A 3D FEM–BEM rolling contact formulation for unstructured meshes

International Journal of Solids and Structures 47 (2010) 330–353

Contents lists available at ScienceDirect

International Journal of Solids and Structures

journal homepage: www.elsevier .com/locate / i jsolst r

A 3D FEM–BEM rolling contact formulation for unstructured meshes

L. Rodríguez-Tembleque, R. Abascal *

Escuela Técnica Superior de Ingenieros, Camino de los descubrimientos s/n, 41092 Sevilla, Spain

a r t i c l e i n f o

Article history:Received 20 June 2009Received in revised form 4 September 2009Available online 13 October 2009

Keywords:Rolling contact mechanicsUnstructured meshesBoundary element methodFinite element method

0020-7683/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.ijsolstr.2009.10.008

* Corresponding author. Tel.: +34 954 487486; fax:E-mail address: [email protected] (R. Abascal).

a b s t r a c t

This work presents a new formulation for solving 3D steady-state rolling contact problems. The convec-tive terms for computing the tangential slip velocities involved in the rolling problem, are evaluatedusing a new approximation inspired in numerical fluid dynamics techniques for unstructured meshes.Moreover, the elastic influence coefficients of the surface points in contact are approached by meansof the finite element method (FEM) and/or the boundary element method (BEM). The contact problemis based on an Augmented Lagrangian Formulation and the use of projection functions to establish the con-tact restrictions. Finally, the resulting nonlinear equations set is solved using the generalized Newtonmethod with line search (GNMls), presenting some acceleration strategies as: a new and more simplifiedprojection operator, which makes it possible to obtain a quasi-complementarity of the contact variables,reducing the number of contact problem unknowns, and using iterative solvers. The presented method-ology is validated solving some rolling contact problems and analyzed for some unstructured meshexamples.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The necessity of having (analytical and numerical) designingtools for mechanical elements under rotation, such as rotary ma-chine components, bearings and rollers, as well as wheels, rollingpaths for cars or railways, makes the rolling contact problems havean important place in the technical literature relative to mechani-cal engineering and computational mechanics.

The earliest rolling contact problems were related to the rail-way. The pioneering work in this direction was developed by Car-ter (1926) and later by Bentall and Johnson (1967, 1968) andNowell and Hills (1987a,b). Also it has to be mentioned the ap-proaches, formulations and algorithms developed by Kalker(1967, 1971, 1975, 1982, 1986, 1988). Two essential books in roll-ing contact mechanics are: Kalker’s book (Kalker, 1990), where hesummarized all his work, and Johnson’s book (Johnson, 1985),which collects most of the analytical developments in contactand rolling contact mechanic together with his works in rollingcontact.

In the numerical area of rolling contact mechanics, besides theKalker’s contributions, it has to be mentioned the works based onfinite element method (FEM) of Singh and Paul (1974), Batra(1981), Zeid and Padovan (1981), Padovan and Zeid (1984) andBass (1987), as well as Oden and his collaborators in the field thevariational inequalities and FEM, collected in the book of Kikuchiand Oden (1988). Also it has to be cited the works of Panagiotopo-

ll rights reserved.

+34 954 487295.

ulos (1985), Klarbring (1986, 1987, 1993) and Klarbring and Björk-man (1988) in mathematical programming.

More recently it has to be mentioned the works of Hu and Wrig-gers (2002), Nackenhorst (2004) and Ziefle and Nackenhorst(2008), who present an Arbitrary Lagrangian Eulerian (ALE) formu-lation of the problem, and Strömberg (2005), who suggests a for-mulation for contact and impact problems including rolling.

Between the rolling contact works related with the boundaryelement method (BEM), we emphasize the ones of Anderson(1982), París and Garrido (1989), Man et al. (1993a,b), Martínand Aliabadi (1993), González and Abascal (1998, 2000a,b, 2002)and Abascal et al. (2007).

The methodology suggested in this work is based on an Aug-mented Lagrangian Formulation whose precursors are Landersand Taylor (1985), Wriggers et al. (1985), Simo et al. (1985), Alartand Curnier (1991) and Simo and Laursen (1992), and it was ex-tended by Strömberg (1997) and Christensen et al. (1998). Thebooks of Laursen (2002) and Wriggers (2002) compile all this for-mulations together with the main strategies for numerical contactproblems.

The present work proposes a new methodology based on Kal-ker’s Eulerian formulation (Kalker, 1990) for solving 3D frictionalrolling contact problems, using structured or unstructured mesheson the solids interface contact zone. There are some difficultieswhen it has to be evaluated the convective terms which arise fromthe state of strain at the surface. When the contact meshes are reg-ular and the nodes positions are equidistant at the contact surfaces,the derivative associates to the convective terms can be evaluatedusing a finite difference forward technique, like (González and

http://dx.doi.org/10.1016/j.ijsolstr.2009.10.008

mailto:[email protected]

http://www.sciencedirect.com/science/journal/00207683

http://www.elsevier.com/locate/ijsolstr

Fig. 1. Contact pair I of points Pa 2 Xa ða ¼ 1;2Þ.

Fig. 2. Eulerian description for a wheel-road rolling contact problem.

L. Rodríguez-Tembleque, R. Abascal / International Journal of Solids and Structures 47 (2010) 330–353 331

Abascal, 1998, 2000a,b). In case the solid meshes are unstructured,it is necessary to resort to more complex approximations. The ex-tended Godunov scheme (Godunov, 1959; Hirsch, 1990) or least-squares gradient approximation, in numerical fluid dynamics usingfinite volume method (FVM) on structures and unstructuredmeshes (Blazek, 2001) are redefined in this work for computingthe convective terms in the rolling contact Kalker’s formulation.

A resolution framework for 3D frictional contact problems ispresented in this work, which is valid for both FEM and/or BEM sol-ids modeling, based on the generalized Newton method (GNM)proposed by Pang (1990). The contact problem is formulated bymeans of the Augmented Lagrangian similarly to Alart and Curnier(1991), and imposes the contact restrictions with projection func-tions. The contact operators approximation carried out in this workin the Newton process, allows for a reduction in the number of un-knowns in the contact zone, as well as a simplification in theexpressions of the Jacobian matrices associated to the contactrestrictions, while maintaining a good convergence ratio. Further-more, the line search calculation in the Newton’s method is carriedout by means of direct resolution strategies, such as factorization,and iterative ones, such as generalized minimum residual(GMRES), thus allowing for a CPU time comparative analysis. A ser-ies of numeric examples will be presented to validate the formula-tion and the algorithm proposed.

2. Kinematic equations

2.1. Contact kinematic equations

We are going to consider the contact between two solidsXa ða ¼ 1;2Þ, with boundaries Ca, and defined with respect to aCartesian reference system: xi � fx1; x2; x3g in R3. In order to knowthe relative position between both bodies at all times (s), a gap var-iable is defined for the pair I � fP1; P2g of points ðPa 2 Xa; a ¼ 1;2Þ,as

g ¼ BTðx2 � x1Þ ð1Þ

where xa is the position of Pa at every instant, defined as:xa ¼ Xa þ ua

o þ ua (Xa: global position; uao : rigid body global dis-

placement; ua: elastic displacement expressed in the global sys-tem). Matrix B ¼ t1jt2jn½ �, is a base change matrix expressing thepair I gap in relation to the local orthonormal base ft1; t2;ng associ-ated to every I pair. The unitary vector n is normal to the contactsurfaces with the same direction as the normal to C1 and expressedin the global system. Vectors ft1; t2g are the tangential unitarianvectors (see Fig. 1).

The expression (1) can be written as

g ¼ BTðX2 � X1Þ þ BTðu2o � u1

oÞ þ BTðu2 � u1Þ ð2Þ

being BTðX2 � X1Þ the geometric gap between two solids in the ref-erence configuration ðggÞ, and BTðu2

o � u1oÞ the gap originated due

to the rigid body movements ðgoÞ. Therefore, the gap of the I pair re-mains as follows:

g ¼ ggo þ BTðu2 � u1Þ ð3Þ

where ggo ¼ gg þ go. In this work, the reference configuration foreach solid ðXaÞ that will be considered is the initial configuration(before applying load). Consequently, gg may also be termed initialgeometric gap.

In the expression (3) two components can be identified: thenormal gap gn and the tangential gap or slip, gt ,

gn ¼ ggo;n þ u2n � u1

n gt ¼ ggo;t þ u2t � u1

t ð4Þ

being uan and ua

t the normal and tangential components of the dis-placements ua : ua

t uan

� �T ¼ BT ua.

2.2. Rolling kinematic equations

In a rolling contact problem the solid particles are travelingthrough the contact zone because of the solids rotations. Therefore,the contact of a pair I � fP1; P2g has to consider as kinematic vari-ables the normal gap, gn, and the tangential slip velocity, _gt .

