Esteban Ferrer

Soledad Le Clainche, Esteban Ferrer, Sam Gibson et al.

This review covers the new developments in machine learning (ML) that are impacting the multi-disciplinary area of aerospace engineering, including fundamental fluid dynamics (experimental and numerical), aerodynamics, acoustics, combustion and structural health monitoring. We review the state of the art, gathering the advantages and challenges of ML methods across different aerospace disciplines and provide our view on future opportunities. The basic concepts and the most relevant strategies for ML are presented together with the most relevant applications in aerospace engineering, revealing that ML is improving aircraft performance and that these techniques will have a large impact in the near future.

Juan Manzanero, Gonzalo Rubio, Esteban Ferrer et al.

We analyse instabilities due to aliasing errors when solving one dimensional non-constant advection speed equations and discuss means to alleviate these types of errors when using high order discontinuous Galerkin (DG) schemes. First, we compare analytical bounds for the continuous and discrete version of the PDEs. Whilst traditional $L^2$ norm energy bounds applied to the discrete PDE do not always predict the physical behaviour of the continuous version of the equation, more strict elliptic norm bounds correctly bound the behaviour of the continuous PDE. Having derived consistent bounds, we analyse the effectiveness of two stabilising techniques: over-integration and split form variations (conservative, non-conservative and skew-symmetric). Whilst the former is shown to not alleviate aliasing in general, the latter ensures an aliasing-free solution if the splitting form of the discrete PDE is consistent with the continuous equation. The success of split form de-aliasing is restricted to DG schemes with the summation-by-parts simultaneous-approximation-term (SBP-SAT) properties (e.g. DG with Gauss-Lobatto points). Numerical experiments are included to illustrate the theoretical findings.

Andrés M. Rueda-Ramírez, Juan Manzanero, Esteban Ferrer et al.

High-order DG methods have become a popular technique in computational fluid dynamics because their accuracy increases spectrally in smooth solutions with the order of the approximation. However, their main drawback is that increasing the order also increases the computational cost. Several techniques have been introduced in the past to reduce this cost. On the one hand, local mesh adaptation strategies based on error estimation have been proposed to reduce the number of degrees of freedom while keeping a similar accuracy. On the other hand, multigrid solvers may accelerate time marching computations for a fixed number of degrees of freedom. In this paper, we combine both methods and present a novel anisotropic p-adaptation multigrid algorithm for steady-state problems that uses the multigrid scheme both as a solver and as an anisotropic error estimator. To achieve this, we show that a recently developed anisotropic truncation error estimator [\textit{Rueda-Ramírez et al., Truncation Error Estimation in the p-Anisotropic DGSEM, J. of Scientific Computing}] is perfectly suited to be performed inside the multigrid cycle with negligible extra cost. Furthermore, we introduce a multi-stage p-adaptation procedure which reduces the computational time when very accurate results are required. The proposed methods are tested for the compressible Navier-Stokes equations, where we investigate two cases: the 2D flow on a flat plate is studied to assess accuracy and computational cost of the algorithm, where a speed-up of 816 is achieved compared to the traditional explicit method; and the 3D flow around a sphere is simulated and used to test the anisotropic properties of the proposed method, where a speed-up of 152 is achieved compared to the explicit method. The proposed multi-stage procedure achieved a speed-up of 2.6 in comparison to the single-stage method in highly accurate simulations.

Juan Manzanero, Gonzalo Rubio, David A. Kopriva et al.

We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn-Hilliard equation that satisfies the summation-by-parts simultaneous-approximation-term (SBP-SAT) property. The latter permits us to show that the discrete free-energy is bounded, and as a result, the scheme is provably stable. The scheme and the stability proof are presented for general curvilinear three-dimensional hexahedral meshes. We use the Bassi-Rebay 1 (BR1) scheme to compute interface fluxes, and an IMplicit-EXplicit (IMEX) scheme to integrate in time. Lastly, we test the theoretical findings numerically and present examples for two and three-dimensional problems.

Juan Manzanero, Esteban Ferrer, Gonzalo Rubio et al.

