Design of Dielectric Resonator Antennas Using Surrogate Optimization

S. Koziel1 S. Ogurtsov2

1 Engineering Optimization & Modeling Center, School of Science and Engineering, Reykjavik University, Menntavegur 1, 101 Reykjavik, Iceland, e-mail: koziel@ru.is, tel.: +354 599 6376. 2 Engineering Optimization & Modeling Center, School of Science and Engineering, Reykjavik University, Menntavegur 1, 101 Reykjavik, Iceland, e-mail: stanislav@ru.is, tel.: +354 599 6539.

Abstract − Design optimization of dielectric resonator antennas (DRAs) is a challenging task that normally involves electromagnetic (EM) simulation. Automated simulation-based DRA design realized as an optimization problem with a discrete EM solver directly embedded into the optimization algorithm is often very time consuming due to computational demands of discrete high-fidelity simulations. Here, we demonstrate a computationally efficient optimization approach to DRA design that exploits space mapping as the optimization engine, and kriging interpolation, used to create a fast surrogate model of the DRA under design. The kriging model is set up using simulation data of the coarse-discretization EM model which is evaluated by the same solver as the one that simulates the high-fidelity model of the DRA. Two DRA cases are demonstrated; in each case the optimal design is found at the computational cost corresponding only to a few high-fidelity simulations of the antenna structure.

1 INTRODUCTION

Dielectric resonator antennas (DRAs) are interesting options for antennas installed at the printed circuit boards due their small footprint, high radiation efficiency, mechanical reliability, and wide temperature range of operation [1], [2].

In general DRA design involves time-demanding electromagnetic (EM) simulations. Initial designs can be figured out using simple models, and EM-analysis-based equations [2]. Typically these designs need further adjustment. Proper adjustment based on accurate (therefore, computationally expensive) discrete simulations can be challenging if the environment (feeding circuit, dielectric resonator fixture, antenna housing) has to be taken into account. Such impacts can only be properly captured through full-wave simulation.

In practice, DRA geometry is adjusted with a parametric study [2] what is tedious and does not ensure optimum results. Straightforward ways to automate the design process with the EM solver plugged into the optimization loop are—in most cases—impractical due to their high computational costs. Metaheuristic algorithms applied to simulation-driven antenna problems, e.g. [3] and [4], do not alleviate the issue of high computational costs of evaluation of the accurate antenna model, and these algorithms typically involve hundreds or even thousands of simulations of the antenna model.

Rapid simulation-driven design of DRAs can be realized using a surrogate-based optimization (SBO)

concept [5]. The most successful SBO approaches used in microwave engineering include space mapping (SM) [6]-[8], tuning [9] and tuning space mapping [7]. Unfortunately, their applicability to antenna design is limited. SM typically requires a fast circuit equivalent [6] which is not available for DRAs; tuning is not directly applicable to DRAs.

We discuss an efficient DRA design approach using coarse-discretization simulations; such models can be used—after suitable correction—as reliable surrogates [10]. Here, the correction is realized using SM with the coarse model implemented as kriging interpolation of the coarse model response: the latter cannot be used directly in the SM optimization because of being relatively expensive [10]. It is essential that the combination of SM, coarse-discretization EM simulations, and kriging allows us to carry out the DRA optimization at a low computational cost as demonstrated through two designs.

2 SURROGATE-BASED OPTIMIZATION USING SPACE MAPPING AND KRIGING-BASED COARSE MODELS

2.1 DRA Design Problem Formulation

Let Rf denote the response vector of a fine model of the DRA under design. Rf is obtained with high-fidelity simulation. Our goal is to solve

( )* arg min ( )f fU=x

x R x (1)

where U is an objective function, e.g., minimax. Since Rf is computationally expensive, a direct (e.g., gradient-based) optimization of U(Rf(x)) is hardly feasible.

