Arc-Length Control
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:
Code Developed by: fmk