24th Annual Review of Progress in Applied Computational Electromagnetics

March 30 - April 4, 2008 - Niagara Falls, Canada '2008 ACES

Neural Network Technique Simultaneously Optimizing Antenna Return Loss and Radiation Pattern Ethan K. Murphy1,2 and Vadim V. Yakovlev2 1

Department of Mathematics Colorado State University, Fort Collins, CO 80521 USA [email protected] 2

Department of Mathematical Sciences Worcester Polytechnic Institute, Worcester, MA 01609 USA [email protected] Abstract: This paper outlines a new algorithm of neural optimization for viable CAD of complex antenna structures. The input impedance (return loss) and radiation pattern are optimized with respect to the same design variables by a decomposed RBF neural network capable of dealing with various antennas accessible for 3D FDTD analysis. The algorithm features the dynamic generation of as much FDTD data as the network needs to find a solution satisfying the constraints. The performance of the optimization technique is illustrated by its application to a lens antenna constructed as an open-end circular waveguide inserted into a dielectric sphere. Keywords: FDTD analysis, lens antenna, neural network, optimization, radial basis function, radiation pattern, return loss. 1. Introduction The present significant interest in microwave (MW) optimization and computer-aided design (CAD) is strongly motivated by practical needs. In [1], we have given a comprehensive review of techniques of modern MW optimization and suggested the algorithm of radial basis function (RBF) network optimization backed by 3D FDTD simulation and suitable for viable CAD of complex systems. It has also been shown that this technique is characterized by excellent generalizing capabilities with the use of relatively small data sets and by the potential to find the “best” local optimum in the specified domain. In this paper, we further explore the resources of the FDTD-backed artificial neural network (ANN) approach [1] in efficient optimizing complex antennas. Existing optimization methods either use properties of particular antennas (and are thus applicable only to them, like in [2]), or feature general algorithms based on, e.g., ANNs [3], evolutionary algorithms [4, 5], space mapping [6], etc. In any case, the known techniques deal with optimization of either the input impedance (or return loss (RL)), or of a radiation pattern (RP). Physically, these characteristics usually depend on different parameters and may be in a conflicting relationship. One of the popular approaches to practical CAD of antennas is therefore based on optimization of the RL’s frequency response and subsequent computation of the related radiation properties; if the latter turns out to be acceptable, the design is considered optimal. This contribution presents a new ANN procedure which optimizes both RL and RP simultaneously. The algorithm follows the major concepts and techniques used in [1]. The decomposed RBF network operating in conjunction with dynamic generation of FDTD data is built in accordance with the objective function combining the ones formulated separately for RL and RP. The technique finds antenna parameters corresponding to an optimal (or best possible) characteristic of RL in a pre-specified

528

24th Annual Review of Progress in Applied Computational Electromagnetics

March 30 - April 4, 2008 - Niagara Falls, Canada '2008 ACES

Fig. 1. Architecture of a decomposed RBF ANN with η hidden neurons.

frequency range and an optimal (or best possible) characteristic of the RP in pre-specified angle intervals. The performance of our algorithm of multiobjective optimization is illustrated by its application to a lens antenna found to be convenient for testing the technique with minimal number of design variables. 2. Optimization Algorithm A. RBF Network Model We introduce the RBF ANN (Fig. 1) working with input vectors X i = [ x1 x 2 ... x n ] , where

x1 , ..., x n are system parameters (design variables), and output vectors

Yij = [ f (S11 , A)]ij =

c j S11 ( f j1 ≤ f ≤ f j2 ), X i , if 1 ≤ j ≤ ns11 c j A(φ c j A(φ

1 j − n s 11

1 j − n s 11

≤φ ≤φ

≤φ ≤φ

2 j − n s 11

2 j − n s 11

2

), X i , if ns11 + 1 ≤ j ≤ nmin + ns11 2

(1)

), X i , if ns11 + nmin + 1 ≤ j ≤ nmax + nmin + ns11 2

where i = 1, ..., P , Yij is a matrix of size P × ns11 + nmin + nmax that measures the objective of RL (the reflection coefficient S11) and the objective of RP (the field magnitude A, | Eθ | or | Eϕ | ), ns11 is the number of intervals we are interested with minimizing S11 in, nmin and nmax are the numbers of intervals we are interested in minimizing and maximizing the RP, respectively, P is the number of input-output pairs of FDTD modeling data. The constant vector c is used in general to weight each goal equally, but can be used to emphasize one goal over another. Lastly, the intervals for f and φ determine frequency and angle intervals of interest for S11 and A respectively. The RBF in our algorithm is chosen to be a local −2

(i) Gaussian function, i.e., ϕ l = exp( − rl X i − c l the algorithm are essentially the same as in [1].

