Feature-Mapping Topology Optimization with Neural Heaviside Signed Distance Functions

Thank you for your thorough and insightful review! We are grateful for your detailed comments, which have not only highlighted the strengths of our methodology but also pointed out key areas for improvement, ultimately helping us enhance the clarity and impact of our research.

Refactoring mechanism. During topology optimization, reconstructed geometry can significantly deviate from the Heaviside function boundaries because the latent representation $Z$ extends beyond the learned shape distributions. To address this, we initially reprojected latent vectors into the correct regions using an additional objective function, but this hindered convergence. Our refactoring mechanism instead steers latent vectors back to their learned distribution and potentially allows for geometric corrections during optimization, such as enforcing minimum radii or correcting overlapping arcs.

Of course, we did not try every possible variant that might address the shortcomings of our method. In the future, we plan to explore alternative approaches to improve the algorithm.

Heaviside-“SDF” representation. We chose to use the Heaviside-SDF representation instead of a direct SDF during model training for the following reasons:

1. The Heaviside transformation produces a sharper, more distinct boundary—exactly what we need for our application.

2. This transformation enhances FMTO's robustness to neural network noise. Using a direct SDF requires applying the Heaviside function later, which significantly amplifies boundary noise occurring in the model.

Motivation for the method. Our method is motivated by the common industry practice of creating parts by extruding sketches—typically polygons with rounded corners—for milling and casting. We developed a framework that allows polygons to evolve from an initial circle during optimization, producing more natural and manufacturable designs. Additionally, we plan to extend the framework to include other common geometric features, such as polygons with arc segments.

Essential References. Thank you for the note. These directions are indeed closely related to our work, and we would be happy to include them in our review.

Comparison with other methods. We can provide more comparisons with other methods, including one recently published work $`TreeTOp`$ [1], which directly relates to FMTO.

1. K. Padhy, R., Thombre, P., Suresh, K. et al. Treetop: topology optimization using constructive solid geometry trees. 2025.

Table: Methods comparison for Example 3: MBB beam half

Table: Methods comparison for Example 4: Beam Distributed Load

$**Note:**$ Since the NTopo method is quite limited in altering the parameters of the initial conditions, we had to adjust the parameters of the other methods to match those of the NTopo method. Therefore, the metric results differ from those in the original manuscript.

Our approach achieves the best Compliance metric values among all FMTO methods while using less material. Additionally, in some experiments, our method is comparable to the NTopo method (see Example 3).

“Cantilever” is misspelled, incorrect citation, “linear elasticity” keyword. Thank you for pointing this out. We will correct the spelling error, add the “linear elasticity” keyword, and fix the citation.

What is $\xi_e$ and how does it differ from $\xi$ in Eqs. (7-8). $\xi$ represents any point within the design domain and is used to train the model. In contrast, $\xi_e$ denotes the center point of element $e$.

Why are there 6 entries in $\xi$ in Fig. 3? Isn’t $\xi$ a (x,y,z) coordinate? We apologize for the confusion caused by our previous depiction of a batch size of 3. We have updated the scheme to display a batch size of 1. Our experiments focus on 2D problems, so only two coordinates (x, y) are used.

Shape code $X$. We apologize for the missing definitions. Inconsistencies in the method description have been corrected, and an updated version—including details on the shape code $\chi$—will appear in the main text, as well as a revised Figure 4. Additionally, a table of notations will be added to the appendix.

Thank you very much for taking the time to provide such a detailed and insightful review. Your comment has been extremely valuable in enabling us to carefully analyze various aspects of our method's implementation, including the integration of the Kreisselmeier-Steinhauser (KS) function.

Using Sigmoid instead of Heaviside. The use of the term "sigmoid" is, of course, acceptable. However, the Heaviside function is a more traditional term in the context of topology optimization. We, like many other authors in this field, prefer to retain the possibility of using various smooth approximations of the Heaviside function without changing the name of the method.

Higher Degrees of Complexity. Adding new primitives forces an increase in latent space dimensions to keep accuracy, which can harm convergence. However, since our latent space captures overall contours rather than exact parameters, we believe that there's a threshold beyond which more dimensions are unnecessary. We will investigate this further.

Kreisselmeier-Steinhauser (KS) function. The current implementation includes several additional details that are not covered in the main text.

