Displacement Control
In an analysis step with Displacement Control we seek to determine the time step \(\Delta \lambda\) 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 $DeltaUmin$ $DeltaUmax$ >|
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node whose response controls solution |
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degree of freedom at the node, valid options: 1 through |
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first displacement increment \(\Delta U_{\text{dof}}\) |
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the number of iterations the user would like to occur in the solution algorithm. Optional, default = 1.0. |
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\(\Delta U\text{min}\) |
the min stepsize the user will allow. optional, defualt = \(\Delta U_{min} = \Delta U_0\) |
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\(\Delta U \text{max}\) |
the max stepsize the user will allow. optional, default = \(\Delta U_{max} = \Delta U_0\) |
Examples
displacement control algorithm seking constant increment of 0.1 at node 1 at 2’nd dof.
integrator DisplacementControl 1 2 0.1; {"integrator": ["DisplacementControl", 1, 2, 0.1]}Theory
If we write the governing finite element equation at \(t + \Delta t\) as:
\[ R(\boldsymbol{u}_{n}, \lambda_{n}) = \lambda_{t+\Delta t} \boldsymbol{p}_f - \boldsymbol{p}_{\sigma}(\boldsymbol{u}_{n}) \]
where ({}({n})) are the internal forces which are a function of the displacements (_{n}), (_f) is the set of reference loads and () is the load multiplier. Linearizing the equation results in:
\[ \mathbf{K}_{n}^{*i} \Delta \mathbf{u}_{n}^{i+1} = \left( \lambda^i_{n} + \Delta \lambda^i \right) \boldsymbol{p}_f - \boldsymbol{p}_{\sigma}(\boldsymbol{u}_{n}) \]
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:
\[ \Delta u_\text{dof} = \text{incr} \]
Incrementation
In Displacement Control \(\Delta u_{\text{dof}}\) is incremented at \(t +\Delta t\) to
\[ \Delta u_\text{dof}^{t+1} = \max \left( \Delta U_{\text{min}}, \min \left( \Delta U_{\text{max}}, \frac{\text{numIter}}{\text{lastNumIter}} \Delta U_\text{dof}^{t} \right) \right) \]
Code Developed by: fmk