A Dirichlet-Neumann type algorithm for contact problems with friction

ROLF H. KRAUSE 

AND BARBARA I. WOHLMUTH y

Abstra t. Domainde ompositionte hniquesprovideapowerfultoolforthenumeri al

approx-imationofpartialdierentialequations. Weintrodu eanewalgorithmforthenumeri alsolutionof

anonlinear onta tproblemwithCoulombfri tionbetweenlinearelasti bodies.Thedis retization

ofthe nonlinear problemisbasedon mortarte hniques. Weusea dualbasis Lagrangemultiplier

spa eforthe oupling ofthe dierentbodies. Theboundary data transferatthe onta tzone is

essential forthe algorithm. It isrealized bya s aledmass matrixwhi h resultsfromthe mortar

dis retizationonnon-mat hingtriangulations. Weapplyanonlinearblo kGau{Seidel methodas

iterativesolverwhi h anbeinterpretedasaDiri hlet{Neumannalgorithmforthenonlinear

prob-lem. Inea hiterationstep,wehaveto solvealinearNeumannproblemand anonlinearSignorini

problem.ThesolutionoftheSignoriniproblemisrealizedintermsofmonotonemultigridmethods,

[Kor97a,Kra01℄. Numeri alresultsillustratetheperforman eofourapproa hin2Dand3D.

Keywords. mortarniteelements,dualspa e,Diri hlet{Neumannalgorithm, non-mat hing

triangulations,multigridmethods, onta tproblems,linearelasti ity

AMSsubje t lassi ations.65N30,65N55,74B10

1. Introdu tion. Wepresentdomainde ompositionmethodswithinthe

frame-workofmortarte hniques[BMP93,BMP94℄. Originallyintrodu edasa

non onform-ing method for the oupling of spe tral elements, these te hniques an be used in

a large lass of situations. The oupling of dierentphysi al models, dis retization

s hemes or non-mat hing triangulationsalong interiorinterfa es of the domain an

be analyzed by mortar methods. These domain de omposition te hniques provide

amore exible approa h than standard onformingformulations, andare of spe ial

interest for time dependent problems, rotating geometries, inhomogeneous

materi-als,problemswithlo alanisotropies, ornersingularities, onta tproblemsandwhen

dierenttermsdominateindierentregionsofthesimulationdomain. Onemajor

re-quirementtoobtainoptimaldis retizations hemesisthattheinterfa esbetweenthe

dierentregionsarehandledappropriately,see,e.g.,[BD98,Ben99,BMP93,BMP94℄.

Veryoften,suitablemat hing onditionsattheinterfa es anbeformulatedasweak

ontinuity onditions. Here,we onsidermortarniteelementformulationsbasedon

a dual basis for the Lagrange multiplier spa e, see [Woh00℄, with spe ial emphasis

onnonlinear onta tproblems. Asa onsequen eofthebiorthogonalityrelationand

in ontrasttothestandardmortarmethods, thelo alityofthesupportof thenodal

basisfun tionsofthe orresponding onstrainedspa eispreserved.

We fo us on a nonlinearproblem modeling the onta tof linear elasti bodies.

The a tual zone of onta t is not known in advan e and has to be identied

dur-ingtheiterationpro ess. Alotofworkhasbeendoneon onta tproblems,see,e.g.,

[DNS99,WG97,HH80,HH81,ESW99℄and[Wri95,IHL88,KO88a℄forsurveypapers.

TwomaindiÆ ultieso urinthenumeri alsimulationof onta tproblems. Therst

isthehandlingoftheboundarydatatransferattheinterfa ebetweenthetwobodies.

Inoursetting,thisinformationtransferisrealizedintermsofthes aledmassmatrix

fromthemortarformulation. These onddiÆ ultyistheintrinsi nonlinearityofthe



Institut furMathematikI,Freie Universitat Berlin,Arnimallee2, D{14195Berlin, Germany

Email:krausemath.fu-berlin. de, http://www.math.fu-berli n.de /~kr aus e

y

Math.Institut,UniversitatAugsburg,Universitatsstr.14,D{86159Augsburg,Germany.

problem at the onta t boundary. Toover ome this diÆ ulty, we use a monotone

multigrid method as a subdomain solver, see [Kor97a, KK99, KK00, Kra01℄. This

method providesaneÆ ientiteratives hemeforellipti obsta leproblemsin luding

theSignorini problem. Wereferto [Kra01℄for atheoreti alandnumeri alanalyzis.

However, it annot be applied dire tly to multi body problems with non-mat hing

triangulations. Usingmortarte hniquesforthedis retizationandamonotone

multi-grid method as subdomain solver, we introdu e a newalgorithm for the numeri al

solutionof onta tproblems. It anbeinterpretedasanonlinearDiri hlet{Neumann

typepre onditioner.

Therestofthepaperisorganizedasfollows:InSe tion2,we onsideranonlinear

onta tproblem. Wefo usontheelasti onta twithoutfri tionbetweendeformable

bodies. Thedis retizationattheinterfa eisbasedonamortar ouplingintermsofa

dualLagrangemultiplierspa e. UsingtheroleoftheLagrangemultiplier,we

formu-late anonlinearDiri hlet{Neumannalgorithm in Se tion 3. InSe tion 4,numeri al

resultsin 2D are presentedillustrating the onvergen e rates of ouralgorithm. We

extendourapproa h in Se tion5to onta tproblems withCoulomb fri tion.

Com-pared to Se tion 3, no additional outer iteration is required. Finally in Se tion 6,

numeri alresultsareshownin2Dand3DillustratingtheeÆ ien y and exibilityof

ourproposedalgorithm.

2. A nonlinear fri tionless onta t problem. In this se tion, we onsider

a non onforming approa h for the elasti onta t between deformable bodies. One

of themajordiÆ ulties in thenumeri al simulationof onta tproblems isthe

non-dierentiability of the asso iated energy fun tional at the onta t boundary. Very

oftenregularizationte hniques;see,e.g.,[CSW99,ESW99℄,oraugmentedLagrangian

methods;see, e.g.,[Tal94,PC99℄areused.

Material 1

Material 2

free displacement

no penetration

L

₂

u

₂

= f

2

L

1

u

1

= f

1

Fig.2.1.Nonlinear onta tproblem

Figure 2.1 illustrates thesituation at the onta tzone betweentwobodies. No

penetrationbetweenthebodieso ursbut freetangentialdispla ementispermitted.

