Geometrically Exact Theory of Contact Interactions–Further Developments and Achievements

doi:10.4236/ojapps.2013.31B1004

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Open Journal of Applied Sciences, 2013, 3, 15-20

doi:10.4236/ojapps.2013.31B1004 Published Online April 2013 (http://www.scirp.org/journal/ojapps)

Geometrically Exact Theory of Contact Interactions

– Further Developments and Achievements

Alexander Konyukhov1, Karl Schweizerhof2

1Department of Mechanical Engineering, University of Nottingham-Ningbo China, Ningbo, People’s Republic of China

2Institute of Mechanics, Karlsruhe Institute of Technology, Karlsruhe, Germany

Email: [email protected], [email protected]

Received 2013

ABSTRACT

The focus of the current contribution is on the development of the unified geometrical formulation of contact algorithms

in a covariant form for various geometrical situations of contacting bodies leading to contact pairs: surface-to-surface,

line-to-surface, point-to-surface, line-to-line, point-to-line, point-to-point. The construction of the corresponding com-

putational contact algorithms are considered in accordance with the geometry of contact bodies in a covariant form.

These forms can be easily discredited within finite element methods independently of order of approximation and,

therefore, the result is straightforwardly applied within iso-geometric finite element methods. This ap proach is recently

became known as geometrically exact theory of contact interaction [10]. Application for contact between bodies with

iso- and anisotropic surface, for contact between cables and curvilinear beams as well as recent development for contact

between cables and bodies is straightforward. Recent developments include the improvement of the curve-to-surface

(deformable) contact algorithm.

Keywords: Contact Finite Element; Surface-to-Surface Contact; Beam-to-Beam Contact; Curve-to-Surface Contact;

Covarian t Approach; Geo metrically Exact Contact D es criptio n

1. Introduction

Computational contact mechanics has become a sepa-

rated branch of computational mechanics during the last

decades. A fairly large number of publications including

several monographs on computational contact mechanics

have been devoted to this development. Modelling of

contact interactions became fairly standard in numerous

finite element software packages available for engineers.

Various aspects of the numerical solution such enforce-

ment of contact conditions, possibility to apply high or-

der and iso-geometric type of approximation has been

considered. One of the important aspects, even though

being obvious for everyone, – the geometrical treatment

of the contact – is often remains hidden inside the com-

putational algorithm. Contact interaction from a geomet-

rical point of view can be seen as interaction between

deformable surfaces possessing various geometrical fea-

tures such as surfaces, edges and vertexes, therefore,

geometrical approaches can be exploited. During the last

ten years these approaches has formed a basis of the

geometrically exact theory of contact interaction, recently

published in monograph of Konyukhov and Schweizerhof

[10] by Springer.

Current contribution is aimed on the overview of this

theory with concentration on recent developments.

2. Geometrical Approaches in Computa-

tional Contact Mechanics

Only a very few publications are devoted to geometrical

issues of contact interaction aiming at the final computa-

tional models. Gurtin, Wiessmueller and Larche [2] (1998)

considered surface tractions on curvilinear interfaces

describing them from a geometrical point of view. Jones

and Papadopoulos [5] (2006) considered contact de-

scribing various mappings from the reference configura-

tion employing the Lie derivative. Laursen and Simo [12]

(1993) described some contact parameters via geometri-

cal surface parameters.Heegaard and Curnier [3] (1996)

considered geometrical properties of slip operators.

Consistent Linearization

The iterative solution of Newton type is a standard way

to obtain the solution in the computational contact me-

chanics. However, one of the difficult parts is to obtain

the full derivative of the functional which is necessary

for the fast Newton solver – this procedure is known as

linearization. Two approaches for linearization of the

final functional representing the work of contact tractio ns

can be distinguished in order to obtain co nsistent tangent

matrices. The direct approach follows the following se-

A. KONYUKHOV, K. SCHWEIZERHOF

quence: functional – discretization – linearization and the

covariant approach follows the rule: functional – lin-

earization – discretization.The direct approach, histori-

cally motivated by the development of the finite element

method, assumes that the discretization is then involved

in the process and the linearization is provided with re-

gard to the displacement vector u and, therefore, of the

discretized system. This leads to the final results con-

taining a set of approximation matrices: for surface to

surface contact it is described in Wriggers and Simo [16]

(1985), Parisch and Luebbing [14](1997), Peric and Owen

[15] (1992), Laursen and Simo[12] (1993), for anisot-

ropic friction in Alart and Heege [1 ] (199 5), for beam-to-

beam type contact in Zavarise and Wriggers [17](2000),

Litewka and Wriggers [13] (2002).The complexity in the

derivation for curved contact interfaces led to the use ofa

code containing an automatic derivation with mathe-

matical software, see Heegeand Alart [4] (1996), Krstu-

lovic-Opara,Wriggers and Korelc [11] (2002) and other

researchers.

