Assessment of Statistical Moments of a Performance Function for Robust Design of Electromagnetic Devices

Abstract

This paper proposes a statistical probability approach to evaluate the quality loss function of electromagnetic design problems, which is expressed in terms of the first two statistical moments, mean and variance. A univariate dimension reduction method is employed to calculate the statistical moments of a performance function and their sensitivities accurately, and efficiently. Finally, the method is integrated into the robust design optimization algorithm. The proposed method explores an optimum design with statistical information so that it can provide a more accurate solution than other robust design methods. The proposed method is tested with a mathematical model and a blushless DC motor, and its numerical accuracy and efficiency are examined by comparison with existing methods.

A Comparative Study on Estimation Methods for Statistical Moments of Electromagnetic Performance Functions

“…The univariate DRM additively decomposes any Ndimensional performance function into one-dimension ones on the input design domain [5,7]: (2) where  i is the mean value of the ith random variable x i , and N is the number of random variables. The onedimensional numerical integration can be computed by using the moment-based integration rule (MBIR) [5], which is similar to Gaussian quadrature.…”

“…Using (2) and 3, the mean m h and variance of h(x) is expressed as (4) 5where and mean the jth weight factor and quadrature point for the ith random variable x i , respectively. In case of implementing robust design optimization (RDO), not only t the first two statistical moments but also their sensitivities are needed [5], [7]. By applying the partial derivative to (4) and 5, the sensitivities of two statistical moments at the kth design variable (d k = μ k ) are derived as follows:…”

“…Therefore, it is necessary to develop accurate and efficient numerical methods, which can estimate the two statistical moments of the performance function and their sensitivities. In accordance with this demand, various attempts such as worst-case scenario, experimental design technique, Monte Carlo Simulation (MCS), dimension reduction method (DRM) and so on have been made in our community to date [1][2][3][4][5][6][7][8][9]. They all focus on improving the product quality through minimizing variability of the output performance function.…”

“…Among them, it has been revealed that only two numerical techniques of MCS and DRM can quantitatively assess the first two statistical moments of EMrelated performance functions [4][5][6][7]. However, both of them have individual merits and demerits when applied to real-world engineering problems.…”

“…The MCS could be accurate for the moment estimation, but it requires a huge number of function evaluations, which is apt to cause an unacceptable computation cost [5]. To alleviate the above difficulty, the univariate DRM in [7] was introduced to evaluate the statistical moments and their sensitivities of EM performance functions. Therein any n-dimensional function is additively decomposed into one-dimension ones, and then moment-based numerical integration is conducted on the input design domain (i.e.…”