Kalker (1990) formulates the tangential kinematical variable ofthe rolling contact problem from an Eulerian point of view (seeFig. 2). Consequently, it is defined the contact pairs relative tangen-tial slip velocities with respect to a system of reference which trav-els with the contact zone. Thus the tangential slip velocity can beexpressed as

_gt ¼dgt

ds¼ _ggo;t þ ð _u2

t � _u1t Þ ð5Þ

where _ggo;t is the creep or rigid bodies tangential velocities ðvat Þ slip:

_ggo;t ¼ v2t � v1

t ð6Þ

and _uat ða ¼ 1;2Þ is the displacement field ðua

t Þ material derivative:

_uat ¼

ouat

osþ va

t � ruat ð7Þ

332 L. Rodríguez-Tembleque, R. Abascal / International Journal of Solids and Structures 47 (2010) 330–353

Therefore, the tangential slip velocity can be written as:

_gt ¼ v2t � v1

t þoðu2

t � u1t Þ

osþ v2

t � ru2t � v1

t � ru1t ð8Þ

According to the rolling situation, the expression (8) could besimplified:

� Rolling with partial slip between the two surfaces. In this situa-tion the relative velocities are similar, so ðv2

t ’ v1t Þ. Therefore,

kv2t � v1

t k � kv2t þ v1

t k (being k � k the Euclid’s norm), and theexpression (7) could be approximated as:

_uat ¼

ouat

osþ vt � rua

t ð9Þ

being vt the mean rolling velocity: vt ¼ ðv2t þ v1

t Þ=2, so the tangentialslip velocity (8) can be expressed as:

_gt ¼ v2t � v1

t þoðu2

t � u1t Þ

osþ vt � rðu2

t � u1t Þ ð10Þ

� Rolling with total slip between the two surfaces: In this situationthe relative velocities are not similar, and kv2

t � v1t k � k _u2

t�_u1

t k, so the tangential slip velocity is,

_gt ¼ v2t � v1

t ð11Þ

However, for technical applications the case of partial slip is ofmost importance, so Eq. (10) will be considered in this formulationand it will be written in a compact form as

_gt ¼ cþ Drðu2t � u1

t Þ ð12Þ

In the expression below c is the rigid body tangential slip velocity orcreep:

c ¼ _ggo;t ¼ kvtknt ð13Þ

being nt the non-dimensional creep: nt ¼ ðv2t � v1

t Þ=kvtk. The opera-tor Drð�Þ is defined as:

Drð�Þ ¼oð�Þosþ v t1

oð�Þoxt1

þ v t2

oð�Þoxt2

ð14Þ

and for a steady state rolling contact situation, as:

Drð�Þ ¼ v t1

oð�Þoxt1

þ v t2

oð�Þoxt2

ð15Þ

3. Rolling contact laws

3.1. Unilateral contact law

� No interpenetration of the bodies of domain Xa and boundaryCa ða ¼ 1;2Þ: X1 \X2 ¼£. Therefore, the surface of each bodycan be divided in three regions depending on whether it is incontact Ca

c

� �, with tractions imposed Ca

�t

� �or with displacements

imposed Ca�u

� �, so that: Ca ¼ Ca

c [ Ca�u [ Ca

�t and Cac \ Ca

�u \ Ca�t ¼£,

with a ¼ 1;2. Moreover, it is possible to denote the Contact Zoneas Cc , since: Cc ¼ C1

c ¼ C2c .

� The solids are in contact without cohesion, they can be separated,therefore for each pair I � fP1; P2g 2 Cc : gn P 0 and tn 6 0. Thevariable gn is the pair I normal gap, and tn is the normal contacttraction defined as: tn ¼ BT

nt1 ¼ �BTnt2, where ta is the traction

of point Pa 2 Cac expressed in the global system of reference,

and Bn ¼ ½n� is the third column in the change of base matrix:

B ¼ ½Bt jBn� ¼ ½t1jt2jn�, as defined in Section 2. The normal trac-tions acting upon the pair I points are of the same value andopposite signs, in accordance with Newton’s third law.

� The variables gn and tn are complementary: gntn ¼ 0.

3.2. Friction law

The law of friction is defined for the tangential contact vari-ables: tangential slip velocity, _gt , and tangential traction, tt:tt ¼ BT

t t1 ¼ �BTt t2, where ta is the traction of point Pa 2 Cc ex-

pressed in the global axes, and Bt ¼ ½t1jt2�.Thus, the law of friction for each pair I � fP1; P2g 2 Cc , is ex-

pressed as follows:

� Tangential contact tractions module fulfills the law of friction:kttk 6 ljtnj, being l the friction coefficient, and j � j the absolutevalue.

� The tangential slip velocity and the tangential traction comply withthe maximum energy dissipation principle: _gt ¼ �kttðk P 0Þ,where k is a real positive number.

� There is a complementarity in the variables: tangential slipvelocity module, k _gtk, and ðkttk � ljtnjÞ : k _gtkðkttk � ljtnjÞ ¼ 0.

3.3. Rolling contact laws

To sum up, the unilateral contact condition and the law of frictiondefined for any I pair of points in contact can be compiled as fol-lows, according to their contact status:

Contact-adhesion : tn 6 0; gn ¼ 0; _gt ¼ 0

Contact-Slip :tn 6 0; gn ¼ 0kttk ¼ ljtnj; _gt � tt ¼ �k _gtkkttk

�No contact : tn ¼ 0; gn P 0; tt ¼ 0

ð16Þ

4. Rolling contact constraints

For the rolling contact tractions of any I pair of points in contact,the contact laws (16) define an admissible convex region in R3;Cf ,called Contact Cone or Friction Cone. In order to guarantee that con-tact traction values remain in Cf , and that (16) is fulfilled, a refor-mulation of the restrictions is carried out by means of thefollowing contact operators:

4.1. Normal operator

Unilateral contact constrains can be shortened as

Unðtn; gnÞ ¼ tn � PR� ðtnÞ ¼ 0 ð17Þ

The normal contact operator or normal projection function:PR� ð�Þ : R! R�,

PR� ðxÞ ¼minðx; 0Þ ð18Þ

projects the variable x in R�, the admissible region of the contactnormal tractions. The mixed variable or augmented normal tractiontn is defined as,

tn ¼ tn þ rgn ð19Þ

where the parameter r is a dimensional positive penalizationparameter ðr 2 RþÞ.

4.2. Tangential operator

Similarly to the normal case, the tangential contact restrictionsmay also be summarized as


Ut tt ; _gtð Þ ¼ tt � PCg ðtt Þ ¼ 0 ð20Þ

The tangential contact operator, or tangential projection function,PCg ð�Þ : R2 ! R2,

PCg ðxÞ ¼x; if kxk < g

get ; if kxkP g

�ð21Þ

(where et ¼ x=kxk is the projection direction) ensures that the valueof the variable x 2 R2 remains within the Coulomb disk, Cg , of ra-dium gðg ¼ jltnjÞ. In the plane tt1 —tt2 , Cg represents the tangentialtractions admissible region for a given normal traction value ðtnÞ.

In this case, the augmented tangential traction is defined as,

tt ¼ tt � r _gt ð22Þ

where the parameter r is a dimensional positive penalizationparameter ðr 2 RþÞ, whose value may differ from that of the aug-mented normal variable (19). The negative sign in (22) is due tothe fact that tt and _gt present opposite directions.

4.3. Normal-tangential operator

The constraints of the combined normal-tangential contactproblem can be formulated as

Uðt; gn; _gtÞ ¼ t� PCfðtÞ ¼ 0 ð23Þ

The contact operator PCfis defined as

PCfðtÞ ¼

PCg tt� �

PR� tn� �( )

ð24Þ

PR� being the normal projection function defined in (18), and PCg atangential projection function similar to (21), where the region Cg

radium is jlPR� ðtnÞj. The Augmented Lagrangian of the contact trac-

tions is: ðtÞT ¼ tt� �T tnh i

.

Eq. (23) compiles the law of unilateral contact and the law offriction, taking the following values depending on the contact sta-tus of the I pair of points:

� ðtnÞI > 0 (No contact): ðtÞI ¼ 0� tn� �

I 6 0 (Contact) and:– kðtt ÞIk < jlPR� tn

� �I

� �j (Adhesion):

_gt

gn

� �I¼ 0

– kðtt ÞIkP jlPR� ððtnÞIÞj (Slip): tt þ ltnxt

gn

� �I

¼ 0

5. Variational formulation

Let us consider the contact between two rolling solids Xa ða ¼1;2Þ. The total virtual work for the problem can be expressed as thesum of the virtual over each solid Xa ðdPa; a ¼ 1;2Þ, plus the inter-face contact virtual work, dPc , so

dPtotal ¼ dP1 þ dP2 þ dPc ð25Þ

The contact virtual work, dPc , can be expressed as

dPcðu1;u2; tc;gÞ ¼Z

Cc

dtc BTðu1 � u2Þ � ggo þ gn o

dC

þZ

Cc

ðdu1 � du2ÞT Btc dC

þZ

Cc

dgT tc � PCfðtcÞ

n odC ð26Þ

being ðtcÞT ¼ ðtt Þ

T tnh i

the augmented contact tractions (19) and(22). The first term in (26), on the right-hand side, is the weak form

of the contact kinematic equation (3), the second one representsthe work of the tractions over each solid’s contact interface,and the third term expresses, in a weak form, the contact con-straints (23).

6. Discrete equations

6.1. FEM equations

To formulate the virtual work of a three-dimensional domain X,this one is divided in a set of elemental sub-domains Xe 2 X, so:X ¼

SNee¼1X

e andTNe

e¼1Xe ¼ Ø, being Ne the number of elements of

X.The total work of the solid can be expressed as

dP ¼XNe

e¼1

dPeðuÞ ð27Þ

where the variation of potential energy of element Xe can bewritten as

dPeðuÞ ¼Z

XedeTrdX�

ZXe

duT bdX�Z

CeduT�tdC ð28Þ

Solid X has null contact boundary conditions on (27), so the contactvirtual work is considered on (26).

Using iso-parametric elements and the classical FEM formula-tion (Zienkiewicz, 1977), the displacement field is approximated,on Xe, using its nodal values ðdeÞ in the following way: u ’ u ¼Nde. N is the shape function approximation matrix, whose dimen-sion is 3 3ne (ne: number of nodes per element).

