A node-to-face contact element consists of a slave node connected to a master face (cf. Figure 130). Therefore, it consists of 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.
Denoting the position of the slave node by and the position of the projection onto the master face by , the vector connecting both satisfies:
The clearance at this position can be described by
where 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 , one can write:
is the Jacobian vector on the surface. The internal force on node is now given by
where is the pressure versus clearance function selected by the user and is the slave area for which node is representative. If the slave node belongs to contact slave faces with area , this area may be calculated as:
The minus sign in Equation (166) 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 (166) by the Jacobian vector devided by its norm and taking the derivative w.r.t. , where can be the slave node or any node belonging to the master face one obtains:
the above equation can be rewritten as
Consequently, the derivatives which are left to be determined are , and .
The derivative of is obtained by considering Equation (164), which can also be written as:
Derivation yields (notice that and are a function of , and that ) :
The derivatives and on the right hand side are unknown and will be determined later on. They represent the change of and 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 and is obtained by orthogonal projection of the slave node on the master face.
Combining Equations (161) and (163) to obtain , the derivative w.r.t. can be written as:
where 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:
The only derivatives left to determine are the derivatives of and w.r.t. . Requiring that is the orthogonal projection of onto the master face is equivalent to expressing that the connecting vector is orthogonal to the vectors and , which are tangent to the master surface.
can be rewritten as
Differentation of the above expression leads to
where is the derivative of w.r.t. . The above equation is equivalent to:
One finally arrives at:
and similarly for the tangent in -direction:
From this , and so on can be determined. Indeed, suppose that all and . Then, the right hand side of the above equations reduces to and and one ends up with two equations in the two unknowns and . Once is determined one automatically obtains since
and similarly for the other derivatives. This concludes the derivation of .
for the derivatives of the forces in the master nodes.