TransverseIsotropicFlowModel: consistent (Newton) tangent inconsistent with the residual when anisotropy is active

[lmoresi]

Evidence (PETSc -snes_test_jacobian, native essential-BC path — no rotated machinery involved)

Annulus, power-law background (eta_0 = eps_II^(1/3-1)), localised weak-plane band (eta_1 = eta_0*(1-0.99*band), radial director), consistent_jacobian=True. Checker run at bounded curvature (regularisation 1e-4 inside the sqrt) so the finite-difference reference is valid:

state	ViscousFlowModel (control)	TI isotropic limit	TI with anisotropy
rest / first iterate	1.4e-10	1.4e-10	1.4e-10
early	9.2e-7	9.2e-7	9.2e-7
developed flow	4–5e-6 (FD-limited)	—	1.8e-3 → 4.5e-3

The isotropic power-law tangent is correct (FD-limited). The defect appears only when eta_1 != eta_0 — the anisotropy part of the consistent tangent (the dC/d(eps_II) director-coupled terms, C(eps_II) = eta_0*I + (eta_0-eta_1)*[n-projector terms]) is missing or wrong. The band is localised, so the global Frobenius mismatch is small (~0.2–0.5% of ||J||) while being locally O(1) — which matches the observed behaviour: TI solves under consistent_jacobian=True converge at Picard pace (24 iterations on the #438 TI smoke vs ~11 for the isotropic power-law), and early trajectories are near-identical to the frozen tangent.

Methodological note (matters for anyone re-running this)

With the customary 1e-12 strain-rate regularisation the checker is invalid for shear-thinning rheology: the viscosity curvature at the floor (~eps_II^(-8/3)) breaks the FD reference, and every model — including the known-good isotropic Newton tangent — reads ||J-Jfd||/||J|| ~ 0.013–0.17 identically. Bound the curvature (regularisation ~1e-4 in the sqrt) before trusting the numbers.

Impact

• TI + consistent_jacobian=True silently runs at (roughly) Picard convergence rates; no wrong answers observed (the residual is the true one), just lost Newton performance exactly where fault/anisotropy physics needs it.
• Blocks trusting Newton-rate claims for TI in the free-surface/fault programme (workstream C inherits TI rheology in 3D).
• The rotated free-slip machinery is exonerated: the defect reproduces on the native path, and the isotropic Newton tangent is superlinear through the rotated loop.

Reproduce

Harness: ~/+Simulations/FreeSurface/annulus_fs_convection/b_snes_datum_acceptance/ti_tangent_check.py (parts native/rotated, -iso 1 control); bounded-curvature logs ti_tangent_bounded_aniso.log alongside. Found during the #438 workstream-B session (hypothesis: Louis, 2026-07-28); TODO(BUG) marker on the class points here.

Underworld development team with AI support from Claude Code