RCS Special Analysis Method for Non-Cooperative Aircraft Based on Inverse Reconfiguration Coupled with Aerodynamic Optimization

Abstract

To address the challenge of evaluating a radar cross-section (RCS) for a non-cooperative aircraft with limited aerodynamic shape information, this paper presents a multi-source, data-driven inverse reconstruction method. This approach integrates data fusion techniques to facilitate an initial shape reconstruction, followed by an iterative optimization process that utilizes computational fluid dynamics (CFD) to enhance the shape, accounting for the aerodynamic performance. Additionally, an inverse deduction analysis is effectively employed to ascertain the characteristics of the power system, leading to the design of a double S-curved tail nozzle layout with stealth capabilities. An aerodynamic analysis demonstrates that at Mach 0.6, the lift-to-drag ratio peaks at 27.3 for the attack angle of 4°, after which it declines as the angle increases. At higher angles of attack, complex flow separation occurs and expands with the increasing angle. The electromagnetic simulation results indicate that under vertical polarization, the omnidirectional RCS reaches its peak as the incident angle is deflected downward by 10° and reduces with the growth of the angle, demonstrating angular robustness. Conversely, under horizontal polarization, the RCS is more sensitive to edge-induced rounding. The findings illustrate that this methodology enables accurate shape modeling for non-cooperative targets, thereby providing a fairly solid basis for stealth performance evaluation and the assessment of surprise effectiveness.

1. Introduction

In modern combat environments, the continuous advancement of stealth technology presents increasing challenges in accurately identifying and assessing defense capabilities against non-cooperative aircraft. Stealth aircraft significantly enhance their survival probability by effectively lessening the radar cross-section (RCS), making its assessment a critical area of focus in air and space defense [1,2,3,4,5]. However, non-cooperative targets often lack publicly available design information, and their aerodynamic profiles typically depend on remotely sensed imagery, RCS characteristics, or profile data for inverse modeling and geometric reconstruction [6,7]. Consequently, the development of methods for aerodynamic shape reconstruction and RCS analysis has emerged as a core technology for identifying and evaluating the surprise capabilities of non-cooperative aircraft, thereby playing a vital role in enhancing threat assessment and defensive capabilities for combat systems [8,9,10].

In recent years, several innovative methodologies have been proposed to tackle the challenges of inverse design and aerodynamic shape reconstruction [11,12]. Bui-Thanh et al. [13] employed a classical 3D inverse method for target shape construction. While this approach has demonstrated some success under particular conditions, it is constrained by mesh dependence and convergence issues. Sun et al. [14] utilized neural networks to address the inverse problem of airfoils, leading to improved computational efficiency. However, this method struggles with stability issues in unconventional layouts due to the absence of physical constraints. Chen et al. [15] pointed out that traditional parameterization techniques, such as Bezier, PARSEC, and B-spline, often fail to strike an effective balance between design flexibility and stability when navigating high-dimensional design spaces, which limits their effectiveness in complex inverse problems. Moreover, methods based on reversible generative networks (CINNs) have emerged recently, capable of mapping high-dimensional, nonlinear challenges. Nonetheless, their practical applications are frequently restricted by the sparse training data available in non-cooperative scenarios, resulting in an inadequate generalization performance. Consequently, these existing methods face substantial difficulties when applied to non-cooperative aircraft due to their heavy dependence on rich and complete target data.

With advancements in computational fluid dynamics (CFD) technology and improvements in computational hardware performance, the analysis of aerodynamic performance has achieved significant breakthroughs in both accuracy and engineering efficiency [16,17,18]. Using the discrete concomitant method, Martins et al. [19] applied a RANS solver, resulting in a stable and efficient high-dimensional sensitivity transfer that considerably enhanced the aerodynamic analysis accuracy. Tao et al. [20] implemented a multi-fidelity nested optimization strategy that synergistically combines high-fidelity and low-fidelity models, thereby enhancing design efficiency under complex geometrical conditions. Li et al. [21] integrated machine learning with CFD inversion technology, proposing a dimensionality reduction method that ensures physical consistency while boosting optimization efficiency and accuracy. Additionally, Zhou et al. [22] harmonized concomitant analyses to enhance both aerodynamic and stealth performance in complex wing layouts. Alba et al. [23] developed an agent-model-driven, multi-disciplinary design framework, effectively improving the optimization efficiency under intricate constraints. Taj et al. [24] merged radar cross-section (RCS) minimization with aerodynamic performance optimization to introduce an integrated design approach. Leng et al. [25] presented an integrated design method by combining deep learning-based alternative models with migration learning technologies, thereby extending the modeling generalization for complex UAV systems. The application of CFD for efficient design and rapid optimization has become a central focus of research in this domain.

By integrating computational fluid dynamics (CFD) with multi-source data, the challenges of the sparse data associated with non-cooperative aircraft can be effectively addressed. Through the application of constraints and iterative refinements on aerodynamic shape performance, we can converge on a shape reconstructed through inverse methods that meets high precision and design requirements. On this basis, this paper essentially aims to present a multi-source, data-driven inverse reconstruction method that enhances the accuracy of aerodynamic shapes for non-cooperative aircraft, thereby providing more reliable technical support for evaluating their surprise defense capabilities.