2

) . The training of the network and other details of

B. Minimization Scheme In our analysis, certain forms of a frequency characteristic of S11 and of an angular characteristic of A are considered comprising a multiobjective function of the optimal design. The objective function is defined by Y in (1); possible forms of optimal regions are shown in Fig. 2. We consider a solution to be optimal if the curves in question fall completely within the optimality zones. It is supposed that for any allowable set of design variables, corresponding characteristics can be obtained with 3D FDTD analysis. With dynamic generation of the DB [1], we begin with a wittingly small data set. The procedure constructs an RBF network, finds its minimum, and we check if the solution does fit the criteria of being

529

24th Annual Review of Progress in Applied Computational Electromagnetics

March 30 - April 4, 2008 - Niagara Falls, Canada '2008 ACES

(a)

(b)

Fig. 2. The optimal zones that define the objective functions of RL (a) and RP (b).

optimal. If it does, the procedure stops; otherwise, we add the candidate minimum along with several more data points to the DB and repeat the process. We define each run through this process to be an iteration. The idea is to find the solution of the optimization problem with as little FDTD simulations as possible because these simulations for complex antennas may be computationally expensive. 3. Numerical Results and Discussion We consider an optimization of a lens antenna shown in Fig. 3. It is built as an open-end circular waveguide inserted into a dielectric sphere; the waveguide is completely filled with a dielectric of the same permittivity as of the sphere ε1 = ε1′– j0. This design is chosen for two reasons: (1) it is too complex to be represented by any empirical/analytical model, but is accessible for 3D FDTD analysis; (2) it is characterized by the distinct system parameters influencing mostly S11 (the sizes of the tuner, b1 and b2) and A (the depth of the waveguide insertion in the sphere, t) – this allows us to clearly illustrate functionality of the algorithm with minimum number of design variables. Looking for the best configuration of the antenna, we solve a two-parameter two-objective optimization problem for 3 ≤ b2 ≤ 11 mm and 5 ≤ t ≤ 40 mm. One objective is to minimize S11 in the range of 5.35 to 5.45 GHz, and another one is to minimize | Eθ | from 0o to 30o and from 160o to 180o, and

maximize | Eθ | from 75o to 115o. The tolerances Ti are set to T = [0.3 0.5 0.5 2]Τ . Numerical data for the neural network are generated by the full-wave 3D conformal FDTD simulator QuickWave-3D v. 6.5 [7]. In the model of the antenna, the mesh consists of 1.2-mm cells with the smallest cell of 0.65×0.6 mm around the coaxial feed in the yz-plane. The model contains 717,600 cells. Steady state is reached nearly within 8,000 time steps, and it takes about 4.4 min on a Xeon 3.2 GHz PC. The solution of the optimization problem took 21 iterations (after which the final DB size was 89) and 11 h of CPU time. The obtained RL and RP characteristics (the output of the 16th iteration) are presented in Fig. 4. They correspond to b2 = 8 mm and t = 5 mm. It is seen that we did not find a solution that strictly fulfill our requirements. However, it appears that the set of constraints chosen in this example specifies the domain where the optimal solution does not exist. Indeed, the optimization ended by alternating between two solutions, so it implies that our technique does approach the goal fairly well and has found the “best” possible solution. The above results therefore demonstrate a functionality of the new algorithm by showing simultaneous improvement in the return loss and the radiation pattern in accordance with their specified objective functions. The presented multiobjective optimization procedure may be very convenient in CAD of realistic complex antenna systems. 5. Acknowledgment This work was supported in part by the EADS Company Foundation. The authors are grateful to Vadim A. Kaloshin for a fruitful discussion of the design of the lens antenna.

