Comput Mech (2012) 49:1–20DOI 10.1007/s00466-011-0623-4

ORIGINAL PAPER

A mortar formulation for 3D large deformation contactusing NURBS-based isogeometric analysis and the augmentedLagrangian method

L. De Lorenzis · P. Wriggers · G. Zavarise

Received: 3 June 2011 / Accepted: 23 June 2011 / Published online: 14 July 2011© Springer-Verlag 2011

Abstract NURBS-based isogeometric analysis is appliedto 3D frictionless large deformation contact problems. Thecontact constraints are treated with a mortar-based approachcombined with a simplified integration method avoiding seg-mentation of the contact surfaces, and the discretization ofthe continuum is performed with arbitrary order NURBS andLagrange polynomial elements. The contact constraints aresatisfied exactly with the augmented Lagrangian formula-tion proposed by Alart and Curnier, whereby a Newton-likesolution scheme is applied to solve the saddle point prob-lem simultaneously for displacements and Lagrange mul-tipliers. The numerical examples show that the proposedcontact formulation in conjunction with the NURBS dis-cretization delivers accurate and robust predictions. In bothsmall and large deformation cases, the quality of the con-tact pressures is shown to improve significantly over thatachieved with Lagrange discretizations. In large deformationand large sliding examples, the NURBS discretization pro-vides an improved smoothness of the traction history curves.In both cases, increasing the order of the discretization iseither beneficial or not influential when using isogeometricanalysis, whereas it affects results negatively for Lagrangediscretizations.

Keywords Contact · Isogeometric analysis ·Large deformation · Mortar method · NURBS

L. De Lorenzis (B) · G. ZavariseDepartment of Innovation Engineering,University of Salento, Lecce, Italye-mail: laura.delorenzis@unisalento.it

P. WriggersInstitute for Continuum Mechanics,Leibniz Universität Hannover, Hanover, Germany

1 Introduction

The numerical solution of large deformation, large slip multi-body contact problems with the finite element method (FEM)presents several difficulties, including high non-linearity andnon-smoothness, potential ill-conditioning, and heavy com-putational costs associated with contact detection. Althoughseveral improvements have been achieved in the past fewyears, contact problems still represent a significant challengefor the analyst and cannot yet be considered solved with thesame level of robustness and accuracy of many other prob-lems in non-linear mechanics [10].

The research related to computational contact mechan-ics has taken several directions. One of these has been thedevelopment of smoothing techniques, aimed at reducingthe drawbacks associated with the non-smooth discretiza-tion of the master surface. Several techniques are avail-able in the literature, including Bézier, Hermitian or otherspline interpolations, Gregory patches, subdivision surfaces[12,23,26,28,31,42], and more recently also NURBS inter-polations [24,36]. These procedures generally improve theperformance of the contact algorithms by enhancing the con-tinuity of the contact master surface, whereas, being typ-ically associated to node-to-surface contact formulations,they leave the geometrical smoothness of the slave sur-face unaltered. Due to the interaction of the bulk andsurface discretizations in determining the smoothness ofthe traction history curves for large deformation and largesliding problems, the observed improvement in the qual-ity of the contact response is limited by the fact thatthe higher-order approximation does not involve the bulkbehavior of the solid. Moreover, the introduction of asmoothened master surface in addition to the existing finiteelement mesh yields additional complications in the imple-mentation and data management, and can in some cases

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2 Comput Mech (2012) 49:1–20

even compromise the banded structure of the stiffnessmatrix [28].

A second direction has been the improvement of the man-ner in which the contact surfaces are parameterized, incor-porated into a variational framework, and discretized. Thewell-known node-to-surface formulation, while simple andcomputationally inexpensive, is affected by several patholo-gies [44], and has been shown not to satisfy the contact patchtest [11,38,45], which implies that the errors introduced atthe contacting surfaces do not necessarily decrease with meshrefinement. On the other hand, formulations based on theenforcement of the contact constraints at an arbitrary num-ber of integration points along the contact surface [13,14],while significantly improving the patch test performance,have been shown to lead to an overconstrained problem andhence to LBB-instability. A perturbed Lagrangian formula-tion first introducing integration over contact segments wasproposed by Simo et al. [35]. Here a piecewise constantapproximation of the contact pressure, discontinuous acrosscontact segments, led to the enforcement of the contact con-straint in an average sense on each contact segment. Segment-to-segment approaches were subsequently pursued by e.g.Zavarise and Wriggers [46] and El Abbasi and Bathe [11]). Afurther improvement has been more recently introduced withthe advent of the mortar methods [16–18,32,33,39], whichsimultaneously satisfy patch test and LBB stability require-ments, albeit at a higher computational cost. Based on thedemonstrated accuracy and robustness of mortar methods, amortar-based approach is pursued in this work as extensivelyillustrated later.

The recent advent of isogeometric analysis [19] has pro-vided a framework in which an exact description of the geom-etry is combined with the possibility to achieve the desireddegree of continuity at the element boundaries, as well aswith additional advantageous features including variationdiminishing and convex hull properties, and inherent non-negativeness of the basis functions. In the last few years thisnew computational mechanics technology has been success-fully applied to a large variety of problems (see e.g. [2–4,6,7,48]). Its potential superiority to traditional finite elementsbased on Lagrange discretizations in the area of contact mod-eling was already suggested in the original paper by Hugheset al. [19].

The first investigations on contact treatments in the frame-work of isogeometric analysis were recently conducted byTemizer et al. [40], Lu [27], and De Lorenzis et al. [8].Temizer et al. [40] applied NURBS-based isogeometric anal-ysis to thermomechanical frictionless contact problems in a3D large deformation setting with the penalty method. Theisogeometric framework was combined both with a mortar-based contact formulation, and with a non-mortar one basedon the enforcement of the contact constraint at an arbitrarynumber of Gauss points on the contact surface. Qualita-

tive analyses on large deformation frictionless sliding indi-cated a better iterative convergence behavior of the NURBSdiscretizations over the Lagrange ones. The quantitative anal-ysis of the classical Hertz problem demonstrated the supe-riority of the NURBS discretization in terms of quality androbustness of results. The non-mortar contact formulationwas also found unsuitable for an accurate and robust con-tact treatment, due to its overconstrained nature. Lu [27],by using an interference fitting example, demonstrated thatgeometric smoothness alleviated the non-physical contactforce oscillations which stem from the traditional approachof enforcing contact conditions through faceted surfaces. Theisogeometric setting was combined with a surface-to-surfacecontact formulation proposed by Papadopoulos and Taylor[29] and the penalty method was adopted. It was also sug-gested that the NURBS geometry can be efficiently usedto describe the mechanics of intrinsically smooth materi-als such as fabrics. The related example showed that thesmooth motion of the fabric, including some large wrin-kles, could be captured with a relatively small number ofdegree of freedoms. De Lorenzis et al. [8] presented botha non-mortar and a mortar-based isogeometric contact for-mulation, in a 2D large deformation setting with Coulombfriction, using the penalty method. Quantitative analysis ofthe Hertz problem with friction evidenced a clear superior-ity of the NURBS discretizations over the Lagrange ones interms of quality of the normal and frictional contact stressdistributions for different discretization orders. Results inTemizer et al. [40] regarding the drawbacks of the non-mortar formulation were further confirmed in the frictionalsetting. An ironing problem demonstrated as the magni-tude of the non-physical oscillations in the traction vs. timecurve decreased with increasing order of NURBS discretiza-tions, whereas it tended to increase with increasing order ofLagrange polynomials eventually leading to loss of conver-gence.