2. Methods

2.1. Perspective Reverse 3D Recovery

The key feature points are identified from a given image, and the local features that remain invariant to scale changes and rotation are extracted via the Scale Invariant Feature Transform (SIFT) algorithm to detect extreme points in the multiscale space [26]. The multi-scale processing of the image is then executed through a Gaussian pyramid, with the degree of blurring controlled by the standard deviation, σ, in the following form:

$$G(x,y,\sigma )={\displaystyle \frac{1}{2\pi {\sigma }^2}}\exp \left(-{\displaystyle \frac{x^2+y^2}{2{\sigma }^2}}\right)$$

(1)

where G(x,y,σ) represents the Gaussian kernel, (x,y) denote the pixel coordinates, and σ signifies the standard deviation of the Gaussian function.

The RANSAC algorithm is also utilized to match feature points from different viewpoints to determine the geometric relationships between images [27].

The initial reconstruction of the geometry is performed using feature points. To ensure the accuracy and smoothness of the geometry, a non-uniform, rational B-spline (NURBS) curve is effectually employed for modeling the problem per the following relationship:

$$S(u)={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{P}_i}{N}_{i,k}(u)}{{\displaystyle \sum _{i=1}^n{N}_{i,k}}(u)}}$$

(2)

where P_i denotes the control point, N_i,k(u) represents the basis function, k is the order of the basis function, and u is the curve’s parameter.

Limited by the characteristics of B-spline curves, in the distribution area of complex data points, B-spline curve fitting produces a certain deviation. The control points are optimized via the least squares method, with the optimization objective of minimizing the error between the model and the feature points, as per the following formula:

$${\min }_{P_i}{\displaystyle \sum _{i=1}^m\left(\Vert P_i-S(u_i){\Vert }^2\right)}$$

(3)

where S(u_i) denotes the curve value of the NURBS curve at the parameter point u_i, P_i represents the control point, and m is the number of feature points.

After constructing the initial aerodynamic shape, the aerodynamic performance can be evaluated by CFD to perform an iterative optimization procedure, as per the Figure 1:

2.2. CFD Numerical Computation of Control Equations

When developing the numerical calculation model for the flow field around an aircraft, we focus solely on aerodynamic data during the cruise state, excluding factors related to take-off, landing, or maneuvering processes. Consequently, we adopt a steady-state assumption, which means we ignore the time derivative term. The control equations can be stated in the following form [28]:

$$\begin{array}{c}\nabla ·(\rho u)=0\\ \nabla ·(\rho u\otimes u)=-\nabla p+\nabla ·\left[(\mu +{\mu }_t)\left(\nabla u+\nabla u^{\top }\right)-{\displaystyle \frac{2}{3}}(\mu +{\mu }_t)(\nabla ·u)I\right]\\ \nabla ·[(\rho E+p)u]=\nabla ·[(\mu +{\mu }_t)\nabla u·u-k_{eff}\nabla T]\end{array}$$

(4)

where $\rho$ denotes the density (kg/m³), $V$ represents the velocity vector (m/s), and $\nabla$ stands for the Hamiltonian operator. In addition, the factor μ_t reads as follows:

$${\mu }_t=\rho \tilde{\nu }{f}_{v1}$$

(5)

in which $f_{v1}={\displaystyle \frac{{\chi }^3}{{\chi }^3+C_{v1}^3}},\hspace{1em}\chi ={\displaystyle \frac{\tilde{\nu }}{\nu }},\hspace{1em}\nu ={\displaystyle \frac{\mu }{\rho }}$. Herein, the suppression function, $f_{v1}$, is utilized for near-wall region correction; $\nu$ denotes the molecular kinematic viscosity; $\mu$ signifies the molecular dynamical viscosity; and $C_{v1}$ represents the model constant. Additionally, $\tilde{\nu }$ is then solved based on the Spalart–Allmaras turbulence model [29]:

$$\nabla ·(\rho \tilde{\nu }u)=\nabla ·[(\mu +\rho \tilde{\nu })\nabla \tilde{\nu }]+\rho \left(C_{b1}\tilde{S}\tilde{\nu }-C_{w1}{f}_w{\left({\displaystyle \frac{\tilde{\nu }}{d}}\right)}^2+{\displaystyle \frac{C_{b2}}{\sigma }}|\nabla \tilde{\nu }{|}^2\right)$$

(6)

The system of equations is closed by the ideal gas equation of state [30]:

$$p=\rho RT$$

(7)

where R is the gas constant.

2.3. Engine Thrust Calculation

If the B-21 adopts the unboosted combustion chamber version of the F-135-PW-100 model [31], the turbofan engine’s thrust-to-weight ratio (TWR) time series model is developed based on the historical data (

Table 1. Summary of the engine parameters.

Model	Maximum Thrust (KN)	Maximum Diameter (m)	Length (m)	Service Time	Aircraft Type
TF30-P-100	111.665	1.242	6.139	1964	F-111
F100-PW-229	132.27	1.181	4.856	1989	F-15, F-16
F414-GE-400	98	0.89	3.912	1999	F/A-18E/F
F119-PW-100	156	1.13	4.826	2007	F-22A
F135-PW-100	191.3	1.3	5.59	2009	F-35

), and the fitting results are shown in Figure 2a:

As illustrated in Figure 2a, the turbofan engine thrust-to-weight ratio rises almost linearly with the year, based on the following fitting equation: $y=-195.34949+0.10238x$ (goodness-of-fit R² = 0.94791). Figure 2b demonstrates that after the year 2000, the engine volume variation correlates identically with thrust change.

