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Feature-Mapping Topology Optimization with Neural Heaviside Signed Distance Functions
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Proposes Neural Heaviside SDFs: a VAE-style network that approximates the Heaviside of feature SDFs in a shared latent space, used as a differentiable decoder for feature-mapping topology optimization of ellipses, triangles and quadrangles.
Problem
Feature-mapping topology optimization is limited by the small set of geometric features, their fixed parameterizations, and the need for differentiable SDFs, which leads to manufacturing-friendly but restricted designs.
Approach
Train an encoder and Heaviside decoder (DeepSDF-style) to map a one-hot shape code of several feature types into a unified latent space, then optimize latent codes with adjoint gradients against compliance, with a refactoring step to keep latents feasible.
Key results
- On Task 1, the method reaches compliance 3.91e-5 at volume fraction 0.255, versus 4.08e-5 for NTopo and 5.85e-5 for TreeTOp.
- Among feature-mapping methods, the method achieves the lowest compliance on most tasks, e.g. Task 2 compliance 1.98e-3 vs 2.22e-3 for the ellipse baseline with less material (0.354 vs 0.368).
- Training the latent via the Heaviside decoder with a VAE and DeepSDF decoder gives MSE_Hv of 0.000277 versus 0.00128 for a symmetric decoder variant.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use Neural Heaviside SDF instead of Ellipses FMTO | FMTO topology optimization · Task 1 | compliance: 4.39 (0.297) -> 3.91 (0.255) | Helped |
| use Neural Heaviside SDF instead of Ellipses FMTO | FMTO topology optimization · Task 2 | compliance: 2.22 (0.368) -> 1.98 (0.354) | Helped |
| use Neural Heaviside SDF instead of Ellipses FMTO | FMTO topology optimization · Task 3 | compliance: 1.74 (0.449) -> 1.63 (0.437) | Helped |
| use Neural Heaviside SDF instead of Ellipses FMTO | FMTO topology optimization · Task 4 | compliance: 1.55 (0.345) -> 1.52 (0.337) | Helped |
| use Neural Heaviside SDF instead of TreeTOp | FMTO topology optimization · Task 5 | compliance: 1.57 (0.449) -> 1.55 (0.409) | Helped |
| use Neural Heaviside SDF instead of TreeTOp | FMTO topology optimization · Task 6 | compliance: 3.09 (0.439) -> 2.34 (0.401) | Helped |
| replace DeepSDF decoder with symmetric decoder (ablation) | VAE Heaviside SDF approximator · Neural Heaviside SDF test dataset | MSEHv: 0.000277 ± 1.07e-05 -> 0.00128 ± 0.000201 | Hurt |
| replace DeepSDF decoder with symmetric decoder (ablation) | VAE Heaviside SDF approximator · Neural Heaviside SDF test dataset | Smoothness: 0.00575 ± 7.11e-05 -> 0.0163 ± 0.000574 | Hurt |
| train the latent representation via Reconstruction decoder instead of Heaviside decoder (ablation) | VAE with DeepSDF decoder · Neural Heaviside SDF test dataset | MSEHv: 0.000277 ± 1.07e-05 -> 0.000475 ± 3.51e-05 | Hurt |
| train the latent representation via Reconstruction decoder instead of Heaviside decoder (ablation) | VAE with DeepSDF decoder · Neural Heaviside SDF test dataset | MSEχ: 0.000202 ± 2.82e-05 -> 7.42e-05 ± 1.09e-05 | Helped |
Surprising findings
- Reconstruction decoder corrupts shape transitions between feature types in the latent space, while the Heaviside decoder yields accurate intermediate contours.
Limitations
- Poor transitions between different geometric feature types due to the reconstruction decoder.
- Many conflicting variables during optimization cause oscillation of offset variables; requires slowed optimization.
- Results depend on initialization; only ellipses, triangles, and quadrangles supported and only 2D problems.
Experimental setup
Datasets: synthetically generated geometric feature shape codes (ellipses, triangles, quadrangles) with Heaviside values
Benchmarks: six classical topology optimization tasks (Tasks 1-6)
Metrics: compliance volume fraction MSE of Heaviside MSE of shape code smoothness (gradient norm of Heaviside error)
Baselines: SIMP NTopo TopoDiff TreeTOp Ellipses FMTO (Kumar & Saxena, 2022)
Builds on: DeepSDF SIMP TreeTOp variational autoencoder
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