In this paper, NURBS-based isogeometric analysis isadopted to model 3D large deformation frictionless contactproblems. In view of the ascertained drawbacks of the non-mortar approach, a mortar-based approach as formulated inTemizer et al. [40] and De Lorenzis et al. [8] is directlyadopted herein, following earlier developments in mortarapproaches for Lagrange discretizations. Unlike in the previ-ous contributions, an exact enforcement of the contact con-straints is obtained by using the augmented Lagrangian (AL)method proposed by Alart and Curnier [1], which is charac-terized by a remarkable degree of robustness and yields anasymptotically quadratic convergence rate.

The paper is organized as follows: Sect. 2 describes the 3Dlarge deformation formulation of the contact problem usingthe AL method for frictionless conditions; Sect. 3 illustratesthe NURBS discretization; Sects. 4 and 5 present the con-tact formulation in detail, up to the full linearization which is

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Comput Mech (2012) 49:1–20 3

deferred to the Appendix; finally, in Sect. 6 some numericalexamples are presented and discussed.

2 3D large deformation contact problem

This section summarizes the theoretical and algorithmicbackground of frictionless contact between two deformablebodies undergoing finite deformations in a 3D setting. Moredetails can be found e.g. in Laursen [25] and Wriggers [41].

2.1 Contact constraints

Two 3D elastic bodies are assumed to come into contactundergoing large deformations. The bodies are denoted asslave (or non-mortar), Bs , and master (or mortar), Bm . Therelation between the initial (reference) configuration X, thedisplacement u and the current configuration x of the genericpoint of each body is given by

xi = Xi + ui (1)

where the superscript i = {s, m} denotes the slave and masterbodies, respectively.

The master surface is parameterized via convective coor-dinates ξα, α ∈ {1, 2}, that define the covariant vectors τα =xm,α . Using the metric mαβ := τα · τβ with inverse com-

ponents mαβ , the contravariant vectors τα := mαβτβ areinduced. The components of the symmetric curvature tensorfollow from kαβ = xm

α,β ·n, where n = nm is the normal unitvector.

In the current configuration, the contact interface is�c := �s

c = �mc , �i

c being the current contact boundary onbody Bi . For its determination, a function is introduced whichdescribes the distance between a given point of position xs on�s

c and an arbitrary point located at xm = xm(ξ1, ξ2) on �mc

d :=‖ xs − xm(ξ1, ξ2) ‖ (2)

The projection of each point of the slave surface onto themaster one is carried out by minimizing such distance. Thisclosest point projection defines a residual

fα(ξ1, ξ2) = τα(ξ1, ξ2) ·[xs − xm(ξ1, ξ2)

](3)

that vanishes at the projection point corresponding to {ξ1, ξ2}for all α, i.e. fα(ξ1, ξ2) = 0. The closest projection point andthe related variables are often identified in the literature withthe (•) notation, such as xm = xm(ξ1, ξ2). In the following,to simplify the notation the bar will mostly be omitted, and allthe quantities related to the master surface will be implicitlyintended as evaluated at the projection point.

The contact interface is pulled back to �c0 := �sc0 �=

�mc0, where �i

c0 is the contact boundary in the reference

configuration on body Bi . In the present formulation all con-tact integrals will be evaluated on �s

c0.The normal gap, gN , between the two bodies is defined as

gN = (xs − xm) · n (4)

Note that, with this definition, the gap is positive if the con-tact is open and negative when penetration of the bodies takesplace.

The normal contact traction λN is defined as the normalcomponent of the Piola traction vector t = tm = −ts

t = λN n λN = t · n (5)

Considering unilateral contact without adhesion, theKuhn–Tucker conditions for impenetrability on �c0 are

gN ≥ 0, λN ≤ 0, gN λN = 0 gN λN = 0 (6)

For further use, the variation and the linearization of thenormal gap are now introduced. The variation of the nor-mal gap is as follows, for details see e.g. Laursen [25] andWriggers [41],

δgN = (δxs − δxm) · n (7)

The linearized increment is simply obtained by substitutingthe symbol for virtual variation, δ, with that for linearizedincrement, �. Finally, the linearization of the normal gap isgiven by

�δgN = (δxm

,α�ξα + �xm,αδξ α

) · n + δξαkαβ�ξβ

+gN δn · �n (8)

where

δn = − (δxm

,α · n + kαβδξβ)τα (9)

and

δξβ = Aαβ[(

δxs − δxm) · τα − gN n · δx,αm] (10)

In Eq. (10) , Aαβ are the inverse components of

Aαβ = mαβ − gN kαβ (11)

2.2 Contact virtual work

The frictionless contact problem between two deformablehyperelastic bodies can be formulated as a constrained min-imization problem as follows

minϕW (ϕ) subject to gN ≥ 0 on �c (12)

where W is the potential energy functional, and ϕ is the map-ping describing the deformation, x = ϕ(X, t).

Among the different possible methods available for thesolution of constrained minimization problems of this type,we adopt henceforth the AL method as formulated byAlart and Curnier [1], see also Pietrzak and Curnier [31],Stupkiewicz et al. [37]. In this method, a dual field of

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4 Comput Mech (2012) 49:1–20

Lagrange multipliers λN is defined on the contact surface�c, and the AL functional is constructed as

L (ϕ, λN ) = W (ϕ) +∫

�c0

lN (gN , λN ) d� (13)

where

lN (gN , λN ) ={(

λN + εN2 gN

)gN , λN ≤ 0

− 12εN

λ2N , λN > 0

(14)

In the previous definition, εN > 0 is an arbitrary penaltyparameter and the AL multiplier λN = λN + εN gN is usedto discriminate between contact (λN ≤ 0) and separation(λN > 0) states. As further discussed later, the key advan-tage of this formulation over its penalty and Lagrange mul-tiplier alternatives is that lN and L are C1-differentiable. Inlight of the above, the constrained minimization problem inEq. (12) can be reformulated as the following unconstrainedsaddle-point problem

minϕmaxλN L (ϕ, λN ) (15)

The necessary condition of the saddle point takes the form

δL = δW + δWc = 0 (16)

where

δWc =∫

�c0

[λ

e f fN δgN + CN δλN

]d� (17)

is the contact contribution to the virtual work, and the fol-lowing notation has been introduced [37]

λe f fN = ∂lN

∂gN={

λN , λN ≤ 00 λN > 0

CN = ∂lN

∂λN={

gN , λN ≤ 0− 1

εNλN , λN > 0

(18)

The AL multiplier λe f fN defined by Eq. (18) is the state-

dependent normal contact traction which is work-conju-gate to δgN , whereas CN introduces constraints that areactive depending on the contact state. Thanks to the C1-differentiability of L, the change of the contact state fromcontact to separation preserves the continuity of both λ

e f fN

and CN , thereby enabling an efficient solution of the equa-tions with Newton’s method upon discretization. Note thatthe contact constraint is enforced exactly regardless of thevalue of the penalty parameter, which can be then kept con-veniently low to improve the convergence behavior.