Open questions and drawbacks of the direct approach

can be summarized as follows:

 A closed form for tangent matrices is available only

for linear approximations of surfaces.

 The structure of the derived matrices is very com-

plicated and often not transparent. There is no clear in-

terpretation of each part possible.

 A specification of complex contact interface laws

with properties explicitly depending on the surface ge-

ometry (e.g. arbitrary a ni s otropy) is not p ossible.

 A contact description of many geometrical features

(curved line-to-curved line, curved line-to-surface) is

almost not possible because of the necessity of convec-

tive surface coordinates.

The fully covariant approach, however, assumes only

a local coordinate system associated with the deformed

continuum (convective coordinates) and requires exten-

sive application of covariant operations (derivatives etc.).

This approach historically appeared with the considera-

tion of convective variables arising from the surface ap-

proximations directly for contact traction and displace-

ments: see Simoand Laursen and Simo [12] (1993). Two

convective variables1

, 2

in a surface covariant basis

are used as tangential measure.

This approach has many advantages:

 objectivity is straightforwardly observed because

the surface coordinates are used;

 geometrical interpretation of a measure – line on a

surface; geometrical interpretation of a line arized meas-

ure – relative tangent velocity of a contact point;

 the number of history variables is minimal (two for

surface interaction);

 A complex constitutive law for tangent interaction

can be easily formulated in a robust form for computa-

tion.

 Expressions for contact tangent matrices are by far

less complex within the fully covarian t approach than for

direct approach.

A fully covariant approach, though, is in tended for the

finite element method, but does not assume approxima-

tions from the beginning and it serves to describe all n-

ecessary for solution parameters based on the geometry

of the contacting bodies in the local coordinate system.

The method, however, requires a lot of preliminary tran-

sformations based on differential geometry of contacting

objects (surfaces or even curves) and extensive applica-

tion of the tensor analysis especially for differential op-

eration and linearization.

3. Development of the Geometrically Exact

Theory

The development of the ideas started from formulation of

contact algorithms in covariant form for non-frictional

contact in [6], then for frictional contact in [7]. The

basement of the geometrically exact theory of contact

interaction with more references is summarized in

monograph of Konyukhov and Schweizerhof [10]. In

order to formulate goals and describe the development of

geometrically exact theory we consider a model contact

problem with two bodies possessing smooth surfaces as

well as various geometrical features such as edges and

vertexes – an example of this is a banana and a knife

shown in Figure 1. Considering all possible geometrical

situations in which knife and banana can contact each

other, the following hierarchical sequence of contact

pairs is appearing:

1. Point to point contact pair

2. Point to curve contact pair

3. Point to surface contact pair

4. Curve to curve contact pair

5. Curve to surface contact pair

6. Surface to surface contact pair

Figure 1. Contact between bodies with complex geometry.

Various geometrical situations are possible: Surface-To-

Surface, Curve-To-Surface, Point-To-Curve, Curve-To-

Curve and Point-To-Point.

A. KONYUKHOV, K. SCHWEIZERHOF 17

3.1. Construction of Kinematics and Numerical

Algorithms for Various Contact Pairs

geom

riant form for various geo-

) procedures. Thus, fundamental

ection routines leads to “projection do-

conditions:

r independent of approxima-

ariant form for arbitrary

set of relative motions

h many publications, they can be sum-

nding to a

n “master-slave” contact al-

The following open problems are stated as goals for the

etrically exact theory

 Development of the unified geometrical formulation

of contact conditions in cova

metrical situations of contacting bodies leading to contact

pairs: surface-to-surface, curve-to-surface, point-

to-surface, curve-to-curve, point-to-curve, point-to-point

(joint). The description will be fully based on the differ-

ential geometry of specific features forming a continuum,

because it is carried out in the local coordinate systems

attached to this feature: this is the Gaussian surface coor-

dinate system in the case of surface; the Serret-Frenet

basis in the case of a curved line; the coordinate system

standard for rigid body rotation problem (e.g. via the

Euler angles) the case of a point in. This general de-

scription is forming a geometrically exact theory for

contact interaction.