2.2 Surrogate-Based Optimization. Space Mapping

SBO [6]-[8] does not handle the problem (1) directly. Instead, a sequence of approximate solutions to (1), x(i), i = 0, 1, 2, …, is generated as

( )( 1) ( )arg min ( )i isU+ =

xx R x (2)

Here, x(0) is the initial design; Rs(i) is a surrogate

representation of Rf, created using available fine model data, and iteratively updated.

Normally, Rf is evaluated once per iteration. The surrogate is assumed to be much faster than Rf. For a

978-1-61284-978-2/11/$26.00 ©2011 IEEE

136

well performing SBO algorithm, the number of iterations (2) necessary to find a satisfactory design is small [7] so that the total cost of the optimization process is low.

SM [7] uses the scheme (2) and constructs the surrogates based on the coarse model Rc. Let Rs be a generic SM surrogate model, i.e., Rc composed with suitable (usually linear) transformations. At the i-th iteration Rs

(i) is defined as ( ) ( )( ) ( , )i is s=R x R x p (3)

where ( ) ( ) ( )

0argmin || ( ) ( , ) ||ii k k

f sk== −∑p

p R x R x p (4) is a vector of the SM model parameters.

The popular SM models include the input SM [7], where Rs(x,p) = Rs(x,B,c) = Rs(B ⋅x + c), with the parameters B and c obtained by solving a parameter extraction problem (4); the output SM [7] with Rs(x,p) = Rs(x,d) = Rs(x) + d, where d is a correction term at the i-th iteration, d(i) = Rf(x(i)) – Rc(x(i)).

2.3 Kriging-Based Coarse Models

Coarse model Rc is a critical component of successful SM optimization. It should be physics-based, i.e., describing the same phenomena as Rf. Therefore, Rs configured from Rc is expected to have reliable prediction capability [6]. Also Rc should be much cheaper computationally than Rf so that the overhead due to numerous Rc runs, during optimization (2) and parameter extraction (4), would be negligible.

Because of the underlying EM phenomena the only candidate for Rc of the DRA is coarse-discretization EM model, hereafter Rcd. Normally, this kind of model is too expensive for the direct use in the SM algorithm because its numerous runs are necessary to solve (4); therefore, surrogate optimization (3) would contribute too much to the total design costs. As a workaround, we construct a coarse model for the SM algorithm as a response surface approximation of the sampled Rcd data. We use kriging as an approximation [10].

Also, in order to avoid building the response surface model for the entire design space (which would be impractical, particularly if the number of design variables is large), we only build the coarse model in the vicinity of the (approximate) optimum of Rcd, which is our best initial guess of the solution of (1).

The design procedure is the following: 1. Starting from the initial design xinit, find an

approximate optimal design x(0) of the coarse-discretization model Rcd. In this work, we use a pattern search algorithm [11].

2. Sample Rcd in the vicinity of x(0) and construct an approximation model Rc (here, kriging).

3. Find a high-fidelity model optimum by applying the SM algorithm (3), (4) with Rc as an underlying coarse model.

Once established, the kriging coarse model is very fast; except the initial computational effort (Step 2), Rcd is not evaluated in the SM optimization stage (Step 3).

3 DRA DESIGN EXAMPLES

3.1 DRA Design Example 1

Consider a rectangular DRA [2] shown in Fig. 2. It comprises a rectangular dielectric resonator (DR), RO4003C [12] support slabs, and polycarbonate housing. The housing is fixed to the circuit board with four through M1 bolts. The DRA is energized with a 50 ohm microstrip through a slot made in the metal ground. Substrate is 0.5 mm thick RO4003C. Design specifications are |S11| ≤ –15 dB for 5.1–to–5.9 GHz; also the antenna gain over bandwidth of interest should be better than 5 dBi for the zero zenith angle.

