DisplacementControl Command
---------------------------

This command is used to construct a DisplacementControl integrator object. In an analysis step with Displacement Control we seek to determine the time step that will result in a displacement increment for a particular degree-of-freedom at a node to be a prescribed value.


.. function:: integrator DisplacementControl $node $dof $incr <$numIter :math`\Delta U \text{min}` $:math:\Delta U \text{max}`>


.. csv-table:: 
   :header: "Argument", "Type", "Description"
   :widths: 10, 10, 40

   $node, |integer|, node whose response controls solution

   $dof, |integer|, degree of freedom at the node; valid options: 1 through ndf at node.
   $incr, |float|, first displacement increment <math>\Delta U_{\text{dof}}</math>
   $numIter, |integer|, the number of iterations the user would like to occur in the solution algorithm. Optional; default = 1.0.
   $dUmin, |float|,   the min step size the user will allow. optional; default :math:`\Delta U_{min} = \Delta U_0`
   $dUmax, |float|, the max step size the user will allow. optional: default :math:`\Delta U_{max} = \Delta U_0`
   
:math:`f(x_n+\Delta x) = 0`



integrator DisplacementControl 1 2 0.1; # displacement control algorithm seeking constant increment of 0.1 at node 1 at 2'nd dof.


Theory
=======

If we write the governing finite element equation at :math:`t + \Delta t\` as:

.. math::

    R(U_{t+\Delta t}, \lambda_{t+\Delta t}) = \lambda_{t+\Delta t} F^{ext} - F(U_{t+\Delta t}) \!`

where :math:`F(U_{t+\Delta t})\!` are the internal forces which are a function of the displacements :math:`U_{t+\Delta t}\!`, :math:`F^{ext}\!` is the set of reference loads and :math:`\lambda\!` is the load multiplier. Linearizing the equation results in:

.. math::

   K_{t+\Delta t}^{*i} \Delta U_{t+\Delta t}^{i+1} = \left ( \lambda^i_{t+\Delta t} + \Delta \lambda^i \right ) F^{ext} - F(U_{t+\Delta t})

This equation represents n equations in :math:`n+1`` unknowns, and so an additional equation is needed to solve the equation. For displacement control, we introduce a new constraint equation in which in each analysis step we set to ensure that the displacement increment for the degree-of-freedom <math>\text{dof}</math> at the specified node is:

.. math::

   \Delta U_\text{dof} = \text{incr}\!</math>

MORE TO COME:


In Displacement Control the :math:`\Delta_U\text{dof}` set to :math:`t + \lambda_{t+1}` where,


:math:`\Delta U_\text{dof}^{t+1} = \max \left ( \Delta U_{min}, \min \left ( \Delta U_\text{max}, \frac{\text{numIter}}{\text{lastNumIter}} \Delta U_\text{dof}^{t} \right ) \right )`