A node-to-face contact element consists of a slave node connected to a master
face (cf. Figure 114). Therefore, it consists of ![]() nodes, where
 nodes, where ![]() is the number of
nodes belonging to the master face. The stiffness matrix of a finite element
is the derivative of the internal forces in each of the nodes w.r.t. the
displacements of each of the nodes. Therefore, we need to determine the
internal force in the nodes.
 is the number of
nodes belonging to the master face. The stiffness matrix of a finite element
is the derivative of the internal forces in each of the nodes w.r.t. the
displacements of each of the nodes. Therefore, we need to determine the
internal force in the nodes.
Denoting the position of the slave node by 
![]() and the position
of the projection onto the master face by
 and the position
of the projection onto the master face by 
![]() , the vector
connecting both satisfies:
, the vector
connecting both satisfies:
The clearance ![]() at this position can be described by
 at this position can be described by
| (112) | 
where 
![]() is the local normal on the master
face. Denoting the nodes belonging to the master face by
 is the local normal on the master
face. Denoting the nodes belonging to the master face by
![]() and the local coordinates within the face by
 and the local coordinates within the face by ![]() and
and ![]() , one can write:
, one can write:
and
| (115) | 
![]() is the Jacobian vector on the surface. The internal
force on node
 is the Jacobian vector on the surface. The internal
force on node ![]() is now given by
 is now given by
where ![]() is the pressure versus clearance function selected by the
user and
 is the pressure versus clearance function selected by the
user and
![]() is the slave area for which node
 is the slave area for which node ![]() is representative. If the slave
node belongs to
 is representative. If the slave
node belongs to ![]() contact slave faces
 contact slave faces ![]() with area
 with area ![]() , this area may be calculated as:
, this area may be calculated as:
|  | (117) | 
The minus sign in Equation (116) stems from the fact that the
internal force is minus the external force (the external force is the force
the master face exerts on the slave node). Replacing the normal in  Equation
(116) by the Jacobian vector devided by its norm and taking the
derivative w.r.t. 
![]() , where
, where ![]() can be the slave node or any node belonging
to the master face one obtains:
 can be the slave node or any node belonging
to the master face one obtains:
|  | (118) | 
Since
|  | (119) | 
the above equation can be rewritten as
Consequently, the derivatives which are left to be determined are
![]() ,
, 
![]() and
 and 
![]() .
.
The derivative of 
![]() is obtained by considering Equation
(114), which can also be written as:
 is obtained by considering Equation
(114), which can also be written as:
| ![$\displaystyle \boldsymbol{m}= \sum_j \sum_k \frac{\partial \varphi _j}{\partial...
...partial \varphi_k}{\partial \eta } [ \boldsymbol{q_j} \times \boldsymbol{q_k}].$](img673.png) | (121) | 
Derivation yields (notice that ![]() and
 and ![]() are a function of
 are a function of
![]() , and that
, and that 
![]() )  :
)  :
|  | ![$\displaystyle \left [ \frac{\partial^2 \boldsymbol{q} }{\partial \xi^2} \times ...
...partial \eta } \right] \otimes \frac{\partial \xi }{\partial \boldsymbol{u_i} }$](img676.png) | |
| ![$\displaystyle \left[ \frac{\partial \boldsymbol{q} }{\partial \xi } \times \fra...
...partial \eta} \right] \otimes \frac{\partial \eta }{\partial \boldsymbol{u_i} }$](img677.png) | ||
| ![$\displaystyle \sum_{j=1}^{n_m} \sum_{k=1}^{n_m} \left[ \frac{\partial \varphi_j...
...right] (\boldsymbol{I} \times \boldsymbol{q_k}) \delta_{ij}, \:\:\: i=1,..n_m;p$](img678.png) | (122) | 
The derivatives 
![]() and
 and
![]() on the right hand side are
unknown and will be determined later on. They represent the change of
 on the right hand side are
unknown and will be determined later on. They represent the change of ![]() and
 and ![]() whenever any of the
 whenever any of the 
![]() is changed, k being the
slave node or any of the nodes belonging to the master face. Recall that the
value of
 is changed, k being the
slave node or any of the nodes belonging to the master face. Recall that the
value of ![]() and
 and ![]() is obtained by orthogonal projection of the slave
node on the master face.
 is obtained by orthogonal projection of the slave
node on the master face. 
