Zero-shot rib design: merging training-free generative prior with topology optimization

Yongmin Kwon; Namwoo Kang

Source-linked AI summary

Zero-shot rib design: merging training-free generative prior with topology optimization

Yongmin Kwon, Namwoo Kang

arXiv:2609.10643v1Sep 9, 2026cs.LGphysics.app-ph

TL;DR

Classical topology optimization rarely reaches nature-inspired rib geometries and offers limited ways to express design intent in machine-interpretable form. This work couples a frozen text-to-image diffusion prior to density-based topology optimization through score distillation and FEA sensitivities. Across 245 runs, 38 of 49 prompt–domain combinations significantly reduced compliance, while dead-end suppression and scheduled projection explain key improvements and limitations.

Problem
Classical topology optimization rarely reaches nature-inspired rib geometries, while conventional methods lack machine-interpretable representations of engineers’ structural design intent.
Method
A frozen Stable Diffusion 2.1 supplies a training-free SDS gradient that is combined with FEA sensitivities during density-based topology optimization.
Results
38 of 49 prompt–domain combinations achieved statistically significant compliance reductions across 245 primary SDS runs, outperforming gradient-based baselines.
Takeaways & Limitations
The framework provides a reusable training-free prior whose benefits depend on load-path alignment and physics filtering of prompt-induced morphology.
Takeaways & Limitations
The study is limited to 2D density-to-image coupling, and prompt selection remains manual because no single prompt is universally optimal.

Abstract

from arXiv · show

Natural load-bearing patterns such as leaf venation, trabecular bone, and spider webs achieve high stiffness per unit mass, yet classical topology optimizers rarely reach such geometries, and few let engineers express structural design intent through natural language. This work treats a frozen text-to-image diffusion model as a training-free source of design knowledge and distills it into the physics loop of density-based topology optimization via score distillation sampling, so that a text prompt becomes an explicit, machine-interpretable representation of engineer intent. The prompt-induced generative gradient and the finite element sensitivity are combined at every iteration, letting physics decide which prompt-induced features survive. In 245 primary SDS runs spanning four geometric domains and two physics regimes, 38 of 49 prompt--domain combinations achieved statistically significant compliance reductions (up to $-31.5\%$ mechanical and $-23.0\%$ thermoelastic), outperforming gradient-based baselines. Cross-domain morphological analysis identifies a recurring structural signature of improvement: in most domains the generative prior suppresses dead-end branches in the rib skeleton, with endpoint--compliance correlation $r = +0.56$ to $+0.99$. A Heaviside projection with $β$-continuation resolves a pronounced intermediate-density tendency in this diffusion--physics coupling ($42.6\%$ to $<3\%$), and an automated skeleton-based pipeline converts optimized density fields into \rev{candidate geometry ready for computer-aided design. By retargeting the generative prior across domains, loading conditions, and physics objectives through a change of text prompt, with each new problem's physics setup specified separately, the framework uses a pretrained generative model as a reusable, training-free prior for engineering design.

1. Introduction

Classical topology optimization struggles with local optima and lacks machine-interpretable representations of engineers’ nature-inspired rib-design intent. The framework uses a frozen diffusion prior with physics coupling to guide designs while identifying structural mechanisms and binarization remedies.

Motivation: SIMP’s penalized non-convex landscape creates local minima, making solutions dependent on initialization and offering no guarantee of global optimality.Multi-start strategies mitigate but do not remove the structured exploration problem.
Motivation: Rib design is especially challenging because rib number, orientation, height, spacing, and branching topology form a combinatorial design space.Nature-inspired patterns such as leaf venation and trabecular bone may lie outside local optima reached by SIMP.
Contributions: A frozen Stable Diffusion 2.1 provides training-free, prompt-controlled design knowledge that couples to SIMP without task-specific training or fine-tuning.Retargeting changes the prompt, while each engineering problem still requires its own geometry, physics, constraints, and numerical setup.
Mechanism: Dead-end suppression is the predominant structural signature of improvement, with endpoint–compliance correlations of r = +0.56 to +0.99 across domains.The reported mechanism is removal of non-load-bearing dead ends rather than addition of branches.
Contributions: 38 of 49 prompt–domain combinations achieved statistically significant compliance reductions, with improvements linked to dead-end suppression and reduced gray density.The framework combines a training-free diffusion prior with topology optimization and reports reductions from 42.6% to below 3% in intermediate densities.
Numerical remedy: Heaviside projection with β-continuation addresses the SDS–SIMP gray-density tendency, reducing intermediate densities from 42.6% to below 3%.The schedule synchronizes binarization with generative guidance.