The strain pseudo-vector approximation can be written as:e ¼ Du ’ Bue, being B the resulting matrix after applying the sym-metric gradient operator, D, to N. The stress pseudo-vector is ob-tained as: r ¼ Ee, where E is the constitutive matrix. Substitutingthese interpolations into (28) produces the variation of the discretefunctional for a finite element

dPeðdeÞ ¼ ðddeÞTfKede � feg ð29Þ

where Ke is the element stiffness matrix and fe is the consistentelement nodal force vector. Considering (27), the total virtual workof the system can be expressed as

dP ¼ ðddÞTfKd� Fg ð30Þ

where K and F are constructed from the assembly of Ke and fe,respectively. Eq. (30) express the virtual work of the solid X.

In contact problems, the nodal displacement vector (d) can be

rearranged as: ðdÞT ¼ ðdqÞTðdcÞTh i

, where dc compiles the displace-

ments of the nodes belonging to Cc , and dq the exterior nodaldisplacements. In the same way, the stiffness matrix (K) is orga-nized in columns: K ¼ KqKc

� �.

6.2. Contact virtual work equations

The contact interface virtual work (26), can be approximateddiscretizing the contact tractions ðtcÞ, the gap (g), the tangentialslip velocity ð _gtÞ, and the solids displacements ðua; a ¼ 1;2Þ, over

the contact interface ðCcÞ. To that end, Cc is divided into Nfe ele-

mental surfaces ðCecÞ, thus: Cc ¼

SNfe

e¼1Cec and

TNfe

e¼1Cec ¼£. These

elements ðCecÞ constitute a contact frame.

Considering the discretization on the contact frame, the contactvirtual work discrete expression can be written as:


dPc u1; u2; tc; g� �

¼XNf

e

e¼1

ZCe

c

dtTc BT u1 � u2� �

� ggo þ gn o

dC

þXNf

e

e¼1

ZCe

c

ðdu1 � du2ÞT Btc dC

þXNf

e

e¼1

ZCe

c

dgT tc � PCftc� �n o

dC ð31Þ

being ua ða ¼ 1;2Þ the displacement approximation of Xa over Cc; tc

the discrete contact tractions over the frame, and g the discrete gap.The contact tractions are discretized over the contact frame as:

tc ’ tc ¼XNf

i¼1

dPiki ð32Þ

where dPiis the Dirac’s delta on each contact frame node i, and ki is

the Lagrange multiplier on the node ði ¼ 1; . . . ;Nf Þ.The gap (g) is approximated in the same way:

g ’ g ¼XNf

i¼1

dPiki ð33Þ

where ki is the nodal value. The tangential slip velocity approxima-tion will be described in details in the next section: _g ’ st .

The contact is node-to-node, therefore the solids’ interfacemeshes and the contact frame mesh are node coincident, so Eq.(31) can be expressed as:

dPc ¼XNp

I¼1

dkTI BT

I d1I � d2

I � kgoI þ kI

n o

þXNp

I¼1

dd1I � dd2

I

� TBIkI þ

XNp

I¼1

dkTI kI � P kI

� �� ð34Þ

being I the index for the Np contact pairs in the potential contactzone, BT

I the change of base matrix, dI nodal displacements, kgoI

the geometric gap of solids’ rigid body displacements, kI the I paircontact gap, and kI the contact Lagrange multipliers on the local sys-tem of reference, and kI , the augmented contact traction:kI� �T ¼ kt

� �Tkn

h iI, with

kt ¼ kt � rtst ; kn ¼ kn þ rnkn ð35Þ

To manage the interface discrete variables of each solid ðXaÞand the frame Lagrange multipliers, we introduce the assembling

Boolean operators: Lai and Lf

i . Lai allows to extract the node i

displacement ðdai Þ, from vector da ðda

i ¼Lai daÞ, and Lf

i extractsfrom the multipliers vector (K), the variable associated to node

i ðkiÞ; ðki ¼Lfi KÞ. According to the organization of the displace-

ment vector: ðdaÞT ¼ daq

� Tðda

c ÞT

�, the assembling matrix has the

following distribution: Lai ¼ La

iqLaic

h i.

Defining the matrices:

Ca ¼XNa

i¼1

XNf

j¼1ði;jÞ�I

ðLai Þ

T BjLfj Cg ¼

XNf

i¼1

XNf

j¼1ði;jÞ�I

ðLfi Þ

TLf

j ð36Þ

where Ca has the following structure: ðCaÞT ¼ 0 eCa� T

�, the expres-

sion (34) can be written as:

dPc ¼ ðdKÞT ðC1ÞT d1 � ðC2ÞT d2 � Cgkgo þ Cgkn o

þ dd1� T

C1K� ðdd2ÞT C2K

þ ðdkÞTðCgÞT K� PCfðKÞ

n oð37Þ

being k the nodal gap vector and K the Lagrange multipliers vector.The normal-tangential contact operator, PCf

, is defined for the aug-mented nodal multipliers ðKÞ, and eCa establishes the relation be-tween the contact degrees of freedom relation of each solid Xa

and the Lagrange multipliers. The matrix Cg is equal to the identitymatrix: Cg ¼ I.

7. Slip velocity approximation

Considering a node to node resolution scheme for the contactproblem, the tangential slip velocity equation (12) could be dis-cretized and expressed as:

st ¼ �cþ Drkt ð38Þ

where st is a vector which stores the slip velocity of every contactpair, �c is a vector which stores the creep velocities, kt is a vectorwhich stores the tangential gap (or slip) of every contact pair:

st ¼

ðstÞ1...

ðstÞI...

ðstÞNp

2666666664

3777777775; �c ¼

�c1

..

.

�cI

..

.

�cNp

2666666664

3777777775; kt ¼

ðktÞ1...

ðktÞI...

ðktÞNp

2666666664

3777777775ð39Þ

and finally, Dr is a matrix operator which approximate the convec-tive term in (12), for the steady-state case.

The tangential slip vector ðktÞ is computed as: k ¼ ðC2ÞT d2

�ðC1ÞT d1 þ kgo, compiling the normal and tangential componentsof every I pair:

kI ¼kt

kn

�I

¼d2

t

d2n

" #I

�d1

t

d1n

" #I

þ0

kgo;n

�I

ð40Þ

The convective term in (12) could be approximated, Drkt , usingtwo numerical schemes: Godunov approximation technique or Least-squares technique. The first scheme is based on Godunov methodol-ogy (Godunov, 1959; Hirsch, 1990), and the second scheme, on thedisplacement gradient reconstruction by Least-squares. Both tech-niques are inspired on fluid-dynamic numerical methodologies(see Blazek, 2001) using the finite difference method (FDM) orthe finite volume method (FVM), for structured or unstructuredmeshes (Fig. 3). The formulation presented is valid for any methodused for solid modeling: FEM or BEM, and for lineal or quadraticcontact frame elements.

The control volumes for lineal elements are computed followinga median-dual cell-vertex scheme associated with the grid nodes(see Fig. 4). Median-dual control volumes are formed by connect-ing the elements centroid and edge-midpoints of all elements shar-ing the particular node I. If the contact frame mesh is discretizedusing quadratic elements, they have to be subdivided in triangles,as Fig. 5 shows. Therefore, we have a like linear-triangles mesh,and the median-dual mesh is formed the ‘‘new” elements centroid.

7.1. Godunov approximation

This scheme defines a control cell which surrounds each contactpair I. It is assumed a piece-constant distribution of the tangentialslip velocity, uniform on every control volume or cell (this is equiv-alent a one-point quadrature integration). The perimeter of eachcell or control volume is formed by straight segments, joining theelements centroid with the edge-midpoints of all elements (seeFig. 4), so each frame node is encapsulated by a control cell consti-tuting a dual grid.

The flux of solid particles which cross the contact zone is com-puted on each edge as a function of the tangential slip, to avoid

Fig. 5. Triangular and quadrilateral quadratic elements subdivision.

Fig. 3. Flux of solid particles traveling through an unstructured contact zone mesh.


having a multi-valued numerical solution on each control cell edge.Therefore, for each cell centered on pair I, the material derivative isapproximated as

ðDrktÞI ¼XNJ

J¼1

ðdrÞIJðktÞJ ¼1XI

XNJ

J¼1

X2

k¼1

fkð1� signðfkÞÞ

2½ðktÞJ � ðktÞI�

ð41Þ

where XI is the area of the control volume, NJ is the number of con-tact pairs J sharing a cell edge with I, and k is the index for the num-ber of common edges between the pairs I and J. The mean flux fk isdefined as:

fk ¼ ðvtÞTk nkLk ð42Þ

being Lk; ðvtÞk and nk: the edge k length, the mean rolling velocity,and the unitary vector normal to the edge k. ðktÞI and ðktÞJ are thetangential slip values of contact pairs I and J, respectively. Eq. (41)considers only the upwind flux using the term: ð1� signðfkÞÞ=2,which is null for the downwind flux, and equal to one, for an up-wind one (see Fig. 6(a)).

7.2. Least-squares approximation

The evaluation of gradients at pair I by the least-squares ap-proach is based upon the use of a first-order Taylor series expan-sion for each element edge that is incident to the pair. The

Fig. 4. Cell-vertex median-dual m

approach of the gradient is obtained as a weighted sum of all thisvalues:

ðDrktÞI ¼XNJ

J¼1

ðdrÞIJðktÞJ ¼XNJ

J¼1

ðvtÞTI wI;J½ðktÞJ � ðktÞI� ð43Þ

where wI;J are the weights for the element edge lIJ defined by thepairs I and J, NJ is the number of element edges that containsI, and ðvtÞI is the mean rolling velocity on I.

