EM‐simulation‐driven design optimization of compact microwave structures using multi‐fidelity simulation models and adjoint sensitivities

Reliable design of miniaturized microwave structures requires utilization of full‐wave electromagnetic (EM) simulation models because other types of representations such as analytical or equivalent circuit models are of insufficient accuracy. This is primarily due to considerable cross‐coupling effects in tightly arranged layouts of compact circuits. Unfortunately, high computational cost of accurate EM analysis makes the dimension adjustment process challenging, particularly for traditional methods based on parameter sweeps, but also for conventional numerical optimization techniques. In this article, low‐cost simulation‐driven designs of compact structures were demonstrated using gradient search with adjoint sensitivities as well as multi‐fidelity EM simulation models. The optimization process was arranged sequentially, with the largest steps taken at the level of coarse‐discretization models. Subsequent fine tuning was realized with the models of higher fidelity. Switching between the models was realized by means of adaptively controlled termination conditions. This allowed for considerable reduction of the design cost compared with single‐level optimization. The approach was illustrated using a compact microstrip rat‐race coupler with two cases considered, that is, (i) bandwidth enhancement, and (ii) minimization of the structure size. In both cases, the optimization cost corresponded to a few high‐fidelity EM simulations of the coupler structure. © 2016 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:442–448, 2016.     

Rapid design and size reduction of microwave couplers using variable‐fidelity EM‐driven optimization

In this article, fast electromagnetic (EM) simulation‐driven design optimization of compact microwave couplers is addressed. The main focus is on explicit reduction of the circuit footprint. Our methodology relies on the penalty function approach, which allows us to minimize the circuit area while ensuring equal power split between the output ports and providing a sufficient bandwidth with respect to the return loss and isolation around the operating frequency. Computational efficiency of the design process is achieved by exploiting variable‐fidelity EM simulations, local response surface approximation models, as well as suitable response correction techniques for design tuning. The technique described in this work is demonstrated using two examples of compact rat‐race couplers. The size‐reduction‐oriented designs are compared with performance‐oriented ones to illustrate available design trade‐offs. Final design solutions of the former case illustrate ～92% of miniaturization for both coupler examples (with corresponding fractional bandwidths of 16%). Alternative design solutions pertaining to the latter case show a lesser size reduction (～90% for both examples), but present a much wider bandwidths (～25% for both couplers). The overall computational cost of the design procedure corresponds to about 20 and 10 high‐fidelity coupler simulations for the first and second design example, respectively. Numerical results are also validated experimentally. © 2015 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:27–35, 2016.     

Fast simulation‐driven feature‐based design optimization of compact dual‐band microstrip branch‐line coupler

Design of miniaturized microwave components is a challenging task. On one hand, due to considerable electromagnetic (EM) cross‐couplings in highly compressed layouts full‐wave EM analysis is necessary for accurate evaluation of the structure performance. Conversely, high‐fidelity EM simulation is computationally expensive so that automated determination of the structure dimensions may be prohibitive when using conventional numerical optimization routines. In this article, computationally efficient simulation‐driven design of a miniaturized dual‐band microstrip branch‐line coupler is presented. The optimization methodology relies on suitably extracted features of a highly nonlinear response of the coupler structure under design. The design objectives are formulated in terms of the feature point locations, and the optimization is carried out iteratively with the linear model of the features utilized as a fast predictor. The entire process is embedded in the trust‐region framework as convergence safeguard. Owing to only slightly nonlinear dependence of the features on the geometry parameters of the circuit at hand, the optimized design satisfying prescribed performance requirements is obtained at the low computational cost of only 24 high‐fidelity EM simulations of the structure. Experimental validation of the fabricated coupler prototype is also provided. © 2015 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:13–20, 2016.     

