Multi-Objective Design of Antennas Using Surrogate Models

The following sections are included:

• Dedication
• Preface
• About the Authors
• Acknowledgments
• Contents

Antennas are essential components of wireless communication systems such as radio and television broadcasting, radar, mobile phones, satellite communications, airborne navigation, Bluetooth devices, RFID tags, wireless computer networks, to name just a few (Kraus and Marhefka, 2002). Design of modern antennas is a challenging task where an important step is adjustment of geometry and material parameters of the structure so as to satisfy given performance requirements concerning return loss, radiation pattern, gain, etc. (Gautam et al., 2013; Nguyen et al., 2013; Kuwahara, 2005). In many cases, geometrical constraints have to be taken into account because — for some applications (e.g., handheld and wearable devices; Guo et al., 2012; Chahat et al., 2011) — achieving compact designs might be of primary concern. For the sake of reliability, the process of dimension adjustment (also referred to as design closure; Koziel and Ogurtsov, 2014a) has to be based on accurate evaluation of antenna performance. The latter can only be ensured by full-wave electromagnetic (EM)-simulation. This is particularly important for compact antennas as well as other cases where EM couplings between the antenna itself and its environment (connectors, housing, feeding circuitry) may affect the device’s operation. However, EM analysis of realistic and finely discretized antenna models might be computationally expensive. Typically, this is not a problem for design validation, but it may be prohibitive from the point of view of parametric optimization which requires multiple simulations of the antenna structure at hand…

In this chapter, formulation and challenges of antenna design are discussed from the perspective of simulation-driven optimization. The emphasis is put on explaining the role of computer simulations in modern antenna engineering, formulating the design task as an optimization problem, as well as introducing the notation used throughout the book. Furthermore, we outline typical antenna design objectives. We also provide a brief discussion of computational models utilized in antenna optimization with the emphasis on a distinction between high- and low-fidelity electromagnetic (EM)-simulations, the latter utilized as auxiliary models to speed up the EM-driven design process. The chapter is concluded with a discussion of the challenges related to simulation-based design of antenna structures.

The primary topic of this book is surrogate-assisted of antenna structures; however, for the convenience of the reader, we provide a brief introduction to numerical optimization. In this chapter, we cover selected conventional single-objective optimization techniques, both gradient-based and derivative-free. Chapter 4 outlines global optimization using population-based metaheuristics. An important terminological distinction should be made here between conventional (or direct) and surrogate-based techniques. In direct methods, the expensive electromagnetic (EM)-simulation antenna model is handled directly in the optimization process, whereas in surrogate-assisted algorithms, majority of the operations are carried out using a fast replacement model (the surrogate) with only occasional reference to the expensive simulations. Figure 3.1 shows a general flow of the direct simulation-driven optimization process. It can be observed that each candidate solution (design) produced by the algorithm is evaluated using a computationally expensive high-fidelity simulation. Clearly, this might become a bottleneck if the number of such evaluations is large…

Optimization methods outlined in Chapter 3 are local ones, i.e., they normally allow for finding an optimum that is located in the vicinity of the initial solution. Unfortunately, in many practical problems, objective functions with multiple optima have to be handled. Furthermore, the functional landscape of the problem at hand is often unknown in terms of the nonlinearity of the objective function, importance of particular variables, and also the number and the location of the optima. At the same time, estimating a reasonably good starting point is often very difficult. In all these cases, utilization of local methods usually leads to unsatisfactory results and global search may be necessary…

In the context of electromagnetic (EM)-simulation-driven design of antenna structures, the major disadvantage of conventional numerical optimization techniques outlined in Chapters 3 and 4 is their high computational cost, typically measured in hundreds (for local methods) or thousands and tens of thousands of objective function evaluations (for population-based metaheuristics). The exception is gradient-based search with adjoint sensitivities (Ghassemi et al., 2013; Koziel and Ogurtsov, 2012c; Koziel et al., 2013c, 2014d; Koziel and Bekasiewicz, 2015a, 2016f), where the optimization process can be conducted in reasonable time even for relatively large number of designable parameters. Nevertheless, it seems that the most promising approaches in terms of expedited simulation-driven design are those exploiting surrogate models. In this chapter, we provide a brief introduction to surrogate-based optimization (SBO). In particular, we outline the SBO concept, discuss various surrogate modeling techniques, as well as overview surrogate-assisted optimization methods both approximation- and physics-based. The algorithms for cost-efficient multi-objective optimization of antenna structures considered later in this book are largely based on SBO paradigm and exploit specific SBO methods as the building blocks of the antenna design frameworks.