2. Related work

Prior generative topology-optimization methods rely on domain-specific datasets or provide limited text and physics coupling. Score distillation supplies a training-free route toward spatially detailed, prompt-guided structural optimization.

Data-driven models: TopoDiff trained on approximately 33,000 topology-optimization solutions and reported an eightfold reduction in compliance error versus earlier generative adversarial network approaches.The comparison illustrates both the promise and dataset dependence of data-driven methods.
Score distillation: Score distillation transfers gradients from a frozen text-to-image model into an optimization loop without training data.The approach was introduced for text-to-3D generation and later applied to inverse problems.
Text-driven design: CLIP-based topology optimization imposes text-based styles but provides global image–text similarity rather than spatially detailed pixel-level generative gradients.This distinguishes CLIP guidance from the gradient-level coupling targeted here.
Research gap: Existing data-driven methods require large domain-specific datasets, generalize poorly beyond training conditions, and lack intrinsic guarantees of physical feasibility or optimality.Their encoded design knowledge is implicit and often inaccessible to engineers at design time.
Research gap: The paper positions frozen-model gradient coupling as a way to combine transferable semantic guidance with active FEA-based physical optimization.This addresses the separation between generated designs and physical analysis in prior approaches.

3. Methodology

The method combines a Mindlin–Reissner or plane-stress FEA gradient with an SDS gradient from frozen Stable Diffusion, updates filtered and projected densities, and converts converged fields into CAD-convertible geometry.

Iterative optimization: Each iteration combines the FEA compliance gradient and SDS text gradient in a scheduled density-field update.Warmup stabilizes physics, cosine cooldown removes SDS influence, timestep annealing provides coarse-to-fine guidance, and EMA reduces stochastic noise.
Post-processing: The optimizer uses Adam with a bisection volume constraint and Heaviside projection, then processes converged fields by binarization, thinning, Bézier fitting, and extrusion.The resulting candidate geometries are intended to be CAD-convertible.
Problem formulation: The density field is filtered to suppress checkerboards and passed through a smoothed Heaviside projection toward near-binary physical densities.The formulation distinguishes raw, filtered, and physical densities while enforcing a target volume fraction.
Physics model: Mindlin–Reissner plate elements model the three plate-bending domains, while the industrial link domain uses plane-stress bilinear elements.The plate model includes bending and transverse shear effects relevant to rib-reinforced regions.
Score distillation sampling: SDS renders physical density as an image, encodes it with a frozen VAE, adds diffusion noise, and backpropagates guided noise discrepancies to the density field.Classifier-free guidance combines text-conditioned and unconditional U-Net predictions before the gradient is combined with physics.
Optimization update: The SDS–physics update is a heuristic gradient blend rather than the exact gradient of a single scalar objective.Its behavior is controlled empirically through gradient scaling and the scheduled SDS weight.

4. Experimental setup

Experiments cover four geometric domains, two physics regimes, ten prompts, and five random seeds per comparison. The setup measures statistical significance and practical effect sizes while keeping SDS runs computationally close to pure SIMP.

Domains and physics: The study evaluates three plate-bending domains and one industrial link domain under specified volume fractions and distinct finite-element physics.Plate domains use Mindlin–Reissner theory, while the link uses plane stress with combined tension–shear loading.
Text prompts: Ten prompts span biomimetic, engineering, and geometric categories, with a visual prefix aligning density images to the diffusion model’s training distribution.Negative prompts suppress blur, color, 3D cues, and details below the filter radius.
Compute: Each run uses 1000 iterations and takes approximately 44 minutes, adding about 3 minutes or 7% over the pure-SIMP baseline.Runs use an NVIDIA RTX 4090 with 24 GB VRAM.
Evaluation: Significance testing uses a one-sided binomial test across five seeds, while Cohen’s d measures practical effect relative to the baseline.The minimum p-value for all five seeds beating baseline is 0.031, and significant combinations have median |d| = 1.47.

5. Results

Across four domains and two physics regimes, SDS-guided topology optimization often found lower-compliance designs than SIMP and alternative gradient-based baselines. Improvements depended on prompt–load-path alignment, with the largest mechanical gains in the holed domain and significant thermoelastic gains across most settings.

