Data-Driven Multi-Fidelity Modelling of Large Aspect Ratio Wings with Distributed Propellers

Data-Driven Multi-Fidelity Modelling of Large Aspect Ratio Wings with Distributed Propellers Zachary K Ciera∗, Bryn Jones†, Marco Fossati‡ University of Strathclyde, Glasgow, G1 1XJ, United Kingdom Strut-braced large aspect ratio wings (LARW) with distributed hybrid-electric propulsion (DHEP) show promise in reducing noise and emissions from aircraft. The adoption of these technologies will further aviation sustainability and will improve the quality of life of communities near to airports. Existing low-fidelity models used for optimizing these configurations often neglect crucial flow physics typical of such complex and unconventional configurations, limiting their accuracy. This work explores two methods to enhance prediction capabilities suitable for early design phases: correcting the prediction of an existing low-fidelity model using an error-based approach and developing a stand-alone reduced order model based on high-fidelity Reynolds-Averaged Navier-Stokes (RANS) simulations. An error-based model aims to correct outputs from the low-fidelity surrogate, while a pure high-fidelity surrogate is constructed using the same samples. This work aims at understanding if the behaviour of the error between lowand high-fidelity predictions will have less nonlinearity than the flow itself, allowing for a more efficient and more effective building of the reduced order model. Both methods are evaluated by predicting the performance of an untrained configuration within the training space to determine which approach more effectively predicts lift and drag coefficients. I. Nomenclature AoA = Angle of Attack 𝐶𝐷= Drag coefficient 𝐶𝐿= Lift coefficient QRSM = Quadratic Response Surface Method RBF = Radial Basis Function GP = Gaussian Processing II. Introduction and Motivation F or several years there has been a decided effort to reduce the overall impact of aviation on the environment, aiming to achieve net-zero emissions by 2050 [ 1 , 2 ]. Whilst much of the discussion has been focused on the global impact of the aviation sector, there is also a focus on local communities near to airports. In this case, the goal has been to reduce noise and improve air quality specifically in this smaller environment [ 3 , 4 ]. This has led to the drive for researchers to develop and explore new design concepts. One such concept is to employ the use of Large Aspect Ratio Wings (LARW) to reduce induced drag, improving aircraft efficiency and thus reducing fuel consumption and emissions [ 5 , 6 ]. Furthermore, the employment of distributed propulsion has allowed the leveraging of blown wing designs to enhance local lift and reduce propulsive power requirement [ 7 ]. These concepts proved to be effective in improving aircraft efficiency with the ability to reduce emissions, noise, and reduce take-off/landing lengths – improving aircraft versatility [ 8 , 9 ]. With the understanding that these technologies show promise, there has been a concentrated effort to develop a better understanding of how LARW and DHEP could be integrated to define a highly-effective sustainable aircraft for the future. Preliminary work had already been completed as part of an optimisation study to produce an airframe that would work within set constraints. That study relied on the use of a surrogate models derived on the basis on samples obtained with vortex lattice methods. This modelling technique was useful as a rapid design and optimisation tool to create results on the order of seconds rather than minutes if using low-fidelity tools or hours in the case of high-fidelity RANS studies. ∗PhD Student, Department of Mechanical and Aerospace Engineering, 75 Montrose St, Glasgow G1 1XJ, AIAA Student Member †Post-doctoral researcher, Department of Mechanical and Aerospace Engineering, 75 Montrose St, Glasgow G1 1XJ,AIAA Member ‡Reader, Department of Mechanical and Aerospace Engineering, 75 Montrose St, Glasgow G1 1XJ,AIAA Senior Member 1 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 AIAA AVIATION FORUM AND ASCEND 2025 21 - 25 July 2025, Las Vegas, Nevada 10.2514/6.2025-3657 Copyright © 2025 by Zachary K Ciera, Bryn Jones, Marco Fossati. