If the current step is a steady-state dynamic analysis, the value of an output variable such as stress (S) or displacement (U) can be a complex number with both real and imaginary components. Complex number results are saved during the analysis as pairs of real and imaginary data. 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.

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 is the only exception; it is 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, and , are related to the magnitude, , and phase angle, , as follows:

and The complex results represent a time-domain variation of the form:where is the excitation frequency. This time variation is shown when you request a harmonic animation. The value of the result at a selected angle, , is obtained by using so that The magnitude, , for a given variable is displayed when Stepping from –180° to 180° corresponds to one full animation cycle.

For information on related topics, click any of the following items:

• “Selecting complex results,” Section 42.5.8

• “Controlling the form of complex results,” Section 42.6.9

• “Harmonic animation,” Section 49.1.3