Gappy POD model for Structural Compliance of the Wing of a Civil Tilt-rotor

The selection of the most critical load conditions is an extremely important topic as it allows the design and structural verification to be limited to a limited subset of conditions that envelop the operational thousands of the Design Limit Load (DLL) book without necessarily having to apply a direct approach. Often, thousands of analyses are performed by a Detailed Finite Element Model (DFEM, including model loading, post-processing and full stress and buckling analysis) taking a long time with a higher probability of error. This work describes the effort to set up a surrogate model of a composite wing, which aims to supply, accurately and efficiently, the structural response of the wing to several solicitation loads that populate the DLL. The surrogate model is defined by employing the Proper Orthogonal Decomposition technique (POD), trained on partial knowledge of data. The snapshots collected are composed of load solicitations and, limited to a subset of wing elements, stress and strain states resulting from the structural response to the relative load solicitation in a numerical analysis. Gappy-POD algorithm is set up to deal with this lack of data in the querying stage when only the solicitations are known. The surrogate model training starts with a selection of loads from the DLL, while its reliability is increased by iteratively adding further FEM analyses based on error maps opportunely defined.


Introduction
Most design and modeling goals require accurate predictions of numerical results to gain good design information for a proper insight into complex systems.The need to analyze systems with greater refinement and sophistication requires the support of methodologies, for instance, Finite Element Methods (FEM), able to cover complex configurations and analyses with major precision but resulting in more computational effort.Typically, stress analysis is crucial during the aircraft sizing phase to validate the preliminary design and perform design modification loops to recover from the lowest or negative margin of safety.Aircraft structures are sized to resist a huge amount of load conditions (of the order of tens of thousands) expected during their operative life.The Loads Department defines these loads and stores them in the Design Limit Load Book (DLL).In a certification context, it is expected that all these loads will be analyzed to provide structural safety justification in front of the Authority.In an experimental prototype context, such as the context of Collaborative Research projects funded by the European Commission, to substantiate the structural safety for a first Permit to Flight, the investigation on the DLL has to be kept to a minimum due to time constraints of the project.This could mean to analyze, among the thousands of Loads, only the most critical subset.This task is not straightforward because a significant amount of engineering judgment is needed to select critical loads.One of the methodologies to sort out the most critical loads is based on maximum Shear Forces Bending Moment (SFBM) cross-plot diagrams along the wingspan.An alternative methodology, fed only in part by Finite Element Analysis (FEA), can contribute to optimize the job of the structural department to sort out the critical loads subset to be then stress analyzed in detail.This could be the case with surrogate modeling methods.The principle of surrogate modeling is to replace a high-fidelity system model (e.g., a detailed FEM, DFEM) with a model (surrogate) that is less demanding from a computational standpoint while showing at the same time an acceptable representation of the original governing equations.In the framework of Horizon2020 Clean Sky 2 T-WING project, a methodology to build, train, assess and use a surrogate model of the wing structure has been set up.The aim is to aid the structural department in down-selecting the most demanding loads among the thousands of DLL.It is based on a well-established methodology within the computational fluid dynamics domain, namely Gappy -Proper Orthogonal Decomposition (G-POD).It consists of building a number of POD-based "surrogate models" of the wing structure that need limited knowledge from FE calculations.A restricted set of design limit SFMB conditions has been selected thanks to a preparatory clustering process, based on the maximum value of shear, bending, and torsion quantities in order to start the surrogate model building process, i.e., to choose the number and type of loading conditions to be run in the first iteration.At the beginning of the whole activity, a number of FEM control points are chosen based on the most critical locations to monitor.These critical locations are decided in concurrence with stress analysts since they represent the critical locations of the structure (e.g., wing root rib, front and rear spars, upper and lower skin, and wing-fuselage interface links).FE simulations in correspondence with the first set of loads are then run to estimate the structural stress state in the control locations.These simulations will act as a model set-up at the first iteration.After the first iteration, an assessment based on the definition of an empirical norm gives the degree of reliability of the model in predicting the outputs.Based on this assessment, in an iterative manner, additional FE runs can be performed to enhance the surrogate model reliability.Once built and properly trained, the surrogate POD-based model can be used to predict, in a predefined subset of FE elements, the strains, the stresses, and the internal forces with a controlled degree of reliability.Knowing the structural safety limits (allowable stress, for instance), the prediction of the surrogate model will identify the DLLs that give rise to negative safety margins.These load conditions will be considered critical in order to perform a detailed stress verification.In this preliminary work, the main pillars on which the methodology is based are reported, focusing on the error's management and the potentiality of the surrogate models of being adaptive, with an advantage in terms of prediction reliability.The obtained results will be validated with a cross-check process based on FEA available results.