There are nine design variables: x = [ax ay az ay0 us ws ys g1 y1]T, where ax, ay, and az are dimensions of the DR; ay0 stands for the offset of the DR center relative to the slot center (marked by black dot in Fig. 2 (b)) in Y-direction; us and ws are the slot dimensions; ys is the length of the microstrip stub; and g1 and y1 are for the support slabs. Relative permittivity and loss tangent of the DR ceramic core are 10 and 1e-4 respectively at 6.5 GHz. Relative permittivity and loss tangent of the polycarbonate housing are 2.8 and 0.01 at 6.5 GHz respectively. The width of the microstrip signal trace is 1.15 mm. Other dimensions are fixed: hx = hy = hz = 1, bx = 7.5, sx = 2, and ty = ay ‒ ay0 ‒ 1, all in mm. DRA models simulated with the CST MWS [13].

The initial design is xinit = [8.0 14.0 9.0 0.0 1.75 10.0 3.0 1.5 6.0]T mm. The fine (~1,000,000 mesh cells) and coarse-discretization (~27,000 mesh cells) antenna models are evaluated with the CST MWS transient solver in 2,175 and 42 seconds, respectively.

(a) (b)

(c)

Figure 1: DRA: (a) 3D view; (b) top view; and (c) front view; substrate and housing shown transparent in the top and front views.

137

An Rcd optimum is found at x(0) = [7.444 13.556 9.167 0.25 1.75 10.5 2.5 1.5 6.0]T mm. Figure 2 shows the fine model reflection response at the initial design as well as that of the fine and coarse-discretization model Rcd at x(0). The kriging coarse model is set up using 200 samples of Rcd allocated in the vicinity of x(0) of the size [0.5 0.5 0.5 0.25 0.5 0.25 0.25 0.25 0.5]T mm.

The final design, x(4)=[7.556 13.278 9.630 0.472 1.287 10.593 2.667 1.722 6.482]T mm (Figs. 3 and 4), is obtained after four SM iterations. For the bandwidth of interest, the peak gain is above 5 dBi, and the back radiation is under –14 dB (relative to the maximum). The surrogate model of the optimization algorithm exploited both input and output SM, namely, Rs(x) = Rc(x + c) + d. The total design time corresponds to 11 evaluations of the fine model (Table 1).

3.2 DRA Design Example 2

Consider a DRA shown in Fig. 5. It comprises two coupled rectangular DRs [14] installed above a 2.5 mm thick RT6010 [15] layer with upper and lower metal grounds. The DRs are covered by polycarbonate housing. Feeding of the DRA is with a 50 ohm grounded coplanar waveguide (GCPW) terminated by two symmetrical slots exiting the TEx

δ11 mode in the DRs. The relative permittivity and loss tangent of the DR cores are 36 and 10-4, respectively. Two slots shown in Fig. 5(b) energize the DRs directly [14]. However, parallel plate modes can be launched in the substrate; that results in increase of the board noise and the drop of the antenna gain and efficiency. Parasitic emission can be suppressed with the use of vias connecting the upper and lower grounds and forming a substrate integrated cavity (Fig. 5).

There are 11 design variables: x = [x0 y0 xd yd zd s1 x1 xv yv sx sy]T. Dimensions of the input GCPW are the trace width, w0, of 1.5 mm and spacing, s0, of 1mm. Diameter of the vias, dv, is 1.5 mm. Thicknesses of the housing, xh, yh, and zh, are 2 mm. The frequency band of interest is 2.4-to-2.5 GHz. Design requirements are: |S11| ≤ –20 dB, and gain is to be higher than 3dBi for θ = 00 (Z-direction). The radiation pattern is expected to be omnidirectional in the upper YZ-plane (cf. Fig. 5) due to the TEx

δ11 DR mode excited. Design starts from xinit = [x0 y0 xd yd zd s1 x1 xv yv sx sy]T=

[7.75 5 6 16.5 18 2 10.75 6 14 4 6]T mm. The final design was found to be x* = [7.62 5.70 6.2 16.43 17.9 1.9 10.45 6.08 13.83 4.37 6.03]T mm.

