Design-Oriented Modeling of High-Frequency Structures

Abstract

Design-oriented modeling of high-frequency structures requires identification of a region of the parameter space containing the designs that are “good” in a particular sense. The latter is determined by the set of figures of interest, which might be the operating conditions of the structure at hand, its material parameters, or other factors that are of interest for the designer. The rationale behind it is that good-quality designs normally occupy a very small portion of the parameter space. Identification of such a subset and restricting the modeling process to it allow for a significant reduction of the number of training data samples required to set up a reliable surrogate without formally reducing the parameter ranges. Clearly, the fundamental question is how the aforementioned “promising” region can be found. This is achieved by considering a set of reference designs pre-optimized with respect to the performance figures that are of interest in a given context. This chapter discusses several straightforward techniques following the concept outlined above.

Surrogate modeling, as demonstrated in this book so far, offers a practical way of handling computationally expensive simulation models. This might be especially convenient when massive evaluations thereof are required, for example, for the purpose of design optimization or uncertainty quantification (Bandler et al. 2008; Koziel and Bandler 2015). Chapters 2 and 3 outlined a number of modeling approaches concerning both data-driven and physics-based surrogates. Their major advantages include low evaluation cost and versatility (approximation models) as well as good generalization (physics-based models). Yet, as mentioned on various occasions, surrogate modeling exhibits some fundamental issues that limit its applicability in a significant manner. These include the curse of dimensionality, difficulties in constructing the models over wide ranges of parameters, or—in the context of physics-based surrogates—potential problems in finding and setting up low-fidelity models. In this book, we are mostly interested in modeling of high-frequency structures, routinely featuring highly nonlinear and vector-valued responses, handling of which incurs additional challenges.

Several methods of alleviating the difficulties pertinent to conventional modeling have been discussed (e.g., high-dimensional model representation, HDMR; Foo and Karniadakis 2010; various model order reduction (MOR) methods; Baur et al. 2014; or the orthogonal matching pursuit, OMP; Tropp 2004), but these techniques are typically designed for handling particular classes of problems (e.g., underdetermined regression tasks in the case of OMP). Here, the main focus is on constructing reusable surrogates that are valid for wide ranges of parameters and that can be utilized for a variety of purposes including optimization (also multi-objective) or robust design. In order to achieve this goal, a specific approach is taken, the main component of which is identification of a region of the parameter space containing the designs that are “good” in a particular sense. The latter is determined by the set of figures of interest, which might be the operating conditions of the structure at hand, its material parameters, or other factors that are of interest for the designer. The rationale behind it is that good-quality designs normally occupy a very small portion of the parameter space. Identification of such a subset and restricting the modeling process to it allow for a significant reduction of the number of training data samples required to set up a reliable surrogate without formally reducing the parameter ranges. Clearly, the fundamental question is how the aforementioned “promising” region can be found. In this book, this is achieved by considering a set of reference designs pre-optimized with respect to the performance figures that are of interest in a given context. This chapter discusses several rather straightforward techniques following the concept outlined above. The remaining chapters of this book describe more systematic approaches to performance-driven modeling as well as their design applications.

4.1 Data-Driven Modeling by Constrained Sampling

Here, a simple approach to surrogate modeling via constrained sampling is presented and illustrated using two examples of antenna structures.

4.1.1 Uniform Versus Constrained Sampling

In general, data-driven models are most often constructed based on uniform sampling of the design space (Couckuyt et al. 2010), the latter being an interval delimited using the lower and upper bounds on the parameters. On the other hand, in most practical cases, the parameter sets corresponding to the designs of sufficient quality with respect to the typical performance specifications (such as good matching at the specific frequency bands in the case of antennas) exhibit a relatively high level of correlation. In other words, usable designs are normally allocated along a specific path, a surface (or, more generally, along a manifold) within the design space. Consequently, uniform (and unconstrained) sampling leads to wasting majority of the samples because they correspond to designs that are of poor quality.

Here, for the sake of reducing the cost of training data acquisition, the aim is to focus the modeling process only on these parts of the design space that correspond to potentially useful designs. A particular approach to defining the interesting part of the space is described below (Koziel 2017).

Let us denote by x⁽¹⁾, … , x^(K), the set of the reference designs of the system at hand. In this section, for illustration purposes, the focus is on high-frequency structures designed for various operating frequencies, which is a quite typical situation in the areas of microwave and antenna engineering. In this case, the reference designs correspond to the system optimized for particular values of the operating frequency within the range of interest. Let d = [d₁ … d_n]^T be the deviation vector. The samples are only allocated in the part X_s of the original design space X that is within the distance d (component-wise) from the piecewise linear path connecting the reference designs, i.e.,

$$ {\boldsymbol{v}}^{(k)}=\alpha {\boldsymbol{x}}^{(k)}+\left(1\hbox{--} \alpha \right){\boldsymbol{x}}^{\left(k+1\right)},\kern1em 0\le \alpha \le 1, $$

(4.1)

k = 1, … , K – 1. As an example, K = 3 is assumed. This is not critical but a lower K is computationally cheaper because it requires a smaller number of optimizations of the system of interest.

