AI 中文总结
针对无矩阵三维SIMP拓扑优化中求解器收敛误判问题，提出防护型下限选择策略，提升求解正确性并控制成本，验证了其有效性与可解释性。
AI 中文摘要
在用于三维SIMP拓扑优化的无矩阵几何多重网格FGMRES求解器中，已报告的收敛求解结果未必是真正的收敛结果。在102个保留状态中，有4个状态的用于停止的投影残差低于10⁻⁶，而重新计算的真实残差是容差的1.35至49.5倍；在无防护的优化轨迹中，40个状态求解中有22个达到迭代上限却未报错。我们将下限选择构建为一个可验证的控制问题：在原始下限处探测冻结状态，利用两个残差特征选择首次尝试的下限，且在重新计算的||f-Ku||/||f||≤10⁻⁶之前不接受任何解。该双特征规则匹配了102个参考分类中的98个；残差防护检测到了4个遗漏的升级，且所有102个选定求解均满足容差。相较于始终使用10⁻³下限，该策略在24个可容许状态上保留了原始算子，且在那些严重随机状态上避免了31.0%的平均柔度和0.340的梯度变化，在7个优化设计上避免了0.48%和0.008的变化，平均 wall time 为原来的2.5倍。在保留状态的12状态子集中，有8个在传统下限10⁻⁶时仍需要升级。在禁用预条件子自适应组件的9状态对照中，所有故障均可见且无错误接受，将停止估计漂移与依赖迭代的预条件子关联起来。重新计算的残差是正确性保障；探测和下限阶梯控制特定于实现的成本-保真度权衡。
英文摘要
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.
Comments31 pages; 8 main figures and 4 supplementary figures. Code: https://github.com/nbbllxx0/solver-admissibility-and-guarded-floor-selection