The linearization of δWc follows immediately fromEq. (17) as

�δWc =∫

�c0

[λ

e f fN �δgN + �λ

e f fN δgN + �C N δλN

]d�

(19)

3 NURBS discretization

This section describes briefly the 3D NURBS discretizationused for the continuum and the ensuing 2D NURBS dis-cretization of the contact surfaces. In what follows, standardNURBS terminology is employed. Further details and exten-sive references can be retrieved in Piegl and Tiller [30] andCottrell et al. [7], see also Temizer et al. [40] and De Lorenziset al. [8].

Let �i be the open non-uniform knot vector associatedwith the i th dimension of a patch, i = {1, 3}�i =

{ξ i

1, . . ., ξini +pi +1

}(20)

The first pi +1 terms in �i are equal, and so are the last pi +1terms. Here, pi are the polynomial orders of the accompa-nying B-spline basis functions, ξ i

j is the j th knot and ni isthe number of accompanying control points in the i th dimen-sion. Moreover, ni − pi is the number of elements in the samedimension. A 3D domain is parameterized by

S(ξ1, ξ2, ξ3

)=

n1∑d1=1

n2∑d2=1

n3∑d3=1

Rd1d2d3

(ξ1, ξ2, ξ3

)Xd1d2d3

(21)

where Xd1d2d3 are the control point coordinates and Rd1d2d3

≥0 are the rational B-spline (NURBS) basis functions. Thelatter are defined via a tensor product in a four-dimensionalspace based on homogeneous coordinates [30]. The projectedform in the three-dimensional space is

Rd1d2d3

(ξ1, ξ2, ξ3

)= wd1d2d3

W(ξ1, ξ2, ξ3

)

×B1d1

(ξ1)

B2d2

(ξ2)

B3d3

(ξ3)

(22)

with Bidi

as a nonrational B-spline basis function. The nor-malizing weight W is given in terms of the weights wd1d2d3

and of the functions Bidi

via

W(ξ1, ξ2, ξ3

)=

n1∑d1=1

n2∑d2=1

n3∑d3=1

wd1d2d3

×B1d1

(ξ1)

B2d2

(ξ2)

B3d3

(ξ3)

(23)

The knot vectors together with the associated controlpoints and the accompanying weights constitute a patch. Thecontinuity and order of Bi

didepends on �i only. If �i has no

repeated interior knot ξ ij , j ∈ [pi + 1, ni ], then the order-pi

basis function Bidi

has continuity C pi −1. Every repetition ofa knot decreases the continuity by one order at this knot.

The counterparts of the h- and p-refinement proceduresfor FEM discretizations based on Lagrange polynomials arethe knot insertion and order elevation procedures in theNURBS setting. While p-refinement preserves the number

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Comput Mech (2012) 49:1–20 5

of nodes, order elevation leads to a slight increase in thenumber of control points. When the two must be conductedtogether, the k-refinement procedure will be employed whereknot refinement precedes order elevation [7]. This has theadvantage that a higher degree of smoothness can be achievedwithin the patch across non-repeated knot entries and the finalnumber of control points is less compared to the case whereknot refinement precedes. For the numerical evaluation ofthe weak forms emanating from Lagrange or NURBS baseddiscretizations, 2p Gauss points will be employed withineach element for order-p approximations. This ensures aconverged quadrature. See Hughes et al. [20] for a recentdiscussion of efficient quadrature schemes appropriate forisogeometric analysis.

The contact discretized surface is a NURBS surface thatis directly inherited from the 3D NURBS discretization. Forinstance, let ξ3− := ξ3

1 . By construction [30]

S(ξ1, ξ2, ξ3−

)=

n1∑d1=1

n2∑d2=1

R−d1d2

(ξ1, ξ2

)X−

d1d2(24)

where X−d1d2

:= Xd1d21 and, including the weighting factor

R−d1d2

(ξ1, ξ2

):= wd1d21 B1

d1

(ξ1)

B2d2

(ξ2)

∑n1d1=1

∑n2d2=1 wd1d21 B1

d1

(ξ1)

B2d2

(ξ2)

(25)

Hence, only the knowledge of the knot vectors �1 and �2 anda reduced set of control points together with the accompany-ing weights are sufficient to characterize the surface associ-ated with ξ3−. The same principle applies for ξ3+ = ξ3

n3+p3+1and for the other dimensions. Hence, in general, a 2D patch(in particular, a contact patch) is directly inherited from the3D continuum patch and has its same parameterization butonly with two dimensions α ∈ {1, 2} that correspond to anytwo of the three dimensions. The corresponding knot vec-tors are �α with associated B-spline basis functions Bα

dαand

parametric space coordinates ξα that are conveniently cho-sen as the convective coordinates for contact computations.The surface parameterization is therefore

S(ξ1, ξ2

)=

n1∑d1=1

n2∑d2=1

Rd1d2

(ξ1, ξ2

)Xd1d2 (26)

Adopting the isoparametric concept, an analogous inter-polation is used for the unknown displacement field, its var-iation and the current coordinates. For conciseness, theseparameterizations will be expressed as follows as

S =ncp∑A=1

RAXA u =ncp∑A=1

RAuA δu =ncp∑A=1

RAδuA

x =ncp∑A=1

RAxA (27)

where ncp is the number of control points associated withthe surface (i.e. the product of the number of control pointsin the two parametric directions associated with the surface),RA is the rational basis function corresponding to controlpoint A, whereas XA, uA, δuA and xA are the related refer-ence coordinate, displacement, displacement variation andcurrent coordinate vectors. Eq. (27) can also be used for theLagrange polynomial discretization, provided that the stan-dard Lagrangian shape functions are used in place of RA, andpoints A are interpreted as nodal points.

It is also useful to reconsider the above parameterizationas a collection of local mappings, each defined over one indi-vidual element of the contact surface. The parameterizationover an element e is

Se =nnes∑a=1

RaXa ue =nnes∑a=1

Raua δue =nnes∑a=1

Raδua

xe =nnes∑a=1

Raxa (28)

where nnes = ∏i=1,2 (pi + 1) is the number of control

points whose basis functions have support on a single ele-ment of the contact surface. Once again, Eq. (28) can be alsoapplied to a Lagrangian interpolation, in which case nnes issimply the number of nodes per element of the contact sur-face. As follows, we will refer the above global and localparameterizations to the slave and the master surfaces byadding the appropriate superscript s or m, respectively.

