Curved-element meshing and Palace 3D photonics example

[mdmaas]

This PR implements spline-fitting to replace curve loops, which enables curved-element meshing.

This is potentially interesting for several PDKs with curved features (like QPDK) but maybe most interesting for 3D photonics, for which Palace could lead to better performance than Meep.

[gemini-code-assist]

Code Review

This pull request introduces support for patterned dielectric layers and high-order geometric mesh elements in Palace simulations. Key features include the ability to disable synthetic dielectric regions, the implementation of spline and bspline curve fitting for dielectric boundaries, and the integration of high-order elements into the meshing and visualization pipelines. A new example notebook for a directional coupler is provided; however, feedback indicates that the current validation logic incorrectly triggers a ValueError for purely dielectric simulations, which is a critical issue for photonic component modeling.

[gemini-code-assist]

This cell fails with a ValueError during mesh validation, specifically complaining about the absence of conductor surfaces. This seems to indicate that the validation logic in PalaceSimMixin.validate_mesh() doesn't correctly handle purely dielectric simulations, which is common for photonic components like this directional coupler.

Since this notebook serves as an example for the new curved-element meshing feature, it's critical that it runs successfully. The validation should likely be relaxed for simulations that only use wave ports on dielectric materials.

[joamatab]

LGTM - curved-element meshing and Palace 3D photonics

[codecov]

Codecov Report

❌ Patch coverage is 64.05405% with 133 lines in your changes missing coverage. Please review.
✅ Project coverage is 58.97%. Comparing base (b94e0c3) to head (b44ff2e).
⚠️ Report is 3 commits behind head on main.

Files with missing lines	Patch %	Lines
src/gsim/palace/base.py	24.67%	48 Missing and 10 partials ⚠️
src/gsim/palace/mesh/geometry.py	63.38%	16 Missing and 10 partials ⚠️
src/gsim/palace/mesh/gmsh_utils.py	73.03%	13 Missing and 11 partials ⚠️
src/gsim/palace/mesh/generator.py	51.11%	18 Missing and 4 partials ⚠️
src/gsim/viz.py	96.36%	1 Missing and 1 partial ⚠️
src/gsim/common/stack/extractor.py	83.33%	0 Missing and 1 partial ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #131      +/-   ##
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+ Coverage   58.17%   58.97%   +0.79%     
==========================================
  Files          61       61              
  Lines        7778     8087     +309     
  Branches     1487     1570      +83     
==========================================
+ Hits         4525     4769     +244     
- Misses       2804     2826      +22     
- Partials      449      492      +43     

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[mdmaas]

We didn't get better performance than Meep because the resulting FEM linear system turns out to require a huge number of GMRES iterations (I think it is ill-conditioned due to high dielectric constrast). Using direct solvers (MUMPS, also supported by this PR) is a promising alternative, and results in very good performance for small examples, but RAM memory gets depleted very quickly, so this would need to run on a larger server (perhaps HPC for large photonic problems).

[mdmaas]

I think we can merge this PR, even if the Photonic example didn't turn out to work right now on the current hardware, because there are some features that could benefit other working cases, and we might want to try again running 3d FEM photonics in the future, on a system with massive RAM.