A framework for accelerated optimization of antennas using design database and initial parameter set estimation

Abstract

Purpose

The purpose of this paper is to exploit a database of pre-existing designs to accelerate parametric optimization of antenna structures is investigated.

Design/methodology/approach

The usefulness of pre-existing designs for rapid design of antennas is investigated. The proposed approach exploits the database existing antenna base designs to determine a good starting point for structure optimization and its response sensitivities. The considered method is suitable for handling computationally expensive models, which are evaluated using full-wave electromagnetic (EM) simulations. Numerical case studies are provided demonstrating the feasibility of the framework for the design of real-world structures.

Findings

The use of pre-existing designs enables rapid identification of a good starting point for antenna optimization and speeds-up estimation of the structure response sensitivities. The base designs can be arranged into subsets (simplexes) in the objective space and used to represent the target vector, i.e. the starting point for structure design. The base closest base point w.r.t. the initial design can be used to initialize Jacobian for local optimization. Moreover, local optimization costs can be reduced through the use of Broyden formula for Jacobian updates in consecutive iterations.

Research limitations/implications

The study investigates the possibility of reusing pre-existing designs for the acceleration of antenna optimization. The proposed technique enables the identification of a good starting point and reduces the number of expensive EM simulations required to obtain the final design.

Originality/value

The proposed design framework proved to be useful for the identification of good initial design and rapid optimization of modern antennas. Identification of the starting point for the design of such structures is extremely challenging when using conventional methods involving parametric studies or repetitive local optimizations. The presented methodology proved to be a useful design and geometry scaling tool when previously obtained designs are available for the same antenna structure.

Citation

Koziel, S. and Bekasiewicz, A. (2020), "A framework for accelerated optimization of antennas using design database and initial parameter set estimation", Engineering Computations, Vol. 37 No. 7, pp. 2487-2500. https://doi.org/10.1108/EC-06-2019-0267

1. Introduction

The topological complexity of contemporary antenna structures has been steadily increasing over the years to meet more and more stringent specifications imposed on their electrical and field characteristics (Li et al., 2017; Deng et al., 2017). Satisfying various demands concerning broadband (Saini and Dwari, 2016) or multi-band operation (Guo et al., 2017), circular polarization (Mener et al., 2017; Feng et al., 2017), stable gain response (Tang et al., 2016), pattern stability (Antonino-Daviu et al., 2017), time-domain performance [i.e. pulse fidelity (Chen and Chiu, 2016)] and additional functionalities [e.g. band notches (Wen et al., 2018), tunable performance (Simsek and Aoad, 2017)], while maintaining small physical dimensions (Xu et al., 2017; Cheng et al., 2018), requires development of unconventional geometries described by a large number of parameters that need to be tuned to obtain the best possible performance. At the same time, full-wave electromagnetic (EM) analysis is necessary to provide a reliable evaluation of antenna characteristics. Both of these factors make the design closure process challenging. This is not only due to sheer computational effort required to handle multi-dimensional parameter spaces but also difficulties in identifying reasonable initial designs for further (local) optimization. The latter call for global search methods (Lakbakhsh et al., 2017; Goudos et al., 2011), which increase the cost of parameter adjustment tremendously.

The challenges related to the high-cost of antenna design can be addressed to some extent using surrogate-based optimization (SBO) methods. The basic SBO concept assumes that direct handling of the high-fidelity EM model is replaced by iterative construction and re-optimization of its faster representation referred to as a surrogate model (Su et al., 2017). The surrogate – represented in the form of a coarsely discretized EM simulation (Wang et al., 2018) or an approximation model (Koziel and Ogurtsov, 2014) along with an appropriate correction layer – is used as a prediction tool that facilitates the identification of a better design. Meanwhile, the data accumulated from evaluations of the high-fidelity EM model featuring fine discretization help improving the surrogate model during the design process.