Forsimpli ity, werestri tourselvestothe aseoftwodeformablebodies in onta t.

Thetwobodiesin theirreferen e ongurationareidentiedwiththedomains

k 

IR d

, k=1;2,d=2;3,and we de omposethe solutionuin u=(u

1 ;u 2 ),and write (u k ) n := u k
n k , k = 1;2, where n k

is the outer unit normal on 

k

. The

non-mortar side is asso iated with subdomain

1

. We startwith the de omposition of

theboundaryofintothreedisjointparts,

D

istheDiri hletpart,

N

denotesthe

Neumann part and

C

stands for the onta t boundary. The a tual onta t zone

betweenthetwobodiesisaprioriunknownandisassumedtobeasubsetof

C . We

denotetensorandve torquantitiesbyboldsymbols,e.g.,andv ,andits omponents

by

ij and v

i

, 1i;j d. Thepartial derivativewith respe t to x

j

is abbreviated

withtheindex

;j

. Furthermore,weenfor ethesummation onventiononallrepeated

indi esrangingfrom1tod,andwedenotebyÆ

ij

nonlinear onta tproblem anbewrittenasaboundaryvalueproblem. Inaddition

totheequilibrium onditionsin

1 and

2

andtheboundary onditionson

 ij (u) ;j = f i ; in 1 [ 2 ; u = 0; on D ;  ij (u)
n j = p i ; on N ; (2.1)

wehavethefollowing onditionsonthepossible onta tboundary

C  T (u 1 ) =  T (u 2 ) = 0 ;  n (u 1 ) =  n (u 2 )  0 ; (2.2)

andthelinearized onta t onditionon

C t  (u 1 ) n +(u 2 ) n ; 0 = ((u 1 ) n +(u 2 ) n t) n (u 1 ) ; (2.3)

wherethefun tiont:

C IR

d

!IRisthedistan ebetweenthetwobodiesin

nor-maldire tion takenwithrespe t tothereferen e onguration;see [HH80,BGK87℄.

We assume that t is ontinuous. The system (2.1) is obtained by the equation of

equilibrium, the strain-displa ement relation and the onstitutive law. In the ase

of alinearelasti material, the stresstensor  depends linearlyonthe innitesimal

straintensor(u):=1=2(ru+ru T

). Thestresstensor isgivenbyHooke'slaw

 ij (u):=E ijlm u l;m ;

where Hooke's tensor E := (E

ijlm ) d ijlm=1 , E ijlm 2 L 1 (), is assumed to be

suÆ- ientlysmooth,symmetri anduniformlypositivedenite. Inthe aseofa

homoge-neousisotropi material,Hooke'stensorhasthesimpleform

E ijlm = E  (1+)(1 2) Æ ij Æ k l + E 2(1+) (Æ ik Æ jl +Æ il Æ jk ) ;

where E > 0is Young's modulus and  2 (0;1=2) is the Poisson ratio. Figure 2.2

illustratesthenormalstressat the onta tboundary.

σ

_Ω

Ω

σn

n

(u )

₁

2

(u )

1

2

Fig.2.2.Normalstressatthe onta tboundary

Here,we onsidera onta tproblemwithoutfri tion. Thus,thetangential

om-ponentofthestresstensorvanishesatthe onta tboundary,andissettozerointhe

rstequation of (2.2). Wehaveonly onta tpressure at

C

. Ifthere isno onta t

between the twobodies, the boundary stresses at

C

are zero; see (2.2) and (2.3).

Thebilinearforma(
;
)is denedby

a(v ;w ):= 2 X Z k E ijlm w i;j v l;m dx; w ;v2 K Y H 1 ( k ) ;

whereE

ijlm

isassumedtobe onstantonea hsubdomainandH 1 ( k ):=(H 1 ( k )) d . Wewritef(v ):=(v ;f) 0; +(v ;p) 0; N anddenotebyf k (
)anda k (
;
)therestri tion off(
)anda(
;
)to k ;k=1;2,respe tively.

Theweaksolutionofthe nonlinear onta tproblem anbeobtainedby a

mini-mizationproblemona onvexset. Wedenethe onvexsetKofadmissible

displa e-mentsby K = fv2H 1  ( 1 )H 1  ( 2 )j (v 1 ) n +(v 2 ) n tg ; where H 1  ( k )  H 1 ( k

) satises homogeneous Diri hlet boundary onditions on



k \

D

. Then,theweaksolutionof(2.1){(2.3)isdenedby: Findu2Ksu hthat

J(u)  min

v 2K

J(v ) ; (2.4)

where the energy fun tional J(
) is given by J(v ) := 1

2

a(v ;v ) f(v ) on K ; see,

e.g.,[HH80,BGK87℄. Theminimization problem(2.4) isequivalent toavariational

inequality: Find u2K su hthat

a(u;v u)  f(v u); v2K :

Our approa h on the dis rete level is basedon aNeumann{Diri hletalgorithm

andinexa tsolvers.Inea hstep,alinearinhomogeneousNeumannproblemhastobe

solved. Thisisdonebystandardmultigridte hniques. Furthermore,anonlinear

one-sided onta tproblemhastobesolved. Here,weusemonotonemultigridmethods;see

[Kor97b,KK00,Kra01℄. Theinformationtransferatthe onta tboundaryisrealized

intermsofthes aledmassmatrix. Themajoradvantagesofthisnewapproa harethe

eÆ ien y oftheiterativesolver,andtheaprioriestimatesfortheboundarystresses

at the a tual onta t zone. Introdu ing the boundary stress formally as Lagrange

multiplier,theNeumann{Diri hletformulation anbeinterpretedasamortarsetting.

In ontrasttopenaltymethods,thedis retizationerroroftheboundarystressesdoes

notdependonregularizationparameters.

