Simulation-Based Optimization of Antenna Arrays

The following sections are included:

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

Computer-aided design (CAD) of antenna arrays is becoming a common practice these days. In a significant part, this is due to development of computational electromagnetics which delivers reliable tools for setting up and simulation of models of antenna array elements, feeds, and array apertures. The development of computers also tremendously contributes to the state of the art of antenna array CAD so that antenna array models comprising millions and tens of millions of unknowns (mesh cells, knots, etc.) can be evaluated on regular desktops in a reasonable time. The use of a graphics processing unit (GPU) for certain discrete types of electromagnetic solvers can speed up simulations in a few folds. Therefore, configured designs of array components and even the entire antenna array structures, to certain extend, including their installation environment, such as connectors, radomes, covers, finite grounds of printed apertures, can be simulated by full-wave discrete solvers prior manufacturing at a high-fidelity level of description…

The following sections are included:

• Introduction
• Antenna Array Radiation Figures
• Array Factor Analysis
• Mutual Coupling and Antenna Array Reflection Response
• Mutual Coupling and Antenna Array Design Validation
• Simulation-Based Antenna Array Design
• A Concept of Antenna Array Simulation-Based Optimization
• Challenges of Antenna Array Simulation-Based Optimization

The main topic of this book is automated, simulation-driven design of antenna arrays. The primary tools utilized in the design process are therefore numerical optimization algorithms. In the following three chapters including this one, we discuss the fundamentals of numerical optimization. The material covered here is not supposed to be a systematic and exhaustive presentation of the subject (interested readers may refer to numerous textbooks such as Nocedal and Wright, 2006; Yang, 2010a; Conn et al., 2009) but it should give the reader sufficient background information necessary to understand the remaining parts of the book. In this chapter, we cover conventional methods, including gradient-based algorithms for both unconstrained and constrained optimization, as well as derivative-free techniques such as pattern search. Chapter 4 is a brief exposition of global optimization using population-based metaheuristics. Methods of this class are probably the most popular algorithms for design optimization of antenna arrays these days, although, as pointed out later, their application is very limited for handling realistic design problems involving expensive simulated electromagnetic models. As already mentioned in Chapter 2, design of contemporary antenna arrays heavily relies on full-wave electromagnetic (EM) analysis which is typically carried out using methods of computational electromagnetics. Because of considerable CPU cost of accurate EM simulations, direct optimization of electromagnetic models might be prohibitive. Surrogate-assisted methods alleviate this difficulty by shifting the majority of operations to a fast replacement model (also referred to as the surrogate). This class of techniques is discussed in Chapter 5.

From practical perspective (with our main focus being design of antenna arrays), one of the most important features of the optimization methods presented in Chapter 3 is their locality. Gradient-based algorithms as well as pattern search derivative-free techniques normally allow for finding an optimum that is located in the vicinity of the initial solution, or, more generally, in the region of attraction of the nearest local optimum. Unfortunately, the objective functions that one needs to deal with in a large number of real-world engineering problems are characterized by multiple local optima. Moreover, the functional landscape of the problem at hand is often unknown in terms of the nonlinearity of the objective function, importance of particular variables, but also the number and the location of the optima. In many cases, finding a reasonably good starting point is a non-trivial task by itself. In such situations, straightforward application of local algorithms often leads to unsatisfactory results; therefore, global search may be necessary. Design optimization of antenna arrays, in particular, radiation pattern synthesis (e.g., for minimum sidelobe level) or design of sparse arrays, normally requires global optimization. Still, local methods are useful when good initial point has been identified…

Design optimization of antennas and antenna arrays is challenging for several reasons. One of these is that the process involves high-fidelity EM simulation models necessary for reliable performance evaluation of the structure under design. Another reason is a large number of designable parameters: these might be dimensions of radiating elements and feeds, parameters of substrates as well as excitation amplitudes and/or phases. Other challenges might include typically graded composition of the array structures, presence of connectors and radomes. Furthermore, global optimization is often necessary because functional landscapes pertinent to antennas and antenna arrays tend to be multi-modal with multiple local optima. All these challenges make conventional numerical optimization techniques (see Chapters 3 and 4 for details) unsuitable for handling real-world antenna design problems. In particular, the computational cost of such algorithms is typically measured in hundreds (for local methods) or thousands and tens of thousands (for population-based metaheuristics) of objective function evaluations. A notable exception is gradient-based search with adjoint sensitivities (Ghassemi et al., 2013; Koziel and Ogurtsov, 2012c; Koziel et al., 2013b, 2014b; Koziel and Bekasiewicz, 2015a, 2016b) where the optimization process can be conducted in reasonable time even for relatively large number of designable parameters. Unfortunately, adjoints are not yet widespread in computational electromagnetics, especially in terms of availability of this technology through commercial simulation software packages (CST, 2016; HFSS, 2016)…

In this chapter, we discuss low-fidelity EM-simulated models as fundamental components of antenna and antenna array design processes which are performed by means of simulation-based surrogate-assisted optimization. These low-fidelity models are also referred as coarse models in the broader context of surrogate-based microwave circuit design optimization (Bandler et al., 1994, 1995, 2003, 2004a; Koziel and Ogurtsov, 2014; Leifsson and Koziel, 2015). We list and discuss general requirements the low-fidelity models need to satisfy as well as overview typical low-fidelity model setups. Numerical examples emphasize the trade-off between the low-fidelity model’s accuracy and its computational cost.

