Optimizing Initial Feature-Mapping Variables from Given Designs via Tracking

TL;DR

The paper presents a pill-based framework to reconstruct density fields from density-based topology optimization by representing designs with capsules defined by endpoints P,Q and radius r, mapped to a fixed grid via differentiable signed distances and transition functions. It derives closed-form first- and second-order derivatives for all primitives, enabling Newton-type optimization with exact Hessians and introducing an asymmetric transition and a reward-driven exploration stage to improve robustness. A staged optimization pipeline (exploration, bridging, convergence) with pruning, merging, and iterative refinement yields high-fidelity reconstructions on canonical benchmarks (e.g., five-bar, cantilever) using a moderate number of pills, and demonstrates advantages of exact Hessians over quasi-Newton methods. The work also analyzes aggregation operators (p-norm, softmax, Sum-Softcap), discretization effects, initialization strategies, and multi-pill interactions, offering practical insights into achieving accurate, interpretable, CAD-friendly reconstructions from voxel-based outputs. The methods show potential for extension to 3D, tapered primitives, and CAD-ready pipelines, establishing a bridge from density fields to explicit parametric models with rigorous analytic sensitivities.

Abstract

A feature-mapping framework for inverse reconstruction of density-based topology optimization results is proposed. Unlike SIMP, whose voxelized outputs are hard to interpret or reuse, the method represents designs with high-level geometric primitives mapped to a fixed analysis grid. Capsule-shaped bars (endpoints plus radius) are used, with closed-form signed distances and smooth transition functions providing derivatives up to second order. Differentiable pseudo-densities are aggregated with smooth operators, enabling gradient-based optimization with exact Hessians. Robustness is improved through asymmetric transition functions that propagate sensitivities into void regions, a reward-only objective for initialization, and geometric safeguards against degenerate configurations. Reconstruction is performed in stages (exploration, bridging, convergence) with optional refinement that can add, remove, or merge features based on residuals and geometric criteria. Experiments on canonical SIMP benchmarks, including five-bar and cantilever layouts, show high-fidelity reconstructions using a moderate number of features. p-norm and softmax aggregation yield sharp results; pruning removes redundant features and additive refinement restores coverage. Exact Hessians accelerate convergence and improve robustness compared to quasi-Newton updates, providing a bridge from voxel-based outputs to explicit parametric models.

Figures (87)

• Figure 1: Illustration of pill feature $\mathcal{F}(P,Q,r)$.
• Figure 2: Left: partition of the plane into regions where $d_{\mathrm{seg}}$ (blue) or the endpoint distances $d_{\mathrm{pt}}(\cdot,P)$, $d_{\mathrm{pt}}(\cdot,Q)$ (red/green) are minimal. Right: distance decomposition for a sample point $\boldsymbol{x}$, showing $d_{\mathrm{seg}}$, $d_{\mathrm{pt}}(\boldsymbol{x},P)$, $d_{\mathrm{pt}}(\boldsymbol{x},Q)$ together with the segment direction $u_0$ and normal $n$.
• Figure 3: Signed distance field of a pill, with negative values(blue) inside, zero on the boundary and positive values(red) outside.
• Figure 4: Tanh-based transition functions $\phi_\delta(d)$ for different steepness values $\beta$.
• Figure 5: Polynomial smoothstep functions $\phi_\delta(d)$ for different regularity classes $C^k$.