
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

72710
10.1038/s41598-024-72710-7
Article
Performance-constrained multi-objective optimization of antennas for miniaturization design
Yang Qi yangqi08@nudt.edu.cn

1
Wang Hongqiang 1
Peng Xin 2
1 https://ror.org/05d2yfz11 grid.412110.7 0000 0000 9548 2110 College of Electronic Science and Technology, National University of Defense Technology, Changsha, 410073 China
2 https://ror.org/04w9fbh59 grid.31880.32 0000 0000 8780 1230 School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing, 100876 China
14 9 2024
14 9 2024
2024
14 214978 7 2024
10 9 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
The miniaturization of antennas is crucial as it improves the integration of wireless communication system. In order to achieve miniaturization of antennas, a performance-constrained multi-objective optimization method (PCMOM) considering the size and return loss is proposed. In the PCMOM method, an optimization strategy based on a multi-port network model is introduced, enabling the formation of antennas with various structures. Furthermore, we integrate the constraint of return loss performance into the non-dominated sorting genetic algorithm II (NSGA-II), eliminating solutions that do not meet performance requirements. Three pixel antennas are designed using the PCMOM method and two of them are fabricated. Experimental results demonstrate that the proposed PCMOM method can effectively address the complex trade-off issues in antenna miniaturization design.

Keywords

Antenna design
Miniaturization
Multi-objective optimization
Multi-port network modeling
Subject terms

Electrical and electronic engineering
Aerospace engineering
National Natural Science Foundation of China under Grant622011591 61921001 62105363 62035014 issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

The miniaturization of antennas significantly reduces the overall footprint of communication systems, thereby enabling higher integration densities and greater flexibility1–3. Nevertheless, achieving antenna miniaturization without compromising its excellent performance remains a challenging task4,5. In the miniaturization design of antennas, there exists a delicate balance between the antenna’s size and its key performance indicators, including return loss, bandwidth, and gain6,7. As the antenna’s size diminishes, its capacity to efficiently radiate and receive electromagnetic (EM) waves may be hindered, resulting in a degradation of performance.

Multi-objective optimization algorithms strive to simultaneously optimize multiple, often conflicting, objectives8–10. These algorithms have potential applications in the antenna design. The Pareto front11 of multi-objective optimization is derived from the intricate interrelationships among multiple objective functions and represents the set of non-dominated solutions. In antenna design, improving one objective function often necessitates a compromise in another, and the Pareto front offers designers a diverse combination of trade-off options12,13. By leveraging the Pareto front, designers can select the most suitable antenna design that aligns with their specific specifications and constraints. Therefore, the utilization of multi-objective optimization algorithms holds promise in achieving miniaturization while maintaining optimal performance, offering a comprehensive approach to strike the desired balance between size reduction and performance enhancement.

The non-dominated sorting genetic algorithm (NSGA-II) is a widely used multi-objective optimization algorithm14,15. However, the application of the NSGA-II algorithm in antenna design faces two major challenges. Firstly, the optimization process necessitates EM simulations, which can be computationally demanding and time-consuming16,17. It poses a significant hurdle in the optimization process, especially when dealing with complex antenna geometries and multiple objectives. Secondly, the Pareto front, while offering a range of trade-off solutions, may not always yield solutions that perfectly meets design specifications. That is to say, multi-objective optimization algorithms only emphasize the tradeoff relationship between objective functions, without ensuring whether the obtained Pareto solutions meet practical requirements. For instance, if the return loss of all solutions in the front exceeds − 10 dB, it indicates insufficient radiation of electromagnetic (EM) waves. Therefore, it is difficult to directly apply the NSGA-II algorithm for antenna design.

To address these issues, an efficient physical model is required to replace intensive EM simulation models for multi-objective optimization of antennas. Moreover, it is crucial to incorporate design specifications into the updating of individuals so that all solutions along the Pareto front satisfy the design specification of antennas. In this work, we propose a performance-constrained multi-objective optimization (PCMOM) method for the miniaturization of antenna design. The specific contributions are as follows:

In the proposed method, a multi-port network model18 is utilized to create a cost-effective physical model for the multi-objective design of antennas. This approach significantly reduces the dependency on resource-intensive EM simulations.