530

24th Annual Review of Progress in Applied Computational Electromagnetics

March 30 - April 4, 2008 - Niagara Falls, Canada '2008 ACES

Fig. 3. Schematic view of the considered lens antenna; optimization is performed for ε1′ = 1.7; D1 = 20 mm, D2 = 60 mm, l = 2 mm, c = 20 mm, d = 5 mm, b1 = 30 mm, and two design variables b2 and t.

Fig. 4. Solution of both RL and RP optimizations; the non-optimized design represent the geometry for the design variables in the mid-points of their respective intervals.

6. References [1] E.K. Murphy and V.V. Yakovlev, “RBF network optimization of complex microwave systems represented by small FDTD modeling data sets,” IEEE Trans. Microwave Theory Tech., Vol. 54, No. 7, pp. 3069-3083, 2006. [2] N. Telzhensky and Y. Leviatan, “Planar differential elliptical UWB antenna optimization,” IEEE Trans. Antennas Propag., Vol. 54, No 11, pp. 3400-3406, 2006. [3] H.J. Degado, M.H. Thursby and F.M. Ham, “A novel neural network for the synthesis of antennas and microwave devices,” IEEE Trans. Neural Networks, Vol. 16, No. 6, pp. 1590-1600, 2005. [4] D. Boeringer and D. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag., Vol. 52, No. 3, pp. 771-779, 2004. [5] A. Hoorfar, “Evolutionary programming in electromagnetic optimization: a review,” IEEE Trans. Antennas Propagat., Vol. 55, No. 3, pp. 523-537, 2007. [6] J. Zhu, J.W. Bandler, N.K. Nikolova and S. Koziel, Antenna optimization through space mapping, IEEE Trans on Antennas Propag., Vol. 55, No. 3, pp. 651-658, 2007. [7] QuickWave-3DTM, QWED Sp. z o. o., ul. Nowowiejska 28 lok. 32 02-010 Warsaw, Poland, http: //www.qwed.com.pl/.

531

March 30 - April 4, 2008 - Niagara Falls, Canada '2008 ACES

Neural Network Technique Simultaneously Optimizing Antenna Return Loss and Radiation Pattern Ethan K. Murphy1,2 and Vadim V. Yakovlev2 1

Department of Mathematics Colorado State University, Fort Collins, CO 80521 USA [email protected] 2

Department of Mathematical Sciences Worcester Polytechnic Institute, Worcester, MA 01609 USA [email protected] Abstract: This paper outlines a new algorithm of neural optimization for viable CAD of complex antenna structures. The input impedance (return loss) and radiation pattern are optimized with respect to the same design variables by a decomposed RBF neural network capable of dealing with various antennas accessible for 3D FDTD analysis. The algorithm features the dynamic generation of as much FDTD data as the network needs to find a solution satisfying the constraints. The performance of the optimization technique is illustrated by its application to a lens antenna constructed as an open-end circular waveguide inserted into a dielectric sphere. Keywords: FDTD analysis, lens antenna, neural network, optimization, radial basis function, radiation pattern, return loss. 1. Introduction The present significant interest in microwave (MW) optimization and computer-aided design (CAD) is strongly motivated by practical needs. In [1], we have given a comprehensive review of techniques of modern MW optimization and suggested the algorithm of radial basis function (RBF) network optimization backed by 3D FDTD simulation and suitable for viable CAD of complex systems. It has also been shown that this technique is characterized by excellent generalizing capabilities with the use of relatively small data sets and by the potential to find the “best” local optimum in the specified domain. In this paper, we further explore the resources of the FDTD-backed artificial neural network (ANN) approach [1] in efficient optimizing complex antennas. Existing optimization methods either use properties of particular antennas (and are thus applicable only to them, like in [2]), or feature general algorithms based on, e.g., ANNs [3], evolutionary algorithms [4, 5], space mapping [6], etc. In any case, the known techniques deal with optimization of either the input impedance (or return loss (RL)), or of a radiation pattern (RP). Physically, these characteristics usually depend on different parameters and may be in a conflicting relationship. One of the popular approaches to practical CAD of antennas is therefore based on optimization of the RL’s frequency response and subsequent computation of the related radiation properties; if the latter turns out to be acceptable, the design is considered optimal. This contribution presents a new ANN procedure which optimizes both RL and RP simultaneously. The algorithm follows the major concepts and techniques used in [1]. The decomposed RBF network operating in conjunction with dynamic generation of FDTD data is built in accordance with the objective function combining the ones formulated separately for RL and RP. The technique finds antenna parameters corresponding to an optimal (or best possible) characteristic of RL in a pre-specified