Accelerated multiobjective design of miniaturized microwave components by means of nested kriging surrogates

Design of microwave components is an inherently multiobjective task. Often, the objectives are at least partially conflicting and the designer has to work out a suitable compromise. In practice, generating the best possible trade‐off designs requires multiobjective optimization, which is a computationally demanding task. If the structure of interest is evaluated through full‐wave electromagnetic (EM) analysis, the employment of widely used population‐based metaheuristics algorithms may become prohibitive in computational terms. This is a common situation for miniaturized components, where considerable cross‐coupling effects make traditional representations (eg, network equivalents) grossly inaccurate. This article presents a framework for accelerated EM‐driven multiobjective design of compact microwave devices. It adopts a recently reported nested kriging methodology to identify the parameter space region containing the Pareto front and to render a fast surrogate, subsequently used to find the first approximation of the Pareto set. The final trade‐off designs are produced in a separate, surrogate‐assisted refinement process. Our approach is demonstrated using a three‐section impedance matching transformer designed for the best matching and the minimum footprint area. The Pareto set is generated at the cost of only a few hundred of high‐fidelity EM simulations of the transformer circuit despite a large number of geometry parameters involved.     

Inverse modeling for fast design optimization of small‐size rat‐race couplers incorporating compact cells

In the paper, a framework for computationally‐efficient design optimization of compact rat‐race couplers (RRCs) is discussed. A class of hybrid RRCs with variable operating conditions is investigated, whose size reduction is obtained by replacing ordinary transmission lines with compact microstrip resonant cells (CMRCs). Our approach employs a bottom‐up design strategy leading to the development of compact RRCs through rapid design optimization of its building blocks and a subsequent fine tuning to account for parasitic cross‐coupling effects. The fundamental component of the proposed method is an inverse CMRC surrogate model, covering a wide range of cell electrical parameters, and enabling a convenient adjustment of coupler bandwidth. Having the surrogate model established, it is possible to produce close‐to‐optimum CMRC dimensions at a negligible computational cost. The subsequent correction step requires only up to two electromagnetic simulations of the CMRC. The proposed method is demonstrated by designing an RRC for several operational bandwidths. Experimental results are also provided.     

Surrogate‐assisted EM‐driven miniaturization of wideband microwave couplers by means of co‐simulation low‐fidelity models

This article proposes a methodology for rapid design optimization of miniaturized wideband couplers. More specifically, a class of circuits is considered, in which conventional transmission lines are replaced by their abbreviated counterparts referred to as slow‐wave compact cells. Our focus is on explicit reduction of the structure size as well as on reducing the CPU cost of the design process. For the sake of computational feasibility, a surrogate‐based optimization paradigm involving a co‐simulation low‐fidelity model is used. The latter is a fundamental component of the proposed technique. The low‐fidelity model represents cascaded slow‐wave cells replacing the low‐impedance lines of the original coupler circuit. It is implemented in a circuit simulator (here, ADS) and consists of duplicated compact cell EM simulation data as well as circuit theory‐based feeding line models. Our primary optimization routine is a trust‐region‐embedded gradient search algorithm. To further reduce the design cost, the system response Jacobian is estimated at the level of the low‐fidelity model, which is sufficient due to good correlation between the low‐ and high‐fidelity models. The coupler is explicitly optimized for size reduction, whereas electrical performance parameters are controlled using a penalty function approach. The presented methodology is demonstrated through the design of a 1‐GHz wideband microstrip branch‐line coupler. Numerical results are supported by experimental validation of the fabricated coupler prototype.     

Efficient multi‐fidelity design optimization of microwave filters using adjoint sensitivity

A simple and robust algorithm for computationally efficient design optimization of microwave filters is presented. Our approach exploits a trust‐region (TR)‐based algorithm that utilizes linear approximation of the filter response obtained using adjoint sensitivity. The algorithm is sequentially executed on a family of electromagnetic (EM)‐simulated models of different fidelities, starting from a coarse‐discretization one, and ending at the original, high‐fidelity filter model to be optimized. Switching between the models is determined using suitably defined convergence criteria. This arrangement allows for substantial cost reduction of the initial stages of the optimization process without compromising the accuracy and resolution of the final design. The performance of our technique is illustrated through the design of a fifth‐order waveguide filter and a coupled iris waveguide filter. We also demonstrate that the multi‐fidelity approach allows for considerable computational savings compared to TR‐based optimization of the high‐fidelity EM model (also utilizing adjoint sensitivity). © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:178–183, 2015.     