We propose the PCMOM method for incorporating constraints into multi-objective optimization algorithms, which is based on multi-port networks and the NSGA-II algorithm. Compared with19,20, each solution in the optimized Pareto front of our proposed PCMOM method fulfills the requirements of antenna manufacturing, a crucial aspect for engineering applications in antenna design.

The proposed method is used to design three pixel antennas, with two of them being fabricated and measured. And the performance of the antennas is verified through EM simulation and measurement.

The proposed method

In this section, we introduce the PCMOM approach, consisting of three main elements. Initially, we introduce a design process of antennas using multi-port network. Next, we outline the various objective functions for antenna design. Lastly, we describe the performance-constrained NSGA-II (PC-NSGA-II) algorithms.

Design process of antennas using multi-port network

To reduce the time cost of antenna design using EM simulation, we establish the initial multi-port structure of antennas composed of multiple discrete metal pixel patches, as shown in Fig. 1a. It is worth noting that the size of the entire radiator is frequency dependent, usually around 1/2 wavelength, the size of each pixel patch is between 1/10 wavelength and 1/20 wavelength, and the distance between pixels is between 1/10 and 1/20 of the wavelength. These metal pixels are connected by multiple ports, which are divided into N internal ports and one external port (feeding port). The relationships between the feeding antenna and the surrounded elements can be represented by a (N + 1)×(N + 1) complex impedance matrix of the ports as Eq. (1).1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Z}}\left( \omega \right) = \left[ {\begin{array}{*{20}l} {Z_{{00}} \left( \omega \right)} & {{\mathbf{Z}}_{{0N}} \left( \omega \right)} \\ {{\mathbf{Z}}_{{N0}} \left( \omega \right)} & {{\mathbf{Z}}_{{NN}} \left( \omega \right)} \\ \end{array} } \right]$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Z_{00}}$$\end{document} denotes the self-impedance for the antenna feeding port. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf{Z}}_{0 N}} \in {{\mathbb{C}}^{1 \times N}}$$\end{document} is a vector which express the mutual impedance between the feeding port and each internal port. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Z}}_{{N0}} \in \mathbb{C}^{{1 \times N}}$$\end{document} is the transpose of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf{Z}}_{0N}}$$\end{document}. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf{Z}}_{NN}} \in {{\mathbb{C}}^{N \times N}}$$\end{document} is a sub-matrix and denotes the impedance between all internal ports. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega$$\end{document} represents the antenna operating frequency.

Then, the relationship between the load impedance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf{Z}}_L}$$\end{document}= [ZL1, ZL2, …, ZLN] of the internal ports and the input impedance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf{Z}}_{{\text{in}}}}$$\end{document}of the antenna is calculated by Eqs. (2) and (3)18,19. Each element in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf{Z}}_L}$$\end{document}has two possibilities: “open circuit” and “short circuit”, represented by 1 and 0, with corresponding impedances of infinity and 0, respectively.2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf{Z}}^L}={\text{diag}}\left( {{{\mathbf{Z}}_L}} \right)$$\end{document}

3 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{{{\text{in}}}} \left( \omega \right) = Z_{{00}} \left( \omega \right) - {\mathbf{Z}}_{{1N}} \left( \omega \right) \cdot \left( {{\mathbf{Z}}^{L} \left( \omega \right) + {\mathbf{Z}}_{{NN}} \left( \omega \right)} \right)^{{ - 1}} \cdot {\mathbf{Z}}_{{N1}} \left( \omega \right)$$\end{document}