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. AIAA Aviation Forum and ASCEND co-located Conference Proceedings As such designers are then tasked with analysing a trade-off between highly accurate results vs time and resource cost. Methods exist to build models of potential performance by predicting performance from a set design space. For designers there is a valid use for the low-fidelity approach to do initial concept design and then use results to refine set design spaces and reduce the overall search requirements. The question of whether this pre-established model should be discarded, or developed upon is discussed here. The two approaches a designer could take would be to use a refined design space to sample from and build a new high-fidelity model to try and predict performance, or conversely use an error-based scheme that takes the difference between the low-fidelity model predictions and the high-fidelity samples then build a so-called “corrective” model that aims to rapidly emulate a high-fidelity model by adding a delta to the lift and drag coefficients. The primary notion behind this approach is that a low-fidelity model will be able to capture core flow physics, providing a valuable base that can be improved upon. As such, this study aims to compare and evaluate the efficacy of the approaches by taking several levels of sampling and comparing the accuracy of the two approaches compared to out-of-sample data as the number of training data points is increased. A. Evaluation of the Initial Surrogate The initial surrogate was developed with two distinct flight regimes in mind. The low-speed condition, set to evaluate performance at take-off or landing; and the cruise condition aimed to evaluate performance throughout the majority of flight. The atmospheric conditions of these two regimes are listed below. Environment Alt (m) Pressure (Pa) Temperature (K) Density (𝑘𝑔/𝑚3) Mach AoAs (°) High-Speed Cruise 8000 35600 236.15 0.5253 0.6 [-5:5] Low-Speed Take-off / Landing 1000 89880 281.65 1.1119 0.25 [-5:5] Table 1 Flight Environment Details Therefore, for analysis of the initial surrogate both conditions must be simulated on the reference aircraft. For the sake of this portion of the study it will be useful to see how the surrogate differs from higher fidelity simulation on a presented optimised design. This configuration was the result of a preliminary global optimisation study that used the aerodynamic low-fidelity surrogate as an integral part of the process. This optimisation study aimed to leverage LARW and DHEP technologies to reduce noise and emissions close to airports. As such it will be useful to conduct an analysis of the predicted lift and drag polars from the low-fidelity surrogate and then compare the results to that from high-fidelity RANS. Listed below are the resulting parameters from the optimisation study: AR Wing Taper Root Twist (°) Rel Tip Twist (°) Strut Twist (°) Strut Chord (m) t/c Root t/c Tip 21.617 0.42 -3 -5 1.64 1.5965 0.18 0.09 Table 2 Reference Aircraft Geometry In Diameter (m) Out Diameter (m) In Thrust (N) Out Thrust (N) In Tip Mach Out Tip Mach No. Props 5 3.979 5820 3550 0.59468 0.5885 5 Table 3 Reference Aircraft Propeller Parameters The aircraft CAD model resulting from these parameters is presented below; with the propellers represented by 2D disk surfaces in the relevant locations: 2 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 Fig. 1 Reference aircraft generated from low-fidelity optimisation study This aircraft uses the Airbus A320 wing area of 122.4 𝑚2 to determine the design of the wing, using a fixed area along with the parameters in Table 2 allows for a full definition of the wing. With the aircraft fully defined an initial comparison of the ROM predictions and the true performance can occur for both flight regimes. Fig. 2 Reference Case Comparison for RANS and ROM at Low-Speed: Lift Curve (Left), Drag Curve (Right) Fig. 3 Reference Case Comparison for RANS and ROM at Cruise: Lift Curve (Left), Drag Curve (Right) From Figures 2 & 3, the necessity for improved modelling techniques is clearly apparent. The model has the tendency to over predict lift at positive AoAs and is completely unable to capture near-stall behaviour as the lift curve no longer behaves in a linear manner. When comparing the drag predictions, it is clear that the performance is significantly worse. With the model severely under predicting drag and having a reduced rate of drag gradient increase leading to the difference between the prediction and reality increasing as the AoA magnitude increases. This clearly outlines the necessity for enhanced models that better capture the aerodynamic behaviour of these configurations. 