Performance-Driven Yield Optimization of High-Frequency Structures by Kriging Surrogates

: Uncertainty quantiﬁcation is an important aspect of engineering design, as manufacturing tolerances may affect the characteristics of the structure. Therefore, the quantiﬁcation of these effects is indispensable for an adequate assessment of design quality. Toward this end, statistical analysis is performed, for reliability reasons, using full-wave electromagnetic (EM) simulations. Still, the computational expenditures associated with EM-driven statistical analysis often turn out to be unendurable. Recently, a performance-driven modeling technique has been proposed that may be employed for uncertainty quantiﬁcation purposes and can enable circumventing the aforementioned difﬁculties. Capitalizing on this idea, this paper discusses a procedure for fast and simple surrogate-based yield optimization of high-frequency structures. The main concept of the approach is a tailored deﬁnition of the surrogate domain, which is based on a couple of pre-optimized designs that reﬂect the directions featuring maximum variability of the circuit responses with respect to its dimensions. A compact size of such a domain allows for the construction of an accurate metamodel therein using moderate numbers of training samples, and subsequently, it is employyed to enhance the yield. The implementation details are dedicated to a particular type of device. Results obtained for a ring-slot antenna and a miniaturized rat-race coupler imply that the cost of yield optimization process can be reduced to few dozens of EM analyses.


Introduction
High-frequency systems are normally designed in the nominal sense, with possible deviations of geometry and material parameters (e.g., due to fabrication inaccuracies) being generally neglected. Yet, uncertainties may have detrimental effects on the performance. As a consequence, it is important to develop procedures for their evaluation. Among possible types of uncertainties, those pertinent to imperfect manufacturing [1,2] play the most important role in practice. They are stochastic, which makes statistical analysis imperative for their evaluation [3][4][5]. Diminishing the impact of parameter deviations requires a maximization of appropriately defined figures of merit such as response variance or the yield [6][7][8]. In the case of high-frequency systems, yield seems to be more suitable because performance specifications are often expressed in a minimax form, particularly for lower/upper acceptance thresholds for S-parameters, etc. [9][10][11][12].
Reducing the effects of manufacturing tolerances is even more essential than their evaluation. The relevant procedures are often referred to as robust design (also yielddriven design, tolerance-aware optimization, etc.) [1,[44][45][46][47]. In practical terms, this can be accomplished by improving the statistical merit functions of choice, such as the yield. Unfortunately, yield maximization is a CPU-heavy task to the extent of being prohibitive when directly executed at the level of EM simulation models. Surrogate-assisted procedures offer viable workarounds [6,[24][25][26][27][28]. Some of the most popular modeling methods utilized in this context are polynomial approximations [24], space mapping [48], NNs [49], and PCE [50]. As for statistical analysis, the bottleneck is a potentially high cost of the surrogate model setup, related to the dimensionality of the parameter space and parameter ranges. A partial mitigation has been offered by sequential approximation optimization (SAO) [51], where the metamodel is rendered along the optimization path, within the limited-volume domains centered at the current design produced by the robust design procedure. Another option is to employ response-feature technology [52]. This method capitalizes on the reduced nonlinearity of the relationship between the appropriately selected characteristic points of the system's responses and geometry parameters of the structure under design. The latter enables the construction of accurate metamodels using small training datasets [53].
In this work, we discuss a method for reduced-expense yield maximization of antenna and microwave components. Our approach involves the recently introduced performancedriven (or constrained) modeling [54][55][56], which addresses the surrogate construction task from the perspective of the model domain. More specifically, the model is only rendered in the vicinity of the region containing high-quality designs, which prevents wasting computational resources in parameter space regions containing uninteresting designs. This approach enables the construction of reliable metamodels over broad ranges of parameters and operating conditions using limited numbers of training data points [57][58][59]. Here, this paradigm is employed to construct surrogate models for robust design purposes. In particular, the domain of such a surrogate is extended along the directions possessing a major impact on the system's yield (the directions are found using a separate optimization sub-problems) and restricted along the remaining directions. Low domain volumes allow us to construct reliable surrogates, which are sufficient for conducting the yield maximization process without the necessity to rebuild the model. Our methodology is demonstrated using a ring-slot antenna and a microstrip coupler. In both cases, the robust design process is accomplished at a cost corresponding to less than a hundred of EM simulations. Reliability is corroborated by using EM-driven Monte Carlo analysis.

Yield Optimization Problem and Benchmark Algorithms
This section recalls the yield optimization problem statement, illustrated using two specific cases, multi-band antennas, and equal power split coupler. Subsequently, two basic state-of-the-art surrogate-assisted yield optimization algorithms are described. These will be used as benchmark methods in Section 4.