4 Variations and linearizations of the contact variablesin discretized form

By substituting the interpolations in Eq. (28) into Eq. (4), thenormal gap becomes

gN =⎡⎣

nsnes∑

a=1

Rsa

(ξ1

s , ξ2s

)xs

a −nm

nes∑a=1

Rma

(ξ1, ξ2

)xm

a

⎤⎦ · n

(29)

In Eq. (29), {ξ1s , ξ2

s } are the parametric coordinates of thegeneric point on �c0, whereas {ξ1, ξ2} are the parametriccoordinates of the corresponding projection point on the mas-ter surface. The virtual variation follows from Eq. (7) as

δgN =⎡⎣

nsnes∑

a=1

Rsa

(ξ1

s , ξ2s

)δus

a −nm

nes∑a=1

Rma

(ξ1, ξ2

)δum

a

⎤⎦ · n

(30)

The same substitution can be carried out for the other equa-tions in Sect. 2.1. In order to formulate the problem in matrixform, the following auxiliary vectors are introduced

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6 Comput Mech (2012) 49:1–20

δu =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

δus1

...

δusns

nes

δum1

...

δumnm

nes

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

�u =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�us1

...

�usns

nes

�um1

...

�umnm

nes

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(31)

N =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Rs1

(ξ1

s , ξ2s

)n

...

Rsns

nes

(ξ1

s , ξ2s

)n

−Rm1

(ξ1, ξ2

)n

...

−Rmnm

nes

(ξ1, ξ2

)n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tα =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Rs1

(ξ1

s , ξ2s

)τα

...

Rsns

nes

(ξ1

s , ξ2s

)τα

−Rm1

(ξ1, ξ2

)τα

...

−Rmnm

nes

(ξ1, ξ2

)τα

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Nα =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0...

0−Rm

1,α

(ξ1, ξ2

)n

...

−Rmnm

nes ,α

(ξ1, ξ2

)n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(32)

D1 = 1

det [A][A22 (T1 − gN N1) − A12 (T2 − gN N2)]

(33)

D2 = 1

det [A][A11 (T2 − gN N2) − A12 (T1 − gN N1)]

(34)

N1 = N1 − k11D1 − k12D2 N2 = N2 − k12D1 − k22D2

(35)

where matrix A has been defined in Eq. (11).With the above positions, Eq. (30) (and the dual one giving

the linearized increments) can be cast in matrix form as

δgN = δuT N (36)

�gN = NT �u (37)

and the linearization �δgN can be expressed as

�δgN = δuT{−gN

[m11N1NT

1 + m12(

N1NT2 + N2NT

1

)

+m22N2NT2

]− D1NT

1 − D2NT2 − N1DT

1 − N2DT2

+k11D1DT1+k22D2DT

2+k12

(D1DT

2 +D2DT1

)}�u

(38)

A similar vectorial notation for the above geometrical quan-tities in a Lagrange polynomial discretization is adopted inLaursen [25] and Wriggers [41].

5 Mortar-based contact algorithm

The algorithm used in this work is based on the mortarmethod, as formulated in Temizer et al. [40] and De Lorenziset al. [8]. With this approach, the contact constraints are onlyenforced at the control points in the NURBS discretization,and at the nodes in the Lagrange discretization. Like in Turet al. [39], integration is here carried out without segmenta-tion of the contact surfaces, in order to avoid the associatedcomputational cost and implementation burden. For generaldiscretizations this procedure will introduce an error, whichcan however be reduced by increasing the number of inte-gration points on the contact surface, see also Fischer andWriggers [13].

In the spirit of the mortar method, the contact contributionto the virtual work is expressed as

δWc =∑

A

(λ

e f fN AδgN A + CN AδλN A

)AA (39)

where

AA =∫

�c0

RAd� (40)

and

λe f fN A =

{λN A = λN A + εN gN A, λN A ≤ 00 λN A > 0

CN A ={

gN A, λN A ≤ 0− 1

εNλN A, λN A > 0

(41)

are the control point counterparts of the quantities introducedin Eq. (18). Note that, unlike in the mortar algorithm basedon the penalty method (see e.g. [8]), the summation is hereextended to all control points and not only to the active ones(i.e. those with λN A ≤ 0).

Note that, formally, Eq. 17 closely resembles the expres-sion of the contact virtual work used in classical node-to-surface algorithms, where the contact constraint is enforcedat the nodes using a collocation approach. Similarly, Eq. 40,here referred to a control point, is equivalent to the tributaryarea of a slave node introduced in area-regularized node-to-surface approaches (see e.g. [47,44]).

The notable difference of the current approach from anode-to-surface one is introduced by the following defini-tion of the control point normal gap as the weighted averageof the corresponding “local” gaps, with the basis functionsas weights

gN A =∫�c0

RAgN d�∫�c0

RAd�(42)

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Comput Mech (2012) 49:1–20 7

The virtual variation of Eq. (42) gives

δgN A =∫�c0

RAδgN d�∫�c0

RAd�(43)

Substituting the above definitions into Eq. (39) yields

δWc =∑

A,act.

⎛⎜⎝λN A

∫

�c0

RAδgN d� + gN AδλN A AA

⎞⎟⎠

+∑

A,inact.

(− 1

εNλN AδλN A AA

)

=∫

�c0

λNintδgN d� +∑

A,act.

(gN AδλN A AA)

+∑

A,inact.

(− 1

εNλN AδλN A AA

)(44)

where

λNint =∑

A,act.

λN A RA (45)

is the interpolated value of the AL multiplier over the activecontrol points.

For the further developments it is convenient to definethe vectors of the virtual variations and linearizations of thecontrol point Lagrange multiplier unknowns

δλ =⎡⎢⎣

δλ1...

δλncp

⎤⎥⎦ �λ =

⎡⎢⎣

�λ1...

�λncp

⎤⎥⎦ (46)

and the additional vector

Nλ =⎡⎢⎣

Nλ1...

Nλncp

⎤⎥⎦ (47)

whose terms are given by

NλA ={

gN A AA, λN A ≤ 0− 1

εNλN A AA, λN A > 0

(48)

The combination of Eq. (44) with Eq. (36) and with the def-initions (46) and (47) results in

δWc = δuT∫

�c0

λNint Nd� + δλT Nλ (49)

From Eq. (49) the expression of the residual for the Newton-Rapson iterative scheme is immediately obtained as

R =[

Ru

Rλ

]=[∫

�c0λNint Nd�

Nλ

](50)

As mentioned earlier, numerical integration is conducteddirectly on �c0 without segmentation of the contact surfaces,so that Ru in Eq. (50) is numerically evaluated as follows

Ru =∑G P

(λNintg Ng

)wg jg (51)

where the subscript g refers to the dependence on the Gausspoint coordinate, wg and jg are respectively the weight andthe jacobian associated to the g-th integration point on �c0,and

λNintg =∑

A,act.

λN A RAg (52)

In Eq. (52), RAg is the value at the g-th integration point of thebasis function associated with control point A. The completelinearization and the consequent expression of the consistenttangent stiffness matrix are reported in the Appendix.

