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

Summarize

In the next table are presented the different opctions available in order to solve the structural elements as truss, beam, etc.


Linearization Type Dof .. Solver
Newton-Raphson Dynamic Displacement structural_solver_dynamic
Newton-Raphson Dynamic Displacement and rotation structural_solver_dynamic_rotation
Newton-Raphson Static Displacement structural_solver_relaxation
Newton-Raphson Static Displacement and rotation structural_solver_relaxation_rotation
LineSearch Dyamic Displacement structural_solver_dynamic_general
LineSearch Dynamic Displacement and rotation structural_solver_dynamic_rotation_general
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