Both in case of direct optimization of the high-fidelity EM antenna model and surrogate-assisted procedures, especially those involving coarse-discretization low-fidelity models, lowering the computational cost of the optimization process requires reduction of the number of EM analyzes (at either level of discretization). A possible way of achieving this is to use already existing designs of the same antenna structure, e.g. obtained for different operating conditions (such as operating frequencies) or implemented on different dielectric substrates. The purpose of this paper is to propose and demonstrate a conceptually simple but efficient technique that allows reusing such designs and, consequently, expedite the optimization process of antenna structures. The main contributions of the work include development of the approach for determination of a good starting point for antenna optimization, which is based on the analysis of the base set of pre-existing designs. Furthermore, the work introduces a modification to the trust-region-based gradient optimization process that substantially reduces the number of EM simulations required for antenna optimization. The latter is achieved by estimating antenna response sensitivities using the same set of pre-existing designs. The proposed methodology has been validated using a unidirectional quasi-Yagi antenna optimized for a maximum in-band gain within a range of operating frequencies and substrate permittivity values. The results indicate low design cost while ensuring reliability and satisfactory control over the performance figures of interest.

2. Related work

A considerable research effort has been directed toward the development of techniques that would speed-up the optimization processes. In the context of local optimization, utilization of adjoint sensitivities may bring significant benefits (Wang et al., 2018; Koziel and Bekasiewicz, 2016), however, adjoints are not widely supported in commercial EM simulation packages. Over recent years, the growing popularity of SBO methodologies has been observed (Koziel and Ogurtsov, 2014; Koziel and Unnsteinsson, 2018). The recent SBO include physics-based approaches such as space mapping (Zhu et al., 2007), response correction methods (Koziel and Leifsson, 2016) or feature-based optimization (Koziel, 2015) and various algorithms involving data-driven (approximation) surrogates such as kriging (de Villiers et al., 2017), Gaussian process regression (Jacobs, 2016) or even polynomial-based response surfaces (Easum et al., 2017). For statistical analysis and robust design applications, surrogate models exploiting polynomial chaos expansion have been gaining popularity (Du and Roblin, 2017; Rossi et al., 2014). Proper tailoring of the SBO framework components to a particular problem may result in a dramatic reduction of the optimization costs (Koziel and Ogurtsov, 2014; Koziel, 2015). In the case of antennas, many surrogate-assisted methods involve underlying low-fidelity models obtained from coarse-discretization EM simulations (Koziel and Ogurtsov, 2014), which is due to the lack of reliable alternatives (e.g. equivalent network representations).

Low-cost design optimization of contemporary antennas can be realized using approximation models or suitably corrected coarsely-discretized EM simulations (Forrester and Keane, 2009; Koziel and Ogurtsov, 2012a). Approximation (or data-driven) models are particularly attractive for handling numerically demanding tasks such as multi-objective optimization (Koziel and Ogurtsov, 2014; de Villiers et al., 2017; Easum et al., 2017; Koziel and Ogurtsov, 2013). However, they also found applications for statistical analysis and material characterization (Nguyen and Chung, 2015; Koziel and Bekasiewicz, 2017). In Nguyen and Chung (2015), kriging-based modeling was used for determination antenna substrate parameters based on the measurements of its impedance and bandwidth. A variation of the approach, oriented toward utilization of approximation-based modeling to obtain EM properties of textile materials, was proposed in (Declercq et al., 2013). Data-driven models are also useful for the optimization of reflector antennas (Mokhupuki and de Villiers, 2019) or design of frequency-reconfigurable structures (Simsek and Aoad, 2017). On the other hand, construction of accurate data-driven model requires a large number of training samples. Responses of the latter are typically expensive to obtain, even when evaluated at the low-fidelity level. Another problem is that the number of data samples required for model construction grows exponentially with the problem dimensionality (Forrester and Keane, 2009; Koziel and Ogurtsov, 2013). This hinders application of data-driven surrogates for design of multi-parameter antennas. Low-fidelity EM models proved to provide an interesting alternative for approximation-based design of high-frequency structures (Koziel and Ogurtsov, 2012a; Koziel and Ogurtsov, 2015; Koziel and Ogurtsov, 2018). In Koziel and Ogurtsov (2012a), corrected physics-based surrogates were used for low-cost optimization of modern antennas. Rapid design of antenna arrays using a coarse model combined with response refinement method was considered in (Koziel and Ogurtsov, 2015). Also, design schemes that involve optimization of antenna responses expressed in terms of so-called response features proved to be useful for rapid design (Koziel and Ogurtsov, 2018). Regardless advantages, the main prerequisite for applying these design tools is that the region of interest for the problem or a starting point for optimization are available. In reality, however, such information may not be available, which increases the complexity of design problem.

