Stochastic modeling in computational electromagnetics by coordinate transformations

Computational Electromagnetics (CEM) involves the process of modeling the interaction of electromagnetic fields with physical objects and environment through numerical solutions of Maxwell’s equations. The study of such interactions is crucial in the analysis and design of electromagnetic systems or devices. For example, in electromagnetic compatibility (EMC) applications, it is essential to characterize the immunity of devices to parasitic or intentional electromagnetic fields emanating from internal components of integrated circuits or from external electromagnetic sources. Although computational models that are used to model electromagnetic interactions are getting more and more accurate everyday, there is still an unavoidable gap between numerical and real/physical experiments in various cases. This is mainly due to the fact that the models are parametrized by a set of inputs, and in practice these input parameters may exhibit variability due to, e.g., manufacturing defects in the geometry or material properties, noise in the measurement of the input parameters, and uncontrolled environmental effects. Typically, a stochastic analysis can be used to quantify such uncertainties, which models the unknown input parameters (observables) as random variables and employs the principles of probability theory to obtain their distributions that show the dependence of the observables on the configuration parameters. However, it is usually too complicated to mathematically formulate the probability distributions especially if the variations are not “small”. One intuitive way is to use the Monte Carlo approach where the variations are modeled by a repeated simulation of the numerical model for each possible configuration. However, it is evident that the repeated execution of the numerical method might be tedious especially due to high computational load.This study presents various computational models that are designed to alleviate the difficulties in repeated realizations of the Monte Carlo approach by finite methods (i.e., finite element method, finite difference methods). These models are developed for the analysis of electromagnetic wave interactions in the following problems: (a) Scattering from rough surfaces[1-3], (b) scattering from a random array of objects[4], and (c) propagation in a waveguide with rough surface or with ridges whose position and geometry are random[5-6]. These models utilize the principles of transformation electromagnetics which is based on the form-invariance property of Maxwell’s equations under coordinate transformations. When the spatial domain of a O. Ozgun is with the Departmentof Electricaland ElectronicsEngineering, HacettepeUniversity,Beytepe, Ankara,Turkey. (e-mail: ozlem@ee.hacettepe.edu.tr)M. Kuzuoglu is with the Departmentof Electricaland ElectronicsEngineering, Middle EastTechnical University, Ankara, Turkey. (e-mail: kuzuoglu@metu.edu.tr)medium is modified by defining a coordinate transformation, this medium equivalently turns into an anisotropic medium in which Maxwell’s equations keep the same mathematical form. The constitutive parameters of the anisotropic medium are determined by the Jacobian of the transformation. The principle concept that underlies all of the models in this study is tomodify the computational domain and to place suitable transformation media inside the domain to devise simple and efficient computational tools. To the contrary of the standard approach where multiple meshes are generated, a single and simple mesh is usedandonly the material parameters of the transformation media are modified with respect to each configuration. Hence, mesh generation process is simplified to a great extent and computational load decreases. The models are validated through various numerical simulations