Poisson’scoefficient t according to:

k ¼ Etð1þ tÞð1� 2tÞ and l ¼ E

2ð1þ tÞ

Then, the equivalent displacement deq(x, y, z) of the two solids 1 and2 under the same concentrated point force F acting normally to thesurface z = 0 at the origin is given by:

deqðx; y; zÞ ¼ d1ðx; y; zÞ þ d2ðx; y; zÞ

F4pleq

z2

r3 þðkeq þ 2leqÞF

4pleqðkeq þ leqÞ1r¼ F

4pl1

z2

r3 þðk1 þ 2l1ÞF

4pl1ðk1 þ l1Þ1r

þ F4pl2

z2

r3 þðk2 þ 2l2ÞF

4pl2ðk2 þ l2Þ1r

Fig. B1. Point loading of an elastic half-space.

W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18 17

After simplification, the previous equation becomes:

1leq

z2

r2 þ 2ð1� teqÞ�

¼ 1l1

z2

r2 þ 2ð1� t1Þ�

þ 1l2

z2

r2 þ 2ð1� t2Þ�

ðB:1Þ

Any couple of material properties (leq, teq) that satisfies the previ-ous equation can be used to define the equivalent elastic problem.Here, we consider the particular case where:

1leq¼ 1

l1þ 1

l2ðB:2Þ

) 1þ teq

Eeq¼ 1þ t1

E1þ 1þ t2

E2ðB:3Þ

Eq. (B.1) becomes after simplification and replacing 1/leq by itsexpression given in (B.2) and the Lamé constant l by its expressionas a function of E and t:

1� t2eq

Eeq¼ 1� t2

1

E1þ 1� t2

2

E2ðB:4Þ

Thus, solving the system of equations formed by (B.3) and (B.4), onegets the equivalent material properties Eeq and teq:

Eeq ¼E2

1E2ð1þ t2Þ2 þ E22E1ð1þ t1Þ2

½E1ð1þ t2Þ þ E2ð1þ t1Þ�2and teq

¼ E1t2ð1þ t2Þ þ E2t1ð1þ t1ÞE1ð1þ t2Þ þ E2ð1þ t1Þ

In order to validate the equivalent problem’s theory, a test case iscarried out with a dry glass-on-steel circular contact with a Hertz-ian pressure distribution applied in the contact region.Fig. B2 showsthe non-dimensional elastic deflection curves on the central line inthe X-direction obtained by both the equivalent problem defined in

Fig. B2. Total elastic deflection of a dry glass-on-steel circular contact.

this section and the sum of the elastic displacements of the twocontacting elements. It is clear that the two curves show a perfectmatch, revealing thus the equivalence between the two approaches.

Appendix C. Dimensionless parameters

H ¼ hRa2 X ¼ x

aY ¼ y

aZ ¼

z=a in the solid bodiesz=h in the lubricant film

�

�q ¼ qqR

�l ¼ llR

�g ¼ glR

P ¼ pph

where E0 ¼ 21�t2

1E1þ

1�t22

E2

a ¼ffiffiffiffiffiffi3FR2E0

3q

ph ¼ 3F2pa2

References

[1] Oh KP, Rohde SM. Numerical solution of the point contact problem using thefinite element method. Int J Numer Methods Eng 1977;11:1507–18.

[2] Holmes MJA, Evans HP, Hughes TG, Snidle RW. Transient elastohydrodynamicpoint contact analysis using a new coupled differential deflexion method. Part1: theory and validation. Proc Inst Mech Eng J Eng Tribol 2003;217(J):289–303.

[3] Lubrecht AA, Ten Napel WE, Bosma R. Multigrid, an alternative method forcalculating film thickness and pressure profiles in elastohydrodynamicallylubricated line contacts. ASME J Tribol 1986;108(4):551–8.

[4] Venner CH. Multilevel solution of the EHL line and point contact problems.Ph.D. Thesis, University of Twente, Encchede, The Netherlands; 1991.

[5] Habchi W, Eyheramendy D, Vergne P, Morales-Espejel G. A full-systemapproach of the elastohydrodynamic line/point contact problem. ASME JTribol 2008;130(2). doi:10.1115/1.284224.

[6] Habchi W, Eyheramendy D, Bair S, Vergne P, Morales-Espejel G. Thermalelastohydrodynamic lubrication of point contacts using a newtonian/generalized newtonian lubricant. Tribol Lett 2008;30(1):41–52.

[7] Hertz H. Uber die Berührung fester Elastischer Körper. Journal fur die reine undangewandte Mathematik 1882;92:156–71.

