Rapid multi-objective optimization of antennas using nested kriging surrogates and single-fidelity EM simulation models Open Access

Performance requirements imposed on contemporary antenna systems are multifold and pertinent to both electrical and field characteristics (Stutzman and Thiele, 2012), multi-band and multi-functional operation (Borhani et al., 2016; Soltani et al., 2017), as well as geometrical constraints (primarily compact size, Bhattacharya et al., 2016). The need for fulfilling stringent specifications necessitates the development of antennas that feature more and more complex geometries and are described by a large number of variables (Matekovits et al., 2007; Szini et al., 2015; Buckley et al., 2016). Both factors create the need for handling/optimizing several design objectives at the same time while operating within parameter spaces of ever-increasing number of dimensions. Usage of full-wave electromagnetic (EM) simulation tools, computationally expensive but mandatory to ensure sufficient evaluation reliability, is yet another design challenge. Many of the design objectives are at least partially conflicting, i.e. improvement of one implies certain degradation of others. Perhaps the most recognized example is antenna miniaturization: reduction of the size normally leads to problems with ensuring sufficient impedance matching as well as affects other characteristics, such as efficiency or pattern stability (Liu et al., 2014; Koziel et al., 2018). Consequently, any practical design must be a compromise (or trade-off) between the objectives of interest.

The most reliable way of handling multiple goals (as well as just one objective for that matter) is numerical optimization (Deb and Gupta, 2006). Majority of practically used algorithms are single-objective routines (Tian et al., 2010; Koziel and Ogurtsov, 2014). These, however, require a scalar cost function. To control several design goals, some sort of aggregation (e.g. using a weighted sum approach) or constrained optimization with acceptance levels set for all but the primary objective, need to be performed. In either case, the outcome is a single design that qualitatively (but not necessarily quantitatively) represents the designer’s preferences about the goals. On the other hand, genuine multi-objective optimization (MO) procedures are capable of identifying the entire set of trade-off designs (the Pareto set, Koziel and Bekasiewicz, 2016). The most popular class of MO algorithms involve population-based metaheuristics (Darvish and Ebrahimzadeh, 2018; Goudos et al., 2011; Bauernfeind et al., 2017; Jayaprakasam et al., 2014; Aravanis et al., 2015; Rinaldo et al., 2005), which exhibit global search capabilities (important in some cases), but come at the expense of a tremendous computational cost (Easum et al., 2018; Nagar and Werner, 2018). The latter is normally unmanageable if the antenna performance is evaluated through EM simulation. A new approach to expedited global design optimization of antennas with evolutionary algorithm by means of small-scale parallelizing computations has been reported recently (Akinsolu et al., 2019; Liu et al., 2018), yet the method has been applied to single-objective optimization problems only.

It seems that the usage of surrogate models is currently the most promising approach to render EM-driven MO procedures computationally feasible (Koziel and Bekasiewicz, 2016). In relatively simple cases (in terms of the parameter space dimensionality), construction of global surrogates may be sufficient (Chen, 2015; de Villiers et al., 2017; Easum et al., 2017). For other situations (>10 geometry parameters) more needs to be done. Some of the recently proposed techniques (Koziel and Ogurtsov, 2013; Koziel et al., 2014a), suggest usage of variable-fidelity EM simulations along with the initial design space reduction. The latter aims at identifying the region of the parameter space that contains the Pareto front and at constructing the surrogate therein. This has been demonstrated to yield considerable computational savings even though the applied space reduction methods only give a very rough estimation of the Pareto front geometry.