3 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 B. Objectives and Manuscript Structure This will require two comparisons of each approachone for each flight environment. As such the overall objectives are listed below: 1) Choose at least 2 design features that can be varied to create minimum and maximum bounds for the design space. 2) Create a well defined design space of the two chosen parameters, ensuring there will be no conflicts as samples are generated 3) Complete a mid-fidelity analysis of the design space, utilising Euler simulations to run the corners and central cases. Then proceeding to populate the design space with 27 Latin-hypercube samples to ensure proper spacing. This will define two further cases: the fully sampled and the sparsely sampled case. 4) Build surrogate models for each case, evaluating metrics such as leave-one-out error for each approach. 5) Complete a model analysis of how a range of approaches can predict the performance of the central case for out-of-sample angles of attack. Analysing individual errors between points and overall accuracy. 6) Using the best performing models from the mid-fidelity analysis a further comparison will be made, using high-fidelity RANS data to evaluate how the corrective approach performs against a model built off of the RANS samples alone. The number of design features to be altered has be set to 2 as to keep the number of total samples required to build a reliable model to a manageable number. The minimum number of samples grows with 2𝑛+1 where 𝑛 represents the number of design parameters, along with the AoA. As this study will aim to validate and evaluate the approach and not create a model for a conceptual aircraft, a smaller design space will be suitable. The structure of this manuscript will be as follows; SectionIII will discuss the tools utilised to complete the study. Firstly evaluating the tools used to gather data for both the low and high fidelity models. Then proceeding with an in-depth description of the possible approaches for surrogate creation. Section IV contains the bulk of the analysis. Beginning with a preliminary analysis of parameters to ensure that the design variables selected provide an appropriate degree of non-linearity that will be challenging to model. After the parameters are selected the full analysis will be completed at the mid-fidelity level; predicting lift and drag coefficients in the low-speed and high-speed cases with differing levels of training data. Finally this section will conclude with analysis of the high-fidelity approach, comparing a pure RANS model, a corrective VSPAero model, and a corrective Euler model which utilises a developed model from the mid-fidelity analysis. Section V will summarise the full findings of the study and propose areas for future investigation. III. Methodology A. Aerodynamic Data Creation and Methods 1. VSPAero VSPAero is the foundation for the low-fidelity surrogate. It is a well-established software package that allows for the simulation of aircraft designs. Instead of high-fidelity RANS or lower fidelity Euler simulations, VSPAero employs a vortex-lattice method (VLM) with further capabilities to improve accuracy. The VLM approach uses a lattice of horseshoe vortices to predict the pressure on the wing surface and is based on potential flow theory. As such VLM makes several assumptions regarding the flow that may limit the accuracy of results. 1) The fluid flow is incompressible 2) The fluid flow is inviscid 3) The fluid flow is irrotational As such VSPAero results may deviate significantly compared to CFD results. However VSPAero does not solely rely on VLM, the use of panel methods are also included to model flow effects due to aerofoil thickness and estimations of parasite drag. [10] 2. RANS Simulation To complete the high-fidelity simulations, Stanford University Unstructured (SU2) is used to complete steady RANS simulations. In this case, the one-equation Spalart-Allmaras (SA) model is used and the second order Jameson-Schmidt-Turkel (JST) scheme is used for discretising convective fluxes. [11] 4 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 The meshing of the aircraft configuration involves primary generation of an inviscid mesh via the use of the open-source