Optimization Algorithms Developed in EOMC
Optimization of CPU-Intensive Structures
Contemporary engineering design is heavily based on computer simulation. Traditional optimization techniques directly utilize the simulated responses and possibly available derivatives to make the responses to satisfy given design specifications.
In many cases, the computational cost of such an optimization process may be prohibitive, especially for complex problems. This is not only because of the long simulation time for a single design, which, depending on the system complexity and required accuracy, may be as long as several hours or even a couple of days, but also because the number of simulations required to obtain convergence of the optimization algorithm is usually large (e.g., hundreds) for traditional direct methods.
On the other hand, sensitivity information is normally unavailable, whereas sensitivity estimation using, e.g., finite differences, may be inaccurate due to various reasons, such as discontinuity of the system response as a function of the design parameters, related to re-meshing the structure inside the simulator whenever the design variables are being modified. Even in cases, when using finite differences is numerically feasible, it is usually computationally too expensive.
Another problem is that objective functions may be quite nonlinear and very often one has to face multiple local optima, which normally requires restarting of the optimization process with different (e.g., randomly selected) initial designs. This, obviously, increases the computational cost of the optimization process.
All of these problems are especially visible in microwave engineering, one of the focus areas of EOMC, because microwave structures are normally evaluated through accurate but CPU intensive full-wave electromagnetic simulations.
Surrogate-Based Optimization for Microwave Engineering
One of EOMC's endeavors is the development of computationally-efficient design optimization algorithms for microwave/RF engineering. Of particular interest are the methods exploiting so-called surrogate-based optimization (SBO) principle . The main idea is that the direct optimization of the microwave structure with a CPU-intensive objective function is replaced by the iterative updating and re-optimization of a so-called surrogate model. The surrogate model is a simplified but computationally cheap and analytically tractable representation of the original structure subsequently called the fine model.
The most successfull SBO approach in microwave engineering is space mapping (SM) , . According to SM, the surrogate model is constructed using a physically-based and computationally cheap representation of the structure being optimized, so-called coarse model (typically, circuit equivalent), and auxiliary (usually linear) mappings. Some introductory material regarding space mapping as well as review of recent developments can be found in Koziel et al. 2008, Koziel et al. 2006 and Bandler et al. 2006.
EOMC is working on all aspects of space mapping optimization, including theory, algorithm development, robustness, convergence, surrogate model selection, and software implementation.
A well performing SBO procedure, in particular, space maping algorithm, requires only a few evaluations of a CPU-intensive structure being optimized to yield a satisfactory design. Most of the optimization burden is shifted to a computationally cheap surrogate model.
Illustration: Design optimization of the third-order Chebyshev bandpass filter . The fine model is implemented in Sonnet em:
The coarse model is a circuit equivalent implemented in Agilent ADS:
Space mapping algorithm requires four iterations (i.e., five electromagnetic simulations of the filter structure) to find the optimal design. The plot below shows the fine model responses corresponding to the initial design (dashed line) and the final design (solid line):
Space Mapping with Functional Approximation
Normally, space mapping requires fast coarse model (typically, equivalent circuit), however, EOMC works on alternative formulations of the technique that allows us to use SM even in cases when the computationally cheap coarse model is not available (e.g., antennas or waveguide structures). One possible way, explored in EOMC, is to build a coarse model using coarse-mesh EM simulations and available classical function approximation techniques such as kriging.
Illustration: The use of functional and physical surrogate modeling in microwave CAD can be illustrated using the example of the microstrip fed monopole antenna:
The fine model of the antenna is evaluated in CST Microwave Studio (simulation time about 2.5 hours). The design specifications are |S11| ≤ –10 dB for 3.1 GHz to 10.6 GHz. We want to optimize the antenna using space mapping, however, no equivalent-circuit coarse model is available in this case. Instead, we use a coarse-discretization CST model (evaluation time 2 minutes and 15 seconds). This model is still computationally too expensive to be used directly as a coarse model in the SM optimization process. Therefore, the coarse model is created in the neighbourhood of the starting point (here, the approximate optimum of the coarse-discretization model), using kriging interpolation of the coarse-discretization model data. The coarse model created this way is computationally cheap, easy to optimize, and yet retains the features of a physically-based model.