In this paper, a novel multi-optimization framework is proposed for a cost-efficient EM-driven antenna design. Our methodology adopts a recently reported nested kriging surrogate modeling technique (Koziel and Pietrenko-Dabrowska, 2019). The technique uses two kriging surrogates. The first-level model is used to define the domain of the surrogate, containing the designs that are of high quality from the point of view of the performance figures relevant to the considered system. The second-level model, set up in that domain, is the actual surrogate representing the system outputs. The fundamental benefit of constraining the surrogate domain is a notable reduction of the required training data set as compared to the traditional kriging approach over a box-constrained domain. Here, nested kriging methodology is used to identify the parameter space region containing the Pareto set (design space reduction) as well as to construct the surrogate that allows us to produce the initial approximation of the trade-off designs through metaheuristic optimization. Because of a good precision of representing the Pareto front geometry by the nested kriging framework, the MO process can be executed in a computationally efficient manner despite merely using high-fidelity EM simulations of the antenna at hand. For the sake of demonstration, two structures are optimized, a planar Yagi antenna (for best matching and maximum gain) and a wideband monopole (for best matching, minimum gain variability and minimum size). In both cases, the Pareto set is obtained at the cost of only a few hundred of EM simulations, which is almost 80 per cent less than for the surrogate-assisted framework involving a simpler design space reduction technique.

To the best of the authors’ knowledge, this paper is the first attempt to use the nested kriging method for MO. The major novelty is in adopting the nested kriging technique for design space confinement and providing a fast surrogate for low-cost generation of the initial Pareto set. For that purpose, a rigorous mathematical formalism is developed in Section 2.4, which permits the establishment of the surrogate in an arbitrary subset of the objective space instead of the box-constrained space as in Koziel and Pietrenko-Dabrowska (2019). The presented methodology is demonstrated to efficiently solve MO of real-world antenna design tasks. In addition, its superiority over state-of-the-art surrogate-assisted procedures is validated. The proposed method is generic in the sense of its capability of handling any number of design objectives.

This section provides a necessary background material concerning MO by means of surrogate modeling. It also briefly recalls the concept of nested kriging, as well as describes how to incorporate it into the multi-objective design framework. The main goal is the reduction of the computational overhead of multi-objective design beyond what is offered by the existing approaches.

For the purpose of further considerations, the design objectives will be denoted as F_k, k = 1, …, N_obj, where N_obj stands for the overall number of the goals. MO aims at finding a Pareto set of designs that are globally non-dominated (Deb, 2001). A design is globally non-dominated – within the considered parameter space – if there is no other design simultaneously better with respect to all objectives. The Pareto set represents the best possible trade-offs with respect to [F₁ F₂ … F_Nobj].

The antenna under design is evaluated using EM analysis and the respective computational model is denoted as R(x), where x is a vector of adjustable (typically, geometry) parameters. As explained in Section 1, it is impractical or even infeasible to carry out MO directly on R when using population-based metaheuristics. According to the recent literature (de Villiers et al., 2017; Easum et al., 2017; Koziel and Ogurtsov, 2013; Koziel et al., 2014a), the most promising way of speeding up the optimization process is the usage of an auxiliary surrogate model R_s (a faster representation of the antenna at hand). In most cases, the surrogate is an approximation model, typically a kriging (An et al., 2018) or Gaussian process regression one (Lyu et al., 2018). Other approximation methods may also be used, e.g. artificial neural networks (ANNs; Mishra et al., 2015) or support vector regression (SVR; Jacobs, 2012). A particular selection of the modeling framework is generally of secondary importance, although particular techniques may suffer from certain issues. For example, the generalization ability of ANNs, i.e. the predictive power over the points outside the training set, may proof deficient (Chauhan et al., 2012). Handling this issue generally requires appropriate adjustment of the network architecture (Na et al., 2017). On the other hand, SVR’s generalization capabilities are superior over ANNs. Yet, developing effective antenna models using SVM is not a straightforward process, and the researchers have to use their domain expertise in it (Chauhan et al., 2012). Regardless of the approximation technique used, the surrogate can be optimized directly using, e.g. a multi-objective evolutionary algorithm (MOEA; Fonseca, 1995).

