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Modeling Procedures Developed in EOMC
Surrogate Modeling Modeling is of primary importance for contemporary engineering design process on both device and system levels. Unfortunately, available highfidelity models are typically based on computationally expensive computer simulations (e.g., finite element analysis). High CPU cost is undesirable from the point of view performing tasks such as parametric design optimization, statistical analysis and yield optimization. Also, sensitivity information is normally unavailable and the highfidelity models are often analytically intractable (e.g., nondifferentiable, discontinuous) which excludes using traditional, gradientbased methods for the design optimization. Surrogate modeling [1] is a way of replacing the original, computationally expensive and otherwise unmanageable model, by a computationally fast surrogate for the purpose of performing the design tasks. Typically, the surrogate model is established locally, in the limited region of interest and, to ensure sufficient accuracy, it is based on some amount of data from the original, highfidelity model. Functional versus Physical Models. Space Mapping There is a variety of surrogate modeling techniques that can be categorized into functional and physical ones. Functional surrogate models are based on appropriate function approximation/interpolation of the sampled highfidelity (fine) model data. Popular approaches include polynomial approximation, radial basis functions [2], kriging [3] and neural networks [4]. Physical surrogates, on the other hand, exploit physicallybased lowfidelity models (e.g., coarsemesh simulations or analytical formulas). The surrogate is typically built by enhancing the lowfidelity (coarse) model by appropriate corrections terms derived from a limited amount of the highfidelity model data. Due to the fact that the lowfidelity model encodes certain knowledge about the original structure, physical surrogates typically exhibit better generalization properties than the functional ones. Space mapping (SM) [5], [6] is a notable example of the physical surrogate modeling approach. Space mapping enhances the lowfidelity model by composing it with simple (usually linear) transformations with the parameters of these transformations extracted to minimize misalignment between the surrogate and the highfidelity models at certain (small) number of base points (designs). Formulation of the standard space mapping modeling methodology as well as the exposition of recent advances in SM modeling can be found in Koziel et al. 2008. Space Mapping Modeling for Microwave Engineering In microwave engineering, highfidelity models are normally based on fullwave electromagnetic simulations, whereas lowfidelity models may be using coarsemesh simulations, equivalent circuits or analytical formulas. Illustration: Surrogate modeling of the microstrip bandpass filter [7]. The highfidelity model is implemented in the electromagnetic simulator FEKO: The lowfidelity model is a circuit equivalent model implemented in Agilent ADS: The surrogate model is set up using input and implicit space mapping [5] using a stardistribution base set [6] (here, 11 highfidelity model simulations). Responses of the highfidelity (solid line) and the lowfidelity (dashed line) models at the reference design: Responses of the highfidelity (solid line) and the space mapping surrogate (dashed line) models at the reference design: Application 1: Parametric design optimization. Highfidelity model response at the reference design (dotted line), optimized surrogate model response (dashed line) and the fine model response (solid line) at the optimal surrogate design. Design specifications are denoted as red lines: Application 2: Yield estimation. Surrogate model was used to estimate the yield assuming 1% deviation for all design variables. Estimation performed for 200 random samples. Estimated yield is 62% (left picture). Similar estimation performed directly on the highfidelity model gives 52% (right picture): EOMC Focus The techniques developed in EOMC are mostly based on the physical surrogate modeling principle, particularly space mapping as well as combination of space mapping with various function approximation methods. EOMC focuses on computationally efficient surrogate modeling methods for microwave/RF engineering and their applications for statistical analysis and design optimization of microwave devices and circuits. On the top of standard space mapping, EOMS is exploiting other techniques such as adaptive response correction [8] and shapepreserving response prediction [9]. EOMC works on the development of userfriendly software implementing surrogate modeling techniques with special emphasis on interfacing commercial EM/circuit simulators. References [1] T.W. Simpson, J. Peplinski, P.N. Koch, and J.K. Allen, “Metamodels for computerbased engineering design: survey and recommendations,” Engineering with Computers, vol.17, no. 2, pp. 129 150, July 2001. [2] M.D. Buhmann and M. J. Ablowitz, Radial Basis Functions: Theory and Implementations, Cambridge University, 2003. [3] W.C.M. van Beers and J.P.C. Kleijnen, “Kriging interpolation in simulation: survey,” Proc. 2004 Winter Simulation Conf., pp. 113121, Washington, DC, USA, 2004. [4] Q.J. Zhang, K.C. Gupta, Neural Networks for RF and Microwave Design, Artech Hourse Publishers, Boston, MA, USA, 2000. [5] J.W. Bandler, Q.S. Cheng, S.A. Dakroury, A.S. Mohamed, M.H. Bakr, K. Madsen, and J. Sondergaard, “Space mapping: the state of the art,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 1, pp. 337 361, Jan. 2004. [6] S. Koziel, J.W. Bandler, A.S. Mohamed, and K. Madsen, “Enhanced surrogate models for statistical design exploiting space mapping technology,” IEEE MTTS Int. Microwave Symp. Dig., Long Beach, CA, June 2005, pp. 16091612. [7] A. Hennings, E. Semouchkina, A. Baker and G. Semouchkin, “Design optimization and implementation of bandpass filters with normally fed microstrip resonators loaded by highpermittivity dielectric,” IEEE Trans. Microwave Theory and Tech., vol. 54, no. 3, pp. 12531261, Mar. 2006. [8] S. Koziel, J.W. Bandler, and K. Madsen, ”Space mapping with adaptive response correction for microwave design optimization,” IEEE Trans. Microwave Theory Tech., vol. 57, no. 2, pp. 478486, 2009. [9] S. Koziel, “Efficient Optimization of Microwave Circuits Using ShapePreserving Response Prediction,” to appear, IEEE MTTS Int. Microwave Symp. Dig., Boston, MA, 2009. 