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Comput. Methods Appl. Mech. Engrg. 187 (2000) 69±77
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A solution method for general contact±impact problems
K. Farahani, M. Mo®d *, A. Vafai
Department of Civil Engineering, Sharif University of Technology, PO Box 11365-9313, Tehran, Iran
Received 6 October 1998
Abstract
In this paper, a solution method is presented for the analysis of contact/impact between deformable bodies. The method is di€erent
from Lagrange multipliers and Penalty methods. Having de®ned a body as contactor and the other as target, it is assumed that contact
forces are developed between contactor nodes and target surface. The method is based on sti€ness transformation and eliminating the
normal degree of freedom of contactor node. The method is absolutely general and can be used in static and dynamic nonlinear
problems. All the drawbacks of Lagrange multipliers and Penalty method are vanished in this technique and it can be easily implemented in computer programs. The eciency and applicability of the method is demonstrated by several illustrative examples. Ó 2000
Elsevier Science S.A. All rights reserved.
1. Introduction
The large amount of researches and e€orts devoted to contact/impact problems during the last two
decades reveals the importance of the phenomena. Various numerical methods and computational techniques have been proposed for the di€erent classes of the problem, involving di€erent nonlinearities due to
the material property, ®nite geometry changes, or friction e€ects. However, a contact problem is inherently
a nonlinear problem. The occurrence of contact can be at any location of contactor and target body and the
basic condition of contact is that no material penetration may take place between the contacting bodies.
Here, it is assumed that the contactor body contacts the target surface through its nodes. The present work
deals with the analysis of general three dimensional contact problems which is applicable to both static and
dynamic conditions, with all probable nonlinearities.
Lagrange Multipliers and Penalty methods are two widely used methods for treating contact/impact
problems.
In the Lagrange multiplier method [2±4], the contact forces are Lagrange parameters and contact
conditions are satis®ed exactly. However, increase in the number of unknowns which are displacements and
contact forces simultaneously, and change in the equation solution scheme are two disadvantages of the
method. In the Penalty method [5±7], the contact forces are proportional to the amount of penetration by
introducing a penalty number which is physically equivalent to an additional string between contacted
bodies. The accuracy of solution is strongly dependent on the penalty number [7] and boundary conditions
can only be approximately satis®ed.
However, there are other methods which have been investigated as alternative solution techniques for
contact problems and have shown good eciency in special cases [9±12]. Depending on the problem,
*
Corresponding author. Tel.: +98-21-600-5818; fax: +98-21-601-4828.
E-mail addresses: [email protected] (K. Farahani), [email protected] (M. Mo®d).
0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 1 1 0 - 3
70
K. Farahani et al. / Comput. Methods Appl. Mech. Engrg. 187 (2000) 69±77
choosing a suitable computational technique in nonlinear dynamics and a rate dependent or independent
constitutive law and a proper plasticity rule may a€ect the accuracy and eciency of the method [8].
The aim of this paper is to introduce a general numerical technique which can e€ectively direct the
motion of a contactor node on target surface to meet the contact conditions. The loss of accuracy, increase
in the number of unknowns, and modi®cation in the solution scheme, which are major drawbacks of
Penalty method and use of Lagrange Multipliers, are vanished in this method. The approach is based on
transformation of contactor node degrees of freedom (dof) according to the target surface, and elimination
of normal dof of the node. The contact forces are not computed directly. However, after the system is
solved, they can be achieved from equilibrium equations. The method is absolutely general and can be used
in static or dynamic analyses in presence of any kind of nonlinearities.
2. Contactor node and target surface
As shown in Fig. 1(a), consider a contactor element C from a body contacting the surface of a target
element T as part of a target body, through a contactor node P. Assume that the coordinates of node P on
the target surface ABCD are obtained from a contact detection algorithm for which there are several
methods. The condition is that there should be no overlap between two bodies. Since node P is on the target
surface, its coordinates at time t, that is …t xP1 ; t xP2 ; t xP3 †, can be explained in terms of target surface nodes
coordinates as
t P
xi
ˆ
nX
odes
jˆ1
N j t xi j ;
i ˆ 1; 2; 3;
…1†
where t xi j is the coordinates of node j of the surface in the direction i at time t. Nj Õs are shape functions
evaluated at node P. Assume that ~
n is normal to the surface at node P as shown in Fig. 1(b).
The constraint on the motion of node P is that as long as the reaction of node P on the surface is
compressional, it can not move along the normal ~
n. We adopt the degrees of freedom of node P in the
t2 , instead of the global coordinate
direction of normal ~
n (normal dof) and parallels to the surface ~
t1 and ~
system XYZ. This introduces simplicity in subsequent computations, namely the elimination of normal dof
of the contactor node, because the constraint is to be imposed on the normal dof. Fig. 2(a) and (b) show the
dof of P before and after transformation.