Taking into account the compactness requirements of the engine for stealth configuration, we adjusted the thrust-to-weight ratio (TWR) fitting formula based on the engine data from the B-2 bomber, obtaining the following relationship: $y=-197.63144+0.10238x$. After applying this corrected model to the B-21 project year, we determined that the thrust-to-weight ratio of the B-21 engine is approximately 9.38. The relevant speculative parameters are presented in

Table 2. Power system comparison.

Factor	F-135-PW-100 (No Additive Force)	B-21 Power System Speculation	Relative Deviation
Thrust ratio	8.47	9.38	+10.66%
Single-engine thrust force (kN)	125.0 (sea-level static thrust)	133.86	+7.09%
Volume (m3)	5.12	5.876	+14.8%
Length (m)	4.32	4.78	+10.6%

, which indicates that both the B-21 power system and the F-135-PW-100 (unboosted) engine show positive improvements. Therefore, it can be inferred that the B-21 power system is likely to be an enhanced version of the F-135-PW-100 (unboosted) engine.

The reasonableness of the B-21 engine parameters obtained by reasoning can be verified by calculating the high-altitude hover radius. The engine thrust decay is usually calculated based on the following relationships:

For the case of H < 11,000 m,

$$T_{H,Ma}=T_0(1-0.46Ma+0.44M{a}^2){\Delta }_H^{\zeta }$$

(8)

In the case of H > 11,000 m,

$$T_{H,Ma}=T_0(1-0.46Ma+0.44M{a}^2){\Delta }_H/{0.297}^{(1-\zeta )}$$

(9)

where $T_0$ represents the ground bench thrust; $\zeta$ denotes the engine thrust altitude drop index, turbofan 0.8~0.85; and ${\Delta }_H$ stands for the relative density of the atmosphere at altitude H.

By assuming that the B-21 is fully loaded, a force analysis of the aircraft in the hovering state yields the following:

(i)

Centripetal force in the horizontal direction: This is essentially provided by the horizontal component of the lift force (L). Since the direction of lift is inclined with respect to the direction of gravity, its horizontal component is provided by the following:

$$F_c=L\sin (\mu )$$

(10)

In addition, the centripetal force is related to the turning radius and the flight speed, as follows:

$$F_c={\displaystyle \frac{m{V}^2}{R}}$$

(11)

where m represents the aircraft’s mass, V denotes the aircraft’s velocity, and R is the hovering radius.

(ii)

The equilibrium of forces in the vertical direction: The vertical component of the lift force is in equilibrium with the aircraft’s gravity (W), per the following formulas:

$$L\cos (\mu )=W$$

(12)

Therefore, the turning radius can be stated by the following:

$$R={\displaystyle \frac{V^2}{g\tan \mu }}$$

(13)

where g denotes the gravitational acceleration, and μ represents the inclination angle. Further, an overload factor (n) is introduced to constrain the tilt angle:

$$n={\displaystyle \frac{C_{L}qS}{W}}$$

(14)

The performance data under various conditions are illustrated in Figure 3. At a Mach 0.6 working condition, the engine thrust can effectively balance the flight drag when the angle of attack (AoA) is less than 8° at altitudes of 0 km and 10 km. In the Mach 0.8 working condition, the engine thrust is able to counter the drag when the AoA is less than 9° and 10° at the same altitudes. The primary limiting factor for flight performance in these scenarios is the normal overload limitation corresponding to the specified AoA, meaning that the lift produced must compensate for gravitational forces. Once the equilibrium of external forces is established, it is also crucial to take structural overload limitations into account. For a typical bomber, the maximum allowable overload coefficient is 2.5, and the associated limiting condition can be calculated using Equation (14). Additionally, the corresponding minimum turning radius can be determined by Equation (13). The calculation results indicate that at altitudes of 0 km and 10 km, the minimum circling radius of the B-21 is 2337.86 m and 3390.68 m, respectively. This falls within the acceptable range for the turning radius of conventional bombers, thus validating the reasoning regarding the engine type.

2.4. Inlet and Tailpipe Modeling

The principle of the moderate distribution of the curvature gradient is employed in the intake section to balance the flow distortion control and RCS curtailment. In this regard, the control equation reads as follows:

$$y_{\mathrm{i}}=y_0+\Delta Y_{\mathrm{j}}\times [3\times {({\displaystyle \frac{X_{\mathrm{i}}}{L_{\mathrm{j}}}})}^2-2\times {({\displaystyle \frac{X_{\mathrm{i}}}{L_{\mathrm{j}}}})}^3]$$

(15)

The tail nozzle section adopts a centerline distribution with a nonlinearly decreasing curvature gradient, and its change rule is essentially controlled by the following equations:

$$y_{\mathrm{i}}=y_0+\Delta Y_{\mathrm{j}}\times [-3\times {({\displaystyle \frac{X_{\mathrm{i}}}{L_{\mathrm{j}}}})}^4+4\times {({\displaystyle \frac{X_{\mathrm{i}}}{L_{\mathrm{j}}}})}^3]$$

(16)

$$y_{\mathrm{i}}=y_0+\Delta Y_{\mathrm{j}}\times [3\times {({\displaystyle \frac{X_{\mathrm{i}}}{L_{\mathrm{j}}}})}^4-8\times {({\displaystyle \frac{X_{\mathrm{i}}}{L_{\mathrm{j}}}})}^3+6\times {({\displaystyle \frac{X_{\mathrm{i}}}{L_{\mathrm{j}}}})}^2]$$

(17)

where X_i denotes the axial coordinate of the control point, L_j characterizes the axial spread of the S-shaped intake duct configuration, and ∆Y_j represents the longitudinal offset of each S-shaped duct segment.