According to the input impedance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf{Z}}_{{\text{in}}}}$$\end{document}of the antenna and the impedance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Z_0}$$\end{document} of the EM, we obtain the reflection coefficient of the antenna in Eq. (4)18,19.4 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{{11}} \left( \omega \right) = \frac{{{\mathbf{Z}}_{{00}} \left( \omega \right) - {\mathbf{Z}}_{{1N}} \left( \omega \right) \cdot \left( {{\mathbf{Z}}^{L} \left( \omega \right) + {\mathbf{Z}}_{{NN}} \left( \omega \right)} \right)^{{ - 1}} \cdot {\mathbf{Z}}_{{N1}} \left( \omega \right) - Z_{0} \left( \omega \right)}}{{{\mathbf{Z}}_{{00}} \left( \omega \right) - {\mathbf{Z}}_{{1N}} \left( \omega \right) \cdot \left( {{\mathbf{Z}}^{L} \left( \omega \right) + {\mathbf{Z}}_{{NN}} \left( \omega \right)} \right)^{{ - 1}} \cdot {\mathbf{Z}}_{{N1}} \left( \omega \right) + Z_{0} \left( \omega \right)}}$$\end{document}

Fig. 1 Optimization process based on multi-port network of antennas, (a) Initial structure, (b) Performance-optimized structure, and (c) Topology-optimized structure.

Utilizing the multi-port network, we can regulate the antenna’s performance by altering the load of the internal port \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf{Z}}_L}$$\end{document} in accordance with Eq. (4), without the need for EM simulations, which reduces the time-cost of optimization of antenna performance. The optimized antenna structure is depicted in Fig. 1b. Once \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf{Z}}_L}$$\end{document} is established, the antenna’s structure can be derived. Subsequently, due to the small impact of discrete pixel patches on the current distribution of the antenna, isolated pixels that are not integrated into the overall structure are eliminated, leading to a compact antenna design as illustrated in Fig. 1c.

Multi-objective function formulations for antenna design

In this work, the proposed multi-objective method aims to optimize two objective functions: (a) minimizing the size of the radiator, and (b) minimizing the reflection coefficient, while simultaneously accounting for the performance constraint. The multi-objective optimization problem is defined by the objectives themselves, along with the constraints that must be satisfied during the optimization process to ensure actual fabrication standards are met. The multi-objective optimization problem is defined by5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \mathop {{\text{Minimize}}}\limits_{{\mathbf{x}}} {\text{ }}{\mathbf{F}}\left( {\mathbf{x}} \right)={\left[ {{f_1}\left( {\mathbf{x}} \right),{\text{ }}{f_2}\left( {\mathbf{x}} \right)} \right]^T} \\ & s.t.{\text{ }}g\left( {\mathbf{x}} \right) \leqslant 0 \\ \end{aligned}$$\end{document}

where x = ZL=[ZL1, ZL2,…, ZLN] represents the vector of the states for all internal ports, in which “0” represents “open” and “1” represents “short”. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f_1}\left( {\mathbf{x}} \right)$$\end{document} is the objective function related to antenna radiation size. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f_2}\left( {\mathbf{x}} \right)$$\end{document} is the objective function related to reflection coefficient of the antenna. And \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\left( {\mathbf{x}} \right)$$\end{document} is the inequality constraint function.

The first objective function is to minimize the operating radiation size of the antenna.6 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\left( {\mathbf{x}} \right)=\forall {{\mathbf{x}}_{i,1}}\parallel {{\mathbf{x}}_{i,2}} \cdots {\text{ }}\parallel {{\mathbf{x}}_{i,N}},i=1, \ldots ,N$$\end{document}

7 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\left( {\mathbf{x}} \right)=\forall {{\mathbf{x}}_{1,j}}\parallel {{\mathbf{x}}_{2,j}} \cdots {\text{ }}\parallel {{\mathbf{x}}_{N,j}},j=1, \ldots ,N$$\end{document}

where N is the number of internal ports. C(x) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\left( {\mathbf{x}} \right)$$\end{document} are vectors obtained by performing a custom operation based on the ‘or’ logic between corresponding row (for C(x)) and column (for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\left( {\mathbf{x}} \right)$$\end{document}) elements, respectively.8 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f_1}\left( {\mathbf{x}} \right)={M_c}\left( {C\left( {\mathbf{x}} \right)} \right) \cdot {M_c}\left( {R\left( {\mathbf{x}} \right)} \right)$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_c}\left( \cdot \right)$$\end{document} is the number of elements that are continuously 1 in the vector∙.