Tomotivateourapproa h, letus assumeforthemomentthat the onta tstress



n

is known on

C

. Then, problem (2.1){(2.3) an be de oupled in the following

way: In a rst step, we solve an inhomogeneous Neumann problem on

2 : Find u 2 2H 1  ( 2 )su h that a 2 (u 2 ;v )=f 2 (v )+( n ;v n ) 0; C ; v2H 1  ( 2 ) : (2.5) Havingu 2 2H 1  ( 2 ),u 1 2H 1  ( 1

) anbeobtainedintermsofu

2 j C . Wedenethe onvexsetK v 2

ofadmissibledispla ementsforagivenv

2 2H 1  ( 2 ) K v2 := fv 1 2H 1  ( 1 )j(v 1 ) n t (v 2 ) n on C g :

Then,theone-sided onta tproblemon

1

anbewrittenasavariationalinequality:

Findu 1 2K u2 su hthat a 1 (u 1 ;v u 1 )  f 1 (v u 1 ); v2K u2 : (2.6)

Thedis retizationofthesetK

v 2 isgivenby K h v2 := fv 1 2X 1;h j(v 1 ) n (p)t(p) (v 2 ) n (p)forallp2P C g ; (2.7)

whereX

k ;h

istheniteelementspa eX

h \ H 1  ( k

),k=1;2,ofve torvaluedpie ewise

linearhatfun tionson

k . P

C

denotesthesetof verti esonthenon-mortarsideof

C

, and  is asuitable mappingfrom the mortar side on the non-mortar side. In

the onforming asewhere =Id isthe standard hoi e, apriori estimatesfor the

dis retizationerror anbefoundin,e.g.,[KO88b℄. Wereferto[BHL97,BHL99℄foran

apriorianalysisin thenon onforming ase. Resultsonaposteriorierrorestimation

forunilateral onta tproblems anbe foundin [CHP00℄. Numeri al examplesfora

mortar ouplingwithstandardLagrangemultipliers in2Dwithoutfri tionaregiven

in[Hil00℄. Inthefollowing,wedonotuseanadditionalindexhtodenotethedis rete

approximationu=(u 1 ;u 2 )2X 1;h X 2;h

, and stands forthe dis reteboundary

stress. Here,inanabuseofnotation,wedonotdistinguishbetweenanelementv2X

h

anditsve torrepresentationwithrespe ttothestandardnodalbasis. Inaddition,we

identify thespa esX

k ;h andIR n k ,n k :=dimX k ;h , k=1;2. Fork=1;2,wedenote by A k N

the stiness matrix with respe t to a

k

(
;
) and by f

k

the ve torasso iated

with the right hand side. The index N of A k

N

indi ates that the stiness matrix

orrespondstoNeumanntypeboundary onditionsat theinterfa e.

Before we formulate ouralgorithm, we onsider theinformation transfer at the

interfa e in more detail. We dene the proje tion  in terms of a dual Lagrange

multiplier spa e. Let

j

,1jN

C

,bethestandardpie ewiselinearhatfun tions

asso iatedwiththenon-mortarside. N

C :=#P

C

standsforthenumberofverti eson

thenon-mortarside. Wedenoteby

j

,1jN

C

,asetoflo allydened pie ewise

linearbiorthonormalbasisfun tions,i.e.,

Z C j  l ds=Æ jl ; 1j;lN C :

Moreover,weassumethatP

0 ( C )spanf j ;1jN C g=:M h . Theexisten eof

su hbasisfun tions supportedbytwoedgeshasbeenestablished,see, e.g.,[Woh01℄.

Wenote that in ontrastto astandardmortar approa h with rosspoints,no

mod-i ation of the dual basis fun tions in the neighborhood of the endpoints of

C is

ne essary. Now,wedeneourproje tion:X

2;h !X 1;h , (v ) i := NC X j=1 Z C v i j ds  j ; v2X 2;h ; 1id :

It is lear that  an also be applied to v

2 2 H 1  ( 2

). We denote the algebrai

representationof asfun tionfrom IR n

2

ontoIR n

1

byS,andweobservethatS isa

n

1 n

2

matrix,whi h onsistsoflargezeroblo ksandonenonzeroblo kasso iated

with the verti es on the non-mortar and mortar side. Solving a dis rete Diri hlet

problem on

1

provides an approximation for the orresponding ux  2 M

h := (M h ) d on C

. Withinthemortarapproa hthedis rete uxisuniquelydened by

Z C  vds=a 1 (u 1 ;v ) f 1 (v ); v2X 1;h : Using2M h in (2.5),wendforanelementv 2 in X 2;h Z C v 2 ds= Z C v 2 ds= Z C   v 2 ds ; where  

denotesthe adjointoperator of . Thematrix representationof whi his

given by S T

. Here, we identify M with IR m

obstacle

residual

stress

trace

Ω

1

Ω

2

linear

non linear

Neumann problem

one-sided contact pb.

S

S

T

Fig.2.3.Dis reteDiri hlet{Neumann oupling

embeddingIR m

IR n1

. Figure2.3illustratestheroleofdis retetransferoperatorsS

andS T

.

ThetransferoftheDiri hletvaluesatthe onta tboundaryisrealizedin terms

of the linear operator  and the transfer of the boundary stresses in terms of the

adjointoperator, orrespondingtothedualitybetweendispla ementsandstresses. In

thealgebrai formulation,thematrixS is usedto transferthedispla ementsonthe

mortarsideasDiri hletvalues,ormorepre iselyasanobsta le,ontothenon-mortar

side, and the s aled boundary stresses are transferred from the non-mortar side to

the mortar side in termsof the transposed matrix S T

. The interfa e onditionsof

themortarformulationguaranteethat(2.2)and(2.3)aresatisedinaweakintegral

form.

3. Diri hlet{Neumann algorithm. Now, our nonlinear Neumann{Diri hlet

algorithmisdened intermsoff

1 ,f

2 andS:

Choosedampingparameters: 0<!

D ;! N 1. Initialize: X 1;h 3g 0 =0; X 2;h 3p 1 =0. For =1;:::;N do

Solve linearNeumannproblem: Findu  2 2X 2;h : A 2 N u  2 =f 2 p  :

Transfer ofthe displa ement anddamping:

g  =(1 ! D )g  1 +! D Su  2 :

Solve nonlinearone-sided onta tproblem: Find u  1 2K h g  : (A 1 N u  1 ;v u  1 )(f 1 ;v u  1 ); v2K h g  n :

Compute the residual r  1 2X 1;h : r  1 =A 1 N u  1 f 1 :

Transfer ofthe boundarystressanddamping:

p +1 =(1 ! N )p  +! N S T r  1 :

Inea hstepofouralgorithm,weuseamultigridmethodsassolver. Thevariational

inequality anbesolvedeÆ ientlybymonotonemultigridmethods. Themainideais

tominimizetheenergyfun tionalJ

1

(
)onK h

u

test fun tions. Choosing the multilevel nodal basis of amultigrid hierar hy astest

fun tions,thisturnsouttobea ombinationofaproje tedblo kGau{Seidelonthe

nest gridwith lo ally damped oarse grid orre tions, and an be implementedas

amodied V- y le. Sin e the oarse grid orre tionshave to satisfythe onstraints

given by (2.7) with respe t to the nest triangulation, suitable non-trivial oarse

gridfun tions haveto be onstru ted. The onstru tionofthe modied oarse grid

orre tions an be found in [Kra01℄. It an be shown, that after a nite number

of iterations the dis rete onta t boundary is identied; see [Kor97a℄. Then, the

method degenerates to a standardmultigrid method with spe ial treatment of the

eventually urvilinear onta t boundary. Fordetails and a onvergen e theory, we

referto[KK99,KK00, Kra01℄.

