The dependent variables can be seen as functions of mesh coordinates Xm and time
t. If the mesh is moving, then
Xm(t) is also a function of time when seen from a point
x fixed in space. Therefore
The equation-based interfaces have an Equation form setting that controls the interpretation of time derivatives, depending on whether the equation is seen as a time-domain equation, a frequency domain equation using the harmonic ansatz
where ω = 2πf is the angular frequency, or an eigenvalue equation using
When the equation form is set to Time domain, then the eigenvalue ansatz is used when solving using an Eigenvalue or Eigenfrequency study step. For all other solvers, the time derivative variable
ut evaluates based on the time derivative of the degrees of freedom, which are zero in the stationary solver used by both Stationary and Frequency Domain study steps.
When the equation form is set to Frequency domain, then the harmonic ansatz is always used, meaning that time derivative variables are evaluated as
ut =
iωu for all solvers and study step types. But the angular frequency
ω is still by default defined by the study step. A Frequency Domain study step sets the angular frequency to
ω = 2πf where the frequency
f is given in the study step. An Eigenvalue or Eigenfrequency study step defines
ω = iλ, turning the frequency domain problem into a corresponding eigenvalue problem. It is also possible to explicitly set the frequency in the PDE interface settings.
The default equation form setting when working from the user interface is Study controlled, which means that the frequency-domain interpretation of time derivatives will be used in Frequency Domain study steps with the frequency supplied by the study step. For all other study types, the time-domain interpretation will be used.