 A full set of contact pairs requires various closest

point projection (CPP

problems of existence and uniqueness of closest point

projection routines corresponding to the following situa-

tions are investigated: point-to-surface, point-to-line,

line- to-line.

 A solution of existence and uniqueness problems of

closest point proj

mains” as the “maximal searching domains“.

 Derivation of a unified covariant description of

various applicable methods to enforce contact

Lagrange multipliers methods, penalty methods, aug-

mented Lagrange multipliers method. Consistent tangent

matrices are given in closed covariant form possessing a

clear geometrical structure.

 Description of all geometrical situations in a co-

variant form which is a-prio

tions of these geometrical features leads to straightfor-

ward numerical algorithms for the implementation with

any order of approximation for finite elements including

iso-geometric finite elements.

 Generalization of classical Coulomb law into a

complex interface laws in cov

geometry of the surfaces (e.g. coupled anisotropic fric-

tion and adhesion for surfaces).

 Development of the curve-to-curve contact model

allowing considering the co mplete

between curves including a rotational interaction (this is

a novel in the current theory and has not been possible in

earlier theories).

Though, the specific points of the proposed theory are

developed throug

arized under the unified aim, see more detail in mono-

graph [10]. In order to construct a numerical algorithm

for a certain contact pair, first of all, it is identified that

the closest distance between contacting bodies is a natu-

ral measure of the contact interaction. The procedure is

introduced via the closest point projection procedure

(CPP), solution of which requires the differentiability of

the function represen ting the parameterization of the sur-

face of the contacting body. Analysis of the solvability

for the CPP procedure, see more in [8], allows then to

classify all types of all possible contact pairs discussed

earlier. Starting with a consideration of C2-continuous

surfaces, the concept of the projection domain is intro-

duced as a domain from which any potential contact

point can be uniquely projected, and therefore, the nu-

merical contact algorithm can be further constructed.

This domain can be constructed for utmostC1-continuous

surfaces. If the surfaces contain edges and vertex then the

CPP procedure should be generalized in order to include

the projection onto edges and onto vertexes.

The main idea for application for the contact is then

straightforward – the CPPprocedure correspo

rtain geometrical feature gives a rise to a special, in

general, curvilinear 3D coordinate system. This coordi-

nate system is attached to a geometrical feature and its

convective coordinates are directly used for further defi-

nition of the contact measures. Thus, all contact pairs

listed earlier should be described in the corresponding

local coordinate system. The requirement of the exis-

tence for the generalized CPP procedure leads to the

transformation rule between types of contact pairs ac-

cording to which the corresponding coordinate system is

taken. Thus, the all contact pairs can be uniquely de-

scribed in most situations.

A surface-to-surface contact pair, see Figure 1, is de-

scribed via the well know

rithm based on the CPP procedure onto the surface.

This projection allows defining a coordinate system as

follows:









12123 12

,, ,rx xx xxnxx



 (1)





,rx x

Vector is a vector for the “slave” point,







is a parameterization of the “master” surface,





nx x

, is a normal vector to the surface. Equation (1)

in fact, a coordinate transformation in which

e coordinates 123

describes,

convectiv

are used for measure

of contact interaction: the first two 12

x are meas-

ures for the tangent internd the third coordinate

3action a

is a penetration – the measure of normal

on-penetrability condition. This transformation is valid

lied only if the solution of the corresponding surface

CPP procedure exists. Initially, the computational algo-

rithm is constructed for non-frictional contact interaction

of smooth surfaces. Here the description starts in the co-

ordinate system given in equation (1), however, due to

the small penetration it is mostly falling into the descrip-

app

A. KONYUKHOV, K. SCHWEIZERHOF

tion in the Gaussian surface coordinate system arising

from the surface parameterization. All contact parameters

such as sliding distance and tangent forces are described

then on the tangent plane. The linearization procedure is

given in a form of covariant derivatives. This leads to a

closed form of the tangent matrix subdivided into a main,

a rotational and a curvature parts. The evolution equation

for contact tangent tractions should be taken in a form of

covariant derivatives in order to solve the problem with a

Coulomb friction.