Depending on the problem, the plane ABCD may be considered ¯at. Therefore, normal ~
n can be taken
as constant on the surface. However, in problems with higher order ®nite elements or signi®cant geometrical changes, this is not a good approximation. Assuming a Lagrangian description of motion and
isoparametric ®nite elements we have
t
x i ˆ 0 x i ‡ t ui ;
…2†
Fig. 1. Contractor node and target element.
K. Farahani et al. / Comput. Methods Appl. Mech. Engrg. 187 (2000) 69±77
71
Fig. 2. Contractor node degrees of freedom: (a) before rotation; (b) after rotation.
t P
xi
ˆ 0 xPi ‡ t uPi ˆ
i
Xh Nj P …0 xji ‡ t uji † ;
…3†
j
Nj P ˆ Nj …nP ; gP ; fP †;
…4†
where left superscript shows the time, right superscript shows the node number and right subscript shows
the direction. ui Õs and xi Õs are displacements and nodal coordinates of target surface nodes in the direction i,
respectively. Also Nj Õs are shape functions, which are evaluated at node P through Eq. (4). To calculate nP ,
gP and fP at node P, the system of Eqs. (3) and (4) should be solved.
t2 and ~
nˆ ~
t1 ~
t2 . The
For the transformation of degrees of freedom of node P we need to introduce ~
t1 ;~
t2 ;~
n†, which de®ne a proper orthonormal local coordinate system, can be percalculations of vectors …~
t1 ;~
formed exactly. However, there are some approximate methods [3] for the calculation of ~
n which o€er more
convenience and enough accuracy. The local transformation matrix [a] is de®ned as:
2 D E3
~
2
3
t1
t11 t12 t13
6D E7
6
7
7
~ 7 6
…5†
‰aij Š ˆ 6
6 t2 7 ˆ 4 t21 t22 t23 5:
4 D E 5
n1 n2 n3
~
n
Matrix [a] transforms the incremental displacements of node P (i.e. uPi ) as
0
uP g
f~
uP g ˆ ‰aŠf~
…6†
or in component form
0
uPi ˆ
jˆ3
X
jˆ1
aij uPj ;
i ˆ 1; 2; 3;
…7†
P0 and f~
uP g are the rotated and unrotated incremental displacement vectors of contactor node P,
where ~
u
respectively. Therefore, the transformation matrix [A] is de®ned as
2
3
I1 0 0
…8†
‰Aij Š ˆ 4 0 ‰aŠ 0 5;
0 0 I2
where number of rows and columns of [A] is equal to the dof of contactor element C. I 1 and I 2 are appropriate identity matrices and [a], as de®ned in Eq. (5), is set in the position of node P. Therefore, we can
write
0
‰t K C Š ˆ ‰AŠ ‰t K C Š ‰AŠT
or in component form
XX
t C0
K ij ˆ
Air Ajs t K Crs
r
s
…9†
…10†
72
K. Farahani et al. / Comput. Methods Appl. Mech. Engrg. 187 (2000) 69±77
0
where ‰t K C Š and ‰t K C Š are the original and transformed tangent sti€ness matrices of contactor element C at
time t, respectively. The term tangent implies a general nonlinear problem. It is noted that the transformation process may be performed at the level of element or structure.
3. Uniting the contactor and target elements and eliminating the normal dof of the contactor node P
The movement of contactor node P must be compatible with the shape of target surface. To eliminate
0
t
the normal dof of node P, as shown in Fig. 2, the tangent sti€ness matrix of elements C and T (that is ‰ K C Š
t
and ‰ K T Š) can be united as one stiffness matrix. Because, as long as node P is in contact with element T, the
dof of elements T and C are not independent. That is the reason we would like to call the method the United
Elements. Since node P is on the surface ABCD shown in Fig. 2, the incremental displacement of node P in
n and the incremental displacements of the surface
direction ~
n, i.e. uPn , can be explained as a function of ~
j
nodes ui Õs as
n; uji †;
uPn ˆ F …~
…11†
where right superscript is the node and the right subscript shows the direction. It is noted that in writing
Eq. (11), all the incremental displacements are in®nitesimal.
