Airfoil Design under Uncertainty using Non-Intrusive Polynomial Chaos Theory and Utility Functions

Fast and accurate airfoil design under uncertainty using non-intrusive polynomial chaos (NIPC) expansions and utility functions is proposed. The NIPC expansions provide a means to efficiently and accurately compute statistical information for a given set of input variables with associated probability distribution. Utility functions provide a way to rigorously formulate the design problem. In this work, these two methods are integrated for the design of airfoil shapes under uncertainty. The proposed approach is illustrated on a numerical example of lift-constrained airfoil drag minimization in transonic viscous flow using the Mach number as an uncertain variable. The results show that compared with the standard problem formulation the proposed approach yields more robust designs. In other words, the designs obtained by the proposed approach are less sensitive to variations in the uncertain variables than those obtained with the standard problem formulation

Robust Airfoil Design Optimization using Stochastic Expansions and Utility Theory

Efficient and effective robust airfoil design optimization is proposed by integrating stochastic expansions and utility theory. In this work, the stochastic expansions are generated efficiently using non-intrusive polynomial chaos (NIPC) expansions. The robust design problem is formulated using utility functions which transfer a targeted response using a prescribed mathematical function to represent the designers\u27 risk preferences. The proposed approach is demonstrated using examples of lift-constrained airfoil drag minimization in transonic viscous flow using the Mach number as an uncertain variable in the range of 0.70 to 0.75. The results are compared with the common problem formulation for robust design of the minimizing the sum of the mean and standard deviation of performance metric, as well as with single- and multi-point deterministic optimization. The approach is demonstrated on two numerical test cases, one at relatively low lift coefficient of 0.5, and the other one at a high lift of 0.824. The constraints differ between the cases as well. In both cases, the proposed approach with utility function formulation achieves the most insensitive responses compared with the standard robust problem formulation and the single- and multi-point deterministic problem formulations

Using Stochastic Dominance in Multi-Objective Optimizers for Aerospace Design Under Uncertainty

In optimization under uncertainty for aerospace design, statistical moments of the quan- tity of interest are often treated as separate objectives and are traded off in a multi-objective optimization formulation. However, in many design problems the trade-off between sta- tistical moments can be large and the Pareto front representing this trade-off can include designs with undesirable behavior, such as being robust but being guaranteed to give a worse performance than another design. When a simulation of a system is computation- ally expensive, obtaining the full Pareto front is unfeasible and so spending optimization time obtaining such undesirable designs wastes time that could be spent obtaining more desirable alternatives. As a remedy, we propose an optimization formulation that can use multiple dominance criteria to avoid generating potentially inferior designs. We consider various orders of stochastic dominance as criteria to use alongside statistical moment based Pareto dominance, and illustrate how this gives rise to improved designs using a limited computational budget in an acoustic horn design problem and a transonic airfoil design problem.EPSRC DTA grant, grant number EP/L504920/

Quantifying Initial Condition and Parametric Uncertainties in a Nonlinear Aeroelastic System with an Efficient Stochastic Algorithm

There is a growing interest in understanding how uncertainties in flight conditions and structural parameters affect the character of a limit cycle oscillation (LCO) response, leading to failure of an aeroelastic system. Uncertainty quantification of a stochastic system (parametric uncertainty) with stochastic inputs (initial condition uncertainty) has traditionally been analyzed with Monte Carlo simulations (MCS). Probability density functions (PDF) of the LCO response are obtained from the MCS to estimate the probability of failure. A candidate approach to efficiently estimate the PDF of an LCO response is the stochastic projection method. The objective of this research is to extend the stochastic projection method to include the construction of B-spline surfaces in the stochastic domain. The multivariate B-spline problem is solved to estimate the LCO response surface. An MCS is performed on this response surface to estimate the PDF of the LCO response. The probability of failure is then computed from the PDF. This method is applied to the problem of estimating the PDF of a subcritical LCO response of a nonlinear airfoil in inviscid transonic flow. The stochastic algorithm provides a conservative estimate of the probability of failure of this aeroelastic system two orders of magnitude more efficiently than performing an MCS on the governing equations