Figure 4.1 shows the responses of a dielectric resonator antenna (DRA) considered as one of the illustration examples (Sect. 4.1.3) optimized for three operating frequencies of 4 GHz, 5.5 GHz, and 7 GHz, as well as selected geometry parameters versus frequency. Figure 4.2 shows the uniform sampling versus constrained sampling for four selected two-dimensional projections. The size of the constrained design space is considerably smaller than that of the original space, and the benefits are increasing with the increase of the problem dimensionality. For n = 7 (which is the case for both illustration cases considered in Sect. 4.1.3), the volume-wise reduction of the design space is over three orders of magnitude.

Fig. 4.1

(a) Reference designs of the DRA of Fig. 4.3 (Sect. 4.1.3) optimized for 4 GHz (····), 5.5 GHz (- - -), and 7 GHz (—); (b) geometry parameter values for selected dimensions (—) with the region of interest marked using the dashed line (Koziel 2017)

Fig. 4.2

Uniform versus constrained sampling for the DRA of Fig. 4.3 shown for four selected projections onto (a) a_x–a_y plane, (b) a_y–a_z plane, (c) a_z–u_s plane, and (d) a_c–u_s plane (see Sect. 4.1.3 for symbol explanation) (Koziel 2017)

4.1.2 Modeling Procedure

The overall modeling flow follows the typical data-driven surrogate modeling procedures (cf. Chap. 2) except the particular definition of the model domain as outlined in Sect. 4.1.1. It can be summarized as follows:

• 1. Obtain K reference designs as described in Sect. 4.1.1 (surrogate-based optimization methods are used for the sake of computational efficiency; Koziel and Ogurtsov 2014).

• 2. Define the constrained design space X_s (cf. Sect. 4.1.1).

• 3. Sample X_s and acquire EM simulation data.

• 4. Identify surrogate model s within X_s.

In the examples of Sect. 4.1.3., the surrogate model is constructed using kriging interpolation (Queipo et al. 2005).

4.1.3 Illustration Examples

Two verification examples are presented, a DRA (Koziel and Bekasiewicz 2015) and a planar PIFA (Volakis 2007). Both antennas are narrowband, described by seven geometry parameters each and with highly nonlinear responses. Furthermore, we are interested in setting the surrogate models that would cover a wide range of antenna operating frequencies. All of these make the modeling problem challenging for conventional data-driven methods.

4.1.3.1 Dielectric Resonator Antenna

The first example is a dielectric resonator antenna (DRA) shown in Fig. 4.3 (Koziel and Bekasiewicz 2015). The structure consists of a dielectric resonator (ε_r = 10 and tan δ = 0.0001) situated on the ground plane. The resonator is fed through the ground plane slot by a microstrip line. The substrate material is 0.5-mm-thick Rogers RO4003 (ε_r = 3.3). The DRA is covered by polycarbonate housing (ε_r = 2.8). Design variables are x = [a_x a_y a_z a_c u_s w_s y_s]^T mm. The fixed parameters are d_x = d_y = d_z = 1 mm, w₀ = 1.15 mm. The EM model f(x) of the antenna is implemented in CST (~420,000 cells, simulation time 19 minutes).

Fig. 4.3

DRA: (a) 3D view of its housing and top (b) and front (c) views (Koziel and Bekasiewicz 2015)

The antenna of Fig. 4.3 has been modeled using kriging with constrained sampling. The lower and upper bounds for design variables are l = [9 12 4 0 0.5 7.5 6]^T and u = [14 20 12 4 6 12.5 9]^T, all in mm. The design space defined by these bounds contains the three DRA designs optimized for the operating frequencies 4 GHz, 5.5 GHz, and 7 GHz that form the path for the constrained sampling plan. The reference designs have been obtained using feature-based optimization (Koziel 2015), and the average computational cost of finding each design corresponds to 40 EM simulations of the antenna structure (~760 min of the CPU time). The distance vector is d = [1 1 1 1 1 1 1]^T. The volume-wise reduction of the design space is almost three orders of magnitude. The number of training samples is 800. Table 4.1 shows the modeling accuracy, which is compared to the conventional sampling in the entire space (error measure is RMS averaged over 100 random test points). The minimum number of samples required to achieve the same accuracy (within the constrained space) as that of the conventional model has be found to be only 80 (the cost reduction factor is over 6).