4.5 5 5.5 6 6.5

-20

-10

0

Frequency [GHz]

|S11

| [dB

]

Figure 2: DRA, |S11| versus frequency: fine model Rf at the initial design (- - -), optimized coarse-discretization model Rcd (· · · ·), and Rf at the Rcd optimum (—).

4.5 5 5.5 6 6.5

-20

-10

0

Frequency [GHz]

|S11

| [dB

]

Figure 3: DRA, |S11| versus frequency: Rf at the final design.

4.5 5 5.5 6 6.5-15

-10

-5

0

5

10

Frequency [GHz]

[dB

i]

Figure 4: DRA, realized gain versus frequency: (—) is for the zero zenith angle (θ = 00); (- - -) is for back radiation (θ = 1800). Here, only θ-polarization (ϕ = 900) contributes to the gain for the listed directions.

Table 1: DRA Design Example 1: Optimization Cost Algorithm

Component Number of Model

Evaluations CPU Time

Absolute Relative to Rf

Optimization of Rcd 150 × Rcd 105 min 2.9 Setting up Rc 200 × Rcd 140 min 3.9

Evaluation of Rf 4 × Rf 145 min 4.0 Total cost N/A 390 min 10.8

(a)

(b)

(c)

Figure 5: Double core DRA: (a) 3D view: two rectangular DRs in a housing; feeding is with a GCPW, (b) top view; (b) front view (vias forming substrate integrated cavity not shown).

138

The design response meets the specifications; its |S11| is shown in Fig. 6, the gain versus frequency for θ = 00 is shown in Fig. 7, and the gain pattern cuts are shown in Fig. 8. The design cost corresponds to about 20 evaluations of the high-fidelity model.

For comparison, the DRA without substrate integrated cavity was also designed. There were 7 design variables x*,n.v. = [x0 y0 xd yd zd s1 x1]T. Here the optimum do not satisfy the design requirements (|S11| < –20 dB, gain (θ = 00) > 3dBi, both for 2.4-to-2.5 GHz). Figures 6 and 7 also show responses of the two alternative designs, x*,n.v = [7.65 5.51 5.39 16.20 19.45 0.263 10.05]T mm (|S11| < –11.5 dB, gain (θ = 00) > 2.5dBi) and x**,n.v = [6.79 5.25 5.68 16.22 19.97 0.250 9.46]T mm (|S11| < –13.5 dB, gain (θ = 00) > 0.5 dBi). The values of the gain (θ = 00) at 2.45 GHz are: 3.7 dBi for design x*, 2.8 dBi for design x*,n.v, and 1.4 dBi for design x**,n.v. The peak gain at 2.45 GHz are: 5.4 dBi for design x*, 2.9 dBi for design x*,n.v, and 1.5 dBi for design x**,n.v.. The lower gain of the vialess designs, clearly noticeable over the simulation bandwidth, is due to the parasitic emission into substrate.

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8-30

-20

-10

0

Frequency [GHz]

|S11

| [dB

]

Fig. 6. DRA, |S11| response at the final design: with substrate integrated cavity, x* (—); no vias, x*,n.v (- - -); and no vias, x**,n.v. (⋅ ⋅ ⋅).

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8-5

0

5

10

Frequency [GHz]

[dB

i]

Fig. 7. DRA, gain response in Z-direction at the final design: with substrate integrated cavity, x* (—); no vias, x*,n.v (- - -); and no vias, x**,n.v. (⋅ ⋅ ⋅). Design specifications shown with the horizontal line.

-10-10 00 101090

60

300

30

60

90[dBi] (a)

-10-10 00 101090

60

300

30

60

90[dBi] (b)

Fig. 8. DRA, gain at 2.45 GHz (a) co-pol. in the E-plane (YOZ), the right sector is for the positive Y-direction; (b) x-pol. in the H-plane. Design with substrate integrated cavity, x*, (—); designs without vias, x*,n.v (- - -) and x**,n.v. (⋅ ⋅ ⋅).

