Concurrent two-scale topology optimization of printable bone scaffolds via rectified-flow generation and exact latent-space refinement

Reza T Batley; Sourav Saha

physics.comp-phOct 18, 2026

Concurrent two-scale topology optimization of printable bone scaffolds via rectified-flow generation and exact latent-space refinement

Authors: Reza T Batley, Sourav Saha

Organizations: Kevin T Crofton Department of Aerospace and Ocean Engineering, Virginia Tech, 1600 Innovation Drive, Blacksburg, 24060, VA, United States

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.

Figures & tables

Figure 1: Overview of the proposed framework. a, Offline training of the conditional rectified-flow model. b, Online cell-level design. c, Concurrent two-scale topology optimization with coupled latent and assignment updates.

Figure 2: Autoencoder architecture and unit-cell reconstruction.

Figure 3: Conditional rectified-flow architecture and unit-cell generation.

Figure 4: Validation flow-matching loss curves of the DiT, MM-DiT and U-Net backbones.

Backbone	Parameters	Val. flow loss	Success $\leq 5\%$	Success $\leq 10\%$
DiT	161.6M	0.330	73%	98%
MM-DiT	159.8M	0.366	73%	94%
U-Net	171.0M	0.511	69%	98%

Table 1: Backbone ablation. Success is the fraction of targets for which at least one of 64 samples is printable.

Figure 5: Anatomical configurations and design regions. a, The cancellous distal portion of a rabbit femur. b, Human femur with a cortical mid-shaft defect. c, Human femur with a proximal cancellous replacement.

Figure 6: Representative realized unit cells obtained by the proposed method and baseline approaches. Maximum relative engineering-modulus error $\epsilon_{\max}$ and physical feature size $\mathcal{T}L$ are indicated.

Figure 7: ( left ) Relative errors in the six prescribed engineering moduli for the cancellous bone target; ( center ) relative errors in the target whole-femur compliance under axial, two bending, and torsional loading; ( right ) relative strain-energy deviations. Baselines in the left panel are uniform cells; the center and right show their graded implants; radial scales are logarithmic and reversed.

Figure 8: The realized distal rabbit femur implants constructed from the proposed and baseline lattice designs.

Figure 9: Multiscale visualization of the proposed distal rabbit femur scaffold.

Figure 10: Front and side views of the loading conditions for the human femur model. ( left ) Walking loads ( right ) Stair-climbing loads. Forces are expressed relative to body weight – modeled as 850N.

Figure 11: Concurrent optimization paths ( top ) and verification of the reported designs ( bottom ). Each path traces verified implant porosity against the constrained response under walking (solid) and stair-climbing (dashed) loads, from the initial design (ring) to the reported design (star); the shaded band is the

\pm 2\%

bound and gray segments follow the reported design. Mid-shaft points are colored by the number of active cell classes; the proximal design retains all 12 classes throughout. Bottom: finite-element verification of the reported designs on the 1.5mm design mesh and a refined 1.0mm mesh.

Figure 12: The six active cell classes in the optimized cortical mid-shaft scaffold, showing block populations, feature thickness, and porosity.

Figure 13: Multiscale visualization of the proposed human mid-shaft femoral scaffold.

Figure 14: The twelve active cell classes in the optimized proximal cancellous scaffold, showing block populations, feature thickness, and porosity.

Figure 15: Multiscale visualization of the proposed proximal cancellous femoral scaffold.

Figure 16: Organization of the learned latent space. ( top ) density of 4,000 validation-cell latents in their two leading principal components, shown inverted so that densely populated regions form wells; no labels or properties are used. The tube follows a straight latent path between two flow-generated cells in the two most populated wells, with markers at the decoded frames below. ( bottom ) Decoded cells along the path, with their solid fractions. The cell changes abruptly between two architectural families at the ridge rather than morphing gradually.

Figure 17: Attention patterns learned by the flow transformer. ( top ) Attention of a single query token (white box) on an $8\times 8$ slice of the $8^{3}$ token grid for representative heads, and, for the axis-permutation head, its attention to the five coordinate-permuted copies of the query, viewed along the cube diagonal. ( middle ) Three-dimensional detail of the six nearest-neighbor shift heads, each labeled (block, head), and the three axis-swap heads, each attending to the query’s image under an exchange of two coordinate axes. ( bottom ) Classification of all 128 heads by block and head index.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Symbol	Description
Geometry and learned representations
$X,\tilde{X}$	Binary $64^{3}$ symmetry octant and decoded occupancy-probability field.
$\mathcal{D},\mathcal{R}$	Frozen decoder and binary unit-cell realization operator.
$z,z_{a}$	Decoder-space latent and decoder-space latent of cell class $a$ .
$z_{0},z_{\tau},z_{1}$	Gaussian initial, intermediate, and terminal standardized flow latents.
$\mu_{z},\sigma_{z}$	Scalar mean and standard deviation used to standardize autoencoder latents for flow training.