Table 4.1 Modeling results of DRA

Figure 4.4 shows the responses of the DRA at the selected test designs. Figure 4.5 shows the responses of the antenna optimized (using the surrogate) for the operating frequencies 5.1 GHz, 5.8 GHz, and 6.5 GHz. The optimization routine was Matlab’s fmincon (Matlab Optimization Toolbox, R2016a), and the optimization time is negligible (a few seconds). The quality of the optimized designs is excellent with no further design tuning necessary. Figure 4.6 shows the constrained model domain using four reference designs. The modeling error for this case is 4.1% (very similar to that in Table 4.1; using the same number of training samples), which indicates robustness of the approach.

Fig. 4.4

DRA responses at the selected test designs: high-fidelity EM model (—), proposed surrogate model (o) (Koziel 2017)

Fig. 4.5

EM-simulated DRA responses at the designs obtained by optimizing the proposed surrogate model for operating frequencies 5.1 GHz (····), 5.8 GHz (- - -), and 6.5 GHz (—)

Fig. 4.6

An alternative constrained surrogate model domain setup for the DRA example (here using four instead of three reference designs). The average modeling error for this case is 4.1% (similar to that in Table 4.1, thus indicating robustness of the proposed modeling scheme)

The recommended size of the deviation vector d is a fraction of the range of design variables (corresponding to the reference designs), say 10–20% of these ranges. Here, for simplicity, the vector of ones has been utilized. To check the effect of the vector d setup, the example has been reworked using deviation vectors of 1.5d and d/1.5. The average modeling errors obtained for these two cases, 5.7% and 3.9%, are noticeably different yet comparable to 4.3% obtained for the original vector d.

4.1.3.2 Planar Inverted-F Antenna

The second example is a planar PIFA (Volakis 2007) shown in Fig. 4.7. The design variables are x = [v₀ v₁ v₂ v₃ v₄ v₅ v₆]^T. Fixed parameters are [u₀ u₁ u₂ u₃ u₄ u₅ u₆ u₇ w₀ r₀]^T = [6.15 –50.0 –15.0 10.5 29.35 11.65 5.0 1.0 0.5]^T mm; 0.508 mm substrate, and the boxes are of Rogers TMM4 and TMM6. The EM model f is evaluated in CST (~650,000 mesh cells, simulation time 13 min).

Fig. 4.7

PIFA geometry: (a) top and side view, substrate shown transparent; (b) perspective view (Koziel 2017)

The lower and upper bounds defining the initial design space are l = [–5 2 4.5 2 6 –30 –40]^T and u = [0 10 7 8 15 –8 –30]^T, all in mm. Similarly, as for the first example, three reference designs are used, optimized for the operating frequencies of 1.5 GHz, 2.5 GHz, and 3.5 GHz. The reference designs have been obtained using the method of (Koziel 2015); the average computational cost of finding each design corresponds to 40 EM simulations of the antenna structure (~500 min of the CPU time). The distance vector is d = [0.5 0.5 0.5 1.0 1.5 1.5 1.0]^T. The volume-wise reduction of the design space is over three orders of magnitude. The number of training samples is 800.

Table 4.2 shows the modeling accuracy of the proposed scheme, which is also compared to conventional sampling in the entire space. The minimum number of samples required to achieve comparable accuracy (within the constrained space) as that of the conventional model is only 100 (cf. Table 4.2). Figure 4.8 shows the responses of the PIFA at the selected test designs. As an application example, the antenna was optimized for the operating frequencies of 1.85 GHz, 2.45 GHz, and 2.85 GHz (300 MHz bandwidth required in all cases). The EM-simulated antenna responses at the optimized designs are shown in Fig. 4.9.

Table 4.2 Modeling results of PIFA

Fig. 4.8

PIFA responses at the selected test designs: high-fidelity EM model (—), proposed surrogate model (o) (Koziel 2017)

Fig. 4.9

EM-simulated PIFA responses at the designs obtained by optimizing the proposed surrogate model for operating frequencies of 2.85 GHz (····), 2.45 GHz (- - -), and 1.85 GHz (—) (±150 MHz bandwidth required in each case) (Koziel 2017)

4.2 Design-Oriented Constrained Modeling for Operating Frequency and Substrate Parameters

In this section, constrained modeling of antenna structures with respect to both operating conditions (center frequency) and material parameters (relative permittivity of the dielectric substrate) is considered (Koziel and Bekasiewicz 2017). This is a slight generalization of the concept introduced in Sect. 4.1, which will be further generalized in the subsequent chapters of the book.