It is important for us to maintain sensitivity with respect to the objective function; however, to ensure the method does not fail, we are forced to clamp the $\rho$ parameter because the KS function can produce values greater than 1. Therefore, in addition to equation (11), we use the following formula to scale $\rho$:

$$\\rho_e = \\frac{1 - \\rho_{\\min}}{KS_{\\max} - KS_{\\min}} \\left(KS_{\\max} - \\frac{\\ln \\sum_{m=1}^{M} \\exp(\\gamma_{KS} \\widetilde{H_{m,e}})}{\\gamma_{KS}}\\right) + \\rho_{\\min}$$

where $KS_{\min}$ is the minimum value of the KS function, which is equal to

$$KS_{\min} = \\frac{\\ln P}{\\gamma_{KS}}$$

and $KS_{\max}$ is the maximum value of the KS function, which is equal to

$$KS_{\\max} = \\frac{\\ln (P\\exp(\\gamma_{KS}))}{\\gamma_{KS}}$$

where $P$ is the expected maximum number of geometric primitives intersecting at one point (by default, this is 2).

Therefore, the combined $\rho$ will exceed the range $[\rho_{min}, 1]$ only at intersections where more than $P$ geometric primitives overlap. In these cases, $\rho$ is clamped to the range and becomes insensitive to the objective function.

In our case, the primary shape for topology is a non-overlapping primitive within which a void must form. For large values of P but small values of $\gamma_{KS}$, the value of $\rho$ inside the primitive becomes considerably higher than $\rho_{min}$. Therefore, we choose $\gamma_{KS}$ to be as large as possible, while ensuring that computations do not become impractical due to excessively large values of $\exp(\gamma_{KS})$.

Changing $\gamma_{KS}$ above 10 has little impact on convergence or final topology, whereas lower values worsen convergence because $\rho_e$ remains too high for void formation.

Table: experiments with different $\gamma_{ks}$ values for Example 3: Bracket

B-spline and Bézier-based Feature-Mapping Methods. Unfortunately, most published works in topology optimization come without accompanying code and do not always provide enough details for reproduction. However, we have the opportunity to compare our method with a recently published approach [1], where the main geometric feature is a polygon constructed from half-spaces. These comparisons will be added to the manuscript.

[1] K. Padhy, R., Thombre, P., Suresh, K. et al. Treetop: topology optimization using constructive solid geometry trees.

Table: Methods comparison for Example 3: MBB beam half

$**Note: **$ Since the NTopo method is quite limited in altering the parameters of the initial conditions, we had to adjust the parameters of the other methods to match those of the NTopo method. Therefore, the metric results differ from those in the original manuscript.

Our approach achieves the best Compliance metric values among all FMTO methods while using less material. Additionally, in some experiments, our method is comparable to the NTopo method (see Table with Example 3).

Thank you for your comprehensive and constructive feedback on our work. We value your insights on the VAE encode-decode structure, the Neural Heaviside SDF representation, and the potential challenges related to geometric approximation and boundary precision.

Novelty of the Proposed Method. We acknowledge that VAEs have been widely used for learning shape distributions. However, our work differentiates itself by integrating a learned neural approximation of the Heaviside SDF function within Feature Mapping Topology Optimization (FMTO) tasks. Conventional FMTO approaches rely on explicitly defined SDFs, restricting the geometric diversity that can be effectively represented. Our approach, in contrast, enables a unified latent space where multiple geometric primitives can coexist and transform smoothly, facilitating shape evolution in FMTO beyond conventional methods.

Moreover, although sigmoid and Heaviside functions are commonly used for defining geometric boundaries, our method employs a learned neural surrogate. This allows for an adaptive, data-driven boundary representation, enhancing both optimization flexibility and saving manufacturability.

Smooth Transitions Between Geometric Features. Your concern regarding smooth transitions, particularly between ellipses and polygons, is well-taken.

While our dataset does not explicitly include gradual shape interpolations between such forms, our model demonstrates an ability to approximate smooth deformations even beyond the trained shape classes. As illustrated in Figure 6 of the main text, the learned latent space permits meaningful shape variations, even if exact interpolations between distinct classes are not explicitly encoded. Nevertheless, we recognize that the latent space could be further refined to better support smooth morphing between different geometric classes. Future work may explore improved training strategies, such as incorporating intermediate transition shapes or using additional regularization to better structure the latent space for smooth interpolations.