In addition, the design of the tail nozzle is presented in Figure 4 in order to completely encompass the high-temperature engine components.

2.5. RCS Electromagnetic Calculation Equations

The continuous electric field integral equation [32] can be discretized into the following matrix equation:

$$\mathbf{Z} \mathbf{J}=\mathbf{E},$$

(18)

where Z denotes the N × N-dimensional impedance matrix, N represents the number of unknown quantities, J signifies the N × 1-dimensional current vector to be solved, and E is the excitation vector. The characteristic basis function method first divides the scattering target into M subdomains [33] and then dissects each subdomain to suitably alter Equation (18) into the following form:

$$\left[\begin{array}{ccccc}{\mathit{Z}}_{11}& {\mathit{Z}}_{12}& {\mathit{Z}}_{13}& \cdots & {\mathit{Z}}_{1M}\\ {\mathit{Z}}_{21}& {\mathit{Z}}_{22}& {\mathit{Z}}_{23}& \cdots & {\mathit{Z}}_{2M}\\ {\mathit{Z}}_{31}& {\mathit{Z}}_{32}& {\mathit{Z}}_{33}& \cdots & {\mathit{Z}}_{3M}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ {\mathit{Z}}_{M1}& {\mathit{Z}}_{M2}& {\mathit{Z}}_{M3}& \cdots & {\mathit{Z}}_{MM}\end{array}\right]\left[\begin{array}{l}{\mathit{J}}_1\\ {\mathit{J}}_2\\ {\mathit{J}}_3\\ ⋮\\ {\mathit{J}}_M\end{array}\right]=\left[\begin{array}{l}{\mathit{E}}_1\\ {\mathit{E}}_2\\ {\mathit{E}}_3\\ ⋮\\ {\mathit{E}}_M\end{array}\right]$$

(19)

where Z_ij(i = 1, 2, …, M; j = 1, 2, …, M) denotes the sub-matrix of dimensions N_i × N_j, J_i represents the i-th sub-vector of the total induced current vector to be solved, and E_i signifies the i-th sub-vector of the total excitation vector.

A sufficient number of incident plane wave excitations are uniformly set up above the target, according to ∆θ and ∆ϕ. For electrically large and complex targets, the angular spacing needs to satisfy 3 ≤ (∆θ,∆ϕ) ≤ 10 [34], assuming that N_θ and N_ϕ are the numbers of incident plane wave excitations in the directions of θ and ϕ, respectively. Based on these two types of polarizations, there would be a total of N_pws = 2N_θN_ϕ incident plane wave excitations, which are denoted by V_ii^Npws. Then, the current of the i-th subdomain under plane wave excitation can be analyzed using the following relationship:

$${\mathbf{Z}}_{ii}^e {\mathbf{J}}_{ii}^e={\mathbf{V}}_{ii}^{N_{\mathrm{pws}}},$$

(20)

where ${\mathbf{Z}}_{ii}^e$ represents the N_i^be × N_i^be-dimensional self-impedance matrix after the expansion of the i-th subdomain, ${\mathbf{V}}_{ii}^{N_{\mathrm{pws}}}$ denotes the N_i^be × N_pws-dimensional matrix, and J_ii^e is the current coefficient matrix of the i-th subdomain, and the current response of the i-th subdomain is obtained by removing the expanded part.

The SVD is performed on J_ii^e [35]:

$${\mathbf{J}}_{ii}^e=\mathbf{U}\mathbf{W}{\mathbf{V}}^{\mathrm{T}},$$

(21)

where U represents the orthogonal matrix of dimension N_i^be × N_pws; W denotes the diagonal matrix of N_pws × N_pws, with the diagonal elements being the singular values of J_ii^e; and V signifies the orthogonal matrix of N_pws × N_pws. A suitable threshold is chosen for U, and K_i column vectors larger than the threshold are retained as a set of minimum complete CBFs on the i-th subdomain, which is denoted by J_ii^CBFs. Let us denote the k-th column vector of J_ii^CBFs by J_i^k (i.e., the k-th CBFs on the i-th subdomain). After performing the above operation on each subdomain, assuming that K_i(i = 1, 2, …, M), CBFs are retained for each subdomain. The current on each subdomain can be expressed as a linear combination of the CBFs of the subdomain, as follows:

$$J_i={\displaystyle \sum _{k=1}^{K_i}{a}_i^k}{J}_i^k\hspace{1em}(i=1,2,\cdots ,M)$$

(22)

where $a_i^k$ denotes the corresponding expansion coefficient. Then, the surface current of the target is denoted by the following:

$$\mathit{J}=\left[\begin{array}{c}{J}_1\\ ⋮\\ J_i\\ ⋮\\ J_M\end{array}\right]={\displaystyle \sum _{k=1}^{K_1}{a}_1^k}\left[\begin{array}{c}{J}_1^k\\ [0]\\ ⋮\\ [0]\end{array}\right]+\cdots +{\displaystyle \sum _{k=1}^{K_i}{a}_i^k}\left[\begin{array}{c}[0]\\ ⋮\\ [{\mathit{J}}_i^k]\\ ⋮\\ [0]\end{array}\right]+\cdots +{\displaystyle \sum _{k=1}^{K_M}{a}_M^k}\left[\begin{array}{c}[0]\\ ⋮\\ [0]\\ ⋮\\ [{\mathit{J}}_M^k]\end{array}\right]$$