Expedited simulation‐driven design optimization of UWB antennas by means of response features

In this work, a method for fast design optimization of broadband antennas is considered. The approach is based on a feature‐based optimization (FBO) concept where reflection characteristics of the structure at hand are formulated in terms of suitably defined feature points. Redefinition of the design problem allows for reducing the design optimization cost, because the dependence of feature point coordinates on antenna dimensions is less nonlinear than for the original frequency characteristics (here, S‐parameters). This results in faster convergence of the optimization algorithm. The cost of the design process is further reduced using variable‐fidelity electromagnetic (EM) simulation models. In case of UWB antennas, the feature points are defined, among others, as the levels of the reflection characteristic at its local in‐band maxima, as well as location of the frequency point which corresponds to acceptable reflection around the lower corner frequency within the UWB band. Also, the number of characteristic points depends on antenna topology and its dimensions. Performance of FBO‐based design optimization is demonstrated using two examples of planar UWB antennas. Moreover, the computational cost of the approach is compared with conventional optimization driven by a pattern search algorithm. Experimental validation of the numerical results is also provided.     

Neural inverse space mapping (NISM) optimization for EM‐based microwave design

We present neural inverse space mapping (NISM) optimization for electromagnetics‐based design of microwave structures. The inverse of the mapping from the fine to the coarse model parameter spaces is exploited for the first time in a space mapping algorithm. NISM optimization does not require up‐front EM simulations, multipoint parameter extraction, or frequency mapping. It employs a simple statistical parameter extraction procedure. The inverse of the mapping is approximated by a neural network whose generalization performance is controlled through a network growing strategy. We contrast our new algorithm with neural space mapping (NSM) optimization. © 2003 Wiley Periodicals, Inc. Int J RF and Microwave CAE 13: 136–147, 2003.     

Compact cell topology selection for size‐reduction‐oriented design of microstrip rat‐race couplers

In this work, we address the problem of compact cell topology selection for miniaturization of rat‐race couplers. The principal objective of the design process is to achieve the smallest possible footprint of the coupler, while maintaining the required levels of electrical parameters imposed on its components. Our approach permits identification of the minimum achievable coupler area, provided that the circuit is composed of a given compact cell and folded lines. This allows for the quantitative assessment of a set of considered cells with respect to the miniaturization capabilities they exhibit under certain design specifications. The proposed method is validated using 6 different cells with unified parameterization to identify the smallest rectangular‐like rat‐race coupler described by 2 design specifications. The obtained results attest that circuit topology and electrical parameters of the reference design are critical factors determining the final miniaturization rate. The proof‐of‐concept prototype devices occupy merely 8% of the conventional coupler area, while preserving fractional bandwidths (20% and 13.5%) of their conventional counterparts. The experimental results confirm the claims inferred from the numerical data.     

Analysis of circular polarization antenna design trade‐offs using low‐cost EM‐driven multiobjective optimization

Circular polarization (CP) antennas are vital components of modern communication systems. Their design involves handling several requirements such as low reflection and axial ratio (AR) within the frequency range of interest. Small size is an important criterion for antenna mobility which is normally achieved as a by‐product of performance‐oriented modifications of the structure topology. In this work, multiobjective optimization is used in order to identify and analyze design trade‐offs for miniaturized CP antenna including the antenna capability for maintaining small size while retaining acceptable levels of other performance figures. We use a population‐based metaheuristic algorithm to obtain a set of designs which represent the best attainable compromise between the imposed requirements. To maintain a low optimization cost, the algorithm is executed on a cheap approximation model and the results are further corrected to bring them to the EM model accuracy level. Here, the analysis is carried out for a planar CP antenna. Achievable size reduction of the considered structure—while maintaining acceptable performance—is around 11%. Antenna performance in terms of in‐band reflection and AR varies from ?14 to ?10 dB and from 1.3 to 3 dB, respectively. The numerical results are validated by measurements of fabricated antenna prototypes.     

Computational‐budget‐driven automated microwave design optimization using variable‐fidelity electromagnetic simulations

A robust technique for microwave design optimization is presented. It is based on variable‐fidelity electromagnetic (EM) simulations where the approximate optimum of the “coarser” model becomes an initial design for finding the optimum of the “finer” one. The algorithm automatically switches between the models of different fidelity taking into account the computational budget assumed for the design process. Additional mechanisms enhancing the algorithm include: frequency scaling to reduce the misalignment between the models of different fidelity, as well as the local response surface approximation to reduce the number of EM simulations. The presented technique is particularly suitable for problems where simulation‐driven design is the only option, for example, for wideband antennas and dielectric resonator filters. Our method is demonstrated using two filters and one antenna example. In all cases, the optimal design is obtained at a low computational cost corresponding to a few high‐fidelity simulations of the structure. © 2012 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2013.     

