A BEM Adjoint-Based Differentiable Shape Optimization of a Stealth Aircraft
Abstract
1. Introduction
1.1. Gradient-Based Optimization
1.2. The Boundary Element Method
1.3. Electromagnetic Gradient-Based Optimization
1.4. Contributions
2. RCS, BEM, and Adjoint Formulation
2.1. The Radar Cross-Section
2.2. Far Field
2.3. The Electric Field Integral Equation
2.4. The Boundary Element Method
2.5. The Adjoint Formulation
2.5.1. Continuous Formulation
2.5.2. Discrete Formulation
2.6. Computation of the Gradient
2.7. In Practice
3. Optimization Methodology
3.1. The CAD Modeler
3.2. Computational Electromagnetism Solver
3.3. Optimization Process and Algorithm
4. Validation
4.1. Validation of the BEM Solver
4.2. Validation of the Adjoint Gradients
4.3. Ellipsoid Optimization (Two Variables)
5. Aircraft Shape Optimization
5.1. Case Presentation
5.1.1. Baseline Description
5.1.2. Stealth Analysis of the Baseline
5.2. Optimization Parameters
- The leading edge sweep angle of sections S2 and S4;
- The chord of sections S2, S4, and S6;
- The distance between sections S2 and S6;
- The twist angle of sections S2, S3, S5, and S6,
- The leading edge camber of sections S3, S5, and S6.
5.3. Cost and Constraints
5.3.1. Criterion
Average over the Observation Window
Peak Directions
5.3.2. Constraints
5.4. Results
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
Electric field | |
Magnetic field | |
Density of electric current | |
Potential associated to | |
Density of electric charges | |
Magnetic permeability of vacuum | |
Absolute dielectric permittivity of vacuum | |
Impedance of free space: | |
Linear RCS () | |
Decibel RCS | |
Incident direction | |
Observation direction | |
j | |
k | Wave number: |
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Thoulon, C.; Roge, G.; Pironneau, O. A BEM Adjoint-Based Differentiable Shape Optimization of a Stealth Aircraft. Eng 2025, 6, 147. https://doi.org/10.3390/eng6070147
Thoulon C, Roge G, Pironneau O. A BEM Adjoint-Based Differentiable Shape Optimization of a Stealth Aircraft. Eng. 2025; 6(7):147. https://doi.org/10.3390/eng6070147
Chicago/Turabian StyleThoulon, Charles, Gilbert Roge, and Olivier Pironneau. 2025. "A BEM Adjoint-Based Differentiable Shape Optimization of a Stealth Aircraft" Eng 6, no. 7: 147. https://doi.org/10.3390/eng6070147
APA StyleThoulon, C., Roge, G., & Pironneau, O. (2025). A BEM Adjoint-Based Differentiable Shape Optimization of a Stealth Aircraft. Eng, 6(7), 147. https://doi.org/10.3390/eng6070147