Concurrent two-scale topology optimization of printable bone scaffolds via rectified-flow generation and exact latent-space refinement
Abstract
Bone scaffolds must reconcile a macroscale mechanical response with microscale geometric feasibility, yet conventional approaches to their design restrict attainable constitutive properties. This work presents a concurrent two-scale topology optimization framework; a conditional rectified flow generates periodic lattice microstructures, which are then refined in latent space by exact finite-element homogenization sensitivities. The resulting unit-cell geometries and their spatial assignments are jointly optimized under mechanical and feature-size constraints. In a distal rabbit femur benchmark, all eight flow-started refinements produce printable cells with maximum relative errors under 1\% across six prescribed engineering moduli. Constructing the implant from a single tiled cell reproduces intact whole-femur compliance within 0.1\% and regional strain energy within 0.35\% under four loading conditions. In human femoral reconstruction, multiscale concurrent optimization produces a cortical mid-shaft scaffold with porosity 63\% while satisfying the compliance constraint within 2\% under walking and stair-climbing loads. It achieves an 84\% porosity on a proximal cancellous scaffold with strain-energy deviations below 0.25\% under the same loads. Analysis of the learned representations reveals distinct architectural families in latent space. Further, the attention heads of the rectified-flow transformer specialize for spatial shifts -- learning the analogue of a finite-difference stencil -- and coordinate symmetries, with ablations showing their influence on generated topologies. These results establish a route to printable, mechanically constrained scaffold designs. The framework is general enough for extending to many other multiscale structural design applications including topology-microstructure co-design in metal additive manufacturing.
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BibTeX
@misc{batley2026concurrent,
title = {{Concurrent two-scale topology optimization of printable bone scaffolds via rectified-flow generation and exact latent-space refinement}},
author = {Reza T Batley and Sourav Saha},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25565},
eprint = {2610.25565},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25565},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Concurrent two-scale topology optimization of printable bone scaffolds via rectified-flow generation and exact latent-space refinement
AU - Reza T Batley
AU - Sourav Saha
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25565
UR - https://arxiv.org/abs/2610.25565
AB - Bone scaffolds must reconcile a macroscale mechanical response with microscale geometric feasibility, yet conventional approaches to their design restrict attainable constitutive properties. This work presents a concurrent two-scale topology optimization framework; a conditional rectified flow generates periodic lattice microstructures, which are then refined in latent space by exact finite-element homogenization sensitivities. The resulting unit-cell geometries and their spatial assignments are jointly optimized under mechanical and feature-size constraints. In a distal rabbit femur benchmark, all eight flow-started refinements produce printable cells with maximum relative errors under 1\% across six prescribed engineering moduli. Constructing the implant from a single tiled cell reproduces intact whole-femur compliance within 0.1\% and regional strain energy within 0.35\% under four loading conditions. In human femoral reconstruction, multiscale concurrent optimization produces a cortical mid-shaft scaffold with porosity 63\% while satisfying the compliance constraint within 2\% under walking and stair-climbing loads. It achieves an 84\% porosity on a proximal cancellous scaffold with strain-energy deviations below 0.25\% under the same loads. Analysis of the learned representations reveals distinct architectural families in latent space. Further, the attention heads of the rectified-flow transformer specialize for spatial shifts -- learning the analogue of a finite-difference stencil -- and coordinate symmetries, with ablations showing their influence on generated topologies. These results establish a route to printable, mechanically constrained scaffold designs. The framework is general enough for extending to many other multiscale structural design applications including topology-microstructure co-design in metal additive manufacturing.
ER -