Fast and reliable knowledge-based design closure of antennas by means of iterative prediction-correction scheme

Abstract

Purpose

A novel framework for expedited antenna optimization with an iterative prediction-correction scheme is proposed. The methodology is comprehensively validated using three real-world antenna structures: narrow-band, dual-band and wideband, optimized under various design scenarios.

Design/methodology/approach

The keystone of the proposed approach is to reuse designs pre-optimized for various sets of performance specifications and to encode them into metamodels that render good initial designs, as well as an initial estimate of the antenna response sensitivities. Subsequent design refinement is realized using an iterative prediction-correction loop accommodating the discrepancies between the actual and target design specifications.

Findings

The presented framework is capable of yielding optimized antenna designs at the cost of just a few full-wave electromagnetic simulations. The practical importance of the iterative correction procedure has been corroborated by benchmarking against gradient-only refinement. It has been found that the incorporation of problem-specific knowledge into the optimization framework greatly facilitates parameter adjustment and improves its reliability.

Research limitations/implications

The proposed approach can be a viable tool for antenna optimization whenever a certain number of previously obtained designs are available or the designer finds the initial effort of their gathering justifiable by intended re-use of the procedure. The future work will incorporate response features technology for improving the accuracy of the initial approximation of antenna response sensitivities.

Originality/value

The proposed optimization framework has been proved to be a viable tool for cost-efficient and reliable antenna optimization. To the knowledge, this approach to antenna optimization goes beyond the capabilities of available methods, especially in terms of efficient utilization of the existing knowledge, thus enabling reliable parameter tuning over broad ranges of both operating conditions and material parameters of the structure of interest.

Citation

Koziel, S. and Pietrenko-Dabrowska, A. (2021), "Fast and reliable knowledge-based design closure of antennas by means of iterative prediction-correction scheme", Engineering Computations, Vol. 38 No. 10, pp. 3710-3731. https://doi.org/10.1108/EC-10-2020-0600

1. Introduction

Steadily increasing performance requirements have made the design of contemporary antenna structures a challenging endeavor. These growing demands do not only pertain to electrical or field characteristics (Aldhafeeri and Rahmat-Samii, 2019; Liao et al., 2015) but also the implementation of additional functionalities such as multi-band operation (Deshmukh et al., 2019), pattern/polarization diversity (Saurav et al., 2018), band notches (Xu et al., 2018), circular polarization (Kumar et al., 2019), tunability (Huang et al., 2015), or – more often than not – size reduction (Kim and Nam, 2019; Li et al., 2018). The important stimulus here is the emergence of new application areas including the internet of things (IoT) (Salam et al., 2019), mobile communications (especially 5 G; Zeng and Luk, 2019), medical imaging (Connell and Menon, 2019; Felicio et al., 2019) or wearable/implantable devices (Mendes and Peixeiro, 2018). The need for meeting stringent specifications fosters the development of antennas of increasingly complex geometries. This strongly affects the design practices, in particular, emphasizes the role of full-wave electromagnetic (EM) simulation tools (Wu and Sarabandi, 2017; Han et al., 2017). In many cases, beyond the early conceptual development, the design process is entirely EM-driven, that including topology evolution (Ullah and Koziel, 2018), initial adjustments through parameter sweeping (Bhattacharya et al., 2016) and design closure (Fakih et al., 2019). The latter is more and more frequently realized by means of rigorous numerical optimization. This is virtually imperative when multiple variables, performance figures and constraints need to be handled simultaneously (Dong et al., 2019). Notwithstanding, multiple antenna evaluations required by the optimization algorithms incur considerable computational overhead. This may become a serious bottleneck even for a local search, let alone global optimization (in particular, population-based algorithms such as particle swarm optimization (Lalbakhsh et al., 2017), differential evolution (Goudos et al., 2011) or genetic algorithms (Choi et al., 2016), as well as uncertainty quantification or statistical design (Du and Roblin, 2018; Kouassi et al., 2016).

