Mode solver improvement for simulations at low-frequency containing metals

[weiliangjin2021]

Two main improvements:

• For the eigenvalue problem with the matrix PQ where P = p_mu + p_partial and Q = q_ep + q_partial, where p_mu denotes the matrix containing mu_parallel, and q_ep for epsilon_parallel, and the partial term for the remaining matrix components that contains Grad*ep(mu)^-1*Grad. One observation is that if you write out the block form, one can find that p_partial @ q_partial=0. Since Grad scales 1/(k0* dl), in low-frequency/high-resolution regime, Grad can be huge, not mention that p(q)_partial scales as Grad**2. So even though p_partial @ q_partial is strictly 0, machine error can generate nonzero terms that completely ruin the eigenvalues of matrix PQ. This is responsible for the oscillating phenomenon of effective n as a function of frequency in the low-frequency regime.

  In this PR, we don't compute PQ with the fully assembled matrix P and Q, but PQ = p_mu @ Q + p_partial @ q_ep so that p_partial @ q_partial is for sure dropped.

• Let's take matrix P as an illustration (the same applies to Q), p_partial scales as 1/(k0* dl)**2. So in low-frequency/high-resolution regime, p_partial term might grow so huge that p_mu term becomes negligible. But for PEC, its permittivity value should still be significant. So our default permittivity value for PEC should be scaled by the factor 1/(k0* dl)**2.

(Tests to be added in the solver part in a separate PR.)

[weiliangjin2021]

Here is a benchmark example from @dmarek-flex regarding a micostrip made of PEC. Its effective k should be 0, and effective n approaches a constant around 1.82 as frequency goes to 0. double precision is used.

• Before this PR: significant oscillation in low-frequency region and effective k is huge.
  

• Just scaling pec_val, equally bad.
  

• Just dropping p_partial @ q_partial, the curve becomes smooth, but incorrect
  

• With both:
  

[weiliangjin2021]

@prashkh welcome to test if you have other tests in mind

[dmarek-flex]

Results speak for themselves. Thanks @weiliangjin2021, I tried it out on my own example as well and it worked well down to 1 MHz, which is plenty low enough.

First time I get to remind someone to update the Changelog though!

[weiliangjin2021]

Results speak for themselves. Thanks @weiliangjin2021, I tried it out on my own example as well and it worked well down to 1 MHz, which is plenty low enough.

Well, soon enough we might need to handle <1MHz. But that will probably need a very different formulation.

First time I get to remind someone to update the Changelog though!

Thanks for the reminder. I plan to add it after 1st round of review so that I'm confident enough to say that it works well in low-frequency.

[weiliangjin2021]

@dmarek-flex CHANGELOG added.