Hi folks! After a couple of serious and not-so-serious lifestyle-esque posts, it’s time to get back to work. Mine, to be exact. If you’ve read the previous instalments in this series, you know that I’m currently working on a faster alternative to time-stepping FEM. If you haven’t, I suggest you start from the beginning. Today, it’s the time to discuss some implementation issues.
In the previous post, I described the basics of the harmonic balance method. The time-dependent vector potential is written as a truncated Fourier series
where is a convenience notation for either or . This sum approximation is then substituted into the governing equation
and the unknown coefficients are then solved with the Galerkin’s method.
This gives us a big block matrix equation, with the matrix blocks equal to
where the angle brackets correspond to calculating the time-average. From this matrix system, the coefficients can then be solved.
However, this leaves us with the not-so-trivial problem of actually computing these blocks. As I described earlier, the stiffness matrix can be divided into constant and time-dependent parts, i.e.
Obviously, the static component is no problem to handle. However, this cannot be said for the time-dependent component .
Indeed, the most typical approach to model the rotation of the machine is to change the finite element mesh in the air-gap, where the stator and rotor are moving past each other. In this approach (and most others, for that matter), the entries of will experience sharp, discontinuous changes as the elements of the mesh change. Thus, calculating the time-integral
accurately could be quite complicated indeed.
However, and luckily, this is a problem I have already solved. Remember earlier when I spoke about calculating the Fourier series of the matrix ? In other words, I already know how to write as
and how to compute the coefficients .
Number of required terms
Of course, having to compute a huge number of the terms would not be very practical, would it. I mean, that would be required to present the highly discontinuous entries of accurately, right?
Well, yes and no. Accurate representation of would indeed need quite a many terms. However, we are not interested in an accurate series approximation of – we are interested in calculating the matrix blocks
Thus, let’s now substitute the approximation for here, to get
In other words, we now longer have a time-average of a matrix – we have the time average of the product of three trigonometric functions . And this can be expressed in closed form quite easily.
And even more better, there are only a limited number of terms for which the expression is non-zero. And these values of can also be determined analytically. Thus, it is usually sufficient to compute something like of the air-gap matrix Fourier coefficients.
And this is obviously a much easier task to computing a total of of the matrix block integrals, like we would have been forced to do before.
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