Displacement Control

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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 $minLambda $maxLambda>

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$node	node whose response controls solution
$dof	degree of freedom at the node, valid options: 1 through ndf at node.
$incr	first displacement increment <math>\Delta U_{\text{dof}}</math>
$numIter	the number of iterations the user would like to occur in the solution algorithm. Optional, default = 1.0.
$min \Delta_U	the min stepsize the user will allow. optional, defualt = <math>\Delta U_{min} = \Delta U_0</math>
$max \Delta_U	the max stepsize the user will allow. optional, default = <math>\Delta U_{max} = \Delta U_0</math>

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EXAMPLE:

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

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THEORY:

If we write the governing finite element equation at <math>t + \Delta t\!</math>as:

<math> R(U_{t+\Delta t}, \lambda_{t+\Delta t}) = \lambda_{t+\Delta t} F^{ext} - F(U_{t+\Delta t}) \!</math>

where <math>F(U_{t+\Delta t})\!</math> are the internal forces which are a function of the displacements <math>U_{t+\Delta t}\!</math>, <math>F^{ext}\!</math> is the set of reference loads and <math>\lambda\!</math> 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})</math>

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:

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

MORE TO COME:

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

<math> \Delta^U\text{dof} {t+1} = \max \left ( \DeltaU_{min}, \min \left ( \Delta_U\text{dof}_{max}, \frac{\text{numIter}}{\text{lastNumIter}} \Delta_U\text{dof}^{t} \right ) \right ) </math>

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Code Developed by: fmk