Couckuyt, Ivo

Dhaene, Tom

Koziel, Slawomir

Ogurtsov, Stanislav

English

Ghent University Academic Bibliography

Accurate Modeling of Antennas Using Variable-Fidelity EM Simulations and Co-Kriging  Slawomir Koziel, Stanislav Ogurtsov School of Science and Engineering Reykjavik University Reykjavik, Iceland koziel@ru.is, stanislav@ru.is Ivo Couckuyt, Tom Dhaene Department of Information Technology Ghent University - IBBT Gent, Belgium ivo.couckuyt@ugent.be, tom.dhaene@ugent.be  Abstract—We present an accurate and low-cost modeling of antenna structures using variable-fidelity electromagnetic (EM) simulations. Our approach exploits sparsely sampled high-fidelity (accurate) EM data as well as densely sampled coarse-discretization (low-fidelity) EM simulations that are accommodated into one model using co-kriging technique. By using coarse-discretization simulations, the computational cost of creating the antenna model is greatly reduced compared to conventional approach, where high-fidelity simulations are directly used to set up the model. To our knowledge, this is the first application of co-kriging to antenna modeling. Numerical verification and comparisons with kriging interpolation are given. Keywords-Antenna modeling; electromagnetic (EM) simulation; kriging; co-kriging; computer-aided design (CAD). I.  INTRODUCTION Reliable evaluation of antenna structures can be obtained through electromagnetic (EM) simulation. High-fidelity simulation is CPU intensive, which is a bottleneck for EM-based design tasks such as parametric optimization, statistical analysis, or yield-driven design. Thus, accurate and computationally cheap models of antennas (so-called surrogates) are indispensable.  Cheap antenna models can be obtained using approximation techniques such as polynomial regression [1], radial basis functions [2], kriging [2], [3], support vector regression [4]-[6], artificial neural networks [7]-[10], fuzzy systems [11], or multidimensional Cauchy approximation [12]. However, for good accuracy, these techniques require a large number of training points, particularly if the number of design variables is large.  Here, we consider antenna models constructed using both high- and low-fidelity EM simulations. Simulation of coarsely-discretized antenna structure may not be accurate; however, it is much faster than the high-fidelity one. As we demonstrate, such low-fidelity data can be combined with sparsely sampled high-fidelity simulations using co-kriging [13]. The resulting antenna model is as accurate as the conventional approximation surrogate using much larger number of training data points. The proposed modelling technique is demonstrated using two examples: a ultrawideband planar dipole antenna and a rectangular dielectric resonator antenna. Comparison with conventional kriging interpolation is also given. II. ANTENNA MODELING USING CO-KRIGING A. Antenna Models We consider two types of antenna models. Let Rf(x) denote an EM-simulated high-fidelity model, which is an accurate representation of the antenna structure. Rf is expensive to evaluate (typical simulation time measured in hours). Here, x is a vector of designable (e.g., geometry) parameters. The components of Rf may represent, e.g., the antenna reflection coefficient |S11| over the frequency band of interest. We also consider an auxiliary (low-fidelity) model Rc which may be evaluated using the same EM solver, however, with coarser discretization. The low-fidelity model Rc is much faster than Rf but not as accurate. Therefore, it cannot be normally directly used instead of the high-fidelity model to perform tasks such as design optimization. In this paper, we combine the low- and high-fidelity simulations to create the surrogate model that is almost as accurate as Rf but requires much smaller number of high-fidelity training points than the approximation model created using only Rf samples. B. Kriging Interpolation Kriging is a popular technique to interpolate deterministic noise-free data [2], [14]. These Gaussian Process based surrogate models are compact and cheap to evaluate. Here, we use kriging as a benchmark technique for comparison with the co-kriging of Section II.C. Let XB.KR = {xKR1, xKR2, …, xKRNKR} ⊂ XR be the base (training) set and Rf(XB.KR) the associated fine model responses. Then, the kriging interpolant, also known as the Best Linear Unbiased Predictor (BLUP), is derived as,  1. .( ) ( ) ( ( ) )s KR f B KRM r X Fα α−= + ⋅Ψ ⋅ −R x x R (1)  where M and F are Vandermonde matrices of the test point x and the base set XB.KR, respectively. The coefficient vector α is determined by Generalized Least Squares (GLS). r(x) is an 1×NKR vector of correlations between the point x and the base set XB.KR, where the entries are ri(x) = ψ(x,xKRi), and Ψ is a NKR×NKR correlation matrix, with the entries given by Ψi,j = ψ(xKRi, xKRj). In this work, the exponential correlation function is used, i.e., ψ(x,y) = exp(∑k=1,...,n –θk|xk–yk|), where the parameters θ1, ..., θn are identified by Maximum Likelihood Estimation (MLE). The regression function is chosen constant, F = [1 ... 