Non-Intrusive, High-Dimensional Uncertainty Quantification for the Robust Simulation of Fluid Flows

Uncertainty Quantification is the field of mathematics that focuses on the propagation and influence of uncertainties on models. Mostly complex numerical models are considered with uncertain parameters or uncertain model properties. Several methods exist to model the uncertain parameters of numerical models. Stochastic Collocation is a method that samples the random variables of the input parameters using a deterministic procedure and then interpolates or integrates the unknown quantity of interest using the samples. Because moments of the distribution of the unknown quantity are essentially integrals of the quantity, the main focus will be on calculating integrals. Calculating an integral using samples can be done efficiently using a quadrature or cubature rule. Both sample the space of integration in a deterministic way and several algorithms to determine the samples exist, each with its own advantages and disadvantages. In the one-dimensional case a method is proposed that has all relevant advantages (positive weights, nested points and dependency on the input distribution). The principle of the introduced quadrature rule can also be applied to a multi-dimensional setting. However, if negative weights are allowed in the multi-dimensional case a cubature rule can be generated that has a very small number of points compared to the methods described in literature. The new method uses the fact that the tensor product of several quadrature rules has many points with the same weight that can be considered as on

Uncertainty Quantification of the Effects of Blade Damage on the Actual Energy Production of Modern Wind Turbines

Wind turbine blade deterioration issues have come to the attention of researchers and manufacturers due to the relevant impact they can have on the actual annual energy production (AEP). Research has shown how after prolonged exposure to hail, rain, insects or other abrasive particles, the outer surface of wind turbine blades deteriorates. This leads to increased surface roughness and material loss. The trailing edge (TE) of the blade is also often damaged during assembly and transportation according to industry veterans. This study aims at investigating the loss of AEP and efficiency of modern multi-MW wind turbines due to such issues using uncertainty quantification. Such an approach is justified by the stochastic and widely different environmental conditions in which wind turbines are installed. These cause uncertainties regarding the blade's conditions. To this end, the test case selected for the study is the DTU 10 MW reference wind turbine (RWT), a modern reference turbine with a rated power of 10 MW. Blade damage is modelled through shape modification of the turbine's airfoils. This is done with a purposely developed numerical tool. Lift and drag coefficients for the damaged airfoils are calculated using computational fluid dynamics. The resulting lift and drag coefficients are used in an aero-servo-elastic model of the wind turbine using NREL's code OpenFAST. An arbitrary polynomial chaos expansion method is used to estimate the probability distributions of AEP and power output of the model when blade damage is present. Average AEP losses of around 1% are predicted mainly due to leading-edge blade damage. Results show that the proposed method is able to account for the uncertainties and to give more meaningful information with respect to the simulation of a single test case

Machine Learning in Aerodynamic Shape Optimization

Machine learning (ML) has been increasingly used to aid aerodynamic shape optimization (ASO), thanks to the availability of aerodynamic data and continued developments in deep learning. We review the applications of ML in ASO to date and provide a perspective on the state-of-the-art and future directions. We first introduce conventional ASO and current challenges. Next, we introduce ML fundamentals and detail ML algorithms that have been successful in ASO. Then, we review ML applications to ASO addressing three aspects: compact geometric design space, fast aerodynamic analysis, and efficient optimization architecture. In addition to providing a comprehensive summary of the research, we comment on the practicality and effectiveness of the developed methods. We show how cutting-edge ML approaches can benefit ASO and address challenging demands, such as interactive design optimization. Practical large-scale design optimizations remain a challenge because of the high cost of ML training. Further research on coupling ML model construction with prior experience and knowledge, such as physics-informed ML, is recommended to solve large-scale ASO problems