Choosing Minimal N and P
Discretization Parameter N
For each non-circular shape in a given scattering problem, we must choose the number of discretization nodes 2N that not only fulfills some accuracy requirement, but also is not large enough to slows down the solution process. Although each shape is only solved or once in the pre-processing stage, with $O(N^3)$ time complexity this stage can be slower than the system matrix solution for large values of N.
As the relationship between N and the resulting error depends not only on the geometry and diameter of the shape, but also on the wavelengths inside and outside of it, a general approach to computing N is crucial for dependable results. Moreover, there are many ways to quantify the error for a given discretization. Here we utilize a fictitious-source approach: the potential densities $\sigma$, $\mu$ are solved for assuming (for example) a plane wave outside the particle and a line source inside it. The fields induced by these densities outside the particle should then be equal to that of the line source, up to some error. Specifically, this error is measured on a circle of radius
R_multipole*maximum(hypot.(s.ft[:,1],s.ft[:,2]))as this is the scattering disc on which the translation to cylindrical harmonics (Bessel & Hankel functions) are performed (and beyond which any gain or loss of accuracy due to N is mostly irrelevant).
Above a certain N, this error tends to decay as $O(N^{-3})$, but with a multiplicative factor that is heavily dependent on the particle and wavelength. With minimumN, we first guess a value and then use a binary search to find the minimal N satisfying some error tolerance tol.