1]T and M = (1). C. Co-Kriging Modeling Co-kriging [13] is a type of kriging where the Rf and Rc model data are combined to enhance the prediction accuracy. Co-kriging is a two-steps process: first a kriging model Rs.KRc of the coarse data (XB.KRc,Rc(XB.KRc)) is constructed and on the residuals of the fine data (XB.KRf,Rd) a second kriging model Rs.KRd is applied, where Rd = Rf(XB.KRf) – ρ⋅Rc(XB.KRf). The parameter ρ is included in the MLE. Note that if the response values Rc(XB.KRf) are not available, they can be approximated by using the first kriging model Rs.KRc, namely, Rc(XB.KRf) ≈ Rs.KRc(XB.KRf). The resulting co-kriging interpolant is defined as  1. ( ) ( ) ( )s CO dM r Fα α−= + ⋅Ψ ⋅ −R x x R  (2)  where the block matrices M, F, r(x) and Ψ can be written in function of the two separate kriging models Rs.KRc and Rs.KRd:  2 2 2 2.( ) [ ( ), ( , ) ( )]fc c c c B KR d dr r r X rρ σ ρ σ σ= ⋅ ⋅ ⋅ ⋅ + ⋅x x x x   2 2. .2 2 2. .( , )0 ( , )c ff fc c c c B KR B KRc c B KR B KR d dX XX Xσ ρ σρ σ σ⎡ ⎤Ψ ⋅ ⋅Ψ⎢ ⎥Ψ = ⋅ ⋅Ψ + ⋅Ψ⎢ ⎥⎣ ⎦   0cd dFFF Fρ⎡ ⎤= ⎢ ⎥⋅⎣ ⎦ ,    [ ]c dM M Mρ= ⋅   (3) where (Fc,σc,Ψc,Mc) and (Fd,σd,Ψd,Md) are matrices obtained from the kriging models Rs.KRc and Rs.KRd, respectively (see Section II.B). In particular, σc2 and σd2 are process variances, while Ψc(⋅,⋅) and Ψd(⋅,⋅) denote correlation matrices of two datasets with the optimized θ1, ..., θn parameters and correlation function of the kriging models Rs.KRc and Rs.KRd, respectively. III. VERIFICATION EXAMPLES A. UWB Planar Dipole Antenna Consider the planar dipole antenna [15] (Fig. 1). The design variables are x = [l0 w0 a0 lp wp s0]T. The high-fidelity model Rf (~10 mln mesh cells, evaluation time 44 minutes) is simulated using the CST MWS transient solver [16]. The low-fidelity model Rc is also evaluated in CST (~100,000 mesh cells, evaluation time 43 seconds). The antenna models are set up in the region with the center at x0 = [19 13 0.5 13 6 1]T and size δ = [1 1 0.2 1 1 0.2]T. The kriging and co-kriging models (Rs.KR, Rs.CO) are constructed using various numbers of training points (from NKR = 20 to NKR = 400). Co-kriging models are configured using 400 Rc samples (the CPU cost of which corresponds to around 6 evaluations of Rf). The quality of the surrogate is assessed using a relative error measure ||Rf(x) – Rs(x)||/||Rf(x)|| expressed in percent. The error is averaged over 50 test designs.  The modeling errors are given in Table I (see also Fig. 2). Note that the co-kriging model accuracy obtained with 20 (50) Rf samples is as good as that of the kriging model obtained for 100 (200) samples, which proves that co-kriging and the use of coarse-discretization EM data allows us to greatly reduce the CPU cost of creating accurate antenna model compared to conventional method using solely Rf information.  B. Rectangular Dielectric Resonator Antenna Consider the rectangular suspended DRA [17] (Fig. 3). The design variables are x = [ε1 h1 h2 s1 w1]T. Other parameters are fixed. The high- and low-fidelity models are evaluated in CST [16] with the following evaluation times: Rf  11 minutes, and Rc 20 sec. The antenna models are set up in the region with the center at x0 = [10 8.5 0.5 3 10]T and size δ = [1 1 0.5 1 2]T mm. Similarly as for the previous example, co-kriging allows to substantially reduce the computational cost of creating the accurate antenna model when compared to approximation of the high-fidelity model data only (cf. Table II and Fig. 4).    Fig. 1. Dipole antenna geometry [15]: top and side views. The dash-dot lines show the magnetic (YOZ) and the electric (XOY) symmetry walls. The 50 ohm source impedance is not shown at the figure.  2 4 6 8 10 12-20-15-10-5Frequency [GHz]|S11| [dB] Figure 2. UWB dipole: responses of Rf (—) and co-kriging surrogate model (o) at selected test points. Co-kriging model created using 50 evaluations of Rf and 400 evaluations of Rc.  IV. CONCLUSION We presented an antenna modeling methodology using co-kriging. We demonstrate that by combining the low- and high-fidelity EM simulations, it is possible to create an accurate model of an antenna structure while using limited number of high-fidelity data points.  ACKNOWLEDGMENT This work was supported in part by the Icelandic Centre for Research (RANNIS) Grant 110034021. The authors would like to thank Computer Simulation Technology AG, Darmstadt, Germany, for making CST Microwave Studio available. REFERENCES [1] A.I.J. 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Petosa, Dielectric resonator antenna handbook, Artech House, 2007.  TABLE I.  UWB DIPOLE ANTENNA: MODELING RESULTS Model Average Modeling Error [%] NKR = 20 NKR = 50 NKR = 100 NKR = 200 NKR = 400 Rs.KR 17.5 5.6 4.3 2.8 2.0 Rs.CO 4.2 2.6 2.4 2.0 1.9      GNDε1ε2w1s1ε3                     (a)             (b)  (c) Fig. 1. DRA geometry [17]: (a) 3D view, (b) top view, and (c) side view.   3 3.5 4 4.5 5 5.5 6 6.5 7 7.5-20-15-10-50Frequency [GHz]|S11| [dB] Figure 2. Rectangular DRA: responses of Rf (—) and co-kriging surrogate (o) at selected test points. Co-kriging model created using 50 evaluations of Rf and 400 evaluations of Rc.  TABLE II.  RECTANGULAR DRA: MODELING RESULTS Model Average Modeling Error [%] NKR = 20 NKR = 50 NKR = 100 NKR = 200 NKR = 400 Rs.KR 12.1 8.8 6.9 5.2 3.6 Rs.CO 6.7 5.4 5.0 4.1 3.5  

Accurate modeling of antennas using variable-fidelity EM simulations and co-kriging

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