A BEM Adjoint-Based Differentiable Shape Optimization of a Stealth Aircraft
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis paper presents an adjoint-based boundary element method (BEM) for gradient-driven shape optimization of stealth aircraft, aiming to minimize radar cross-section (RCS). While the methodology demonstrates potential for industrial applications in high-fidelity electromagnetic optimization, the current version requires minor revisions to address critical gaps in validation, clarity, and technical rigor.
1.The missing term in the adjoint gradient computation (Section 2.7) significantly impacts optimization reliability. Provide a quantitative analysis of its contribution (e.g., error bounds, convergence rates) and revise the validation in Section 4.2 to include this term or justify its exclusion rigorously.
2.Key terms (e.g., "inner FD" vs. "outer FD" in Figure 7) and equations (e.g., Equation 13, 35) lack precise definitions. Reorganize Section 2.5–2.6 to explicitly link the continuous and discrete adjoint formulations, ensuring consistency in notation.
3.Avoid vague claims (e.g., "the error is likely due to neglecting term I" in Section 4.2) and replace with quantitative statements. Address grammatical errors (e.g., "he purpose" → "the purpose" in Section 1.3) and inconsistent tense usage (e.g., shifting between past/present in Section 5).
4.Acknowledge the prohibitive cost of BEM for large-scale industrial models (Section 3.2) and propose strategies (e.g., ACA compression, parallelization) to mitigate this. Clarify how the current implementation scales with mesh size and design variables.
5.The conclusion oversimplifies the trade-offs between BEM and PTD. Elaborate on scenarios where BEM’s fidelity justifies its computational cost and outline plans to integrate the missing adjoint term in future work.
6.Include recent studies on BEM adjoint-based methods. Consider these works to introduce the readers a complete and large picture of this dynamic development (e.g., Shape optimization of sound barriers/isogeometric meshless method, ACA-BM-SBM for sensitivity analysis, SBM for the thickness optimization of sound barriers).
Author Response
Comment 1: The missing term in the adjoint gradient computation (Section 2.7) significantly impacts optimization reliability. Provide a quantitative analysis of its contribution (e.g., error bounds, convergence rates) and revise the validation in Section 4.2 to include this term or justify its exclusion rigorously.
Response: Thank you for pointing this out. We are aware that the missing term significantly impacts the reliability, and we are not able at this point to provide a quantitative analysis of how much it does. However, we argue that eventhough this preliminary version of the formulation is less reliable than with a complete adjoint, given that the optimizer was able to reduce the cost while satisfying the constraint, and the given the gain in computational time, it retains some value in an industrial framework.
Comment 2: Key terms (e.g., "inner FD" vs. "outer FD" in Figure 7) and equations (e.g., Equation 13, 35) lack precise definitions. Reorganize Section 2.5–2.6 to explicitly link the continuous and discrete adjoint formulations, ensuring consistency in notation.
Response: Thank you for pointing this out. I have explicited the definitions. In section 2.6, the change in notations was meant to represent the difference between continuous operator and applications in section 2.5 and matrix and vectors in section 2.6. I have tried to explicit the link between the sections and the correspondance between the notations at the beginning of section 2.6.
Comment 3: Avoid vague claims (e.g., "the error is likely due to neglecting term I" in Section 4.2) and replace with quantitative statements. Address grammatical errors (e.g., "he purpose" → "the purpose" in Section 1.3) and inconsistent tense usage (e.g., shifting between past/present in Section 5).
Response: Thank you fro pointing out thiese mistakes. I have corrected the grammatical errors and fixed the tense. As for the phrase, "the error is likely due to neglecting term" I was tyring to explain that more study is needed to be able to estimate quantitatively to what extent the error is due to the neglected term. I have modified the sentense to exlpain this more clearly, hopefully.
Comment 4: Acknowledge the prohibitive cost of BEM for large-scale industrial models (Section 3.2) and propose strategies (e.g., ACA compression, parallelization) to mitigate this. Clarify how the current implementation scales with mesh size and design variables.
Response: Thank you for your comment. I have clarified how the cost of BEM have impacted the study and highlighted more clearly the proposed solutions.
Comment 5: The conclusion oversimplifies the trade-offs between BEM and PTD. Elaborate on scenarios where BEM’s fidelity justifies its computational cost and outline plans to integrate the missing adjoint term in future work.