4.2.1 Modeling Procedure

For the purpose of presentation, operating frequency f and relative dielectric permittivity ε_r of the substrate are considered as the operating condition and material parameter of interest, respectively. The surrogate model is to be reliable for the range of operating frequencies f_min ≤ f ≤ f_max and the range of permittivity ε_min ≤ ε_r ≤ ε_max.

Let f(x) represent a response of an EM-simulated antenna model, where x is a vector of antenna parameters (in general, both geometry and material). The symbol x^∗(f, ε_r) will denote the design optimized for the operating frequency f and the substrate dielectric permittivity ε_r.

The region of the surrogate model validity is defined as a vicinity of the manifold spanned by nine reference designs covering the aforementioned ranges of the operating frequency and ε_r, f_min ≤ f ≤ f_max and ε_min ≤ ε_r ≤ ε_max. These are x^∗(f^#, ε_r^#), for all combinations of f^# ∈ {f_min, f₀, f_max} and ε_r^# $\in \left\{{\varepsilon}_{\mathrm{min}},{\varepsilon}_{r0},{\varepsilon}_{\mathrm{max}}\right\}$, cf. Fig. 4.10.

Fig. 4.10

Reference designs: (a) distribution on the f/ε plane and (b) designs allocated in a three-dimensional space. The shaded area is a manifold that determines the region of interest for surrogate model construction (Koziel and Bekasiewicz 2017)

Let us define vectors v₁ = x^∗(f_min, ε_min) – x^∗(f₀, ε_r0), v₂ = x^∗(f_min, ε_r0) – x^∗(f₀, ε_r0), v₃ = x^∗(f_min, ε_max) – x^∗(f₀, ε_r0), v₄ = x^∗(f₀, ε_max) – x^∗(f₀, ε_r0), v₅ = x^∗(f_max, ε_max) – x^∗(f₀, ε_r0), v₆ = x^∗(f_max, ε_r0) – x^∗(f₀, ε_r0), v₇ = x^∗(f_max, ε_min) – x^∗(f₀, ε_r0), and v₈ = x^∗(f₀, ε_min) – x^∗(f₀, ε_r0) (see also Fig. 4.11a). In addition, let us define a manifold M, which is spanned by eight pairs of vectors [v₁,v₂], [v₂,v₃], … , [v₈,v₁], as

$$ M=\underset{k=1}{\overset{8}{\cup }}{M}_k=\underset{k=1}{\overset{8}{\cup }}\left\{\boldsymbol{y}={\boldsymbol{x}}^{\ast}\left({f}_0,{\varepsilon}_0\right)+\alpha {\boldsymbol{v}}_k+\beta {\boldsymbol{v}}_{k+1}:\alpha, \beta \ge 0,\kern0.5em \alpha +\beta \le 1\right\}. $$

(4.2)

For consistency of notation, let us also define v₉ = v₁. Figure 4.11b shows a point z and its projection P_k(z) onto the hyperplane containing M_k. P_k is defined in a conventional sense (i.e., as the point on the hyperplane that is the closest to z). It corresponds to the expansion coefficients w.r.t. v_k and v_k + 1:

$$ \arg \underset{\overline{\alpha},\overline{\beta}}{\min }{\left\Vert \boldsymbol{z}-\left[{\boldsymbol{x}}^{\ast}\left({f}_0,{\varepsilon}_{r0}\right)+\overline{\alpha}{\boldsymbol{v}}_k+\overline{\beta}{\boldsymbol{v}}_{k+1}^{\#}\right]\right\Vert}^2, $$

(4.3)

where ${{{\boldsymbol{v}}_k}_{+1}}^{\#}={\boldsymbol{v}}_{k+1}\hbox{--} {p}_k{\boldsymbol{v}}_k$ with p_k = v_k^Tv_k + 1(v_k^Tv_k). Thus, v_k + 1^# is a component of v_k + 1 that is orthogonal to v_k. Let us consider

$$ \left[{\boldsymbol{v}}_k\kern0.5em {\boldsymbol{v}}_{k+1}^{\#}\right]{\left[\overline{\alpha}\kern0.5em \overline{\beta}\right]}^T=\boldsymbol{z}-{\boldsymbol{x}}^{\ast}\left({f}_0,{\varepsilon}_{r0}\right). $$

(4.4)

The least squares solution to (4.4) (equivalent to (4.3)) is given as

$$ {\left[\overline{\alpha}\kern0.5em \overline{\beta}\right]}^T={\left({\boldsymbol{V}}_k^T{\boldsymbol{V}}_k\right)}^{-1}{\boldsymbol{V}}_k^T\left(\boldsymbol{z}-{\boldsymbol{x}}^{\ast}\left({f}_0,{\varepsilon}_{r0}\right)\right), $$

(4.5)

where ${\boldsymbol{V}}_k=\left[{\boldsymbol{v}}_k\ {{{\boldsymbol{v}}_k}_{+1}}^{\#}\right]$. For practical reasons, more convenient are the expansion coefficients with respect to v_k and v_k + 1, which are given as

$$ \alpha =\overline{\alpha}-{p}_k\overline{\beta},\kern2.5em \beta =\overline{\beta}. $$

(4.6)

Note that P_k(z) ∈ M_k if and only if α ≥ 0, β ≥ 0, and α + β ≤ 1.