In this chapter, we demonstrate the use of surrogate-assisted optimization techniques for the design of individual printed-circuit antenna structures including those used as elements of antenna arrays. In all cases, we utilize different variations of surrogate modeling methods, variable-fidelity EM simulation models, response correction techniques, some of which have been discussed in the previous parts of this book, particularly, in Chapter 5. We also emphasize a point that depending on particular design objectives, generic optimization methods need to be tailored for a specific design task, e.g., when dealing with explicit reduction of the antenna size, or especially when finding design trade-offs between conflicting antenna figures. Some of the case studies are provided with experimental verification although — from the optimization viewpoint — the ultimate validating model is the high-fidelity EM simulation one.

Subarrays of microstrip antennas are important elements of contemporary low-profile arrays (Mailloux, 2005; Haupt, 2010). The low profile and low fabrication costs of microstrip subarrays justify their utility. Moreover, they can also be produced with the same fabrication techniques that are used to realize elements of amplitude and phase control, amplification, as well as other necessary antenna arrays elements, all on the same substrate (Garg et al., 2000; Waterhouse, 2003). At the same time, microstrip patch antennas are characterized by narrow fractional bandwidth when built on thin substrates and in their basic configurations (Jackson, 2007)…

The optimization-based design of antenna arrays (AAs) is justified, first, by a large number of designable parameters describing composition and geometries of the aperture, elements, and feeding circuit. Second, array radiation and reflection response figures typically exhibit nonlinear behavior as function of the designable parameters. Accurate evaluations of the response figures can only be achieved through full-wave electromagnetic simulations of the discrete models of array circuits, which can be quite computationally expensive. Further, AA design problems are inherently multi-objective ones with several performance requirements to be controlled at the same time…

This chapter addresses design of linear antenna array apertures using conventional and surrogate-based optimization techniques. A number of specific methods and design cases are considered. We start with several cases of end-fire and broadside linear array apertures where radiation patterns are evaluated using the array factor. It is demonstrated, as opposed to common practice, the design optimization process does not need to use population-based metaheuristics with the array factor-based models. Instead, a random-search-based initialization and the conventional gradient search are sufficient for the considered problems…

In this chapter, a technique for rapid simulation-driven design of planar microstrip antenna arrays is discussed. It exploits the surrogate-based optimization (SBO) paradigm and variable-fidelity electromagnetic (EM) simulations. The design process includes radiation pattern optimization and matching in term of active reflection coefficients at the inputs of the elements of the array aperture…

In this chapter, a technique for simulation-driven design of excitation tapers for planar antenna arrays is discussed. The technique exploits models constructed as a superposition of simulated radiation and reflection responses of the array under design. Low computational costs of these models are ensured by using iteratively corrected EM-simulation data computed with coarse-meshes. The technique allows for simultaneous control of the radiation pattern and the reflection coefficients. Numerical efficiency as well as scalability of the technique is demonstrated using the design examples of various sizes and topologies. The examples include 16-element and 100-element microstrip patch antenna arrays of the Cartesian lattice and a 100-element microstrip antenna array of the hexagonal lattice. The technique is versatile as it also can be applied for simulation-based optimization of antenna arrays comprising other types of radiation elements, e.g., wire dipoles or dielectric resonator antennas.

In this chapter, a technique for rapid simulation-based design of planar antenna arrays using radiation response correction is discussed. The method has been introduced in Chapter 10 for linear arrays. Here, it is generalized for more complex cases. In short, the methodology is based on establishing a suitable correction of the analytical array factor model of the array of interest. The corrected array factor is utilized as a fast surrogate allowing us to find the optimum element spacing and phase excitations at low computational cost. The correction terms are iteratively refined to account for design-dependent discrepancies between the array factor model and the EM-simulated model. The methodology is demonstrated with design of a 10-GHz 100-element planar array of microstrip patch antennas with a plastic cover, as well as a 28-element planar microstrip antenna array.