Majority of engineering design problems are of multi-objective nature, i.e., there are several performance figures that have to be controlled at the same time. In antenna design, typical figures include reflection response, gain, radiation pattern, axial ratio (for circular polarization antennas), as well as the structure size (for compact antennas) (Azaro, 2008; Gautam et al., 2013; Ding et al., 2008; Kuwahara, 2005; Bekasiewicz et al., 2016b; Bai et al., 2016; Smolders and Johannsen, 2011; Viani et al., 2008). In most cases, either because the designer’s priorities are known beforehand or just to simplify the design task, the problem can be reformulated into a single-objective one, which can be achieved by selecting the primary goals and handling the remaining objectives by means of constraints (Koziel and Bekasiewicz, 2015e). Another popular technique is aggregation of the objectives into a scalar cost function using a weighted sum approach (Kuwahara, 2005; Lizzi et al., 2009) or penalty functions (Bekasiewicz and Koziel, 2016b). Nevertheless, in some situations, it might be important to acquire more comprehensive information about the antenna at hand, in particular, to identify the best possible trade-offs between the conflicting criteria. If this is the case, defaulting to proper multi-objective optimization becomes necessary. Multi-objective optimization leads to a set of alternative designs representing a so-called Pareto front (Fonseca, 1995). Subsequently, a decision making process allows for selecting one of these designs as a final solution…

Nowadays, the most popular approaches for solving multi-objective optimization problems are population-based metaheuristics. Their important advantage is the ability of generating the entire representation of the Pareto front in a single run of the algorithm. Other advantages include simplicity and availability of numerous (and often reliable) implementations. A brief exposition of these methods was provided in Chapter 6. On the other hand, a serious disadvantage of population-based algorithms is their considerable computational complexity. In a typical multi-objective setup, a large population size is utilized (from a 100 to a few hundreds of individuals) so that the overall number of objective function evaluations during the optimization run might be as high as a few thousands to tens of thousands. This becomes a fundamental bottleneck when applying metaheuristics for multi-objective design of contemporary antenna structures. As explained in Chapter 2, reliable performance evaluation of antennas requires full-wave electromagnetic (EM) analysis. Such analysis may be quite expensive particularly for realistic models (e.g., including connectors and installation fixtures) with typical simulation times of several minutes to a few hours per design. Clearly, straightforward optimization of EM models using population-based algorithms might be prohibitive…

Multi-objective optimization of contemporary antennas is a very challenging task not only due to the necessity of evaluating the structures using computationally expensive full-wave electromagnetic (EM) analysis but also because a typical search space is large both in terms of the number of (usually geometry) parameters to be adjusted and its wide ranges. Performing the optimization process in such a space is a waste of resources because vast majority of the designs are non-optimal and the region of the space containing Pareto-optimal solutions is normally a very small subset of the original space. Therefore, appropriate reduction of the space may be critical for carrying out multi-objective optimization in a computationally feasible manner. In Chapter 7, several surrogate-assisted optimization techniques for multi-objective design optimization of antenna structures have been described, all involving variable-fidelity EM-simulation models. The first and the most generic of these methods exploits population-based metaheuristics (here, a multi-objective evolutionary algorithm, MOEA) for generating the initial approximation of the Pareto front. Because neither low- nor high-fidelity EM model can be directly handled by MOEA, the optimization process is conducted using an auxiliary data-driven surrogate constructed by Kriging interpolation of sampled low-fidelity model data. Construction of the Kriging model in the original space is impossible if the antenna structure is described by more than a few parameters…

The purpose of this chapter is to demonstrate the operation and performance of the surrogate-assisted multi-objective optimization procedures formulated in Chapters 7 and 8. The numerical studies included here are very comprehensive and contain nine antenna structures of various levels of complexity. For some of the test cases, numerical results are supported by experimental data. The number of geometry parameters ranges from five to 24. The considered design objectives include reflection response, antenna size (footprint area or volume), antenna gain, as well as radiation pattern. In vast majority of cases, two-objective problems are solved, which is the most typical situation, especially for compact antennas, where the trade-offs between the antenna size and its electrical performance (specifically, the maximum in-band reflection level) are sought. There are several antenna types considered in this study, including ultra-wideband (UWB) monopoles, planar Yagi antennas, a dielectric resonator antenna, as well as a UWB multi-input multi-output (MIMO) structure…