Response: Thank you for your comment. I have expended upon the tradeoffs between BEM and PTD and elaborated on possible BEM-PTD hybrid scenario in section 6. I have also detailed how we plan to compute and integrate the missing adjoint term.
Comment 6: Include recent studies on BEM adjoint-based methods. Consider these works to introduce the readers a complete and large picture of this dynamic development (e.g., Shape optimization of sound barriers/isogeometric meshless method, ACA-BM-SBM for sensitivity analysis, SBM for the thickness optimization of sound barriers).
Response: Thank you for the suggestion. I was not familiar with meshless methods. I have added a paragraph to section 1.3. You can find it on page 3, lines 106-116.
Reviewer 2 Report
Comments and Suggestions for AuthorsIn general terms, paper entitled A BEM adjoint-based differentiable shape optimization of a stealth aircraft is recommend for publication on this journal, if the following recommendations are set:
Page 1 Line 23. Stealth (with capital letter), word beings a phrase.
Page 13 Figures 6a and 6b, figures caption is incomplete because figures 6c and 6d are covering them.
Page 15 line 343, Citation of figures must be in order or appearance: “ In figure 10a and 10b, …”
Page 15 line 346, correct the bearing angle, it must say “at around 15º ”. Not -15º.
Page 19 Figure 15, please check figure captions, they are cut. Also, I suggest to put different markers for the graphic lines (star, triangle, etc.) so a black & white version of the paper can be readable.
Page 21 figure 17, same remark as figure 15
Page 21 figure 17, for both optimal adjoint BEM and PTD Exact gradiant, there is a pick near 30º bearing angle, please explain the gain of RCS (dB) at this angle. High RCS at 15º is expected on the baseline for the three methods.
Page 22 Figure 18 same remarks as figure 15 and 17
Page 22 Figure 19 same remark as figure 15, 17 and 18, add different markers
Page 24 Figure 21, same remark as last figures. For Baseline at PTD with 1 frequency, two peaks appear 15º (expected) and 48º, please explain the second pick. Then for 21 frequencies, picks at 15º and 48º vanish and a new one, 30º appears, please explain.
Author Response
Comments 1: Page 1 Line 23. Stealth (with capital letter), word beings a phrase.
Comments 4: Page 15 line 346, correct the bearing angle, it must say “at around 15º ”. Not -15º.
Response: Thank you for bringing these typo to my attention. I have corrected them.
Comments 3: Page 15 line 343, Citation of figures must be in order or appearance: “ In figure 10a and 10b, …
Response: Thank you for pointing this out. This has been corrected.
Comments 2 : Page 13 Figures 6a and 6b, figures caption is incomplete because figures 6c and 6d are covering them.
Comments 5 : Page 19 Figure 15, please check figure captions, they are cut.[…]
Response : Thank you for pointing this out. I have corrected the problem, as well as on figure 17.
Comments 5 : Page 19 Figure 15, […] I suggest to put different markers for the graphic lines (star, triangle, etc.) so a black & white version of the paper can be readable.
Comments 6 : Page 21 figure 17, same remark as figure 15.
Comments 8 : Page 22 Figure 18 same remarks as figure 15 and 17.
Comments 9 : Page 22 Figure 19 same remark as figure 15, 17 and 18, add different markers.
Response : Thank you for the suggestion. I have modified the mentionned figures, as well as figure 11 accordingly.
Comments 7 : Page 21 figure 17, for both optimal adjoint BEM and PTD Exact gradiant, there is a pick near 30º bearing angle, please explain the gain of RCS (dB) at this angle. High RCS at 15º is expected on the baseline for the three methods.
Response: Thank you for your question. The peak in the 30° directions on the optimal shapes is due to the leading and trailing edge shifting to face the 30° direction. This is expected because of the chosen cost function. I have explained this better. You can find the revised explaination on parge 19, line 431.
Comments 10: Page 24 Figure 21, same remark as last figures. For Baseline at PTD with 1 frequency, two peaks appear 15º (expected) and 48º, please explain the second pick. Then for 21 frequencies, picks at 15º and 48º vanish and a new one, 30º appears, please explain.
Response: Thank you for your question. The peak at 48° corresponds to the specular reflexion on the baseline leading edge. The baseline RCS is adressed in section 5.1.2. The 30° peak on the 21 frequency optimization corresponds to the superimposition of the. specular reflexion on the leading edge and the diffraction by the leading edge, both of which have shifted to the 30° direction throughout the optimiaztion. I have explicited which peaks correspond to what and added a refernce to the relevant section to better explain the physics. You can find the modification on page 21, line 476 and page 24, line 477.