The plot below shows the fine (dotted line) and coarse (x) model responses at the initial design, as well as fine (solid line) and coarse (dashed line) model responses at the coarse model optimum. The next plot shows the fine model response at the final design obtained after three SM iterations with kriging-based surrogate model. The optimization cost is 245 evaluations of the coarse-discretization model (145 to get its optimized design and another 100 to set up the kriging surrogate) as well as 4 fine model evaluations (including evaluation at the initial design). Thus, the total cost corresponds to only 8 evaluations of the fine model!
Tuning Space Mapping
Tuning space mapping (TSM) is one of the latest developments in space mapping technology. TSM algorithms offer a remarkably fast design optimization with satisfactory results obtained after one or two iterations which amounts to just a few electromagnetic simulations of the optimized microwave structure. According to the TSM approach, the surrogate model’s role is taken by a so-called tuning model, which is constructed by introducing circuit-theory based components (e.g., capacitors, inductors or coupled-line models) into the fine model structure, and parameters of these circuit components are chosen to be tunable. The tuning model is updated and optimized with respect to the tuning parameters. With the optimal tuning parameters thus obtained, a calibration is needed to transform these tuning values into an appropriate modification of the design variables, which are then assigned to the fine model. The calibration process may involve analytical formulas or it may require an auxiliary model, typically a fast space mapping surrogate.
EOMC is working on various aspects of TSM, including automated implementation of TSM algorithms where all the interactions between various models involved in the optimization process is handled by the SMF system and does not require user intervention.
Illustration: Design optimization of the box-section Chebyshev microstrip bandpass filter. The fine model is simulated in Sonnet em. The tuning model is constructed by dividing the polygons corresponding to parameters L1 to L5 in the middle and inserting the tuning ports at the new cut edges as shown below:
Its S28P data file is then loaded into the S-parameter component in Agilent ADS. The circuit-theory coupled-line components and capacitor components are chosen to be the tuning elements and are inserted into each pair of tuning ports. The lengths of the imposed coupled-lines and the capacitances of the capacitors are assigned to be the tuning parameters:
The calibration model is implemented in ADS. It contains the same tuning elements as the tuning model. It basically mimics the division of the coupled-lines performed while preparing the tuning model. The calibration model also contains six (implicit) SM parameters that are be used in the calibration process:
The plots below show the coarse (dashed line) and fine (solid line) model response at the initial design, as well as the fine model response after just one TSM iteration.
Shape Preserving Response Prediction
EOMC is constantly working towards improving efficiency of simulation-based design optimization process. One of the recent developments is shape-preserving response prediction (SPRP). SPRP relies on a physically-based coarse model. The enhanced coarse model is a surrogate that is being optimized instead of the fine model. The surrogate coincides with the fine model at the starting point of any given iteration, and the surrogate model response change is generated based on the translation vectors of a set of characteristic points of the coarse model response. Because of these features, the surrogate model exhibits very good prediction capability and, therefore, permits efficient optimization of the fine model.
The SPRP concept is explained in the pictures below. Picture (a) shows the example coarse model response at the reference design x0 (solid line), the coarse model response at other design x (dotted line), characteristic points of both responses (circles and squares), and the translation vectors (short lines). Picture (b) shows the fine model response at x0 (solid line) and the predicted fine model response at x (dotted line) obtained using SPRP based on characteristic points of picture (a); characteristic points of the fine model response at x0 (circles) and the translation vectors (short lines) were used to find the characteristic points (squares) of the predicted fine model response; coarse model responses at x0 and at x are plotted using thin solid and dotted line, respectively.
Illustration: Optimization of the wideband bandstop filter. The fine model is simulated in FEKO:
The coarse model is implemented in Agilent ADS:
Six characteristic points are selected to set up SPRP surrogate model: two points for which |S21| = –3 dB, two points with |S21| = –20 dB, and the two local |S21| maxima. SPRP optimization process needs only three iterations (i.e., three EM simulation of the structure) to yield a very good design. The plot below shows the fine model (dashed line) and coarse model (thin dashed line) response at the initial design, and the optimized fine model response (solid line):
The algorithms and optimization methods developed in EOMC are exploiting SBO principles, in particular space mapping, as well as related technologies such as tuning  and adaptive response correction .
EOMC addresses all aspects of computationally efficient design optimization including the theory, the algorithm development, and the applications to EM-based design and optimization of RF/microwave components and circuits, as well as to design problems in other fields, particularly, aerospace engineering.
EOMC also works on the development of the user-friendly software implementing the state-of-the-art design optimization algorithms. Special emphasis is put on interfacing commercial EM/circuit simulators, which is a key step in making the optimization process fully automatic.
References and Selected Recent Publications