Efficient Bilevel Surrogate Approach for Optimization Under Uncertainty of Shock Control Bumps

The assessment of uncertainties is essential in aerodynamic shape optimization problems to come up with configurations that are more robust against operational and geometrical uncertainties. However, exploring the stochastic design space significantly increases the computational cost. The aim of this paper is to develop a framework for efficient optimization under uncertainty by means of a bilevel surrogate approach and to apply it to the robust design of a retrofitted shock control bump over an airfoil. The framework combines a surrogate-based optimizer with an efficient surrogate-based approach for uncertainty quantification. The optimizer efficiently finds the global optimum of a given quantile of the quantity of interest through the combination of adaptive sampling and a moving trust region. At each iteration of the optimization, the surrogate-based uncertainty quantification uses an active infill criterion to accurately quantify the quantile requiring a reduced number of samples. Two different quantiles of the drag are chosen for the design of the shock control bump: the 95% to increase the robustness at off-design conditions, and the 50% for a configuration that is preferred for day-to-day operations. In both cases, the optimum bumps are more robust, compared to the one obtained through classical deterministic optimization

Robust Design of Transonic Natural Laminar Flow Wings under Environmental and Operational Uncertainties

The introduction of laminar flow configurations is envisioned to provide new opportunities to further reduce aircraft fuel consumption. The robustness of laminar wings is critical, both against instabilities that can unexpectedly trigger transition and against off-design conditions outside the cruise point. However, current inverse design methodologies not only provide suboptimal configurations, but are unable to come up with robust configurations. The objective of this paper is the development and demonstration of a framework for the robust direct design of transonic natural laminar flow wings using state-of-the-art industrial tools such as computational fluid dynamics, linear stability theory and surrogate models. The deterministic optimization problem, which serves as a baseline, searches for the optimum shape that minimizes drag applying a surrogate based optimization strategy. In that case Cross-Flow and Tollmien-Schlichting critical N-Factors are fixed according to calibration data. For the robust approach, uncertainties in these critical N-Factors as well as operational conditions such as Mach number are considered to account for situations that could prematurely trigger transition and thus significantly decrease performance. The surrogate based optimizer is therefore coupled with a surrogate based uncertainty quantification methodology, following a bi-level approach. The objective function shifts towards the expectation of the drag to minimize average fuel consumption, or the 95% quantile to account for extreme events. The framework is able to come up with state-of-the-art natural laminar configurations for a short-haul civil aircraft configuration. The deterministic optimum is able to delay transition till 60% of the wing upper surface where the shock is present but is highly sensitive to small changes in the predefined critical N-Factors, as minor deviations will lead to fully turbulent configuration and hence an increase in drag. The robust configurations are more balanced, as the transition location smoothly moves upstream as the critical N-Factors are reduced. As a direct consequence, obtained pressure profiles are more resistant against instabilities, extending the current design envelope of natural laminar flow wings

Parametric Compressible Flow Predictions using Physics-Informed Neural Networks

The numerical approximation of solutions to the compressible Euler and Navierstokes equations is a crucial but challenging task with relevance in various fields of science and engineering. Recently, methods from deep learning have been successfully employed for solving partial differential equations by incorporating the equations into a loss function that is minimized during the training of a neural network. This approach yields a so-called physics-informed neural network which does not rely on a classical discretization and can address parametric problems in a straightforward manner. Therefore, it avoids characteristic difficulties of traditional approaches, such as finite volume methods. This has raised the question, whether physics-informed neural networks may be a viable alternative to conventional methods for computational fluid dynamics. Here, we show a new physics-informed neural network training procedure to approximately solve the two-dimensional compressible Euler equations, which makes use of artificial dissipation during the training process. We demonstrate how additional dissipative terms help to avoid unphysical results and how the additional numerical viscosity can be reduced during training while iterating towards a solution. Furthermore, we showcase how this approach can be combined with parametric boundary conditions. Our results highlight the appearance of unphysical results when solving compressible flows with physics-informed neural networks and offer a new approach to overcome this problem. We therefore expect that the presented methods enable the application of physics-informed neural networks for previously difficult to solve problems