Under these circumstances, expediting simulation-based design procedures has been a focus of extensive research over the recent years (Hassan et al., 2015 through Torun and Swaminathan, 2019). Some of the promising directions include algorithmic developments (e.g. incorporation of adjoint sensitivities into gradient search algorithms (Hassan et al., 2015; Wang et al., 2018), sparse sensitivity updates (Koziel and Pietrenko-Dabrowska, 2019a, 2019b), application of fast analysis techniques (typically targeting particular classes of the antenna structures, e.g. Tsukamoto and Arai (2016), Arndt (2012) or machine learning methods (Alzahed et al., 2019; Tak et al., 2018), as well as exploitation of a particular structure of antenna responses (feature-based optimization (Koziel, 2015)). Surrogate-assisted frameworks constitute another branch of quickly growing popularity (Baratta et al., 2018 through Torun and Swaminathan, 2019). These methods may involve both physics-based models (space mapping (Baratta et al., 2018), adaptive response scaling (Koziel and Unnsteinsson, 2018) and data-driven (approximation) surrogates (kriging interpolation, Easum et al. (2018); support vector regression, Jacobs (2012); Gaussian process regression, Jacobs and Koziel (2014) or artificial neural networks, Mishra et al. (2015), also in combination with sequential sampling methods (Torun and Swaminathan, 2019).

One of the scenarios of practical importance is re-design (or dimension scaling) of a particular structure for different operating conditions (e.g. operating frequencies in the case of multi-band antennas) or material parameters (e.g. substrate height or dielectric permittivity in the case of planar structures). When executed without prior problem-specific knowledge, the re-design process faces the same challenges and entails similar CPU costs as those associated with finding the original design (or designs). Using previously obtained designs may expedite the optimization process. One possibility is analytical design curves that determine the relationships between the performance figures and geometry parameters and permit rapid identification of at least a decent initial design (Caenepeel et al., 2016). Recently, the utilization of inverse surrogate models has been fostered for that purpose (Leifsson and Koziel, 2014), also using variable-fidelity EM simulation models (Ullah et al., 2020).

This paper presents a novel knowledge-based methodology for low-cost and reliable optimization of antenna structures. Our procedure incorporates the problem-specific information (in the form of pre-existing designs, e.g. optimized for various sets of performance specifications) into the machine learning framework involving data-driven (here, kriging interpolation) metamodels. These surrogates are used to render the initial point but also to produce an approximation of the sensitivity matrix, which accelerates subsequent design refinement. To further reduce the computational overhead of the optimization process and improve its reliability, an iterative correction procedure is developed that feeds the discrepancies between the actual and the target performance figure values back to the surrogate. The latter returns an improved parameter vector at a cost of a single EM antenna analysis per iteration. The proposed methodology is demonstrated using three-antenna structures: a dual-band dipole, a quasi-Yagi antenna with a parabolic reflector and a wideband monopole, all optimized under various design scenarios. The obtained numerical results corroborate the efficacy of our approach, the importance of using the available problem-specific knowledge, as well as illustrates the advantages of the presented procedure over gradient-based design refinement. The average cost of producing an optimized design is only a few evaluations of the EM antenna model.

The originality and technical contributions of this work include the development of an automated and reliable machine learning optimization framework using pre-existing knowledge on the antenna structure at hand, development of iterative correction procedure for rapid design refinement, demonstration of a possibility of expedited design while handling various performance figures and operating conditions (e.g. operating frequencies, realized gain, substrate permittivity), demonstrated capability of quasi-global design closure using primarily local methods.

2. Fast design closure using machine learning and iterative correction

This section formulates the proposed knowledge-based design closure methodology. One of its distinctive features is the incorporation of available information about the antenna of interest in the form of previously obtained designs, either available from the previous work with the structure or prepared specifically to set up the framework. This information is blended into a kriging surrogate model which yields – for a given target vector of performance figures – a reasonably good initial design (Section 2.2). The second feature is an iterative correction procedure for rapid design refinement. The latter is achieved by feeding the surrogate with the accumulated discrepancies between the target performance figure values and those extracted from the design produced in the previous iteration.