Further acceleration can be achieved by exploiting variable-fidelity EM simulations. This sort of approach has been adopted in Koziel and Ogurtsov (2013) and Koziel et al. (2014a), where the surrogate was constructed at the level of coarse-discretization EM model. Unfortunately, a refinement process is required to accommodate the discrepancies between the low- and high-fidelity simulations. In Koziel and Ogurtsov (2013), a simple response correction in the form of output space mapping (Koziel et al., 2008) has been used. The high-fidelity Pareto designs have been obtained through the following refinement of the selected low-fidelity Pareto-optimal designs x_s^(k):

The correction term R(x_s^(k)) − R_s(x_s^(k)) ensures perfect alignment between the surrogate R_s and the high-fidelity model at x_s^(k) (Koziel and Ogurtsov, 2013). In Koziel et al. (2014b), an alternative refinement procedure based on co-kriging has been proposed.

The fundamental difficulty concerning the surrogate-assisted MO, as described in Section 2.1, is a construction of the surrogate model itself. Apart from low-dimensional cases (up to a few parameters), it is simply not possible to build the model in the entire design space. The reasons include the lack of prior knowledge about the Pareto front location (which implies wide parameter ranges), and the curse of dimensionality. Both issues lead to excessive costs of training data acquisition.

The design space can be reduced in a reasonable manner by considering the single-objective optima x*^(k) = argmin{x: F_k(R(x))}, i.e. the extreme Pareto-optimal designs that determine the span of the Pareto front. Thus, the lower and upper bounds for the antenna parameters can be set as l^* = min{x^*(1),…, x^*(Nobj)} and u^* = max{x^*(1),…, x^*(Nobj)} (Koziel and Ogurtsov, 2013). Apart from some “pathological” cases, the interval [l^*,u^*] contains the vast majority of the Pareto front. A better idea about the Pareto front geometry can be acquired using more involved methods such as the rotational space reduction (Koziel and Bekasiewicz, 2015), which is, however, applicable to two-objective problems only.

This work aims at the development of an improved surrogate-assisted MO procedure, which permits further computational savings over the framework described in Sections 2.1 and 2.2. To this end, we adopt a recently reported nested kriging modeling (Koziel and Pietrenko-Dabrowska, 2019). It is used to implement a more efficient design space reduction scheme and to construct the surrogate model for initial Pareto set identification. The term “nested” refers to performing kriging interpolation consecutively twice: first to identify the surrogate model domain, then to establish the actual surrogate model itself in the constrained domain. The difference between the nested and traditional kriging is that the latter operates in an interval-like domain, defined by lower and upper parameter bounds. Because of complex interactions between geometry parameters, designs that are optimal with respect to the performance figures are focused within small subsets of such a domain (Koziel and Sigurdsson, 2017). Thus, constraining the surrogate model domain with the nested kriging, leads to significant cost savings. In this section, we briefly recall the nested modeling concept, whereas its adaptation for MO purposes is explained in Section 2.4.

The figures of interest selected for the design process (e.g. operating frequency, bandwidth and maximum/average in-band gain) are denoted by f_k, k = 1, …, N. These are identical to the MO design goals F_k of Section 2.1. The foundation of the nested kriging are the reference designs x^(j) = [x₁^(j) … x_n^(j)]^T, j = 1, …, p, optimized with respect to the performance vectors f^(j) = [f₁^(j) … f_N^(j)]. The vectors f^(j) are allocated within the objective space F defined using the ranges f_k.min ≤ f_k^(j) ≤ f_k.max, k = 1, …, N, to be covered by the surrogate model (Koziel and Pietrenko-Dabrowska, 2019). Clearly, the coverage of the objective space should be possibly uniform. Typically, the number of the reference designs is small, e.g. around 10 or so (Koziel and Pietrenko-Dabrowska, 2019) because the dimensionality of F is low.