Fig. 4.11

Auxiliary components of the region of validity of the surrogate model: (a) the manifold of Fig. 4.10b with the spanning vectors v_k marked, with: v₁ = x^∗(f_min, ε_min) – x^∗(f₀, ε_r0), v₂ = x^∗(f_min, ε_r0) – x^∗(f₀, ε_r0), v₃ = x^∗(f_min, ε_max) – x^∗(f₀, ε_r0), … , v₈ = x^∗(f₀, ε_min) – x^∗(f₀, ε_r0); (b) manifold M_k with its spanning vectors and a point z and its projection onto the hyperplane containing M_k (Koziel and Bekasiewicz 2017)

Let us define x_max = max {x^∗(f₀, ε_r0) + v₁, … , x^∗(f₀, ε_r0) + v₈} and x_min =  min {x^∗(f₀, ε_r0) + v₁, … , x^∗(f₀, ε_r0) + v₈}. The vector dx = x_max – x_min is the range of variation of antenna geometry parameters within the manifold M. The surrogate model domain X_s is defined as a vector y ∈ X_s if and only if:

• 1. The set K(y) = {k ∈ {1, … , 8} : P_k(y) ∈ M_k} is not empty;

• 2. min{‖(y – P_k(y)) ÷ dx‖ : k ∈ K(y)} ≤ d_max, where ÷ denotes component-wise division (d_max is a user-defined parameter).

The first condition ensures that y is sufficiently close to M in a “horizontal” sense. In the second condition, the user-defined d_max is compared to the normalized distance between y and its projection onto that M_k to which the distance is the shortest. Due to normalization w.r.t. the parameter ranges dx, d_max determines the “perpendicular” size of the surrogate model domain (as compared to the “tangential” size given by dx). Therefore, a typical value of d_max would be 0.2 or so.

By definition, all the reference designs and the manifold M belong to X_s. The size of X_s is dramatically smaller (volume-wise) than the size of the hypercube containing the reference designs (i.e., x such that x_min ≤ x ≤ x_max). It should be mentioned that because the number of reference designs grow very quickly (exponentially) with the number of operating conditions considered (in this case, two), a practical application of the propose approach is limited to a few operating conditions.

The surrogate model is constructed using kriging interpolation of the EM model response f based on the training data sampled within X_s (Queipo et al. 2005). A separate kriging model is constructed for each frequency in the frequency spectrum at which the response f is evaluated. The design of experiment technique is random sampling within the interval [x_min, x_max] assuming uniform probability distribution. The samples allocated outside [x_min, x_max] are rejected.

4.2.2 Case Study: Ring Slot Antenna

The modeling technique is demonstrated using a ring slot antenna shown in Fig. 4.12 (Sim et al. 2014). The structure comprises a microstrip line that feeds a circular ground plane slot with defected ground structure (DGS). The low-pass properties of the DGS allows for suppression of the antenna harmonic frequencies. The thickness and loss tangent of the substrate are 0.762 and 0.0018, respectively. The parameter set is x = [l_f l_d w_d r s s_d o g ε_r]^T; ε_r represents relative permittivity of the substrate. The feed line width w_f is computed for each ε_r to ensure 50 ohm input impedance. The computational model of the antenna is implemented in CST (~300,000 cells, simulation 90 s).

Fig. 4.12

Geometry of the ring slot antenna with a microstrip feed (dashed line) (Sim et al. 2014)

The modeling problem is already difficult due to a large number of parameters. To make it even more challenging, a wide range of operating frequencies of interest, f_min = 2.5 GHz to f_max = 6.5 GHz, and a wide range of substrate permittivity, ε_min = 2.0 to ε_max = 5.0, were assumed. The reference designs have been obtained by optimizing the structure of Fig. 4.12 for all combinations of f ∈ {2.5, 4.5, 6.5} GHz and ε_r ∈ {2.0, 3.5, 5.0} using feature-based optimization (FBO) (Koziel 2015). Optimization is understood as minimizing the antenna reflection at f₀. In general, in case of possible nonuniqueness of the optimization result, regularization can be used, e.g., by introducing a penalty factor that enforces extension of the antenna bandwidth. The cost of FBO was 40–50 antenna simulations (per design). Antenna responses at all nine reference designs are shown in Fig. 4.13.