5.1. SDS-guided optimization discovers superior topologies: 9 of 10 prompts significantly improved DCircle, while only 5 of 10 did so in DRect, whose efficient SIMP solution left less headroom.Diamond led DCircle at −15.2%, whereas Spiderweb led DRect at −11.9%.
5.1. SDS-guided optimization discovers superior topologies: −31.5% was the largest mechanical improvement, achieved by Spiderweb in DHole, where all ten prompts attained 5/5 wins.Spiderweb led the domain at −28.4% by five-seed mean; the best single run reached compliance 43.20.
5.2. Comparison with alternative baselines: The fixed-target image-guided variant could not recover SDS’s improvement because it lacked the adaptive topology reshaping of the iterative, state-dependent gradient.Multi-start remained limited to basins reachable from unstructured Gaussian-noise initializations.
5.2. Comparison with alternative baselines: The best SDS run reached lower compliance than MMA in DCircle (2.522 vs. 2.598) and DHole (43.20 vs. 48.85), but not DRect (7.886 vs. 7.721).The comparison uses matched single-run results; five-seed SDS means vary across prompts.
5.6. Generalization and scalability: Retargeting to a new domain, boundary condition, or physics objective required a single optimization run without task-specific retraining or labeled data.The stated cost contrast is with data-driven alternatives requiring large solution datasets and thousands of GPU-hours.
5.4. Load-path alignment: Prompt performance followed load-path alignment: Spiderweb led radially loaded domains, whereas Grid degraded radial cases but improved to −9.1% in asymmetric DLink.Spiderweb ranked ninth in DLink at −2.6%, while Grid ranked fourth there.
5.5. Generalization to thermoelastic loading: Across thermoelastic settings, 7 of 9 prompt–domain combinations achieved 5/5 wins and the remaining two achieved 4/5.The domain difficulty ranking reversed relative to mechanical loading, with DRect becoming more responsive under uniform thermal loading.

6. Mechanisms of improvement

SDS-guided improvement was associated with simpler rib skeletons, especially through removal of non-load-bearing dead ends, while prompt–load-path alignment determined how effective that mechanism was. Scheduling and projection choices were necessary to couple generative exploration with physical optimization and obtain manufacturable density fields.

6.1. Structural signature: dead-end suppression: SDS-guided designs usually removed non-load-bearing dead ends rather than adding branches, simplifying the rib skeleton.The baseline retained free-ending ribs that did not reach supports, whereas the SDS-guided network rerouted or removed them.
6.1. Structural signature: dead-end suppression: Endpoint count correlated positively with compliance in every tested domain, with Pearson r = +0.56 to +0.99 and median r = +0.97.Branch-point count showed no consistent sign across domains.
6.2. Physical interpretation: Removing dead ends frees material for primary load paths because such branches carry near-zero stress while still consuming the volume budget.This provides the physical rationale for endpoint count tracking compliance more consistently than branch count or connectivity alone.
6.2. Optimization-level consequence: The SDS gradient altered the effective energy landscape, enabling transitions to basins inaccessible under the physics landscape alone.The convergence histories separate SDS-active and physics-only phases while comparing best-prompt, worst-prompt, and SIMP trajectories.
6.3. Boundary condition: load-path alignment: Dead-end suppression was most effective when prompt geometry aligned with dominant load paths; mismatch produced the +6% degradations of Diamond and Grid in DRect.Load-path alignment determined the effectiveness of the common pruning mechanism.
6.4. Resolving the gray-density tendency: Without correction, SDS produced a 42.6% gray-density ratio versus approximately 20% for pure SIMP.The continuous pixel-space score function biases the density field toward intermediate values.
6.4. Resolving the gray-density tendency: Synchronizing Heaviside β-continuation with SDS scheduling reduced gray density to 2.5% at negligible compliance cost.β doubled every 50 iterations from β0 = 1 to βmax = 64, allowing early exploration and later binarization.
6.5. Scheduling component ablation: Gradient-weight scheduling had the largest ablation impact, with λ_sds = 1 causing +21.4% degradation and λ_sds = 20 causing +6.9%.The optimum λ_sds = 10 balanced the SDS and physics gradients; timestep annealing was the second most influential component.

7. Discussion

The discussion frames SDS as a physics-filtered design prior whose benefits depend on non-convexity, prompt–load-path alignment, and suppression of non-load-bearing skeleton dead ends. It also identifies gray-density control, prompt screening, and dimensionality as practical boundaries.

