# Dymamic

## Time integration

The temporal integration of the governing equation is carried out using the Newmark-Bossaq method. The equation integrated in time gives

Last equation together with the Newmark-Bossaq formulae for the acceleration and velocity

defines a non-linear system.

The stability conditions are satisfied at ; ;

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 and the tangent stiffnesses need to be established.

By definition \ the tangent stiffnes is the derivative of the residual with respect to the primary variable

The Newton method can be summarized as follows:

Solve for

update

Go to 1 until convergence in

where 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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