Fig. 4.13

Reflection responses of the antenna of Fig. 4.12 for nine reference designs: (a) ε_r = 2.0, (b) ε_r = 3.5, (c) ε_r = 5.0; (····) f₀ = 2.5 GHz, (- - -) f₀ = 4.5 GHz, (—) f₀ = 6.5 GHz

The modeling approach has been verified for d_max = 0.2 by setting up the kriging surrogate with 100, 200, 500, and 1000 random samples. The test set contained 100 random points. For benchmarking, the kriging model was also constructed using 1000 training points allocated in a conventional (unconstrained) manner. Table 4.3 shows the average RMS errors for all considered models. Selected two-dimensional projections of the training sets for uniform and constrained sampling are shown in Fig. 4.14. It can be observed that the latter allows for 3.5-fold improvement of the predictive power of the surrogate. At the same time, comparable modeling error is achieved with tenfold reduction of the number of training samples . Figure 4.15 shows the surrogate and EM model responses at the selected test designs.

Table 4.3 Ring slot antenna: modeling results

Fig. 4.14

Uniform versus constrained sampling for selected two-dimensional projections onto (a) l_f -w_d plane, (b) l_f -s_d plane, (c) s-o plane, and (d) o-ε_r plane (Koziel and Bekasiewicz 2017)

Fig. 4.15

Responses of the ring slot antenna of Fig. 4.14 at the selected test designs for N = 1000: high-fidelity EM model (—), proposed surrogate model (o) (Koziel and Bekasiewicz 2017)

4.2.3 Application Examples and Experimental Validation

To demonstrate practical application of the proposed modeling approach, the antenna of Fig. 4.12 was designed—by optimizing the devised surrogate—for various substrate permittivity and operating frequencies (see Table 4.4).

Table 4.4 Ring slot antenna: optimized antenna designs

Figure 4.16 shows the optimization results for the designs of Table 4.4. An excellent agreement between the surrogate and EM model can be observed. Also, the antenna responses are well centered at the requested operating frequencies.

Fig. 4.16

Surrogate (- - -) and EM-simulated responses (—) of the ring slot antenna of Fig. 4.12 at the designs obtained by optimizing the proposed surrogate model for (a) ε_r = 2.2 and f₀ = 3.4 GHz, (b) ε_r = 2.6 and f₀ = 4.8 GHz, (c) ε_r = 4.3 and f₀ = 3.75 GHz, (d) ε_r = 4.1 and f₀ = 5.3 GHz, (e) ε_r = 2.6 and f₀ = 5.8 GHz, and (f) ε_r = 3.5 and f₀ = 5.8 GHz. Requested operating frequencies marked using vertical lines (Koziel and Bekasiewicz 2017)

The antenna designs with ε_r = 2.2 and f₀ = 3.4 GHz, as well as with ε_r = 3.5 and f₀ = 5.8 GHz, have been fabricated and measured. Figure 4.17 shows the photographs of the manufactured prototypes. The agreement between the simulated and measured characteristics is very good as indicated in Fig. 4.18. Slight discrepancies between the responses are due to electrically large measurement setup which was not accounted for in the EM simulation model.

Fig. 4.17

Antenna prototypes: ε_r = 2.2, f₀ = 3.4 GHz (left) and ε_r = 2.6, f₀ = 5.8 GHz (right) (Koziel and Bekasiewicz 2017)

Fig. 4.18

Simulated (gray) and measured (black) characteristics of the antennas of Fig. 4.17. Solid and dashed lines in radiation pattern plots represent H- and E-plane responses, respectively (Koziel and Bekasiewicz 2017)

4.3 Constrained Feature-Based Modeling of Compact Microwave Structures

In this section, a technique of Sect. 4.1 is combined with response features as well as nonuniform sampling approach in order to construct a low-cost design-oriented surrogate of a miniaturized microwave coupler (Koziel and Bekasiewicz 2016).

4.3.1 Case Study. RRC and Response Features

The modeling concept will be explained and demonstrated using the example compact equal-split rat-race coupler (RRC) composed of two vertical and four horizontal slow-wave resonant structures (Bekasiewicz et al. 2015) as shown in Fig. 4.19. The RRC is implemented on a Taconic RF-35 dielectric substrate (h = 0.762 mm, ε_r = 3.5, tanδ = 0.0018). The design variables are x = [l₁ l₂ l₃ w₁ l₄ l₅ l₆]^T, whereas the dimension w₀ = 1.7 is kept constant in order to ensure 50 ohm input impedance. The dependent variables are w₂ = w₁, w₃ = 20w₁ + 19l₁, and w₄ = 6w₂ + 7l₅. All dimensions are in mm.