[8] Reynolds O. On the theory of the lubrication and its application to Mr.Beauchamp tower’s experiments, including an experimental determination ofthe viscosity of olive oil. Philos Trans Roy Soc London 1886;177:157–234.

[9] Wu SR. A penalty formulation and numerical approximation of the Reynolds–Hertz problem of elastohydrodynamic lubrication. Int J Eng Sci 1986;24(6):1001–13.

[10] Dowson D, Higginson GR. A numerical solution of the elastohydrodynamicproblem. J Mech Eng Sci 1959;1(1):6–15.

[11] Evans HP, Snidle HP. The isothermal elastohydrodynamic lubrication ofspheres. ASME J Lubric Technol 1981;103:547–57.

[12] Hamrock BJ, Dowson D. Isothermal elastohydrodynamic lubrication of pointcontacts, Part I – theoretical formulation. ASME J Lubric Technol 1976;98(2):223–9.

[13] Lubrecht AA. The numerical solution of the elastohydrodynamically lubricatedline and point contact problem using multigrid techniques. Ph.D. Thesis,University of Twente, Enschede, The Netherlands, 1987.

[14] Hughes TG, Elcoate CD, Evans HP. A novel method for integrating first- andsecond-order differential equations in elastohydrodynamic lubrication for thesolution of smooth isothermal, line contact problems. Int J Numer MethodsEng 1999;44:1099–113.

[15] Brooks AN, Hughes TJR. Streamline-upwind/Petrov–Galerkin formulations forconvective dominated flows with particular emphasis on the incompressibleNavier–Stokes equations. Comput Methods Appl Mech Eng 1982;32:199–259.

[16] Galeão AC, Almeida RC, Malta SMC, Loula AFD. Finite element analysis ofconvection dominated reaction-diffusion problems. Appl Numer Math2004;48:205–22.

[17] Hughes TJR, Franca LP, Hulbert GM. A new finite element formulation forcomputational fluid dynamics: VII. The galerkin-least-squares method foradvective–diffusive equations. Comput Methods Appl Mech Eng 1989;73:173–89.

[18] Hughes TJR. Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins ofstabilized methods. Comput Methods Appl Mech Eng 1995;127:387–401.

[19] Oñate E. Derivation of stabilized equations for numerical solution ofadvective–diffusive transport and fluid flow problems. Comput MethodsAppl Mech Eng 1998;151:233–65.

[20] Johnson C, Nävert U, Pitkäranta J. Finite element methods for linear hyperbolicproblems. Comput Methods Appl Mech Eng 1984;45:285–312.

[21] Johnson C, Pitkäranta J. An analysis of the discontinuous Galerkin method for ascalar hyperbolic equation. Math Comput 1986;46:1–26.

[22] Dowson D, Higginson GR. Elastohydrodynamic lubrication. The fundamental ofroller and gear lubrication. Oxford: Pergamon; 1966.

[23] Yasutomi S, Bair S, Winer WO. An application of a free-volume model tolubricant rheology, (1). Dependence of viscosity on temperature and pressure.ASME J Tribol 1984;106:291–312.

18 W. Habchi et al. / Advances in Engineering Software 46 (2012) 4–18

[24] Zienkiewicz OC, Taylor RL. The finite element method. Fluid dynamics, 5th ed.,vol. 3. England: Butterworth & Heinmann; 2000.

[25] Roelands CJA. Correlational aspects of the viscosity-temperature-pressurerelationship of lubricating oils. Ph.D. Thesis, Technische Hogeschool Delft, TheNetherlands, 1966.

[26] Venner CH, Lubrecht AA. Multilevel methods in lubrication. Tribol Ser, vol.37. Amsterdam: Elsevier; 2000.

[27] Bair S. A rough shear-thinning correction for EHD film thickness. STLE TribolTrans 2004;47:361–5.

[28] Najji B, Bou-Said B, Berthe D. New formulation for lubrication with non-newtonian fluids. ASME J Tribol 1989;111:29–33.

[29] Doolittle AK. Studies in newtonian flow, the dependence of the viscosity ofliquids on free-space. J Appl Phys 1951;22:1471–5.

[30] Hogenboom DL, Webb W, Dixon JD. Viscosity of several liquid hydrocarbons asa function of temperature, pressure and free volume. J Chem Phys1967;46(7):2586–98.

[31] Yang P, Wen S. A generalized reynolds equation for non-newtonian thermalelastohydrodynamic lubrication. ASME J Tribol 1990;112:631–6.

[32] Love AEH. A treatise on the mathematical theory of elasticity. 4th ed. NewYork: Dover; 1944.