Mechanical deformation analysis

The form of the array surface is referenced to that of the wing-integrated array antenna. Due to the similarity in structure, the array surface is equivalently transformed into a cantilever plate on which the antenna elements are uniformly distributed.

Figure 1 shows the schematic diagram of the antenna surface deformation. During the working process, the array surface may be subjected to bending, torsion, or combined bending and torsion loads. Taking the bending deformation as an example, the array surface is bent and deformed by a uniform load. Referring to the deflection function under small deformation hypothesis in [Reference Zienkiewicz24], with different values of loads on the array surface, the phase center of array elements after surface deformation in the z-direction is calculated by:

(1)\begin{equation}{z_n}\left( P \right) = - {{P{x_n}^2} \over {24EI}}\left( {{x_n}^2 + 6{l^2} - 4l{x_n}} \right),{\ }n = 1,2,3, \cdots ,N\end{equation}

Figure 1.

Design model of the pattern synthesis of phased array with unsteady surface deformation. (a) The schematic diagram of the antenna surface deformation. (b) The schematic diagram at x-z plane.

where $N$ is the total element number, $E$ is the elastic modulus, $EI$ is the bending stiffness based on the structure size and $E$, $l$ is the length of the cantilever plate, and P represents any form of external load, here it is the intensity of the uniform surface load.

Optimization formulation and sensitivity analysis

For the theoretical synthesis considering the above bending deformation of a cantilever plate, the following assumptions are made: (1) The configuration (e.g., length and width) of each element remains unchanged during the bending deformation. (2) The change in the mutual coupling among elements caused by the bending deformation is ignored. Then the possible deformed arrays can be simply reconstructed according to the varying centroid $\left( {{x_n},{y_n},{z_n}\left( P \right)} \right)$ of each array element. The far-field pattern of array antenna is derived according to the superposition principle of phased array antenna [Reference Mailloux8], which is expressed as:

(2)\begin{equation}AF\left( {\theta ,\phi ,P} \right) = \mathop \sum \limits_{n = 1}^{N} {\alpha _n}{e^{j{\psi _n}}}{e^{jk\left( {{x_n}{\textrm{sin}}\theta {\textrm{cos}}\phi + {y_n}{\textrm{sin}}\theta {\textrm{sin}}\phi + {z_n}\left( P \right){\textrm{cos}}\theta } \right)}}\end{equation}

where ${\alpha _n}$ is the amplitude of $n$th array element, ${\psi _n}$ is the phase of $n$th array element, $k$ is the wave number, and $\left( {\theta ,\phi } \right)$ is the polar variables.

The synthesis problem is defined as follows: optimizing the excitation coefficients of each element simultaneously to suppress the peak sidelobe level (PSLL) when the array surface deforms with the varying loads. The optimization formulation is given as:

(3)\begin{equation} \left\{ \begin{matrix} {find: \left\{ {\bf{\alpha}, {\bf{\psi}}} \right\} } \\ {min: \Psi = PSLL\left( {AF} \right)} \\ {s.t.: \xi \le \alpha \le D\xi } \\ { 0 \le \psi \le 2\pi } \\ \end{matrix} \right. \end{equation}

where PSLL is the peak sidelobe level, $\xi \le \bf{\alpha} \le D\xi $ denote the DRR constrains, $\xi $ is an auxiliary variable, and $D$ is the upper threshold of DRR. In order to solve the optimization problem efficiently with the gradient-based algorithm, the objective PSLL(AF) should be defined as a continuous function, and its gradient with respect to the design variables $\left\{ {\bf{\alpha}, {\bf{\psi}}} \right\}$ should be calculated analytically. For the sidelobe suppression problem, the Kreisselmeier–Steinhauser function is utilized to smooth the objective function [Reference Wang, Zheng, Yu, Gao and Liu19], and according to the chain rule, the sensitivities are expressed as:

(4)\begin{equation}{{\partial {{\Psi }}} \over {\partial \alpha }} = {{\partial {{\Psi }}} \over {\partial {\textrm{PSLL}}}}{{\partial {\textrm{PSLL}}} \over {\partial {\textrm{AF}}}}{{\partial {\textrm{AF}}} \over {\partial \alpha }},\end{equation}

(5)\begin{equation}{{\partial {{\Psi }}} \over {\partial \psi }} = {{\partial {{\Psi }}} \over {\partial {\textrm{PSLL}}}}{{\partial {\textrm{PSLL}}} \over {\partial {\textrm{AF}}}}{{\partial {\textrm{AF}}} \over {\partial \psi }}.\end{equation}

