Mathematical Problems in Engineering

Volume 2018, Article ID 1917439, 13 pages

https://doi.org/10.1155/2018/1917439

## Stochastic Collocation Applications in Computational Electromagnetics

^{1}FESB, University of Split, 21000 Split, Croatia^{2}Faculty of Maritime Studies, University of Split, 21000 Split, Croatia^{3}Université Clermont Auvergne, CNRS, Sigma Clermont, Institut Pascal, Clermont-Ferrand, France

Correspondence should be addressed to Dragan Poljak; rh.bsef@kajlopd

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.

#### Abstract

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 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 EM-circuit 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 random parameters.

Without loss of generality, the theoretical principles can be presented taking into account a single random parameter . Within this framework, a given mapping through function can be assumed and a random variable (RV) given by an a priori probabilistic distribution ( standing for the relation between the output and input parameter, resp.). Consequently, the expressions for statistical moments, such as mean and variance of RV, can be derived.

First, an approximation of function over th order Lagrange polynomial basis can be written as follows:where Lagrange polynomials are given by

And one has the following property:where denotes Kronecker symbol.

It is worth mentioning that, although Lagrange polynomials are mostly used for the polynomial representation of the stochastic output, other types of basis functions are also possible. In [15] SCM is used to assess the uncertainty quantification for the hybrid EM-circuit systems. Two types of basis functions are used: Lagrange polynomials that have the character of locally global basis functions and piecewise multilinear basis functions that are able to capture the discontinuous issues in stochastic solutions.

Note that collocation points from (1) correspond to the points in Gauss quadrature rule attached to the probability distribution of random inputs (e.g., Legendre polynomials for a uniform distribution and Hermite polynomials for a Gaussian distribution). The choice of an adapted quadrature rule [19] yieldswhere is the probability density of random variable , while represents a function with a sufficient regularity and the coefficients are collocation weights chosen in a way to ensure an exact quadrature rule for polynomials with a degree lower or equal to .

However, the choice of collocation points is not limited to the points that supervene from Gauss quadrature rules. Other formulas may be used to generate the abscissas for the interpolation in (1) such as Clenshaw-Curtis formula with nonequidistant abscissas given as zeros of the extreme points of Chebyshev polynomials as in [15] or equidistant points in [14, 15]. These sets of points exhibit nested fashion unlike Gauss points, which is desirable in certain applications.

Higher dimensions yield multivariate interpolants in (1). The simplest way to interpolate is by using the tensor product in each random dimension. This approach is justified for the cases when random dimension is up to 5 [1]. Successful applications of a fully tensorized SCM model can be found in [7–9]. However, for random dimension > 5, SCM exhibits the “curse of dimensionality” and different techniques need to be considered. One of the most popular approaches is sparse grid interpolation based on Smolyak algorithm [1]. Sparse grid stochastic collocation method (SGSCM) has become an interest of many researchers. In [14] a conventional SGSCM has been applied for stochastic characterization of MEMS. An adaptive approach of SGSCM has been employed in [15] for the stochastic modeling of hybrid EM-circuits while a dimension-reduced sparse grid strategy has been proposed for SCM in EMC software in [16].

Furthermore, different cubature formulas for dealing with multidimensional integrals may be used for the generation of collocation points. One of the examples is Stroud cubature based SC method whose application for the variability analysis of PDNs within SPICE environments is reported in [12] and, for efficient computation of RCS from scatterers of uncertain shapes, in [19].

Once the desired output has been expressed as polynomial dependent on random input variables, the expression for the stochastic moments can be easily derived following their definitions from statistics. The expectation (mean) of is by definitionwhere is a given RV.

Next step is to replace in terms of Lagrange polynomials and expectation of can be written as follows:The Gauss quadrature rule yields

Taking into account the property of Lagrange polynomials (3) one obtains

Combining (6)–(8) leads to an expression for mean of :

Furthermore, the same approach is used to evaluate variance of , which is by definition given byIt can be written as

Again, utilizing the expansion of function over Lagrange polynomial basis (1) yieldswhich can be also written as

Furthermore, integral from the right-hand side of (13) according to Gauss quadrature rule (7) becomesFinally, inserting (14) into (13), one obtains and due to Kronecker property nonzero terms are only with leading to

Now the statistical moments around a considered output can be written as , where stands for a given order of statistical moment; for example, the mean of can then be written as

General expression is given as follows:where represents a mapping pertaining to a certain statistical moment , that is, with such as

Assuming the mutual independence of input random variables the generalization to multi-RVs case is straightforward. Instead of single random variable, one deals with a random vector and th statistical moment of output can be written as [28]where stands for weight of the expansion for random vector of size . Some more details related to this approach are available elsewhere, for example, in [28].

Although the assumption on independence of random input variables is valid in many CEM applications, it is worth noting that correlated random variables can also be taken into account. For example, in [29] collocation points that belong to correlated inputs are computed by using the different orthogonal polynomials, while in [30] the use of partitioning and clustering methods for generation of SC points is proposed which naturally extends to incorporation of correlated inputs.

#### 3. Applications in Computational Electromagnetics

This section outlines the governing equations pertaining to CEM deterministic models reviewed in this work. Thus, first, Section 3.1 deals with transient field generated by GPR dipole antenna. Modeling of induced SAR in the brain and eye is presented in Section 3.2. Sections 3.3 and 3.4 deal with plane wave coupling to buried wire and current injection of horizontal grounding electrode, respectively.

##### 3.1. Transient Field in the Soil Generated by GPR Dipole Antenna

The dipole is characterized by a length and radius and is located at height above an interface as depicted in Figure 1.