528

24th Annual Review of Progress in Applied Computational Electromagnetics

March 30 - April 4, 2008 - Niagara Falls, Canada '2008 ACES

Fig. 1. Architecture of a decomposed RBF ANN with η hidden neurons.

frequency range and an optimal (or best possible) characteristic of the RP in pre-specified angle intervals. The performance of our algorithm of multiobjective optimization is illustrated by its application to a lens antenna found to be convenient for testing the technique with minimal number of design variables. 2. Optimization Algorithm A. RBF Network Model We introduce the RBF ANN (Fig. 1) working with input vectors X i = [ x1 x 2 ... x n ] , where

x1 , ..., x n are system parameters (design variables), and output vectors

Yij = [ f (S11 , A)]ij =

c j S11 ( f j1 ≤ f ≤ f j2 ), X i , if 1 ≤ j ≤ ns11 c j A(φ c j A(φ

1 j − n s 11

1 j − n s 11

≤φ ≤φ

≤φ ≤φ

2 j − n s 11

2 j − n s 11

2

), X i , if ns11 + 1 ≤ j ≤ nmin + ns11 2

(1)

), X i , if ns11 + nmin + 1 ≤ j ≤ nmax + nmin + ns11 2

where i = 1, ..., P , Yij is a matrix of size P × ns11 + nmin + nmax that measures the objective of RL (the reflection coefficient S11) and the objective of RP (the field magnitude A, | Eθ | or | Eϕ | ), ns11 is the number of intervals we are interested with minimizing S11 in, nmin and nmax are the numbers of intervals we are interested in minimizing and maximizing the RP, respectively, P is the number of input-output pairs of FDTD modeling data. The constant vector c is used in general to weight each goal equally, but can be used to emphasize one goal over another. Lastly, the intervals for f and φ determine frequency and angle intervals of interest for S11 and A respectively. The RBF in our algorithm is chosen to be a local −2

(i) Gaussian function, i.e., ϕ l = exp( − rl X i − c l the algorithm are essentially the same as in [1].

2

) . The training of the network and other details of

B. Minimization Scheme In our analysis, certain forms of a frequency characteristic of S11 and of an angular characteristic of A are considered comprising a multiobjective function of the optimal design. The objective function is defined by Y in (1); possible forms of optimal regions are shown in Fig. 2. We consider a solution to be optimal if the curves in question fall completely within the optimality zones. It is supposed that for any allowable set of design variables, corresponding characteristics can be obtained with 3D FDTD analysis. With dynamic generation of the DB [1], we begin with a wittingly small data set. The procedure constructs an RBF network, finds its minimum, and we check if the solution does fit the criteria of being

529

24th Annual Review of Progress in Applied Computational Electromagnetics

March 30 - April 4, 2008 - Niagara Falls, Canada '2008 ACES

(a)

(b)

Fig. 2. The optimal zones that define the objective functions of RL (a) and RP (b).