2.1 Design closure task. Objective space

Let x ∈ X denote a vector of adjustable parameters, where X is the parameter space of the antenna of interest. A certain number of performance figures, F_k, k – 1,…, N, is considered. The typical figures may be target operating frequencies, minimum acceptable fractional bandwidth, minimization of the sidelobe level, maximization of gain but also other conditions, e.g. parameters of the dielectric substrate (thickness, permittivity). The intended ranges F_k.min ≤ F_k ≤ F_k.max for F_k determine the objective space F, i.e. the region in which the design closure framework is supposed to operate.

We denote by R(x) the output of the EM simulation antenna model, typically a set of vectors containing the relevant characteristics such as reflection, gain, etc. A scalar merit function U (R(x)) measures the quality of the design x with respect to the objective vector F = [F₁…F_N]^T. An example follows. Suppose that the goal is to allocate the resonances of a dual-band antenna at the target frequencies f_0.1 and f_0.2 in the sense of maintaining |S₁₁(x, f)|≤ – 10 dB within the fractional bandwidths B (symmetric w.r.t. f_0.k, k = 1,2) and to maximize the average realized gain G(x, f) within the same bandwidths. In this case, the figures of interests are F_k = f_0.k, k = 1, 2, whereas the function U may be defined as:

(1)U(R(x),F)=−∑k=12(Bf0.k)−1∫f0.k(1−B/2)f0.k(1+B/2)G(x,f)df++∑k=12βk[max⁡{max⁡{f∈Bk:|S11(x,f)|}+1010,0}]2

where B_k = [f_0.k(1 – B/2), f_0.k(1 + B/2)] and β_k are the penalty factors. Here, the first component is the average in-band gain, whereas the second component constitutes penalty functions activated if the condition |S₁₁(x, f)| ≤ –10dB is violated in any of the operating bands.

Having defined the objective space and the merit function, the design optimality can be formulated as a solution to the following nonlinear minimization problem:

(2)x*=arg⁡min⁡xU(R(x),F)

2.2 Incorporating problem-specific knowledge

The assumption is that a certain number of designs are already available from the previous work with the same antenna structure. These designs are denoted as xb(j), j = 1,…, p and optimized for the objective vectors F(j)=[F1(j) … FN(j)]T according to (2). The symbol Jb(j)=J(xb(j)), j = 1,…, p, will stand for the Jacobian matrices of the antenna characteristics at the designs xb(j). This data is normally available as a by-product of solving (2). From the point of view of balanced coverage of the objective space, if possible, the vectors F^(j) should be allocated uniformly within the objective space F.

2.3 Initial design rendition by machine learning

Given the pre-existing data as described in Section 2.2, the following kriging interpolation (Queipo et al., 2005) models are constructed:

• s_x (·), mapping the objective space F into the parameter space X; identification of the surrogate s_x (·) is based on the data pairs {F(j),xb(j)}j=1,…,p;

• s_J (·), mapping the objective space F into the space of the system response sensitivities; this model is rendered using {F(j),Jb(j)}j=1,…,p as the training set.

The purpose of the model s_x is to approximate the region of the parameter space X containing the designs optimal with respect to F in the sense of (2). Because a limited amount of data is used to identify the surrogate (i.e. p optimum points xb(j)), the design s_x (F) does not generally coincide with arg min{x: U (R(x), F)}, i.e. the design optimum w.r.t. the objective vector F. Notwithstanding, this is the best information we have at this stage of the process. Consequently, the initial design corresponding to a given target objective vector F_t will be assigned as:

(3)x(0)=sx(Ft)

Similarly, the approximate sensitivity matrix of the antenna characteristics at the design x⁽⁰⁾ is obtained as:

(4)J(0)=sJ(Ft)

Figure 1 provides a graphical illustration of the considered concepts for N = 2 (two-dimensional objective space F) and n = 3 (three-dimensional parameter space X).