The second objective function is to minimize the S11 of the working frequency band, which is defined by9 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f_2}\left( {\mathbf{x}} \right)=\mathop {\hbox{max} }\limits_{\omega } {\left| {{S_{11}}\left( {{\mathbf{x}},\omega } \right)} \right|_{{\text{dB}}}}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left| {{S_{11}}\left( \cdot \right)} \right|_{{\text{dB}}}}$$\end{document} refers to the reflection coefficient in dB. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega$$\end{document} represents the antenna operating frequency.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\left( {\mathbf{x}} \right)$$\end{document} refers to the inequality constraint function of the minimum required return loss, which is defined by10 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{g}}\left( {\mathbf{x}} \right)={\left| {{S_{11}}\left( {{\mathbf{x}},\omega } \right)} \right|_{{\text{dB}}}} - \gamma$$\end{document}

where the constraint value of the reflection coefficient, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document}, is set to − 10 dB in this work. By incorporating this inequality constraint, we ensure that the reflection coefficient S11 remains below − 10 dB for all solutions obtained through multi-objective optimization.

Performance-constrained NSGA-II algorithm

We proposed multi-objective optimization algorithm, called Performance-Constrained NSGA-II (PC-NSGA-II), designed to address the multi-objective optimization challenge outlined in Eq. (7). PC-NSGA-II algorithm is based on a low computational cost multi-port model, so we use the traditional NSGA-II to perform optimization. The PC-NSGA-II algorithm not only provides a Pareto frontier where two objectives mutually constrain each other. And all solutions within the resulting Pareto front meet the requirement of a reflection coefficient below − 10dB.

In the PC-NSGA-II algorithm, performance constraints are integrated into the selection process of the NSGA-II algorithm. Since most individuals in the population do not meet the condition of S11 being less than − 10dB during the initial phase of the NSGA-II algorithm, incorporating the constraints at this early stage would make it challenging to achieve the desired outcomes. Therefore, we allow the algorithm to run for a certain number of iterations before introducing the constraint condition.

Once the algorithm reaches a predefined iteration threshold (Nc), individuals exhibiting reflection coefficients that fail to meet the specified specification are downgraded to the lowest priority level. Consequently, every solution within the optimized Pareto front meets the design specification, enabling their application for fabrication. Flowchart of the performance-constrained multi-objective optimization method is shown in Fig. 2. The specific steps of the proposed PC-NSGA-II method are outlined as follows:Step 1: Establish the initial antenna structure and obtain the multi-port network model of the antenna.

Step 2: Formulate the objective functions of the size of antenna (as Eq. 8) and the reflection coefficient (as Eq. 9), and the constraint function (as Eq. 10).

Step 3: Set the population size, the total number of iterations (Nt), and the number of iterations for adding the constraint function (Nc), and relevant parameters for crossover and mutation operators.

Step 4: Generate the initial population, and conduct non-dominated sorting and crowding distance calculation on the population to establish its hierarchical structure.

Step 5: Perform selection operation.

Step 6: crossover operation.

Step 7: Perform mutation operation.

Step 8: Merge the offspring with the parent for performing non-dominated sorting and crowding calculation, and establish the hierarchical structure.

Step 9: If the predefined iteration threshold for adding performance constraints are reached, downgrade any solution with a reflection coefficient exceeding − 10 dB to the lowest priority level. Otherwise, proceed to the next step.

Step 10: Perform elite selection to identify a predetermined number of outstanding individuals.

Step 11: If the total number of iterations are met, preserve the Pareto front. Otherwise, go to Step 6.

Fig. 2 Flowchart of the performance-constrained multi-objective optimization method.

