Adding a separate factor for NFP to calculate umbilic configurations

[rahulgaur104]

An umbilic torus is a 3D shape with a boundary characterized by a closed torus with a continuous sharp edge that goes around three times toroidally before meeting itself. This shape can be thought of as a stellarator with a rational field period because the cross-section is only exactly identical (each sharp point completes a full poloidal rotation) after three toroidal turns.

Practically, these configurations are useful for resonant divertor design and may simplify divertor placement.

The objective of this PR is to add an additional factor that adds an integer NFP_umbilic_factor so that NFP -> NFP/NFP_umbilic_factor and make sure that the code runs without any issues.

To create an umbilic edge, we take the boundary of the stellarator and a 3D umbilic curve on the boundary and vary both the curve and the surface to obtain a stellarator that resembles an umbilic-torus-like shape.

[github-actions]

Remaining comments which cannot be posted as a review comment to avoid GitHub Rate Limit

flake8

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desc/geometry/core.py|367 col 89| line too long (107 > 88 characters)
desc/geometry/core.py|416 col 89| line too long (109 > 88 characters)

[codecov]

Codecov Report

Attention: Patch coverage is 51.72414% with 98 lines in your changes missing coverage. Please review.

Project coverage is 95.01%. Comparing base (b1900b3) to head (90f0cf8).

Files	Patch %	Lines
desc/objectives/_geometry.py	13.51%	96 Missing ⚠️
desc/geometry/curve.py	93.75%	1 Missing ⚠️
desc/objectives/normalization.py	75.00%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #819      +/-   ##
==========================================
- Coverage   95.42%   95.01%   -0.41%     
==========================================
  Files          89       89              
  Lines       22434    22578     +144     
==========================================
+ Hits        21407    21453      +46     
- Misses       1027     1125      +98     

Files	Coverage Δ
desc/basis.py	98.20% <100.00%> (+0.01%)	⬆️
desc/coils.py	97.26% <100.00%> (+<0.01%)	⬆️
desc/compat.py	83.96% <100.00%> (ø)
desc/equilibrium/equilibrium.py	95.71% <100.00%> (ø)
desc/examples/__init__.py	100.00% <100.00%> (ø)
desc/geometry/core.py	91.82% <100.00%> (+0.16%)	⬆️
desc/geometry/surface.py	96.85% <100.00%> (+<0.01%)	⬆️
desc/grid.py	92.98% <100.00%> (+0.08%)	⬆️
desc/io/hdf5_io.py	81.96% <ø> (ø)
desc/io/optimizable_io.py	86.30% <ø> (ø)
... and 9 more

[dpanici]

I think the objective you added makes sense, it tries to align the alpha=0 field line with the umbilic curve. Something I am not sure of is, do your curves have a given helicity to them? that is, as phi increases do they move clockwise/counterclockwise poloidally? might have to make sure that if they do, that it is consistent with the helicity of the equilibrium field as determined by the sign of iota

[rahulgaur104]

I think the objective you added makes sense, it tries to align the alpha=0 field line with the umbilic curve. Something I am not sure of is, do your curves have a given helicity to them? that is, as phi increases do they move clockwise/counterclockwise poloidally? might have to make sure that if they do, that it is consistent with the helicity of the equilibrium field as determined by the sign of iota

Excellent question! Yes, there can be a sign issue but helicity of the curve = |curve.NFP/curve.NFP_umbilic_factor|. The sign decides clockwise/anticlockwise. I think the sign should be the same as the sign of iota.
The way the surface and the curve are initialized, the curve corresponds to alpha = 0.
Here's what I get for a 3/5 umbilic curve.

The objective tries to align the field line (red line) with the umbilic curve (blue line)