2.4 Gradient-based design refinement

As mentioned above, the initial design (3) has to be improved because s_x (F) ≠ O(F) (the set of optimum designs w.r.t. all F ∈ F). Here, the basic refinement procedure involves local optimization realized using the trust-region gradient-based algorithm (Conn et al., 2000). The algorithm produces a series of approximations x⁽ⁱ⁾, i = 0,1,…to x* as:

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

where L⁽ⁱ⁾(x) = R(x⁽ⁱ⁾ + J (x⁽ⁱ⁾) ·(x – x⁽ⁱ⁾) is the first-order Taylor expansion of R (x) obtained using the Jacobian J at x⁽ⁱ⁾.

The computational cost of (5) is at least n + 1 EM antenna simulations per iteration assuming that the Jacobian is obtained using finite differentiation. Here, this cost is reduced to one EM analysis per iteration by jump-starting the process using the sensitivity matrix estimation with (4) and updating the Jacobian using the Broyden formula (Broyden, 1965) in subsequent iterations.

A practical issue of the refinement through (5) is that it might be prone to a failure if the initial design produced by (3) is away from O (F), i.e. the true optimum or the quality of Jacobian estimation (4) is poor. Also, (5) may not be capable of correcting large frequency shifts (e.g. inaccurate allocation of operating frequencies obtained by (3)).

2.5 Iterative correction procedure

Given the deficiencies of the gradient-based refinement procedure as outlined in Section 2.4, an alternate scheme is proposed. It is a derivative-free correction process that involves multiple evaluations of the surrogate model s_x.

We denote by F_tmp.0 the objective vector that has been extracted from the EM-simulated antenna response at the design x⁽⁰⁾ of (3). For example, if the figures of interest are the target center frequencies of a multi-band antenna, F_tmp.0would consists of the actual center frequencies of R (x⁽⁰⁾). The design “inaccuracy” in terms of the assumed figures of interest can be then quantified as ΔF ⁽⁰⁾ = F_tmp.0 – F_t. The surrogate s_x can be reused to produce the corrected design:

(6)x(1)=sx(Ft−ΔF(0))

which brings us closer to the optimum design manifold O(F). Clearly, the dependence between the antenna parameters and its figures of interest is not linear; therefore, the process (6) has to be iterated by taking into account the accumulated errors. The sequence of corrected designs is generated accordingly as:

(7)x(i+1)=sx(Ft−∑k=0iΔF(k))

where ΔF ^(k) = F_tmp.k – F_t and F_tmp.k denotes the vector containing the objectives extracted from the EM-simulated antenna response for the design x^(k). Figure 2 shows the graphical illustration of the first two iterations of (7).

Because (7) is capable of making relatively large design relocations, therefore, typically, the process converges after a few iterations. Furthermore, (7) permits correcting considerable frequency shifts, which is difficult (or slow at the least) by means of (5), mostly due to using approximate Jacobians. For this very reason, (7) will likely be computationally cheaper than the gradient-based optimization. On the other hand, re-evaluation of the surrogate s_x primarily affects those figures of interest that define the objective space F (e.g. the location of the resonances of multi-band antennas but not the resonance depths). Consequently, further parameter tuning that operates at the level of antenna characteristics (i.e. as in (2)), might be required. Here, it is realized using the trust-region algorithm (5) as a follow-up procedure.

The flowchart of the entire optimization framework has been shown in Figure 3. It should be emphasized that because the domain of the models s_x and s_J is the entire objective space F, the framework exhibits global search properties within the relevant ranges of performance figures.

3. Verification examples

This section provides numerical and experimental verification of the presented knowledge-based design closure framework. It is based on three antenna structures: a uniplanar dual-band dipole, a quasi-Yagi antenna and a wideband monopole antenna. In each case, a different design scenario is considered as discussed in Sections 3.1 through 3.3. The simulation models of all the antennas from the benchmark set have been implemented in CST Microwave Studio. Notwithstanding, the actual choice of the simulation package is of secondary importance and virtually any available commercial solver such as HFSS or ADS may be used as a simulation tool within the proposed framework.