The weights wI;J are computed using the least-squares method,obtaining the expressions presented in Blazek’s book (Blazek,2001):

ðwt1ÞI;J ¼ ðDðxt1ÞIJÞ=r211 � r12 r11Dðxt2ÞIJ � r12Dðxt1ÞIJ

� r2

11r222

� ��ðwt2ÞI;J ¼ r11Dðxt2ÞIJ � r12Dðxt1ÞIJ

� r11r2

22

� ��ð44Þ

being Dðxt1ÞIJ and Dðxt2ÞIJ : DðxtÞIJ ¼ ðxtÞJ � ðxtÞI , and the expressionsfor r11; r12 and r22:

r11 ¼XNJ

J¼1

ðDðxt1ÞIJÞ2

!1=2

r12 ¼XNJ

J¼1

Dðxt1ÞIJDðxt2ÞIJ

!,r11

r22 ¼XNJ

J¼1

ðDðxt2ÞIJÞ2 � r2

12

!1=2

ð45Þ

In the expressions above, NJ is the number of upwind J pairs. Thusthe approximation follows an upwind scheme computing the tan-

esh for linear approximation.

Fig. 6. Upwind approximation using: (a) median-dual cell vertex Godunov scheme and (b) least squares.


gential slip gradient. In Fig. 6(b) we can see that the number upwindJ pairs is three.

7.3. Tangential slip velocity approximation

Eq. (38) can be written in a compact form as:

st ¼ �cþ eDrk ð46Þ

being eDr a square 2Np 3Np matrix, as:

eDr ¼

ðdrÞ11½I 0� � � � ðdrÞ1Np½I 0�

..

. . .. ..

.

ðdrÞNp1½I 0� � � � ðdrÞNpNp½I 0�

266664377775 ð47Þ

where I is a 2 2 unitary matrix, and 0 is a 2 1 null vector. Thecoefficients ðdrÞIJ computed on (41) or (43).

8. FEM–FEM rolling contact equations system

Making null the total potential variation (25), considering (30)and (37), and including Eq. (46) for computing the tangential slipvelocity, the following nonlinear equations set is obtained:

HðzÞ ¼ RðzÞ þ F ¼ 0

RðzÞ ¼

K1 0 C1 0 0

0 K2 �C2 0 0

ðC1ÞT �ðC2ÞT 0 Cg 0

0 0 0 eDr �I

0 0 Pk PgnPst

266666666664

377777777775

d1

d2

K

k

st

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

z ¼

d1

d2

K

k

st

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;F ¼

�F1

�F2

�Cgkgo

�c

0

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

ð48Þ

Matrix eDr was defined in (47). Matrices Pk;Pgnand Pst are the non-

linear rolling contact terms, obtained assembling: ðPkÞI; ðPgnÞI

and ðPst ÞI , whose values depend on the rolling contact state of pairI:

Pk ¼XNf

i¼1

Xj¼1

ði;jÞ�INf

Lfi

� TðPkÞIL

fj ;

Pgn¼XNf

i¼1

XNf

j¼1ði;jÞ�I

Lfi

� TðPgnÞIL

fj ;

Pst ¼XNf

i¼1

XNf

j¼1ði;jÞ�I

Lfi

� TðPst ÞIL

fj ð49Þ

� No-contact: ðk Þ P 0
n I
ðPkÞI ¼

1 0 0

0 1 0

0 0 1

26643775

I

; ðPgnÞI ¼

0 0 0

0 0 0

0 0 0

26643775

I

; ðPst ÞI ¼

0 0 0

0 0 0

0 0 0

26643775

I

ð50Þ

� Contact-adhesion: kn� �

I < 0 and kt� �

I

< l kn� �

I

��

ðPkÞI ¼

0 0 0

0 0 0

0 0 0

26643775

I

; ðPgnÞI ¼

0 0 0

0 0 0

0 0 �rn

26643775

I

; ðPst ÞI ¼

rt 0 0

0 rt 0

0 0 0

26643775

I

ð51Þ

� Contact-slip: kn� �

I < 0 and kt� �

I

P l kn� �

I

��

ðPkÞI ¼

1 0 lxt1

0 1 lxt2

0 0 0

26643775

I

; ðPgnÞI ¼

0 0 0

0 0 0

0 0 �rn

26643775

I

;

ðPst ÞI ¼

0 0 0

0 0 0

0 0 0

26643775

I

ð52Þ

being ðxtÞI ¼ kt� �

I

�kt� �

I

.

9. BEM–BEM rolling contact equations system

9.1. BEM equations

The BEM formulation for elastic continua X with boundary C iswell known and can be found in many classical texts such as Breb-bia and Domínguez (1992).


For a boundary point ðP 2 CÞ, the Somigliana identity can bewritten as:

CuðPÞ þ CPVZ

CtudC

� �¼Z

XubdXþ

ZC

utdC ð53Þ

where u, t and b are, respectively, the displacements, the boundary

tractions and the body forces of X. u ¼ uijðP;QÞn o

is the funda-

mental solution tensor for displacement, and t ¼ tijðP;QÞn o

for

tractions. Both are solution of Navier’s equation at point Q inthe ith direction due to a unit load applied at point P in the jthdirection. The matrix C is equal to 1

2 I for a smooth boundaryC, and CPVfIg is called the Cauchy Principal Value of the inte-gral I.

Dividing the boundary C, into Ne elements, Ce 2 C, so:C ¼

SNee¼1C

e andTNe

e¼1Ce ¼ Ø, the integral equation (53) can be

written as follows:

CuðPÞ þXNe

e¼1

ZCe

tudC� �

¼XNe

e¼1

ZCe

utdC� �

ð54Þ

in case of absence of body loads (b=0).The fields u and t are approximated over each element Ce using

shape functions, as a function of the nodal values ðde and peÞ:u ’ u ¼ Nde and t ’ t ¼ Npe, being N the shape function approxi-mation matrix.

After the discretization, Eq. (54) can be written as

Ciui þXN

j¼1

Hei de ¼

XN

j¼1

Gei pe ð55Þ

being Hei ¼

RCe tNdC; Ge

i ¼R

Ce uNdC, the integrals over the ele-ment e when the collocation point is the node i. Finally, the contri-bution for all i nodes can be written together in matrix form to givethe global system of equations,

Hd ¼ Gp ð56Þ

where d and p are the displacements and tractions nodal vectors,respectively. Matrices G and H are constructed collecting the termsof matrices He

i and Gei .

Assuming that the displacement or the traction is known oneach node and direction, the boundary conditions can be imposedrearranging the columns in H and G, and passing all the unknownsto vector x on the left-hand side. This gives the final system ofequations:

Ax ¼ F ð57Þ

9.2. BEM–BEM formulation

In case of modeling the solids Xa ða ¼ 1;2Þ using the BEM (57),matrix Aa is organized for contact problems as: Aa ¼½Aa

xAap � Aa

x ¼ AaqAa

c

h i� , being Aa

p the columns of Aa which affect

the nodal contact tractions, Aac the columns which affect the con-

tact nodes displacements, and Aaq the columns which affect the

exterior nodes unknowns. Therefore, the terms of (48) for aBEM–BEM formulation are:

RðzÞ ¼

A1x 0 A1

peC1 0 0

0 A2x �A2

pfPC2 0 0


0 0 0 eDr I

0 0 Pk PgnPst

266666666666664

377777777777775

x1

x2

K

k

st

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

z ¼

x1

x2

K

k

st

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;F ¼

�F1

�F2

�Cgkgo

�c

0

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

ð58Þ

being vector K the nodal contact tractions, so that: p1c ¼ eC1K

and p2c ¼ �eC2K.

10. FEM–BEM rolling contact equations system

Contact problems simulation using different numericalmethods (FEM and BEM) for solids modeling has to consider theFEM–BEM coupling techniques. These methodologies follow twostrategies which were started by the works (Zienkiewicz et al.,1977; Brebbia and Georgiou, 1979). The FE variables are: displace-ments and forces, and BE variables are: displacements andtractions. Because of that it is necessary for one of the formula-tions to make it compatible with the other one. In this sense,the work of Zienkiewicz et al. (1977) formulate the problemconsidering the BE region as a FE region. On the other hand,Brebbia and Georgiou (1979) formulate the FE region as a BEone.

Following the second strategy, if solid X1 is modeled using theFEM, the term of dPc corresponding to the tractions virtual workis computed as:

XNfe

e¼1

ZCe

c

ðdu1ÞT Btc dC ¼XNf

e

e¼1

XN1c

i¼1

XNf

j¼1

dd1i

� TZ

Cec

NiNj dC

( )BT

j kj ð59Þ

where the index e represents the frame elements, i the contactnodes of X1, and j the contact frame nodes. Ni and Nj are the shapefunctions associated to these nodes.

Eq. (59) can be written as

XNfe

e¼1

ZCe

c

ðdu1ÞT Btc dC ¼ ðdd1ÞT M1K ð60Þ

defining:

d1 ¼XN1

i¼1

L1i

� �Td1

i K ¼XNf

i¼1

Lfi

� Tki

M1 ¼XN1

i¼1

XNf

j¼1

ðL1i Þ

T MijBjLfj Mij ¼

ZCe

c

NiNj dC

ð61Þ

Therefore, considering (48) and (62), in case of modeling one solidby the FEM and the other one by the BEM, we have:


RðzÞ ¼

K1 0 M1 0 00 A2

x �A2peC2 0 0


0 0 0 eDr I0 0 Pk Pgn

Pst

266666664

377777775

d1

x2

K

kst

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

z ¼

d1

x2

K

kst

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;F ¼

�F1

�F2

�Cgkgo

�c0

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð62Þ

11. Solution procedure

The non-liner equations set (48) is going to be solved using thegeneralized Newton method with line search (GNMls).

