When a positive SIMP density floor is not enough: solver admissibility and guarded floor selection in matrix-free 3D topology optimization

In a matrix-free geometric-multigrid FGMRES solver for three-dimensional SIMP topology optimization, a reported converged solve is not always a converged solve.

On four of 102 held-out states, the projected residual used for stopping falls below $10^{-6}$ while a recomputed true residual is 1.35 to 49.5 times the tolerance; in an unguarded optimization trajectory, 22 of 40 state solves reach the iteration cap without raising an error.

We formulate floor selection as a verified control problem: probe the frozen state at the original floor, use two residual features to choose the first attempted floor, and accept no solution until $||f-Ku||/||f||\le10^{-6}$ is recomputed.

The two-feature rule matches 98 of 102 reference classifications; the residual guard detects the four missed escalations, and all 102 selected solves satisfy the tolerance.

Relative to always using a $10^{-3}$ floor, the policy preserves the original operator on 24 admissible states and avoids mean compliance and gradient changes of 31.0% and 0.340 on those severe random states, and 0.48% and 0.008 on seven optimized designs, at 2.5 times the mean wall time.

In a 12-state subset of the held-out states, eight still require escalation at the conventional floor $10^{-6}$.

In a nine-state control with the preconditioner's adaptive components disabled, every failure is visible and no false acceptance occurs, tying the stopping-estimate drift to the iterate-dependent preconditioner.

The recomputed residual is the correctness safeguard; the probe and floor ladder govern an implementation-specific cost-fidelity tradeoff.