3.1.1. Developments within the Theory – Possibility of

the Iso-geometric Modeling

therefor stem and, of course,

ry – Possibility of

Anisotropic Contact Interfaces

from nisotropic region

If the projection onto the surface does not exist the

ce of low

Since all algorithms are formulated in covariant and,

e, independent on coordinate sy

on types of approximations, the high-order and iso-geo-

metric formulation is just straightforwardly applicable.

Both Mortar methods with penalty regularization and

with Lagrange multipliers are applied. Even the anisotropic

enrichment of the approximation keeping mixed linear

and high order approximation in one finite element is

possible. As a result a contact layer element allowing

anisotropic p−refinement is created. A good correlation

with the analytical Hertz problem is achieved even

within a single contact layer element.

3.1.2. Developments within the Theo

A systematic generalization of a contact interface law

the Coulomb friction law into the a

in a covariant form including various knownvisco-elasto-

plastic mechanical models is derived. Thus, a coupled

model including anisotropy for tangential adhesion and

for friction is obtained. These models formulated via the

principle of maximum dissipation in a rate form. Finally,

the computational model is derived via the application of

the return-mapping scheme to the incremental form. As a

result a frictional force is derived in a closed form in-

cluding both, the adhesion and the friction tensors. The

structure of structural and friction tensors are derived for

various types of anisotropy: a uniform orthotropic of a

plane given by the spectral decomposition, a non-uni-

form orthotropic of a plane inherited with the polar coor-

dinate system and a spiral orthotropic of a cylindrical

surface. The update algorithm for history variables is

developed for the arbitrary coupled anisotropy. The

geometrical interpretation of the return-mapping and the

update algorithm is considered via the ellipse on the tan-

gent plane onto which a contact slave point is projected

in the case of elastic sticking behaviour.

3.1.3. Curve - T o- Cur ve Contact Algoritn it is

necessary to consider step-by-step the existener

in hierarchy CPP i.e. onto the curve and then onto the

point. The solution of generalized CPP including all geo-

metrical exists in case of regular geometry. Consideration

of the existence of the CPP procedure for curve allows

defining then the point-to-curve contact algorithm used

for the curve-to-surface contact pair in the corresponding

curve Serret-Frenet coordinate system, which is con-

structed as follows:





,,() ,rsrs res





 (2)

Here, the vector





,,rsr



sis a param

ector des

is describing

from the surface, eterization o

a “slave” point

f the “master”

rve edge; a unit vcribing the shortest distance









,()cos ()sines ss





 is written via the unit

normal







and bi-normal ()



of the curve. The

sures: r – for normal

interacti – for tangential interaction;

convective coordis mea

on; s

nates used a



– for rota-

tional interaction. The Curve-To-Curve contact pair re-

quires the projection on both curves, therore, there is

no classical “master” and “slave” and both curves are

equivalent. For the description one of two coordinate

systems is taken assigned to the I-th curve:



21 111111

(,, )(),srsre s



 

 (3)

Here, the vector





21 1

,,sr





e second curve is a vector d

contact point of th,escribing a

is a pa

tio rameteriza-

n of the first curvtor describing the short-

est distance e; a unit vec





111

,es



is written agaivia the unit normal

and bi-normal vector s of the first curve as in equation (2).

Equation (3) es the motion of the second contact

point in the coordinate system attached to the first curve.

Description is symmetric with respect to the choice of the

curve choice 1 to 2.

The Point-To-Point contact pair is described then in a

coordinate system sta

describ

ndard for rigid body rotation prob-

the Closest Point Projection

n curves can be considered con-

m (e.g. via the Euler angels), however in the contact

situation is very seldom case, and in computations it is

rather improbable unless specially treated, and therefore,

because of the numerical rounding error would fall into

other contact pair types.

The construction of the curve-to-curve contact pair

begins consistently with

PP) procedure providing a shortest distance between

curves as a natural measure of normal contact interaction.

The CPP procedure leads to a special local coordinate

system in which convective coordinates are used directly

as measures of contact interaction between curves: nor-

mal, tangential and rotational. Several achievements appear

to be novel for the curve-to-curve contact description:

 consideration of any relative motion separately for

each curve is possible;

 Rotational interactions including corresponding ro-

tational moments betwee

sistently.

