FDTD-Backed RBF Neural Network Technique for Efficiency - CiteSeerX

Proceedings of the 9th AMPERE Conf. on Microwave & RF Heating, Loughborough, U.K., September 2003

FDTD-Backed RBF Neural Network Technique for Efficiency Optimization of Microwave Structures Ethan K. Murphy and Vadim V. Yakovlev Worcester Polytechnic Institute, Worcester, MA, USA Abstract A novel approach to computational microwave optimization is presented, namely, a neural network procedure controlling full-wave FDTD simulation of the structure. A radial-basis-function network is trained by geometrical parameters and the computed frequency responses of S-parameters. Computational efficiency of the procedure is illustrated for a waveguide T-junction with a post, a slotted waveguide radiating in free space, and a microwave oven with a sausage on the shelf. Introduction Although traditionally extensive experimentation was the major technique exploited in the development of microwave (MW) applicators, with appearance of new generation of modeling techniques, it has been recently realized that advanced computer simulation could make the design of the MW heating systems more intelligent and thoughtful, shorten the development time, and reduce the project’s cost. Modern modeling tools allow one to get valuable characteristics of the structure prior to constructing a physical prototype. However, routine analysis of the system may not result in many direct instructions for better design. The emerging practicability of inclusion of resourceful full-wave simulators in optimization and automated design of MW structures has been emphasized in [1], but this has not become a standard practice yet as traditionally considered unfeasible due to the high computational cost. Meanwhile, recent extraordinary growth of productivity and capabilities of computer hardware has made possible a comprehensive, fast and reliable numerical modeling of microwave circuits. Microwave Studio (MWS) by CST, the EM code based on Finite Integration Method, and QuickWave-3D (QW3D) by QWED, the conformal FDTD 3D electromagnetic (EM) simulator, have been recently identified among the most efficient full-wave simulators in the market [2]. The present paper proposes, for the first time, a simple yet efficient optimization technique based on artificial neural networks (NN) controlling 3D full-wave FDTD analysis performed by QW3D (http://www.qwed.com.pl). We show that, given the resources of today’s computers, such an approach can be reasonably productive and serve as a competent optimization tool in designing various MW systems. Background In this analysis we understand optimization of MW structure as traditional circuit optimization. While possible approaches to improving MW heating may include special findings in the applicator design, manipulation of the heat cycle, ingredient formulation, design of the package, etc. (see, e.g., [3, 4]), here we deal with parameters computable in the EM modeling. The use of time-domain simulation in design of practical systems has recently made it evident that the applicator’s efficiency (usually associated with energy coupling) can be controlled when one can compute a frequency characteristic of the magnitude of the reflection coefficient |S11| in the range adjacent to the operating frequency f0 [5]. So we interpret the problem of efficiency optimization as that of an appropriate optimization of the frequency response of |S11|. Neural networks were introduced in the computational electromagnetics in the 1990s, and, since then, their typical application has been associated with the networks representing (or directly imitating) the modeled devices and dealing with their 197

physical/electrical characteristics [6]. When developed appropriately, these models are convenient and accurate, but relate only to particular devices, so their usefulness is rather limited. We suggest using a NN technology with a FDTD full-wave simulator as a backing analytical tool: the network processes its input and output data and is capable of optimizing those MW systems to which the software is applicable. We state the optimization problem as follows: find a configuration of the structure such that a magnitude of an S-parameter under consideration is less (or larger) than the assigned level (S0) in the frequency range (f1, f2) around the operating frequency f0 (f1 < f0 < f2). Being a multivariable function of frequency f and n system parameters X = [X1 X2 … Xn]T, |Sij| is an objective function of the optimal design, and S0 and (f1, f2) are interpreted as the relevant constraints. Neural Model We have constructed the Radial Basis Function (RBF) network with the Gaussian activation function. Each sub-net in the NN architecture shown in Fig. 1 possesses n inputs in accordance with the number of the system parameters to be optimized and one output associated with the value of Sij(fk) obtained from the EM solver. The entire network consists of K distinct NNs corresponding to a particular frequency with K determined by the number of approximating points in (f1, f2). Frequency responses of S-parameters obtained in FDTD simulations compose the network database. In order to have the optimization procedure suitable for a variety of MW systems, we use uniform grid sampling which does not give preference to any particular sub-regions of the input space. Making the problem better conditioned for training and helping the network with learning process, linear scaling/de-scaling is applied to the input parameters. Network training is reduced to determination of the network weights such that the error between the actual and the desired NN outputs is minimized. The algorithm has been implemented in MATLAB 6 R12 environment. The master program controls the operation of QW3D, manages processing and transferring data, communicates with appropriate procedures, and conducts required computations. Examples of Optimization To illustrate applicability of the procedure to optimization of different MW structures, we present the results for a 80 x 40 mm waveguide-backed slotted element radiating in free space and a WR75 waveguide T-junction with a post (Fig. 2). In each case, the goal was to minimize |S11|, a function of frequency and three geometric parameters. Quality of network training was checked for different numbers of samples in the database’s training set. It was shown that in both problems training and testing errors were fairly low, even for a quite small number of samples. The optimal solutions obtained with the databases of different sizes are shown in Fig. 3. The goal function was minimized in the frequency range specified by f1 and f2 (whose values are marked in this figure), whereas the limiting value S0 was not explicitly assigned; rather, the characteristic was forced to be as small as possible within the range of (f1, f2). When applied to these structures, the procedure shows superior results in comparison with the RSM-SQP-algorithm developed in [5]: even trained with a 27-sample database, the RBF network generates an improved solution. Corresponding optimal configurations of these structures are given in Tables 1 and 2. In the next example (Fig. 4, a), we optimize the efficiency of much more complex scenario – the 800 W Daewoo microwave oven (a = 320 mm, b = 330 mm, c = 214 mm) excited by a rectangular waveguide (80 × 40 mm) and containing a sausage (complex permittivity 40 – i22) on a glass (6 – i0) shelf (t = 20mm). Assuming that the sausage is always shelf-centered, we look for the best geometry of the problem, 198