Physics-based Regularization of Neural Networks for Aerodynamic Flow Prediction

Aerodynamic data plays a central role in the process of aircraft design, optimization and certification. For these processes a vast amount of data is required for various flight conditions throughout the flight envelope. Currently this data is commonly produced using Computational Fluid Dynamics (CFD). However, such simulations based on the Reynolds-averaged Navier-Stokes equations are computationally expensive and become prohibitive for tasks such as load analysis and shape optimization. During the last decades, this has motivated research focusing on the use of data-driven models with lower evaluation times than the full-order model to replace high-fidelity CFD simulations. More recently, deep learning approaches have gathered significant interest in the aerodynamic community. For the task of predicting surface pressure coefficient distributions, one of the proposed models consists of a multilayer perceptron that for each node in the mesh outputs a prediction of the local coefficient based on the node coordinates and the global operational conditions. If required, known integration formulas are used to compute integral quantities, such as the lift and pitching moment coefficients, based on the previously obtained distribution. In this paper we The method is tested for the NASA Common Research Model transport aircraft with an underlying mesh consisting of around 500,000 surface points. Results show that, when using the mentioned approach for the fine-tuning of a trained multilayer perceptron, physical knowledge can be explicitly revealed to the deep learning model but only limited improvements are achieved in the predictions of the lift and pitching moment coefficients

Linear Frequency Domain Method For Aerodynamic Applications

Applications such as load alleviation involve a wide range of parameters including a multitude of different Mach numbers, angles of attack and load cases and therefore create the demand for rapid prediction of unsteady air loads. First, there is a need for an enhanced prediction accuracy including viscous effects and shocks for a more reliable judgement of aerodynamic behavior compared to the classically-used methods based on the potential theory for such applications. Secondly, short turnaround times must be guaranteed, and this in turn means to find a suitable replacement of the tedious and time-consuming un- steady Navier-Stokes solvers. Driven by these requirements for accurate and fast prediction of air loads, a time-linearized unsteady Navier-Stokes method was developed also known as linear frequency domain method (LFD). The LFD in the DLRs TAU suite is based on the modeling of a damped harmonic oscillator, and it has been shown to be accurate and efficient for the evaluation of unsteady air loads at transonic and partly separated flow conditions [1]. Since then, the LFD method has been continuously extended and applied for various applications. The scope of target applications of the LFD has been growing consistently including different topics in aeroelasticity, where the determined surface pressure and surface skin friction distributions make an important contribution. Moreover, the time-linearized method can also be used for the efficient evaluation of flight dynamic (flight mechanical) characteristics relevant for the stability and control behavior of an aircraft. Furthermore, gust loads were successfully predicted which are important for structural and control surface design [2, 3] as well as control system performance. A recent and demanding application was the extension of the LFD method for fluidic actuators. Thus, the LFD was adopted for simulating pulsating blowing to avoid the enormously long transient phase inherently occurring during time-marching Navier-Stokes simulations. Several applications involving industrial relevant configurations are presented and discussed to outline the maturity of the method and to demonstrate the versatility of the technique

Adjoint high-dimensional aircraft shape optimization using a CAD-ROM parameterization

