Robust Airfoil Optimization and the Importance of Appropriately Representing Uncertainty

Robust optimization and stochastic programming approaches for medium-term production scheduling of a large-scale steelmaking continuous casting process under demand uncertainty

The importance of designing airfoils to be robust with respect to uncertainties in operating conditions is well recognized. However, often the probability distribution of such uncertainties does not exist or is unknown, and a designer looking to perform a robust optimization is tasked with deciding how to represent these uncertainties within the optimization framework. This paper asks ``how important is the choice of how to represent input uncertainties mathematically in robust airfoil optimization?", specifically comparing probabilistically based aleatory uncertainties and interval based epistemic uncertainties. This is first investigated by considering optimizations on several algebraic test problems, which illustrate the mechanisms by which the representation of uncertainty becomes significant in a robust optimization. This insight is then used to predict and subsequently demonstrate that for two airfoil design problems the advantage of doing a robust optimization over a deterministic optimization is similar regardless of how the input uncertainties are represented mathematically. The benefit of this is potentially eliminating the time required to establish an accurate representation of the uncertainties from the preliminary stage of design, where time is a valuable resource.

In order to improve the aerodynamic stability of a two-dimensional airfoil with deformable trailing edge during the change of external conditions,a robust optimization method considering the uncertainty was proposed. Based on class-shape function transformation( CST) method,a parametric model was established to represent the geometry of a two-dimensional airfoil with a deformable trailing edge. The differences between robust optimization method and deterministic optimization method were discussed. Taking into account the uncertain airfoil geometry and inlet Mach number,the robust optimization method was applied to maximize the mean value of the lift-to-drag ratio and minimize its standard deviation. The actuation power requirement of the optimized airfoil with a deformable trailing edge was calculated. The results show that the robust optimization method can help to improve the aerodynamic performance of morphing airfoil and reduce the sensitivity of this performance to inlet Mach number simultaneously and to reduce the actuation power requirement of the robust optimal airfoil.

A robust optimization framework for energy management of CCHP users with integrated demand response in electricity market

In electric machine design, efficient methods for the optimization of the geometry and associated parameters are essential. Nowadays, it is necessary to address the uncertainty caused by manufacturing or material tolerances. This work presents a robust optimization strategy to address uncertainty in the design of a three-phase, six-pole permanent magnet synchronous motor (PMSM). The geometry is constructed in a two-dimensional framework within MATLAB®, employing isogeometric analysis (IGA) to enable flexible shape optimization. The main contributions of this research are twofold. First, we integrate shape optimization with parameter optimization to enhance the performance of PMSM designs. Second, we use robust optimization, which creates a min–max problem, to ensure that the motor maintains its performance when facing uncertainties. To solve this bilevel problem, we work with the maximal value functions of the lower-level maximization problems and apply a version of Danskin’s theorem for the computation of generalized derivatives. Additionally, the adjoint method is employed to efficiently solve the lower-level problems with gradient-based optimization. The paper concludes by presenting numerical results showcasing the efficacy of the proposed robust optimization framework. The results indicate that the optimized PMSM designs not only perform competitively compared to their non-robust counterparts but also show resilience to operational and manufacturing uncertainties, making them attractive for industrial applications.

Application of deep learning based multi-fidelity surrogate model to robust aerodynamic design optimization

To succeed in a demanding and competitive market, great attention needs to be given to the process of product design. Incorporating optimization into the process enables the designer to find high-quality products according to their simulated performance. However, the actual performance may differ from the simulation results due to a variety of uncertainty factors. Robust optimization is commonly used to search for products that are less affected by the anticipated uncertainties. Changeability can improve the robustness of a product, as it allows the product to be adapted to a new configuration whenever the uncertain conditions change. This ability provides the changeable product with an active form of robustness. Several methodologies exist for engineering design of changeable products, none of which includes optimization. This study presents the Active Robust Optimization (ARO) framework that offers the missing tools for optimizing changeable products. A new optimization problem is formulated, named Active Robust Optimization Problem (AROP). The benefit in designing solutions by solving an AROP lies in the realistic manner adaptation is considered when assessing the solutions' performance. The novel methodology can be applied to optimize any product that can be classified as a changeable product, i.e., it can be adjusted by its user during normal operation. This definition applies to a huge variety of applications, ranging from simple products such as fans and heaters, to complex systems such as production halls and transportation systems. The ARO framework is described in this dissertation and its unique features are studied. Its ability to find robust changeable solutions is examined for different sources of uncertainty, robustness criteria and sampling conditions. Additionally, a framework for Active Robust Multi-objective Optimization is developed. This generalisation of ARO itself presents many challenges, not encountered in previous studies. Novel approaches for evaluating and comparing changeable designs comprising multiple objectives are proposed along with algorithms for solving multi-objective AROPs. The framework and associated methodologies are demonstrated on two applications from different fields in engineering design. The first is an adjustable optical table, and the second is the selection of gears in a gearbox.