3.1 Case 1: Uniplanar dipole antenna

The first example is a dual-band uniplanar dipole antenna (Antenna I), shown in Figure 4 Chen et al. (2006), implemented on RO4350 substrate (ε_{r =} 3.5, h = 0.76 mm) and fed by a 50 Ohm coplanar waveguide (CPW). The adjustable parameters are x = [l₁ l₂ l₃ w₁ w₂ w₃]^T. Other parameters are fixed: l₀ = 30, w₀ = 3, s₀ = 0.15 and o = 5 (all dimensions in mm). The EM model R (∼100.000 cells; simulation time 60 s) is implemented in CST Microwave Studio and evaluated using its time-domain solver.

The design problem is to allocate the antenna resonances at the required operating frequencies f₁ and f₂ and, simultaneously, to improve the impedance matching at and around both f₁ and f₂. The ranges of interest are 2.0 GHz ≤ f₁ ≤ 3.0 GHz (lower band) and 4.0 GHz ≤ f₂ ≤ 5.5 GHz (upper band). The surrogate models s_x and s_J are constructed using seven reference designs x_b^(j) corresponding to the pairs {f₁, f₂} {2.0, 4.0}, {2.2,5.5}, {2.0, 5.5}, {2.3,4.5}, {2.4,5.5}, {2.6,4.0}, {2.7,3.5}, {2.8,4.7}, {3.0,4.0} and {3.0,3.5} (frequencies in GHz).

For verification, the procedure of Section 2 has been executed by optimizing the antenna of Figure 4 for the several target objective vectors F_t specified in Table 1. The initial design was obtained using (3). The refinement was conducted by means of the iterative scheme of Section 2.5 (cf. (7)) followed by the gradient search of (5). For benchmarking, gradient-based refinement (5) was carried out and a stand-alone procedure.

Table 1 gathers the numerical results, whereas geometry parameter values of the optimized antenna can be found in Table 2. It can be observed that in all considered cases the initial designs produced by the inverse surrogate are of good quality, which demonstrates that incorporation of problem-specific knowledge (here, in the form of available pre-optimized designs) is critical to ensure the quality and reliability of the parameter adjustment process. Furthermore, it can be noted that the iterative correction yields good designs, with reflection responses well centered at the target operating frequencies for all the considered test cases. The design quality is better than that obtained using solely the gradient search. Furthermore, iterative correction is much faster: its average cost is almost three times lower than that of (5). In addition, the follow-up gradient search (also realized using (5)) does not bring much improvement as shown in Figure 5. It is only for some cases, where the additional gradient optimization slightly improves the resonance depth (especially for Case 3 and 6). This demonstrates that (7) may actually be used as the exclusive refinement technique in some cases.

3.2 Case 2: Quasi-Yagi antenna

The second verification example is a quasi-Yagi antenna (Hua et al., 2015) featuring a parabolic reflector (see Figure 6) and implemented on a 1.5 mm-thick substrate. The structure is simulated in CST Microwave Studio. The geometry parameter vector is x = [W L L_m L_p S_d S_rW₂ W_a W_d g]^T (all dimensions in mm). To ensure 50 ohm input impedance, the feed line width W₁ is computed for a given substrate permittivity.

For this case, the design task is formulated as follows: optimize the antenna structure for a given substrate permittivity ε_r and selected target center frequency f₀ while maintaining at least 8% fractional bandwidth that is symmetric w.r.t. f₀. Additionally, the average realized gain maximization within the same 8% bandwidth is required (cf. (1)). The surrogate is to be valid for the following ranges 2.5 GHz ≤ f₁ ≤ 5.0 GHz and 2.5 ≤ ε_{r ≤ 4}.5. Only six reference designs x_b^(j) are assigned: {f₁, ε_r}: {2.5,4.5}, {5.0,4.5}, {2.5,2.5}, {5.0,2.5}, {4.5,3.5} and {3.0,3.5}(frequencies in GHz). This design problem is far more challenging than the previous one due to a considerably larger number of the parameters (ten to six for the first demonstration case).