So far we discussed surrogate-assisted techniques for optimization of array apertures’ dimensions and excitation tapers. Circuits realizing the required excitation taper, the feeds, stayed out of the scope of this book (i.e., the feeds were assumed to be separate design problems). At the same time, in modern antenna systems, in particular in printed-circuit arrays, the feeds are typically integrated with the aperture so that the post-manufacturing adjustment of the structure is hardly feasible or even impossible. In Chapter 8 we already demonstrated, for a case of microstrip subarrays, the importance and benefits of considering the aperture-feed structure of interest within a single simulation-based projects so that the contribution of the feed to the radiation response and mutual interactions of the elements as well as elements-to-feed interactions, all affecting the radiation and reflection responses of the structure, could be reliably evaluated and, thus, subsequently adjusted…

In this chapter, we discuss and illustrate a simulation-driven surrogate-assisted approach to design of phase excitation tapers for linear phased array apertures. The design process is realized by a mean of surrogate-based optimization (SBO). SBO utilizes two surrogate models: a coarse-mesh surrogate of the array element for adjusting the array’s active reflection responses, and a fast surrogate of the radiation pattern which is a corrected array factor pattern. Iterative referencing to high-fidelity simulated responses allows us to conduct the design process with respect to the near-field interactions, e.g., coupling, and other undesirable wave phenomena within the aperture. At the same time, the design process is made fast due to the utilized SBO technique…

In this chapter, we discuss a surrogate-assisted technique for detecting faulty elements in a small linear microstrip array of patch antennas from samples of the array’s far-field magnitude radiation pattern (here represented by realistic EM simulations). It is important that regardless of the array size, the method requires only one expensive full-wave entire-array simulation. This one simulation gives the accurate far-field magnitude pattern of the original defect-free array, and is used in conjunction with the defect-free array’s analytical array factor to formulate a response correction function. This response correction function can then be used to construct an accurate approximation of the EM-simulated pattern of any arbitrary faulty array at very low cost. The low cost and high accuracy of approximations make possible an enumeration strategy for identifying the faulty elements, which would have been computationally prohibitive were EM-simulated patterns to be used. Furthermore, partial faults and measurement noise are addressed. Accuracies in detecting up to three faults (including partial ones) in arrays of 16 and 32 elements exceeded 97% under noise-free conditions, and were above 93% in the presence of 2 dB measurement noise.

In this chapter, we address the problem of fast statistical analysis of microstrip linear antenna arrays with corporate feeds using surrogate-assisted methods. A realistic design has to take into account manufacturing tolerances that normally lead to a degradation of array performance in terms of its circuit and radiation responses. Tolerances may be pertinent to array radiating elements, element spacing, geometry parameters of the corporate feed (resulting, among others in deviation of the excitation amplitudes and phases from the required nominal values), as well as deviations of parameters of the substrate, e.g., substrate local thickness and permittivity. The main performance figure affected by these parameter deviations is the sidelobe level (SLL) in case of low-sidelobe patterns. Estimation of the SLL spread requires statistical analysis that takes into account all of the aforementioned factors. Accurate evaluation of SLLs requires full-wave electromagnetic (EM) analysis of the array (and the feed if the entire system is of interest) which is very expensive in computational terms. Consequently, performing statistical analysis directly at the level of the EM array model is impractical. Here, we discuss simple techniques that permit speeding up the process of antenna array statistical analysis. These techniques include various surrogate modeling methods as well as correction techniques that are used to align responses of fast surrogates with responses of the respective EM models.

Full-wave electromagnetic (EM) analysis has become a fundamental tool in the design of antennas and antenna arrays. Contemporary EM solvers feature great flexibility and versatility in terms of structure meshing, analysis setup, result post-processing, and are capable of accounting for various EM effects such as coupling of elements of apertures and feeds, undesirable radiation from feeds, excitation of higher-order modes, leakage of fields into substrates, environmental interactions (e.g., due to the presence of radomes, housing, installation fixtures, connectors), as well as dispersion, anisotropy, and nonlinear properties of materials. All of this is necessary to permit reliable evaluation of the array performance. However, accuracy comes at the expense of considerable computational costs. A single simulation may be as long as a few dozen of minutes or even a few hours depending on the array complexity and the modeling and simulation arrangements. Clearly, it is manageable in case of design verification but becomes a bottleneck for simulation-driven design, especially when a large number of analyses are necessary. Typical examples of such tasks include parametric optimization, statistical analysis, yield-driven design, fault detection, etc. For practical reasons (primarily, reduction of the computational overhead), various simplifications are exercised, such as utilization of analytical models (e.g., array factors assuming isotropic radiators), decomposition (e.g., separate treatment of the feed and the aperture), or leaving out some of design objectives at certain design stages (e.g., controlling the reflection level while designing the elements, without further correction upon assembling the aperture). All of these simplifications help maintaining the design time within acceptable frames but also lead to suboptimal performance of the final array designs. On the other hand, to yield the best possible design, accounting for all relevant effects (element coupling, non-perfect matching, feed radiation, etc.) is mandatory. Yet, the computational cost of such comprehensive treatment within the EM-based design frameworks is prohibitive when using conventional methods…

The following sections are included:

• References
• Index