Aftre the contact takes place, a constraint arises. That is, the contactor node must not penetrate the
target surface. A mathematical form of this constraint is the function F in Eq. (11) for in®nitesimal incremental displacements. The problem is how to impose Eq. (11) to the motion of the system. It can be
added to the system through a penalty number or use of Lagrange multipliers. However, there are other
ways, such as treating the elements T and C as one element. Writing the static equilibrium equations of both
elements simultaneously before contact, we have
"
0
t
‰ KC Š
0
0
‰ K TŠ
#(
t
uC
uT
)
(
ˆ
fC
fT
)
;
…12†
where u and f are the incremental displacements and unbalanced force vectors, respectively. The right
superscript show the corresponding element and the o€-diagonal terms are zero which implies that the two
elements are independent. However, after contact and imposing Eq. (11) nonzero o€-diagonals will be
generated. It is noted that since in a real nonlinear problem, the incremental displacements are ®nite, it is
preferred to use Eq. (11) to de®ne a single tangent sti€ness matrix for the elements T and C as a united
element. The concept of stiffness matrix helps to establish the stiffness matrix of the united element. That is,
K ij is the reaction at dof i due to unit displacement at dof j with other dof being restrained. Since the normal
dof is to be eliminated, it has neither independent movement nor can be restrained. However, in order to
avoid signi®cant changes in sti€ness matrix map and to avoid singularity, the o€-diagonals of united
sti€ness matrix corresponding to normal dof are set to zero. After the system is solved, uPn can not be
obtained from Eq. (11) because this equation is written for in®nitesimal incremental displacements.
However, uPn can approximately be achieved from the ®nal con®guration of target surface and displacements of node P parallel to the surface. Assuming that node P is still on the surface and the reaction of the
node is compressional, the position of node P on the target surface can be estimated from the ®nal geometry of the system or equilibrium equations. The accuracy of the solution is improved by performing
iterations to remove the unbalanced forces.
Example (1) manifests the details of this procedure clearly.
4. Illustrative examples
Example 1. Consider two 2D ®nite elements as shown in Fig. 3. Their tangent sti€ness matrices are t K 1 and
t 2
K , respectively, where left superscript shows the time and right superscript shows the element number.
K. Farahani et al. / Comput. Methods Appl. Mech. Engrg. 187 (2000) 69±77
73
Fig. 3. The united element method: (a) elements dof before contact; (b) elements dof after contact; (c) dof of the united element with
eliminated normal dof.
Fig. 3 (a) shows the elements dof and Fig. 3 (b) shows node numbers and rotated dof of the contactor node
number 6 according to the target surface 1±2. The nodes have only translational displacements. It is desired
to ®nd a single sti€ness matrix for the elements. To unite the elements, it is necessary to eliminate the
normal dof number 12 shown in Fig. 3(c).
The element shape functions are:
1
N1 ˆ …1 ‡ n†…1 ‡ g†;
4
1
N3 ˆ …1 ÿ n†…1 ÿ g†;
4
1
N2 ˆ …1 ÿ n†…1 ‡ g†;
4
1
N4 ˆ …1 ‡ n†…1 ÿ g†:
4
0
The rotated sti€ness matrix of element number 2 is t K 2 ˆ T t K 2 T T where
2
6
6
6
6
T ˆ6
6
6
6
4
3
1
7
7
7
7
7
7
7
7
5
1
cosa
sin a
ÿ sin a
cosa
..
.
1
^ From the de®nition of
and a is shown in Fig. 3(b). Let the sti€ness matrix of the united element be t K.
t^
sti€ness matrix, Kij is the reaction at dof i due to a unit displacement at dof j with other freedoms being
restrained. The important point is that since the normal dof number 12 shown in Fig. 3(c) is to be eliminated, it can not be restrained.
^ i1 . The reactions of supports are given in the ®rst
As an example Fig. 4 demonstrates how to calculate t K
t^
column of Kij given below.
^ can be obtained from the de®nition of t K
^ ij as
Thus the united sti€ness matrix t K
a
a
sin a; C2 ˆ cosa;
L
L
a
a
C3 ˆ 1 ÿ
sin a; C4 ˆ 1 ÿ
cosa;
L
L
C1 ˆ
74
K. Farahani et al. / Comput. Methods Appl. Mech. Engrg. 187 (2000) 69±77
^ ij .