(23)

where $a_i^k$ (i = 1, 2, …, M; k = 1, 2, …, K_i) represent the expansion coefficients of the CBFs to be solved. Substituting Equation (22) into Equation (19) and multiplying both sides of the equation by the same transpose of all the column vectors of the J_ii^CBFs, we can obtain a system of linear equations with respect to $a_i^k$ in descending order:

$$Z^{\mathrm{R}}·a=V^{\mathrm{R}}$$

(24)

$$\left[\begin{array}{l}\begin{array}{cccc}{({\mathit{J}}_1^1)}^{\mathrm{T}}{\mathit{Z}}_{11}{\mathit{J}}_1^1& {({\mathit{J}}_1^1)}^{\mathrm{T}}{\mathit{Z}}_{11}{\mathit{J}}_1^2& \cdots & {({\mathit{J}}_1^1)}^{\mathrm{T}}{\mathit{Z}}_{11}{\mathit{J}}_1^{K_1}\\ ⋮& ⋮& & ⋮\\ {({\mathit{J}}_1^{K_1})}^{\mathrm{T}}{\mathit{Z}}_{11}{\mathit{J}}_1^1& {({\mathit{J}}_1^{K_1})}^{\mathrm{T}}{\mathit{Z}}_{11}{\mathit{J}}_1^2& \cdots & {({\mathit{J}}_1^{K_1})}^{\mathrm{T}}{\mathit{Z}}_{11}{\mathit{J}}_1^{K_1}\\ ⋮& ⋮& & ⋮\\ {({\mathit{J}}_M^1)}^{\mathrm{T}}{\mathit{Z}}_{M1}{\mathit{J}}_1^1& {({\mathit{J}}_M^1)}^{\mathrm{T}}{\mathit{Z}}_{M1}{\mathit{J}}_1^2& \cdots & {({\mathit{J}}_M^1)}^{\mathrm{T}}{\mathit{Z}}_{M1}{\mathit{J}}_1^{K_1}\\ ⋮& ⋮& & ⋮\\ {({\mathit{J}}_M^{K_M})}^{\mathrm{T}}{\mathit{Z}}_{M1}{\mathit{J}}_1^1& {({\mathit{J}}_M^{K_M})}^{\mathrm{T}}{\mathit{Z}}_{M1}{\mathit{J}}_1^2& \cdots & {({\mathit{J}}_M^{K_M})}^{\mathrm{T}}{\mathit{Z}}_{M1}{\mathit{J}}_1^{K_1}\end{array}\\ \begin{array}{cccc}\dots {({\mathit{J}}_1^1)}^{\mathrm{T}}{\mathit{Z}}_{1M}{\mathit{J}}_M^1& {({\mathit{J}}_1^1)}^{\mathrm{T}}{\mathit{Z}}_{1M}{\mathit{J}}_1^2& \cdots & {({\mathit{J}}_1^1)}^{\mathrm{T}}{\mathit{Z}}_{1M}{\mathit{J}}_M^{K_M}\\ ⋮& ⋮& & ⋮\\ \dots {({\mathit{J}}_1^{K_1})}^{\mathrm{T}}{\mathit{Z}}_{1M}{\mathit{J}}_M^1& {({\mathit{J}}_1^{K_1})}^{\mathrm{T}}{\mathit{Z}}_{1M}{\mathit{J}}_1^2& \cdots & {({\mathit{J}}_1^{K_1})}^{\mathrm{T}}{\mathit{Z}}_{1M}{\mathit{J}}_M^{K_M}\\ ⋮& ⋮& & ⋮\\ \dots {({\mathit{J}}_M^1)}^{\mathrm{T}}{\mathit{Z}}_{MM}{\mathit{J}}_M^1& {({\mathit{J}}_M^1)}^{\mathrm{T}}{\mathit{Z}}_{MM}{\mathit{J}}_1^2& \cdots & {({\mathit{J}}_M^1)}^{\mathrm{T}}{\mathit{Z}}_{MM}{\mathit{J}}_M^{K_M}\\ ⋮& ⋮& & ⋮\\ \dots {({\mathit{J}}_M^{K_M})}^{\mathrm{T}}{\mathit{Z}}_{MM}{\mathit{J}}_M^1& {({\mathit{J}}_M^{K_M})}^{\mathrm{T}}{\mathit{Z}}_{MM}{\mathit{J}}_M^2& \cdots & {({\mathit{J}}_M^{K_M})}^{\mathrm{T}}{\mathit{Z}}_{MM}{\mathit{J}}_M^{K_M}\end{array}\end{array}\right]$$

(25)

The reduced-order matrix has a small dimension, and the coefficient matrix (a) can be solved directly by LU decomposition, and then the target surface current (J) can be obtained by substituting a into Equation (23) [36].

3. Results

3.1. Grid-Independent Verification

3.1.1. The Validation of the Pneumatic Mesh Irrelevance

The CHN-F1 Standard Aerodynamic Benchmark Model (SABM) was utilized to conduct the validation experiments. The experimental data is sourced from the Verification and Validation Database of computational fluid dynamics, and the experimental environment parameters are detailed in

Table 3. CHN-F1 scale model test data.