Similarly as for the first antenna, six target objective vectors F_t were selected to carry out numerical verification. The numerical results are summarized in Tables 3 and 4. Figure 7 shows the antenna responses at the initial and the optimized designs. Despite a considerably higher complexity, also, in this case, the procedure of Section 2 was able to identify the designs satisfying the prescribed specifications. It should be noted that the quality of the initial design produced by the surrogate model s_x (cf. (3)) is not as good as for the first example. In particular, relatively considerable frequency shifts between the actual and the target operating frequencies can be observed.

Nevertheless, the proposed procedure allows for accommodating these discrepancies for all considered cases. The average cost of the iterative correction (7) is only three EM simulations, whereas the average cost of the follow-up gradient-based refinement (5) is four EM analyzes. At the same time, gradient-based only refinement is more expensive (nine EM antenna evaluations on the average) and the design quality is not as good (in Table 3, presented in terms of the average in-band gain). Also, the maximum number of iterations of (5) was set to 10 so that in most cases, the process was terminated before full convergence.

3.3 Case 3: Wideband monopole antenna

The last example is a monopole antenna (Antenna III) shown in Figure 8. The structure uses a quasi-circular radiator and modified ground plane for bandwidth enhancement (Alsath and Kanagasabai, 2015). The independent geometry parameters are x = [L₀ dR Rr_rel dL d_w L_g L₁ R₁dr c_rel]^T. The feed line width is computed for given substrate parameters (permittivity and height) to ensure 50 Ohm impedance. The computational model is implemented in CST Microwave Studio and evaluated using its transient solver (∼380.000 mesh cells, simulation time 1 min). The model incorporates the SMA connector. The antenna is supposed to operate within the UWB frequency range from 3.1 GHz to 10.6 GHz. Design optimality is understood as the minimization of the maximum in-band reflection.

The objective is to re-design the antenna for any dielectric substrate within the following ranges: permittivity 2.0 ≤ ε_{r ≤} 5.0 and height 0.5 mm ≤ h ≤ 1.5 mm. There are five reference designs pre-optimized for the following pairs {ε_r,h} = {3.5,1.0},{2.0,0.5}, {2.0,1.5}, {5.0,0.5} and {5.0,1.5}.

Tables 5 and 6 gather the numerical results obtained for six target substrate parameter vectors, whereas Figure 9 shows the antenna responses. In this case, the initial designs of (3) are of high quality so that gradient-based refinement (5) is sufficient. As a matter of fact, the application of the iterative scheme (7) is not possible here because the substrate parameters can be considered as operating conditions rather than figures of interest. This example was included specifically to indicate potential limitations of the scope of the presented method. Notwithstanding, indirect application of (7) is possible by translating potential frequency shifts into the appropriate (overall) scaling of the antenna parameters; however, there is no need to do that for Antenna III due to the aforementioned high quality of the initial designs.

4. Experimental validation

Selected optimized designs, one for each of Antennas I through III, have been fabricated for additional validation. For Antenna I, this is a design corresponding to the target operating frequencies f₁ = 2.45 GHz and f₂ = 5.1 GHz, implemented on 0.76 mm-thick RO4350 substrate. In the case of Antenna II, the design optimized for f₀ = 4.2 GHz and ε_{r =} 2.5 has been fabricated on a 1.5 mm thick AD250 substrate. For Antenna III, the design optimized for ε_{r =} 4.4 and h = 1.5 mm has been selected and implemented on FR4 laminate.

Figures 10–12 show the photographs of the antenna prototypes, as well as the comparison of simulated and measured characteristics. The agreement between both data sets is generally good for all considered structures. Small discrepancies can be attributed to the measurement setup (e.g. 90-degree bends used to mount the antennas for E-plane pattern evaluation) or the lack of SMA connectors in computational models (except for Antenna III).

5. Conclusions

The paper presented a knowledge-based framework for accelerated design optimization of antenna structures. One of the fundamental components of our procedure is machine learning tools in the form of the inverse and forward kriging metamodels, which are set up using the available designs of the antenna at hand and used to yield an initial design for further refinement. While the second component is an iterative correction scheme, which permits rapid accommodation of the initial design discrepancies (with respect to the target performance figure values), primarily the frequency shifts. The gradient-based optimizer is also incorporated to accomplish fine-tuning of the parameter in the final stage of the procedure.