4 CONCLUSION

Simulation-driven design of dielectric resonator antennas is presented. Our approach combines coarse-discretization EM simulations, space mapping and kriging interpolation. This combination within an automated SBO procedure allows us low-cost adjustment of DRA geometry. Two dielectric resonator antenna design examples are demonstrated. The effect of the feeding mechanism, fixture, and housing cap on the antenna reflection and radiation responses and, therefore, their impact on the final designs were taken into account in the optimization process with discrete EM simulations of the coarse and fine antenna models. In both design examples, the total design optimization time corresponds to a few high-fidelity full-wave EM simulations of the respective antenna structure, typically 1÷2 times the number of design variables in a particular problem.

References [1] A.A. Kishk and Y.M.M. Antar, “Dielectric Resonator Antennas,”

in Antenna Engineering Handbook, J.L. Volakis, Editor, 4th ed., McGraw-Hill, 2007.

[2] A. Petosa, Dielectric resonator antenna handbook, Artech House, 2007.

[3] F.J. Villegas, T. Cwik, Y. Rahmat-Samii, and M. Manteghi, “A Parallel Electromagnetic Genetic-Algorithm Optimization (EGO) Application for Patch Antenna Design,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2424–2435, Sep. 2004.

[4] N. Jin and Y. Rahmat-Samii, “Parallel particle swarm optimization and finite- difference time-domain (PSO/FDTD) algorithm for multiband and wide-band patch antenna designs,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3459–3468, Nov. 2005.

[5] N.V. Queipo, R.T. Haftka, W. Shyy, T. Goel, R. Vaidynathan, and P.K. Tucker, “Surrogate-based analysis and optimization,” Progress in Aerospace Sciences, vol. 41, no. 1, pp. 1-28, Jan. 2005.

[6] S. Koziel, Q.S. Cheng, and J.W. Bandler, “Space mapping,” IEEE Microwave Magazine, vol. 9, no. 6, pp. 105-122, Dec. 2008.

[7] S. Koziel, J.W. Bandler, and K. Madsen, “A space mapping framework for engineering optimization: theory and implementation,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 10, pp. 3721-3730, Oct. 2006.

[8] S. Koziel, J. Meng, J.W. Bandler, M.H. Bakr, and Q.S. Cheng, “Accelerated microwave design optimization with tuning space mapping,” IEEE Trans. Microwave Theory and Tech., vol. 57, no. 2, pp. 383-394, 2009.

[9] J.C. Rautio, “EM-component-based design of planar circuits,” IEEE Microwave Magazine, vol. 8, no. 4, pp. 79-90, Aug. 2007.

[10] S. Koziel, “Surrogate-based optimization of microwave structures using space mapping and kriging,” European Microwave Conference, Sep. 28 – Oct. 2, Rome, Italy, pp. 1062-1065, 2009.

[11] T.G. Kolda, R.M. Lewis and V. Torczon, “Optimization by direct search: new perspectives on some classical and modern methods,” SIAM Rev., vol. 45, no. 3, pp. 385–482, 2003.

[12] RO4000 series high frequency circuit materials, data sheet, publication no. 92-004, Rogers Corporation, 2010.

[13] CST Microwave Studio, ver. 2010, CST AG, Bad Nauheimer Str. 19, D-64289 Darmstadt, Germany, 2010.

[14] M. Deng, C.L. Tsai, C.W. Chiu, and S.F. Chang, “CPW-fed dual rectangular ceramic dielectric resonator antennas through inductively coupled slots,” in Proc. IEEE Antennas Propag. Soc. Int Symp., vol. 1, 2004, pp. 1102-1105.

[15] SRT/duroid® 6006/6010LM high frequency laminates, data sheet, Rogers Corporation, USA, 2009.

139