The modeling procedure involves two surrogates. The first-level model s_I(f) maps F into the design space X. The model s_I interpolates the training data set {f^(j), x^(j)} using kriging (Simpson et al., 2001) (Figure 1). The image of F through s_I, s_I(F) ⊂ X, provides the first approximation of the surrogate model domain. This is the best information available from the reference points about the designs optimal with respect to f ∈ F. Yet, this information is limited, and s_I(F) needs to be orthogonally extended toward its normal vectors to ensure that all of the optimum designs are enclosed in the model domain. The normal extension vectors are denoted as v_n^(k)(f), k = 1, …, n − N, and they are the functions of the performance vector f (Koziel and Pietrenko-Dabrowska, 2019). Let us also define: x_max = max{x^(k), k = 1, …, p}, x_min = min{x^(k), k = 1, …, p}, x_d = x_max − x_min, as well as the extension coefficients:

where D is a user-defined thickness parameter determining the orthogonal extension. It has been shown in Koziel and Pietrenko-Dabrowska (2019), that the value of D affects the model accuracy to a certain extent; however, the predictive power dependence on D is not significant.

The model domain X_S is defined by the coefficients α_k, and it is located between the surfaces (or manifolds) M₊ and M_– [Figure 1(b)]:

The surrogate model domain is then given as:

The final (second-level) kriging surrogate is subsequently established within X_S with the training data set {x_B^(k), R(x_B^(k))}_{k = 1, …, NB}.

It should be noted that the above definition of the model domain permits straightforward realization of uniform sampling within X_S. This is of fundamental importance for ensuring the best possible predictive power of the surrogate.

The exposition of the nested kriging modeling provided in this section is necessarily brief and only covers the aspects relevant to this work. More details can be found in Koziel and Pietrenko-Dabrowska (2019).

The major contributor to the computational cost of the surrogate-assisted framework of Section 2.1 is the acquisition of the training data for surrogate model construction (Koziel et al., 2014a). It might be responsible for as much as 80 per cent of the overall cost, which creates a room for improvements. In this work, we use the nested kriging approach outlined in Section 2.3 to implement a more efficient design space reduction (as compared to that of Section 2.2), but also to construct the surrogate model itself. The primary challenge of applying the nested kriging technique for MO is that the objective space constitutes merely a small part of an interval, e.g. a one-dimensional object (curve) in the case of N_obj = 2, or a fragment of a two-dimensional surface in the case of N_obj = 3. This section explains how the technique can be adopted for our purposes while retaining its fundamental advantages mentioned in Section 2.3.

In the context of MO, the figures of interest f_k of the nested model are identical with the design objectives F_k of Section 2.1, and, consequently, N = N_obj. For “general” modeling, the reference designs were to be uniformly distributed within the interval-like objective space F. For the purpose of MO, they need to account for the Pareto front geometry (include the extreme designs x^*(k) = argmin{x: F_k(R(x))} and additional designs if more detailed information is desired, e.g. the front curvature). Let us define F_k = [F₁(x^*(k)) … F_N(x^*(k))]^T, k = 1, …, N. The generic way of obtaining the reference designs is to solve:

under the constraints:

where w = [w₁ … w_N]^T is a vector of weights representing a convex combination, i.e.:

Note that the extreme Pareto-optimal designs x^*(k) correspond to the weighting vectors w = [0 … 1 … 0]^T with 1 on the kth position. Other arrangements would distribute the reference designs along the Pareto front; for example w = [1/N 1/N … 1/N]^T corresponds to the front center. Below, F(w) refers to the objective vector corresponding to the reference design x^w.

Let us now establish a mapping between the weighting vectors w [specifically, those satisfying equation (7)] and the part of the objective space corresponding to the Pareto front. First, for constructing the surrogate model and allocation of the training data, it is convenient to use an auxiliary transformation from a unit N − 1 simplex S^N−1 onto the space of the weights w, defined as:

where

Figure 2 shows some illustrative examples of these concepts for the two- and three-objective cases. The thick dotted lines indicate the parts of the objective spaces where the nested kriging model is to be established. A certain extension is necessary to accommodate the fact that only a rough approximation of the Pareto front geometry is given by the available reference designs, the number of which is very limited in practice. The extended region O (the actual domain of the first-level surrogate) is defined as the set of all points of the form:

where d_w is the extension factor (here, we use d_w = 0.05).