A gradient-based aeroelastic shape optimization framework making use of a reduced order model to substitute a parameterization based on computer-aided design software is presented. This parameterization concept is not novel in principle, but it is embedded here in a complex high-fidelity optimization process and proven for a high-dimensional design space. The design software is used initially to generate a parametric model of a three-dimensional transport aircraft configuration. To streamline the actual optimization process, the computer-aided design model is replaced with a parametric reduced order model based on proper orthogonal decomposition that is capable of predicting discrete surface displacement fields as a function of the design parameters. During the optimization, surface displacements are computed according to the current design parameters and applied on the baseline shape. In every optimization step, the aircraft's steady-state equilibrium of forces and moments are satisfied by a trimming algorithm and the Reynolds-averaged Navier-Stokes solver TAU is coupled with a linear structural finite-element method model. Gradients are computed analytically using geometric sensitivities provided by the reduced order model and by applying the adjoint method to the flow solver and the mesh deformation tool. The workflow is embedded within FlowSimulator, a multiphysics environment for high performance computing. The optimization process is demonstrated for a high-dimensional wing parameterization with 126 degrees of freedom. The aircraft cruise drag could be significantly reduced by 6% on a series of three continuously refined meshes for the aerodynamic analysis. For an accurate representation of the optimal shape by the computer-aided design software after the optimization, the approximation error introduced by the reduced order modelling approach must be sufficiently small. Therefore, the accuracy of the predictions was analyzed. The results identify the main source of the geometric error and quantify their effect on the drag reduction gained by the optimization. We dedicate this article to the memory of our colleague and friend Arno Ronzheimer, whose devotion to CAD modeling was unsurpassed

A Machine Learning based Expert System for Optimizing CFD Solver Parameters

Computational Fluid Dynamics is a viable tool in the field of aerodynamics enabling to reduce time, effort and budget required for experimental testing. Although powerful and established for various years, it remains a complex tool calling for experienced users to ensure consistent high-quality results. This complexity primarily stems from the underlying model, namely the Navier-Stokes equations typically combined with a set of equations resolving the effects of turbulence. Additionally, to obtain accurate high-fidelity result appropriate meshes are required. As a consequence, a substantial number of parameters needs to be selected carefully and the quality of a result often highly depends on individual knowledge and experience of a user. Hence, a strong desire exists to reduce the number of input parameters without causing a loss of accuracy and efficiency. Such reduction of parameters might be viewed as a prerequisite to CFD as a tool in process chains for multidisciplinary applications where typically no user interaction is possible. In this article we propose a machine-learned Expert System for CFD to provide guidance for users in selecting optimal or at least near optimal parameter combinations. The proposed Expert System is divided into two macro steps, the surrogate model and a genetic algorithm to determine from the surrogate model the parameters. Numerical examples are presented to demonstrate the approach

Data-driven Bayesian inference of turbulence model closure coefficients incorporating epistemic uncertainty

We introduce a framework for statistical inference of the closure coefficients using machine learning methods. The objective of this framework is to quantify the epistemic uncertainty associated with the closure model by using experimental data via Bayesian statistics. The framework is tailored towards cases for which a limited amount of experimental data is available. It consists of two components. First, by treating all latent variables (non-observed variables) in the model as stochastic variables, all sources of uncertainty of the probabilistic closure model are quantified by a fully Bayesian approach. The probabilistic model is defined to consist of the closure coefficients as parameters and other parameters incorporating noise. Then, the uncertainty associated with the closure coefficients is extracted from the overall uncertainty by considering the noise being zero. The overall uncertainty is rigorously evaluated by using Markov-Chain Monte Carlo sampling assisted by surrogate models. We apply the framework to the Spalart-Allmars one-equation turbulence model. Two test cases are considered, including an industrially relevant full aircraft model at transonic flow conditions, the Airbus XRF1. Eventually, we demonstrate that epistemic uncertainties in the closure coefficients result into uncertainties in flow quantities of interest which are prominent around, and downstream, of the shock occurring over the XRF1 wing. This data-driven approach could help to enhance the predictive capabilities of computational fluid dynamics (CFD) in terms of reliable turbulence modeling at extremes of the flight envelope if measured data is available, which is important in the context of robust design and towards virtual aircraft certification. The plentiful amount of information about the uncertainties could also assist when it comes to estimating the influence of the measured data on the inferred model coefficients. Finally, the developed framework is flexible and can be applied to different test cases and to various turbulence models