Robust Airfoil Optimization Using a Conditional Expected Maximum Value Approach

Deterministic engineering design often leads to unexpected or physically unrealizable results. This is due to the fact that deterministic design is not able to capture the effects of even slight natural fluctuations of parameters. Deterministic transonic shape optimization is no exception: deterministic designs can result in dramatically inferior performance when the actual operating conditions are different from the design conditions used during a deterministic optimization procedure. The goal of this paper is to overcome the off-design performance degradation of deterministic transonic shape optimization by using two different optimization approaches to produce robust designs. Two criteria, the well-known maximum/minimum expected value criterion (MEV) and the alternative expected maximum/minimum value criterion (EMV), are studied and applied to improve an initial RAE 2822 design. It turns out that EMV is much easier to implement than MEV, given a deterministic optimization code, and may provide a promising method for optimizing design shapes under uncertainty.

Incorporate robust optimization and demand defense for optimal planning of shared rental energy storage in multi-user industrial park

Commitment and dispatch of heat and power units via affinely adjustable robust optimization

Robust optimization of a marine current turbine using a novel robustness criterion

Mars entry trajectory planning using robust optimization and uncertainty quantification

The coupling of finite element simulations to mathematical optimization techniques has contributed significantly to product improvements and cost reductions in the metal forming industries. The next challenge is to bridge the gap between deterministic optimization techniques and the industrial need for robustness. This paper introduces a generally applicable strategy for modeling and efficiently solving robust optimization problems based on time consuming simulations. Noise variables and their effect on the responses are taken into account explicitly. The robust optimization strategy consists of four main stages: modeling, sensitivity analysis, robust optimization and sequential robust optimization. Use is made of a metamodel-based optimization approach to couple the computationally expensive finite element simulations with the robust optimization procedure. The initial metamodel approximation will only serve to find a first estimate of the robust optimum. Sequential optimization steps are subsequently applied to efficiently increase the accuracy of the response prediction at regions of interest containing the optimal robust design. The applicability of the proposed robust optimization strategy is demonstrated by the sequential robust optimization of an analytical test function and an industrial V-bending process. For the industrial application, several production trial runs have been performed to investigate and validate the robustness of the production process. For both applications, it is shown that the robust optimization strategy accounts for the effect of different sources of uncertainty onto the process responses in a very efficient manner. Moreover, application of the methodology to the industrial V-bending process results in valuable process insights and an improved robust process design.

<div class="section abstract"><div class="htmlview paragraph">Existing multi-axle steering system designs generally use the deterministic optimization method without considering the uncertainties during the design process; therefore an actual steering movement may deviate from the ideal movement calculated by some mathematical models. In order to make design results have less sensitive to the uncertainties in the design process, some uncertainties need be taken into account at the early design stage. This paper proposes a robust optimization design method for a double front axle steering system (DFASS) of heavy trucks based on Monte Carlo method. The DFASS consists of two trapezoidal steering mechanisms (TSM) and one rocker system, and the optimization objectives of DFASS include the minimum mean value and variance of the maximum turning angle error of the TSM and rocker system. In addition, the robust optimization model includes 13 design variables which are all geometry parameters of DFASS and represented by normal distribution. Through the orthogonal experiment, we obtain the important factors affecting optimization objectives and build the response surface models of optimization objective. Based on the response surfaces, robustness design of DFASS is performed. By simulating analysis, the improved robustness of DFASS based on robust design method is approved.</div></div>