11.1. Generalized Newton method

The GNMls is an effective extension of the Newton’s method forB-differentiable functions proposed by Pang (1990). This methodis based on the computation of the non-linear directional deriva-tive of the objective function.

Fig. 7. (a) Two rings rolling. (b) Meshes details. (c) Nondi

The application of the GNMls to solve the non-linear system,HðzÞ ¼ 0, being HðzÞ a B-differentiable function, can be summa-rized in the following steps:

(I) Let zð0Þ be an arbitrary initial vector, and let q; b; r and e bepositive scalars: q > 0; b 2 ð0;1Þ; r 2 ð0;1=2Þ; and e > 0.

(II) Given zðnÞ with H zðnÞ� �

–0, the direction DzðnÞ is obtainedsolving:� � � �

mension

H zðnÞ þBH zðnÞ;DzðnÞ ¼ 0 ð63Þ

where BH zðnÞ;DzðnÞ� �

is the function B-derivative.ðnÞ

(III) Let aðnÞ ¼ bm q, mðnÞ be obtained as the first non-negativeinteger so: W zðnÞ þ aðnÞDzðnÞ

� �6 1� 2raðnÞ� �

W zðnÞ� �

, beingW zðnÞ� �

¼ 12 kH zðnÞ

� �k2 an error function, whose minimum is

zðnÞ þ aðnÞDzðnÞ� �

.(IV) Set the new solution vector: zðnþ1Þ ¼ zðnÞ þ aðnÞDzðnÞ.(V) Finish if W zðnþ1Þ� �

6 e1; otherwise return to (II) and iterateuntil convergence is reached.

The main difficulty is to compute BH zðnÞ;DzðnÞ� �

. The system ofequations H zðnÞ

� �is divided in two parts:

H zðnÞ� �

¼ HLD zðnÞ� �

þHNLD zðnÞ� �

ð64Þ

where HNLDðzÞ is the part whose directional derivative is non-linearin some points, and HLDðzÞ is the rest, having directional derivativeat all its points. Thus, the B-derivative can be expressed as:

al rolling contact magnitudes on x ¼ 0 plane.

Fig. 8. (a) Rolling contact tractions surfaces. (b) Tangential slip velocity vectors, st , (top panel) and tangential traction vectors, kt , over the normal contact pressure, kn , contour(bottom panel).


BH zðnÞ;DzðnÞ� �

¼ rHLD zðnÞ� �

þ oHNLD zðnÞ� ��

DzðnÞ ð65Þ

being rHLD zðnÞ� �

the linear part of the Jacobian matrix, andoHNLD zðnÞ

� �is the non-linear one:

rHLD zðnÞ� �

¼

K1 0 M1 0 00 A2

x �A2peC2 0 0


0 0 0 eDr �I0 0 0 0 0

266666664

377777775 ð66Þ

and

oHNLD zðnÞ� �

¼

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 Jk Jgn

Jst

26666664

37777775

ðnÞ

ð67Þ

The matrices JðnÞk ; JðnÞgnand JðnÞst

are constructed from the assembly of

the matrices: JðnÞk

� I; JðnÞgn

� I

and JðnÞst

� I, which are associated to each

I pair, like (49). The value of these matrices depends on the I pair

augmented contact variable KðnÞn

� �I ¼ KðnÞn

� �I þ rn kðnÞn

� I

and�

KðnÞt

� I¼ KðnÞt

� I� rt sðnÞt

� IÞ states:

– No-contact: KðnÞn

� �I P 0

JðnÞk

� I¼

1 0 0

0 1 0

0 0 1

2666437775

I

; JðnÞgn

� I¼

0 0 0

0 0 0

0 0 0

2666437775

I

; JðnÞst

� I¼

0 0 0

0 0 0

0 0 0

2666437775

I

ð68Þ

– Contact-adhesion: KðnÞn

� �I < 0 and KðnÞt

� I

< l KðnÞn

� �I

��

JðnÞk

� I¼

0 0 0

0 0 0

0 0 0

2666437775

I

; JðnÞgn

� I¼

0 0 0

0 0 0

0 0 �rn

2666437775

I

; JðnÞst

� I¼

rt 0 0

0 rt 0

0 0 0

2666437775

I

ð69Þ

– Contact-slip: KðnÞn


� I

P l KðnÞn

� �I

��


Jkð ÞðnÞI ¼

W11 W12 lxt1

W21 W22 lxt2

0 0 0

264375ðnÞ

I

; JðnÞgn

� I¼

0 0 00 0 00 0 �rn

264375

I

;

JðnÞst

� I¼�rt

eW11 �rteW12 0

�rteW21 �rt

eW22 00 0 0

264375ðnÞ

I

ð70Þ

being W ¼ ð1þ aÞI� bR, eW ¼ W� I, xðnÞt ¼ KðnÞt

� I

.KðnÞt

� I

,

R ¼ KðnÞt

� I� KðnÞt

� I, a ¼ l KðnÞn

� �I

�KðnÞt

� I

, and

b ¼ l KðnÞn

� �I

�KðnÞt

� 3.

11.2. Computing DzðnÞ direction

To compute the direction DzðnÞ, the system of linear equations(SLEs) (63) has to be solved. For this purpose, three different meth-ods are going to be used in this work: direct resolution by factoriza-tion, iterative resolution by means of GMRES (Saad and Shultz,1986), and iterative resolution using GMRES with adaptative error(Kelley, 1995).

The GMRES iterative process finishes when:


�H zðnÞ� � 6 g2 H zðnÞ

� � ð71Þ

being g2 the tolerance of the process.

Fig. 9. (a) Wheel rolling over a half-space. (b) FE and BE Meshes. (c) BE mesh

Usually, the value of g2 is constant and proportional to the New-ton’s tolerance. Kelley (1995) suggests choosing an appropriate tol-erance according to the problem we are solving. If g2 is too small,too much computational effort can be spent during the startingNewton iterations, because each step is solved to a precision far be-yond what is needed to correct the non-linear iteration. To avoidover-solving of the SLE, an appropriate tolerance is used for eachiteration n gðnÞ2

� , as a function of the ‘‘proximity” between zðnÞ

and the solution.A measurement of the proximity to the real solution, is:

gðnÞ1 ¼ cH zðnÞ� � 2

Hðzðn�1ÞÞk k2 ð72Þ

where c 2 ð0;1�. Thus, Kelley (1995) suggests the following criteriafor g2:

gðnÞ2 ¼g2;max n ¼ 0

min g2;max;gðnÞ1

� n > 0

8<: ð73Þ

being g2;max a limit superior of gðnÞ2 . For practical reasons, we canconsider gðn�1Þ

1 , consequently (73) is expressed as:

gðnÞ2 ¼g2;max n ¼ 0

min g2;max;gðn�1Þ1

� n > 0

8<: ð74Þ

details. (d) Nondimensional rolling contact magnitudes on x ¼ 0 plane.

Fig. 10. (a) Rolling contact tractions surfaces. (b) Tangential slip velocity vectors, st , (top panel) and tangential traction vectors, kt , over the normal contact pressure, kn ,contour (bottom panel).

Table 1Meshes data.

Mesh type Number of nodes Number of elements

Linear quadrilateral elements 673 640Quadratic quadrilateral elements 1985 640Linear triangular elements 703 1340Quadratic triangular elements 2745 1340


In case zðnÞ were far from the solution and gðnÞ2 were too small during

some consecutive iterations, gðnÞ2 cannot be allowed to reduce its va-

lue, thus: gðnÞ2 ¼min g2;max;max gðn�1Þ1 ; c gðn�1Þ

2

� 2� ��

.

Therefore, (74) is written:

– gðnÞ2 ¼ g2;max if n ¼ 0,

– gðnÞ2 ¼min g2;max;gðn�1Þ1

� if n > 0 and c gðn�1Þ

2

� 26 0:1,

– gðnÞ2 ¼minðg2;maxÞ;max gðn�1Þ1 ; c gðn�1Þ

2

� 2� �

if

n > 0 and cðgðn�1Þ2 Þ2 P 0:1, with c ¼ 0:1.

The choice of adaptative error for GMRES resolution allows toreduce the CPU times (CPUT) in contact problems resolution.

11.3. Projection function

The main computational effort on the Newton’s Method is dueto the SLE resolution (63), in obtaining the direction DzðnÞ

� �for

every iteration (n). One strategy for effort decreasing is to reduce

the number of equations and unknowns. Applying the contactoperators (18) and (21), it is possible to know the value ofsome variables ðzcÞ at instant ðnþ 1Þ. Therefore, for the I pair incontact:

� If KðnÞn

� �I > 0) Kðnþ1Þ

n

� �I ¼ 0 and Kðnþ1Þ

t

� I¼ 0.

� If KðnÞn

� �I 6 0) kðnþ1Þ

n

� I¼ 0 and:

– If KðnÞt

� I

< �l KðnÞn

� �I ) kðnþ1Þ

t

� I¼ 0.

– If KðnÞt

� I

P �l KðnÞn

� �I ) Kðnþ1Þ

t

� I¼ �l Kðnþ1Þ

n

� �I eðnþ1Þ

t

� I,

being eðnþ1Þt ¼ x

ðnþ1Þt

� .

Fig. 11. (a) Rolling sphere over a rigid flat. (b) Halfspace triangular mesh details. (c) Halfspace quadrilateral mesh details. (d) Nondimensional rolling contact magnitudes onx ¼ 0 plane.