Our methodology has been comprehensively validated using three-antenna examples of various characteristics: narrow-band, dual-band and wideband, which, in turn, have been optimized under different scenarios. The numerical results obtained for a number of target objective vectors are consistent throughout the considered test set. The presented framework is demonstrably capable of yielding optimized antenna designs at the cost of just a handful of EM simulations. The practical importance of the iterative correction procedure has been corroborated through a comparison with gradient-only refinement. It has been found that the incorporation of the problem-specific knowledge into the optimization framework, encoded using machine learning techniques, is a critical component that both facilitates the parameter adjustment process and renders it considerably more reliable.

The proposed approach can be a viable tool to expedite the antenna design closure if either a certain number of previously obtained designs are available or the designer finds the initial effort of preparing the design database justifiable (e.g. by the intended multiple re-uses of the procedure).

Figures

Figure 1.

Basic concepts of the presented design closure framework: (top picture) the objective space F and the exemplary target objective vector F_t; (bottom picture): s_x(F) (solid lines), the set of optimum designs O(F) (dashed lines) and the exemplary initial design s_x(F_t) (gray-shaded circle) corresponding to the target objective vector F_t. Designs x_b^(j) and s_x(x_b^(j)) are marked using black circles. As s_x(F) is an approximation of x^* = argmin{x: U(R(x),F_t)}, x^* (white circle) does not coincide with s_x(F_t)

Figure 2.

Conceptual illustration of the iterative correction procedure (here, shown for N = 2 and n = 1)

Figure 3.

Flow diagram of the proposed knowledge-based design closure framework

Figure 4.

Geometry of a dual-band dipole antenna (Antenna I) (Chen et al., 2006)

Figure 5.

Dual-band antenna responses at the initial design (…), the design obtained using the iterative correction scheme (o) and the final design obtained using the follow-up gradient search (5) ( – )

Figure 6.

Quasi-Yagi antenna with a parabolic reflector (Hua et al., 2015)

Figure 7.

Responses of Antenna II at the initial design (…), the designs optimized using the iterative correction scheme (…) and the final design obtained using the gradient search (5) (as a follow up) ( – )

Figure 8.

Geometry of the UWB monopole antenna (Alsath and Kanagasabai, 2015); light gray shades show ground plane

Figure 9.

Responses of Antenna III at the initial design (…) and the final design obtained using the gradient search (5)

Figure 10.

Antenna I optimized for f₁ = 2.45 GHz and f₂ = 5.1 GHz, implemented on RO4350 substrate

Figure 11.

Antenna II optimized for f₀ = 4.2 GHz and ε_{r = 2}.5; the structure implemented on AD250 substrate

Figure 12.

Antenna III optimized for ε_{r = 4}.4 and h = 1.5 mm, implemented on FR4 substrate

Table 1.

Antenna I: Optimization results

Target operating conditions Ft		Gradient-based optimization (5)		Iterative correction scheme (7)b
f1 [GHz]	f2 [GHz]	Objective function value*	Optimization costa	Objective function value*	Optimization costa
2.10	4.80	–15.5 dB	9	–15.5 dB	3$
2.30	4.00	–17.7 dB	11	–17.5 dB	3$
2.45	5.10	–11.6 dB	6	–20.9 dB	3$
2.50	4.75	–10.6 dB	10	–21.1 dB	3$
2.70	4.50	–18.9 dB	11	–18.8 dB	3$
3.00	5.00	–22.1 dB	11	–21.3 dB	3$

Notes:

*Objective function defined as the maximum reflection at both operating frequencies including their ±20 MHz vicinity. ^aThe cost in terms of the number of EM simulations required by the optimization process. ^bThe scheme complemented by gradient search according to (5)

Table 2.