software GMSH. This software provides a fully tetrahedral mesh that is fine enough to properly resolve wing and fuselage curvature whilst allowing mesh sizing to be controlled across the domain, preventing unnecessarily detailed mesh in the far field [ 12 ]. Once the inviscid mesh has been created, a hybrid boundary layer utilising both tetrahedral and prismatic elements is inserted via the advancing normal method to generate the boundary layer to create the viscous mesh [ 13 ]. For all cases, an appropriate first cell height is estimated, and an appropriate layer growth rate is chosen to ensure that the boundary layer mesh is appropriate to capture the fluid boundary layer. 3. Actuator Disk The aerodynamic effects of the DHEP system are modelled via the Actuator Disk method. This involves defining the propeller positions, radii, and advance ratios, along with thrust and power coefficients per unit radius. Within the SU2 solver, propellers are modelled by proving momentum and pressure increase directly downstream of the selected surface elements. Rather than applying the properties to an internal boundary face, the forces are applied directly to refined regions that were generated as part of the inviscid meshing process. These forces are applied as such 𝐹𝑎(𝑟)= 2𝜌∞𝑉2 ∞ 𝐽2𝜋𝑟 𝑑𝐶𝑇 𝑑𝑟 (1) Where 𝜌∞ is the free-stream density; 𝑉∞ is the free-stream velocity; 𝐽 is the advance ratio; 𝐶𝑇 is the coefficient of thrust; and 𝑟 is the radial coordinate. 𝐹𝑎(𝑟) is the force per unit area, which is equivalent to the static pressure jump across the disk. 4. Mesh Adaptation To ensure accurate results and to maintain computational efficiency several levels of mesh adaptation are used to ensure that relevant flow features are properly captured. The alternative to utilising mesh adaptation would be to utilise progressively finer and finer mesh options until the solution is grid converged. This method is just as effective when it comes to ensuring that solutions are grid-independent, however, this untargeted approach will lead to mesh sizes that would be impractically larger. This would lead to a massively increased computational cost and would impact the range and/or number of parameters that could be explored as part of a design study. The use of mesh adaptation aims to solve this by taking a suitably refined initial mesh and results in resizing the mesh, inserting, moving, and deleting mesh points where necessary. [14] Fig. 4 Resulting Meshes: Un-adapted (Left) and Adapted (Right) The meshes above in Figure 4 are an example of a single level of mesh adaptation on a viscous mesh of the reference case at cruise conditions flying at an AoA of 0°. It can be seen how the mesh refinement has refined the mesh in areas deemed important, such as the wake region. Subsequent levels of adaptation can be performed to further refine the mesh until a suitable detail is reached. B. Surrogate Generation Methods There are a variety of approaches that can be taken when trying to create a surrogate model of a set system. To properly evaluate the performance of a corrective approach, a range of approaches must be defined. 5 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 1. Quadratic RSM A quadratic response-surface model (RSM) represents an unknown smooth response y(x) by the second-order Taylor expansion of that response about the centre of a suitable design space. Firstly inputs need to be mapped to dimensionless variables 𝑥𝑖∈ [−1,1], the approximation is; ˆ𝑦(x)=𝛽0+ 𝑑 ∑︁ 𝑖=1 𝛽𝑖𝑥𝑖+ 𝑑 ∑︁ 𝑖=1 𝛽𝑖𝑖𝑥2 𝑖+∑︁ 1≤𝑖< 𝑗 ≤𝑑 𝛽𝑖 𝑗 𝑥𝑖𝑥𝑗(2) This second-degree formulation acts as the basis for the response surface methodology due to ease of estimation, interpretability, and ability to capture the dominant linear and non-linear effects throughout the design space. The basis contains 𝑝=(𝑑+1) (𝑑+2) 2 unknown coefficients; where d represents number of input factors. After evaluation of suitably spaced design points, the coefficient vector 𝛽 is obtained by the least-squared method. As the fitted curve is a fully analytical global polynomial; predictions, along with first and second order sensitivities, are easily and rapidly obtained. This combination of closed-form fitting, speed, and direct physical interpretability is why the Quadratic RSM offers an attractive solution to developing surrogate models. [15, 16] 2. Radial Basis Functions A radial-basis-function (RBF) surrogate builds a smooth surface through