Figure 3.1illustratesthe stepsof ourNeumann{Diri hletalgorithmfor !

D =1.

On the left, the rst step is shown. The hoi e p 1

= 0 implies that a

homoge-neous Neumann problem has to be solved for u 1

2

. In the ase that we have a full

symmetri problem,it anbeeasilyseenthatthe hoi e!

N

=1doesnotyielda

on-vergents heme. Theiteratesos illatebetweenthetworstiterates,i.e.,u 2m+1 2 =u 1 2 , u 2m+2 2 =u 2 2 ,m1.

u

u

2

1

1

1

u

u

2

2

2

1

ω

2

0<

ω

2

<1

ω

2

=1

= 0

∝∝

∝∝

u

u

2

1

Fig.3.1.Firstiterates(u 1

1 ;u

1

2

)(left),se ond iterates(middle)andsolution(right)

Remark 3.1. If the a tual onta t zone is known, problem (2.1){(2.3) will be

linear. In this ase, we an expe t the sameorder of onvergen e as for a standard

Neumann{Diri hlettypepre onditioner formortars, see[Dry99,Dry01 ℄.

Inthemortarsetting,theLagrangemultiplierplaystheroleofNeumann

bound-ary onditions. The ombinationofmortarniteelements,monotonemultigrid

meth-ods and domain de omposition te hniques denes in a naturalway a new solution

algorithm forelasti onta tproblems. Thedis rete boundary stressin the -th

it-erationstep  istheresidualr  1 restri tedon C

.,Moreover,weobtainthenormal

stress 

n

andthe tangentialstress 

T

by alo alrotationfrom thenal  

. We

re-mark,that ourapproa h satises

T

=0,although wedonotenfor ethis ondition

onthedis retespa eM

h .

Remark3.2. Usingtheve torvaluedapproa hfortheLagrangemultiplierspa e,

fri tionterms anbeeasily in luded. Therstequationin (2.2)hastoberepla edby

somefri tionlaw, e.g., theCoulomb fri tion.

For fri tionless onta t, the rst equation in (2.2) an also be satised in its

strongform.Then,theLagrangemultiplier spa eisas alarfun tionandthemortar

approa hhastobemodied. Inparti ularatthe onta tboundary,wehavetoimpose

Neumanntypeboundary onditionsintangentialdire tionandanobsta leinnormal

4. Numeri alresults. Finally,wepresentnumeri alexamplesfortheproposed

algorithm. Allournumeri alresultsare arriedoutwithintheframeworkofthenite

element toolbox UG,[BBJ +

97℄. Ourrst test problem isthe Hertzian onta tof a

linearelasti ir lewith alinearelasti plane. In this example,the onta tstresses

anbe omputedanalyti ally[Her82℄. Totesttheperforman eofouralgorithm,we

omparethe omputedboundarystresseswiththeanalyti alones. For omparability,

we hoosethesameproblemdataandgeometryasin[CSW99℄. We onsideranelasti

ir le withs aled materialparametersE =7000,  =0:3and radius r=1,pressed

byapointloadF=100toaplanewithmaterialparametersE=10 6

, =0:45.

Asisdonein[CSW99℄,weapplythesingleloadassurfa eloadtoavoida

singular-ity. Weusebilinearfun tionsonquadrilaterals.Figure4.1illustratestheperforman e

ofour method. In theleft, themaximal onta tstresson ea h levelis given, in the

middle the onta tstresses and tangentialstresses are shown, and in theright, the

omponent

22

(u)ofthestresstensorisdepi ted. Theanalyti alvalueof max

n

=495

isalreadyrea hedonlevel5. Here,only5nodesofthe ir learea tualin onta twith

theplane. Todemonstratethe exibilityofourapproa h,wedonotenfor e

T =0

on the spa e. The Lagrange multiplier of the mortar method plays the role of the

boundarystressesat

C

. Thus,theboundarystressesarehandledasadditional

un-knownswhi hareobtainedbyrestri tingtheresidual. Thisobservationpredestinates

ouralgorithmfor onta tproblemswithfri tion.

1

2

3

4

5

6

7

150

200

250

300

350

400

450

500

PSfragrepla ements  max n =495  maxn ( u l ) Levell x-Co ordinate onta tstresses

−0.2 0

0.2 0.4 0.6 0.8

1

1.2

−100

0

100

200

300

400

500

σ

σ

T

n

PSfragrepla ements  max n =495  max n (u l ) Levell x-Co ordinate on ta t stresses PSfragrepla ements  max n =495  max n (u l ) Levell x-Co ordinate onta tstresses

Fig.4.1. Maximal onta tstresses(left), onta tstresses(middle)and22 (right)

Aslongasthedis rete onta tboundaryisnotfullyre ognized,the onvergen e

of themonotonemultigrid method might beslow. This isdue to thesear hfor the

onta t boundary. In this example, the onta t boundary is dete ted after at most

threeinneriterations,i.e.,threeiterationsofthemonotonemultigridmethod,andno

slowdowno urs.

Wedene thestopping riteriaforouriterativesolverin termsofthe Lagrange

multiplier. Observingthat the hoi eofourstartve torsguaranteesA 1 N u n 1 f 1 =0

forallinteriornodeson

1 , wendk(A 1 N u n 1 f 1 ) C k=kA 1 N u n 1 f 1 k. Moreoverp 

anbeinterpretedas boundarystress onthemortar sidein the -th iterationstep.