When the position of array elements changes, the pattern can also be synthesized by compensating the wave path difference [Reference Kobayashi17]. For the mechanical deformation in equation (1), the array factor with compensation is given by:

(6)\begin{equation}E\left( {\theta ,\phi ,P} \right) = \mathop \sum \limits_{n = 1}^{N} {\alpha _n}{e^{ - jkD{D_n}}}{e^{jk\left( {{x_n}{\textrm{sin}}\theta {\textrm{cos}}\phi + {y_n}{\textrm{sin}}\theta {\textrm{sin}}\phi + {z_n}\left( P \right){\textrm{cos}}\theta } \right)}}\end{equation}

(7)\begin{equation}D{D_n}\left( P \right) = {x_n}\sin {\theta _0}\cos {\phi _0} + {y_n}\sin {\theta _0}\sin {\phi _0} + {z_n}\left( P \right)\cos {\theta _0},\end{equation}

where $D{D_n}\left( P \right)$ denotes the path-length, (${\theta _0},{\phi _0}$) is the beam direction. The optimized excitations can be treated as a better initial solution, and phase excitation coefficients can be derived according to equation (6) for different deformation. It is worth mentioning that we can directly use the optimized amplitude and phase excitations for the compensation, or we can choose to use the amplitude excitation as the initial solution and then use equation (6) to derive the phase excitation under different deformations.

Procedure of the proposed method

The general procedure of the proposed method is given as follows:

1. (1) Preparation for the mechanical deformation analysis where the array surface size, initial element locations, the bending stiffness $EI$, and the loads P are set.

2. (2) Calculating the mechanical deformation form as equation (1), and reconstructing the deformed array and modifying the array factor in equation (2). It is worth mentioning that, when the value of P changes, the deformation form (the relationship among the changed locations of different elements) remains, and the deformation value (changed locations) is associated with P.

3. (3) Preparation for the pattern synthesis, where ${\bf{\alpha}_{ini}}$ and $ {{\bf{\psi}}_{ini}}$ are set.

4. (4) For a given P, calculating the radiation properties to obtain the design objective in equation (3).

5. (5) Calculating the sensitivities of the design objective with respect to $\bf{\alpha}$ and ${\bf{\psi}}$ as equations (4) and (5).

6. (6) Updating $\bf{\alpha}$ and ${\bf{\psi}}$ by the use of a gradient-based algorithm. If $\bf{\alpha}$ and ${\bf{\psi}}$ have converged, the synthesis stops; if not, the synthesis goes to (4).

7. (7) For different P, repeat (4)–(6) to obtain different solutions or use one of these solutions (i.e., optimized amplitude solution) and the phase compensation excitation as equation (6) to realize the compensation for different P (within a feasible range of P value).

Case 1

We consider a linear array that has 30 elements with inter-element spacing $d = 0.5\lambda $, where $\lambda $ is the wavelength of the simulation frequency. The material of the cantilever plate is aluminum 6061 with elastic modulus $E$ of 68.9 GPa, and the length of the cantilever plate is 30$\lambda $. The array surface is bent and deformed by a uniform load. The design objective is to suppress the PSLL while steering the main beam at $ {\theta _0} = 0^\circ $ and ${\phi _0} = 0^\circ $ with a main beam width $\left| \theta \right| \le 5^\circ $.

First, the radiation performance without deformation (i.e., ${z_{{\textrm{max}}}}\left( P \right) = 0$, where ${z_{{\textrm{max}}}}\left( P \right)$ denotes the maximum deformation of array elements under $P$ load) under different DRR is studied. The DRR threshold is set to be 2, 3, and 4. The calculation time for the optimization design is about 10 s. Figure 2 shows the normalized pattern and the amplitude distribution of each element with different DRR. It is seen that the PSLL increases from −27.93 dB to −24.59 dB as the DRR decreases from 4 to 2, which illustrates that the DRR constraint is activated during the optimization and the DRR value affects the radiation performance.

Figure 2.

Results of the array antenna without deformation under different DRR: (a) normalized pattern and (b) amplitude excitation.

Case 2

In the second case, we use the amplitude obtained from the Chebyshev [Reference Mailloux8] with and without the phase compensation in equation (6) as a comparison. Herein, the excitation coefficient obtained by Chebyshev with a sidelobe level equal to −30 dB is applied to the deformed array. ${z_{{\textrm{max}}}}\left( P \right)$ is taken as $0.5\lambda $, $\lambda $, and $1.5\lambda $, respectively. The normalized pattern with ${z_{{\textrm{max}}}}\left( P \right) = \lambda $ is given in Fig. 3, the excitation coefficient is given in Fig. 4, and the radiation properties are listed in Table 1.

