Simulation-Based Optimization of Antenna Arrays

Slawomir Koziel, Stanislav Ogurtsov

• Publisher: WSPC (EUROPE)
• eBook ISBN: 9781786346001
• Topic: Technology & Engineering
• Subtopic: Electrical Engineering & Telecommunications
• Index: Technology & Engineering

The book addresses surrogate-assisted design of antenna arrays, in particular, how surrogate models, both data-driven and physics-based, can be utilized to expedite procedures such as parametric optimization, design closure, statistical analysis, or fault detection. Algorithms and design frameworks are illustrated using a large variety of examples including real-world printed-circuit antenna and antenna array structures.

This unique compendium contains introductory materials concerning numerical optimization, both conventional (gradient-based and derivative-free, including metaheuristics) and surrogate-based, as well as a considerable selection of customized procedures developed specifically to handle antenna array problems. Recommendations concerning practical aspects of surrogate-assisted multi-objective antenna optimization are also given. The methods presented allow for cost-efficient handling of antenna array design problems (involving CPU-intensive EM models) in the context of design optimization and statistical analysis, which will benefit both researchers, designers and graduate students.

Contents:

• Introduction
• Antenna Array Fundamentals
• Fundamentals of Numerical Optimization
• Global Optimization: Population-Based Metaheuristics
• Fundamentals of Surrogate-Based Modeling and Optimization
• Antenna Models for Simulation-Based Design
• Element Design: Case Studies
• Microstrip Antenna Subarray Design Using Simulation-Based Optimization
• Antenna Array Models for Simulation-Based Design and Optimization
• Design of Linear Antenna Array Apertures Using Surrogate-Assisted Optimization
• Design of Planar Microstrip Antenna Arrays Using Variable-Fidelity EM Models
• Design of Planar Microstrip Array Antennas Using Simulation-Based Superposition Models
• Design of Planar Arrays Using Radiation Response Surrogates
• Simulation-Based Design of Corporate Feeds for Low-Sidelobe Microstrip Linear Arrays
• Design of Linear Phased Array Apertures Using Response Correction and Surrogate-Assisted Optimization
• Fault Detection in Linear Arrays Using Response Correction Techniques
• Surrogate-Assisted Tolerance Analysis of Microstrip Linear Arrays with Corporate Feeds
• Discussion and Recommendations: Prospective Look

Readership: Primary readership: antenna engineering and microwave engineering (graduate students, researchers, and designers); Secondary readership: electrical engineering, mechanical engineering (graduate students, researchers, designers).Antenna Arrays;Printed-Circuit Arrays;Simulation-Driven Design;Surrogate Modeling;Surrogate-Based Optimization00

Chapter 1

Introduction

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.

Antenna theory provides overall understanding of electromagnetic interactions within antenna structures, behavior of basic antenna array quantities, including their dependency on geometry parameters, properties of materials, and frequency. Importantly, antenna theory also provides design guidelines and working formulas for setting up prototypes of array elements, feeds, and for dimensioning radiating apertures. On the other hand, computationally inexpensive analytical models can only provide — in most cases — an approximate estimation of the optimum design. This is particularly the case when certain interactions within the array elements, feed networks, and with the array environment (e.g., housing, installation fixture, connectors) have to be taken into account. Reliable evaluation of design characteristics and sensitivity to tolerances before manufacturing are particularly important for integrated antenna array structures. For these reasons, full-wave electromagnetic (EM) simulation plays an essential role in antenna array verifications.

An important part of the antenna array design process is the adjustment of geometry and material parameters to ensure that the array responses satisfy design specifications in terms of certain quantities such as sidelobe level, beamwidth, locations of pattern nulls, peak realized gain, reflection loss, efficiency, etc. (Balanis, 2005; Mailloux, 2005; Hansen, 2009). For these purposes, reliable knowledge of antenna array quantities at different points of the design space is necessary, which requires high-fidelity modeling and simulation of antenna array models.

