Arc-Length Control

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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	<math>s</math> the arcLength.
$alpha	<math>\alpha</math> a scaling factor on the reference loads.

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EXAMPLE:

integrator ArcLength 1.0 0.1;

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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> \!</math>

MORE TO COME:

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Code Developed by: fmk