Limited Displacement
In some physics interfaces, displacements can be prescribed not only to a certain value, but also to move freely within given limits. This is specified using the Limited option in the Prescribed Displacement or Prescribed Displacement/Rotation nodes.
The limit conditions are implemented as a weak inequality constraint. For each displacement component ui, a gap distance is computed as
(3-158)
where u0i,max and u0i,min are the maximum and minimum limits. Given Equation 3-158, the weak inequality constraint is formulated as
(3-159)
and is subjected to the Kuhn-Tucker conditions
where fi is the constraint force (or contact reaction force).
Using the penalty method to regularize the constraint, the contact reaction force is defined as
where kp is the penalty factor. With the above definition of fi, Equation 3-159 is added as a weak contribution to the model to implement the constraint.
The augmented Lagrangian implementation of the inequality constraint is based on the following augmentation of Equation 3-159
(3-160)
where flm,i is a Lagrange multiplier that is added as an extra degree of freedom to the model. The penalized contact reaction force fi is for the augmented Lagrangian method defined as
and the constraint is implemented by adding the weak contributions defined in Equation 3-160 to the model.