3. Antenna optimization framework with design database

This section provides the details of the proposed accelerated antenna optimization using pre-existing design database. As mentioned in the introduction, the base designs might be available because of the previous design work with the same antenna structure (e.g. optimized for various operating frequencies or substrates). These can also be obtained specifically to accelerate future optimization runs. The following subsections introduce appropriate notation for the proposed methodology, describe the analysis of the base designs geometry leading to generating a starting point for further optimization with respect to given operating conditions and explain the accelerated optimization procedure. Illustration examples are provided in Section 4.

3.1 Design database

Let F_k, k = 1, …, N, denote the antenna operating conditions, material parameters or performance figures of interest (here, together referred to as objectives). Examples include operating frequency (or frequencies for multi-band antennas), operating bandwidth, permittivity and height of the substrate the antenna is implemented on, etc. A vector of adjustable parameters of the antenna will be denoted as x, whereas x_b^(j), j = 1, …, p, represents the database designs. These designs are optimized for objective vectors F^(j) = [F₁^(j), …, F_N^(j)]^T. For practical purposes, it is desired that the vectors F^(j) more or less fill in the interval defined by the intended objective ranges F_k.min ≤ F_k ≤ F_k.max, i.e. a region in which the entire optimization framework is supposed to operate. Other than that, allocation of the base designs can be arbitrary.

3.2 Initial design by exploring database geometry

Let F_t = [F_t.1, …, F_t.N]^T be a target vector, i.e. a set of objectives the antenna is to be optimized for. The first step of identifying the starting point for antenna optimization with respect to F_t is to arrange the base designs {F^(j)}_{j = 1, …, p} into affinely independent subsets

S^(j) = [F^(j.1), …, F^(j.N+1)]. The affinely independent designs are those for which the matrix [F^(j.2) – F^(j.1), …, F^(j.N+1) – F^(j.1)] is of full rank (in other words, the vectors F^(j.l) – F^(j.1), l = 2, …, N + 1 are linearly independent) (Horn and Johnson, 1985). Geometrically, the subsets S^(j) is N + 1 simplexes in the objective space.

For any k = 1, …, K, the target vector can be uniquely represented as:

(1) Ft=∑k=1N+1αk(j)F(j.k)

where α_k^(j) are expansion coefficients such that Σ_{k=1, …, N+1}α_k^(j) = 1.

The foundation for finding the starting point is the identification of the simplex S^(j*⁾ that satisfies the following conditions (Rudin, 1976):

• The target vector is possibly close to the center of S^(j*).

• The target vector F_t is possibly close to the vertices of S^(j*) (i.e. smaller simplexes are preferred).

Clearly, these conditions are not rigorous to the extent that would permit a unique selection of S^(j*). A possible quantification can be realized by defining the index j^* as:

(2) j*=min⁡{j=1,…,K:std([α1(j) … αN+1(j)])∏k=1Ndk(j)}

Here, std() stands for the standard deviation and d_k^(j) = ||F_t – F^(j.k)||. Figure 1 shows several illustrative cases of the target vector F_t and S^(j*) for an exemplary three-dimensional objective space.