Reviewer 3 Report
Comments and Suggestions for AuthorsA BEM adjoint-based differentiable shape optimization of a stealth aircraft is studied. Due to the need for efficient and accurate methods to compute the gradient of high-fidelity radar cross-section computation methods concerning shape parameters. The authors employ an adjoint formulation, which allows for efficient computation of these gradients. According to the authors, this is particularly beneficial in shape optimization problems, where accurate and efficient methods are crucial for designing modern fighter aircraft with stealth capabilities.
The work is interesting. The presented results seem useful and correct. The overall article is good and publishable.
The paper needs revision, like:
- The work uses BEM, but the method was not well addressed and described in the introduction, especially regarding references.
- The motivation is clear, but what is the novelty or originality of the work?
- In line 121. Does the Green kernel for the Helmholtz equation refer to the Hankel function? What is the reference?
- In equation 17a, it was not clear how you went from the domain integral to the boundary integral.
- In the Green kernel for the Helmholtz equation, a complex term appears. How was this handled? At high frequencies, wouldn't this be a problem?
- The use of techniques such as dual reciprocity (DRM), radial integration method (RIM), and direct interpolation boundary element technique (DIBEM), which use a simple Green kernel for the Helmholtz equation, could it be applied?
Author Response
Comment 1: The work uses BEM, but the method was not well addressed and described in the introduction, especially regarding references.
Response1: Thank you for your comment. I had written an introductory section presenting and discussing BEM, but I had cut it for the sake of brevity. I have added a subsection to the introduction that is a summarized version of this. You can find on page 2, line 67 to 94.
Comment 2: The motivation is clear, but what is the novelty or originality of the work?
Response: Thank you for your comment. I have added a sentence to section 1.4 to explicit the originality of the work.
Comment 3: In line 121. Does the Green kernel for the Helmholtz equation refer to the Hankel function? What is the reference?
Response: Thank you for your question. In paper, I describe the derivation of the 3D Helmoltz equation, in which the Green kernel appears. A second kind Hankel function is used when deriving a 2D Helmoltz equation.
Comment 4: In equation 17a, it was not clear how you went from the domain integral to the boundary integral.
Response: Thank you for your question. The Stratton-Chu formula is a classic result, and the demonstration is quite lengthy, so I decided not to include it in this paper. I did however include a reference to Stratton and Chu's original paper. To summarize, the switch from domain integral to boundary integral is done using Green's formula. This does however imply inverting the derivation and sum signs, which means that the formula is not valid for points on the boundary. For more details, you may refer to Stratton's original paper : Diffraction theory of electromagnetic waves (1939) or his recently republished book Electromagnetic Theory (2007).
Comment 5: In the Green kernel for the Helmholtz equation, a complex term appears. How was this handled? At high frequencies, wouldn't this be a problem?
Response: I am sorry I am not sure I understood your question. The complex term does not appear in the Green kernel. The Green kernel is complex because the Helmotlz equation is itself complex, which in turn it is because the formulation of the Maxwell equations I used is inherently complex. It is a common practice to use complex valued functions to represent the magnetic and electric fields inherently wavy nature. The complex term does not pause any problem, however there are an infinite number of values for the frequency for which the BEM problem is singular. These values correspond to resonance frequencies. However, this is not a problem in the case of RCS computation. I elaborate on the matter on page 6, lines 154 through 157, at the end of section 2.3.
I hope that I understood your question and that you'll find my answers satisfactory.
Comment 6: The use of techniques such as dual reciprocity (DRM), radial integration method (RIM), and direct interpolation boundary element technique (DIBEM), which use a simple Green kernel for the Helmholtz equation, could it be applied?
Response: Thank you for this question. We are not familiar with these methods. From what I gather, their goal seems to be to reduce any domain integral left in a BEM problem into a surface integral, as BEM is very inefficient when domain integrals are involved. These domain integrals may be due in some applications to some body force terms for example. In electromagnetisms, these terms don't apply as far as I am aware. In the case of inhomogeneous, non-perfectly conductor materials, domain integrals may be used. This is outside my area of expertise so I don't know if theses methods could be applied to such a case, but a finite element method or a finite difference time domain method is usually prefered in this case anyway.