Yield Optimization Problem
In antenna and microwave designs, performance requirements are frequently formulated in a minimax form, i.e., by setting upper/lower acceptance levels on the electrical characteristics of the device at hand. These may include maximum in-band reflection (antennas) or maximum power split error within the operating band (couplers) [9]. As a consequence, a commonly utilized figure of merit is the yield [4], i.e., the percentage of designs for which the target specifications are met given the assumed deviations of the parameters. Figures 1 and 2 show two examples of the design optimization tasks of high-frequency devices: a multi-band antenna and a microwave coupler. The figures provide the design specifications, as well as the formulations of the respective objective functions. For the antenna, the nominal design has to ensure the best possible matching within the operating bands of interest, whereas in the case of the coupler, the aim is to ensure equal power split and to improve the bandwidth. Typically, the nominal designs serve as a starting point for yield optimization.  The deviations dx (e.g., manufacturing tolerances) are characterized by the specified probability distributions (e.g., uniform of maximum deviation δ max or joint Gaussian N(0,σ)). The deviations may be correlated [60], yet we assume here that they are statistically independent. Yield Y(x) at a certain the design x may be assessed by using Monte Carlo analysis.
In (1), . . , p, denote the observables, dx (k) are the random deviations, and H(x) is given by the following.
We employ the following formulation of the yield optimization task.
Commonly, nominal design x (0) is used as a starting point for solving (3), which, in turn, is rendered by solving, e.g., problem (E1.2) (for the example of the multi-band antenna) or (E2.2) (for the example of the microwave coupler).

Yield Optimization. Benchmark Surrogate-Assisted Algorithms
This section delineates the benchmark surrogate-assisted techniques: one-shot approach (Algorithm 1) and sequential approximate optimization SAO (Algorithm 2), the main features of which are juxtaposed in Table 1. Algorithms 1 and 2 exploit kriging datadriven surrogates [61], yet the actual choice of metamodeling technique is of secondary importance (other possibilities are, e.g., RBF [62] or PCE [29]). Table 1. Surrogate-assisted yield optimization: one-shot approach and sequential approximate optimization.

Method
One-shot approach Sequential approximate optimization Solving optimization task Solve a single task x * = argmin{x ∈ X S : -Y(x)} within the surrogate domain X S Render series x (i) , i = 0, 1, . . . , of approximations to x * by solving

Pros
Simple to apply Reduced cost of setting up the surrogate (smaller domain)

Cons Expected high cost of surrogate construction in a larger domain
Iterative process involving domain relocation and constructing several surrogates # δ max -maximum deviation for uniform distribution (or 3σ for Gaussian distribution of variance σ).
The overall idea of Algorithm 1 is to build a single surrogate for which its domain is spread over an ample neighborhood of the nominal design solution, within which the yield may be estimated in a reliable manner. This method is unsophisticated; however, the training data acquisition cost may be sizeable (proportional to the domain size). On the other hand, in Algorithm 2, yield estimation is replaced by an iterative process. Here, the aim is to set up the metamodel over a domain of a lower size, and then reposition it from iteration to iteration. Consequently, the CPU cost of the surrogate setup is reduced (as compared to Algorithm 1); however, the optimization process typically takes several iterations to converge.

Surrogate-Based Yield Optimization with Domain Confinement
This section discusses the main components of the considered optimization framework that preserves the simplicity of the one-shot approach (Algorithm 1), while keeping the surrogate setup cost at a reasonable level. This is achieved by exploiting the performancedriven modeling paradigm [54,55]. The employed procedure for locating the directions that span the metamodel domain is introduced in the context of multi-band antenna designs (Section 3.1) and microwave coupler designs (Section 3.2), along with the respective domain definitions. These two methods share the same underlying idea, yet they are tailored to the particular sets of performance specifications, as delineated in Section 2.1.

Yield Optimization of Multi-Band Antennas
The directions that affect the antenna characteristics to the highest degree are selected by pre-optimizing two supplementary designs: (i) the design that maximizes the antenna symmetrical fractional bandwidths, and (ii) the design that minimizes the antenna reflection at f 0.k , k = 1, . . . , N, (i.e., the resonant frequencies). These designs are rendered by solving the following [63].
x (1) = argmin . . , N, (symmetric portion of the bandwidth), whereas f 1k and f 2k denote the frequencies for which |S 11 | assumes -10 dB level (the lower and higher frequencies around the kth resonance). The trustregion gradient algorithm [64] is employed to solve problems (4) and (5), and the antenna response sensitivities are updated by applying the Broyden formula [65]. Consequently, the optimization cost equals approximately 1.5n EM analyses (n being the number of designable parameters). Figure 3 describes surrogate domain X S and its establishment with the use of the reference designs, whereas Figure 4 illustrates this process graphically. The surrogate domain X S is small; still, it encompasses the most consequential directions of antenna response variations that directly affect the yield, thereby allowing for significant cost reductions. The yield is optimized directly by solving (3) (i.e., the surrogate is used instead of EM simulations), similarly as in Algorithm 1.   (1) , and best matching (at f 0 ) design x (2) . These designs determine the directions of the most significant response changes (from the point of view of the target operating bandwidth); (b) The reference designs x (0) through x (2) form a path (a parameterized curve s(t)). The union of intervals S(t) (cf. (15)) form the surrogate model domain X S .

