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.
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:
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 .... .... ....