Arc-Length Control

This command is used to construct an ArcLength integrator object. In an analysis step with ArcLength we seek to determine the time step that will result in our constraint equation being satisfied.

integrator ArcLength $s $alpha

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s	\(s\) the arcLength.
alpha	\(\alpha\) a scaling factor on the reference loads.

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Examples

integrator ArcLength 1.0 0.1;

{"integrator": ["ArcLength", 1.0, 0.1]}

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Theory

If we write the governing finite element equation at \(t + \Delta t\) as:

\[ \boldsymbol{r}(\boldsymbol{u}_{n}, \lambda_{n}) = \lambda_{n} \boldsymbol{p}_f - \boldsymbol{p}_{\sigma}(\boldsymbol{u}_{n}) \]

where \(\boldsymbol{p}_{\sigma}(\boldsymbol{u}_{n})\) are the internal forces which are a function of the displacements \(\boldsymbol{u}_{n}\), \(\boldsymbol{p}_f\) is the set of reference applied loads and \(\lambda\) is the load multiplier. Linearizing the equation results in:

\[ \boldsymbol{K}_{n}^{*i} \Delta \boldsymbol{u}_{n}^{i+1} = \left( \lambda^i_{n} + \Delta \lambda^i \right) \boldsymbol{p}_f - \boldsymbol{p}_{\sigma}(\boldsymbol{u}_{n}) \]

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 dof at the specified node is:

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On the first equilibrium iteration within a load step, both arc length integrators compute the change in load factor, \(\Delta \lambda^1\), from the “tangent displacements”, \(\mathbf{U}_{f T}=\mathbf{K}_T \backslash \mathbf{P}_{r e f}\) , and the user specified arc length and \(\alpha\). \[ \Delta \lambda^1=\sqrt{\frac{s^2}{\mathbf{U}_{f T} \cdot \mathbf{U}_{f T}+\alpha^2}} \]

Code Developed by: fmk