Large 3D SIMP studies require repeated elasticity solves for density-dependent operators whose finest matrices are expensive to assemble and whose conditioning degrades under high contrast. We study this linear-solver layer rather than claiming end-to-end optimization acceleration. The solver builds a matrix-free Galerkin geometric multigrid (GMG) hierarchy around a fused fine operator: the finest level remains matrix-free, the first coarse level is assembled by local Galerkin aggregation, and deeper levels use sparse Galerkin products. The practical default is FP32-GMG; BF16 is evaluated as a guarded mixed-precision variant and diagnostic stress test, not as the main speed mechanism. In a 27-case heterogeneous cantilever sweep, pass rates under a 200-iteration budget are 7/9, 4/9, and 1/9 at 64k, 216k, and 512k elements; converged-only mean iteration counts are about 112, 134, and 146. On uniform rho=0.5, p=3 solves, FP32-GMG gives 1.62x, 1.75x, and 3.12x wall-time ratios relative to the capped flat Jacobi-PCG baseline at the same sizes; that non-converged baseline reaches the 200-iteration cap in all timed trials. BF16-GMG is not faster than FP32-GMG. In 18 fixed-seed heterogeneous BF16 validation cases, 7/18 converge, matching the FP64 count, and 11 cases that pass the spectral screen still fail the 500-iteration cap; the screen is therefore diagnostic rather than a convergence certificate. The largest reported solve is a 1M-element uniform-modulus system solved in 1.50+/-0.58 s with an 8.66 GiB hierarchy-allocation delta during setup, not a peak-memory trace; this point is reported as uniform scaling, not heterogeneous robustness evidence. The contribution is therefore a bounded single-GPU solver result built on an inherited Level 0 matrix-free operator: a Galerkin GMG hierarchy, direct BF16 guard evidence, and an explicit failure-mode screen for structured 3D SIMP linear systems.

翻译：大型三维SIMP研究需对密度相关算子进行重复弹性求解，其最细层矩阵的组装代价高昂，且高对比度条件下条件数恶化。本研究聚焦于该线性求解器层面，而非声称端到端优化加速。求解器围绕融合精细算子构建无矩阵Galerkin几何多重网格（GMG）层次结构：最细层保持无矩阵形式，第一粗层通过局部Galerkin聚合组装，更深层则采用稀疏Galerkin乘积。实际默认配置为FP32-GMG；BF16作为受保护的混合精度变体与诊断性应力测试进行评估，而非主要加速机制。在27例异质悬臂梁扫描算例中，以200次迭代预算通过率在64k、216k和512k单元数下分别为7/9、4/9及1/9；收敛算例的平均迭代次数约为112、134和146。在均匀ρ=0.5、p=3求解中，FP32-GMG相对于相同尺寸下有上限的平坦Jacobi-PCG基准，实现1.62倍、1.75倍和3.12倍的运行时间比；该未收敛基准在所有计时测试中均达到200次迭代上限。BF16-GMG性能未超越FP32-GMG。在18例固定随机种子的BF16异质验证算例中，7/18收敛，与FP64算例数量一致；11例通过谱筛选的算例仍无法满足500次迭代上限，故该筛选方法仅为诊断性工具而非收敛保证。最大规模求解报告为100万单元均匀模量系统，求解时间1.50±0.58秒，设置阶段层次分配增量为8.66 GiB（非峰值内存轨迹）；该数据作为均匀缩放案例报告，而非异质鲁棒性证据。因此，本成果基于继承的Level 0无矩阵算子提供有限单GPU求解器结果：包含Galerkin GMG层次结构、直接BF16防护证据，以及面向结构化三维SIMP线性系统的显式失效模式筛选方法。