As demonstrated in Chapter 9, surrogate-assisted algorithms of Chapters 7 and 8 can be utilized for expedited multi-objective optimization of antenna structures. The same methods can also handle expensive computational models in other engineering disciplines (cf. Chapter 11). Here, we discuss several aspects and practical issues of these techniques. In particular, we look into their scalability properties. More specifically, we are interested in the relationship between dimensionality of the design space and the computational cost of the optimization algorithm (Sec. 10.1). Another issue is statistical analysis of surrogate-assisted multi-objective optimization algorithm of Sec. 7.1, where several components are of stochastic nature. In particular, the initial Pareto front approximation is obtained using multi-objective evolutionary algorithm (MOEA). In Sec. 10.2, we investigate the influence of MOEA operation on the quality of the final Pareto set representation found by the optimization procedure. Finally, in Sec. 10.3, we study the effect of various patch size setups on the cost and performance of sequential domain patching (SDP) algorithm of Sec. 7.3. The studies presented in this chapter lead to certain conclusions concerning the robustness of the considered optimization procedures, indicate potential applicability for solving more challenging problems (than those presented so far in the book), as well as give some guidelines and recommendations in terms of the algorithm setup.

In this chapter, we discuss applications of surrogate-assisted optimization methods considered in this book for solving multi-objective design problems in other engineering areas. A common issue is a high cost of computational models that prevents the straightforward applications of the off-the-shelf algorithms such as population-based metaheuristics. On the other hand, utilization of the methods described in Chapters 7 and 8 along with variable-fidelity simulations allows for yielding the Pareto-optimal designs in practically acceptable timeframes. Three specific case studies are presented, including multi-objective design of impedance matching transformer using sequential domain patching (SDP), optimization of compact microstrip coupler by means of Pareto front exploration, and two-objective design of transonic airfoils using response surface surrogates and co-Kriging.

The goal of multi-objective optimization — as understood in this book — is to find comprehensive information about a given structure in the form of a set of alternative designs representing the best possible trade-offs between conflicting objectives. The numerical methods presented in Chapters 7 and 8, and demonstrated in Chapters 9 and 11, allow for finding such Pareto sets in reasonable timeframes, even for relatively expensive high-fidelity electromagnetic (EM)-simulation models. In this chapter, we demonstrate utilization of Pareto sets for comprehensive comparison of various antenna structures and microwave circuits. Three case studies are considered. The first one involves three structures of compact ultra-wideband (UWB) antennas, where information about available size-performance trade-offs allow for conclusive comparison of the competing antenna topologies. The second case is a comparison of two structures of UWB monopole antennas that are topologically similar, yet a small modification leads to considerable performance improvement in terms of attainable miniaturization rate. The last study investigates selection of the best possible architecture for compact impedance matching transformers, specifically, the optimum type and arrangement of the compact cells utilized as the basic building blocks of the transformer structure. In all cases, the knowledge of Pareto fronts is indispensable to make application-driven design decisions. At the same time, it is pointed out that the antenna dimensions reported in the literature are often far from the optimum ones; consequently, various comparisons of antenna structures provided in the published works might be of limited use.

Simulation-driven design of antenna structures is an exciting area of research yet of paramount importance for engineering practice. One of its fundamental challenges is high computational cost of full-wave electromagnetic (EM) antenna analysis. High-fidelity EM simulation models are necessary to ensure accurate evaluation of the antenna performance but their optimization using conventional numerical techniques is impractical or — in extreme cases — even prohibitive. Multi-objective optimization, the main focus of this book, is considerably more challenging because the aim is to find the entire set of designs that represent the best possible trade-offs between several (and often conflicting) objectives. In multi-objective design, the high cost of evaluating an antenna structure becomes a more serious bottleneck than for single-objective optimization. Another important issue is multi-dimensionality of the design space and potentially wide ranges of geometry parameters of the antenna. In particular, global search needs to be involved, first to identify a relevant portion of the design space (the one that contains Pareto-optimal designs), then to find a representation of the Pareto front. None of these can be executed directly at the level of EM-simulation models, especially if optimization is conducted using population-based metaheuristics…

The following sections are included:

• References
• Index