Displacement Control
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>U </math> \(<math>\Delta U \text{max}</math>></strong></p></td> </tr> </tbody> </table> <hr /> <table> <tbody> <tr class="odd"> <td><p><code class="parameter-table-variable">node</code></p></td> <td><p>node whose response controls solution</p></td> </tr> <tr class="even"> <td><p><code class="parameter-table-variable">dof</code></p></td> <td><p>degree of freedom at the node, valid options: 1 through ndf at node.</p></td> </tr> <tr class="odd"> <td><p><code class="parameter-table-variable">incr</code></p></td> <td><p>first displacement increment <math>\Delta U_{\text{dof}}</math></p></td> </tr> <tr class="even"> <td><p><code class="parameter-table-variable">numIter</code></p></td> <td><p>the number of iterations the user would like to occur in the solution algorithm. Optional, default = 1.0.</p></td> </tr> <tr class="odd"> <td><p><strong>\)<math>U </math> |
the min stepsize the user will allow. optional, defualt = <math>U_{min} = U_0</math> |
$<math>U </math> |
the max stepsize the user will allow. optional, default = <math>U_{max} = U_0</math> |
Examples
integrator DisplacementControl 1 2 0.1; # displacement control algorithm seking constant increment of 0.1 at node 1 at 2’nd dof.
Theory
If we write the governing finite element equation at <math>t + t!</math>as:
\[ R(U_{t+\Delta t}, \lambda_{t+\Delta t}) = \lambda_{t+\Delta t} F^{ext} - F(U_{t+\Delta t}) \!\]
where <math>F(U_{t+t})!</math> are the internal forces which are a function of the displacements <math>U_{t+ t}!</math>, <math>F^{ext}!</math> is the set of reference loads and <math>!</math> is the load multiplier. Linearizing the equation results in:
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\[K_{t+\Delta t}^{*i} \Delta U_{t+\Delta t}^{i+1} = \left ( \lambda^i_{t+\Delta t} + \Delta \lambda^i \right ) F^{ext} - F(U_{t+\Delta t})\]
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 \(\text{dof}\) at the specified node is:
\[ \Delta U_\text{dof} = \text{incr}\!\]
MORE TO COME:
In Displacement Control the \(\Delta_U\text{dof}\) set to <math>t + _{t+1}</math> where,
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\[\Delta U_\text{dof}^{t+1} = \max \left ( \Delta U_{min}, \min \left ( \Delta U_\text{max}, \frac{\text{numIter}}{\text{lastNumIter}} \Delta U_\text{dof}^{t} \right ) \right ) \]
Code Developed by: fmk