Results

Parameters setting of optimization

In this section, to validate the performance of the PCMOM method, we use the PCMOM method to design antennas. The proposed PCMOM method is implemented on the platform of Matlab 2019b. As illustrated in Fig. 1a, an initial structure composed of 10 × 4 pixels has been designed. Each pixel has a length and width of 3.1 mm, with a distance of 3 mm between adjacent pixels. Additionally, the distance between the ground plane and the nearest pixel is 5 mm. The ground plane itself measures 55 mm in length and 62 mm in width, respectively. The initial structure incorporates 67 internal ports, with an external port (feed port) situated at the fifth pixel of the first row. The dielectric substrate used is FR4, possessing a dielectric constant of 4.4 and a thickness of 1 mm. A full-wave EM simulation is conducted to ascertain the impedance matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Z}}(\omega )$$\end{document} in Eq. (1) of the initial structure using the CST MWS 2019.

Results of multi-objective optimization

The two optimization objective functions of the antenna are shown in formula (8) and (9), respectively. In the second objective functions, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega$$\end{document}is the vector of the 10 frequency points from 2 to 3 GHz, with a spacing of 0.1 GHz between each frequency point. The design specification is defined as an antenna that operates across the band of [2.4 GHz 2.6 GHz]. The PC-NSGA-II algorithm executes for a total of 1000 evolutionary iterations with a population size of 300 individuals. The crossover probability of NSGA-II algorithm is set at 0.6, while the mutation probability of NSGA-II algorithm is established at 0.1. During the selection process of NSGA-II algorithm, elite selection is performed based on ranking. The iteration threshold for adding the constraint is set at 300. Through trials, we have found that when the threshold falls within the range of 30–60% of the total evolutionary iterations, the algorithm can achieve good results.

Figure 3 illustrates the population’s evolution throughout the optimization process. Figure 3a displays the objective function values of the initial population. By the 300th iteration, many individuals cluster near the Pareto front, as seen in Fig. 3b. A constraint is introduced to the NSGA-II algorithm after the 300th iteration. From the 315th generation onward, individuals with an S11 value above − 10 dB are gradually removed, as shown in Fig. 3c. After 1000 iterations, the final Pareto front is depicted in Fig. 3d. All designs belonging to the Pareto front are superior in terms of one or two objectives. However, empirical considerations lead us to select a design that all of its parameters remain below an acceptable value. Three solutions from Pareto front namely Solution 1, Solution 2, and Solution 3 are selected for analysis and comparison, which are highlighted in Fig. 3d.

Fig. 3 Multi-objective optimization process of PC-NSGA-II algorithm, (a) Initial population, (b) 300 iteration (without performance constraints), (c) 315 iteration (with performance constraints), and (d) Pareto front.

Figure 4 presents the geometry of and the current distribution of these three antennas of Solution 1, Solution 2, and Solution 3. The Solution 1 (Antenna 1) exhibits the largest S11 (reflection coefficient) within the target frequency points, accompanied by a working size function value of 18. The Solution 2 (Antenna 2), on the other hand, demonstrates a lower S11 in the target points, with a corresponding working size function value of 21. The Solution 3 (Antenna 3) stands out with the lowest S11 in the target points, achieving a working size function value of 24. Depending on specific user requirements, any of these three solutions can be chosen as the optimal antenna design. The far-field radiation patterns of the three antennas are shown in Fig. 5.

Fig. 4 Current distribution of the pixel antenna, (a) Antenna 1, (b) Antenna 2, and (c) Antenna 3.

Fig. 5 Far-field pattern of the pixel antenna, (a) Antenna 1, (b) Antenna 2, and (c) Antenna 3.

Antennas 1 and 2 are prototyped and subsequently measured. Figure 6 presents the prototype of Antenna 1 along with the results of the measurement and simulation. As shown in Fig. 6b, the measured S-parameters agree well with the simulation, and the measured − 10 dB impedance bandwidth of Antenna 1 is 0.6 GHz (from 2 to 2. 6 GHz). The bandwidth fulfills the design specifications. Similarly, Fig. 7 shows the prototype of Antenna 2 along with its measurement and simulation results. As can be seen in Fig. 7b, the measured curves closely match the simulation, revealing a favorable impedance bandwidth in the target frequency band (the measured − 10 dB impedance bandwidth of Antenna 2 is 0.86 GHz from 2.4 GHz to 3.26 GHz), thereby completely satisfying the design specifications. The two examples effectively demonstrate the efficiency and feasibility of the proposed design method.