Numerically efficient algorithm for compact microwave device optimization with flexible sensitivity updating scheme

An efficient trust‐region algorithm with flexible sensitivity updating management scheme for electromagnetic (EM)‐driven design optimization of compact microwave components is proposed. During the optimization process, updating of selected columns of the circuit response Jacobian is performed using a rank‐one Broyden formula (BF) replacing finite differentiation (FD). The FD update is omitted for directions sufficiently well aligned with the recent design relocation. As the algorithm converges, the alignment threshold is gradually reduced so that the condition for using BF becomes less stringent. This allows for further reduction of the number of EM simulations involved in the optimization process. The presented flexible Jacobian update scheme allows for considerable reduction of the computational cost with only slight degradation of the design quality. Robustness of the presented algorithm is validated through multiple optimization runs from random initial designs. The verification experiments are conducted for a range of microwave components, including a compact microstrip coupler as well as a three‐section compact microwave resonant cell‐based impedance transformer. The effects of the alignment threshold value on the computational efficiency of the algorithm and the design quality are investigated. Significant savings reaching 50% as compared to the reference algorithm are demonstrated.     

Response correction techniques for surrogate‐based design optimization of microwave structures

Simulation‐based optimization has become an important design tool in microwave engineering. However, using electromagnetic (EM) solvers in the design process is a challenging task, primarily due to a high‐computational cost of an accurate EM simulation. In this article, we present a review of EM‐based design optimization techniques exploiting response‐corrected physically based low‐fidelity models. The surrogate models created through such a correction can be used to yield a reasonable approximation of the optimal design of the computationally expensive structure under consideration (high‐fidelity model). Several approaches using this idea are reviewed including output space mapping, manifold mapping, adaptive response correction, and shape‐preserving response prediction. A common feature of these methods is that they are easy to implement and computationally efficient. Application examples are provided. © 2011 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2012.     

Computationally efficient design closure of miniaturized impedance matching transformers using response features

In this article, we describe a procedure for reliable and computationally efficient design optimization of miniaturized impedance matching transformers. Our approach exploits a concept of feature‐based optimization (FBO). According to FBO, considerable reduction of the computational cost of the simulation‐driven design process can be achieved—compared to conventional methods—by reformulating given performance requirements (typically, minimization of reflection over a frequency range of interest) in terms of suitably defined response features. For impedance transformer circuits, the feature points are defined as local maxima of the reflection characteristic, as well as the points defining the ?20 dB bandwidth. As the feature point coordinates (i.e., their frequencies and levels) depend on the geometry parameters of the structure in less nonlinear manner than the original responses (S‐parameters versus frequency), the optimization algorithm exhibits faster convergence. Further reduction of the optimization cost is obtained by utilization of variable‐fidelity electromagnetic simulations. Our technique is demonstrated using two design cases of an example miniaturized three‐section 50‐to‐100 ohm microstrip transformer. © 2016 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:396–401, 2016.     

Surrogate modeling of impedance matching transformers by means of variable‐fidelity electromagnetic simulations and nested cokriging

Accurate performance evaluation of microwave components can be carried out using full‐wave electromagnetic (EM) simulation tools, routinely employed for circuit verification but also in the design process itself. Unfortunately, the computational cost of EM‐driven design may be high. This is especially pertinent to tasks entailing considerable number of simulations (eg, parametric optimization, statistical analysis). A possible way of alleviating these difficulties is utilization of fast replacement models, also referred to as surrogates. Notwithstanding, conventional modeling methods exhibit serious limitations when it comes to handling microwave components. The principal challenges include large number of geometry and material parameters, highly nonlinear characteristics, as well as the necessity of covering wide ranges of operating conditions. The latter is mandatory from the point of view of the surrogate model utility. This article presents a novel modeling approach that incorporates variable‐fidelity EM simulations into the recently reported nested kriging framework. A combination of domain confinement due to nested kriging, and low‐/high‐fidelity EM data blending through cokriging, enables the construction of reliable surrogates at a fraction of cost required by single‐fidelity nested kriging. Our technique is validated using a three‐section miniaturized impedance matching transformer with its surrogate model rendered over wide range of operating frequencies. Comprehensive benchmarking demonstrates superiority of the proposed method over both conventional models and nested kriging.     

