Open Journal of Applied Sciences, 2013, 3, 15-20
doi:10.4236/ojapps.2013.31B1004 Published Online April 2013 (
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
Received 2013
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-
Copyright © 2013 SciRes. OJAppS
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
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-
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
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.
Copyright © 2013 SciRes. OJAppS
3.1. Construction of Kinematics and Numerical
Algorithms for Various Contact Pairs
riant form for various geo-
) procedures. Thus, fundamental
ection routines leads to “projection do-
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
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
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-
Copyright © 2013 SciRes. OJAppS
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 prefinement 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
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
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
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
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
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
Copyright © 2013 SciRes. OJAppS
The Coulomb friction law for tangential interaction and
the Teresa friction law for rotational interaction are eas-
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
A set of contact points (integration points) is set on
the curve
sidered then in the Serret-Frenet curve coordinate sys-
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.
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
dragging relative velocity
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-
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
Copyright © 2013 SciRes. OJAppS
Copyright © 2013 SciRes. OJAppS
rs Involving Anisotropic Fric-
tion,” Revue Europeenne des Elements Finis, Vol. 4, 1995
pp. 183-207.
[2] M. E. Gurtin, che, “A General
[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.
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.
[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.
[5] R. E. Jones and P. Papadopoulos, “A Geometric Interpre-
tation of Frictional Contact Mechanics,” Zeitschriftf¨
2006, pp. 1025-1041.
[6] A. Konyukhov and K.
atik und Physik, Vol. 57, No. 6,
Schweizerhof, “Contact Formula-
tion Via a Velocity Description Allowing Efficiency Im-
provements in Frictionless Contact Analysis,” Computa-
tional Mechanics, Vol. 33, No. 3, 2004, pp. 165-173.
[7] A. Konyukhov and K. Schweizerhof, “Covariant Descrip-
tion for Frictional Contact Problems,” Computational
Mechanics, Vol. 35, No. 3, 2005, pp. 190-213.
[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
[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.
[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.
[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.
[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,