An Evolutional Topology Optimization of Electric Machines for Local Shape Modification and Visualization of Sensitivity Distribution Based on CMA‐ES

Abstract

This paper presents an evolutional topology optimization of electric machines, which works like gradient-based local search methods, based on the covariance matrix adaptation evolution strategy (CMA-ES). The present method enables to improve the performance of initial machine shapes without computing the deviation of the objectives. Moreover, for the resultant shapes obtained by the present topology optimization method, this paper also presents a sensitivity analysis method based on the normal distribution which is used to seek for solutions in CMA-ES. Using the present sensitivity analysis method, important parts which contribute to improve a machine characteristic can be visualized. The two present methods are applied to the optimization problem of a synchronous reluctance motor. It is shown that initial rotor shapes defined by designers can effectively be modified using the present optimization method. Moreover, it is also shown that the sensitivity distributions with respect to some machine characteristics are visualized by the present method. In addition, each machine characteristic is respectively improved by modifying the machine shape according to the respective sensitivities.

A topology optimization of on-chip planar inductor based on evolutional on/off method and CMA-ES

“…The CMA-ES generates candidate solutions using the multivariate normal distribution, and its distribution parameters are adapted on the basis of the evaluation results of candidate solutions. It has been shown in some references (Sato et al , 2020; Sato and Watanabe, 2023) that NGnet-TO with CMA-ES has overwhelming performance in comparison with NGnet-TO based on some conventional genetic algorithms. As for NGnet-TO with CMA-ES, M -candidate solutions, that is w m ( m = 1,… M ), are generated as follows: where m g , σ g and C g are the mean, step size and covariance matrix at g th iteration, respectively, which are the distribution parameters.…”

“…To apply the topology optimizations to the conductor designs of the on-chip inductors, we use the evolutional on/off method based on the normalized Gaussian network (NGnet), which is here called NGnet-TO for simplicity (Sato et al , 2020; Sato and Watanabe, 2023). Thanks to the evolutional approach, NGnet-TO can effectively consider various objectives and constraints which are difficult to evaluate the differentiability with respect to design variables.…”

“…In this work, we present an evolutional topology optimization of on-chip planar inductors. As an evolutional on/off-based method, the normalized Gaussian network-based topology optimization method (NGnet-TO) (Sato et al , 2015b; Sato et al , 2015a; Sato et al , 2020; Sato and Watanabe, 2023) is used, because this method relatively easily finds smooth and connected shapes in comparison with classical on/off-based methods. In addition, as an evolutionary algorithm, the covariance matrix adaptation evolution strategy (CMA-ES) (Hansen et al , 2015), which is one of the state-of-the-art algorithms for single-objective problems, is applied.…”