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

2026-07-28

AI Summary

This study addresses the issue of spurious convergence in matrix-free 3D SIMP topology optimization, which often compromises solution reliability. The authors formulate the selection of the density lower bound as a verification control problem and propose a controlled lower-bound strategy incorporating residual validation. The approach determines an initial value by detecting frozen states and analyzing dual residual characteristics, and enforces recomputation of the true residual to guarantee convergence accuracy. This method enables, for the first time, a reliable assessment of solver acceptability. Integrated with a matrix-free geometric multigrid FGMRES solver, residual feature analysis, an adaptive preconditioner, and a true residual recomputation mechanism, the proposed strategy satisfies convergence tolerances in all 102 test cases. Compared to fixed lower-bound strategies, it reduces average compliance error by 31.0% under severe random conditions, achieves optimization errors below 0.5%, and incurs only a 2.5× increase in average runtime.

Abstract

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.

Problem

• topology optimization
• matrix-free solver
• convergence verification
• residual guard
• density floor

Innovation

• matrix-free topology optimization
• SIMP density floor
• residual guard
• verified control
• FGMRES solver