We analyse numerical errors (dissipation and dispersion) introduced by the discretisation of inviscid and viscous terms in energy stable discontinuous Galerkin methods. First, we analyse these methods using a linear von Neumann analysis (for a linear advection-diffusion equation) to characterise their properties in wave-number space. Second, we validate these observations using the 3D Taylor-Green Vortex Navier-Stokes problem to assess transitional/turbulent flows. We show that the dissipation introduced by upwind Riemann solvers affects primarily high wave-numbers. This dissipation may be increased, through a penalty parameter, until a critical value. However, further augmentation of this parameter leads to a decrease of dissipation, reaching zero for very large values. Regarding the dissipation introduced by second order derivatives, we show that this dissipa- tion acts at low and medium wave-numbers (lower wave-numbers compared to upwind Riemann solvers). In addition, we analyse the Spectral Vanishing Viscosity (SVV) technique, previously used in continuous discretisations (e.g. Fourier), to find that with an appropriate kernel (which damps selected modes) it is possible to control the amount of dissipation introduced in the low and medium wave-number range. Combining these ideas, we finally propose a DG-SVV approach that uses a Smagorinsky model to compute the numerical viscosity. This DG-SVV approach is tested in an isotropic laminar/turbulent under-resolved scenario. Combining the SVV technique with a low dissipation Riemann solver, we obtain a scheme capable of maintaining low dissipation levels for laminar flows, whilst providing the correct dissipation for all wave-number ranges in turbulent regimes.

Andrés M. Rueda-Ramírez, Gonzalo Rubio, Esteban Ferrer et al.

In the context of Discontinuous Galerkin Spectral Element Methods (DGSEM), $τ$-estimation has been successfully used for p-adaptation algorithms. This method estimates the truncation error of representations with different polynomial orders using the solution on a reference mesh of relatively high order. In this paper, we present a novel anisotropic truncation error estimator derived from the $τ$-estimation procedure for DGSEM. We exploit the tensor product basis properties of the numerical solution to design a method where the total truncation error is calculated as a sum of its directional components. We show that the new error estimator is cheaper to evaluate than previous implementations of the $τ$-estimation procedure and that it obtains more accurate extrapolations of the truncation error for representations of a higher order than the reference mesh. The robustness of the method allows performing the p-adaptation strategy with coarser reference solutions, thus further reducing the computational cost. The proposed estimator is validated using the method of manufactured solutions in a test case for the compressible Navier-Stokes equations.

David Huergo, Gonzalo Rubio, Esteban Ferrer

Reinforcement learning (RL) has emerged as a promising approach to automating decision processes. This paper explores the application of RL techniques to optimise the polynomial order in the computational mesh when using high-order solvers. Mesh adaptation plays a crucial role in improving the efficiency of numerical simulations by improving accuracy while reducing the cost. Here, actor-critic RL models based on Proximal Policy Optimization offer a data-driven approach for agents to learn optimal mesh modifications based on evolving conditions. The paper provides a strategy for p-adaptation in high-order solvers and includes insights into the main aspects of RL-based mesh adaptation, including the formulation of appropriate reward structures and the interaction between the RL agent and the simulation environment. We discuss the impact of RL-based mesh p-adaptation on computational efficiency and accuracy. We test the RL p-adaptation strategy on a 1D inviscid Burgers' equation to demonstrate the effectiveness of the strategy. The RL strategy reduces the computational cost and improves accuracy over uniform adaptation, while minimising human intervention.

Andrés Mateo-Gabín, Kenza Tlales, Eusebio Valero et al.

We present a novel unsupervised machine-learning sock sensor based on Gaussian Mixture Models (GMMs). The proposed GMM sensor demonstrates remarkable accuracy in detecting shocks and is robust across diverse test cases with significantly less parameter tuning than other options. We compare the GMM-based sensor with state-of-the-art alternatives. All methods are integrated into a high-order compressible discontinuous Galerkin solver, where two stabilization approaches are coupled to the sensor to provide examples of possible applications. The Sedov blast and double Mach reflection cases demonstrate that our proposed sensor can enhance hybrid sub-cell flux-differencing formulations by providing accurate information of the nodes that require low-order blending. Besides, supersonic test cases including high Reynolds numbers showcase the sensor performance when used to introduce entropy-stable artificial viscosity to capture shocks, demonstrating the same effectiveness as fine-tuned state-of-the-art sensors. The adaptive nature and ability to function without extensive training datasets make this GMM-based sensor suitable for complex geometries and varied flow configurations. Our study reveals the potential of unsupervised machine-learning methods, exemplified by this GMM sensor, to improve the robustness and efficiency of advanced CFD codes.

David Huergo, Laura Alonso, Saumitra Joshi et al.

We explore a reinforcement learning strategy to automate and accelerate h/p-multigrid methods in high-order solvers. Multigrid methods are very efficient but require fine-tuning of numerical parameters, such as the number of smoothing sweeps per level and the correction fraction (i.e., proportion of the corrected solution that is transferred from a coarser grid to a finer grid). The objective of this paper is to use a proximal policy optimization algorithm to automatically tune the multigrid parameters and, by doing so, improve stability and efficiency of the h/p-multigrid strategy. Our findings reveal that the proposed reinforcement learning h/p-multigrid approach significantly accelerates and improves the robustness of steady-state simulations for one dimensional advection-diffusion and nonlinear Burgers' equations, when discretized using high-order h/p methods, on uniform and nonuniform grids.