optimal. If it does, the procedure stops; otherwise, we add the candidate minimum along with several more data points to the DB and repeat the process. We define each run through this process to be an iteration. The idea is to find the solution of the optimization problem with as little FDTD simulations as possible because these simulations for complex antennas may be computationally expensive. 3. Numerical Results and Discussion We consider an optimization of a lens antenna shown in Fig. 3. It is built as an open-end circular waveguide inserted into a dielectric sphere; the waveguide is completely filled with a dielectric of the same permittivity as of the sphere ε1 = ε1′– j0. This design is chosen for two reasons: (1) it is too complex to be represented by any empirical/analytical model, but is accessible for 3D FDTD analysis; (2) it is characterized by the distinct system parameters influencing mostly S11 (the sizes of the tuner, b1 and b2) and A (the depth of the waveguide insertion in the sphere, t) – this allows us to clearly illustrate functionality of the algorithm with minimum number of design variables. Looking for the best configuration of the antenna, we solve a two-parameter two-objective optimization problem for 3 ≤ b2 ≤ 11 mm and 5 ≤ t ≤ 40 mm. One objective is to minimize S11 in the range of 5.35 to 5.45 GHz, and another one is to minimize | Eθ | from 0o to 30o and from 160o to 180o, and

maximize | Eθ | from 75o to 115o. The tolerances Ti are set to T = [0.3 0.5 0.5 2]Τ . Numerical data for the neural network are generated by the full-wave 3D conformal FDTD simulator QuickWave-3D v. 6.5 [7]. In the model of the antenna, the mesh consists of 1.2-mm cells with the smallest cell of 0.65×0.6 mm around the coaxial feed in the yz-plane. The model contains 717,600 cells. Steady state is reached nearly within 8,000 time steps, and it takes about 4.4 min on a Xeon 3.2 GHz PC. The solution of the optimization problem took 21 iterations (after which the final DB size was 89) and 11 h of CPU time. The obtained RL and RP characteristics (the output of the 16th iteration) are presented in Fig. 4. They correspond to b2 = 8 mm and t = 5 mm. It is seen that we did not find a solution that strictly fulfill our requirements. However, it appears that the set of constraints chosen in this example specifies the domain where the optimal solution does not exist. Indeed, the optimization ended by alternating between two solutions, so it implies that our technique does approach the goal fairly well and has found the “best” possible solution. The above results therefore demonstrate a functionality of the new algorithm by showing simultaneous improvement in the return loss and the radiation pattern in accordance with their specified objective functions. The presented multiobjective optimization procedure may be very convenient in CAD of realistic complex antenna systems. 5. Acknowledgment This work was supported in part by the EADS Company Foundation. The authors are grateful to Vadim A. Kaloshin for a fruitful discussion of the design of the lens antenna.

530

24th Annual Review of Progress in Applied Computational Electromagnetics

March 30 - April 4, 2008 - Niagara Falls, Canada '2008 ACES

Fig. 3. Schematic view of the considered lens antenna; optimization is performed for ε1′ = 1.7; D1 = 20 mm, D2 = 60 mm, l = 2 mm, c = 20 mm, d = 5 mm, b1 = 30 mm, and two design variables b2 and t.

Fig. 4. Solution of both RL and RP optimizations; the non-optimized design represent the geometry for the design variables in the mid-points of their respective intervals.

6. References [1] E.K. Murphy and V.V. Yakovlev, “RBF network optimization of complex microwave systems represented by small FDTD modeling data sets,” IEEE Trans. Microwave Theory Tech., Vol. 54, No. 7, pp. 3069-3083, 2006. [2] N. Telzhensky and Y. Leviatan, “Planar differential elliptical UWB antenna optimization,” IEEE Trans. Antennas Propag., Vol. 54, No 11, pp. 3400-3406, 2006. [3] H.J. Degado, M.H. Thursby and F.M. Ham, “A novel neural network for the synthesis of antennas and microwave devices,” IEEE Trans. Neural Networks, Vol. 16, No. 6, pp. 1590-1600, 2005. [4] D. Boeringer and D. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag., Vol. 52, No. 3, pp. 771-779, 2004. [5] A. Hoorfar, “Evolutionary programming in electromagnetic optimization: a review,” IEEE Trans. Antennas Propagat., Vol. 55, No. 3, pp. 523-537, 2007. [6] J. Zhu, J.W. Bandler, N.K. Nikolova and S. Koziel, Antenna optimization through space mapping, IEEE Trans on Antennas Propag., Vol. 55, No. 3, pp. 651-658, 2007. [7] QuickWave-3DTM, QWED Sp. z o. o., ul. Nowowiejska 28 lok. 32 02-010 Warsaw, Poland, http: //www.qwed.com.pl/.

531