Antenna I: Geometry parameter values

Target operating conditions		Geometry parameter values [mm]
f1 [GHz]	f2 [GHz]	l1	l2	l3	w1	w2	w3
2.10	4.80	37.2	8.220	18.7	0.29	3.62	2.39
2.30	4.00	33.5	7.12	21.3	0.60	4.68	1.11
2.45	5.10	34.0	9.64	18.9	0.34	3.06	1.75
2.50	4.75	33.1	9.20	19.8	0.45	3.57	1.28
2.70	4.50	31.1	9.71	20.8	0.36	3.89	0.71
3.00	5.00	30.1	11.1	20.4	0.50	2.85	0.85

Table 3.

Antenna II: Optimization results

Target operating conditions Ft		Gradient-Based Optimization (5)		Iterative Correction Scheme (7)b
f0 [GHz]	εr	Average realized gain*	Optimization costa	Average realized gain*	Optimization costa
4.2	2.5	6.4	11	6.4	3c
2.45	3.2	7.0	6	7.0	3c
5.0	3.2	6.3	7	6.4	3c
3.8	3.5	5.6	11	6.1	3c
3.8	4.1	5.5	11	5.9	3c
3.0	4.4	5.5	9	5.5	3c

Notes:

*Average realized gain within the fractional bandwidth of interest (here, 8%). ^aThe cost in terms of the number of EM simulations required by the optimization process. ^bThe scheme complemented by gradient search according to (5). ^cCost of the iterative correction scheme (7). The average cost of the follow-up gradient search is less than four EM simulations

Table 4.

Antenna II: Geometry parameter values

Target vector Ft		Geometry parameter values [mm]
f0 [GHz]	εr	W	L	Lm	Lp	Sd	Sr	W2	Wa	Wd	g
4.2	2.5	124.9	63.6	23.2	21.6	9.1	12.1	2.7	13.7	23.2	0.75
2.45	3.2	113.7	80.0	15.9	16.5	18.3	16.9	3.8	19.7	36.5	0.86
5.0	3.2	132.8	57.7	26.4	23.6	8.2	10.7	2.7	9.4	16.9	0.74
3.8	3.5	129.7	67.4	25.5	18.3	17.8	12.2	3.6	11.3	24.3	0.50
3.8	4.1	130.8	64.3	23.2	21.3	15.4	14.5	3.1	10.9	23.9	0.50
3.0	4.4	122.9	71.2	20.5	16.9	19.5	18.2	2.7	15.0	28.3	1.00

Table 5.

Antenna III: Optimization results

Target operating conditions Ft
εr	h [mm]	Objective function value*	Optimization costa
2.2	0.76	–12.8 dB	6
3.0	0.76	–13.3 dB	9
4.4	1.5	–15.6 dB	7
2.5	1.5	–19.7 dB	8
3.38	0.76	–13.7 dB	9
3.38	1.5	–18.1 dB	8

Notes:

*Maximum reflection within the frequency range from 3.1 GHz to 10.6 GHz. ^aThe cost in terms of the number of EM simulations required by the optimization process

Table 6.

Antenna III: Geometry parameter values

Target vector Ft		Geometry parameter values [mm]
εr	h [mm]	L	dR	R	rrel	dL	dw	Lg	L1	R1	dr	crel
2.2	0.76	12.2	0.06	6.91	0.21	3.26	5.37	12.4	3.45	3.06	0.29	0.71
3.0	0.76	11.8	0.11	6.43	0.28	3.85	5.97	11.7	2.01	2.34	0.42	0.20
4.4	1.5	12.8	0.00	5.48	0.10	4.54	6.96	12.3	1.56	2.51	0.29	0.90
2.5	1.5	12.2	0.02	6.76	0.13	5.00	5.69	12.4	1.96	3.84	0.20	0.52
3.38	0.76	11.8	0.12	6.09	0.16	3.95	6.14	11.6	2.29	2.16	0.43	0.39
3.38	1.5	12.1	0.01	5.71	0.29	4.86	6.03	11.9	2.08	3.65	0.24	0.87

Note:

*Variables with subscript rel are relative, i.e. unitless