Uncertainty Quantification for the Transient Response of Human Equivalent Antenna Using the Stochastic Collocation Approach

Hindawi International Journal of Antennas and Propagation Volume 2019, Article ID 4640925, 7 pages https://doi.org/10.1155/2019/4640925 Research Article Uncertainty Quantification for the Transient Response of Human Equivalent Antenna Using the Stochastic Collocation Approach Anna Šušnjara and Dragan Poljak Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Department of Electronics, University of Split, 21000 Split, Croatia Correspondence should be addressed to Dragan Poljak; Received 12 October 2018; Accepted 30 December 2018; Published 28 March 2019 Guest Editor: Sandra Costanzo Copyright © 2019 Anna Šušnjara and Dragan Poljak. 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 deals with the uncertainty quantification of the transient axial current induced along the human body exposed to electromagnetic pulse radiation. The body is modeled as a straight wire antenna whose length and radius exhibit random nature. The uncertainty is propagated to the output transient current by means of the stochastic collocation method. The stochastic approach is entirely nonintrusive and serves as a wrapper around the deterministic code. The numerical deterministic model is based on the time domain Hallen integral equation solved by means of the Galerkin-Bubnov indirect boundary element method (GB-IBEM). The stochastic moments, i.e., the mean and the variance of the transient current, are calculated. Confidence margins are obtained for the whole duration of the transient response as well as for the maximal current value. The presented approach enables the estimation of the probability for the induced current to exceed the basic restrictions prescribed by regulatory bodies. The sensitivity analysis of the input parameters indicates to which extent the variation of the input parameter set influences the output values which is particularly interesting for the design of the human equivalent antenna. 1. Introduction Axial current distribution has been one of the quantities of interest not only to quantify human exposure to lowfrequency fields by determining the current density/induced fields but also to compute specific absorption (SA) for the case of human exposure to transient radiation [1]. Thus, these quantities are the first step in quantifying the effects of EM exposure. On the other hand, biological effects depend on the frequency of the EM field. Due to the absence of resonance effects at low frequencies (LF), the thermal effects are negligible while the nonthermal effects could possibly have severe effects on membrane cells [2, 3]. On the contrary, in a high-frequency (HF) range, where the body dimensions are comparable to external field wavelength and resonances become significant, thermal effects are dominant [4, 5]. In any case, there is no way to directly measure the induced electric fields in humans and related biological effects; hence, the use of reliable computational models is mandatory. However, computational models used in EM dosimetry, simplified or anatomically realistic ones, are subjected to variation of input parameters’ values. The morphology (dimensions), the tissue conductivity and relative permittivity, and other constants related to the model description are often partially or even entirely unknown. In the past decade, some efforts have been made to provide the means to include the parameter variability into the model and propagate it to the output value of interest. A term “stochastic dosimetry” has been coined under the idea that by using only average values, the computational models are rough approximation of the real scenarios in EM dosimetry [6]. Some examples of stochastic dosimetry simulations are presented in [7–10]. Such an approach becomes even more important when it comes to international and national guidelines and standards, respectively, which ought to account for the stochastic nature of the input variables thus providing certain expected values for the restrictions along with the confidence margins and worst-case predictions [11, 12]. 2 Generally, the Monte Carlo simulation (MCS) method is considered to be the most reliable and robust stochastic method [13]. However, relatively slow convergence makes this somewhat unattractive and sometimes inconvenient even for the validation of results with respect to other methods. Among various alternative methods reported in the literature, the generalized polynomial chaos expansion (gPCE) emerged as the most often used approach in the stochastic computational electromagnetics (SCEM). This technique for solving stochastic equations is based on spectral discretization, and it comprises the stochastic Galerkin method (SGM) and stochastic collocation method (SCM) [13]. The main difference between the two methods is their intrusive/nonintrusive approach to computational models. The intrusive nature of SGM implies a more demanding implementation since the development of new codes is required. On the other hand, the nonintrusiveness of SCM enables the use of previously validated deterministic models as black boxes. Still, both approaches exhibit fast convergence and high accuracy under different conditions. The analysis regarding the applications in computational electromagnetics may be found elsewhere, e.g., in [14]. The properties of the field induced in the human body due to EM exposure could be studied by means of a simple but rather useful cylindrical model of the human body [1]. This model has been widely used for LF ranges. First, King and Sandler proposed the parasitic-antenna model of the human body exposed to extremely low-frequency (ELF) and very-low-frequency (VLF) sources providing some closed-form expressions for the induced current [15]. An overview of some numerical methods for human exposure from ELF to a microwave region is reported in [16]. Furthermore, for the case of ELF range, the loaded thick-wire model of human body based on Pocklington’s integrodifferential equation in FD has been proposed by Poljak and Rashed [17]. The model is based on the solution of Pocklington’s equation via the Galerkin-Bubnov boundary element method (GB-BEM). Although FD techniques enable the use of simpler formulations and therefore the numerical treatment is less complicated, when the human body is exposed to transient radiation, a direct time domain approach offers a better insight into the physical behavior of the phenomena [18]. One possible approach would be to use an indirect approach, i.e., to implement the FD-based models along with the IFFT algorithm. However, the thick-wire model from [17] suffers from some numerical instabilities when used for TD response, and moreover, coupling of such approach with nonintrusive and sampling-based stochastic methods would imply a significant burden on computational resources. A human e (...truncated)