6 Numerical examples

This section presents some examples to demonstrate theaccuracy and quality of the proposed contact formulation.The first example considers infinitesimal deformations inorder to enable the comparison of numerical results with theavailable analytical solution. Here the primary focus of theanalysis is on the quality of the contact stress distributions.The second example demonstrates the capability of the pro-posed formulation to deal with large deformations of bothmaster and slave bodies. Attention is paid also in this case tothe three-dimensional contact stress distributions. Moreover,the iterative convergence performance and the robustness ofAL and penalty methods are comparatively evaluated. Thesubsequent examples involve large deformation and largesliding conditions. In these cases the major interest lies onthe reaction history curves, which not only reflect the qualityof the stress distributions at the contact interface over time,but also result from the interaction between the contact andthe bulk behavior in the discretized model [8]. This aspect isdiscussed later in greater detail.

For comparison purposes, not only NURBS but alsoLagrange discretizations are employed. In the latter case,generation of the geometry and refinement are first conductedon the exact NURBS parameterization, and then convertedto the Lagrange one. Although the number of control vari-ables (control points or nodes) is different for the two dis-cretizations, the same resolution is maintained at the contactsurfaces, thereby ensuring a meaningful direct comparisonof the contact behavior. The order of NURBS and Lagrangeparameterization will be respectively denoted by Np and L p.The approximation spaces based on N1 and L1 are identicalunder the condition that the weights be uniform. In all exam-ples, higher order parameterizations are only used in the two

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8 Comput Mech (2012) 49:1–20

(a) (b)

Fig. 1 Hertz contact problem: mesh

parametric directions of both contact surfaces, whereas firstorder basis/shape functions are always used in the third para-metric direction. Numerical trials have shown that order ele-vation in this direction yields no appreciable differences inresults for the considered examples.

In all examples, 2p Gauss points in each parametric direc-tion are employed within each element for Np and L p cases.For the integration of the contact contribution, 6 Gauss pointsare adopted on each contact facet in each of the two surfaceparametric directions. A contact facet is here intended as thephysical mapping of the region of the parametric space boundby one non-zero knot span in each of the surface parametricdirections. As facet-wise integration is carried out for bothNURBS and Lagrange discretizations, the same number ofquadrature points is maintained in all cases.

The penalty parameter used in the AL formulation is takenequal to εN = 10. As previously remarked, such value bearsno influence on results as the contact constraint enforcementis exact. The number of time steps is chosen in order to obtainconvergence in maximum 10 iterations.

6.1 Hertz contact problem

The first example deals with frictionless contact of a sphereof radius R = 1 on a rigid plane. The material of the sphere islinearly elastic with Young’s modulus E = 1 and Poisson’sratio ν = 0.3. The natural NURBS description of the geome-try includes an internal spherical surface whose inner radiusis chosen equal to 0.8. However this value was verified toexert no influence on results. The original Hertz problem,where the sphere would be loaded with a concentrated forceon top, is substituted with the similar problem of a half-sphere loaded with a uniform pressure on its equatorial plane.This approximation has been widely used in previous com-parisons, and has been proved sufficiently accurate from the

standpoint of the evaluation of the contact stresses. Thus, onlyone eighth of a sphere is analyzed by exploiting symmetryconditions, see Fig. 1a. The pressure applied on the equato-rial plane in one time step is taken equal to pe = 0.0001,and is suitably modified in the region bound by the innerradius to account for the local slopes of the tangent plane.The analytical solution [21] gives a circular contact area ofradius a = 3

√3P R/4E∗ and a maximum contact pressure

p0 = 3P/2πa2, where P = π R2 pe is the total applied loadand E∗ = E/(1 − ν2) is the effective elastic modulus. In thepresent case a = 0.0599 and p0 = 0.0418.

As shown in Fig. 1b, the mesh is refined in the vicinityof the contact region by redistributing the knot vector entriesas already described in Temizer et al. [40] and De Lorenziset al. [8]. The chosen amount of redistribution of the knotvector entries is such that 75% of the elements are locatedwithin 10% of the total length of the knot vector in the radialand inclination parametric directions, whereas a uniform dis-tribution of the inner knot vector entries is retained in theazimuthal parametric direction.

While the contact stresses for this problem can be easilyfound in a 2D setting by exploiting the axial symmetry ofgeometry and loading, this example is intended as a bench-mark to evaluate the performance of the 3D NURBS-basedmortar formulation. The analysis focuses in particular on theeffect of the order of the interpolation on the quality of thecontact stress distribution. Figure 2a illustrates the resultsobtained from linear (Lagrange) interpolations. The dimen-sionless contact pressure p/p0 is plotted versus the dimen-sionless coordinate r/a, r being the distance from the centerof the contact area. A two-dimensional plot is presented dueto the axial symmetry of results. The agreement is gener-ally very good, although the solution near the edge of thecontact region is affected by the elements that lie across thecontact /non contact zone. This is a well-known problem for

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Comput Mech (2012) 49:1–20 9

Fig. 2 Hertz contact problem:contact stresses for varyingNURBS and Lagrangediscretization order

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

p/p0

x/a

theo.L1

(a)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.2 0.4 0.6 0.8 1 1.2

p/p0

x/a

theo.L2

0

(b)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.2 0.4 0.6 0.8 1 1.2

p/p0

theo.N2

0

r/a

(c)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.2 0.4 0.6 0.8 1 1.2

p/p0

x/a

theo.L3

0

(d)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

p/p0

theo.N3

r/a

(e)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.2 0.4 0.6 0.8 1 1.2

p/p0

x/a

theo.

0.2 0.4 0.6 0.8 1 1.2

L4

0

(f)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.2 0.4 0.6 0.8 1 1.2

p/p0

theo.

0.2 0.4 0.6 0.8 1 1.2

N4

0

r/a

(g)

which solutions based on a better local resolution of the con-tact interface have been proposed (see e.g. [22,15]).

Results obtained with higher-order NURBS and Lagrangediscretizations are shown in Figs. 2b–g. For the NURBS dis-

cretizations, the quality of the results seems not to be affectedby the order to an appreciable extent. Moreover, the oscilla-tions of the contact pressure at the edge of the contact regionare quite limited, due to the variation diminishing property

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10 Comput Mech (2012) 49:1–20

of the NURBS basis functions, and all distributions featurenon-negative values, due to the inherent non-negativenessof these functions. It is evident that Lagrange discretizationsproduce results of inferior quality. The distribution of the con-tact stresses is quite irregular as a result of the interpolatory(and thus oscillatory) nature of the Lagrange basis functions,and of their C0 continuity at the inter-element border. Resultscould be probably enhanced by introducing an averaging ofthe normal vectors at the inter-element nodes such as donein Yang et al. [43], Puso and Laursen [32], Puso et al. [33].This is obviously not needed for NURBS, as they guaranteethe desired degree of continuity between adjacent contactelements. Increasing the order of the Lagrange discretiza-tion is evidently unfavorable, as even larger oscillations areproduced. These oscillations may result in negative valuesat the edge of the contact region which are not physicallymeaningful. These results closely resemble those presentedin Temizer et al. [40] and De Lorenzis et al. [8] for a 2Dfrictionless and a 2D frictional Hertz problem, respectively,and extend them to the 3D frictionless case.