Ma	Re	T (K)	Static Pressure (Pa)	PTotal (Pa)	Dynamic Pressure (Pa)	AoA (°)
0.600	6.15 × 106	300	92,872.05	118,458.65	20,192.87	−2
0.598	6.15 × 106	300	93,292.80	118,995.32	20,256.93	0
0.599	6.18 × 106	300	93,281.09	118,980.39	20,153.9	2
0.600	6.20 × 106	300	93,753.83	119,583.36	20,251.86	4
0.598	6.20 × 106	300	93,980.78	119,872.83	20,268.12	6
0.600	6.22 × 106	300	94,030.65	118,458.65	20,205.14	8
0.600	6.22 × 106	300	94,396.93	120,403.64	20,266.35	10

:

The numerical calculation accurately reproduces the experimental boundary conditions. The turbulence model employs the Spalart–Allmaras approach, with the ideal gas assumption, and the viscosity coefficient is defined according to the Sutherland formula. The outer boundary is defined as the far field of pressure, while the model wall is treated as a non-slip wall. Both pressure and density terms are implemented using second-order schemes. The convergence residual is maintained at a threshold of 10⁻⁴.

Figure 5 illustrates the comparison of the lift coefficient distribution between experimental and numerical simulations. It is evident that the maximum deviation between the experimental and numerical results in the aerodynamic coefficients is less than 5%. This confirms the reliability of the algorithm presented in this paper for simulating complex flows at supersonic speeds.

To quantitatively assess the mesh sensitivity, the numerical simulations are performed with the B-21 aircraft half-model, where the boundary conditions are set as those above. For this purpose, we constructed a hybrid mesh system with six topologically equivalent groups (see

Table 4. Table of different grid parameters.

Grid Number	Number of Units (×10⁶)	Minimum Number of Boundary Layers	Height of the First Layer of the Mesh (m)	Boundary Layer Growth Rate
G1	2.3	22	0.0000063	1.20
G2	3.5	22	0.0000063	1.20
G3	4.5	22	0.0000063	1.20
G4	7.5	22	0.0000063	1.20
G5	9.0	22	0.0000063	1.20
G6	11.1	22	0.0000063	1.20

for the used factors).

Over the full range of operating conditions, the average relative error of the numerical solution is about 1%, and the peak error is controlled within 2% (see Figure 6), which verifies the reliability of the computational model.

3.1.2. Electromagnetic Mesh Independence Verification

For mesh convergence verification, the ideal conductor flat plate with no thickness is selected as the reference model, and its geometric parameters and computational conditions are taken in the following form:

Model specification: Square flat plate, side length = 1 m.

Incident wave frequency: f = 3 GHz (corresponding to a free space wavelength of λ₀ = c/f ≈ 0.1 m).

Incident wave parameters: The plane wave incident along the +z axis, horizontally polarized.

Calculation objective: Evaluating the value of the radar scattering section (RCS) as a function of azimuth.

Mesh division strategy: Use triangular cell division and set four groups of mesh sizes.

Grid properties are shown in

Table 5. Electromagnetic domain grid parameters.

Grid Number	Mesh Size	How It Is Divided
Δ1	λ0/10	Triangular sectioning
Δ2	λ0/6	Triangular sectioning
Δ3	λ0/3	Triangular sectioning
Δ4	2λ0/5	Triangular sectioning

.

Figure 7 illustrates the results of the RCS calculations with various mesh sizes. The plotted results show that when the mesh size is set equal to Δ ≤ λ₀/6 (Δ₁, Δ₂), the maximum difference in the RCS curves in the range of θ ∈ [0°,60°] is less than 1 dBsm, indicating that the computational results tend to converge. The RCS curves for the coarse mesh, Δ₄, exhibit significant deviations around angles of θ = 20–90°. These discrepancies arise from two main factors: (1) Numerical dispersion effects, where discrete errors cause phase accumulation distortion. (2) The insufficient resolution of fringe currents, as the coarse mesh is unable to accurately represent the singular distribution of the induced currents. Therefore, when conducting electromagnetic meshing, it is essential to ensure that the mesh size is comparable to or finer than the free-space wavelength. This algorithm has been validated using the NASA Almond model, as illustrated in Figure 8. Compared to the experimental data [37], the algorithm presented in this paper provides a more accurate computation of the true RCS of the target.

3.2. B-21 Three-Dimensional Model Display

Based on the reverse engineering framework established in Section 2.1, the 3D geometric reconstruction of the target was performed, and its reconstruction model is demonstrated in Figure 9.

The relevant aerodynamic performance is depicted in Figure 10, following iterations with the initial model until the aerodynamic performance aligns with the specified usage conditions. Figure 10b illustrates that the drag coefficient (C_D) monotonically decreases as the headway angle reduces from −2° to 0°, while the lift coefficient (C_L) increases with an increase in the headway angle (as presented in Figure 10a). This dual mechanism, characterized by an increase in lift and a decrease in drag, effectively enhances the lift-to-drag ratio, with the differing growth rates of the lift coefficient contributing to a gradual rise in the lift differential. This phenomenon is primarily attributed to the compressibility effect, which influences the interaction characteristics of the surge and attachment layers.

As demonstrated in Figure 10c, the zero-lift headway angle demonstrates Mach number insensitivity within the subsonic regime (Mach 0.6 vs. Mach 0.8). At Mach 0.6, the lift-to-drag ratio (K) displays a nonlinear variation as the headway angle increases, reaching a maximum value of 27 (K_max) at an angle of attack (AoA) of 4°, after which it gradually declines due to flow separation. In contrast, a broader range of favorable pressure gradients is observed for the Mach 0.6 scenario, and the difference in K values is notably more significant than that observed for the Mach 0.8 case.

