Solutions computed for a harmonic perturbation study step (for example, the second step of a small-signal analysis or prestressed analysis) consist of two separate parts: the linearization point and the perturbation solution. These parts can be combined in a number of different ways when evaluating an expression. This is controlled by the Expression evaluated for list, which appears in many postprocessing nodes if the selected dataset is based on the result of a harmonic perturbation:

Select an option from the Expression evaluated for list: Static solution, Harmonic perturbation, Total instantaneous solution, Average for total solution, RMS for total solution, or Peak value for total solution. Each option is described below.

The expression is evaluated by taking the values of any dependent variables from the linearization point part of the solution. This is achieved by wrapping the expression in the linpoint operator.

If Harmonic perturbation is selected, the Compute differential check box is also available. If the check box is not selected, the expression is evaluated by taking the values of any dependent variables from the harmonic perturbation part of the solution.

If the Compute differential check box is selected (the default), the differential of the expression with respect to the perturbation is computed and evaluated at the linearization point. This is achieved by wrapping the expression in the lindev operator. For expressions that are linear in the solution, the two options are the same.

The expression is evaluated by adding the linearization point and the harmonic perturbation and taking the real part of this sum. This is achieved by wrapping the expression in the lintotal operator. The phase and amplitude of the harmonic perturbation part can be set in the corresponding dataset.

This is the same as evaluating for Total instantaneous solution and then averaging over all phases of the harmonic perturbation. This is achieved by wrapping the expression in the lintotalavg operator.

This is the same as evaluating for Total instantaneous solution and then taking the RMS over all phases of the harmonic perturbation. This is achieved by wrapping the expression in the lintotalrms operator.

This is the same as evaluating for Total instantaneous solution and then taking the maximum over all phases of the harmonic perturbation. This is achieved by wrapping the expression in the lintotalpeak operator.

To illustrate the effect of the different Expression evaluated for settings, consider a prestressed solid mechanics model. The solution has been obtained by first computing a prestressed state, in a Stationary study step, and then adding a small harmonic perturbation on top of the stationary solution, using a Frequency Domain, Perturbation study step. The linearization point solution, u0, is therefore the stationary displacements from the prestress step, and the harmonic perturbation part of the solution, up, is the small complex-valued harmonic displacements induced by the harmonic perturbation step. The instantaneous displacements are therefore

A common measure of the equivalent stress level in a solid is the von Mises stress, σe. This is a nonlinear function of the strain and therefore of the displacements. Assuming that the harmonic perturbation is small, the von Mises stress can be linearized around the linearization point

where the gradient is taken with respect to the solution degrees of freedom. Now suppose you evaluate the von Mises stress using each of the available options for Expression evaluated for:

Evaluating using the static solution returns the von Mises stress from the prestress step, σe0.

Evaluating using the harmonic perturbation solution with the Compute differential option selected returns the complex-valued amplitude of the linearized change in von Mises stress, σep, due to the harmonic perturbation. For a perturbation that is small compared to the linearization point, the von Mises stress varies harmonically

Evaluating using the harmonic perturbation solution with the Compute differential option cleared returns the von Mises stress expression evaluated by substituting the complex perturbation solution, up. This number, σe(up), is essentially a nonsensical value because it involves products of complex-valued amplitudes.

Evaluating using the total instantaneous solution option returns the von Mises stress expression evaluated using the total solution at the phase, θ, specified in the current dataset:

The average for the total solution is computed over one period by varying the phase, θ, in the interval [0, 2π]:

The RMS value for the total solution is computed over one period by varying the phase, θ, in the interval [0, 2π]:

The peak value for the total solution is computed as the maximum of σe(θ) for any phase θ in the interval [0, 2π].