The first-level surrogate for MO is established using the reference designs. It is a composition of two transformations:

1. the mapping from the Cartesian product of S^N−1 × [−d_w, d_w] onto the objective space region O; and

2. the “conventional” first-level model s_I of Section 2.3 from O into X.

The former transformation is merely used for the sake of convenience: it is easier to implement uniform data sampling on S^N−1 × [−d_w, d_w] rather than directly within O.

The second-level surrogate is subsequently set up in the domain constructed as in Section 2.3 but with s_I(O) being orthogonally extended instead of s_I(F). It should be emphasized that the Pareto front geometry is accurately identified through the variation of the nested modeling technique introduced in this section. The number of reference designs required for obtaining a reasonable Pareto front representation is up to three or four for N_obj = 2 [cf. Figure 2(a)], and up to six or seven for N_obj = 3 [cf. Figure 2(b)]. Clearly, identification of the reference designs incurs certain computational expenses. However, the cost of finding additional reference designs (other than the extreme ones) is usually low because good initial points for finding x^w can be established as:

where x^*(l), l = 1, …, N, are, as before, the extreme design corresponding to w = [0 … 1 … 0]^T with 1 on the lth position.

Figure 3 shows the flow diagram of the nested surrogate modeling process for MO. According to the surrogate-assisted procedure of Section 2.1, the second-level model is optimized using MOEA to yield the initial Pareto set, further refined using equation (1). Here, only high-fidelity EM simulations are used to proof the computational efficiency of the proposed methodology even without the usage of variable-fidelity simulations.

A few comments should be made about the surrogate model optimization. Formally, the surrogate is optimized in the original domain of the first-level surrogate, that is, S^N−1 × [−d_w, d_w]. For the sake of antenna evaluation, the designs y ∈ S^N–1 × [−d_w, d_w] are transformed into the weight vectors w using equation (10), and then into x ∈ X using the s_I. This permits us to operate within a geometrically simple domain defined by the lower/upper bounds 0 ≤ y_k ≤ 1, k = 1, …, N − 1, −d_w ≤ y_N ≤ d_w, and a linear constraint ∑_{k = 1,…,N−1} y_k ≤ 1, despite the actual shape of the Pareto front. The process of surrogate model evaluation has been shown in Figure 4.

The MO framework proposed in this paper is demonstrated here using two examples, a planar Yagi antenna and an ultra-wideband monopole antenna. For the first example, there are two objectives considered: improvement of the in-band matching and gain maximization. For the second example, we have three goals: improvement of the impedance matching, reduction of the in-band gain variability and reduction of the antenna size. Our framework is compared to the benchmark surrogate-assisted MO algorithm of Sections 2.1 and 2.2.

It should be noted that only high-fidelity EM simulations are used, therefore a direct comparison with variable-fidelity frameworks is not possible, although the proposed technique can be generalized to accommodate low- and high-fidelity EM simulations in a straightforward manner.

The first demonstration example is a planar Yagi antenna shown in Figure 5 (Kaneda et al., 2002). The structure is implemented on RT6010 substrate (ε_r = 10.2, h = 0.635 mm) and comprises a driven element fed by a coplanar strip-line, director and a microstrip balun. The antenna is fed with a 50 ohm microstrip. Design variables are x = [s₁ s₂ v₁ v₂ u₁ u₂ u₃ u₄]^T. Other parameters are fixed as follows: w₁ = w₃ = w₄ = 0.6, w₂ = 1.2, u₅ = 1.5, s₃ = 3.0 and v₃ = 17.5, all in mm. The computational model R is simulated in CST Microwave Studio (approximately 600,000 mesh cells and simulation time of 4 min) using its time domain solver.

The antenna is supposed to operate within the frequency range 10-11 GHz. The figures of interest are minimization of in-band reflection (F₁) and maximization of the average end-fire gain (F₂), both within the operating bandwidth.