In summary, the performance parameters of the inversely reconstructed B-21 aircraft meet the expected requirements, demonstrating that the method employed possesses high accuracy and could effectively support the subsequent RCS characteristic calculations for the B-21.

3.3. B-21 Aerodynamic Characterization

As shown in Figure 11, the lower leading-edge surface exhibits a high-pressure coefficient distribution in conjunction with the cockpit fuselage. As the angle of approach increases, the pressure coefficient on the lower surface of the leading edge rises linearly, while the pressure coefficient at the cockpit fuselage bond decreases linearly. A notable development of the negative pressure core area is observed around the inlet of the B-21 airplane. Furthermore, as the angle of approach continues to increase, this low-pressure region expands along the upper surface attachment layer toward the fuselage edge.

As shown in Figure 12, the pressure coefficient distribution observed on the upper airfoil of the B-21 airplane closely mirrors that of Mach 0.6. A negative pressure core region develops behind the inlet duct; as the angle of approach increases, this negative pressure zone progressively expands along the upper airfoil in the direction of flow.

As shown in Figure 13, a low-momentum fluid region was observed in the lip region of the intake tract, suggesting enhanced viscous drag near the intake tract.

As shown in Figure 14, the upper airfoil of the B-21 aircraft displays primarily subcritical flow characteristics. As the AoA increases from 1° to 8°, the extent of the airfoil separation region exhibits a non-monotonic evolution: it initially expands to approximately 65% of the wing chord length at AoA = 4°, before contracting to around 45% of the wing chord length at AoA = 8°. Notably, a conical, localized, supersonic region (Ma > 0.92) emerges near the forward fuselage along the symmetry plane, indicating the presence of excitation-induced separation on the upper fuselage surface.

As illustrated in Figure 15a, the vortex field is predominantly concentrated in the wingtip region and the trailing edge separation zone of the wing–body fusion, while the intensity of vortices in the leading-edge region remains low. Under Mach 0.6 operating conditions, the evolution of vortex intensity in the wing–body fusion region demonstrates significant nonlinear characteristics as the headway angle increases from −2° to 4°. Within the range of small headway angles, the separation strength from the lower surface high-pressure flow is weakly coupled with the upwash effect; however, once the critical headway angle is surpassed, a three-dimensional flow separation results in a sharp increase in vortex strength.

Under the supercritical flow condition of Mach 0.8, the trend of vortex evolution resembles that observed under subsonic conditions; however, the absolute intensity of the vortices increases by approximately 18–22% at the same headway angle. These results indicate that the transonic compression effect amplifies excitation and boundary layer interference, which, in turn, enhances the unfavorable pressure gradient on the wing’s upper surface and intensifies the flow separation. This mechanism may partially elucidate the synergistic growth of wave resistance and frictional drag during the transonic flight phase.

3.4. B-21 Radar Characterization

The P-band is a well-established anti-cloaking band, known for its ability to significantly diminish the effectiveness of the wave-absorbing materials utilized on the surfaces of stealth aircraft. This characteristic enables the more-objective and direct evaluation of other stealth measures implemented in the aircraft. It is posited that the B-21 relies on a high-altitude penetrating air defense warning radar network during its combat missions. In this paper, we calculate the RCS characteristics of the B-21 aircraft using a monostatic configuration at a frequency of 425 MHz within the P-band range. A high-precision RCS numerical simulation is conducted by integrating the intake and exhaust systems of the B-21 with its aerodynamic profile, incorporating the engine geometry model into the analysis. As shown in Figure 16, Following the convergence of the electromagnetic calculations, we perform a quantitative analysis based on omnidirectional RCS distribution maps under both horizontal and vertical polarization conditions. This is further enhanced by employing a dynamic azimuthal scattering characteristic evaluation method, allowing for the systematic assessment of the low-detectability characteristics of the B-21.

Assuming that the incident wave is aligned with the body axis of the no-heading-angle aircraft, the single-station RCS of the B-21 stealth aircraft reaches its maximum at a 10° downward deflection of the incidence under VV-polarized irradiation conditions. As the incident angle transitions gradually from 10° downward to 20° upward, the RCS distribution exhibits a monotonically decreasing trend. Within this range, the HH-polarized configuration records the smallest RCS value at 20° upward, approaching −20 dBsm. The scattering characteristics under both of the described polarization (VV and HH) conditions demonstrate a similar angular distribution pattern.

Four distinct specular reflection flaps emerge at azimuth angles of 37°, 143°, 217°, and 323°, where the peak RCS for the HH polarization surpasses that of the VV polarization. The entire azimuthal spectrum is categorized into five characteristic scattering regions: the forward zone (0–30°), the oblique forward zone (45–70°), the lateral zone (70–110°), the oblique backward zone (110–135°), and the backward zone (150–180°). When averaging the scattering characteristics of each region, the average RCS in the forward region is approximately −9.2 dBsm, representing the lowest value across all directions, while the average RCS in the oblique forward region under VV polarization is around 12.3 dBsm, reflecting the highest value in this analysis.

$$\overline{X}={\displaystyle \frac{{\displaystyle \sum _{i=1}^{n}1}{0}^{(x_i/10)}}{n}}$$

(26)

$$\overline{\sigma }={10}^{*}\log \overline{X}$$

(27)

Within the physical optical scattering region, the electromagnetic scattering from the B21 aircraft is primarily influenced by its macroscopic geometry, which results in a dominant specular reflection effect. This effect adheres to the fundamental principle that the angle of incidence equals the angle of reflection, establishing the main energy distribution. On typically smooth and continuous surfaces, the contribution of specular reflection significantly surpasses that of the fringe-winding effect, which only causes localized scattering at specific geometrical discontinuities, such as the tips of the wings or the trailing edge perturbations.

