## Discusses piecewise linear functions in Optistruct

When an enforced displacement is used, a piecewise linear function is referenced from the curve/TABLED, this means that if we want to apply a half sine or quarter sine, we will get a piecewise linear approximation of that curve as can be seen in Figure 1. When Optistruct solves for the displacement of the excitation point, we should see the same enforced displacement at the excitation point output.

The velocity will be piecewise constant if the time step is smaller than two intermediate values on the curve due to the constant derivative of the linear function in between those two points. The velocity will be piecewise linear if the time step is larger than two intermediate values on the curve and appear smoother, this is because the derivative is no longer constant in between two points since between multiple points there are more slopes than one. Figure 2 shows an example of velocity output with the same excitation displacement solved with two different time steps.

The acceleration will be zero if the time step is smaller than two intermediate values on the curve because the velocity is piecewise constant for all time steps. In some cases, there can be acceleration spikes when we switch from one data point to another, this is because of the almost infinite slopes between the constant velocity values. Figure 3 shows the acceleration from the same models as in Figure 2 for the two different time steps.

The above states that there is a relative dependence on output when a displacement is used as enforced motion, if the time step is smaller than intermediate values of data points, we can use finer data points to capture the velocities and accelerations or we can use larger time steps. The best option would be to use acceleration as input since integration works better with piecewise linear functions and as always, use good validation measures, for example, reaction forces. The user can also assign an initial velocity with the displacement curve.

To determine the correct time step to be used, a proper convergence study of the response should be made, using a finer time step is not always the best solution as it can introduce numerical difficulties and convergence issues for non-linear transient analysis.