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}`>

$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. $:math:Delta U text{min}, float, the min step size the user will allow. optional: default :math`= Delta U_{min} = Delta U_0` $<math>Delta U text{max}</math>, float, the max step size the user will allow. optional: default \(= \Delta U_{max} = \Delta U_0\)

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 <math> n+1</math> 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 <math>Delta_Utext{dof}</math> set to <math>t + lambda_{t+1}</math> 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 ) </math>