Thisobservationmotivatesourstopping riteria

kp +1 p  k kp  k TOL kp 3 p 2 k kp 2 k (4.1) whi hisequivalentto kp  S T r  1 k=kp  kTOLkp 2 S T r 2 1 k=kp 2

k. Theuse ofthe

Eu lidean ve tor normis motivated by themesh dependent norm hk
k 2

0;

C

for the

Lagrange multiplier. We note that if the dis rete boundary stress pis equal zero,

thenthe onta tproblemisdegeneratedandtwolinearproblemson

1 and

2 with

sin ep 1

=0,g 0

=0,weobtainthesolutionafteronestep. Moreoverifp6=0,!

D =1 andp 0 =0, 0

2,thealgorithmdoesnot onvergen eandthedampingparameter

!

N

is too large. Table4.1 shows thenumber of requirediteration steps depending

on the damping parameter and the renement level. We set TOL = 10 4

. If the

dampingparametersaresmall enoughthenumberofrequirediterationsteps anbe

bounded independentlyof the renementlevel. Here we useuniform renement on

all levels. Weobservea onsiderably smaller number of required iterationsteps on

Level2andLevel3for!

D

=1and!

N

=0:4;0:5;seealso Figure4.3.

Table4.1

Numberofiterationsteps,(Lagrangemultipliernorm)

lev. 0 lev. 1 lev. 2 lev. 3 lev. 4 lev. 5 lev. 6

! D =1,! N =0:5 11 11 5 6 13 10 12 ! D =1,! N =0:4 14 14 8 9 18 15 16 ! D =0:5,! N =1 12 11 6 6 6 8 11 ! D =0:4,! N =1 16 16 9 10 8 11 11 ! D =0:6,! N =0:8 9 8 8 9 7 8 9 ! D =0:8,! N =0:6 6 7 9 9 10 8 9 ! D =0:7,! N =0:7 8 7 10 9 9 8 9

Figure4.2illustratesthein uen eofthe hoi eofthedampingparameters. The

errorredu tiong():=kp +2 S T r +2 1 kkp 2 k=kp +2 kkp 2 S T r 2 1 kisshownversus

the number  of iteration steps. If the damping parameter is small enough level

independentupperboundsforthe onvergen erates anbeobserved.

1e-10

1e-08

1e-06

0.0001

0.01

1

0

5

10

15

20

Reduction in Lagrange multiplier

Number of iteration steps

ThetaD = 0.5 and ThetaN = 1

level0

level1

level2

level3

level4

level5

level6

1e-10

1e-08

1e-06

0.0001

0.01

1

0

5

10

15

20

Reduction in Lagrange multiplier

Number of iteration steps

ThetaD = 1.0 and ThetaN = 0.5

level0

level1

level2

level3

level4

level5

level6

1e-10

1e-08

1e-06

0.0001

0.01

1

0

5

10

15

20

Reduction in Lagrange multiplier

Number of iteration steps

ThetaD = 0.8 and ThetaN = 0.6

level0

level1

level2

level3

level4

level5

level6

Fig.4.2.Errorredu tionfordierentdampingparameters!

N ,!

D

Figure4.3illustratesthein uen eofsmallandtoolargedampingfa tors. Small

damping parameterslead to a slow onvergen e,see theleft and middle pi ture in

Figure 4.3. On theother hand, thealgorithm does not onvergefor higherlevels if

thedampingparameteristoolarge,see therightpi turein Figure4.3.

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0

5

10

15

20

Reduction in Lagrange multiplier

Number of iteration steps

ThetaD = 1.0 and ThetaN = 0.3

level0

level1

level2

level3

level4

level5

level6

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0

5

10

15

20

Reduction in Lagrange multiplier

Number of iteration steps

ThetaD = 0.3 and ThetaN = 1

level0

level1

level2

level3

level4

level5

level6

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

100

0

5

10

15

20

Reduction in Lagrange multiplier

Number of iteration steps

ThetaD = 1.0 and ThetaN = 0.7

level0

level1

level2

level3

level4

level5

level6

Tobeonthesafeside,onehasto hoseasmalldampingfa tor. Unfortunately,the

optimaldampingparameterisin generalnotknown. Adaptivestrategies ontrolling

thedampingparametermightyield onsiderablybetterresults. Adierentpossibility

toimprovetheperforman eistheuseofouralgorithmaspre onditionerforaKrylov

subspa emethod. As soonas thea tual zone of onta t is dete ted, weare in the

linearsetting. Then,ouralgorithmfor!

D

=1isequivalenttoapre onditionedS hur

omplementsystem,andwe anapplya onjugategradientmethod.

Inournextexample, we onsider theelasti onta tofawren h and anut. At

the interior boundary of the nut, i.e., the partof the boundary with outer normal

pointingtowardsthe enterofgravityofthenut,weimposeDiri hletboundary

on-ditions orrespondingtoarotation. HomogeneousDiri hletboundary onditionsare

applied at thehandle of thewren h andon allremaining parts ofthe boundarywe

impose homogeneousNeumann onditions. Weuselinearelementsontriangles,and

renementis doneadaptively. As an be seenin therightof Figure 4.4, the a tual

onta t zone is only a small part of the onta t boundary

C

. We remark,that a

morerealisti modelwouldin lude fri tionattheinterfa e.

Fig.4.4.Initialtriangulation(left),deformationandnaltriangulation(middle)andzoomat

the onta tzone(right)

5. A onta t problemwith Coulombfri tion. Inthis se tion,we onsider

anonlinear onta tproblem withfri tion. Forsimpli itywerestri tourselvesto the

lassi al Coulomb fri tion, and we do not onsider more general non lo al fri tion

laws; see [KO88b℄. Let us onsider for themoment theSignorini problem where an

elasti body is in onta t with a rigid foundation. In that situation, the Coulomb

law anbedes ribedasfollows: Aslongasthenormofthetangentialstressissmall

enough, no sliding o urs, and the tangential displa ement is zero. If the norm of

thetangentialstress

T

rea hesa riti allimit,whi hisproportionaltotheabsolute

valuesof thenormalstress, sliding in theopposite dire tion of

T

anbeobserved;

see[KO88b℄. Figure5.1illustratestherelationbetweentangentialstressandnormal

stressforasliding andasti kynode.

tan

α

ν

sticky

σ

σ

α

n

T

=

tan

α

ν

slippy

σ

σ

α

n

T

=

u

_T

The Coulomb law an be applied for two linearelasti bodies in onta t if we

repla ethetangentialdispla ementbytherelativetangentialdispla ement;see[IW92,

E k96℄. Then,theequilibrium onditionsatthe onta tboundaryandtheCoulomb's

lawreadasfollows:

 T (u 1 ) =  T (u 2 );  n (u 1 ) =  n (u 2 )  0 ; j T (u 1 )j  j n (u 1 )j;  T (u 1 )[u T ℄+j n (u 1 )jj[u T ℄j=0 ; (5.1)

where>0isthefri tion oeÆ ientandthejumpisdenedby[u

T ℄:=(u 1 ) T (u 2 ) T .