Full-wave EM simulations of realistic and finely discretized antenna array models are computationally expensive: evaluation for a single combination of design parameters may take up to several hours. While such computational costs are acceptable from the design validation standpoint, it is usually prohibitive for tuning or optimization which normally requires a large number of EM simulations of the structure of interest. Such a large number of EM simulation is, in part, a consequence of the fact that realistic models of antenna array apertures and array feeds are typically described by many design variables. As far as accurate simulated data about antenna array figures are expected, e.g., simulated excitation amplitudes at the inputs of low-sidelobe apertures, radiation patterns with sidelobe levels of 30 dB and lower, the use of circuit decomposition approaches to relief a computational load in this situation is quite restricted, unfortunately, due to underlying EM interactions within the structures.

Significant numbers of designable parameters in the antenna array models, especially, at the stage of implementation, and essentially multi-objective nature of array design problems motivate the use of numerical optimization for solving antenna array design tasks. At the stage of prototyping, the objective function is usually computed using explicit formulas, e.g., an array factor, while at the stage of implementation, the objective function is evaluated by means of computational electromagnetics.

Automation of the antenna array design process can be realized by formulating an adjustment of array components or/and an adjustment of the entire aperture-feed circuit as an optimization problem with the objective function supplied by an EM solver (Special Issue, 2007; Bandler et al., 2006). Unfortunately, most of the conventional optimization techniques, including gradient-based methods (Nocedal and Wright, 2006), e.g., conjugate-gradient, quasi-Newton, sequential quadratic programming, etc., and derivative-free (Kolda et al., 2003), e.g., Nelder–Mead, pattern search techniques, require a large number of objective function evaluations to converge to an improved design. For many realistic EM models, where evaluation time per design reaches a few hours with contemporary computing facilities, the cost of such an optimization process may be unacceptably high.

Another practical problem of conventional optimization techniques, dealing with discrete EM models, is numerical noise of the simulated data. This noise is partially a result of adaptive meshing techniques used by most contemporary discrete solvers: even a small change of design variables may result in a change of the mesh topology, and, consequently, discontinuity of the EM-simulated responses as a function of designable parameters. Such numerical noise is particularly an issue for gradient-based methods that normally require smoothness of the objective function.

The aforementioned challenges result in a situation where a common approach to simulation-driven antenna array design is based on repetitive parameters sweeps (usually, one parameter at a time) about an initial design which is set up using other means such as formulas, analytical and circuit models, etc. This approach is usually more reliable in case of simulated responses of antenna arrays than a brute-force optimization using built-in optimization capabilities of commercial simulation tools; however, it is also very laborious, time consuming, and demanding significant supervision of the designer. Moreover, such a parameter-sweep-based optimization process does not guarantee optimal results because only a limited number of parameters can be handled that way. It is also difficult to utilize correlations between the parameters properly. Finally, truly optimal values of the designable variables can be quite counter-intuitive.

In recent years, population-based search methods (also referred to as metaheuristics) (Yang, 2005, 2008) have gained considerable popularity. This group of methods includes, among others, genetic algorithms (GA) (Back et al., 2000), particle swarm optimizers (PSO) (Kennedy, 1997), differential evolution (DE) (Storn and Price, 1997), ant colony optimization (Dorigo and Stutzle, 2004). Most of metaheuristics are biologically inspired algorithms designed to alleviate certain difficulties of the conventional optimization methods, in particular, handling problems with multiple local optima (Yang, 2010a).

Probably the most successful application of the metaheuristic algorithms for antenna arrays resided so far in array factor optimization, e.g., Ares-Pena et al. (1999), Haupt (1995), Jin and Rahmat-Samii (2005, 2006, 2007), Petko and Werner (2007), Bevelacqua and Balanis (2007), Selleri et al. (2008), Li et al. (2010), Rajo-Iglesias and Quevedo-Teruel (2007) and Roy et al. (2011). In these problems, the cost of evaluating the single element response is not of the primary concern or the response of a single element is already available, e.g., with pre-assigned (e.g., isotropic) or pre-simulated array elements. However, application of metaheuristics to EM-simulation-driven design of array apertures and feeds is not practical because corresponding computational costs would be tremendous: typical GA, PSO or DE algorithms require hundreds, thousands or even tens of thousands objective function evaluations to yield a solution (Ares-Pena et al., 1999; Haupt, 1995; Jin and Rahmat-Samii, 2007; Petko and Werner, 2007; Bevelacqua and Balanis, 2007; Pantoja et al., 2007; Selleri et al., 2008; Li et al., 2008; Rajo-Iglesias and Quevedo-Teruel, 2007; Roy et al., 2011).