The initial design corresponding to F_t is obtained as:

(3) x(0)=∑k=1N+1αk(j*)xb(j*.k)

where x_b^(j*.k) is parameter vectors corresponding to the vertices F^(j*.k) of the simplex S^(j*) selected by equation (2). The expansion coefficients α_k^(j*) are obtained from equation (1). Note that the equation (3) assumes linear dependence between the geometry parameters and the objective values. While this is obviously not the case in practice – as the relationship between the design and feature spaces is nonlinear – it gives the best initial design that can be found using N + 1 base designs. It should be noted that the accuracy of prediction from equation (3) is expected to increase for smaller simplexes, which is one of the motivations for choosing the particular form of equation (2).

3.3 Optimization procedure

Starting from the design x⁽⁰⁾ obtained from equation (3), the antenna is further optimized by solving the minimization problem of the form as follows:

(4) x*=arg⁡min⁡xU(R(x))

where R(x) is the EM antenna model. A definition of the cost function U depends on a particular design task. For example, when optimizing for best matching over a specified frequency range, say, from f_min to f_max, one may have U(x) = max{f_min ≤ f ≤ f_max: |S₁₁(x, f)|}. Whatever design constraints are imposed, they can be handled either explicitly or implicitly (e.g. using penalty functions) (Bertsekas, 1996). An illustrative example of the cost function definition will be given in Section 4 for the case study considered therein.

Because the initial design provided by equation (3) is normally good, local optimization is sufficient to carry out design closure. Here, a trust-region (TR) gradient search algorithm is used, which produces a series of approximations x⁽ⁱ⁾, i = 0, 1, …, to the optimum design x^*. The optimization task is given as (Conn et al., 2000):

(5) x(i+1)=arg⁡min⁡x; −d(i)≤x−x(i)≤d(i)U(L(i)(x))

The linear model L⁽ⁱ⁾(x) = R(x⁽ⁱ⁾) + J_R(x⁽ⁱ⁾)⋅(x – x⁽ⁱ⁾) is obtained using antenna responses and sensitivities evaluated at the current point x⁽ⁱ⁾. The parameter d⁽ⁱ⁾ is a trust region size vector updated based on the gain ratio being the ratio of the actual and predicted improvement of the objective function (Conn et al., 2000):

(6) ρ=U(R(x(i)))−U(R(x(i−1)))U(L(i)(x(i)))−U(L(i)(x(i−1)))

The initial size vector d⁽⁰⁾ = 0.1(u – l), where l and u are the lower and upper bounds of the parameter space. It is updated using standard rules, i.e. for ρ < 0.3, d⁽ⁱ⁺¹⁾ = d⁽ⁱ⁾/3 and d⁽ⁱ⁺¹⁾ = 2d⁽ⁱ⁾ for ρ > 0.75. The algorithm is terminated when either ||x⁽ⁱ⁺¹⁾ – x⁽ⁱ⁾|| < ε or ||d⁽ⁱ⁺¹⁾|| < ε, where ε is an user defined threshold (here, ε = 10⁻³) (Conn et al., 2000). The discussed setup proved to be relevant for a variety of test problems considered in the available literature (Koziel and Ogurtsov, 2014; Koziel and Unnsteinsson, 2018; Koziel and Ogurtsov, 2018).

Under typical circumstances (no adjoint sensitivities available), the Jacobian J_R is estimated using finite differentiation (Conn et al., 2000), which incurs the extra cost of n EM simulations (per iteration). In this work, the following two means are applied to reduce this cost [both justified by the presumably good quality of the initial design used for solving equation (4)]:

• The Jacobian is initialized by using J_R from the closest base point.

• Jacobian updates for subsequent iterations are obtained by using a rank-one Broyden formula (Koziel et al., 2010).

Using the Jacobian from the closest base point saves considerable computational effort otherwise necessary to perform finite differentiation. On the other hand, because the initial design is already good, updating Jacobian with the rank-one formula is sufficient.

Assuming the availability of the design database, the proposed framework can be summarized as follows (Figure 2 for a flowchart):

• set the target vector F_t;

• arrange base designs into affinely independent subsets S^(j) and find a simplex S^(j*), which is suitable for the determination of expansion coefficients α_k^(j):

• evaluate equation (3) to obtain the initial design x⁽⁰⁾ corresponding to F_t;

• refine the design by executing equation (5).

