Understanding complex results

When you use Abaqus/CAE to display complex results, you can choose any of the following forms to view the analysis data:

Magnitude displays the combined magnitude of both the real and imaginary portions of the result value.
Phase angle displays the angle between the positive horizontal axis and the plotted point (real, imaginary). This selection is valid only for scalars or for vector and tensor components.
Real displays only the real component. This option is the default setting.
Imaginary displays only the coefficient of the imaginary component of the complex result.
Value at angle displays the combined value of the real and imaginary portion of the result at an angle that you specify. When the angle is 0°, the real part of the result is displayed; when the angle is −90°, the imaginary part of the result is displayed.
Envelope: Min displays the minimum of the combined value of the real and imaginary portion of the result by varying the angle from 0° to 360°.
Envelope: Max displays the maximum of the combined value of the real and imaginary portion of the result by varying the angle from 0° to 360°.
Envelope: Max absolute displays the maximum absolute of the combined value of the real and imaginary portion of the result by varying the angle from 0° to 360°.

The Magnitude is calculated after any invariants, such that the magnitude value is the square root of the sum of the squares of the real and imaginary invariant components. Similarly, the other complex forms are also calculated after invariants. Value at angle and the complex envelope values are exceptions; they are calculated prior to invariants to preserve the proper physical meaning.

You can also choose to animate complex results using harmonic animation. This technique animates complex field output by displaying the Value at angle through a sequence of angles. Abaqus/CAE generates angles ranging from 0° to 180° or from −180° to 180°, according to your specification, and displays the value of the complex results at each angle.

The magnitude and phase angle are related to the real and imaginary components with the usual expressions. For example, the real and imaginary displacement components, $U_{r}$ and $U_{i}$ , are related to the magnitude, $\bar{U}$ , and phase angle, $ϕ$ , as follows:

$U_{r} = \bar{U} c o s ϕ$

and

$U_{i} = \bar{U} s i n ϕ .$

The complex results represent a time-domain variation of the form:

$\begin{matrix} U (t) & = U_{r} c o s Ω t - U_{i} s i n Ω t, \\ = \bar{U} c o s (Ω t + ϕ), \end{matrix}$

where $Ω$ is the excitation frequency. This time variation is shown when you request a harmonic animation. The value of the result at a selected angle, $θ$ , is obtained by using $θ = Ω t$ so that

$\begin{matrix} U (θ) & = U_{r} c o s θ - U_{i} s i n θ, \\ = \bar{U} c o s (θ + ϕ) . \end{matrix}$

The magnitude, $\bar{U}$ , for a given variable is displayed when $θ = - ϕ .$ Stepping $θ$ from −180° to 180° corresponds to one full animation cycle.