scattered data by placing a “bump” at every training point and letting the weighted sum of those bumps pass exactly through the known values. For an input x the prediction is; ˆ𝑦(x)= 𝑁 ∑︁ 𝑘=1 𝜆𝑘𝜑∥x−x𝑘∥(3) Where 𝑥𝑘 are the N sample locations, 𝜑(𝑟) is a radially symmetric kernel such as the Gaussian, and the weights 𝜆𝑘 are chosen so the model reproduces every training value. As the kernel depends only on the distance from each centre, the surface adapts steadily to data that are irregularly spaced and to design spaces of any dimensionality. A single shape parameter controls how wide each bump spreads: small 𝜖 gives a tight fit that can capture sharp features; large 𝜖 yields a smoother, more global response. The shape function can be optimised by a one-dimensional sweep, retaining the value that minimises error based on a leave-one-out analysis. Training reduces to solving the dense linear system 𝐾𝝀=y , where 𝐾𝑖 𝑗 =𝜑∥x𝑖−x𝑗∥ ; this O(𝑁3) step is inexpensive for the few dozen high-fidelity samples typical of aerodynamic design studies. Once the weights are known, predictions involve 𝑁 kernel evaluations and a dot product, so evaluation time remain low. The combination of flexibility, intuitive control, and modest computational cost explains why RBF interpolation is a staple surrogate option. [17] 3. Gaussian Processing A Gaussian-process (GP) surrogate treats an unknown response 𝑦(x) not as a single best-fit curve but as a distribution over possible functions whose shapes are controlled by a covariance function. For any two input vectors x and x′ the kernel 𝑘(x,x′) , typically a squared-exponential or Matérn kernel, returns the expected similarity between their responses; nearby points (small distance relative to the length-scale parameters ℓ𝑖 ) therefore correlate strongly, while distant points correlate weakly. After observing 𝑁 sample pairs {(x𝑘, 𝑦𝑘)}𝑁 𝑘=1 , the GP yields a predictive mean ˆ𝜇(x) and a predictive variance ˆ𝜎2(x)at any new x. The hyper-parameters: signal variance 𝜎2 ; the set of length-scales {ℓ𝑖} ; and a small noise term that absorbs numerical errors are selected by maximising the marginal likelihood of the data. Training then requires inverting the 𝑁×𝑁 kernel matrix, an O(𝑁3) cost that is negligible for the few dozen CFD samples common in aerodynamic studies. Prediction is O(𝑁) ; batched queries therefore run in milliseconds and provide both the point estimate and a statistically grounded uncertainty band that can guide adaptive sampling. Thanks to this probabilistic framework, automatic smoothness control, and built-in error mapping, GP regression is often the surrogate of choice when data is sparse and assessing model confidence is as important as the prediction itself. [18] 6 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 IV. Results A. Non-Linearity Investigations and Parameter Selection Parameter Title Upper Bound Lower Bound Aspect Ratio 25 15 Wing Taper Ratio 0.42 0.32 Root Twist (Degrees) 3 -3 Relative Tip Twist (Degrees) 1 -5 Strut Twist (Degrees) 3 -3 Strut Chord (m) 1.597 0.906 t/c Root 0.18 0.12 t/c Tip 0.14 0.09 Table 4 Reference Aircraft Geometry Parameter Space Parameter Title Upper Bound Lower Bound Inboard Diameter (m) 6 4 Outboard Diameter (m) 4 2 Inboard Tip Mach 0.8 0.4 Outboard Tip Mach 0.8 0.4 Table 5 Reference Aircraft Propeller Parameter Space To properly verify the effectiveness of the corrective approach, appropriate terms must be selected to provide a sufficient level of nonlinearity. Parameters that are too linear will be easily corrected by an addition of a corrective delta or a scaling factor. For a rigorous approach parameters must be evaluated as they are individually varied from the minimum value to the maximum value. This is done via Euler simulations of low-speed configurations, a central case is generated, and a fixed lift investigation is completed for each parameter. This data will aim to present an overview of the linearity of selected design variables, with the bounds of each variable appearing in Table 4 and Table 5. 