Transient Analysis

class xara.TransientAnalysis(model, integrator=None, system=None, test=None, numberer=None, hooks=None, constraints=None)

• Newmark
• Generalized Alpha
• TRBDF2

Theory

In a nonlinear transient finite element analysis we seek a solution (\(\boldsymbol{u}\), \(\dot{\boldsymbol{u}}\), \(\ddot{\boldsymbol{u}}\)) to the nonlinear residual equation

\[\boldsymbol{r}({\boldsymbol{u}},\dot{\boldsymbol{u}}, \ddot{\boldsymbol{u}}) = \boldsymbol{p}_f(t) - \boldsymbol{p}_{\mathrm{i}}(\ddot{\boldsymbol{u}}) - \boldsymbol{p}_{\sigma}({\boldsymbol{u}}, \dot{\boldsymbol{u}}) = \boldsymbol{0}\]

The most widely used technique for solving the transient non-linear finite element equation is to use an incremental direct integration scheme. In the incremental formulation, a solution to the equation is sought at successive time steps \(\Delta t\) apart.

\[\boldsymbol{r}({\boldsymbol{u}}_{n \Delta t},\dot{\boldsymbol{u}}_{n \Delta t}, \ddot{\boldsymbol{u}}_{n \Delta t}) = \boldsymbol{p}_f(n \Delta t) - \boldsymbol{p}_{\mathrm{i}}(\ddot{\boldsymbol{u}}_{n \Delta t}) - \boldsymbol{p}_{\sigma}({\boldsymbol{u}}_{n \Delta t}, \dot{\boldsymbol{u}}_{n \Delta t})\]

For each time step, \(t\), the integration schemes provide two operators, \(\operatorname{I}_1\) and \(\operatorname{I}_2\), to relate the velocity and accelerations at the time step as a function of the displacement at the time step and the response at previous time steps:

\[\dot {\boldsymbol{u}}_{t} = {\mathrm{I}}_1 ({\boldsymbol{u}}_t, {\boldsymbol{u}}_{t-\Delta t}, \dot {\boldsymbol{u}}_{t-\Delta t}, \ddot {\boldsymbol{u}}_{t - \Delta t}, {\boldsymbol{u}}_{t - 2\Delta t}, \dot {\boldsymbol{u}}_{t - 2 \Delta t}. ..., ) %\label{I1}\]

\[\ddot {\boldsymbol{u}}_{t} = {\mathrm{I}}_2 ({\boldsymbol{u}}_t, {\boldsymbol{u}}_{t-\Delta t}, \dot{\boldsymbol{u}}_{t-\Delta t}, \ddot{\boldsymbol{u}}_{t - \Delta t}, {\boldsymbol{u}}_{t - 2\Delta t}, \dot{\boldsymbol{u}}_{t - 2 \Delta t}. ..., ) %\label{I2}\]

These allow us to rewrite equation fullTimeForm, in terms of a single response quantity, typically the displacement:

\[\boldsymbol{r}({\boldsymbol{u}}_t) = \boldsymbol{p}_f(t) - \boldsymbol{p}_{\mathrm{i}}(\ddot{\boldsymbol{u}}_t) - \boldsymbol{p}_{\sigma}({\boldsymbol{u}}_t, \dot{\boldsymbol{u}}_t) %\label{genForm}\]

The solution of this equation is typically obtained using an iterative procedure, i.e. making an initial prediction for \({\boldsymbol{u}}_{t}\), denoted \({\boldsymbol{u}}_{t}^{(0)}\) a sequence of approximations \({\boldsymbol{u}}_{t}^{(i)}\), \(i=1,2, ..\) is obtained which converges (we hope) to the solution \({\boldsymbol{u}}_{t}\). The most frequently used iterative schemes, such as Newton-Raphson, modified Newton, and quasi Newton schemes, are based on a Taylor expansion of equation genForm about \({\boldsymbol{u}}_{t}\):

\[\boldsymbol{r}({\boldsymbol{u}}_{t}) = \boldsymbol{r}({\boldsymbol{u}}_{t}^{(i)}) + \left[ {\frac{\partial \boldsymbol{r}}{\partial {\boldsymbol{u}}_t} \vert}_{{\boldsymbol{u}}_{t}^{(i)}}\right] \left( {\boldsymbol{u}}_{t} - {\boldsymbol{u}}_{t}^{(i)} \right)\]

\[\boldsymbol{r}({\boldsymbol{u}}_{t}) = \boldsymbol{p}_f (t) - \boldsymbol{p}_{\mathrm{i}} \left( \ddot {\boldsymbol{u}}_{t}^{(i)} \right) - \boldsymbol{p}_{\sigma} \left( \dot {\boldsymbol{u}}_{t}^{(i)}, {\boldsymbol{u}}_{t}^{(i)} \right)- \left[ \boldsymbol{M}^{(i)} {\mathrm{I}}_2' + \boldsymbol{C}^{(i)} {\mathrm{I}}_1' + \boldsymbol{K}^{(i)} \right] \left( {\boldsymbol{u}}_{t} - {\boldsymbol{u}}_{t}^{(i)} \right) %\label{femGenFormTaylor}\]

To start the iteration scheme, trial values for \({\boldsymbol{u}}_{t}\), \(\dot {\boldsymbol{u}}_{t}\) and \(\ddot {\boldsymbol{u}}_{t}\) are required. These are obtained by assuming \({\boldsymbol{u}}_{t}^{(0)} = {\boldsymbol{u}}_{t-\Delta t}\). The \(\dot {\boldsymbol{u}}_{t}^{(0)}\) and \(\ddot {\boldsymbol{u}}_{t}^{(0)}\) can then be obtained from the operators for the integration scheme.