Yield Optimization of Microwave Couplers
In this case, the design requirements include the power split error and the bandwidth.  [66].
As before, we also have two supplementary designs x (1) and x (2) yielded by solving the following.
x (2) = argmin Both (7) and (8) are subject to ||xx (0) || ≤ D (D being selected by the user; here, we set D = 0.5 mm). Solving (7) and (8) requires merely n EM analyses of the coupler (it is the cost of estimating J F in (6)). These designs serve to find the directions corresponding to the maximal variations of the coupler power split and bandwidth, respectively. Figure 5 provides a brief description of the surrogate domain definition using these reference designs, whereas a graphical illustration of these procedure is provided in Figure 6.  Surrogate domain X S is small volume-wise, yet it is ample enough to encompass the directions of essential changes of the coupler responses affecting its yield. As in the previous case (Section 3.1), a small volume of the domain allows a significant reduction in the cost of setting up the surrogate, and also-due to the mentioned coverage of important directions-there is no need to iterate the entire process (as in Algorithm 2). The yield optimization process is conducted by solving (3), i.e., similarly as in the algorithm of Section 3.1.

Demonstration Case Studies
This section provides the results obtained using the algorithms of Sections 3.1 and 3.2 using a ring-slot antenna, and a miniaturized rat-race coupler as verification structures. The procedure is benchmarked against the surrogate-assisted approaches of Section 2. In order to verify the reliability of the considered methodology, a Monte Carlo analysis has been executed at the nominal design and the yield maximizing design. Figure 7 shows our first verification structure: a ring slot antenna, for which its circular ground plane featuring a slot with defected ground structure is excited through a microstrip line [67], whereas all the pertinent details are provided in Figure 8. We assumed independent uniformly distributed parameter deviations with maximum deviation δ max = 0.05 mm. The yield optimization results are provided in Table 2 for the algorithm of Section 3.1 and the benchmark surrogate-assisted procedures of Table 1. For the discussed approach, the size of the metamodel domain was set to δ c.k = 2δ max (see Figure 3); the training data set for constructing the surrogate contained 35 samples; and its relative RMS error equals 0.5%.   The details pertaining to the benchmark procedures are as follows. For Algorithm 1, the metamodel of relative RMS error 0.7% was constructed using 400 samples within the domain of size 10δ max . For Algorithm 2, the surrogate models were built using 50 training data samples over the domain of size 3d max . Here, the first metamodel (of the domain focused around x (0) ) featured a relative RMS error of 0.4%. In both cases, the aim was to render the metamodels of similar accuracy to less than one percent to ensure the reliability of the yield estimation. The following yield maximizing design w rendered x * = [20.18 6.43 0.21 11.85 2.95 6.78 7.90 2.31] T . Figure 9 visualizes the results of Monte Carlo analysis at x (0) and at x * (carried out with the use of 500 samples). The results of Table 2 may be summarized as follows. The metamodel domain confinement according to the methodology described in Section 3.1. results in dramatic cost savings, which is mainly due to the decreased volume size. Still, as the domain is spanned over the significant directions of the design space (those representing the maximum variations of the antenna response), the yield optimization process may successfully be concluded in one stage.