Fig. 4.19

Microstrip rat-race coupler constructed of slow-wave resonant structures—geometry (Bekasiewicz et al. 2015)

The RRC is implemented in CST Microwave Studio (CST 2018) and simulated using its frequency domain solver with ~800,000 mesh cells. The simulation time is about 75 min. Thus, the considered coupler is a representative example of a topologically complex structure with expensive computational model.

A typical response of the RRC of Fig. 4.19 is shown in Fig. 4.20. The responses correspond to the design optimized for bandwidth at the operating frequency of around 0.8 GHz. It can be observed that S-parameters are highly nonlinear functions of frequency and therefore difficult to be modeled. At the same time, only a few points extracted from the response are necessary for design purposes, specifically to determine the figures of interest such as the operating frequency of the circuit, the power split error, or the −20 dB bandwidth. These points include the frequency and levels of the minima of ∣S₁₁∣, ∣S₃₁∣, and ∣S₄₁∣, maximum of ∣S₂₁∣, as well as the points corresponding to −20 dB level of ∣S₁₁∣, and ∣S₄₁∣, all shown in Fig. 4.20. As indicated in Fig. 4.21, the dependence of these characteristic point coordinates on the geometry parameters of the coupler is only slightly nonlinear and therefore easier to model. Thus, we choose to model only the characteristic points rather than the entire responses as long as the figures of interest the coupler is designed for can be restored from these points.

Fig. 4.20

Compact RRC: typical responses of the structure tuned to around 0.8 GHz operating frequency and the characteristic points as described in the text

Fig. 4.21

Compact RRC: dependence of the selected characteristic points on geometry parameters l₁ and l₂ of the structure: (a) frequency of ∣S₁₁∣ minimum, (b) level of ∣S₂₁∣ maximum (corresponding to the circuit operating frequency) (Koziel and Bekasiewicz 2016)

4.3.2 Modeling Methodology

In the case of high-frequency structures, including compact microwave circuits, the design variables and their changes have to be well correlated in order to produce designs that are acceptable with respect to typical performance specifications (cf. Fig. 4.22). This means that uniform sampling across an interval-type of a domain (Koziel et al. 2018; Koziel 2017) leads to a situation where majority of the samples are useless and the resources utilized to acquire the corresponding EM data are essentially wasted. Similarly, sequential sampling schemes (Crombecq et al. 2011; Liu et al. 2016) are of not much help either because the infill criteria are normally based on model accuracy rather than the quality of the samples from the design requirement standpoint.

Fig. 4.22

Dependence of geometry parameters (here, l₁, l₂, and l₅) on the operating frequency f₀ of the RRC (here, for bandwidth-optimized designs) (Koziel and Bekasiewicz 2016)

Here, the modeling procedure similar to that of Sect. 4.1 is adopted, enhanced by nonlinear domain scaling and employment of response features (Koziel and Bekasiewicz 2016). Let x⁽¹⁾, … , x^(K), be the set of the reference (e.g., bandwidth-optimized) designs of the structure at hand that correspond to a range of operating frequencies being of interest for the modeling purposes.

Furthermore, let d = (u – l)/M be a deviation vector where l =  min {x⁽¹⁾, … , x^(K)}, u =  max {x⁽¹⁾, … , x^(K)} (lower and upper bounds for the design variables). The samples are only allocated in the part of the space X_s that is within the distance d from the piecewise linear path connecting the reference designs, i.e., v^(k) = αx^(k) + (1 – α)x^(k + 1), 0 ≤ α ≤ 1, k = 1, … , K – 1. In this case, K = 3 and M = 5 have been assumed (neither is critical but lower K reduces the cost of reference design acquisition, cf. Sect. 4.1).

Because of a nonlinear relationship between the circuit’s operating frequency and its dimensions (cf. Fig. 4.22), uniform sampling in the constrained domain, as described in the previous paragraph, would lead to having majority of samples corresponding to lower operating frequencies. From the point of view of design-oriented modeling, we are more interested in obtaining the training set in which all operating frequencies are represented in a relatively uniform fashion. Toward this end, the following nonuniform probability distribution is employed:

$$ {\boldsymbol{x}}^{\mathrm{tmp}}=\boldsymbol{l}+\left(\boldsymbol{u}-\boldsymbol{l}\right)\circ {\boldsymbol{r}}^{\boldsymbol{\rho}}, $$

(4.7)

where x^tmp is a candidate sample (accepted if it is in X_s or rejected otherwise), r is a vector of uniformly distributed random numbers from [0,1] interval, and ρ = [ρ₁ … ρ_n] is a scaling vector so that ρ_k > 1 (typically between 1.2 and 3). Here, multiplication ° and r ^ρ are understood as component-wise operations. Setting ρ_k > 1 implies that designs corresponding to higher values of the operating frequencies will be better represented in the training pool. The specific values of the coefficients can be estimated based on the reference designs (details are omitted here for the sake of brevity). Figure 4.23 shows the examples of uniform sampling in the entire design space, restricted to X_s, and nonuniform sampling as described above. The plots are for the RRC of Fig. 4.19 and correspond to the 2D projections onto the l₁–l₂ plane. Overall, both space restriction and nonuniform sampling lead to considerable reduction of the number of samples required for the surrogate model setup as demonstrated in Sect. 4.3.3.