6.2 Elastic ring problem

In the second example, illustrated in Fig. 3a, an elasticring (radius 0.5, width 0.8, thickness 0.05, enclosed angle160 degrees) is pressed onto an elastic slab (dimensions1.2 × 1.2 × 0.3). The ring consists of an inner and an outerlayers of equal thickness made of different materials. Hy-perelastic neo-Hookean behaviour is assumed for all bodies,with material properties E = 10 and ν = 0.3 for the outerring, E = 100 and ν = 0.3 for the inner ring, and E = 0.1and ν = 0.3 for the slab. The slab is fixed at its lower side anda uniform downward displacement Uz = −0.25 is applied tothe upper surface of the ring in 25 time steps, causing bothring and slab to undergo large deformations. The lower sur-face of the ring is treated as slave. Figure 3b, c depicts thedeformed shape obtained with the N2 discretization at oneintermediate stage and after the final time step.

A comparison between the three-dimensional contactstress distributions obtained after the first loading step fromNURBS and Lagrange discretizations of varying order ispresented in Fig. 4. The observations stemming from theseresults are very similar to those made earlier for the Hertzcontact problem. The NURBS discretizations deliver quitesmooth contact pressure distributions, which do not vary sig-nificantly with the interpolation order. Note that the steepincrease in contact pressure towards the front and backedges of the ring surface has a physical ground, due tothe simultaneous large deformation of the master surface.Moreover, all NURBS distributions feature non-negative val-ues, which automatically stems from the non-negativenessof the basis functions. Conversely, results obtained from theLagrange interpolations are quite sensitive to the interpo-

(a)

(b)

(c)

Fig. 3 Elastic ring problem: mesh and deformed shape

lation order, whose increase leads to a more pronouncedoscillatory behavior. The oscillations in turn lead the contactpressure to assume quite large, unphysical negative valuesespecially at the edges of the contact region, see Figs. 4band d.

123

Comput Mech (2012) 49:1–20 11

(a)

(b) (c)

(d) (e)

Fig. 4 Elastic ring problem: contact stresses for varying NURBS and Lagrange discretization order

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12 Comput Mech (2012) 49:1–20

Table 1 Comparison betweenAL and penalty (P) methods:maximum step size forconvergence

AL P (εN = 1E + 01) P (εN = 1E + 02) P (εN = 1E + 03) P (εN = 1E + 04)

0.0208 0.0139 0.0083 0.0071 0.0036

Table 2 Comparison betweenAL and penalty (P) methods:iterative convergence behaviorat a representative time step

Iteration AL P (εN = 1E + 01) P (εN = 1E + 02) P (εN = 1E + 03)

0 0.10000E+01 0.10000E+01 0.10000E+01 0.10000E+01

1 0.25055E−02 0.19865E−02 0.67944E−02 0.30642E−02

2 0.34670E−04 0.20946E−04 0.47549E−02 0.10909E−02

3 0.69893E−09 0.26029E−09 0.10462E−02 0.24880E−02

4 0.17084E−04 0.75532E−05

5 0.38659E−09 0.17246E−09

Fig. 5 Block ironing problem:mesh

(a) (b)

As one of the features of the proposed formulation is theuse of the AL method, it is of interest to evaluate its per-formance in comparison with the penalty method used inprevious related investigations. To this end, the maximuminitial step size (i.e., the maximum magnitude of the dis-placement imposed at the first step) for which convergencewas achieved in the current problem is reported in Table 1.The better robustness of the AL is evident, as well as themonotonic decrease in robustness of the penalty method asthe value of penalty parameter is increased. A subsequentcomparison is made by setting the step size to 0.0071, i.e.the value giving convergence for the penalty method withεN = 1E +03 (excluding the largest penalty value for whichan impractically low step size is required), and evaluatingthe iterative convergence history for a representative timestep as shown in Table 2. While AL and penalty with εN =1E + 01 appear here substantially equivalent, the penaltysolution with this low εN is affected by relatively large pen-etrations, which may be deemed unacceptable depending onthe required accuracy. Larger values of the penalty parameterdo not only require more iterations, but also cause the analy-sis to stop converging at steps 11 or 6 for εN = 1E + 02 and1E +03, respectively, following modifications in the contactactive set.

6.3 Block ironing problem

In the third example, a square block (side dimension 4.0) ispressed into an elastic slab (dimensions 20.0x10.0x5.0) byapplying a uniform downward displacement Uz = −1.0 toits upper surface in 2 time steps. This displacement is thenmaintained constant while the upper block surface is draggedby Ux = 12.5 along the longitudinal direction in further 50time steps. The lower surface of the slab is restrained in alldirections. Hyper-elastic Neo-Hookean material behavior isassumed for both bodies, with material parameters E = 1and ν = 0.3 for the slab, and E = 10 and ν = 0.3 for theupper block. The lower surface of the upper block is treatedas slave.

The problem is solved using both NURBS and Lagrangediscretizations. In this case the L1 and N1 approximationspaces are coincident, being the weights of the basic NURBSdiscretization uniform. Figure 5a illustrates the mesh forall NURBS discretizations (including N1 = L1), whereasFig. 5b pertains to the L2 case. A coarse mesh is chosenin order to amplify the differences between results in termsboth of deformed shape and of reaction vs. time behavior,as will be better discussed in the following. As previouslymentioned, the comparison between NURBS and Lagrange

123

Comput Mech (2012) 49:1–20 13

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Fig. 6 Block ironing problem: deformed shape

cases is always made for a given resolution of the contactsurface, which implies that the number of elements in theparametric directions of the contact surfaces are half in theL2 than in the NURBS cases. For NURBS parameterizationsorder elevation does not alter the number of elements, as thisis determined by the number of non-zero knot spans whichis unaffected by p-refinement.

Figure 6 shows the deformed mesh for first- and second-order NURBS and Lagrange discretizations at different timesteps, along with the contours of the vertical displacements.Figure 7 further provides a close-up of the contact region atthe final step. Here the stress contour highlights the stressconcentrations at the block edges, which make this prob-lem particularly challenging. Finally, the reaction history isreported in Fig. 8.