Methodological significance: The method couples a frozen visual prior to finite-element sensitivities at every iteration, allowing physics to filter prompt-induced morphology instead of accepting visually plausible but physically suboptimal designs.This extends text-guided design intent beyond fixed parametric bases and avoids task-specific training or fine-tuning.
Limitations and deployment: Prompt selection remains manual because mismatched prompt morphology can oppose the physics gradient and increase compliance by 6% for Diamond/Grid in DRect.The study therefore motivates screening several structural prompt categories before full evaluation.
Practical stabilization: Heaviside projection with synchronized β-continuation reduces the SDS-induced intermediate-density ratio from 42.6% to below 3%.The discussion presents this as a way to reconcile continuous generative guidance with binary density fields.
Limitations and deployment: The framework is currently 2D, and extending it to 3D requires resolving the mismatch between 3D density fields and 2D diffusion priors, including discretization artifacts.The paper also identifies retuning of SDS weight across physics regimes and β-continuation as transferable implementation ideas.

8. Conclusions

The paper concludes that a frozen diffusion prior can be coupled with density-based topology optimization to turn natural-language intent into physics-arbitrated rib geometry. Across broad experiments, the approach improves many prompt–domain combinations, while its scope is bounded by the physics landscape and current 2D formulation.

Framework: The framework couples frozen Stable Diffusion 2.1 to a SIMP solver through score distillation sampling, using text prompts as explicit design-intent representations filtered by finite-element sensitivity.Heaviside projection and β-continuation reduce gray densities, while skeleton processing produces CAD-convertible candidate STEP geometry.
Overall findings: 38 of 49 prompt–domain combinations achieved statistically significant compliance reductions across 245 primary SDS runs, outperforming multi-start, perturbation, and image-guided baselines.The gains remained significant after Benjamini–Hochberg correction and had median Cohen’s |d| = 1.47.
Implications: The results support using pretrained generative models as reusable design interfaces, especially for physics problems where gradient-based optimizers are severely trapped by local optima.The paper proposes future extensions to 3D priors, textual inversion, fluid–structure interaction, and metamaterial design.

Data availability

The study is training-free and uses a publicly available pretrained diffusion model rather than generating or using a training dataset.

Data availability: No training datasets were generated or used; all numerical results are computational outputs determined by the code, inputs, and documented random seeds.The framework relies on the publicly available pretrained Stable Diffusion 2.1 model.

CRediT authorship contribution statement

The contribution statement assigns the project’s conceptual, technical, analytical, and writing roles across Yongmin Kwon and Namwoo Kang.

Contributions: Yongmin Kwon contributed across conceptualization, methodology, software, validation, visualization, analysis, investigation, and original-draft writing.He also handled data curation and resources.
Contributions: Namwoo Kang contributed project administration, supervision, investigation, funding acquisition, and review and editing.

A. Finite element and SIMP modeling details

The model combines Mindlin–Reissner plate analysis with modified SIMP interpolation, filtering, projection, and sensitivity calculations to optimize rib-reinforced plates under physical and manufacturing constraints.

Finite-element model: Mindlin–Reissner theory captures transverse shear deformation in rib regions, while MITC4 elements avoid shear locking in the finite-element implementation.The formulation uses three degrees of freedom per node: transverse displacement and two rotations.
SIMP interpolation: Modified SIMP interpolates bending and shear stiffness between rib-supported and base-plate regions, preserving nonzero stiffness where ribs are absent.This avoids treating zero rib density as a physically empty plate region.
Manufacturing constraints: Density filtering suppresses checkerboards and controls minimum feature size, while a local-volume penalty limits excessive rib thickness and disperses concentrated material.The study uses r_min = 5 and r_max = 10 to balance manufacturability against design freedom.
Binarization: Heaviside projection with β-continuation converts filtered densities toward binary designs while avoiding the instability caused by starting with a sharp projection.β starts at 1, doubles every 50 iterations, and reaches β_max = 64; SDS forms global patterns before later binarization.
Optimization sensitivities: The compliance sensitivity is propagated through stiffness interpolation, Heaviside projection, and density filtering to form the physics gradient used in optimization.Self-adjoint compliance enables sensitivity computation without an additional linear-system solve, while volume enforcement uses bisection compatible with Adam and stochastic SDS gradients.

B.4. Optimizer: Adam vs. OC

Adam is selected over OC because it handles arbitrary SDS gradients and remains stable with sharp Heaviside projections. OC exhibits oscillation and degraded baseline designs, whereas Adam provides the conventional SIMP reference.

