Dymamic

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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 .... .... ....

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