The N1/L1 parameterization, as expected, results in anunsatisfactory resolution of the contact surface for this coarse

discretization. The abrupt changes in slope in both para-metric directions at the element junctions, clearly visible inFig. 7a, are responsible for pathologic behavior and con-vergence difficulties. Also, they produce large non-physicaloscillations in the reaction versus time response any time theslave surface traverses a master element boundary (Fig. 8).Similar oscillations for the ironing problem treated with bothnode-to-segment and mortar approaches have already beenreported in the literature, see e.g. Puso and Laursen [32],and result from the limited ability of the master surface dis-cretization to accommodate the deformation profile imposedby the displacement of the slave body. As already noted inDe Lorenzis et al. [8], the parameterization exerts a clearinfluence on the magnitude and regularity of these oscilla-tions. The L2 parameterization delivers a better quality ofthe deformed shape for the same number of degrees of free-dom (Fig. 7b). However, the effects of the C0 continuity

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14 Comput Mech (2012) 49:1–20

(a) (b) (c)

Fig. 7 Block ironing problem: close-up of the deformed shape at step 52

at the inter-element boundaries are quite evident and deter-mine large spurious stress concentrations in the vicinity ofthe highly stressed zone corresponding to the current masterelements and at the boundary between this and the adjacentregions (Figs. 6d–f). The reaction versus time response fea-tures also in this case quite large oscillations with a moreirregular but still repetitive pattern (Fig. 8). These againreflect the transitions between master elements and the conse-quent difficulty in the match of the deformed profile betweenslave and master surfaces. Finally, in the N2 discretization,the C1 continuity of the contact surface determines a well-resolved deformed contact interface, with smooth transitionsbetween adjacent master elements (Fig. 7c) and lower stressconcentrations (Figs. 6g–i). Also, the non-local support ofthe basis functions extends the region of the slab potentiallyinvolved by the strong deformation regime, thereby provid-ing to the master surface a greater degree of flexibility toaccommodate the imposed deformation profile. Correspond-ingly, the reaction versus time response features oscillationsof reduced magnitude (Fig. 8). Increasing the order to N3

further smoothens the reaction vs. time response, yieldinga macroscopically smooth curve despite the coarseness ofthe mesh. These results are in good agreement with thosereported by De Lorenzis et al. [8] in a 2D frictional setting.Note that the small shift in the average level of the curvesis due to the slight increase in the number of control pointswith respect to the reference L1/N1 case as a result of theorder elevation procedure.

6.4 Rotating block ironing problem

Herein the same geometry and material properties of the pre-vious example are maintained, with the only variant that theupper block, after having been displaced by −1.0 in the ver-tical direction, is simultaneously subjected to a translationof 12.5 in the longitudinal direction and to a counterclock-wise rotation of 45 degrees. Figure 9 provides snapshots ofthe deformed shape at the intermediate and final configura-

Fig. 8 Block ironing problem: reaction-time response

tions, whereas Fig. 10 shows a detail of the contact surface atthe last time step. Deformed shapes and displacement/stresscontours are similar to those observed in the previous exam-ple and the related comments are not repeated herein. Thereaction vs. time response is reported in Fig. 11, and showsin this case an ever-increasing trend due to the effect of thesimultaneous translation and rotation. The added complexityof the deformation breaks the repetitive oscillation patternsobserved in the previous examples. Nevertheless, a number ofoscillations of variable magnitude corresponding to the maintransitions between master elements are still clearly visibleand are successfully alleviated by NURBS discretizations ofincreasing order.

6.5 Cylinder ironing problem

In the last example, a cylindrical die (radius 0.5, wall thick-ness 0.1, and width 0.6), is pressed into an elastic slab(dimensions 2.5x1.0x0.6) by applying a uniform downwarddisplacement Uz = −0.2 to its upper surface in 10 timesteps. The die is then dragged in the longitudinal directionby Ux = 1.5 in further 100 time steps while the vertical

123

Comput Mech (2012) 49:1–20 15

(a) (b)

(c) (d)

(e) (f)

Fig. 9 Rotating block ironing problem: deformed shape

123

16 Comput Mech (2012) 49:1–20

(a) (b) (c)

Fig. 10 Rotating block ironing problem: close-up of the deformed shape at step 52

Fig. 11 Rotating block ironing problem: reaction-time response

displacement is maintained constant. The lower surface ofthe slab is restrained in all directions. Neo-Hookean hyper-elastic material behavior is assumed for both bodies, withmaterial parameters E = 1 and ν = 0.3 for the slab, andE = 100 and ν = 0.3 for the die. The lower surface of thedie is treated as slave. Figure 12a illustrates the discretization.

The main aim of this example is to further demonstrate theperformance of the presented mortar method with NURBSdiscretizations for contact problems with very large defor-mations. For the given values of imposed displacements, noconvergence is obtained with Lagrange discretizations, dueto the C0 continuity at the inter-element boundaries whichproduces pathologies in the projection of the contact inte-gration points onto the master surface. As mentioned previ-ously, these problems have been alleviated in previous mortarimplementations by introducing an averaging of the normalvectors at the inter-element nodes [32,33,43].

NURBS second and third order discretizations, featuringrespectively C1 and C2 inter-element continuity, are adoptedherein. The deformed mesh is shown in Figs. 12b–d forsecond-order NURBS discretizations at different stages ofthe dragging phase. The reaction versus time response in

Fig. 13 features very small oscillations already with the N2

parameterization. Increasing the order to N3 leads to a fur-ther improvement, yielding a macroscopically smooth curve.Also in this case, the small shift in the level of the plateauis due to the slight increase in the number of control pointsfollowing order elevation.

7 Conclusions

This work presents a three-dimensional mortar formulationfor frictionless contact between deformable bodies undergo-ing large deformations, following on from previous recentstudies [8,40]. NURBS-based isogeometric analysis andstandard C0-continuous Lagrange finite element interpola-tions were adopted for the discretization of the continuum,and the same parameterization was inherited by the contactsurfaces in a straightforward manner. The proposed mortarapproach is based on the enforcement of the contact con-straints at the control points (respectively at the nodes forLagrange discretizations), combined with a non-local evalu-ation of the control point /node quantities as weighted aver-ages of the corresponding local ones. A simple integrationscheme is deployed which does not involve segmentation ofthe contact surfaces and thus alleviates the implementationburden and the computational cost. The contact constraintsare satisfied exactly by using the AL formulation proposed byAlart and Curnier, in which a Newton-like solution schemeis applied to solve the saddle point problem simultaneouslyfor the displacements and the Lagrange multipliers.

Based on results obtained in this investigation, it canbe concluded that the proposed contact mortar formulationusing NURBS-based isogeometric analysis displays a sig-nificantly superior performance with respect to the sameformulation using standard Lagrange polynomials. In thebenchmark small deformation case, as well as in the examplefeaturing large deformations of both contacting bodies, thequality of the contact pressures is shown to improve clearly

123

Comput Mech (2012) 49:1–20 17

(a) (b)

(c) (d)

Fig. 12 Cylinder ironing problem: mesh and deformed shape

over that achieved with Lagrange discretizations. The contactpressure distributions stemming from the NURBS parame-terizations are always non-negative and are practically insen-sitive to changes in the interpolation order. The respectivedistributions obtained from Lagrange parameterizations aresensitive to the interpolation order, display spurious oscilla-tions and may attain significant non-physical negative val-ues. In large deformation and large sliding examples, thetime histories of the reactions obtained from the NURBS dis-cretizations are remarkably smooth even for coarse meshesand improve in quality with increasing order of the param-eterization. Conversely, the curves obtained from Lagrangeparameterizations display non-physical oscillations whosemagnitude is not alleviated by the interpolation order andwhich may even prevent convergence.

From the standpoint of the solution method, a comparisonbetween AL and penalty confirms the clear superiority of

the former in terms of robustness and iterative convergencebehavior, in addition to the obvious advantage of the exactcontact constraint enforcement.