Figure 3.

Performance comparison with ${z_{{\textrm{max}}}}\left( P \right) = \lambda $.

Figure 4.

Excitation coefficient after optimization with ${z_{{\textrm{max}}}}\left( P \right) = \lambda $: (a) amplitude excitation and (b) phase excitation.

Table 1.

Radiation properties of the synthesis

When the surface deforms with different ${z_{{\textrm{max}}}}\left( P \right)$, the main beam deviation of the array with the excitation coefficient obtained by Chebyshev without any compensation (Chebyshev-woc) increases gradually as ${z_{{\textrm{max}}}}\left( P \right)$ increases (from $2.12^\circ $ to $6.41^\circ $). After the optimization, the PSLLs are suppressed successfully and the main beam is adjusted to the desired direction. The PSLL is optimized to −28.85 dB, −28.89 dB, and −28.87 dB, respectively. The optimized normalized pattern closes to the pattern obtained by Chebyshev with phase compensation (Chebyshev-wc) and the DRR is optimized to 3, which decreases by 1.04 compared with the Chebyshev results.

Case 3

In this case, the linear array in the first example is deformed by a concentrated load, and all parameters of cantilever plate remain unchanged. The bending deformation satisfies:

(8)\begin{equation}{z_n} = - {{P{x_n}^2} \over {6EI}}\left( {3l - {x_n}} \right),{\ }n = 1,2,3, \cdots ,N\end{equation}

where $P$ is the concentrated load on the free end of the plate. ${z_{{\textrm{max}}}}$ takes $\lambda $, and the DRR threshold is set to be 3. The calculation time for the optimization design is about 10 s. The PSLL is suppressed to −28.87 dB, and the gain is optimized to 17.26 dB, which is close to the pattern obtained by Chebyshev without any deformation with lower DRR. The patterns and excitation coefficient are given in Fig. 5.

Figure 5.

Pattern and excitations: (a) normalized pattern and (b) excitations.

Then, we consider another deformation type where the array surface is equivalently transformed into a plate fixed at both ends and is deformed by a concentrated load. All parameters of plate remain unchanged, and the bending deformation satisfies:

(9)\begin{equation}{z_n}\left( P \right) = - {{P{x_n}} \over {48EI}}\left( {3{l^2} - 4{x_n}^2} \right),{\ }n = 1,2,3, \cdots ,N\end{equation}

where $P$ is the concentrated load and ${z_{{\textrm{max}}}}\left( P \right)$ takes $\lambda $. The PSLL is suppressed to −28.85 dB, which decreases by about 6.13 dB compared with the pattern obtained by Chebyshev-woc (DRR = 4.04) with lower DRR (DRR = 3). The patterns and excitation coefficients are given in Fig. 6. The results show that different types of array deformations cause different forms of distortion in the array pattern, and the proposed approach can recover the radiation pattern from a deteriorate pattern caused by the unsteady deformations.

Figure 6.

Pattern and excitations: (a) normalized pattern and (b) excitations.

Case 4

Finally, we study the regularity of the excitation coefficients. In this case, we use the linear array in the first example, and the deformation form of the array follows (1). The radiation performance for the deformation of ${z_{{\textrm{max}}}}\left( P \right) = 0.1\lambda $ to $0.9\lambda $, with an interval of $0.1\lambda $, is calculated. We use the optimized amplitude and the derived phase excitations obtained by equation (6). Figure 7 shows the optimized amplitude and phase excitation coefficients (obtained by equation (6)) of each element, and Fig. 8 shows the normalized pattern (for each case, the optimal PSLLs are around −26.33 dB). It is worth mentioning that, when ${z_{{\textrm{max}}}}\left( P \right) = 0.5\lambda $, we compare the calculation results with those in the electromagnetic simulation software FEKO. The result is also plotted in Fig. 8, and it shows an agreement.

Figure 7.

Excitations with different ${z_{{\textrm{max}}}}\left( P \right)$: (a) amplitude excitation and (b) phase excitation.

Figure 8.

Normalized pattern with different ${z_{{\textrm{max}}}}\left( P \right)$.

It is seen that as the deformation increases, the amplitude excitations obtained from the optimization do not change much. So, the set of optimized solutions (amplitude excitations) is treated as an initial solution, and phase excitation coefficients are obtained by equation (6) (the phase excitations are within one group [see Fig.7(b)]), which can effectively reduce the amount of equipment data stored. For practical applications, it provides a feasible way to reduce the complexity of beamforming network through determining a set of excitations.