Fig. 4. Calculation of t K
2
3
0
1
2
K11
‡ C12 K44
6 K 1 ‡ C1 C2 K 20
44
6 21
6 K 1 ‡ C C K 20
1 3
6 31
44
6 1
20
6 K41 ‡ C1 C4 K44
6
1
6
K51
6
1
6
K61
6
6
1
K
6
71
6
1
K81
6
6
20
6
C1 K14
6
20
6
C1 K24
6
20
6
C1 K34
6
6
0
6
6
20
6
C1 K54
6
20
6
C
K
1
64
6
20
4
C1 K74
20
C1 K84
1
20
K22
‡ C22 K44
1
20
K32 ‡ C1 C4 K44
1
20
K42
‡ C2 C4 K44
1
K52
1
K62
1
K72
1
K82
20
C2 K14
20
C2 K24
20
C2 K34
0
20
C2 K54
20
C2 K64
20
C2 K74
20
C2 K84
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
0
1
2
K33
‡ C32 K44
1
20
K43
‡ C3 C4 K44
1
K53
1
K63
1
K73
1
K83
20
C3 K14
20
C3 K24
20
C3 K34
0
20
C3 K54
20
C3 K64
20
C3 K74
20
C3 K84
1
20
K44
‡ C42 K44
1
K54
1
K64
1
K74
1
K84
20
C4 K14
20
C4 K24
20
C4 K34
0
20
C4 K54
20
C4 K64
20
C4 K74
20
C4 K84
1
K55
1
K65
1
K75
1
K85
0
0
0
0
0
0
0
0
1
K66
1
K76
1
K86
0
0
0
0
0
0
0
0
1
K77
1
K87
0
0
0
0
0
0
0
0
1
K88
0
0
0
0
0
0
0
0
0
2
K11
20
K21
20
K31
0
20
K51
20
K61
20
K71
20
K81
0
2
K22
20
K32
0
20
K52
20
K62
20
K72
20
K82
0
2
K33
0
20
K53
20
K63
20
K73
20
K83
0
2
K44
0
0
0
0
0
2
K55
20
K65
20
K75
20
K85
0
2
K66
20
K76
20
K86
0
2
K77
20
K87
0
2
K88
which is symmetric. As can be seen, the o€-diagonal members of dof 12 are intentionally set to zero.
Therefore, it is independent of other degrees of freedom. Since the diagonal member is not zero, the sti€ness
matrix is not singular. Therefore, the number of unknowns is not increased and the solution scheme does
not change. After the equilibrium equations are solved, the answer for normal dof is ignored and it is
computed from equilibrium equations or the displacement of dof number 11 and ®nal position of the target
surface 1±2. Fig. 5 depicts the approximate situation of contactor node as discussed above. The ®nal position of the target surface at time t ‡ Dt and the increment in dof number 11, u11 , are achieved after the
system is solved. The position of contactor node P, is obtained from the intersection point of a line parallel
to ~
n (normal to the target surface at time t), and target surface at time t ‡ Dt.
Fig. 5. Approximate situation of contactor node P.
K. Farahani et al. / Comput. Methods Appl. Mech. Engrg. 187 (2000) 69±77
75
The process of uniting can be performed at the level of structure instead of element to minimise the
numerical e€ort and ease of computer programming. Also the procedure can be employed for any number
of contactor nodes.
Example 2. Two bodies shown in Fig. 6 are pushed against each other. The Total Lagrangian (TL) formulation is used for geometric nonlinearities. When using the TL formulation, special attention must be
paid to large strains, and it is based on ignoring the second and third order terms of incremental displacement in virtual work equations [13,14]. The bodies are discretized to 4 node plane strain elements. The
contact is static and frictionless.
As shown in Fig. 6 the initial position of contactor nodes are at the middle of target elements and sliding
of contactor nodes has occurred without friction.
Example 3. This example shows a special case of contact-impact problem, that is deformable-rigid case. As
demonstrated in Fig. 7, a ring collides an inclined frictionless rigid surface. The ring is built from 16 solid
elements and the inner and outer radii are 20 and 30, respectively. The normal reaction of rigid surface is
calculated based on linear theory and also including geometric nonlinearity. The TL formulation is used
and the contact-impact is frictionless. According to the united element procedure, the degrees of freedom of
contactor node must rotate according to the rigid surface. The dof normal to the rigid surface must be
restrained by setting all the o€-diagonal members of this dof equal to zero [1]. After the solution of dynamic
equilibrium equations, the incremental displacement of normal dof must be set to zero, because no penetration must take place. In Fig. 7, V is initial velocity and M is mass and subscripts show the direction.
E and m are modulus of elasticity and PoissonÕs ratio, respectively.
Fig. 6. Static contact with united element method.
Fig. 7. Deformable±rigid, contact±impact problem.
76
K. Farahani et al. / Comput. Methods Appl. Mech. Engrg. 187 (2000) 69±77
Fig. 8. Reaction of rigid surface.
Fig. 8 shows the normal reaction of the rigid surface with and without geometric nonlinear e€ects.
The comparison between the eciency of the methods reveals that for linear analysis of the above example, the united element method is approximately 1.7 times faster than the Lagrange multipliers method.
5. Conclusions
A numerical method is presented in detail for body to body contact/impact problems. The method is
absolutely general and can be used with all kinds of nonlinearities. It has not the drawbacks of Lagrange
multipiers family and Penalty methods. That is, the number of unknowns does not increase and the solution
scheme needs no modi®cation. Besides, the boundary conditions are satis®ed exactly. However, it needs
some matrix transformation and modi®cation in the structure or elements sti€ness matrix. The idea is to
unite the contactor and target elements. Illustrative examples are given to demonstrate the capability and
eciency of the method.
Acknowledgements
Special thanks to Dr. S. Vahdani from Department of Civil Engineering, Tehran University, Tehran,
Iran, for his useful discussions and considerations.
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