Optimizer choice: Adam is used instead of OC because adaptive moments accommodate arbitrary SDS gradients and damp Heaviside-induced oscillations.The learning rate decays from 0.04 to 0.01 using cosine annealing.
Optimizer choice: OC combined with Heaviside projection exhibits period-2 oscillation at high β values because its bisection update interacts with the projection nonlinearity.Adam’s momentum and learning-rate decay stabilize the update.
Baseline comparison: OC produces poor no-SDS baselines: compliance is 2.36× worse than Adam in D_Hole and 1.84× worse in D_Rect, while D_Circle collapses to volume fraction V = 0.13.The D_Circle field becomes nearly empty, and D_Hole is severely fragmented.
Baseline comparison: The baseline comparison is reported in Table B.1 for Adam and OC under the same SIMP setting and target volume fraction V* = 0.20.

C. Post-processing details: density field to CAD geometry

A six-stage pipeline converts optimized continuous density fields into CAD-convertible rib geometries through binarization, skeleton-based graph extraction, Bézier fitting, thickness recovery, and extrusion.

Pipeline overview: The post-processing pipeline transforms a density field into CAD geometry through binarization, skeletonization, graph construction, Bézier fitting, and STEP export.The resulting candidate geometry is intended for computer-aided design and engineering workflows.
Binarization: Binarization thresholds the density field at ρ_th = 0.5, while β = 64 already reduces gray density below 3%, eliminating the need for morphological cleanup.
Skeleton and graph extraction: Skeletonization extracts the rib network’s medial axis, and graph extraction classifies branch points and endpoints before tracing paths into graph edges.
Curve fitting: Each graph edge is approximated by piecewise cubic Bézier curves, with corners detected by direction changes exceeding 30° and control points fitted by least squares.
CAD export: Distance-transform widths expand Bézier centerlines into closed polygons, which are extruded into 3D solids and exported as AP214 STEP files.A default extrusion height of 5 mm is used, with an optional base plate.
Evaluation support: Matched-budget evaluations compare 50-run SIMP baselines with five-run SDS subsets, while seed-level plots expose individual compliance outcomes across prompts and domains.
Gray-density ablation: Without Heaviside projection, SDS produces a 42.6% gray-density ratio; β-continuation reduces it below 3% without a compliance penalty at β_max = 64.The reported compliance is C = 9.29 at β_max = 64 versus 9.93 at β_max = 32.
Failure modes: SDS can degrade performance when prompts over-constrain load paths, with Diamond and Grid prompts worsening D_Rect compliance by +6.0% and +5.9%.The SDS gradient then opposes the physics gradient, producing a compromise optimized for neither objective.

D.5. Prompt ablation: isolating the semantic contribution

Matched-seed prompt ablations show that structural semantics and endpoint count track compliance, while supplementary tests support robustness across mesh resolution, 3D re-analysis, and dead-end interventions.

D.5. Prompt ablation: isolating the semantic contribution: −11.9% from the Spiderweb prompt outperformed empty and prefix-only controls at −3.6% and −4.5%, while endpoint count and compliance change correlated at r = 0.87.The dot prompt reached +7.3% with 0/5 winning seeds and n_end = 49, whereas Spiderweb had n_end = 19.
D.5. Prompt ablation: isolating the semantic contribution: The thermoelastic SDS-weight sweep produced a U-shaped response, selecting λ0_sds = 100 before mild saturation at −20.0% for λ0_sds = 200.The optimum shifted toward larger weights than in the mechanical setting, consistent with the larger and more diffuse thermal physics gradient.
D.7. Mesh refinement and the filter length scale: Fixing filter radii as rmin = N/30 and rmax = N/15 preserves physical feature size, whereas fixed element-count radii would reintroduce mesh-dependent detail.The refinement study used DHole because its four-sided distributed loading provides a well-conditioned convergence setting.
D.7. Mesh refinement and the filter length scale: SDS improvement persisted across N = 100, 150, and 200 when the filter radius was fixed physically, yielding −19.3%, −28.4%, and −10.0% with the Spiderweb probe.The baseline topology remained qualitatively consistent across resolutions, supporting refinement robustness.
D.8. 3D re-analysis of the protruding-rib geometry: 2D and 3D compliance rankings were largely preserved, with cross-model correlations of r = 0.92, 0.82, and 0.98 for DRect, DCircle, and DHole.Spiderweb ranked first in both models for DRect, while the best 2D prompts remained among the strongest 3D designs in the other domains.
D.9. Within-design causal test of dead-end removal: Dead-end pruning raised compliance only +5.9% on average, compared with +61.5% after pruning one load-bearing rib and less than 1% after grafting a dead end.The intervention preserved surrounding topology and attributed SDS improvement to suppressing low-load dead ends rather than removing load-bearing ribs.