Combining Equations (111) and (113) to obtain 
![]() ,
the derivative w.r.t.
,
the derivative w.r.t. 
![]() can be written as:
 can be written as:
| ![$\displaystyle \frac{\partial \boldsymbol{r} }{\partial \boldsymbol{u_i} }= \del...
...rtial \boldsymbol{u_i} } + \varphi_i (1 - \delta _{ip}) \boldsymbol{I} \right],$](img682.png) | (123) | 
where ![]() represents the slave node.
 represents the slave node.
Finally, the derivative of the norm of a vector can be written as a function of the derivative of the vector itself:
|  | (124) | 
The only derivatives left to determine are the derivatives of ![]() and
 and ![]() w.r.t.
 w.r.t. 
![]() . Requiring that
. Requiring that 
![]() is the orthogonal
projection of
 is the orthogonal
projection of 
![]() onto the master face is equivalent to
expressing that the connecting vector
 onto the master face is equivalent to
expressing that the connecting vector 
![]() is orthogonal to the
vectors
 is orthogonal to the
vectors 
![]() and
 and 
![]() , which are tangent to the master surface.
, which are tangent to the master surface.
Now,
|  | (125) | 
can be rewritten as
|  | (126) | 
or
| ![$\displaystyle \left [\boldsymbol{p} - \sum_i \varphi_i (\xi ,\eta ) \boldsymbol...
...ft[ \sum_i \frac{\partial \varphi_i}{\partial \xi } \boldsymbol{q_i} \right]=0.$](img688.png) | (127) | 
Differentation of the above expression leads to
| ![$\displaystyle \left[ d \boldsymbol{p} - \sum_i \left( \frac{\partial \varphi_i}...
...ta + \varphi_i d \boldsymbol{q_i} \right) \right] \cdot \boldsymbol{q_{\xi }} +$](img689.png) | ||
| ![$\displaystyle \boldsymbol{r} \cdot \left[ \sum_i \left( \frac{\partial^2 \varph...
...\frac{\partial \varphi_i}{\partial \xi } d \boldsymbol{ q_i} \right) \right] =0$](img690.png) | (128) | 
where 
![]() is the derivative of
 is the derivative of 
![]() w.r.t.
w.r.t. ![]() . The above equation is equivalent to:
. The above equation is equivalent to:  
|  | (129) | 
One finally arrives at:
| ![$\displaystyle - \boldsymbol{q}_{\xi} \cdot d \boldsymbol{p} + \sum_i \left[ ( \...
...tial \varphi_i}{\partial \xi } \boldsymbol{r}) \cdot d \boldsymbol{q_i} \right]$](img695.png) | (130) | 
and similarly for the tangent in ![]() -direction:
-direction:
| ![$\displaystyle - \boldsymbol{q}_{\eta} \cdot d \boldsymbol{p} + \sum_i \left[ ( ...
...ial \varphi_i}{\partial \eta } \boldsymbol{r}) \cdot d \boldsymbol{q_i} \right]$](img697.png) | (131) | 
From this 
![]() ,
, 
![]() and so on can be determined. Indeed, suppose that all
 and so on can be determined. Indeed, suppose that all
![]() and
 and  
![]() . Then, the
right hand side of the above equations reduces to
. Then, the
right hand side of the above equations reduces to 
![]() and
 and 
![]() and one ends up with two equations in the two unknowns
 and one ends up with two equations in the two unknowns 
![]() and
 and 
![]() . Once
. Once 
![]() is determined one automatically obtains
 is determined one automatically obtains  
![]() since
 since
|  | (132) | 
and similarly for the other derivatives. This concludes the derivation of
![]() .
.  
Since
| (133) | 
one obtains:
for the derivatives of the forces in the master nodes.