To set up the surrogate model we use four reference designs, corresponding to the two extreme designs x^*(1) = [4.43 3.85 8.77 4.28 4.09 4.76 2.08 1.63]^T, x^*(2) = [5.19 6.90 7.10 5.08 3.54 4.78 2.23 0.93]^T, and two more, corresponding to z = 0.33 and z = 0.66 (cf. (6)): x^*(3) = [4.56 4.38 8.56 4.50 3.89 4.93 2.01 1.57]^T and x^*(4) = [4.84 5.00 8.09 4.64 3.98 4.89 2.00 1.50]^T.

The thickness parameter was set to D = 0.05, which allows for achieving good predictive power of the surrogate while using a small number of training data points. The in-depth analysis of the dependence of the modeling error on the parameter D can be found in Koziel and Pietrenko-Dabrowska (2019).

The nested kriging model was set up as outlined in Sections 2.3 and 2.4 using only 100 data samples. The relative RMS error of the surrogate (determined through cross validation) is 5 per cent for the reflection response and 1 per cent for the gain. The relative error is defined as ||R(x) − R_s(x)||/||R(x)||, where R_s stands for the surrogate. The 10-fold cross-validation has been used (Queipo et al., 2005).

To emphasize the benefits of the proposed approach, the surrogate was also constructed within the initially reduced space, i.e. the interval l^* = min{x^*(1), x^*(2)} and u^* = max{x^*(1), x^*(²)}. Despite of using 1,600 training samples, the model error levels were much higher: 9 and 3 per cent for reflection and gain characteristics, respectively.

The initial Pareto set found by optimizing the surrogate using MOEA is shown in Figure 6 along with the selected designs evaluated through EM simulation, before and after the refinement process. The corresponding numerical data has been gathered in Table I. Figure 7 shows the reflection and realized gain characteristics for the selected designs. Table II summarizes the optimization cost which includes finding the reference designs.

Note that the major contributor is the training data acquisition for setting up the surrogate model which is dramatically reduced for the proposed framework (from 1,600 to only 100 samples). This leads to lowering the total expenses by 80 per cent. Also, the overall cost is just around 400 EM simulations despite the fact that single-fidelity models are used. Another advantage of our approach is a more precise identification of the initial Pareto set, which is because of a considerably smaller domain of the nested kriging model and better predictive power of the surrogate. This is illustrated in Figure 8 that shows the initial Pareto set found using the surrogate established in the initially reduced design space. It can be observed that the span of the front obtained using the proposed methodology is considerably wider and of better quality (in terms of the objective function values) than the one yielded by the benchmark method. Even more importantly, significant discrepancies between the initial front, the EM-simulated objectives and the refined designs can be noticed, demonstrating the lack of surrogate model accuracy.

The selected designs have been fabricated and measured for additional verification. Figure 9 shows the photographs of the antenna prototypes, whereas Figure 10 provides a comparison of the simulated and measured reflection and realized gain characteristics (these being design objectives considered in the optimization process). It can be observed that the agreement between the two sets of data is satisfactory.

Our second example is the ultrawideband monopole antenna (Alsath and Kanagasabai, 2015) implemented on Taconic RF-35 substrate (ε_r = 3.5, h = 0.762 mm) and shown in Figure 11. The independent geometry parameters for this structure are x = [L₀ dR R r_rel dL dw L_g L₁ R₁ dr c_rel]^T. The computational model is implemented in CST Microwave Studio and evaluated using its transient solver (approximately 840,000 mesh cells and simulation time of 5 min). The EM model incorporates the SMA connector.

The antenna is to operate within the UWB band of 3.1-10.6 GHz. Three figures of interest are considered: minimization of in-band reflection (F₁), reduction of the realized gain variability within the operating band (F₂) and reduction of the antenna size (F₃). As before, the thickness parameter was set to D = 0.05.