Fig. 6 Fabricated and measured results of antenna 1, (a) Prototype, and (b)S11.

Fig. 7 Fabricated and measured results of antenna 2, (a) Prototype, and (b) S11.

Discussion on the performance of the proposed algorithm

To clearly illustrate the performance of our proposed PCMOM algorithm, we discuss its impact on the miniaturization of antennas, the time cost, and the potential applications.

Miniaturization of the proposed algorithm.

In communication systems, antennas need to be designed together with other RF devices, such as in mobile phone antennas, which have a single large ground plane that supports circuit boards and touch screens. Therefore, we did not optimize the size of the ground plane but the radiators for antennas. The radiation structure of the initial antenna has a length of approximately 0.25 wavelength and a width of 0.5 wavelength, composed of 40 pixels. The radiation structures of these two antennas utilize only 17 and 19 pixels, respectively, representing 42.5% and 47.5% of the initial radiation structure’s pixel count.

To further assess the miniaturization of the proposed algorithm, we compare the antennas in this work with existing pixel antenna structures. Table 1 shows the performance comparison between our designed antennas and others. Our approach excels in miniaturizing antenna radiators, featuring the smallest radiator size among those antennas. Thus, the miniaturization of the radiation structures is achieved through the application of the PCMOM method.

Table 1 Performance comparison with the pixel antennas in other references.

References	Size of the radiator (mm2)	Center frequency (GHz)	Optimization method	Publication year	
[21]	85 × 81.5	2.5	Single objective	2017	
[22]	65 × 65	2.45	Single objective	2022	
[23]	45 × 27	2.6	Single objective	2023	
[20]	Ant. I	46.9 × 27.9	2.45	Multiple objectives	2023	
Ant. II	46.9 × 27.9	2.45	
[24]	75.5 × 36.5	2.45	Single objective	2022	
In this work	Ant. I	33.6 × 15.3	2.45	Multiple objectives	2024	
Ant. II	39.7 × 15.3	2.45	

(2) Time cost of the proposed algorithm.

Moreover, the PCMOM method relies on a multi-network model and EM simulation to extract Z-impedance, requiring low time cost. It takes 75 min, including 300 individuals’ evaluations of the population (each < 0.01 s), 1000 iterations of the population, and one EM simulation for Z-impedance extraction (approximately 15 min). In contrast, if the EM simulation model is directly employed for multi-objective optimization, each EM simulation takes 30 s. The total time cost using the EM simulation model is estimated at 150,000 min, with the same number of populations and iterations as in the PCMOM method. PCMOM reduces total time cost by over 90% compared to EM simulation.

(3) Potential applications of the proposed algorithm.

The key contribution of this work is the miniaturization of antenna radiation structures, which has practical applications. For instance, in mobile communication systems, reducing the antenna radiator’s size allows more space for other RF devices, thereby enhancing system integration25,26. Additionally, in transparent antennas, removing unnecessary pixels can improve transparency.

Conclusions

In this work, we propose a performance-constrained multi-objective optimization method (PCMOM) for the miniaturization of antenna design. In the proposed PCMOM method, an optimization strategy based on a multi-port network model is used to facilitate diverse topological configurations. Two design objective functions are considered: antenna size and return loss. Performance constraints are integrated into the NSGA-II algorithm, thus ensuring that all Pareto solutions adhere to the specified antenna design specifications. To validate the performance of the PCMOM method, three pixel antennas are designed using the PCMOM method, and two of them are fabricated. The measured − 10 dB impedance bandwidths of these two antennas are 0.6 GHz and 0.86 GHz, respectively, both of which meet the design specification. Its performance is demonstrated in the experimental results.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant 62201591, Grant 61921001, Grant 62105363, and Grant 62035014.

Author contributions

Q.Y. is responsible for multi-obejctive algorithm design and antenna design. H.W. is responsible for the framework of the manuscript. X.P. is responsible for revising the manuscript.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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