David Huergo, Martín de Frutos, Eduardo Jané et al.

We present a novel approach to automate and optimize anisotropic p-adaptation in high-order h/p solvers using Reinforcement Learning (RL). The dynamic RL adaptation uses the evolving solution to adjust the high-order polynomials. We develop an offline training approach, decoupled from the main solver, which shows minimal overcost when performing simulations. In addition, we derive an inexpensive RL-based error estimation approach that enables the quantification of local discretization errors. The proposed methodology is agnostic to both the computational mesh and the partial differential equation to be solved. The application of RL to mesh adaptation offers several benefits. It enables automated and adaptive mesh refinement, reducing the need for manual intervention. It optimizes computational resources by dynamically allocating high-order polynomials where necessary and minimizing refinement in stable regions. This leads to computational cost savings while maintaining the accuracy of the solution. Furthermore, RL allows for the exploration of unconventional mesh adaptations, potentially enhancing the accuracy and robustness of simulations. This work extends our original research, offering a more robust, reproducible, and generalizable approach applicable to complex three-dimensional problems. We provide validation for laminar and turbulent cases: circular cylinders, Taylor Green Vortex and a 10MW wind turbine to illustrate the flexibility of the proposed approach.

Martín de Frutos, Oscar A. Marino, David Huergo et al.

We develop a torque-pitch control framework using deep reinforcement learning for wind turbines to optimize the generation of wind turbine energy while minimizing operational noise. We employ a double deep Q-learning, coupled to a blade element momentum solver, to enable precise control over wind turbine parameters. In addition to the blade element momentum, we use the wind turbine acoustic model of Brooks Pope and Marcolini. Through training with simple winds, the agent learns optimal control policies that allow efficient control for complex turbulent winds. Our experiments demonstrate that the reinforcement learning is able to find optima at the Pareto front, when maximizing energy while minimizing noise. In addition, the adaptability of the reinforcement learning agent to changing turbulent wind conditions, underscores its efficacy for real-world applications. We validate the methodology using a SWT2.3-93 wind turbine with a rated power of 2.3 MW. We compare the reinforcement learning control to classic controls to show that they are comparable when not taking into account noise emissions. When including a maximum limit of 45 dB to the noise produced (100 meters downwind of the turbine), the extracted yearly energy decreases by 22%. The methodology is flexible and allows for easy tuning of the objectives and constraints through the reward definitions, resulting in a flexible multi-objective optimization framework for wind turbine control. Overall, our findings highlight the potential of RL-based control strategies to improve wind turbine efficiency while mitigating noise pollution, thus advancing sustainable energy generation technologies

Laura Botero Bolívar, David Huergo, Fernanda L. dos Santos et al.

Fast-turn around methods to predict airfoil trailing-edge noise are crucial for incorporating noise limitations into design optimization loops of several applications. Among these aeroacoustic predictive models, Amiet's theory offers the best balance between accuracy and simplicity. The accuracy of the model relies heavily on precise wall-pressure spectrum predictions, which are often based on single-equation formulations with adjustable parameters. These parameters are calibrated for particular airfoils and flow conditions and consequently tend to fail when applied outside their calibration range. This paper introduces a new wall-pressure spectrum empirical model designed to enhance the robustness and accuracy of current state-of-the-art predictions while widening the range of applicability of the model to different airfoils and flow conditions. The model is developed using AI-based symbolic regression via a genetic-algorithm-based approach, and applied to a dataset of wall-pressure fluctuations measured on NACA 0008 and NACA 63018 airfoils at multiple angles of attack and inflow velocities, covering turbulent boundary layers with both adverse and favorable pressure gradients. Validation against experimental data (outside the training dataset) demonstrates the robustness of the model compared to well-accepted semi-empirical models. Finally, the model is integrated with Amiet's theory to predict the aeroacoustic noise of a full-scale wind turbine, showing good agreement with experimental measurements.

Daniel Soler, Oscar Mariño, David Huergo et al.

We propose a reinforcement learning strategy to control wind turbine energy generation by actively changing the rotor speed, the rotor yaw angle and the blade pitch angle. A double deep Q-learning with a prioritized experience replay agent is coupled with a blade element momentum model and is trained to allow control for changing winds. The agent is trained to decide the best control (speed, yaw, pitch) for simple steady winds and is subsequently challenged with real dynamic turbulent winds, showing good performance. The double deep Q- learning is compared with a classic value iteration reinforcement learning control and both strategies outperform a classic PID control in all environments. Furthermore, the reinforcement learning approach is well suited to changing environments including turbulent/gusty winds, showing great adaptability. Finally, we compare all control strategies with real winds and compute the annual energy production. In this case, the double deep Q-learning algorithm also outperforms classic methodologies.