The surrogate model is set up using seven reference designs. These include three extreme designs (best matching, minimum gain variation and minimum size): x^*(1) = [10.64 0.0 6.00 0.10 1.46 6.20 10.46 4.26 2.00 0.73 0.49]^T, x^*(2) = [8.74 1.55 5.81 0.51 0.016 5.65 8.95 5.47 2.60 0.99 0.84]^T, x^*(3) = [9.51 0.19 4.46 0.27 4.33 1.17 10.05 6.00 2.94 0.99 0.90]^T, and four more, corresponding to z = [0.0 0.5]^T, z = [0.5 0.0]^T, z = [0.5 0.5]^T, and z = [0.333 0.333]^T (cf. (6)): x^*(4) = [10.04 0.43 5.85 0.26 0.0 6.46 10.01 5.49 2.14 1.00 0.83]^T, x^*(5) = [9.58 0.0 5.05 0.28 3.37 4.14 9.68 5.26 2.37 0.85 0.89]^T, x^*(6) = [8.76 0.0 5.62 0.69 2.24 2.92 8.93 5.94 2.58 0.99 0.27]^T, x^*(7) = [9.52 0 0.37 5.08 0.16 2.61 4.85 9.55 5.39 2.25 0.91 0.88]^T.

The nested kriging model was set up using 200 data samples. The RMS error of the surrogate is 7.5 per cent for the reflection response and 4 per cent for the gain. For the sake of comparison, the surrogate was also constructed within the initially reduced space, i.e. the interval l^* = min{x^*(1), x^*(2)} and u^* = max{x^*(1), x^*(2)}, using 1,600 training samples. The obtained error values are much higher: 15 and 11 per cent for reflection and gain characteristics, respectively. It should be noted that this case is considerably more difficult from the standpoint of surrogate model construction because of higher dimensionality of the parameter space (11 variables vs 8 for the Yagi antenna) as well as wider parameter ranges.

Figure 12 shows the initial Pareto set found through MOEA-optimization of the nested kriging surrogate as well as the selected designs evaluated using EM simulation (before and after the refinement). Table III provides the numerical data, whereas Figure 13 shows the reflection and realized gain characteristics for the selected designs. The optimization process cost breakdown has been given in Table IV. Similarly as for the first example, acquisition of the training data plays the major role and the proposed methodology allows us to reduce the overall expenses by over 50 per cent. For this example, the nested kriging surrogate and the surrogate constructed in the initially reduced space lead to similar initial Pareto fronts as illustrated in Figure 14 with the slightly wider span of the front for the latter model (this being a consequence of a larger model domain).

Selected Pareto-optimal designs of the antenna of Figure 11 have been fabricated and measured. Figure 15 shows the photographs of the antenna prototypes. The relevant antenna characteristics, i.e. reflection and realized gain, are shown in Figure 16. The agreement between simulations and measurements is satisfactory.

In the paper, a technique for computationally efficient multi-objective design optimization of antenna structures has been proposed. Our methodology adopts a nested kriging modeling approach to identify a region of the design space that contains the best trade-off designs, and to set up – within that region – a fast surrogate model used to yield an initial approximation of the Pareto set. A rigorous formalism has been introduced to generalize the nested kriging framework and apply it over an arbitrary subset of the objective space instead of the interval (as in the original formulation of the method).

The resulting framework has been comprehensively validated using two antenna examples, a planar Yagi and a UWB monopole, optimized for two and three objectives, respectively. It has been demonstrated that the proposed framework offers significant reduction of the computational cost of up to 80 per cent as compared to the state-of-the-art surrogate-assisted approach while retaining or even improving the quality of Pareto front representation.

In this work, the entire optimization process was intentionally carried out using exclusively high-fidelity EM simulation models. This was to demonstrate that even within this setup, the computational overhead of the MO process can be manageable. The future work will include generalizations of the proposed methodology incorporating variable-fidelity simulations.

The authors would like to thank Dassault Systemes, France, for making CST Microwave Studio available.