In most cases the value is null, thus we can say that a quasi-com-plementarity exists between the contact variables. This fact can beapplied to reduce the size of the equations system, decreasing thenumber of rolling contact unknowns, and consequently accelerat-ing the GNMls.

To approximate the direction et , we can observe the contactprojection function behavior (21) during the Newton iteration.Expanding the tangential component of this operator in Taylor

series, in the contact-slip case, KðnÞt

� I

P �l KðnÞn

� �I ,

PCg Kðnþ1Þt

� I

� ¼ PCg KðnÞt

� I

� þ

oPCg

ozc

�ðnÞI

DzðnÞc

� �I þ � � � ð75Þ

being g ¼ l KðnÞn

� �I

�� ,PCg Kðnþ1Þ

t

� I

� ¼ �l Kðnþ1Þ

n

� �I x

ðnÞt

� I� l

KðnÞn

� �I

KðnÞt

� I

�xt2

xt1

� �ðnÞI

�xt2xt1f gðnÞI DKðnÞt

� Iþ � � � ð76Þ


Taking into account that the second term on the right-hand sidegoes to zero when zðnÞ is converging, the projection function canbe expressed as:

PCg Kðnþ1Þt

� I

� ¼ �l Kðnþ1Þ

n

� �I x

ðnÞt

� I

ð77Þ

Consequently, the direction et for the tangential operator at instantðnþ 1Þ is equal to the previous one,

eðnþ1Þt

� I¼ x

ðnÞt

� I

ð78Þ

Fig. 12. (a) Rolling contact tractions distribution. (b) Tangential slip velocity vectors, s(bottom panel).

This fact allows us to know a half of the rolling contact unknowns inthe step ðnþ 1Þ.

11.4. Matrices for the n-iteration

The system (63) can be expressed for the n-iteration as,

H zðnÞ� �

¼ RðnÞzðnÞ � F ¼ 0 ð79Þ

being

t . (c) Tangential traction vectors, kt , over the normal contact pressure, kn , contour

Fig. 13. Comparison between the results obtained using Godunov upwind approach (left) and least-squares upwind approach (right) for the rolling contact tractions: (a) kx , (b)ky , and (c) kn .


RðnÞ ¼R1 R2 Rk Rg 0

0 0 0 eDr �I0 0 PðnÞk PðnÞgn

PðnÞst

264375

zðnÞ ¼

d1

x2

Kkst

8>>>>><>>>>>:

9>>>>>=>>>>>;

ðnÞ

F ¼F�c0

8><>:9>=>;

ð80Þ

The matrices R1; R2; Rk and Rg , and vector F are defined for theFEM–BEM rolling contact formulation:

R1 ¼K1

0ðC1ÞT

264375; R2 ¼

0A2

x

�ðC2ÞT

264375; Rk ¼

M1

�A2peC2

0

264375;

Rg ¼00Cg

264375; F ¼

F1

F2

Cgkgo

264375 ð81Þ

Fig. 14. Comparison between the results obtained using Godunov upwind approach (left) and least-squares upwind approach (right) for the rolling contact kinematic variables:(a) sx , (b) sy , and (c) kn .


The matrices PðnÞk ; PðnÞgnand PðnÞst

are constructed from the assembling

of the I pair matrices PðnÞk

� I; PðnÞgn

� I

and PðnÞst

� I, respectively, as in

(49). Due to the approximation (77), the expressions (68)–(70) for

the contact-slip situation KðnÞn


� I

P ljðKnÞIj�

are:

PðnÞk

� I¼

1 0 let1

0 1 let2

0 0 0

24 35I

; PðnÞg

� I¼

0 0 00 0 00 0 �rn

24 35I

;

PðnÞst

� I¼

0 0 00 0 00 0 0

24 35I

ð82Þ

con ðetÞI ¼ xðn�1Þt

� I.


11.5. B-derivative expression

The rolling contact restrictions are linearized, considering thedirection ðetÞI is constant for every step. Therefore, the Jacobianmatrices are:

JðnÞk ¼ PðnÞk JðnÞgn¼ PðnÞgn

JðnÞst¼ PðnÞst

ð83Þ

and the expression for the B-derivative:


’ RðnÞDzðnÞ ð84Þ

11.6. Initial solution

The initial solution assumes that all points have zero tractions:Kð0Þ ¼ 0. Using the rigid body motions, ko, the gap variable kð0Þ isobtained, as well as other unknowns, zð0Þ.

11.7. Solution for the next step

The direction DzðnÞ is computed solving Eq. (63):

RðnÞ zðnÞ þ DzðnÞ� �

¼ F ð85Þ

Each iteration computes a tentative solution, ~zðnþ1Þ ~zðnþ1Þ ¼ zðnÞ�

þDzðnÞÞ, instead of direction DzðnÞ, solving:

RðnÞ~zðnþ1Þ ¼ F ð86Þ

Fig. 15. Tangential slip velocity component sy obtained on different meshes: (a) linearelements, and (d) quadratic triangular elements.

The variable ~zðnþ1Þ is the next step solution for aðkÞ ¼ 1. The solution~zðnþ1Þ will be obtained from the scale parameter aðnÞ as:

zðnþ1Þ ¼ aðnÞ~zðnþ1Þ þ 1� aðnÞ� �

zðnÞ ð87Þ

11.8. SLE size reduction

Being Ndofa ða ¼ 1;2Þ the number of degrees of freedom of solidXa and Ndofc the number of contact pairs degrees of freedomðNdofc ¼ 3NpÞ, the number of system (85) unknowns is:Ndof1 þ Ndof2 þ 8Ndofc=3. Some of the contact unknowns are nullsor known, thus, the number of variables is: Ndof1 þ Ndof2 þ Ndofc.Therefore, the SLE to solve in every iteration is:

R1 R2 RðnÞy

h i ~d1

~x2

~y

8><>:9>=>;ðnþ1Þ

¼ �FðnÞ ð88Þ

where matrix RðnÞy is constructed with the columns of matrices:Rk and Rg , depending on the contact situation. Vector �FðnÞ is com-puted as:

�FðnÞ ¼ F�X

I pair incontact-adhesion

ðRgtÞI ~kðnþ1Þ

t

� I

ð89Þ

being ~kðkþ1Þt

� I

the tangential vector estimated for I pair. The col-

umns of RðnÞy are defined according to the contact status on step (n):

quadrilateral elements, (b) quadratic quadrilateral elements, (c) linear triangular


– If KðnÞn

� �I P 0: the column of RðnÞy corresponding to the I pair nor-

mal component, RðnÞyn

� I, is equal to ðRgn

ÞI , and the tangential

components, RðnÞyt

� I, to ðRgt

ÞI . Consequently, the I pair rolling

contact unknowns are: ~yðnþ1Þ� �I ¼

~kðnþ1Þ�

I. The values of the

other variables are: ~Kðnþ1Þ� �I ¼ 0 and ~sðnþ1Þ

t

� I¼ �cð ÞI

þP

J~dr

� IJðkðkÞt ÞJ , being ~dIJ the matrix eDr coefficients relative to

pair J which compute the convective terms for I pair.

– If KðnÞn


� I

< �l KðnÞn

� �I : RðnÞyn

� I¼ ðRkn ÞI and

RðnÞyt

� I¼ ðRkt ÞI . In this case the I pair rolling contact unknowns

are: ~yðnþ1Þ� �I ¼ ~Kðnþ1Þ� �

I . The tangential slip, ~kðnþ1Þ�

I, is com-

puted knowing that ~stð ÞI ¼ 0, from any of the followingapproximations:

Fig. 16. (a) CPU times spent by different methods for solving the SLE in the Newton algormethods for solving the SLE for different linear quadrilateral meshes.

– Jacobi approximation:

~kðnþ1Þt

� I¼ � 1

~d�

II

0B@1CA �cð ÞI þ

XJ–I

~dr

� IJ

kðnÞt

� J

!ð90Þ

– Pseudo-temporal approximation:

~kðnþ1Þt

� I¼ kðnÞt

� I� ðDtÞ �cð ÞI þ

XJ

~dr

� IJ

kðnÞt

� J

!ð91Þ

– Extrapolation:

~kðnþ1Þt

� I¼ ð1� hÞ kðnÞt

� I

� h1

~dr

� II

0B@1CA �cð ÞI þ

XJ

~dr

� IJ

kðnÞt

� J

!ð92Þ

ithm, for the same mesh. (b) CPU times relative to LU factorization spent by different

Fig. 17. Wheel rolling over a railway.


The Pseudo-temporal approximation and the Extrapolationapproximation are variations of Jacobi approximation, trying toaccelerate the problem resolution, choosing appropriate valuesfor parameters: Dt and hðh > 1Þ, respectively. Their values areadjusted by trial-error on each problem. In our case, we have used:Dt ¼ 0:15 and h ¼ 1:2. Choosing a low value for Dt and close to 1for h, we have tested no convergence problems.

– Finally, if KðnÞn


� I

P �l KðnÞn

� �I : RðnÞyn

� I¼

ðRkn ÞI � lðRkt ÞI eðnÞt

� I

h i, and RðnÞyt

� I¼ ðRgt

ÞI . Therefore, the I pair

rolling contact unknowns are: ~yðnþ1Þn

� I¼ ~Kðnþ1Þ

n

� I

and

~yðnþ1Þt

� I¼ ~kðnþ1Þ

t

� I. The values of the other variables are:

eKðnþ1Þt

� I¼l eKðnþ1Þ

n

� IðetÞðnÞI and ~sðnþ1Þ

t

� I¼ �cð ÞIþ

PJ

~dr

� IJ

kðnÞt

� J.

