Comparative study of methods for optimization of electromagnetic devices with uncertainty

Archive ouverte : Article de revue

Deng, Siyang | Brisset, Stephane | Clenet, Stéphane

Edité par HAL CCSD ; Emerald

International audience. This paper compares different probabilistic optimization methods dealing with uncertainties. Reliability-Based Design Optimization is presented as well as various approaches to calculate the probability of failure. They are compared in terms of precision and number of evaluations on mathematical and electromagnetic design problems to highlight the most effective methods. 1 INTRODUCTION In most optimization problems, the variables are usually considered as deterministic, i.e. without any variability. This traditional Deterministic Design Optimization (DDO) addresses only the performances but not the reliability and robustness. Since, the manufacturing process, the characteristics of materials, and the dimensions undergo variability, the device performances are in practice not deterministic but uncertain. Therefore, uncertainties related to design parameters are more and more taken into account in all engineering fields. Various formulations are available in the literature to express optimization problems with uncertainty, which can be mainly divided into Worst-Case Optimization (WCO), Robust Design Optimization (RDO), Reliability-Based Design Optimization (RBDO) and Reliability-Based Robust Design Optimization (RBRDO). WCO is a non-probabilistic approach that is based on minimax problem formulation. For instance, [1] solves a multi-objective problem that aims to minimize the objective function, its maximum feasible value in a surrounding box, and the greatest component of the objective function derivative in the surrounding box. The three others are probabilistic approaches that quantify the uncertainty of quantities of interest by probability distribution functions. RDO minimize a weighted sum of the mean value and variance of the objective function subject to deterministic constraints. RBDO minimize the mean value of the objective function subject to constraints on the probability of failure, i.e. constraint violation. Finally, RBRDO [2] integrates both last formulations by changing the objective function and constraints at the same time. A comparative study [3] of two RBDO Double-Loop Methods (DLM) with Monte Carlo Simulation (MCS) shows the interest of RBDO-DLM compared to MCS that requires a very large sampling to be accurate. However, other RBDO approaches such as Single-Loop (SLM) and Sequential Decoupled Methods (SDM) were not simultaneously investigated and this paper proposes to compare 6 algorithms belonging to the three aforementioned RBDO approaches with MCS in order to highlight the most accurate and the less time consuming. This is performed with a simple mathematical model and the multidisciplinary optimization problem of a safety transformer with uncertainty. The paper is organized into three parts. Chapter 2 introduces the different categories of RBDO approaches. Two examples are detailed in the chapter 3 and used to compare the different methods. Last chapter is the conclusion. 2 RELIABILITY-BASED DESIGN OPTIMIZATION The original formulation without any uncertainty or DDO is expressed as: (1) min í µí± í µí±(í µí±) í µí±. í µí±¡. í µí±(í µí±) ≤ 0 í µí± í µí°¿ ≤ í µí± ≤ í µí± í µí± where í µí± is the input design variable, í µí±(•) and í µí±(•) are the objective function and the inequality constraint, and í µí± í µí°¿ , í µí± í µí± represent the lower and upper bounds of í µí±, respectively. As the variability of the design variables is taken into account, the original deterministic input parameter í µí± should be replaced by a random input parameter í µí±, which follows the normal law in this paper for simplicity. The mean value of í µí± is denoted í µí± and is the unknown of the new design variables, the standard deviation is

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