In this section the theoretical background of the keyword *COUPLING followed by *KINEMATIC or *DISTRIBUTING is covered, and not the keyword DISTRIBUTING COUPLING.

Coupling constraints generally lead to nonlinear equations. In linear calculations (without the parameter NLGEOM on the *STEP card) these equations are linearized once and solved. In nonlinear calculations, iterations are performed in each of which the equations are linearized at the momentary solution point until convergence.

Coupling constraints apply to all nodes of a surface given by the user. In a kinematic coupling constraint by the user specified degrees of freedom in these nodes follow the rigid body motion about a reference point (also given by the user). In CalculiX the rigid body equations elaborated in section 3.5 of [18] are implemented. Since CalculiX does not have internal rotational degrees of freedom, the translational degrees of freedom of an extra node (rotational node) are used for that purpose, cf. *RIGID BODY. Therefore, in the case of kinematic coupling the following equations are set up:

- 3 equations connecting the rotational degrees of freedom of the reference node to the translational degrees of freedom of an extra rotational node.
- per node belonging to the surface at stake, for each degree of freedom specified by the user (maximum 3) a rigid body equation.

This applies if no ORIENTATION was used on the *COUPLING card, i.e. the specified degrees of freedom apply to the global coordinate system. If an ORIENTATION parameter is used, the degrees of freedom apply in a local system. Then, the nodes belonging to the surface at stake (let us give them the numbers 1,2,3...) are duplicated (let us call these d1,d2,d3.....) and the following equations are set up:

- 3 equations connecting the rotational degrees of freedom of the reference node to the translational degrees of freedom of an extra rotational node.
- per duplicated node belonging to the surface at stake, a rigid body equation for each translational degree of freedom (i.e. 3 per duplicated node).
- per node an equation equating the by the user specified degrees of freedom in the local system (maximum 3) to the same ones in the duplicated nodes.

For distributing coupling the area weighted mean translations (all three of them) and the rotations specified by the user are connected to the corresponding translations/rotations of a reference node. In order to take the weighting into account, all nodes belonging to the surface (let us give them the numbers 1,2,3...) are duplicated (let us call these d1,d2,d3....). Now, the following equations are generated:

- 3 equations connecting the rotational degrees of freedom of the reference node to the translational degrees of freedom of an extra rotational node.
- for each node and for each translational degree of freedom in this node
an equation connecting this degree of freedom with the area-weighted
corresponding degree of freedom in the duplicated node. If there are n nodes
in the surface and the area corresponding to node i is then the
equation for the first node and the displacement in the global direction
would run like:
(142)

- 3 equations connecting the mean translation of the duplicated nodes to the translation of the
refererence node, e.g. for :
(143)

- as many equations as rotational degrees of freedom specified by the user
linking the mean rotation of the duplicated nodes to the rotation of the
reference node. For these equations the user is referred to section 3.6 in
[18].

If an ORIENTATION is applied to a distributing coupling definition the mean rotation equations are reformulated for the local rotation axis.