Modeling and optimization of microwave filter by ADS‐based KBNN

When microwave devices are designed by knowledge‐based neural network (KBNN), the empirical formula is always used as priori knowledge. However, it is difficult to derive the corresponding formulas for the most electromagnetic problems, especially for complex electromagnetic problems, the formula derivation is almost impossible. In this article, they combine neural network with simulation software and use results of Agilent ADS as priori knowledge and HFSS as teaching signal to train the neural network by particle swarm optimization (PSO), which solves the difficulty in obtaining priori knowledge and effectively reduces the complexity of the neural network structure. Based on the KBNN, the microwave filters are designed. The results of optimization satisfy the required specifications which show the effectiveness and superiority of the method.     

Simulation‐based optimization for rigorous assessment of ground plane modifications in compact UWB antenna design

Introducing ground plane modifications is a popular approach in the design of compact UWB antennas. Yet, specific topological alterations are normally reported on case to case basis without thorough investigations concerning their general suitability for antenna miniaturization. In particular, detailed performance comparison of different ground plane modifications is lacking in the literature. In this article, the effect of selected ground plane modifications on achievable miniaturization rate is considered based on a set of four UWB antennas. EM‐driven optimization is carried out to minimize the antenna footprints while maintaining acceptable matching within the UWB frequency range. In each case, all geometry parameters of the respective structures are utilized in the design process. For the sake of fair comparison, all antennas are implemented on the same dielectric substrate. Our results indicate a clear performance pattern, here, an advantage of the elliptical ground plane slit below the feed line over the rectangular one (average size reduction ratio of 26% versus 19% across the benchmark set). Our conjectures are confirmed by physical measurements of the fabricated antenna prototypes.     

Improved trust‐region gradient‐search algorithm for accelerated optimization of wideband antenna input characteristics

Full‐wave electromagnetic (EM) simulation models are ubiquitous in carrying out design closure of antenna structures. Yet, EM‐based design is expensive due to a large number of analyses necessary to yield an optimized design. Computational savings can be achieved using, for example, adjoint sensitivities, surrogate‐assisted procedures, design space dimensionality reduction, or similar sophisticated means. In this article, a simple modification of a rudimentary trust‐region‐embedded gradient search with numerical derivatives is proposed for reduced‐cost optimization of input characteristics of wideband antennas. The approach exploits information and history of relative changes of the design (as compared with the trust region size) during algorithm iterations to control the updates of components of the antenna response Jacobian, specifically, to execute them only if necessary. It is demonstrated that the proposed framework may lead to over 50% savings over the reference algorithm with only minor degradation of the design quality, specifically, up to 0.3 dB (or <3%). Numerical results are supported by experimental validation of the optimized antenna designs. The presented algorithm can be utilized as a stand‐alone optimization routine or as a building block of surrogate‐assisted procedures.     

Low‐cost multiband compact branch‐line coupler design using response features and automated EM model fidelity adjustment

Design closure of compact microwave components is a challenging problem because of significant electromagnetic (EM) cross‐couplings in densely arranged layouts. A separate issue is a large number of designable parameters resulting from replacement of conventional transmission line sections by compact microstrip resonant cells. This increases complexity of the design optimization problem and requires employment of expensive high‐fidelity EM analysis for reliable performance evaluation of the structure at hand. Consequently, neither conventional numerical optimization algorithms nor interactive approaches (e.g., experience‐driven parameters sweeps) are capable of identifying optimum designs in reasonable timeframes. Here, we discuss application of feature‐based optimization for fast design optimization of dual‐ and multiband compact couplers. On one hand, design of such components is difficult because of multiple objectives (achieving equal power split and good matching and port isolation for all frequency bands of interest). On the other hand, because of well‐defined shapes of the S‐parameter responses for this class of components, feature‐based optimization seems to be well suited to control multiple figures of interest as demonstrated in this work. Two‐level EM modeling is used for further design cost reduction. More importantly, we develop a procedure for automated determination of the low‐fidelity EM model coarseness that allows us to find the fastest possible model that still ensures sufficient correlation with its high‐fidelity counterpart, which is critical for robustness of the optimization process. Our approach is illustrated using two dual‐band compact couplers. Experimental validation is also provided.