A. KONYUKHOV, K. SCHWEIZERHOF 19

The Coulomb friction law for tangential interaction and

the Teresa friction law for rotational interaction are eas-

ily

Contact Algorithm

duce-To-Surface contact algo-

segment: all kinematical parameters are con-

he surface;

ace coordinate system.

o-Surface con-

considered as examples for constitutive relations be-

tween curves. All necessary linearizations for the itera-

tive solution scheme are provided as covariant derivation

in the introduced coordinate system for arbitrary large

distances between curves. This leads to a closed form of

tangent matrices independent of the approximation used

for the finite elements. The verification of the algorithm

contains the comparison between beam-to-beam and edge-

to-edge finite element models as well as verification with

a famous “Equilibrium of Euler elastic problem” com-

puted via finite difference scheme see details in [9].

4. Further Development – Curve-To-Surfe

The Curve-To-Surface contact pair is constructed in a

al fashion via the Surfa

rithm if we consider a “slave”point on the curve and pro-

ject it onto the “master” surface, see Figure 2. This spe-

cial dual consideration of contact both in the surface co-

ordinate system in equation (1) and in the curve coordi-

nate system in equation (2) allows building the

Curve-To-Surface contact algorithm. In this algorithm all

contact parameters are defined, first, in the local surface

coordinate system equation (1) attached to the surface,

after fulfilling the surface CPP procedure, and then they

should be projected into the curve coordinate system

equation (2), see Figure 2. The kinematics of the

Curve-To-Surface contact interaction is formulated as

follows:

 A set of contact points (integration points) is set on

the curve

sidered then in the Serret-Frenet curve coordinate sys-

tem;

 The contact point (integration points) is projected

onto t

 At each point all kinematical parameters are con-

sidered in the surf

The combination of both Curve-To-Curve and Surface-

To-Surface strategies leads to the Curve-T

ct algorithm which is constructed as follows. The short-

est distance between integration points and the surfaces

Figure 2. Both a surface coordinate system and a cu

coordinate system are employ ed to define all characte

of the Curve-To-Surface contact pair.

rve

stics ri

are considered as penetration. Now the Closest Point

Projection (CPP) procedure as the projection onto the

surface plays the main role. In general, Newton method is

exploited to solve the CPP procedure defining then a

point on the surface and the penetration between this

surface and the selected contact (integration) point S.

Kinematical relations during the contact can be obtained

dually considering the relative velocity of the contact

point during contact:

• normal relative velocity during contact

(_ )vsv



 (4)

• pulling relative velocity

(



)









(5)

• dragging relative velocity

x()









(6)

The corresponding normal, pulling and dragging forces

are formulated in the curve Serret-

The overview of the geometrically exact theory of the

and recent development can be sum-

laws

Frenet coordinate sys-

m. The result of the linearization is taken as if provided

in the surface coordinate system to carry out analysis for

the deformed surface parameters and as if provided in the

curve Serret-Frenet coordinate system to apply for all

curve parameters.

5. Conclusions

contact interaction

marized as follows:

 Consideration of contact between bodies from geo-

metrical point of view allows to study systematically all

possible geometric contact cases: contact between sur-

faces, edges, beams;

 The basis of the theory is the formulation of all pa-

rameters in a local coordinate system inherited with a

corresponding closest point projectio n (CPP ) proce d ure;

 Surface-To-Surface contact pair is considered in the

surface coordinate system of the “master” body.

 Curve-To-Curve contact pair is considered equiva-

lently in both curveSerret-Frenet coordinate systems at-

tached to both curves. There is no specific choice of the

master and the slave in this case.

 A novel Curve-To-Surface is constructed dually in

both surface and curve coordinate system. Normal, pull-

ing and dragging velocities and corresponding forces are

specified in the curve coordinate system. Linearization

result should be transferred to both surface and curve

coordinate system.

 All known constitutive relations (for elasticity and

plasticity) can be carried into metrics giving a rise to a

new contact interface

A. KONYUKHOV, K. SCHWEIZERHOF

rs Involving Anisotropic Fric-

tion,” Revue Europeenne des Elements Finis, Vol. 4, 1995

pp. 183-207.

[2] M. E. Gurtin, che, “A General

418619808239977

REFERENCES

[1] P. Alart and A. Heege, “Consistent Tangent Matrices of

Curved Contact Operato,

J. Weissmueller and F. Lar

Theory of Curved Deformable Interfaces in Solids at

Equilibrium,” Philosophical Magazine A, Vol. 78, No. 5,

1998, pp. 1093-1109.

doi:10.1080/01

3-4, 1996, pp. 263-286.