The problem of high-computational cost of conventional EM-based optimization can be alleviated to some extent by the use of adjoint sensitivity (Director and Rohrer, 1969), which is a computationally cheap way to obtain derivatives of the system response with respect to its design parameters. Adjoint sensitivities can substantially speed up microwave design optimization while using gradient-based algorithms (Bandler and Seviora 1972; Chung et al., 2001). This technology was also demonstrated for antenna optimization (Jacobsson and Rylander, 2010; Toivanen et al., 2009; Zhang et al., 2012). It should be noted, however, that adjoint sensitivities are not yet widespread in commercial EM solvers. To our knowledge this feature is available only in CST Microwave Studio (CST, 2016) and ANSYS (HFSS, 2016). In addition, a reliable utilization of adjoint sensitivities is limited by numerical noise of EM-simulated responses (Koziel et al., 2013b).

Surrogate-based optimization (SBO) (Forrester and Keane, 2009; Queipo et al., 2005) is one of the most recent and promising ways to realize computationally efficient simulation-based design of antenna arrays (Koziel and Ogurtsov, 2014a,b, 2015a–d). The main idea of SBO is to shift the computational burden of the optimization process to a surrogate model. This surrogate model is a cheap representation of the optimized structure (Bandler et al., 1994; Queipo et al., 2005; Koziel et al., 2006; Koziel and Ogurtsov, 2011a). In a typical scenario, the surrogate model is used as a prediction tool to find approximate location of the original (high-fidelity or fine) model. After evaluating the high-fidelity model at this predicted optimum, the surrogate is updated in order to improve its local accuracy. The key prerequisite of the SBO paradigm is that the surrogate is much faster than the high-fidelity model. Also, in many SBO algorithms, the high-fidelity model is only evaluated once per iteration. Therefore, the computational cost of the design process with a well working SBO algorithm may be significantly lower than those with most of conventional optimization methods.

There are two major types of surrogate models. The first one comprises function-approximation models constructed from sampled high-fidelity simulation data (Simpson et al., 2001). A number of approximation and interpolation techniques are available, including artificial neural networks (Haykin, 1998), radial basis functions (Wild et al., 2008), kriging (Forrester et al., 2009), support vector machines (Smola and Schölkopf, 2004), Gaussian process regression (Angiulli et al., 2007; Jacobs, 2012), or multi-dimensional rational approximation (Shaker et al., 2009). If the design space is sampled with sufficient density, the resulting model becomes reliable so that the optimal design can be found just by optimizing the surrogate. In fact, approximation methods are usually used to create multiple use library models of specific components. The computational overhead related to such models may be very high. Depending on the number of designable parameters, the number of training samples necessary to ensure decent accuracy might be hundreds, thousands or even tens of thousands. Moreover, the number of samples quickly grows with the dimensionality of the problem (so-called curse of dimensionality). As a consequence, globally accurate approximation modeling is not suitable for ad-hoc (onetime) antenna array optimization. Iteratively improved approximation surrogates are becoming popular for global optimization (Couckuyt, 2013). Various ways of incorporating new training points into the model (so-called infill criteria) exist, including exploitative models (i.e., models oriented towards improving the design in the vicinity of the current one), explorative models (i.e., models aiming at improving global accuracy), as well as model with balanced exploration and exploitation (Forrester and Keane, 2009).

