Journal of Magnetics 24(3), 408-412 (2019) https://doi.org/10.4283/JMAG.2019.24.3.408 © 2019 Journal of Magnetics Most Probable Failure Point Update Method for Accurate First-Order Reliability-Based Electromagnetic Designs Byungsu Kang 1 , Jaegyeong Mun 1 , Jongsu Lim 1 , K. K. Choi 2 , and Dong-Hun Kim 1 * 1 Dept. of Electrical Eng., Kyungpook National Univ., Daegu 41566, Republic of Korea 2 Dept. of Mechanical and Industrial Eng., Univ. of Iowa, Iowa City, IA 52242-1527, USA (Received 22 April 2019, Received in final form 26 August 2019, Accepted 29 August 2019) A most probable failure point update method is proposed to obtain an accurate reliability-based design of elec- tromagnetic devices or systems in the presence of uncertainties. The first-order reliability method has been recently adopted to solve electromagnetic design problems. However, its result could be very erroneous espe- cially for nonlinear or multi-dimensional performance functions. To overcome the drawback, a three-step com- putational procedure is additionally executed to ensure prescribed design feasibility at an optimum obtained from the conventional first-order reliability method: failure rate calculation, reliability index update, and most probable point update. A mathematical example and a blushless DC motor design problem are provided to demonstrate numerical accuracy of the proposed method by comparison with the conventional method. Keywords : Electromagnetics, optimization, reliability theory, robustness 1. Introduction In recent years, one of probabilistic design methods, called reliability-based design optimization (RBDO), has drawn engineer’s attention in our community because it can systematically incorporate uncertainties into an early design stage. The RBDO formulation involves an objective function as deterministic optimization, and also contains probabilistic constraints for taking into account the prob- ability of the satisfaction/failure of constraint functions. Thus, accurate reliability assessment of performance func- tions concerned in constraint conditions is an essential step in the RBDO process. Up to date, various attempts to quantitatively predict the probability of failure of a performance function have been made in electromagnetic (EM) field analysis and design: first-order reliability method (FORM), moment method, Monte Carlo simulation (MCS) and so on [1-9]. Among them, FORM has been popularly used to evaluate the probability failure rate of an EM performance function because of its simple and efficient implementa- tion [2-4]. In FORM, design random variables are first transformed into the independent and standard normal probability distributions. Then, the performance function is approximated by the first-order Taylor series, and its failure rate is computed by means of a most probable failure point (MPP). According to the MPP search algorithm, there are two different approaches: reliability index ap- proach (RIA) and performance measure approach (PMA). It has been revealed that the FORM-based methods could be very erroneous if the performance function are highly nonlinear [5-7]. That is because FORM approximates the performance function using a liner function, and so it cannot reflect the complexity of nonlinear or high dimen- sional functions. As an effort to alleviate the difficulty in the FORM- based methods, the authors proposed a hybrid reliability analysis method for the failure rate calculation of non- linear or multi-dimensional EM performance functions [8]. Therein, the univariate dimension reduction method (DRM) is incorporated with RIA in order to enhance numerical accuracy of RIA. Through test problems, it has been shown that the method can estimate the probability of failure of a performance function more accurately than RIA and more efficiently than MCS. In this paper, a wealth of information not used in [8] is exploited to improve numerical accuracy of a solution of the conventional FROM-based RBDO. For the purpose of doing this, a so-called MPP-based DRM is newly used for enhanced RBDO design as well as accurate reliability ©The Korean Magnetics Society. All rights reserved. *Corresponding author: Tel: +82-53-950-5603 Fax: +82-53-950-5603, e-mail: dh29kim@ee.knu.ac.kr ISSN (Print) 1226-1750 ISSN (Online) 2233-6656