12. Numerical examples

This section presents some numerical examples that enable usto validate the methodology presented, as well as the algorithmof resolution and the approximation techniques for the materialderivative.

12.1. Example 1: two rings rolling

The first example under study is the rolling contact between twotwin rings of radii: R1 ¼ 25 mm and R2 ¼ 50 mm, and L ¼ 8 mm, asFig. 7(a) shows. The resultant loads are: jFt j=ljFnj ¼ 0:6271 (being

jFnj ¼ 1233:85 N and the coefficient of friction: l ¼ 0:1), the meanangular velocity x ¼ 0:2 s�1, and the material properties are: E1 ¼E2 ¼ 104 MPa and m1 ¼ m2 ¼ 0:3 (rolling contact similar problem).

The two rings are modeled using a boundary element mesh of1474 nodes and 1472 quadrilateral elements (see Fig. 7(b)). Thepotential contact zone is discretized by 12 12 elements. Thisproblem has been solved using both methodologies for materialderivative computing (Godunov scheme and least-squares scheme),obtaining the same results in both cases.

Fig. 7(b) shows the nondimensional rolling contact variables onx ¼ 0. The characteristic magnitudes are the two infinite cylindersHertz maximum normal pressure, po;Hertz ¼ 103:87 MPa, and contactzone width, a ¼ 0:9453 mm. The normal gap and the tangential slipvelocity are non-dimensionalized by their maximum values on thecontact zone: maxðknÞ ¼ 0:0156 mm and maxðstÞ ¼ 0:0428 mm=s,respectively.

Fig. 8(a) presents the traction components distribution, andFig. 8(b) the tangential slip velocity vectors (top panel), and tan-gential traction vectors over the normal contact pressure contour(bottom panel). We can see in Fig. 8 how the tangential contactrestriction is fulfilled.

12.2. Example 2: ring rolling over an elastic halfspace

The next example is the rolling contact of a wheel over an elastichalf-space. The wheel dimensions are: R1 ¼ 25 mm; R2 ¼ 50 mmand L ¼ 8 mm, as Fig. 9(a) shows.

The wheel is discretized by means of a 2016 hexaedrical finiteelement mesh, and the elastic half-space is discretized with 288

Fig. 18. (a) Boundary element mesh. (b) Mesh details. (c) Quadrilateral boundary element mesh on potential contact zone.


quadrilateral boundary elements, with 12 12 elements in the po-tential contact zone (see details in Fig. 9(b) and (c)).

The material properties are: E1 ¼ 104 MPa; E2 ¼ 106 MPa, andm1 ¼ m2 ¼ 0:3. The resultant applied loads are: jFt j=ljFnj ¼ 0:5373,being jFnj ¼ 1345; N and the coefficient of friction: l ¼ 0:1. Theangular velocity is equal to: x ¼ 0:2 s�1.

This problem has been solved using Godunov scheme and least-squares scheme, obtaining the same results in both cases. Fig. 9(b)shows the nondimensional rolling contact variables on x ¼ 0. Inthis case, the characteristic magnitudes are the infinite cylinderover a flat Hertz maximum normal pressure, po;Hertz ¼ 107:9 MPa,and contact zone width, a ¼ 0:99 mm. The normal gap and thetangential slip velocity are nondimensionalized by their maximumvalues on the contact zone: maxðknÞ ¼ 0:0086 mm and maxðstÞ ¼0:0139 mm=s, respectively.

Fig. 10(a) presents the traction components distribution, andFig. 10(b) the tangential slip velocity vectors (top panel), and tan-gential traction vectors over the normal contact pressure contour(bottom panel). We can see maximum energy dissipation principlerestriction fulfilment in Fig. 8.

12.3. Unstructured meshes and CPU times

Two aspects will be dealt with below. The first one pertains thebehavior that the suggested formulations show when differentkinds of elements are used on the contact interfaces: quadrilateraland triangular (linear or quadratic), and different kind of meshes

(structured and unstructured) are considered. Secondly, the focuswill be placed on the CPU times used to solved the problem bythe GNMls with the resolution of the linear system of equations(LSEs) by means of: LU factorization, GMRES iterative resolution,and GMRES resolution with adaptative error. To that end, the follow-ing examples will be solved and commented on:

12.3.1. Example 3: sphere rolling over a rigid planeThis example presents the rolling contact problem of a sphere of

radii R1 ¼ 50 mm, over e rigid flat (see Fig. 11(a)). The resultant ap-plied loads are: jFt j=ljFnj ¼ 0:75, being jFnj ¼ 298:6 N and the coef-ficient of friction: l ¼ 0:1. The angular velocity is x ¼ 0:2 s�1, andthe material properties are: E ¼ 104 MPa and m ¼ 0:3.

The problem is solved using the two methodologies presentedfor material derivative approximation (Godunov scheme and least-squares scheme), different kind of elements and meshes, and differ-ent kind of SLE resolution techniques and acceleration techniques.

The sphere is modeled as an elastic halfspace, using a quadrilat-eral and triangular (linear and quadratic) boundary elements mesh(see Fig. 11(b) and (c)). The meshes details are presented on Table 1.

Fig. 11(d) shows the nondimensional rolling contact variableson x ¼ 0. In this case, the characteristic magnitudes are the sphereover a flat Hertz maximum friction less normal pressure,po;Hertz ¼ 140.

The rolling contact tractions distributions are presented inFig. 12(a), and the tangential slip velocity vectors and the tangen-tial tractions vectors in Fig. 12(b) and (c), respectively.

Fig. 19. Rolling contact variables distribution on Cc: (a) sx , (b) kx , (c) sy , (d) ky , (e) kn , and (f) kn.


Figs. 13 and 14 compares the rolling contact traction vari-ables and the rolling contact kinematic variables, respectively,obtained using the Godunov scheme (left) and Least-squarescheme (right), on an unstructured triangular elements mesh.Both methodologies provide very similar results, even thoughthe Least-square scheme keeps better the symmetries of theresults.

The comparison between the different kind of meshes (struc-tured and unstructured) is presented in Fig. 15, where the tangen-tial slip velocity component, sy, is presented. As Figure shows, theresults presents a high agreement for the linear and quadraticcases, even using different elements density.

Finally, the CPU studies are presented. Fig. 16(a) compares theCPU times spent on problem resolution, using different method

Fig. 21. (a) Error function evolution for the GNMls, using different methods for SLE resolution. (b) CPU times spent by different methods for solving the SLE in the Newtonalgorithm.

Fig. 20. (a) Tangential slip velocity vectors. (b) Tangential traction vectors.


solving the SLE and different slip estimation strategies: Pseudo-temporal, Jacobi and Extrapolation, for the same linear quadrilateralmesh. The CPU times are very similar for the slip estimation strat-egies, but not between the SLE resolution methods. Using an itera-tive GMRES with adaptative error, an important CPU timereduction is obtained. This fact is even more important when themesh density increases the number of degrees of freedom(Fig. 16(b)).

12.3.2. Example 4: wheel rolling over a railwayThe last example presents a wheel rolling over a rail (see

Fig. 17). The wheel is subjected to a rigid body displacement:kzo ¼ 0:015 mm (equivalent to jFzj ¼ 457 N), and jFt j=ljFnj ¼ 0:57,being the coefficient of friction: l ¼ 0:1. The angular velocity is:x ¼ 0:1 s�1. The material that the wheel is made of is characterizedby the values of the elastic constants: E1 ¼ 104 MPa and m1 ¼ 0:3.The rail, in turn, consists of a material whose E2 � E1. Conse-quently, the rail will be taken as infinitely rigid.

Fig. 18(a) shows the region of the wheel discretized using linearquadrilateral boundary elements. The mesh has 1954 nodes and1952 elements. In Fig. 18(b) and (c) it is possible to see the meshdetails next to the potential contact zone of dimensions: A B(A = 5.49 mm and B = 7 mm).

Fig. 19(a) and (b), (c) and (d) and (e) and (f) presents, respec-tively: sx—kx; sy—ky and kn—kn. Fig. 20(a) shows the tangential slipvelocity vectors, and Fig. 20(b) the tangential traction ones.

Finally, Fig. 21(a) presents the error function evolution onthe MNGbd for different methods of SLE resolution, andFig. 21(b) shows the total CPU time comparison. We cansee how the number of Newton iterations is very similar forthis three SLE methods, but the total CPU resolution timespent using GMRES or GMRES with adaptative error is less thana half.

13. Conclusions

This work presents a new formulation for solving 3D rollingcontact problems valid for Finite Elements and/or BoundaryElements techniques. Furthermore, an algorithm of resolutionbased on Generalized Newton Method is presented for solvingthe resulting nonlinear rolling contact equations set.

The rolling contact problem is formulated from an Eulerianpoint of view, presenting two different methodologies for comput-ing the material derivative. Those methodologies are based on fluiddynamic techniques, allowing to model structured and unstruc-tured interface meshes.


The restrictions are defined by projection function acting overthe augmented rolling contact tractions, presenting in this worka new projection function which provides a quasi-complementar-ity on the contact variables. This fact, allows to know one of therolling contact complementarity variables during the iterativeNewton’s process, so the number of contact unknowns have beenreduced.

Some validation examples have been solved using differentmeshes, showing a very good accuracy achieved for different kindof 3D rolling contact problems, different kind of elements, andstructured and unstructured meshes.

In conclusion, the proposed methodology is a new and versatilenumerical tool that allows to solve rolling contact problems usingthe FEM and/or the BEM for solids modeling.