Fig. 1. Structure of the RBF NN

Fig. 2. Geometry of a slotted radiator (top) and a T-junction with a post (bottom)

(a)

(b)

Fig. 3. Optimized characteristics of |S11| for the slotted radiator (a) and the T-junction (b) obtained by the RBF-NN approach and the RSM-SQP procedure [5]. Table 1. Optimal Design of the Radiating Slotted Element (w = 8 mm & l = 64 mm) Geometry

θ, degrees S, mm D, mm

Table 2. Optimal Design of the T-Junction with a Post

Training samples 27 64 125 34.5 18.4 42.9 69.8 70.0 40.3 30.3 44.9 55.1

Geometry r, mm h, mm s, mm

Training samples 27 64 125 0.50 0.82 0.51 4.00 4.01 7.98 6.00 5.97 -2.36

i.e., a position (t, ϕ and h) and dimensions (w and l) of the load guaranteeing the 90% energy coupling (i.e., S0 = 0.3) for (f1, f2) = (2.4, 2.5 GHz). For the design variables, we accepted that 5 ≤ t ≤ 30 mm, 0 ≤ ϕ ≤ 180o, 10 ≤ h ≤ 60 mm, 15 ≤ w ≤ 50 mm, and 50 ≤ l ≤ 150 mm. Meaning to avoid a time-consuming 6-parameter optimization, we split the problem into 3 simpler phases. Firstly, the dimensions (l =70mm, w =20mm) and the angle (ϕ = 0) were fixed, and we found the best thickness to solving a 3-parameter optimization problem for (f, t, h). Then keeping the same l, w and to constant, 199

(a)

(b)

Fig. 4. Geometry of a MW oven with a sausage (a) and its non-optimized (curve A) and optimized |S11| characteristics for (f, t, h)- (B), (f, h, ϕ)- (C), and (f, l, w)-optimizations (D) (b). Table 3. Optimized Oven/Sausage System and CPU Time (Dual Pentium IV 1.8 GHz) Variables, Samples, Graph (f, t, h): (120 samples), B (f, h, ϕ): (248 samples), C (f, l, w): (142 samples), D

t, mm 21.6 21.6 21.6

h, mm 9.1 17.0 17.0

ϕ, deg. 0 67 67

l, mm 70.0 70.0 86.6

w, mm 20.0 20.0 22.9

Time, h 34 72 82

we addressed an optimization problem for (f, h, ϕ) to find the best position ho and ϕo. Finally, with the optimized values of to, ho and ϕo fixed, we determined the best dimensions lo and wo (optimization for (f, l, w)). The optimized configurations and the |S11| curves obtained for each step are presented in Table 3 and Fig. 4, b respectively. In case of complex multiparameter structures, the choice of optimization strategy may be up to the designer. It is worth noting that although a simplified “partial” approach is quicker it looks for local optima whereas the global optimization could be performed as the full 6-parameter optimization. The fact that curve D in Fig. 4, b does not satisfy the applied constraints (|S11| = 0.13 at f0 = 2.45 GHz, but is not less than 0.3 on the interval (f1, f2,)), does not necessarily result from the “partial” optimization: the sought solution may not exist in the considered spaces of the design variables. This may be the case in the last example dealing with strong narrow resonances natural for MW ovens; the procedure, however, still generates the best possible solution. Conclusions Our NN-based optimization technique backed by the full-wave FDTD simulation inherits a major NN capability of accurately approximating functions and is capable of handling systems with strong resonances. The presented examples demonstrate remarkable practical flexibility; efficiency optimization of the loaded microwave oven shows that it is directly applicable to applied structures of MW power engineering. References 1. J.W. Bandler, et al, Int. J. RF and Microwave CAE, 12 [1] 79-89 (2002). 2. V.V. Yakovlev, In: Advances in Microwave and Radio-Frequency Processing, M. Willert-Porada, Ed., Springer (2003), 176-188. 3. J.R. Banga, et al, Proc. 7th Conf. Microwave and HF Heating, Valencia, Spain, 1999, 193-196. 4. D.S. Lee, et al, Food Service Technology, 2 [2] 87-93 (2002). 5. V.A. Mechenova and V.V. Yakovlev, Abstract Book 3rd World Congress on Microwave & RF Processing, Sydney, Australia, Sept. 2002, M4A.24-M4A.25. 6. Q.J. Zhang and K.C. Gupta, Neural networks for RF and microwave design, Artech House (2000). 200