An equivalent formulationof Coulomb's law an be givenby j

T (u 1 )j j n (u 1 )j and if j T (u 1 )j < j n (u 1 )j =) [u T ℄=0 if j T (u 1 )j = j n (u 1 )j =) [u T ℄= s T (u 1 ); s0 ;

seealso[Has92,KB92℄. Then,theequilibrium onditionsatisestheboundaryvalue

problem given by (2.1), (2.3) and (5.1). As in the fri tionless ase, we base our

numeri alapproa honthevariationalformulation. Todoso,weusetheprin ipleof

virtualwork andintrodu eanonlinearfun tionalj(
;
)to des ribethevirtualwork

ofthefri tionalfor e

j(u;v ):= Z C j n (u 1 )jj[v T ℄jds :

Followingthelines of[KO88b, Chapter10℄, avariationalinequality anbeobtained

from(2.1),(2.3)and(5.1)byapplyingGreen'sformula. Theweakformof(2.1),(2.3)

and(5.1)readsasfollows: Find u2K su hthat

a(u;v u)+j(u;v ) j(u;u)f(v u); u2K : (5.2)

Moreoverunder suitable assumptions on the data, (5.2) and (2.1), (2.3) and (5.1)

areequivalent;see[E k96, Satz1.6℄. Wedonotaddressquestionssu hasexisten e,

uniqueness and regularity of asolution. Re ently existen e resultsfor a large lass

of onta t problems with fri tion have been obtained. We refer to [E k96, EJ98,

IHL88, NJH80℄ and the referen es therein. Very often existen e proofs are based

onpenalizationandregularizationte hniques,andboundsfortheadmissible fri tion

oeÆ ient anbeestablished;see[EJ98℄.

Wefollow,thelinesofthepreviousparagraphto motivateouralgorithm. Letus

assumethat theboundarystresses 

T and 

n

are known onthe onta tboundary.

Then,theboundaryvalueproblem(2.1),(2.3)and(5.1) anbede oupled. The

solu-tionon!

N

anbeobtainedasthesolutionofaninhomogeneousNeumannproblem:

Findu 2 2H 1  (! N )su hthat a 2 (u 2 ;v )=f 2 (v )+( n ;v n ) 0; C ;+( T ;v T ) 0; C v2H 1  (! N ) : Toobtainu 1

,wesolveanonlinearone-sided onta tproblem withCoulomb fri tion

on

1

. Then,thevariationalinequality(5.2)redu es tou

1 2K u2 a 1 (u 1 ;v u 1 )+j red n;u2 (v ) j red n;u2 (u 1 )f 1 (v u 1 ); v2K u2 ;

where the redu ed form of the virtual work for a given fun tion s and agiven

dis-pla ementw isdenedby j red s;w (v ):= Z jsjjv T w T jds : (5.3)

Thistypeoffri tionlawwherethe onta tstressisassumedtobeknownisalso alled

Tres afri tion. Now,wepro eed asin thepreviousparagraphand arryoutaxed

point iteration. Our nonlinear Neumann{Diri hletalgorithm for a onta tproblem

withCoulomb fri tionreads as:

Choosedampingparameters: 0<!

D ;! N 1. Initialize: X 1;h 3g 0 :=0; X 2;h 3p 1 :=0. For =1;:::;N do

Solve linearNeumannproblem: Find u  2 2X 2;h : A 2 N u  2 =f 2 p  :

Transferof displa ementsanddamping:

g  =(1 ! D )g  1 +! D Su  2 :

Solve nonlinearone-sided onta tproblem withCoulomb fri tion:

Findu  1 2K h g  : (A 1 N u  1 ;v u  1 )+j red (r  1 )n;g  (v ) j red (r  1 )n;g  (u  1 )(f 1 ;v u  1 ); v2K h g  n :

Computethe residual r  1 2X 1;h : r  1 =A 1 N u  1 f 1 :

Transferof s aledboundarystressesanddamping:

p +1 =(1 ! N )p  +! N S T r  1 : Here,j red
;

(
)isthealgebrai representationofthenonlinearfun tionalj red

;

(
)dened

by(5.3). Apossibleapproa hfortheanalysisofourxedpointiterationisthetheory

ofquasi-variationalinequalities,i.e.,su hinequalitieswherethe onvexsetdependson

thesolution;see[GJT81,BC84℄. Theiterativesolutionofthenonlinearsubproblem

isdis ussedandanalyzedin[Kra01℄.

6. Numeri al results with Coulomb fri tion. In this se tion, we present

somenumeri alresultsin2Dand3Dillustratingthein uen eoftheCoulombfri tion

onthedeformation. Inallour2Dresults,thefri tion oeÆ ientis=0:3. Westart

with theHertzproblem ofSe tion 4. Figure6.1 showstheboundarystressesat the

onta tzone. Theinitialtriangulationhasfourelementsonea hsubdomain.

0.4

0.44

0.48

0.52

0.56

0.6

−100

−50

0

50

100

150

200

250

300

350

normal stresses

tangential stresses

and themaximal normal stressare onsiderably smaller ifCoulombfri tion o urs.

Between theminimum and the maximum of the tangential stress no sliding o urs

at the onta tzone. Sliding nodes an befound intheneighborhood oftheleftand

rightendpointsofthea tual onta t.

As ase ond examplein 2Dwe onsider asymmetri problem. Here due to the

symmetry, we expe t that the tangential stressis zero even iffri tion termsare

in- luded. Figure6.2showstheboundarystressforthefri tionless aseandforthe ase

in luding Coulomb fri tion. Comparing theleft and rightpi ture in Figure 6.2, we

ndexa tlythesamevaluesforthenormalstressandthusthea tualzoneof onta t.

