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A few other parameters to keep in mind when designing an effective PML using the split field PML is your
choice of sigma max, so if you are going to use a grading of the sigma then at a distance D into the
PML, this would be where the PEC backing would be, so that would be the total thickness of your PML.
This would, the -- sigma, -- this would be sigma X or sigma X star.
This would reach a maximum sigma, some sigma max value.
So something to keep in mind, when choosing a sigma max value is that if sigma max is small then the reflection is that
you get, that the reflection that you might get back into the computational grid is mostly due to the PEC backing.
So mostly due to the PEC boundary on the other side of the PML.
If on the other hand, sigma max is too large then the discretization error dominates.
So these are trade-offs.
Actually in this case sigma max the actual reflection error would be even bigger
than that predicted by the reflection error equation that we wrote out previously.
So there is a trade-off.
And the optimal choice of sigma maxes balances both of these errors that can arise or the
dominating error, either the reflection from the PEC boundary or the discretization error.
Now something else to keep in mind is the choice of D which is the thickness of the PML.
There is a trade-off here between the PML effectiveness and computational efficiency.
So here the issue would be small D where you wouldn't have a very effective PML but in this case the problem is
requiring too much memory and time because you have to update all of those fields as well, so memory and time.
And finally, the choice of M. if you're going to use, for example, a polynomial grading and M is the order of the
polynomial that you choose, then the trade-off here is if M is too small, so if it's too small then your PEC is
-- say this is your interface, which I guess we've been calling X equals zero, then the step discontinuity right
at the beginning would be too small, or too large and then you would have you know some number of cells here but
this, you would get reflection from the interface between the main computational grid and the beginning of the PML.
So your reflection would be dominated by the beginning of the PML if your order of your polynomials is too small.
If it's too large then -- so here is your PEC. Here would be your PEC again.
This is X equals zero at our interface.
Then you have, this is not really drawn to scale, but it would, then it would, the order would be too high that
you would actually get too much reflection from like somewhere in the middle of your PML, and so there is a
trade-off here and you want to make sure that you balance this and not get, you know, in this case, it, the
connectivity rose too rapidly and we are going to get discretization error because it rizes too quickly to
have very good resolution of this connectivity check profile and then you are going to get reflection from this.
But if it's too small you will get too many reflections, too much reflections from the beginning.
And this is because, you know, the reflections, the wave is not going
to propagate long enough into the PML to be sufficiently attenuated.
And so as a last note here, polynomial grading M is typically between three and four.
That seems to be a good trade-off between these two issues.