Acknowledgements

This work was co-funded by the DGICYT of Ministerio de Cien-cia y Tecnología, Spain, research Project DPI2006-04598, and bythe Consegería de Innovación Ciencia y Empresa de la Junta deAndalucía, Spain, research Projects P05-TEP-00882 and P09-TEP-03804.

References

Abascal, R., Rodríguez-Tembleque, L., 2007. Steady-state 3D rolling contact usingboundary elements. Communications in Numerical Methods in Engineering 23,905–920.

Alart, P., Curnier, A., 1991. A mixed formulation for frictional contact problemsprone to Newton like solution methods. Computer methods in Appliedmechanics and engineering 92, 353–375.

Anderson, T., 1982. The boundary element method applied to two-dimensionalcontact problems with friction. In: Brebbia, C.A. (Ed.), Boundary ElementMethods. Springer, Berlin.

Bass, J.M., 1987. Three-dimensional finite deformation, rolling contact of ahyperelastic cylinder, formulation of the problem and computational results.Computers and Structures 26, 991–1004.

Batra, R.C., 1981. Quasi-static indentation of a rubber-covered roll by a rigidroll. International Journal for Numerical Methods in Engineering 17, 1823–1833.

Bentall, R.H., Johnson, K.L., 1967. Slip in the rolling contact of two dissimilar elasticrollers. International Journal of Mechanical Sciences 9, 389–404.

Bentall, R.H., Johnson, K.L., 1968. An elastic strip in plane rolling contact.International Journal of Mechanical Sciences 10, 637–663.

Blazek, J., 2001. Computational Fluids Dynamics: Principles and Applications.Elsevier, New York.

Brebbia, C.A., Domínguez, J., 1992. Boundary Elements: An Introductory Course,second ed. Computational Mechanics Publications.

Brebbia, C.A., Georgiou, P., 1979. Combination of boundary and finite elements forelastostatics. Applied Mathematical Modelling 3, 212–220.

Carter, F.W., 1926. On the action of a locomotive driving wheel. Proceedings of theRoyal Society of London Series A 122, 151–157.

Christensen, P.W., Klarbring, A., Pang, J.S., Strömberg, N., 1998. Formulation andcomparison of algorithms for frictional contact problems. International Journalfor Numerical Methods in Engineering 42, 145–173.

Godunov, S.K., 1959. A difference scheme for numerical solution ofdiscontinuous solution of hydrodynamic equations. Math. Sbornik, 47,271–306; Translated US Joint Publications Research Service, JPRS 1969;7226.

González, J.A., Abascal, R., 1998. Using the boundary element method to solverolling contact problems. Engineering Analysis with Boundary Elements 21,392–395.

González, J.A., Abascal, R., 2000a. Solving rolling contact problems using boundaryelement method and mathematical programming algorithms. ComputerModeling in Engineering and Sciences 1, 141–150.

González, J.A., Abascal, R., 2000b. An algorithm to solve coupled 2D rolling contactproblems. International Journal for Numerical Methods in Engineering 49,1143–1167.

González, J.A., Abascal, R., 2002. Solving 2D transient rolling contact problems usingthe BEM and mathematical programming techniques. International Journal forNumerical Methods in Engineering 53, 843–874.

Hirsch, C., 1990. Numerical computation of internal and external flows.Computational Methods for Inviscid and Viscous Flows, vol. 2. Wiley, NewYork.

Hu, G.D., Wriggers, P., 2002. On the adaptative finite element method of steady-state rolling contact for hyperelasticity in finite deformations. ComputerMethods in Applied Mechanics and Engineering 191, 1333–1348.

Johnson, K.L., 1985. Contact Mechanics. Cambridge University Press, Cambridge.Kalker, J.J., 1967. On the rolling contact of two elastic bodies in presence

of dry friction. Ph.D. Thesis, Delft Technical University, Delft, TheNetherlands.

Kalker, J.J., 1971. A minimum principle for the law of dry friction, with applicationsto elastic cylinders in rolling contact. Part1: fundamentals, application to steadyrolling. Journal of Applied Mechanics 38, 875–880.

Kalker, J.J., 1975. The mechanics of the contact between deformable bodies. In:Pater, A.D., Kalker, J.J. (Eds.), Proceedings of the IUTAM Symposium. DelftUniversity Press, The Netherlands.

Kalker, J.J., 1982. A fast algorithm for the simplified theory of rolling contact. VehicleSystem Dynamics 11, 1–13.

Kalker, J.J., 1986. Railway wheel and automotive tyre. Vehicle System Dynamics 15,255–269.

Kalker, J.J., 1988. Contact mechanics algorithms. Communications in AppliedNumerical Methods 4, 25–32.

Kalker, J.J., 1990. Three-Dimensional Elastic Bodies in Rolling Contact. KluwerAcademic Press, Dordrecht.

Kelley, T.C., 1995. Iterative Methods for Linear and Nonlinear Equations. SIAMStudies in Applied Mathematics.

Kikuchi, N., Oden, J.T., 1988. Contact Problems in Elasticity. A Study of VariationalInequalities and Finite Elements Methods. SIAM, Philadelphia, PA.

Klarbring, A., 1986. A mathematical programming approach to three-dimensionalcontact problems with friction. Computer Methods in Applied Mechanics andEngineering 58, 175–200.

Klarbring, A., 1987. Contact problems with friction by linear complementarity. In:Unilateral Problems in Structural Analisys. CSIM Courses and Lectures, vol. II.Springer, New York.

Klarbring, A., 1993. Mathematical programming in contact problems. In: Aliabadi,M.H., Brebbia, C.A. (Eds.), Computational Methods in Contact Mechanics.Computational Mechanics Publications, Southampton.

Klarbring, A., Björkman, G., 1988. A mathematical programming approach tocontact problems with friction and varying contact surface. Computers andStructures 30, 1185–1198.

Landers, J.A., Taylor, R.L., 1985. An Augmented Lagrangian formulation for the finiteelement solution of contact problems. Rept. No. UCB/SESM-85/09, Departmentof Civil Engineering, University of California, Berkeley.

Laursen, T.A., 2002. Computational Contact and Impact Mechanics. Springer,England.

Man, A.W., Aliabadi, M.H., Rooke, D.P., 1993a. BEM frictional contact analysis,modelling considerations. Engineering Analysis with Boundary Elements 11,77–85.

Man, A.W., Aliabadi, M.H., Rooke, D.P., 1993b. BEM frictional contact analysis, loadincremental technique. Computers and Structures 47, 893–905.

Martín, D., Aliabadi, M.H., 1993. Boundary element analysis of two-dimensionalelastoplastic contact problems. Engineering Analysis with Boundary Elements21, 349–360.

Nackenhorst, U., 2004. The ALE-formulation of bodies in rolling contact theoreticalfoundations and finite element approach. Computer Methods in AppliedMechanics and Engineering 193, 4299–4322.

Nowell, D., Hills, D.A., 1987a. Tractive rolling of dissimilar elastic cylinders.International Journal of Mechanical Sciences 30, 427–439.

Nowell, D., Hills, D.A., 1987b. Tractive rolling of tyred cylinders. InternationalJournal of Mechanical Sciences 30, 945–957.

Padovan, J., Zeid, I., 1984. Finite element analysis of steadily moving contact fields.Computers and Structures 18, 191–200.

Panagiotopoulos, P.D., 1985. Inecuality Problems in Mechanics and Applications,Convex and Nonconvex Energy Functions. Springer, Basel.

Pang, J.S., 1990. Newton’s method for B-differentiable equations. MathematicalOperational Research 15, 311–341.

París, F., Garrido, J.A., 1989. An incremental procedure for friction contact problemswith the boundary element method. Engineering Analysis with BoundaryElements 6, 202–213.

Saad, Y., Shultz, M., 1986. GMRES a generalized minimal residual algorithm forsolving nonsymmetric linear systems. SIAM Journal of Scientific Computing 7,856–869.

Simo, J.C., Laursen, T.A., 1992. An augmented Lagrangian treatment of contactproblems involving friction. Computers and Structures 42, 97–116.

Simo, J.C., Wriggers, P., Taylor, R.L., 1985. A perturbed Lagrangian formulation forthe finite element solution of contact problems. Computer Methods in AppliedMechanics and Engineering 50, 163–180.

Singh, K.P., Paul, B., 1974. Numerical solution of non-Hertzian elastic contactproblems. Journal of Applied Mechanics 41, 484–490.

Strömberg, N., 1997. An augmented lagrangian method for fretting problems.European Journal of Mechanics A/Solids 16, 573–593.

Strömberg, N., 2005. An implicit method for frictional contact, impact and rolling.European journal of Mechanics A/Solids 24, 1016–1029.

Wriggers, P., 2002. Computational Contact Mechanics. West Sussex, England.Wriggers, P., Simo, J.C., Taylor, R.L., 1985. Penalty and augmented Lagrangian

formulations for contact problems. In: Proceedings of the NUMETA’85Conference, Swansea.


Zeid, I., Padovan, J., 1981. Finite element modeling of rolling contact. Computers andStructures 14, 163–170.

Ziefle, M., Nackenhorst, U., 2008. Numerical techniques for rolling rubber wheels,treatment of inelastic material properties and frictional contact. ComputationalMechanics 42 (3), 337–356.

Zienkiewicz, O.C., 1977. The Finite Element Method. McGraw-Hill, NewYork.

Zienkiewicz, O.C., Kelly, D.W., Bettes, P., 1977. The coupling of the finite elementmethod and boundary solution procedures. International Journal for NumericalMethods in Engineering 11, 355–375.

Documents

A 3D FEM–BEM rolling contact formulation for unstructured meshes