Fig. 4.23

Design space sampling: (a) uniform sampling in the entire space, (b) uniform sampling restricted to the region X_s, and (c) nonuniform sampling in X_s. The solid line denotes the piecewise linear path connecting the reference designs (Koziel and Bekasiewicz 2016)

The surrogate model is constructed using kriging interpolation (Queipo et al. 2005). As mentioned before, only the characteristic points (as described in Sect. 4.3.1) are being modeled. This means that the surrogate cannot be used to restore the entire coupler response upon evaluating the models. However, it contains sufficient information to carry out the design optimization process (in particular, information about the center frequency, corresponding levels of matching and isolation, bandwidth, as well as power split).

4.3.3 Numerical Verification and Application Case Studies

The training set for constructing the surrogate model contains 1000 samples allocated according to the procedure of Sects. 3.2 and 3.3 with l = [0.11 2.4 2.0 0.18 0.25 3.8 3.8]^T and u = [0.6 4.2 3.4 0.28 0.68 11.5 8.3]^T, which contains the coupler designs corresponding to the wide range of operating frequencies from 0.5 to 2.0 GHz. The scaling vector p determining the data sampling nonuniformity (cf. Sect. 4.3.2) is p = [1.4 1.2 1.6 1.1 2.2 1.3 1.5]^T.

Table 4.5 shows the relative least squares error ‖s(x) – f(x)‖/‖f(x)‖ (averaged for 100 random test designs; s stands for the surrogate model) for the surrogate model described in this section and the standard kriging interpolation of the S-parameter responses. It can be observed that the sampling scheme utilized here has a major impact on the model accuracy and that the proposed model outperforms the conventional one in terms of the predictive power.

Table 4.5 Modeling results of compact microstrip rat-race

As explained earlier, the main purpose of the surrogate model considered here is to expedite the design process. For the sake of illustration, the surrogate model was utilized to design the RRC of Fig. 4.19 for two different scenarios: (i) widening the −20 dB bandwidth for matching ∣S₁₁∣ and isolation ∣S₄₁∣ and (ii) reducing the coupler size while maintaining the minimum level of ∣S₁₁∣ and ∣S₄₁∣ below −20 dB at the operating frequency. In both cases, equal power split (i.e., ∣S₂₁∣  =  ∣S₃₁∣) at the operating frequency is also requested. Figures 4.24, 4.25, and 4.26 show the designs obtained by optimizing the surrogate model under the above scenarios for three different operating frequencies of 0.9 GHz, 1.1 GHz, and 1.5 GHz, respectively. Note that all the designs are obtained by direct optimization of the surrogate model with no further corrections necessary.

Fig. 4.24

RRC designs obtained using the feature-based surrogate model for the operating frequency 0.9 GHz: (a) bandwidth enhancement (BW = 137 MHz, size 520 mm²) and (b) size minimization (BW = 67 MHz, size 482 mm²). ∣S₁₁∣, ∣S₂₁∣, ∣S₃₁∣, and ∣S₄₁∣ are marked using (–––), (– –), (– ∙ –), and (∙ ∙ ∙), respectively (Koziel and Bekasiewicz 2016)

Fig. 4.25

RRC designs obtained using the feature-based surrogate model for the operating frequency 1.1 GHz: (a) bandwidth enhancement (BW = 231 MHz, size 407 mm²) and (b) size minimization (BW = 75 MHz, size 378 mm²). ∣S₁₁∣, ∣S₂₁∣, ∣S₃₁∣, and ∣S₄₁∣ are marked using (–––), (– –), (– ∙ –), and (∙ ∙ ∙), respectively (Koziel and Bekasiewicz 2016)

Fig. 4.26

RRC designs obtained using the feature-based surrogate model for the operating frequency 1.5 GHz: (a) bandwidth enhancement (BW = 275 MHz, size 299 mm²) and (b) size minimization (BW = 136 MHz, size 267 mm²). ∣S₁₁∣, ∣S₂₁∣, ∣S₃₁∣, and ∣S₄₁∣ are marked using (–––), (– –), (– ∙ –), and (∙ ∙ ∙), respectively (Koziel and Bekasiewicz 2016)