7 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 Fig. 5 Normalised Parameter Sweep Value Ranges RT TT ST Mi Mo AR Do Di AoA 4.81006 2.19201 0.86365 0.00265 0.03448 0.76011 0.18741 0.03859 Drag Count 4.04 9.87 6.34 2.08 9.65 194.12 19.31 2.68 Table 6 Resulting Values From Parameter Sweep The data presented shows how the angle of attack and drag varies with changes to the central configuration. When inspecting the range of values, it is seen that the aspect ratio of the wing is the most impactful of the selected variables regarding drag ranges, and the diameter of the outboard propellers the most impactful of the propeller parameters. When coupling this with the normalised graphs in Figure5 it is seen that these parameters are worthwhile to be used when attempting to investigate the corrective model. B. Mid-fidelity Euler Investigation As a starting point it is useful to evaluate how the corrective model compares to a mid-fidelity level with a range of samples. For this a series of Euler simulations are conducted to build a highly sampled surrogate and a sparse surrogate. Evaluating how the method compares as the number of available samples are reduced to only the corners and centre of the design space. The evaluation will come in multiple forms. The first will be pure model analysis, utilising leave-one-out errors and comparing parameter sweeps. Then next step will be applied analysis of the model’s predictive performance compared to an AoA sweep of the central case. A similar method will be applied for the corrective model, however the predictive performance will be analysed on the approach that has the lowest error out of the RSM, RBF, and GP methods specific to the corrective model. The following models were built using the 9 corner and central cases, and 27 Latin hypercube samples to ensure proper design space coverage. Defining two scenarios; the full and sparsely sampled cases 8 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 1. Fully Sampled Low Speed Evaluation Fig. 6 Fully Sampled Low Speed Lift Leave-One-Out Analysis Fig. 7 Fully Sampled Low Speed Lift Leave-One-Out Analysis Histograms Fig. 8 Fully Sampled Low Speed Drag Leave-One-Out Analysis Fig. 9 Fully Sampled Low Speed Drag Leave-One-Out Analysis Histograms 9 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 Model Data Quad RSM RBF GP AoA 𝐶𝐿𝐶𝐷𝐶𝐿𝐶𝐷𝐶𝐿𝐶𝐷 -5 -0.0360 -0.0066 -0.2552 -0.0083 -0.1426 -0.0052 -2.5 0.2107 -0.0050 0.0736 -0.0056 0.1789 -0.0001 2.5 0.8538 0.0052 0.9961 0.0079 0.8220 0.0101 5 1.2501 0.0137 1.3224 0.0148 1.1435 0.0152 Table 19 Sparsely Sampled Low-Speed Euler Models Model Data Low Fidelity Predictions Predicted Error New Corrected Data AoA 𝐶𝐿𝐶𝐷𝐶𝐿𝐶𝐷𝐶𝐿𝐶𝐷 -5 -0.4264 0.0164 0.2648 -0.0190 -0.1616 -0.0026 -2.5 -0.0449 0.0127 0.2144 -0.0191 0.1695 -0.0064 2.5 0.7182 0.0228 0.1049 -0.0194 0.8231 0.0034 5 1.0997 0.0366 0.0458 -0.0195 1.1455 0.0171 Table 20 Sparsely Sampled Low-Speed Euler Corrective Model AoA RSM RBF GP Corrective -5 74.53% 80.38% 0.82% 14.24% -2.5 15.34% 59.70% 2.08% 7.23% 2.5 2.73% 19.85% 1.10% 0.97% 5 8.30% 14.56% 0.93% 0.76% Table 21 Sparsely Sampled Low Speed Lift Error AoA RSM RBF GP Corrective -5 78.30% 122.38% 39.17% 29.85% -2.5 17.78% 33.01% 97.85% 52.94% 2.5 12.21% 32.52% 69.53% 42.41% 5 16.78% 10.37% 7.95% 3.77% Table 22 Sparsely Sampled Low Speed Drag Error 16 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 Compared to the previous cases, it is seen that accurate predictions of lift and drag coefficients are far more challenging. When inspecting the leave-one-out error for each model and comparing it to the highly sampled cases, it is apparent that the minimum sampled scenario leads to poor prediction of the excluded data points. This is further reflected when inspecting the errors between the model predictions and the simulated data. With elevated errors reaching up to 80.38% for the lift coefficient predictions and up to 122.38% for the drag coefficient predictions. However, once again not all models perform equally; the GP method and the corrective approach remain relatively accurate for lift prediction. Furthermore, when inspecting drag prediction both the corrective approach and RSM provide far more accurate results than RBF and GP. Model RSS 𝐶𝐿RSS 𝐶𝐷 QRSM 2.16E-02 1.72E-05 RBF 8.03E-02 2.93E-05 GP 2.15E-04 3.78E-05 Corr 7.23E-04 1.29E-05 Table 23 Sparsely Sampled Low Speed RSS Values The RSS analysis shows a clear difference between the highly sampled scenarios. Showing far higher values than previously. Though two key shifts are noted: RSM approach outperforms the GP approach for drag predictions; and the corrective approach outperforms all other methods for drag prediction. 