Implement _ParametricGeometry

[mikeingold]

Changes

• Implements an internal _ParametricGeometry <: Meshes.Geometry type to serve as a simple wrapper for custom parametric functions _parametric(geometry, ts...).
• Uses this new geometry type to implement new integral methods for BezierCurve and Tetrahedron geometries, closing Integral on Tetrahedron not fully supported #40. This achieved an 80x performance improvement for integrals over a BezierCurve.
• Removes the jacobian(::BezierCurve, ts, ::Analytical) method - performance was inferior to the generic FiniteDifference method.
• Remove extended tag from Tetrahedron tests now that they return in reasonable CI timespans.
• Updated a remnant usage of Base.zeros to _zeros, providing HAdaptiveCubature integrals with an up to 20% performance improvement.

Discussion

This is a follow-up to #131, where the scope of changes had become over-extended. Some of those changes were implemented separately in other PR’s.

I’ve chosen not to also tackle other specialization geometries yet, since those seemed to suffer performance penalties. Those implementations are left for future work.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 100.00%. Comparing base (cba39f0) to head (fa35675).
Report is 2 commits behind head on main.

Additional details and impacted files

@@             Coverage Diff             @@
##             main      #138      +/-   ##
===========================================
+ Coverage   95.86%   100.00%   +4.13%     
===========================================
  Files          17        18       +1     
  Lines         290       284       -6     
===========================================
+ Hits          278       284       +6     
+ Misses         12         0      -12     

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[github-actions]

Benchmark Results

	main	fa35675...	main/fa35675518c10e...
Differentials/Differential	0.239 ± 0.0011 μs	0.239 ± 0.001 μs	1
Differentials/Jacobian	0.205 ± 0.001 μs	0.206 ± 0.001 μs	0.996
Integrals/Segment/Scalar GaussKronrod	1.2 ± 0.0081 μs	1.21 ± 0.011 μs	0.995
Integrals/Segment/Scalar GaussLegendre	8.39 ± 0.18 μs	8.4 ± 0.17 μs	0.999
Integrals/Segment/Scalar HAdaptiveCubature	1.73 ± 0.053 μs	1.43 ± 0.15 μs	1.21
Integrals/Segment/Vector GaussKronrod	3.4 ± 0.056 μs	3.42 ± 0.046 μs	0.995
Integrals/Segment/Vector GaussLegendre	24.1 ± 0.55 μs	24.5 ± 0.49 μs	0.984
Integrals/Segment/Vector HAdaptiveCubature	4.67 ± 0.1 μs	4.42 ± 0.079 μs	1.06
Integrals/Sphere/Scalar GaussKronrod	0.0817 ± 0.00059 ms	0.0821 ± 0.0019 ms	0.995
Integrals/Sphere/Scalar GaussLegendre	2.26 ± 0.009 ms	2.26 ± 0.0089 ms	1
Integrals/Sphere/Scalar HAdaptiveCubature	0.06 ± 0.00021 ms	0.0582 ± 0.00015 ms	1.03
Integrals/Sphere/Vector GaussKronrod	0.115 ± 0.00086 ms	0.117 ± 0.00093 ms	0.981
Integrals/Sphere/Vector GaussLegendre	3.61 ± 0.06 ms	3.6 ± 0.076 ms	1
Integrals/Sphere/Vector HAdaptiveCubature	0.109 ± 0.0011 ms	0.107 ± 0.0011 ms	1.02
Specializations/Scalar GaussLegendre/BezierCurve	21.9 ± 0.059 ms	0.25 ± 0.0033 ms	87.4
Specializations/Scalar GaussLegendre/Line	10.2 ± 0.44 μs	10.1 ± 0.39 μs	1.01
Specializations/Scalar GaussLegendre/Plane	0.515 ± 0.0035 ms	0.515 ± 0.0036 ms	1
Specializations/Scalar GaussLegendre/Ray	6.24 ± 0.11 μs	6.26 ± 0.13 μs	0.996
Specializations/Scalar GaussLegendre/Triangle	0.681 ± 0.018 ms	0.687 ± 0.013 ms	0.991
time_to_load	1.49 ± 0.011 s	1.49 ± 0.016 s	1

Benchmark Plots

A plot of the benchmark results have been uploaded as an artifact to the workflow run for this PR.
Go to "Actions"->"Benchmark a pull request"->[the most recent run]->"Artifacts" (at the bottom).

[JoshuaLampert]

A first review.

[mikeingold]

A first review.

Ah, good catch. I forgot to remove the Functions when I ported the changes over.

[JoshuaLampert]

Thanks!