Case IV: Compact Microstrip Rat-Race Coupler
The second verification structure is a miniaturized microstrip rat-race coupler (RRC) presented in Figure 10 and described in Figure 11 [68]. The compact size of the circuit is a result of folding of the transmission lines that constitute ts interior. Figure 10. Layout of the miniaturized folded rat-race coupler [68]; the numbered circles (1 through 4) mark the structure ports. Parameter deviations are assumed to be independently and uniformly distributed with the maximum deviation of δ max = 0.05 mm. For the discussed algorithm, the metamodel of the relative RMS error of 2.3% was set up within a domain of size δ c.k = 2δ max , and the training data set comprised 72 training samples. In the case of Algorithm 1, the surrogate was constructed with 400 samples allocated over the domain of size 10δ max , and the relative RMS error was equal to 3.4%. For Algorithm 2, the first metamodel set up with 50 samples within the domain encompassing x (0) featured a relative RMS error of 2.2% (the size-defining parameter has been set to 3δ max ). Table 3 provides the relevant results for the discussed and benchmark algorithms. The following optimal design was yielded: x * = [4.65 11.10 21.87 0.71 0.95 0.81] T . Table 3. Yield optimization of the compact coupler of Figure 10. The results of Monte Carlo analysis at the initial and optimal designs are shown in Figure 12 (500 uniformly distributed samples were used with δ max = 0.05 mm). In this case, the discussed approach and Algorithm 2 were able to estimate the yield in a reliable manner. As for Algorithm 1, the values of the yield predicted using the surrogate and EM simulations differed considerably. This is because the surrogate rendered across a domain of a larger size featured worse predictive power.
As in the previous case, it has been verified that the yield optimization within a confined domain enables the obtainment of the benefits of both benchmark procedures. Thus, the entire process may be concluded in one stage due to spanning the domain along the most consequential directions (as opposed to the iterative Algorithm 2). Moreover, as the domain is of reduced size, it was possible to construct the surrogate at a remarkably low cost. The computational savings reached up to 82 and 64 percent (the discussed procedure versus Algorithms 1 and 2, respectively). At the same time, high design quality was maintained in all cases, and the value of the optimized yield is similar for all compared algorithms. Furthermore, the obtained results agree with those of EM-based Monte Carlo analysis. As in the previous case, it has been verified that the yield optimization within a confined domain enables the obtainment of the benefits of both benchmark procedures. Thus, the entire process may be concluded in one stage due to spanning the domain along the most consequential directions (as opposed to the iterative Algorithm 2). Moreover, as the domain is of reduced size, it was possible to construct the surrogate at a remarkably low cost. The computational savings reached up to 82 and 64 percent (the discussed procedure versus Algorithms 1 and 2, respectively). At the same time, high design quality was maintained in all cases, and the value of the optimized yield is similar for all compared algorithms. Furthermore, the obtained results agree with those of EM-based Monte Carlo analysis.

Conclusions
This work discussed the recent techniques for surrogate-based yield maximization of high-frequency components. The main ingredients of the optimization framework are domain-confined metamodels constructed by keeping in mind the parameter-space directions that are the most influential in terms of affecting the system's performance. By maintaining the low overall volume of the domain, it is possible to construct reliable surrogates using a limited number of training data samples, and it is possible to carry out the yield optimization process without the need to rebuild the model. As demonstrated using two microstrip structures (an antenna and a rat-race coupler), the robust design process can be accomplished at remarkably low costs of a few dozens of EM simulations. Thorough benchmarking indicates significant savings, with >80 percent over reference surrogatebased approach and >60 percent over the SAO algorithm. At the same time, reliability has been corroborated using EM-driven Monte Carlo simulations. The optimization techniques considered in the paper offer improved computational efficiency over the state-ofthe-art methods without compromising reliability. They are applicable to a variety of an-

Conclusions
This work discussed the recent techniques for surrogate-based yield maximization of high-frequency components. The main ingredients of the optimization framework are domain-confined metamodels constructed by keeping in mind the parameter-space directions that are the most influential in terms of affecting the system's performance. By maintaining the low overall volume of the domain, it is possible to construct reliable surrogates using a limited number of training data samples, and it is possible to carry out the yield optimization process without the need to rebuild the model. As demonstrated using two microstrip structures (an antenna and a rat-race coupler), the robust design process can be accomplished at remarkably low costs of a few dozens of EM simulations. Thorough benchmarking indicates significant savings, with >80 percent over reference surrogatebased approach and >60 percent over the SAO algorithm. At the same time, reliability has been corroborated using EM-driven Monte Carlo simulations. The optimization techniques considered in the paper offer improved computational efficiency over the state-of-the-art methods without compromising reliability. They are applicable to a variety of antenna and microwave components, and they may be viewed as potential replacements of conventional (also surrogate-assisted) procedures, especially for more complex, e.g., higher-dimensional design scenarios. Since in the considered approach, the surrogate domain is spanned by the directions that influence yield values in the most significant manner, the generalization of the technique would require properly defining these directions so that they reflect the respective design requirements.
In general, the two specific approaches for constructing the domain of the surrogate model (extension along the one-dimensional curve for the antenna example, and along two vectors corresponding to the maximum changes of the considered performance figures for the coupler example) could be applied to either of the case studies. However, maintaining the domain extension's dimensionality as equal to the number of considered performance figures is recommended, which is to provide sufficient room for yield improvement within the domain.