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arXiv · 2604.18020

Matrix-Free 3D SIMP Topology Optimization with Fused Gather-GEMM-Scatter Kernels

Shaoliang Yang·Jun Wang·Yunsheng Wang
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Abstract

The matrix-free gather-batched-GEMM-scatter pattern eliminates global stiffness assembly for three-dimensional SIMP topology optimization, but the conventional three-stage implementation forces avoidable DRAM traffic between stages. We present a single fused CUDA kernel, implemented through CuPy's runtime compilation interface, that performs gather, per-element stiffness multiplication, and scatter accumulation in one pass. On a single RTX 4090 (24 GB), the fused path reaches a problem-size-dependent 4.6-7.3x end-to-end SIMP wall-time speedup across 216k-4.9M cantilever elements and 4.4x on the 499,125-element torsion benchmark. Against the same-precision FP32 three-stage baseline, the fused path still yields 2.3-4.6x on cantilever and 2.8x on torsion. Isolated CUDA-event cantilever-operator measurements reach 8.9-13.8x per matvec call, while separate instrumented board-power traces at 216k and 1M show 3.2-4.9x lower energy than matched FP64 runs. A separate bridge stress test shows the same FP32-versus-FP64 three-stage trend under one distributed-load case; direct fused-kernel bridge benchmarks are not reported. We also evaluate a BF16 WMMA variant: a separate PyTorch BF16 GEMM proxy on matching tensor shapes yields 14.3x, but direct condition-number estimates of 6.1e5-2.3e6 across 64k-512k uniform-density test states imply BF16 conditioning products of 2.4e3-9.1e3, far above the 256 threshold, observed alongside BF16 iterative-refinement stagnation at the two tested inner tolerances.

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Shaoliang Yang, Jun Wang, Yunsheng Wang. 2026-04-20. Matrix-Free 3D SIMP Topology Optimization with Fused Gather-GEMM-Scatter Kernels. https://arxiv.org/abs/2604.18020

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