As already indicated, the computational benefits of the proposed approach stem from the utilization of the pre-existing designs for the determination of the starting point. Furthermore, re-using the available derivative data and Broyden-based Jacobian updates are important factors for obtaining the final design at a limited number of EM simulations. As confirmed by the numerical results of Section 4, the cost of optimization using our method is low compared to conventional approaches.

4. Case study and results

The procedure of Section 3 is demonstrated using a unidirectional quasi-Yagi antenna. The antenna is optimized for maximum in-band gain within a wide range of operating frequencies and substrate permittivity values.

4.1 Antenna structure design problem

Consider a quasi-Yagi antenna shown in Figure 3. The structure is implemented on a 0.81-mm-thick dielectric substrate. The geometry is described by the parameter vector x = [w₀ l_g l_g3r g_f g_c l_c w_c l_r1r o w_dl_dr]^T. The relative variables are l_g1 = 0.4 l_g, l_g2 = 0.6 l_g, l_g3 = l_g2l_g3r, l_f2 = l_g3, l_r1 = 0.5(w₀ – g_c)l_r1r and l_d = w₀l_dr. Dimensions w_f = 1.8 and l_f = 13.7 are constant. The unit of all parameters is mm, except those ending with r, which are unit-less. The EM antenna model is implemented in CST Microwave Studio (CST Microwave Studio, ver., 2013).

4.2 Experimental setup

The purpose is to design the antenna for a substrate with a given relative permittivity ε_r, and to operate with a fractional bandwidth B (defined by |S₁₁| ≤ –10 dB), centered at a required frequency f₀. Within that band, the average realized gain G_A of the antenna is to be maximized. The ranges of interest for the center frequency and permittivity are as follows: 3.5 GHz ≤ f₀ ≤ 5.5 GHz and 2.5 ≤ ε_r ≤ 4.5. For the sake of illustration, B is set to seven percent.

The objective function U defined to handle the design problem posed in the previous paragraph is formulated as:

(6) U(R(x))=−GA(x)+βmax⁡{max⁡{f∈B:|S11(x,f)|}+1010,0}2

The primary objective in equation (6) is the maximization of the average gain, whereas the penalty term allows for maintaining required impedance bandwidth. The penalty coefficient β is set to 10, which is sufficient to control the condition |S₁₁| ≤ –10 dB within the fraction of dB.

Following the methodology of Section 3, a design database has been prepared that corresponds to ten pairs of the operating frequency and substrate permittivity {f₀,ε_r}: {3.0, 2.5}, {3.0, 4.5}, {3.7, 2.9}, {3.6, 3.7}, {4.5, 2.5}, {4.6, 3.3}, {5.5, 3.5}, {6.0, 2.5} and {6.0, 4.5}. Figure 4 shows the antenna responses at the selected reference designs, whereas their corresponding dimensions are gathered in Table I.

4.3 Results and discussion

For verification purposes, the antenna of Figure 3 has been designed for the following operating frequencies and substrate permittivity values as follows: {5.3, 2.5}, {4.8, 4.1}, {3.5, 2.5}, {3.85, 4.3}, {3.5, 3.5} and {4.65, 2.97}. Table II shows the optimized parameter values for all considered cases, whereas Figure 4 shows the antenna responses (reflection and realized gain) at the initial design obtained by solving equation (3) and the optimized design. The average optimization cost is only about five EM simulations of the antenna. It should be noted that the quality of the initial design is already very good, which is an important benefit of the proposed approach.

The proposed design framework has been compared to alternative methods in terms of computational cost. The design goal in this study was to optimize the antenna of Figure 3 for the objective pair {f₀, ε_r} = {5.3, 2.5}. The benchmark procedures include as follows:

• a conventional TR-based algorithm [37] launched from the database design corresponding to the pair {5.5, 3.5} (Table I), and

• the conventional TR algorithm executed from the design obtained as described in Section 3.2.

