Stochastic Collocation Applications in Computational Electromagnetics

Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 1917439, 13 pages https://doi.org/10.1155/2018/1917439 Review Article Stochastic Collocation Applications in Computational Electromagnetics Dragan Poljak ,1 Silvestar ŠesniT ,1 Mario CvetkoviT ,1 Anna Šušnjara ,1 Hrvoje Dodig ,2 Sébastien Lalléchère ,3 and Khalil El Khamlichi Drissi3 1 FESB, University of Split, 21000 Split, Croatia Faculty of Maritime Studies, University of Split, 21000 Split, Croatia 3 Université Clermont Auvergne, CNRS, Sigma Clermont, Institut Pascal, Clermont-Ferrand, France 2 Correspondence should be addressed to Dragan Poljak; Received 29 January 2018; Revised 23 March 2018; Accepted 4 April 2018; Published 14 May 2018 Academic Editor: Francesca Vipiana Copyright © 2018 Dragan Poljak et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The paper reviews the application of deterministic-stochastic models in some areas of computational electromagnetics. Namely, in certain problems there is an uncertainty in the input data set as some properties of a system are partly or entirely unknown. Thus, a simple stochastic collocation (SC) method is used to determine relevant statistics about given responses. The SC approach also provides the assessment of related confidence intervals in the set of calculated numerical results. The expansion of statistical output in terms of mean and variance over a polynomial basis, via SC method, is shown to be robust and efficient approach providing a satisfactory convergence rate. This review paper provides certain computational examples from the previous work by the authors illustrating successful application of SC technique in the areas of ground penetrating radar (GPR), human exposure to electromagnetic fields, and buried lines and grounding systems. 1. Introduction Some areas in computational electromagnetics (CEM) suffer from uncertainties of the input parameters resulting in the uncertainties in the assessment of the related response. These problems could be overcome, to a certain extent, by an efficient combination of well-established deterministic electromagnetic models with certain stochastic methods. Traditional methods rely upon statistical approaches such as brute Monte Carlo (MC) simulations and various sampling techniques like stratified sampling and Latin hypercube sampling (LHS). These methods are easy to implement and do not suffer from “curse of dimensionality” which means that the sample size does not depend on number of random variables. On the other hand the sample size needs to be very high (>100.000), and thus they exhibit very slow convergence. Contrary to statistical approaches, the nonstatistical based techniques aim to represent the unknown stochastic solution as a function of random input variables. Among the various methods available in the literature, the spectral discretization based technique—generalized polynomial chaos (gPCE)—emerged as the most used approach in the stochastic CEM. The gPCE framework comprises stochastic Galerkin method (SGM) and stochastic collocation method (SCM) for solving stochastic equations [1]. The SGMs have been successfully used in recent years in the area of circuit uncertainty modeling both for Signal Integrity (SI) [2] and microwave applications [3] as well as in stochastic dosimetry [4, 5]. The intrusive nature of SGM implies more demanding implementation since a development of new codes is required. On the contrary, the nonintrusive nature of SCM enables the use of reliable deterministic models as black boxes in stochastic computations. Both approaches exhibit fast convergence and high accuracy under different conditions and a detailed comparison of their use in EMC simulation can be found in [6]. The combination of the nonintrusive, sampling based nature of Monte Carlo simulations with the polynomial approximation of output value, which is the characteristic of gPCE methods, made stochastic collocation one of the most researched and applied stochastic approaches [7–9]. The 2 examples of successful coupling of SCM and its variations with different deterministic codes have been reported in stochastic dosimetry [10, 11], in the analysis of complex power distribution networks (PDN) [12], in the design of integrated circuits (ICs), microelectromechanical systems, (MEMSs), and photonic circuits [13, 14], in the simulation of EMcircuit systems [15, 16], and in the area of antenna modeling [17–19] and electromagnetic compatibility (EMC) of space applications [20]. This paper reviews the previous work of the authors pertaining to the use of SC techniques in areas of CEM such as ground penetrating radar (GPR), EM-thermal dosimetry of the eye and brain, and buried lines and grounding systems [21]. Some illustrative computational examples for the transient transmitted field from GPR antenna [22, 23], specific absorption rate (SAR) distribution in the brain and the eye [24, 25], plane wave coupling to buried conductors [26, 27], and transient analysis of grounding electrodes [28] pertaining to certain statistical moments are given in the paper as well. The paper is organized, as follows: first an outline of the SC method is given in Section 2 along with the short overview of its applications in various areas. Section 3 deals with different applications of SCM carried out by authors with related examples. This is followed by some conclusions and guidelines for a future work. 2. An Outline of the Stochastic Collocation Method The uncertainty quantification (UQ) of the unknown stochastic output of the model is preceded by two steps: the UQ of input parameters and uncertainty propagation (UP) of uncertainties present in the model inputs to the output of interest. The UQ of input parameters implies modeling the input parameters as random variables and/or random processes. The UP refers to the choice and implementation of the stochastic method that is capable of solving the stochastic model. The advantage of the stochastic collocation method (SCM) used in this work is its simplicity, a strong mathematical background, and the polynomial representation of stochastic output. The nonintrusive nature of the method enables the use of deterministic models as a black box. This way, previously validated computational models, such as FEM-BEM models described in Section 3, are used at predetermined set of simulation points. This section outlines the fundaments of the mathematical background for the SCM, with the brief mention of some other variants. Once the deterministic modeling of a problem of interest is completed, a stochastic processing of the numerical results can be carried out via the SC method [1, 22–27]. The theoretical basis of SC technique is to use the polynomial approximation of the considered output for a certain number of (...truncated)