The NURBS-based contact mortar formulation thusappears to provide a robust description of large deformationcontact between deformable bodies, which is equally effec-tive and accurate for different interpolation orders. This ulti-mately implies that the unification of the geometric modelingand analysis phases aimed at by the proposers of isogeomet-ric analysis [7] can be effectively extended to mechanicalproblems in which contact phenomena occur. In view of thepossible unified treatment of contact and debonding prob-lems by using, e.g., generalized contact elements embed-ding a mixed-mode cohesive zone model [9] or adhesioneffects [34], the proposed formulation may be able to alle-viate the well-known numerical stability problems inducedby the discretization in the computation of delamination

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18 Comput Mech (2012) 49:1–20

Fig. 13 Cylinder ironing problem: reaction-time response

processes. A further possible step forward is the extensionof the formulation to T-spline discretizations, which enablelocal refinement and waterproof merging of different patches[5]. These research directions are currently being pursued bythe authors.

Acknowledgments The first author gratefully acknowledges the sup-port of the Alexander von Humboldt Stiftung through a “HumboldtResearch Fellowship for Experienced Researchers” for a research stayat the Leibniz Universität Hannover. Further support was provided bythe Italian Ministry of Research through program PRIN2008.

A Appendix

The linearization of Eq. (39) yields

�δWc = �δWc,m + �δWc,g (53)

where

�δWc,m =∑

A

(�λ

e f fN AδgN A + �CN AδλN A

)AA

=∑

A,act.

(εN �gN AδgN A + �λN AδgN A

+�gN AδλN A

)AA

+∑

A,inact.

(− 1

εN�λN AδλN A

)AA (54)

is the “main” component, and

�δWc,g =∑

A

(λ

e f fN A�δgN A

)AA =

∑A,act.

(λN A�δgN A

)AA

(55)

is the “geometric” component.As follows, the main and geometric components of the

tangent stiffness are computed in matrix form starting fromthe above expressions.

A.1 Main component

By combining Eqs. (43) and (40) with Eq. (54), the maincomponent �δWc,m can be expressed as

�δWc,m =∑

A,act.

⎛⎜⎝ εN∫

�c0RAd�

∫

�c0

RAδgN d�

∫

�c0

RA�gN d�

+�λN A

∫

�c0

RAδgN d�+δλN A

∫

�c0

RA�gN d�

⎞⎟⎠

+∑

A,inact.

(− 1

εNAA�λN AδλN A

)(56)

Employment of the matrix expressions in Sect. 4 yields

�δWc,m = δuT

⎡⎢⎣∑

A,act.

εN∫�c0

RAd�

∫

�c0

RANd�

∫

�c0

RANT d�

⎤⎥⎦�u

+δuT

⎡⎢⎣∑

A,act.

⎛⎜⎝∫

�c0

RANd�

⎞⎟⎠�λN A

⎤⎥⎦

+⎡⎢⎣∑

A,act.

δλN A

⎛⎜⎝∫

�c0

RANT d�

⎞⎟⎠

⎤⎥⎦�u

+∑

A,inact.

(− 1

εNAA�λN AδλN A

)(57)

By introducing the additional vectors

Lλ =⎡⎢⎣

Lλ1...

Lλncp

⎤⎥⎦ Aλ =

⎡⎢⎣

Aλ1...

Aλncp

⎤⎥⎦ (58)

whose terms are given by

LλA ={

RA, λN A ≤ 00, λN A > 0

AλA ={

0, λN A ≤ 0− AA

εN, λN A > 0

(59)

Eq. (57) can be rewritten as follows

�δWc,m = δuT

⎡⎢⎣∑

A,act.

εN∫�c0

RAd�

∫

�c0

RANd�

∫

�c0

RANT d�

⎤⎥⎦�u

+δuT[∫

�c0

NLTλ d�

]�λ + δλT

[∫

�c0

LλNT d�

]�u

+δλT AλATλ �λ (60)

Finally, the main component of the stiffness matrix, Km ,is readily obtained from Eq. (60) as

Km =[

Kuum Kuλ

m

Kλum Kλλ

m

](61)

123

Comput Mech (2012) 49:1–20 19

where the term

Kuum =

∑A,act.

⎡⎢⎣ εN∫

�c0

RAd�

∫

�c0

RANd�

∫

�c0

RANT d�

⎤⎥⎦ (62)

mutually connects the displacement degrees of freedom, thetwo terms

Kuλm =

∫

�c0

NLTλ d� Kλu

m =∫

�c0

LλNT d� (63)

connect the displacement and the Lagrange multipliersdegrees of freedom, and finally

Kλλm = AλAT

λ (64)

mutually connects the Lagrange multiplier degrees of free-dom. The integrals in Eqs. (62) and (63) are carried outnumerically as follows

Kuum =

∑A,act.

[εN∑

G P RAg wg jg

∑G P

RAg Ngwg jg

×∑G P

RAg NTg wg jg

](65)

Kuλm =

∑G P

NgLTλg

wg jg Kλum =

∑G P

Lλg NTg wg jg (66)

where “GP” indicates that the summation is extended tothe Gauss points, subscript g refers to the dependence onthe Gauss point coordinate, and wg and jg are respectivelythe weight and the jacobian associated to the g-th integrationpoint on �c0.

A.2 Geometric component

The geometric component in Eq. (55) can be rewritten as

�δWc,g =∑

A,act.

⎛⎜⎝λN A

∫

�c0

RA�δgN d�

⎞⎟⎠=

∫

�c0

λNint�δgN d�

(67)

where the linearization of Eq. (43)

�δgN A =∫�c0

RA�δgN d�∫�c0

RAd�(68)

and Eqs. (40) and (45) have been employed. CombiningEq. (67) with the matrix expression of �δgN in Eq. (38)yields

�δWc,g = δuT∫

�c0

λNint

×{−gN

[m11N1NT

1 + m12(

N1NT2 + N2NT

1

)

+m22N2NT2

]− D1NT

1 − D2NT2 − N1DT

1

−N2DT2 + k11D1DT

1

+k22D2DT2 + k12

(D1DT

2 + D2DT1

)}d��u

(69)

The geometric component of the stiffness matrix results asfollows

Kg =∫

�c0

λNint

{−gN

[m11N1NT

1 +m12(

N1NT2 + N2NT

1

)

+m22N2NT2

]− D1NT

1 − D2NT2 − N1DT

1

−N2DT2 + k11D1DT

1

+k22D2DT2 + k12

(D1DT

2 + D2DT1

)}d� (70)

and mutually connects the displacement degrees of freedoms.Finally, integration is conducted numerically yielding the fol-lowing expression

Kg =∑G P

λNintg

×{−gNg

[m11

g N1gNT1g + m12

g

(N1gNT

2g + N2gNT1g

)

+m22g N2gNT

2g

]−D1gNT

1g −D2gNT2g −N1gDT

1g

−N2gDT2g + k11gD1gDT

1g + k22gD2gDT2g + k12g

×(

D1gDT2g + D2gDT

1g

)wg jg

}(71)

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