The results gathered in Table III indicate that the cost of the proposed approach is substantially lower than that of the benchmark methods. At this point, it should be reiterated that rendering a good initial design (cf. (3)) is an inherent part of the procedure. This – within the objective space region covered by the database designs – brings in quasi-global search capabilities. The latter is to be understood as releasing the user from the necessity of providing a reasonable starting point for further refinement.

5. Conclusion

In the paper, a novel technique for expedited design optimization of antenna structures is proposed. The approach uses a database of reference designs and information therein to find a good initial set of geometry parameter values and accelerate the optimization process. The latter exploits estimated sensitivity extracted from the database along with the updating formulas. The proposed method can be used for rapid and reliable optimization of structures with respect to multiple performance figures. Moreover, the latter can be controlled within a wide range of operating conditions. In this work, the performance of the approach was demonstrated using a quasi-Yagi antenna optimized with respect to center frequency, bandwidth and realized gain. The presented methodology can be a useful design and geometry scaling tool, especially in situations when previously obtained designs are available for the same antenna structure. The future work will include generalization of the method for variable-fidelity simulation models (Figure 5).

Figures

Figure 1.

Design database and objective space (here, assuming three operating conditions)

Figure 2.

A flowchart of the framework for the estimation of the initial design and rapid optimization of antenna structures

Figure 3.

Quasi-Yagi antenna with a director fed through a microstrip-to-coplanar stripline transition balun. Feedline marked using dark gray shade

Figure 4.

Responses of antenna of Figure 3 at selected reference designs, corresponding to – from top-left to bottom-right – the following operating frequency and substrate permittivity pairs {f₀,ε_r}: {5.5, 3.5}, {6.0, 2.5}, {4.5, 2.5} and {3.6, 3.7}

Figure 5.

Responses of the antenna of Figure 3 at the verification designs obtained using the proposed methodology

Table I.

Dimensions of selected base designs

Base designs from database
f0 [GHz]	5.5	6.0	4.5	3.6
εr	3.5	2.5	2.5	3.7
Geometry parameters
w0	25.33	24.38	31.61	36.79
lg	10.03	9.46	12.17	14.95
lg3r	0.95	0.89	0.92	0.93
gf	0.58	0.81	0.46	1.00
gc	0.20	0.20	0.56	0.68
lc	7.66	6.66	8.15	16.35
wc	3.00	2.49	4.15	5.00
lr1r	0.82	0.85	0.97	1.02
o	5.00	5.18	6.59	7.50
wd	5.00	5.63	6.70	7.50
ldr	0.64	0.66	0.70	0.71

Table II.

Quasi-Yagi Antenna: results of design optimization

Operating conditions and geometry parameters		Verification case
Center frequency	f0 [GHz]	5.3	4.8	3.5	3.85	3.5	4.65
Substrate permittivity	εr	2.5	4.1	2.5	4.3	3.5	2.97
Geometry parameters	w0	27.06	28.12	37.66	33.61	37.40	29.37
	lg	10.49	12.84	14.42	14.32	15.41	11.65
	lg3r	0.91	0.91	0.92	0.92	0.93	0.94
	gf	0.68	0.43	0.76	0.56	0.92	0.61
	gc	0.34	0.56	1.08	0.85	0.68	0.45
	lc	7.24	7.71	14.57	11.79	16.72	8.81
	wc	3.09	3.57	3.96	4.09	4.77	3.59
	lr1r	0.89	0.89	1.15	0.96	1.05	0.92
	o	5.72	6.63	8.55	7.73	7.87	6.08
	wd	6.01	6.33	8.59	7.55	7.81	6.11
	ldr	0.68	0.63	0.79	0.68	0.73	0.68

Table III.

Quasi-Yagi Antenna: benchmark

Design method	No of algorithm iterations	No. of EM simulations	Design cost [h]
(i)	16	201	13.4
(ii)	6	64	4.3
This work	4	5	0.3