−0.4

0

0.4

0.8

1.2

−20

0

20

40

60

80

100

120

140

160

180

normal stresses

tangential stresses

−0.4

0

0.4

0.8

1.2

−20

0

20

40

60

80

100

120

140

160

180

normal stresses

tangential stresses

Fig.6.2. Boundary stresses(fri tionless) (left),initial triangulation (middle) and boundary

stresses(Coulombfri tion)(right)

In ontrasttotheunsymmetri probleminFigure6.1,notangentialstresso urs.

Sin e thenormal and tangentialstress isthe sameat

C

, the dis retesolutions for

thetwosituationsarethesame. Figure6.3illustratesthestress omponent

22 lose

to the onta tzone. Although our Diri hletNeumann algorithm is nonsymmetri ,

weobtainasymmetri numeri alapproximation. Thenumeri alresults onrmsthe

exibility andreliability ofthe non- onformingapproa hin terms of dual Lagrange

multipliers.

Fig.6.3.Boundarystress22

Inour last example,we onsider theelasti onta tof three bodies in 3D. Two

ylindersarein onta twithahexahedralbar. Atthetopandbottomof theupper

andlower ylinder, respe tively,adispla ementinverti aldire tiontowardsthebar

is enfor ed. As before in 2D, we start with a very oarse initial triangulation, see

Figure 6.4. On Level 0, we only have 5elements. We use a standard mean value

adaptiverenementstrategy. The lo al renement is ontrolled bya residualbased

lo al aposteriorierrorindi ator onthe subdomains and theinformationtransferat

theinterfa eisrealizedbyadditionaltermsinthedenitionofthelo alerror

indi a-tor. Onthenon-mortarsideofthea tual onta tzone, thelo aljump 1=hk[u℄k 2

0;

On themortar side, weadd lo ally the termhk[℄k 2

0;

C

whi h ontrols the dis rete

equilibrium onditionforthestressatthe onta tboundary. Here,weinterpreteas

adis reteapproximationof thestress. Inthe mortar ontext, thealgebrai relation

betweenthevaluesonmortar

m

andnon-mortar

nm

sidearegivenby

 m =S T  nm ; where  m and  nm

areset to zero in theinterior ofthe subdomains. Wenote that

wehaveuseddis ontinuousdualbasisfun tionsonbothsidesofthe onta tzoneto

approximatethestress.

The lo al weights 1=h and h re e t the duality between the H 1=2

and H 1=2

spa es. In ontrastto onformingmethods,norenementruleshavetobe onsidered

attheinterfa es.

Fig.6.4.Initial(left)andadaptive(right)triangulationofathreebody onta tproblemin3D

Intherightpi turein Figure6.4a utoftheadaptivetriangulationonLevel7is

depi ted,showingthemeshesintheinteriorofthe omputationaldomain. Weobserve

strong renement in the neighborhood of the onta t zone. Instead of 10;485;760

elements in the ase of uniform renement, we have 207;561 elements on Level 7.

Usinga oeÆ ientoffri tionof=0:25,weobtain210non-mortarnodesin onta t

on thenest level and 186sti ky nodes. We note, that no element at the interfa e

hasbeenrenedwithinthelastrenementstep.

Fig.6.5.Tangential(left)andnormal (right)stressatoneoftheinterfa es

asapre onditionerwithin aKrylovsubspa emethod. Finally, Figure6.5 showsthe

oeÆ ients with respe t to the dual basis of the stress in normal dire tion and of

onestress omponentintangentialdire tion. These ondtangential omponentisof

smallersize.The onta tstressesaredepi tedwithrespe ttothesurfa eoftheupper

ylinder. This hoi eisarbitrary,sin etheproblemissymmetri withrespe ttothe

symmetryplaneofthetwo ylinders. Sin ethewidthofthe ylinderindire tionofthe

axisofthe ylinderislargerthanthewidthofthebar,thenormalstressiszeroatthe

partofthe ylinder'ssurfa ebeingontheleftandrightofthebar,respe tively. The

tangentialstresses in rease until their norm rea hesthe riti al value j

n

j. Then,

slidingo ursinoppositedire tiontothetangentialstresses. Allnodeonthe onta t

boundarylyinginbetweentheminimumandmaximumofthetangentialstressesare

sti kynodes,allothersareslidingnodes.

REFERENCES

[BBJ +

97℄ P. Bastian, K. Birken, K. Johannsen, S. Lang, N. Neuss, H. Rentz-Rei hert, and

C. Wieners. UG { a exible software toolbox for solvingpartialdierential

equa-tions. ComputingandVisualizationinS ien e,1:27{40,1997.

[BC84℄ C.Baio hiandA.Capelo.Variationalandquasivariational inequalities.Appli ationsto

free-boundaryvalueproblems. JohnWiley&SonsLtd.,1984.

[BD98℄ D. Braessand W.Dahmen. Stabilityestimatesof themortarnite elementmethodfor

3{dimensionalproblems.East{WestJ.Numer.Math.,6:249{263,1998.

[Ben99℄ F.BenBelga em. ThemortarniteelementmethodwithLagrangemultipliers.Numer.

Math.,84:173{197,1999.

[BGK87℄ P.Boieri,F.Gastaldi,andD.Kinderlehrer.Existen e,uniquenessandregularityresultsfor

the two{body onta tproblem. Applied Mathemati sand Optimzation,15:251{227,

1987.

[BHL97℄ F.BenBelga em,P.Hild,andP.Laborde.Approximationoftheunilateral onta tproblem

bythemortarniteelementmethod. C.R. A ad.S i.,Paris, Ser.I, 324:123{127,

1997.

[BHL99℄ F.BenBelga em,P.Hild,andP.Laborde.Extensionofthemortarniteelementmethod

toavariationalinequalitymodelingunilateral onta t. Math.ModelsMethodsAppl.

S i.,9:287{303,1999.

[BMP93℄ C.Bernardi,Y.Maday,andA.T.Patera. Domainde ompositionbythemortarelement

method.InH.Kaperetal.,editor,In: Asymptoti andnumeri almethodsforpartial

dierentialequationswith riti alparameters,pages269{286.Reidel,Dordre ht,1993.

[BMP94℄ C.Bernardi,Y.Maday,andT.Patera.Anewnon onformingapproa htodomain

de om-position: Themortarelementmethod.InH.BrezisandJ.L.Lions,editors,Nonlinear

PartialDierentialEquationsandTheirAppli ations,pages13{51.Pitman,1994.

