Fix EE-mollifier shape derivative (two bugs)

[Huangzizhou]

Summary

Two bugs in the per-pair NormalPotential::shape_derivative when collision.is_mollified() == true (near-parallel EE pair). Both surface only on the mollified branch, which IPC's existing shape_derivative test (stacked cubes, test_barrier_potential.cpp:248) does not exercise.

1. edge_edge_mollifier_gradient_wrt_x used the wrong second factor

Returns ∇_rest m = ∂m/∂ε · ∇_rest ε. The second factor was using edge_edge_mollifier_gradient(positions, eps_x) (returns ∇_position m) instead of edge_edge_mollifier_threshold_gradient(rest) (returns ∇_rest ε). Sister function edge_edge_mollifier_gradient_jacobian_wrt_x already used the correct factor.

 if (ee_cross_norm_sqr < eps_x) {
     return edge_edge_mollifier_derivative_wrt_eps_x(
                ee_cross_norm_sqr, eps_x)
-        * edge_edge_mollifier_gradient(ea0, ea1, eb0, eb1, eps_x);
+        * edge_edge_mollifier_threshold_gradient(
+              ea0_rest, ea1_rest, eb0_rest, eb1_rest);
 }

2. NormalPotential::shape_derivative was missing one outer-product term

The correct chain-rule expansion of ∇ₓ(f∇ᵤm + m∇ᵤf) — with ∇ₓm = ∇ᵤm + (∂m/∂ε·∇_rest ε), ∇ₓf = ∇ᵤf, ∇ₓ∇ᵤf = ∇ᵤ²f — has five terms:

    f · ∇ₓ∇ᵤm                                       (jac_m)
  + (∂m/∂ε · ∇_rest ε) · (∇ᵤf)ᵀ                   (gradx_m · gradu_fᵀ)
  + (∇ᵤf)(∇ᵤm)ᵀ + (∇ᵤm)(∇ᵤf)ᵀ                    symmetric pair
  + m · ∇ᵤ²f                                        (hessu_f)

The code summed only four (the (∇ᵤm)(∇ᵤf)ᵀ term was missing). That term is the "∇ᵤm contribution" of (∇ₓm)(∇ᵤf)ᵀ — its rest counterpart was being correctly accumulated into gradx_m · gradu_fᵀ, but its position counterpart was being forgotten.

-local_hess = f * jac_m + gradx_m * gradu_f.transpose()
-    + gradu_f * gradu_m.transpose() + m * hessu_f;
+local_hess = f * jac_m + gradx_m * gradu_f.transpose()
+    + gradu_f * gradu_m.transpose()
+    + gradu_m * gradu_f.transpose() + m * hessu_f;

Why it matters

In a polyfem shape-optimization transient adjoint, these two bugs combined produced a ~400× per-pair shape-derivative inflation at the touchdown frame and a ~10⁸× sign-flipped end-to-end gradient. The existing IPC shape_derivative test uses a non-mollified scene so neither bug appeared there.

Test plan

New regression test "Edge-Edge Mollifier Shape Derivative" in tests/src/tests/distance/test_edge_edge_mollifier.cpp constructs a minimal 4-vertex / 2-edge scene with a single mollified EE pair (y_sep = 9.99e-4 just inside dhat = 1e-3, z_twist = 2e-2 so |A × B|² ≪ eps_x). FD-checks the full 12×12 analytic shape_derivative against centered FD of gradient at fixed collision set with eps = 1e-7 (same convention as test_barrier_potential.cpp:377-391).

	Frobenius rel
Without either fix	≥ 2.5 × 10⁻² (FAILS)
With only fix #1	~5 × 10⁻⁵
With both fixes	3.5 × 10⁻⁶ (PASSES)
Threshold	1 × 10⁻⁵

~7000× discrimination, essentially FD precision after both fixes. All 122 existing mollifier-suite assertions still pass.

🤖 Generated with Claude Code

[Copilot]

Pull request overview

This PR fixes two correctness bugs in IPC’s mollified edge-edge (EE) branch for NormalPotential::shape_derivative, and adds a regression test intended to exercise a near-parallel, mollified EE pair that existing shape-derivative tests do not cover.

Changes:

• Fix edge_edge_mollifier_gradient_wrt_x() to use the correct threshold-gradient factor (rest-space) when differentiating through eps_x.
• Fix NormalPotential::shape_derivative() (mollified branch) by adding the missing outer-product term ∇ᵤm (∇ᵤf)ᵀ.
• Add a new FD regression test for mollified EE shape_derivative vs. centered FD of gradient.

Reviewed changes

Copilot reviewed 3 out of 3 changed files in this pull request and generated 1 comment.

File	Description
src/ipc/distance/edge_edge_mollifier.cpp	Corrects the rest-space factor used in the mollifier gradient w.r.t. rest positions.
src/ipc/potentials/normal_potential.cpp	Adds the missing symmetric outer-product term in the mollified shape_derivative expansion.
tests/src/tests/distance/test_edge_edge_mollifier.cpp	Introduces a new regression test intended to validate mollified EE shape derivatives via finite differences.

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[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 96.51%. Comparing base (9a704ea) to head (7b97031).
⚠️ Report is 16 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #239      +/-   ##
==========================================
+ Coverage   95.77%   96.51%   +0.73%     
==========================================
  Files         159      163       +4     
  Lines       16521    16651     +130     
  Branches      927      920       -7     
==========================================
+ Hits        15823    16070     +247     
+ Misses        698      581     -117     

Flag	Coverage Δ
unittests	96.51% <100.00%> (+0.73%)	⬆️

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