Another type of surrogates, so-called physics-based surrogates, are constructed from underlying low-fidelity (or coarse) models or the respective structures. Because the low-fidelity models inherit some knowledge of the system under consideration, usually a small number of high-fidelity simulations are sufficient to configure a reliable surrogate. The most popular SBO approaches using physics-based surrogates that proved to be successful in microwave engineering are space mapping (SM) (Bandler et al., 1994, 1995, 2003, 2004a,b), tuning SM (Cheng et al., 2010; Koziel et al., 2011b), as well as various response correction methods (Echeverria and Hemker, 2005; Koziel and Leifsson, 2012b, 2016). To ensure computational efficiency, it is important to have the low-fidelity model considerably faster than the high-fidelity model. For that reason, circuit equivalents or models based on analytical formulas are preferred (Bandler et al., 2004a). The aforementioned methods (particularly SM) were mostly used for design of microwave filters or transmission-line-based components. Unfortunately, in case of antennas and antenna array apertures, reliable circuit equivalents are rarely available. For radiating structures, a universal way of obtaining their low-fidelity models is with low-fidelity simulations. Such low-fidelity models are relatively expensive even for a single antenna. This poses additional challenges in terms of optimization.

The central topic of this book is the simulation-based antenna array CAD methods where numerical optimization serves as a mean to conduct the automated design process, implement designs, and/or tune already configured designs of antenna array elements, apertures, feeds, and entire aperture-feed circuits. In the view of the outlined challenges faced by simulation-based antenna array design methods, a substantial part of this book is devoted to numerically efficient realization of optimization techniques, in particular, the ones utilizing surrogate models. We believe that the methods described and demonstrated in this book will contribute to the development of practical and reliable antenna array CAD tools. The book provides a useable description of optimization methods which are relevant to EM-simulation-based antenna and antenna array design. We also expect that a variety of application examples help in explaining how the described methods work for different components and at different levels of antenna array circuits. Finally, we hope that this book will help an interested reader in forming his vision about the benefits, complexity, and challenges of the described methods in the context of antenna array CAD.

We begin, in Chapter 2, by giving a motivation for formulating the simulation-based antenna array design task as an optimization problem. We also outline basic antenna array figures of interest as well as relations which, to our experience, are typically used for simulation-based design of antenna arrays.

In Chapter 3, we provide background information about conventional numerical optimization techniques, including both gradient-based and derivative-free methods.

Chapter 4 is focused on introducing numerical methods for global optimization.

Chapter 5 introduces fundamentals of SBO, techniques for surrogate-based modeling, as well as surrogate-based algorithms for solving computationally expensive microwave and antenna engineering problems.

Chapter 6 overviews basic requirements imposed on low-fidelity models of antennas. Basic options for setting such models are described. The accuracy–speed tradeoff which is inherent to simulated low-fidelity models of antennas is illustrated on examples.

Chapters 7 and 8 describe and demonstrate application of the surrogate-based methods, which are described in Chapter 5, to antenna and antenna subarray design ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Dedication
• Preface
• About the Authors
• Acknowledgments
• 1. Introduction
• 2. Antenna Array Fundamentals
• 3. Fundamentals of Numerical Optimization
• 4. Global Optimization: Population-Based Metaheuristics
• 5. Fundamentals of Surrogate-Based Modeling and Optimization
• 6. Antenna Models for Simulation-Based Design
• 7. Element Design: Case Studies
• 8. Microstrip Antenna Subarray Design Using Simulation-Based Optimization
• 9. Antenna Array Models for Simulation-Based Design and Optimization
• 10. Design of Linear Antenna Array Apertures Using Surrogate-Assisted Optimization
• 11. Design of Planar Microstrip Antenna Arrays Using Variable-Fidelity EM Models
• 12. Design of Planar Microstrip Array Antennas Using Simulation-Based Superposition Models
• 13. Design of Planar Arrays Using Radiation Response Surrogates
• 14. Simulation-Based Design of Corporate Feeds for Low-Sidelobe Microstrip Linear Arrays
• 15. Design of Linear Phased Array Apertures Using Response Correction and Surrogate-Assisted Optimization
• 16. Fault Detection in Linear Arrays Using Response Correction Techniques
• 17. Surrogate-Assisted Tolerance Analysis of Microstrip Linear Arrays with Corporate Feeds
• 18. Discussion and Recommendations: Prospective Look
• References
• Index