Robust Airfoil Optimization Using a Conditional Expected Maximum Value Approach

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 the following: “How important is the choice of how to represent input uncertainties mathematically in robust airfoil optimization?” It specifically compares 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.

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.

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.

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.

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.

Surrogate-based Robust Airfoil Optimization under Aleatory Flight Condition and Geometric Uncertainties

Robust design optimizations of an airfoil are conducted to increase the aerodynamic performance and to reduce the performance deterioration due to the manufacturing variations of a human-powered aircraft. Randomly-generated perturbed designs are evaluated for the robustness index. Maximum performance difference is used for robustness definition to prevent the worst performance deterioration. The approach can design less-sensitive airfoil against the manufacturing variations while improving the performance compared to initial airfoil. In addition, the essential parameter to control performance and robustness is discussed.

The optimization of train plans is highly dependent on the space–time distribution of passenger demand in high-speed railway systems. A train plan usually needs to be implemented on multiple operation days, and obviously the amount and space–time distribution of demand over these days has noteworthy differences. To ensure the same train plan is able to be implemented on multiple operation days while effectively satisfying the different levels of demand on those days, a novel robust optimization of a high-speed railway train plan based on multiple demand scenarios is performed in this research. Firstly, the passenger demand of each operation day is described as a demand scenario, and a candidate train set is generated that is able to satisfy the multiple demand scenarios. Then, a regret value corresponding to the total cost, including the train operation cost and passenger travel expense, is proposed to measure the deviation in the costs generated between the robust and the optimal train plan under each demand scenario. Then a robust optimization model for a high-speed railway train plan is constructed to minimize the maximum regret value. Moreover, a simulated annealing algorithm for solving the model is designed by constructing some neighborhood solution search strategies for multiple demand scenarios. Finally, the validity and feasibility of the proposed robust optimization method for train planning are verified on the Shijiazhuang–Jinandong high-speed railway line in China.

In this paper, we present a novel approach to find the robust optimum of integrated photonic devices affected by implementation error. We apply robust optimization on a cheap surrogate model of an expensive integrated photonic device simulation. Robust optimization is the process of finding the best design point, in the presence of uncertainties, by minimizing the maximum realizable value of the objective with respect to the uncertainty set. The interpolation across the sampled values is performed by Kriging, a surrogate modelling technique with a statistical basis, which provides an estimate of the error in the interpolation. Using this error estimate, Kriging provides a metric, known as expected improvement, which indicates where a new sample should be added in order to reach the nominal optimum efficiently. In this work, we combine expected improvement with robust optimization in order to exhibit how a robust optimum on an integrated photonic device can be found. This method is illustrated by robustly minimizing two objectives, excess loss and imbalance, of a 2×2 multimode interference coupler in the presence of uncertainties arising from variations in the fabricated design geometry. We also demonstrate how different weighting factors affect the multi-objective optimization problem.

This chapter describes the use of a multiobjective memetic algorithm in aerodynamic shape optimization applications using computational fluid dynamics(CFD). A systematic overviewon multiobjective optimization is given. It covers parameterized scalarization, population-based Pareto front forming using evolutionary algorithms, formulation of Pareto front improvement using local search methods, and various possible heuristics that can be applied in formulating efficient memetic algorithms. A constrained local search, combined with a non-dominated sorting genetic algorithm with niching in both objective and design space is proposed in the paper. Test function results are given and finally the method is used to solve the robust airfoil design problem using a multi-point formulation.

In this paper, we propose a novel design-for-manufacture strategy for integrated photonics which specifically addresses the commonly encountered scenario in which probability distributions of the manufacturing variations are not available, however their bounds are known. The best design point for the device, in the presence of these uncertainties, can be found by applying robust optimization. This is performed by minimizing the maximum realizable value of the objective with respect to the uncertainty set so that an optimum is found whose performance is relatively immune to fabrication variations. Instead of applying robust optimization directly on a computationally expensive simulation model of the integrated photonic device, we construct a cheap surrogate model by uniformly sampling the simulated device at different values of the design variables and interpolating the resulting objective using a Kriging metamodel. By applying robust optimization on the constructed surrogate, the global robust optimum can be found at low computational cost. As an illustration of the method's general applicability, we apply the robust optimization approach on a 2x2 multimode interference (MMI) coupler. We robustly minimize the imbalance in the presence of uncertainties arising from variations in the fabricated design geometry. For this example device, we also study the influence of the number of sample points on the quality of the metamodel and on the robust optimization process.

飛行中の局所変形を考慮した人力飛行機用翼型のロバスト空力最適化

Cascading failures has seen as deadly on the network that causing sudden great damage in a different area. In the everyday life, different traffic loads can be carried by different networks those traffic loads whose values are changing from the load forwarding in the equivalent network and interruption provoked by interdependent network. Amount of opposed types of interdependencies demonstrate that higher interdependency drives to a smaller possibility of a wide blackout, inter- dependencies between networks are modeled and process of cascade failure is learned regarding their effects on the propagation failure. For then, reinstallation effects count on the activation timing, reinstallation possibility and primacy of the reinstallation actions along with the additional interruption. In this work, interdependencies between networks are modeled and the cascade failure process is learned desiring their effects on failure propagation. Our results demonstrate the necessity of reducing the high cascading failure by designing and controlling interdependent network sufficiently. Besides, when the transmission inability is below the maximum values some of power system can be useful.

Robust optimization of 2D airfoils driven by full Navier–Stokes computations

Optimization is a process for finding maximum or minimum of a function subject to set of functions. In other words optimization finds the most suitable value for a function within a given domain. Global Optimization means finding minimum value of all local minima and maxima value from all local maxima. The procedure to find out the global maxima or minima point is called a Global Optimization. Teaching-Learning-Based Optimization (TLBO) is recently being used as a new, reliable, accurate and robust optimization technique scheme for global optimization over continuous spaces. In this paper we are comparing the performance of various versions of TLBO.