Appendix

Table 19

Topology Optimization

Topology optimization finds material layouts that meet performance and manufacturing constraints across engineering systems.

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Yongmin Kwon, Namwoo Kang

Cho Chun Shik Graduate School of Mobility, Korea Advanced Institute of Science and Technology, 193 Munji-ro, Yuseong-gu, Daejeon, 34051, Republic of Korea · AI Research Team, Narnia Labs, 193 Munji-ro, Yuseong-gu, Daejeon, 34051, Republic of Korea

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Topology optimisation (TO) often requires repeated finite element analysis and sensitivity-based material updates, which can be costly when multiple candidate designs are needed under varying physical and design conditions. Generative TO offers a route to rapid design exploration, but existing models may rely on adversarial training, long reverse-diffusion sampling, or external guidance to maintain structural feasibility and physical consistency. This study develops a flow matching-based topology optimisation (FMTO) framework for conditional topology generation. Linear FMTO is first formulated as an endpoint-based baseline by interpolating between a Gaussian source field and the BESO reference topology. To introduce mechanically meaningful intermediate states, a trajectory-aware FMTO formulation is proposed, where volume-fraction-indexed BESO states are used to construct the probability path and target velocity field. This incorporates physics-guided optimisation history into generative flow learning without adding inference-time optimisation. A path--velocity mismatch analysis explains why moderate trajectory weighting can improve generation stability, whereas excessive guidance may over-constrain the learned transport. Numerical examples show that FMTO generates diverse topology candidates with improved compliance-related performance, volume-fraction satisfaction, topology fidelity, and substantially fewer sampling steps than a diffusion-based baseline. Under limited training data, trajectory-aware FMTO achieves the best overall performance with a moderate trajectory weight. Studies on trajectory-anchor density and three-dimensional topology generation further demonstrate the influence of path design and the applicability of the proposed framework beyond two-dimensional problems.

Shusheng Xiao, Jinshuai Bai, Hyogu Jeong +3

School of Mechanical, Medical and Process Engineering, Queensland University of Technology, Brisbane, 4000, QLD, Australia · Institute of Biomechanics and Medical Engineering, AML, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China · State Key Laboratory of Advanced Environmental Technology, Guangzhou Institute of Geochemistry, Chinese Academy of Sciences, Guangzhou, 510640, China +1

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Oct 20, 2026

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Large-scale topology optimization (TO) remains computationally demanding, especially for stress-constrained problems. To address this challenge, we propose the simultaneous neural topology optimization (SNTO) framework where we represent state, adjoint, and design variables with custom neural networks (NNs) whose parameters are jointly optimized within a single optimization loop that leverages continuous adjoint analysis and automatic differentiation. This optimization minimizes a multi-component loss function that combines the design objective, an aggregation of the local stress constraints, and the potential energy functionals. Our NNs have a unique architecture that accelerates convergence while enabling the networks to capture sharp gradients and boundaries which are critical to stress-constrained TO. To achieve scalability to millions of degrees of freedom (DoF), we develop a hardware-aware training scheme in which the design domain is partitioned into subdomains that are distributed across multiple GPUs. To control peak GPU memory usage, we further slice each subdomain into smaller regions whose loss gradients are evaluated sequentially and accumulated to update the NN parameters. Unlike domain decomposition approaches, our partitioning naturally ensures field continuity across the subdomains and does not require the introduction of additional terms into the loss function. For validation, we first evaluate the proposed framework on various 2D stress-constrained benchmarks and compare our results to a conventional finite element-based approach, showing that SNTO produces super-resolution designs with sharper boundaries, smaller gray areas, and lower volume fractions. We then consider large-scale 3D problems with up to eight million collocation points and distribute the training over up to eight GPUs.

Xiangyu Sun, Shirin Hosseinmardi, Ramin Bostanabad

Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA · Department of Mechanical Engineering, Northwestern University, Evanston, IL, USA · Department of Civil and Environmental Engineering, University of California, Irvine, CA, USA

Topology Optimization

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Abstract

Topology Optimization

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