SDF Precision and Boundary Fitting. We understand the importance of accurately capturing high-curvature regions and sharp edges in SDF representations. To reduce inaccuracies in boundary fitting, we have enhanced our dataset generation strategy as follows:

Each shape type in our dataset now includes 5,000 instances, sampled to ensure comprehensive coverage of geometric variations.

Additionally, we have modified the point generation method. Out of 10,000 points per shape, one third are now generated directly along the shape boundaries, improving precision and ensuring that the trained SDF captures intricate features more effectively. The remaining points are evenly distributed between those drawn from a Gaussian distribution centered around the shape and those concentrated near vertices and edges.

These improvements have significantly enhanced boundary fidelity, minimizing discrepancies between original and reconstructed shapes. The updated dataset and training approach result in more accurate SDF modeling, especially in areas with high curvature and sharp transitions.

Discussion on Related Work. We appreciate the reviewer’s suggestion to better position our contributions within the existing literature. While previous works [1,2,3] have showcased the use of VAEs for shape learning, our study uniquely incorporates this framework into FMTO with a learned Heaviside SDF representation. We will expand our discussion in the manuscript to clearly contrast our method with prior approaches and emphasize our contributions more explicitly.

Once again, we sincerely thank the reviewer for their valuable feedback. We believe that our revisions effectively address the concerns raised and further strengthen our work.

Thank you for your thorough and insightful review!! We deeply appreciate the time and effort you invested in evaluating our work.

Comparison with other methods Our method follows the FMTO approach by creating topology with geometric primitives. Unlike traditional free-form optimization that focuses only on compliance, FMTO constrains voids to specific shapes. Thus, comparing with other FMTO methods is most appropriate, even though many lack available code and full reproducibility details.

Nevertheless, we can still provide more comparisons with four methods, including two new frameworks: the first competitor is a free-form method based on deep learning, $`NTopo`$ [1], and one recently published work $`TreeTOp`$ [2], which directly relates to FMTO. These comparisons will be added to the manuscript.

Table: Methods comparison for Example 3: MBB beam half

Table: Methods comparison for Example 4: Beam Distributed Load

$**Note:**$ Since the NTopo method is quite limited in altering the parameters of the initial conditions, we had to adjust the parameters of the other methods to match those of the NTopo method. Therefore, the metric results differ from those in the original manuscript.

Our approach achieves the best Compliance metric values among all FMTO methods while using less material. Additionally, in some experiments, our method is comparable to the NTopo method (see Example 3).

1. J. Zehnder, Y. Li, S. Coros, and B. Thomaszewski. NTopo: mesh-free topology optimization using implicit neural representations. 2021.

2. K. Padhy, R., Thombre, P., Suresh, K. et al. Treetop: topology optimization using constructive solid geometry trees. 2025.

Training and inference During training, we optimize our model to best approximate the Heaviside SDF for various geometric primitives, independent of topology optimization. At inference, we use the frozen Heaviside decoder for SDF evaluation in FMTO and for geometry reconstruction during post‐processing.

Constraints $V_{max}$, $s_{max}$, and $s_{min}$ are enforced at inference—$V_{max}$ controls the design domain volume via the Lagrangian, while $s_{max}$/$s_{min}$ bound the shape variables via a sigmoid.

We have clarified our method by distinctly separating training from inference. These changes improve clarity and reproducibility.

Table 1, Bolding Best metric values in Table 1 are bolded. Due to layout limitations, the table was split; we apologize for the confusion and will restore the original format.

Statistical Significance We conducted a t-test on 20 independent runs, comparing each model to the best-performing model. The results are presented in the table below and and be provided in the revised appendix, confirming our choice of VAE architecture and training strategy based on $MSE_{sdf}$.

Table: Metric: $MSE_{sdf}$ (p-values computed relative to st1_VAE_DeepSDF)

In the case of Table 2, which presents the comparison results with other methods, we did not perform a t-test because these methods are deterministic with respect to the input parameters.

Notations We apologize for the missing definitions. Inconsistencies in the method description have been corrected, and an updated version—including details on the shape code $\chi$—will appear in the main text, as well as a revised Figure 4. Additionally, a table of notations will be added to the appendix.