17 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 4. Sparsely Sampled High Speed Evaluation Fig. 18 Sparsely Sampled High Speed Lift Leave-One-Out Analysis Fig. 19 Sparsely Sampled High Speed Lift Leave-One-Out Analysis Histograms Fig. 20 Sparsely Sampled High Speed Drag Leave-One-Out Analysis Fig. 21 Sparsely Sampled High Speed Drag Leave-One-Out Analysis Histograms 18 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 Model Data Quad RSM RBF GP AoA 𝐶𝐿𝐶𝐷𝐶𝐿𝐶𝐷𝐶𝐿𝐶𝐷 -5 -0.0740 0.0040 -0.3298 0.0016 -0.1930 -0.0036 -2.5 0.2194 0.0034 0.0627 0.0012 0.1932 0.0015 2.5 0.5747 0.0068 0.6117 0.0068 0.5747 0.0068 5 0.9918 0.0143 1.1745 0.0177 0.9656 0.0124 Table 24 Sparsely Sampled High-Speed Euler Models Model Data Low Fidelity Predictions Predicted Error New Corrected Data AoA 𝐶𝐿𝐶𝐷𝐶𝐿𝐶𝐷𝐶𝐿𝐶𝐷 -5 -0.4076 0.0159 0.2119 -0.0055 -0.1958 0.0103 -2.5 -0.0429 0.0120 0.2291 -0.0069 0.1862 0.0051 2.5 0.6866 0.0209 0.2529 -0.0069 0.5747 0.0068 5 1.0513 0.0336 0.2832 -0.0053 0.9697 0.0156 Table 25 Sparsely Sampled High-Speed Euler Corrective Model AoA RSM RBF GP Corrective -5 61.41% 72.04% 0.69% 2.14% -2.5 23.85% 64.61% 9.04% 5.09% 2.5 1.59% 20.30% 1.10% 0.67% 5 7.28% 14.63% 1.40% 0.03% Table 26 Sparsely Sampled High Speed Lift Error AoA RSM RBF GP Corrective -5 71.51% 88.76% 125.66% 27.18% -2.5 12.65% 68.83% 61.73% 29.46% 2.5 13.22% 7.53% 24.88% 5.03% 5 28.87% 17.82% 50.06% 13.17% Table 27 Sparsely Sampled High Speed Drag Error 19 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 With the same analysis being completed for the high-speed condition. It is seen that the results are similar to the low-speed analysis – only with exacerbated errors in most cases. Lift and drag predictions are once again worse than the highly sampled scenario, with maximum errors reaching up to 44.29% for lift and 125.66% for drag. Once more the GP method remains relatively accurate for lift and drag predictions. With once again the corrective approach outperforming all approaches for the drag prediction. Model RSS 𝐶𝐿RSS 𝐶𝐷 QRSM 2.16E-02 1.72E-05 RBF 8.03E-02 2.93E-05 GP 2.15E-04 3.78E-05 Corr 7.23E-04 1.29E-05 Table 28 Sparsely Sampled High Speed RSS The RSS values reinforce this, whilst also highlighting the performance of the corrective approach. Whilst the GP and RSM approaches consistently have low RSS values, the corrective approach outperforms all methods, with almost a full order of magnitude lower RSS for drag prediction. With the information from all four cases, it becomes clear that there are certain scenarios where the predictive approach outperforms approaches based on pure sampled data. From the trends identified the advantages are particularly apparent when the source problem is highly non-linear and correspondingly poorly sampled to capture that non-linearity. 20 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 C. High-fidelity RANS Investigation With the mid-fidelity Euler investigation completed, it is appropriate to elevate to the highest level of fidelity RANS simulations. Due to the inherent cost of running viscous RANS simulations, the number of cases will be limited to minimum. As seen from the previous investigation it is expected that the corresponding RANS model built off this data will be limited in it’s accuracy. However it presents a unique chance to not only explore the development of the corrective surrogate based on VSPAero data, but also a corrective model based on the most versatile Euler surrogatethe GP method. Likewise, the central case will be simulated with out of sample AoAs, coefficientss generated, and the RSS of the lift and drag evaluated for each approach Model Data True RANS Coefficients Best RANS Model Multi-Fi VSP Multi-Fi Eu AoA 𝐶𝐿𝐶𝐷𝐶𝐿𝐶𝐷𝐶𝐿𝐶𝐷𝐶𝐿𝐶𝐷 -5 -0.2052 0.0376 -0.2080 0.0344 -0.2052 0.0376 -0.2080 -4.9656 -2.5 0.1099 0.0354 0.1038 0.0387 0.1099 0.0354 0.1038 -2.4613 0 0.4252 0.0358 0.4252 0.0358 0.4252 0.0358 0.4252 0.0358 2.5 0.7335 0.0438 0.7273 0.0474 0.7335 0.0438 0.7273 2.5474 5 1.0479 0.0527 1.0390 0.0517 1.0479 0.0527 1.0390 5.0517 Table 29 Pure RANS Model vs Corrective Techniques RANS VSP MultiFi Eu Mulit-Fi 𝐶𝐿𝐶𝐷𝐶𝐿𝐶𝐷𝐶𝐿𝐶𝐷 1.33% 8.44% 10.62% 1.61% 2.84% 3.55% 5.61% 9.35% 18.16% 8.57% 0.39% 3.37% 0.85% 8.09% 1.30% 7.13% 0.40% 1.39% 0.85% 1.90% 0.67% 1.77% 0.27% 0.24% Table 30 RANS Model Lift & Drag Errors From the presented coefficients, all models seem to predict performance remarkably well. Especially when accounting for the sparse sample size. All models show remarkable predictive performance. With majority of errors for both lift