As illustrated in Figure 17b, the anisotropic scatterer RCS demonstrates a pronounced gradient response to the incident wave azimuth. Conversely, the single-site RCS azimuthal sensitivity depicted in Figure 17a under VV polarization shows a reduction of approximately 40%, while its overall amplitude continues to follow a monotonically increasing trend. In the forward region and the side scattering flap under VV polarization, the RCS levels consistently remain below 0 dBsm, and the implementation of wave-absorbing materials could diminish the overall RCS by an order of magnitude.

4. Discussion

This study aims to tackle the challenge of accurately assessing the radar scattering characteristics (RCS) of non-cooperative aircraft, primarily due to insufficient aerodynamic profile data. It is hypothesized that by integrating multi-source, data-driven inverse reconstruction methods, high-precision profiles of non-cooperative targets can be efficiently derived, allowing for precise RCS evaluations and thereby enhancing the analysis of the surprise capabilities of stealth aircraft.

The obtained results confirm that the proposed approach could effectively reconstruct the 3D geometry of the B-21 aircraft and achieve high accuracy in predicting its aerodynamic performance through a CFD analysis. In the subsonic range (Mach 0.6 vs. Mach 0.8), the variation of the zero-lift angle of attack is observed to be less sensitive to the Mach number, while the lift-to-drag ratio reaches a maximum value of 27 at an angle of attack of 4°, subsequently decreasing as the angle of attack increases due to flow separation. These findings align with the aerodynamic characteristics of the flying wing layout reported in existing studies, thereby verifying the reliability and validity of the present method.

In terms of electromagnetic characterization, the P-band RCS simulation results reveal significant polarization sensitivity and orientation dependence. The aircraft demonstrates high sensitivity to the edge rounding caused by the trailing-edge sawtooth structure under horizontal polarization yet exhibits commendable angular robustness under vertical polarization conditions. The flying wing layout shows notable angular stability under specific polarization scenarios, while displaying heightened scattering sensitivity at certain structural feature points.

This study contributes to advancing research in non-cooperative aircraft target identification and surprise capability assessment through the integration of inverse reconstruction, CFD simulation analysis, and electromagnetic scattering property evaluation. The findings highlight the applicability and accuracy of the proposed methodology in practical non-cooperative target modeling.

When the target image is significantly affected by lens distortion, the data represented in the image could become completely distorted, leading to considerable deviations in the initial reconstructed shape, making iterative optimization via CFD impossible. Moreover, if the target parameters are entirely concealed or there is no approximate image, this methodology becomes inapplicable. Additionally, when feature points in the image are sparse or unevenly distributed, the SIFT and RANSAC algorithms may result in an increased error in geometric reconstruction, particularly during the NURBS curve fitting stage.

Future research directions suggest the further enhancement of non-cooperative target data acquisition technology, particularly focusing on effectiveness in scenarios with extremely sparse data. The further exploration of advanced machine learning and data fusion techniques is also recommended to improve profile reconstruction accuracy and computational efficiency, thereby providing robust technical support for target identification and surprise defense performance evaluation in actual combat environments.

The data utilized in this study are sourced from legitimate and public entities and rely solely on publicly accessible information, devoid of any sensitive details. This research adheres to international legal and ethical standards, aiming to enhance technical understanding and foster academic research, rather than contribute to military-use technology or disrupt the existing military balance.

5. Conclusions

In the present investigation, an inverse reconstruction method driven by the integration of multi-source data is proposed to address the challenges associated with accurately assessing the radar scattering characteristics (RCS) of non-cooperative aircraft, particularly in light of the limited access to precise aerodynamic shape information. Through a combination of numerical simulations and experimental validations, the proposed methodology successfully reconstructs the three-dimensional shape of the B-21 aircraft. This achievement highlights the significant theoretical value and practical implications of the study for stealth aircraft identification and the assessment of surprise defense capabilities. The main results obtained from the present study are as follows:

(1)

The aerodynamic performance analysis indicates that the modified B-21 aircraft demonstrates commendable stability within the subsonic domain (Mach 0.6 and Mach 0.8). Furthermore, the zero-lift angle of attack shows a low sensitivity to variations in the Mach number. The lift-to-drag ratio peaks at an impressive 27 at a 4° AoA in the presence of Mach 0.6, gradually declining due to flow separation effects.

(2)

The electromagnetic characterization reveals that the RCS of the B-21 aircraft displays notable variations under different polarization conditions. In vertical polarization (VV) scenarios, the aircraft exhibits reduced angular sensitivity and demonstrates a robust angular performance. Conversely, under horizontal polarization (HH) conditions, the sensitivity to edge rounding induced by trailing edge structures is markedly heightened.

(3)

The methodologies and findings presented in the current investigation not only advance the precision of aerodynamic and RCS characterization techniques for non-cooperative aircraft but also serve as a valuable complement to existing research, validating the efficacy and reliability of the methodology employed in this paper. These results hold significant importance in bolstering the identification and defense capabilities of stealth aircraft in real combat situations.