3.2.6.2. 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.

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

Argument	Type	Description
$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 $\Delta U_{min} = \Delta U_0$
$dUmax	float	the max step size the user will allow. optional: default $\Delta U_{max} = \Delta U_0$

$f(x_n+\Delta x) = 0$

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

3.2.6.2.1. Theory¶

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

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

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

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

This equation represents n equations in $n+1$ 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:

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

MORE TO COME:

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

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