and drag coefficients being below 5%; with respective peak errors of 18.16% and 9.35%. This is marked better performance than the Euler analysis at this sampling level. Out of all the methods the VSPAero corrective model is the poorest for lift prediction whilst the VSPAero and the pure RANS model are comparable for drag prediction. Model RSS 𝐶𝐿RSS 𝐶𝐷 RANS 1.64E-04 3.46E-05 VSP 1.01E-03 2.02E-05 EU 5.08E-05 3.59E-06 Table 31 RANS Model RSS Evaluation of the RSS shows a clear marked improvement when adopting a corrective methodology. Whilst the VSPAero model performs relatively poorly for lift predictions, for drag predictions it slightly outperforms the RANS model. By comparison the purely Euler corrective model is superior in both lift and drag prediction. As to support the indication that the improvement is marked by the comparatively low number of samples compared to the non-linearity of the problem, there is a greater difference between the drag RSS values of the corrective models and the RANS model than the corresponding lift RSS values. 21 Downloaded by Zachary Ciera on November 13, 2025 | http://arc.aiaa.org | DOI: 10.2514/6.2025-3657 V. Conclusion and Future Work Throughout this investigation a corrective methodology has been proposed and utilised to investigate how multifidelity error-based techniques can be used to model the complex aerodynamics of LARW-DHEP configurations. With the development of low fidelity surrogate models that utilise VSPAero samples there has been a demonstrated difference between the low fidelity predictions when compared to high fidelity results. With necessity for model improvement being highlighted, tools have been defined to properly conduct this study. With the focus on two specific branches: the collection of high-quality data for each level of fidelity; and the range of modelling techniques that can be utilised to construct surrogate models. The tools were chosen to be appropriate for the size of a reduced three-dimensional design space consisting of aspect ratio, outboard propeller diameter, and angle of attack. A first pass investigation was conducted comparing the low-fidelity model to a model built on Euler samples, focusing on a highly sampled design space and the minimum sampled space. At this level results were collected for both the low-speed and the high-speed condition. Throughout this investigation GP was seen to be a well performing approach to modelling lift and drag in all cases investigated. The RBF approach consistently performed poorly compared to all other models. Finally, the corrective approach performed reasonably well in predicting lift and drag in the highly sampled case, whilst outperforming all other models in predicting drag for the sparsely sampled scenario. From there the corrective methodology was further investigated by conducting a RANS investigation using the minimum amount of data. Creating three separate models: a model built off the pure RANS data, a corrective model aiming to enhance the VSPAero surrogate, and a corrective model aiming to enhance the GP Euler model. When comparing the RSS values for each approach: the RANS model outperforms the VSPAero model in lift yet performs worse when predicting drag; whilst the Euler corrective model performs far better than both approaches showing a significant reduction when comparing the RSS. Overall this study has highlighted the validity of utilizing various levels of fidelity to create high fidelity models. When considering the context of next generation aircraft design evaluation, the validity of the corrective approach is even more attractive when considering that lower fidelity models would already be created as the design process advances from purely conceptual to a final configuration. For future investigations, it would be appropriate to see how the performance scales as more parameters are added. This would increase the design space complexity and minimum number of samples required. If the trend from this investigation holds, the accuracy of the corrective approach should be far higher than any models built on pure high-fidelity data. Acknowledgments Funding for this work was kindly provided by the UK Research and Innovation council (UKRI) via the guaranteed fund for the Horizon Europe project INDIGO project (UKRI Project No 1006472 and HE grant agreement No 101096055). 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