[CHP00℄ P.Coorevits,P.Hild,andJ.-P.Pelle. Aposteriorierrorestimationforunilateral onta t

withmat hingandnon-mat hingmeshes.Comput.MethodsAppl.Me h.Eng.,186:65{

83,2000.

[CSW99℄ C.Carstensen,O.S herf,andP.Wriggers. Adaptiveniteelementsforelasti bodiesin

onta t. SIAMJ.S i.Comp.,20(5):1605{1626,1999.

[DNS99℄ Z.Dostal,F.A.M.GomesNeto,and S.A.Santos. Solutionof onta tproblemsbyFETI

domainde ompositionwithnatural oarsespa eproje tions.Comp.MethAppl.Me h.

Eng.,1999.toappear.

[Dry99℄ M.Dryja.ADiri hlet{Neumannalgorithmforellipti mortarniteelementproblems.In

W.Ha kbus hand S.Sauter, editors,Numeri al Te hniquesforComposite

Materi-als,Notes onNumeri alFluidMe hani s.Vieweg,Brauns hweig,Submittedto15th

GAMM{Seminar1999.

[Dry01℄ M.Dryja. TheDiri hlet{Neumannalgorithmformortarsaddlepointproblems.BIT,41,

toappear2001.

[E k96℄ C.E k. ExistenzundRegularitatderLosungenf urKontaktproblememitReibung. PhD

thesis,Math.Inst.AderUniversitatStuttgart,1996.

[EJ98℄ C. E k and J.Jaruse . Existen eresults for the stati onta t problemwith Coulomb

onta tproblemswithfri tion.Mathemati sandComputersinSimulation,50:43{61,

1999.

[GJT81℄ R.Glowinski,J.L.Lions,andR.Tremolieres. Numeri alAnalysisofvariational

inequali-ties,volume8ofStudiesinMathemati sanditsappli ations. North{Holland,1981.

[Has92℄ J. Haslinger. Signoriniproblem withCoulomb'slaw of fri tion.Shapeoptimization in

onta tproblems.InternationalJ.fornumeri almethodsinengineering,34:223{231,

1992.

[Her82℄ H.Hertz. 

UberdieBeruhrungfesterelastis herKorper. J.f.Math.,92,1882.

[HH80℄ J.HaslingerandI.Hlava ek.Conta tbetweenelasti bodies.I. ontinuousproblems.Apl.

Mat.,25:324{327,1980.

[HH81℄ J.HaslingerandI.Hlava ek. Conta tbetweenelasti bodies.II.niteelementanalysis.

Apl.Mat.,26:263{290,1981.

[Hil00℄ P.Hild. Numeri alimplementationoftwonon onformingniteelementmethodsfor

uni-lateral onta t. Comput.MethodsAppl.Me h.Eng.,184:99{123,2000.

[IHL88℄ J.Ne asI.Hlava ek,J.HaslingerandJ.Lovsek. Solutionof variationalinequalitiesin

me hani s.Springer,Berlin,1988.

[IW92℄ A.Ibrahimbegovi andE.L.Wilson.Unied omputationalmodelforstati anddynami

fri tional onta t. InternationalJ.fornumeri almethodsinengineering,34:233{247,

1992.

[KB92℄ A.KLarbringandG.Bjorkman.Solutionoflargedispla ement onta tproblemswith

fri -tionusingNewton'smethodforgeneralizedequations. InternationalJ.fornumeri al

methodsinengineering,34:249{269,1992.

[KK99℄ R. Kornhuberand R.Krause. Onmonotone multigridmethods forthe Signorini

prob-lem.InW.Ha kbus handS.A.Sauter,editors,Numeri alTe hniquesforComposite

Materials,Pro eedingsofthe15thGAMMSeminarKiel,1999. inpreparation.

[KK00℄ R.KornhuberandR.Krause.AdaptivemultigridmethodsforSignorini'sprobleminlinear

elasti ity. toappear,2000.

[KO88a℄ N.Kiku hiandJ.T.Oden. Conta tProblemsinelasti ity. SIAM,Philadelphia,1988.

[KO88b℄ N.Kiku hiandJ.T.Oden.Conta tproblemsinelasti ity:Astudyofvariational

inequal-itiesandniteelementmethods. SIAMStudiesinAppliedMathemati s8,

Philadel-phia,1988.

[Kor97a℄ R.Kornhuber.Adaptivemonotonemultigridmethodsfornonlinearvariationalproblems.

Teubner{Verlag,Stuttgart,1997.

[Kor97b℄ R.Kornhuber.Adaptivemonotonemultigridmethodsforsomenon{smoothoptimization

problems.InR.Glowinskietal.,editor,DomainDe ompositionMethodsinS ien es

andEngeneering,pages177{191.Wiley,1997.

[Kra01℄ R.H.Krause. MonotoneMultigridMethodsforSignorini's Problem withFri tion. PhD

thesis,FUBerlin,2001.

[NJH80℄ J.Ne as,J.Jaruse ,andJ.Haslinger. Onthesolutionofthevariationalinequalitytothe

signoriniproblemwithsmallfri tion.BollettinoU.M.I.,17:796{811,1980.

[PC99℄ G. PietrzakandA.Curnier. Largedeformationfri tional onta tme hani s: ontinuum

formulationand augmented Lagrangian treatment. Computer Methods in Applied

Me hani sandEngeneering,177(3{4):351{381,1999.

[Tal94℄ P.LeTalle . Handbook of Numeri alAnalysis,volumeIII, hapter Numeri alMethods

for Nonlinear Three-DimensionalElasti ity. Nort{Holland, 1994. P.G.Ciarlet and

J.L.Lions(Eds.).

[WG97℄ K.WillnerandL.Gaul. Conta tdes riptionbyfembasedoninterfa ephysi s. In

Com-putationalPlasti ityV,Bar elona,Spain,1997.

[Woh00℄ B.I.Wohlmuth. Amortarniteelementmethodusingdualspa esfortheLagrange

mul-tiplier. SINUM,38:989{1014,2000.

[Woh01℄ B.I.Wohlmuth. Dis retizationMethodsandIterativeSolvers Based onDomain

De om-position. SpringerHeidelberg,2001.

[Wri95℄ P.Wriggers.Finiteelementalgorithmsfor onta tproblems. Ar h.Comp.Meth.Engrg.,