[3] J. H. Heegaard and A. Curnier, “Geometric Properties of

2D and 3D Unilateral Large Slip Contact Opera-

tors,”Computer Methods in Applied Mechanics and En-

gineering, Vol. 131, No.

doi:10.1016/0045-7825(95)00977-9

15)39:1<165::AID-

[4] A. Heege and P. Alart, “A Frictional Cont act Element for

Strongly Curved Contact Problems,” International Jour-

nal for Numerical Methods in Engineering, Vol. 39, No. 1,

1996, pp. 165-184.

doi;10.1002/(SICI)1097-0207(199601

NME846>3.0.CO;2-Y

[5] R. E. Jones and P. Papadopoulos, “A Geometric Interpre-

tation of Frictional Contact Mechanics,” Zeitschriftf¨

urAngewandteMathem

2006, pp. 1025-1041.

[6] A. Konyukhov and K.

atik und Physik, Vol. 57, No. 6,

Schweizerhof, “Contact Formula-

3-0515-3

tion Via a Velocity Description Allowing Efficiency Im-

provements in Frictionless Contact Analysis,” Computa-

tional Mechanics, Vol. 33, No. 3, 2004, pp. 165-173.

doi:10.1007/s00466-00

[7] A. Konyukhov and K. Schweizerhof, “Covariant Descrip-

tion for Frictional Contact Problems,” Computational

Mechanics, Vol. 35, No. 3, 2005, pp. 190-213.

doi:10.1007/s00466-004-0616-7

[8] A. Konyukhov and K. Schweizerhof, “On the Solvability

, 2008, pp.

of Closest Point Projection Procedures in Contact Analy-

sis: Analysis and Solution Strategy for Surfaces of Arbi-

trary Geometry,” Computer Methods in Applied Mechan-

ics and Engineering, Vol. 197, No. 33-40

3045-3056.

[9] A. Konyukhov and K. Schweizerhof, “Geometrically

Exact Covariant Approach for Contact between Curves

Representing Beam and Cable Type Structures,” Com-

puter Methods in Applied Mechanics and Engineering,

Vol. 199, No. 37-40, 2010, pp. 2510-2531.

doi:10.1016/j.cma.2010.04.012

[10] A. Konyukhov and K. Schweizerhof, “Co

Contact Mechanics – Geometricmputational

ally Exact Theory for Ar-

r 3D Finite Deformation Fric-

bitrary Shaped Bodies,” Springer, Heidelberg, New York,

Dordrecht, London, 2012.

[11] L. Krstulovic-Opara, P. Wriggers and J. Korelc, “A C1

Continuous Formulation fo

tional Contact,” Computational Mechanics, Vol. 29, No.

1, 2002, pp. 27-42. doi:10.1007/s00466-002-0317-z

[12] T. A. Laursen and J. C. Simo, “A continuum-based Finite

Element Formulation for the Implicit Solution of Multi-

, Vol. 28, No. 1,

body Large Deformation Frictional Contact Problems,”

International Journal for Numerical Methods in Engi-

neering, Vol. 35, 1993, pp. 3451-3485.

[13] P. Litewka and P. Wriggers, “Frictional Contact between

3D Beams,” Computational Mechanics

2002, pp. 26-39. doi: 10.1007/s004660100266

[14] H. Parisch and C. Luebbing, “A Formulation of Arbitrar-

ily Shaped Surface Elements for Three-dimensional Large

Deformation Contact with Friction,” International Jour-

nal for Numerical Methods in Engineering, Vol. 40, No.

18, 1997, pp. 3359-3383.

doi:10.1002/(SICI)1097-0207(19970930)40:18<3359::AI

D-NME217>3.0.CO;2-5

[15] D. Peric and D. R. J. Owen, “Computational Model for

3D Contact Problems wi

th Frictionbased on the Penalty

Method,” International Journal for Numerical Methods in

Engineering, Vol. 35, No. 6, 1992, pp. 1289-1309.

doi:10.1002/nme.1620350609

[16] P. Wriggers and J. C. Simo, “A Note on Tangent Stiff

for Fully Nonlinear Contact Prness

oblems,” Communications

in Applied Numerical Methods, Vol. 1, No. 5, 1985, pp.

199-203. doi:10.1002/cnm.1630010503

[17] G. Zavarise and P. Wriggers, “Contact with Friction be-

tween Beams in 3D Space,” International Journal for

Numerical Methods in Engineering, Vol. 49, No. 8, 2000,

977-1006.

doi;10.1002/1097-0207(20001120)49:8<977::AID-NME9

86>3.0.CO;2-C