8.5 Microwave Simulation-Driven Design Exploiting Physically-Based Surrogates
8.5.3 Shape-Preserving Response Prediction
Shape-preserving response prediction (SPRP) [51] is a response correction tech- nique that takes advantage of the similarity between responses of the high- and low-fidelity models in a very straightforward way. SPRP assumes that the change of the high-fidelity model response due to the adjustment of the design variables can be predicted using the actual changes of the low-fidelity model response.
Therefore, it is critically important that the low-fidelity model is physically based, which ensures that the effect of the design parameter variations on the model re- sponse is similar for both models. In microwave engineering this property is likely to hold, particularly if the low-fidelity model is the coarsely-discretization struc- ture evaluated using the same EM solver as the one used to simulate the high- fidelity model.
The change of the low-fidelidy model response is described by the translation vectors corresponding to a certain (finite) number of characteristic points of the model’s response. These translation vectors are subsequently used to predict the change of the high-fidelity model response with the actual response of Rf at the current iteration point, Rf(x(i)), treated as a reference.
Figure 8.7(a) shows the example low-fidelity model response, |S21| in the fre- quency range 8 GHz to 18 GHz, at the design x(i), as well as the low-fidelity model response at some other design x. The responses come from the double folded stub
bandstop filter example considered in [51]. Circles denote characteristic points of Rc(x(i)), selected here to represent |S21| = –3 dB, |S21| = –20 dB, and the local |S21| maximum (at about 13 GHz). Squares denote corresponding characteristic points for Rc(x), while line segments represent the translation vectors (“shift”) of the characteristic points of Rc when changing the design variables from x(i) to x. Since the low-fidelity model is physically based, the high-fidelity model response at the given design, here, x, can be predicted using the same translation vectors applied to the corresponding characteristic points of the high-fidelity model response at x(i), Rf(x(i)). This is illustrated in Fig. 8.7(b).
Rigorous formulation of SPRP uses the following notation concerning the re- sponses: Rf(x) = [Rf(x,ω1) … Rf(x,ωm)]T and Rc(x) = [Rc(x,ω1) … Rc(x,ωm)]T, where ωj, j = 1, …, m, is the frequency sweep. Let pjf
= [ωjf rjf]T, pjc0 = [ωjc0 rjc0]T, and pjc
= [ωjc rjc]T, j = 1, …, K, denote the sets of characteristic points of Rf(x(i)), Rc(x(i)) and Rc(x), respectively. Here, ω and r denote the frequency and magnitude compo- nents of the respective point. The translation vectors of the low-fidelity model re- sponse are defined as tj = [ωjt rjt]T,j = 1,…, K, where ωjt = ωjc – ωjc0 and rjt = rjc–rjc0.
The shape-preserving response prediction surrogate model is defined as follows
( ) ( ) ( )
( )=[ ( , 1) ... ( , )]
i i i T
s Rs ω Rs ωm
R x x x (8.9) where
( ) ( )
. 1 1
( , ) ( , ( ,{ } )) ( ,{ } )
i i t K t K
s j f i j k k j k k
R xω =R x Fω −ω = +Rω r = (8.10) for j = 1, …, m. Rf.i (x,ω1) is an interpolation of {Rf(x,ω1), …, Rf(x,ωm)} onto the frequency interval [ω1,ωm].
The scaling function F interpolates the data pairs {ω1,ω1}, {ω1f,ω1f–ω1t}, …, {ωKf,ωKf–ωKt}, {ωm,ωm}, onto the frequency interval [ω1,ωm]. The function R does a similar interpolation for data pairs {ω1,r1}, {ω1f,r1f–r1t}, …, {ωKf,rKf–rKt}, {ωm,rm};
here r1 = Rc(x,ω1) – Rc(xr,ω1) and rm = Rc(x,ωm) – Rc(xr,ωm). In other words, the function F translates the frequency components of the characteristic points of Rf(x(i)) to the frequencies at which they should be located according to the transla- tion vectors tj, while the function R adds the necessary magnitude component.
It should be emphasized that shape-preserving response prediction a physically- based low-fidelity model is critical for the method’s performance. On the other hand, SPRP can be characterized as a non-parametric, nonlinear and design-variable dependent response correction, and it is therefore distinct from any known space mapping approaches. Another important feature that differentiates SPRP from space mapping and other approaches (e.g., tuning) is implementation simplicity.
Unlike space mapping, SPRP does not use any extractable parameters (which are normally found by solving a separate nonlinear minimization problem), the prob- lem of the surrogate model selection [38], [39] (i.e., the choice of the transforma- tion and its parameters) does not exist, and the interaction between the models is very simple (only through the translation vectors (8.3), (8.4)). Unlike tuning methodologies, SPRP does not require any modification of the optimized structure (such as “cutting” and insertion of the tuning components [50]).
8 10 12 14 16 18 -50
-40 -30 -20 -10 0
Frequency [GHz]
|S 21| [dB]
(a)
8 10 12 14 16 18
-50 -40 -30 -20 -10 0
Frequency [GHz]
|S 21| [dB]
(b)
Fig. 8.7 SPRP concept: (a) Example low-fidelity model response at the design x(i), Rc(x(i)) (solid line), the low-fidelity model response at x, Rc(x) (dotted line), characteristic points of Rc(x(i)) (circles) and Rc(x) (squares), and the translation vectors (short lines); (b) High- fidelity model response at x(i), Rf(x(i)) (solid line) and the predicted high-fidelity model response at x (dotted line) obtained using SPRP based on characteristic points of Fig. 8.1(a); characteristic points of Rf(x(i)) (circles) and the translation vectors (short lines) were used to find the character- istic points (squares) of the predicted high-fidelity model response; low-fidelity model responses Rc(x(i)) and Rc(x) are plotted using thin solid and dotted line, respectively [51].
If one-to-one correspondence between the characteristic points of the high- and low-fidelity model is not satisfied despite use of the coarse-mesh EM-based low- fidelity model, the sets of corresponding characteristic points can be generated based not on distinctive features of the responses (e.g., characteristic response lev- els or local minima/maxima) but by introducing additional points that are equally spaced in frequency and inserted between well defined points [51]. These addi- tional points not only ensure that the shape-preserving response prediction model (8.3), (8.4) is well defined but also allows us to capture the response shape of the models even though the number of distinctive features (e.g., local maxima and minima) is different for high- and low-fidelity models.