# '''Dynamic'''

## Time integration

The temporal integration of the governing equation is carried out using the Newmark-Bossaq method. The equation integrated in time gives $\text{M } \left( 1 - \alpha \right) a_{n + 1} + \text{M} \alpha a_n + \text{C} v_{n + 1} + \text{K} u_{n + 1} = \text{F}_{n + 1}$

Last equation together with the Newmark-Bossaq formulae for the acceleration and velocity $v_{n + 1} = \left( \frac{\gamma}{\beta} \right) \frac{u_{n + 1} - u_n}{\Delta t} - \left( \frac{\gamma}{\beta} - 1 \right) v_n - \frac{\Delta t}{2} \left( \frac{\gamma}{\beta} - 2 \right) a_n$ $a_{n + 1} = \frac{u_{n + 1 -} u_n}{\beta \Delta t^2} - \frac{1}{\beta \Delta t} v_n - \left( \frac{1 - 2 \beta}{2 \beta} \right) a_n$

defines a non-linear system.

The stability conditions are satisfied at $\alpha \leqslant 1 / 2$ ; $\beta \geqslant \gamma / 2 \geqslant 1 / 4$ ; $\alpha + \gamma \geqq 1 / 4$

In order to solve this system there are two options to linearizate it:a)Newton and b) Line search method.

Newton Method

For this the residual $\mathbf{r}$ and the tangent stiffnesses $\mathbf{H}$ need to be established.

By definition \ the tangent stiffnes is the derivative of the residual with respect to the primary variable $\mathbf{H}= - \frac{\partial \mathbf{r}}{\partial \mathbf{d}}$

The Newton method can be summarized as follows:

Solve $\mathbf{H}d\mathbf{u}=\mathbf{r} \left( \mathbf{u}^i \right)$ for $\mathbf{u}$

update $\mathbf{u}^{i + 1} =\mathbf{u}^i + d\mathbf{u}^i$

Go to 1 until convergence in $d\mathbf{u}$

where $d\mathbf{u}$ is the displacement increment, n stands for the time step and i is a non-linear iteration index.

Line search method .... .... ....