Scenario and Objective
This work has been developed in the framework of the T-WING project (Grant Agreement number: 807090), working on design, manufacturing, qualification, and flight-testing of the wing and movables surfaces of the composite wing NGCTR-TD planned to fly in 2023.
To Asses structural safety aimed at obtaining a first Flight Permit, the investigation of Design Limit Load Conditions can be reduced to a minimum.This could mean analyzing, among the thousands of load cases, the most critical subset.
During the aircraft sizing phase, stress analysis is essential both to validate the preliminary design and to perform design modification cycles to recover any low or negative safety margins.The aircraft structures are sized to withstand the enormous amount of load conditions (in the order of tens of thousands) predicted, during its operational life cycle, by the load department using DLL.In this scenario, stress analysts often try to confine the calculation to a set of more severe load conditions.This task is not simple because it requires a significant amount of engineering judgment and experience.The selection of the most critical load conditions is an extremely important topic as it allows the design and structural verification to be limited to a limited subset of conditions that envelop the operational thousands of the DLL without necessarily having to apply a direct approach.In this case, thousands of analyses are performed DFEM (including model loading, post-processing, and full stress and buckling analysis) taking a long time with a higher probability of error.Furthermore, directly comparing maximum shear, bending, and torsional values is not a comprehensive approach due to the three-dimensional nature of the structure.Hence, the objective of the activity is to build a surrogate model for load analysis aimed at selecting the worst-case conditions and the effects of these conditions on the case of a wing structure of a new generation civilian TiltRotor.The surrogate model will be adaptive and will use data from high-fidelity structural calculations performed on a subset of loads selected from the DLL, such as sizing ones.The surrogate model thus obtained should be able to identify the critical conditions on which NASTRAN will carry out subsequent detailed analyses.At the end, the surrogate model of the structure of the wing allows identifying the response of the structure at a number of limited control points when it takes in input the shear/bending/torsion (SFBM) given for the DLL conditions.The surrogate model is built based on a limited knowledge constituted by a number of relevant FEA results (strains/stresses/int.forces) calculated from a subset of DLL conditions.The selected subset of critical load conditions is subjected to detailed FEM analysis, for example, with NASTRAN.It takes as input wing solicitation Shear/Force/Bending/Moments from Load Book, monitoring a limited group of wing points/elements (i.e.like having strain gauges), it gives as output internal forces/strains/stress, and finally selects a subset of critical conditions and repeat strength and buckling FEA, enriching itself by new FEM solutions, selected by the model itself by criteria like a map of critical conditions foreseen by the model or an error map from model cross-validation.
The flowchart in Figure 2 illustrates the training stage of the surrogate model.The training stage is characterized by the collection of available DFEM results that are opportunely preprocessed in order to be synthesized by the surrogate engine.To select DFEM, a previous activity is foreseen: the Design Load book is scrutinized to select the worst loads subset, analyze them, and collect them.An output of this stage could be the graph in the right part of Figure 3 that illustrates, for each of the load conditions on the horizontal axis, the structural state indicator, with the safety threshold, from which further decisions on training could be taken into account.
The querying stage is the fast population of the whole DLL that will depict the landscape of functions like strain and stress forces estimated in the selected control point (Figure 3).(here loads and stress/and strain status) are characterized by a lack of information.Finally, to enhance the model reliability, a smart sampling of the load space is implemented, performing an online implementation of the surrogate model: here the model is driven by a user-defined objective function which can sometimes reward reliability, sometimes robustness of the model, to enhance POD span.

Training Load selection: Clustering
Clustering is a data-driven procedure that aims to acquire a low-rank knowledge from existing loads.Here is employed to classify the DLL and select the worst load case from clustered groups, in an alternative way with respect to engineering judgment criteria: from each of the clusters resulting from the K-means algorithm, the worst loads have been selected, and, after being evaluated with DFEM analyses, will populate the surrogate model.The selection criteria for Critical Load are the following: The clustering aim is to start with a sampling of the DLL space that should ensure a large diversity of the loads characteristics.A first a-priori division occurs by splitting in two families the loads (Flight Loads, FL, and Ground Loads, GL).K-means Clustering is conducted in the transformed POD space of the loads.As an example, the FL Load Book is composed of 9377 loads for 48 stations along the wing span; hence, the input array 'FL' is 288 × 9377.This array is transformed in a new (Reduced) POD space of solicitations, linearized with ten modes.Follows the clustering process with the K-means algorithm on the two families 'a-priori': FL and GL, resulting in 5 clusters for each family.This has originated two subsets, composed of 25 load conditions for each family, for a total of 50 load conditions, for which DFEM analyses are executed and whose results are used in the surrogate model training.

Gappy POD
The research field that deals with gappy data has been developing for several decades in fields like meteorology, climate [1], aerodynamic measurements [2][3][4] (i.e., PIV), aerodynamic design [5,6], image reconstruction [7] and also some works that deal with partial knowledge can be found in the fields of structural science [8].In this work, the implemented algorithm refers to the study of Gunes et al [9].Here, the input vector s is a contribution from two parameters: loads vector, SFBM "Shear Forces and Bending Moments", q: with (r = 6m), where m is the index for SFBM at selected wing station.The second parameter is the deformation ϵ F EM , obtained from FEM analysis and read from the subset of p control points, corresponding to the conditions selected from clustering: s = [ϵ FEM , q] T .In this application, p = 39 control points (read from CFRP elements).
The Snapshots Matrix S is filled by n input vector, s, resulting from n critical loads.During the querying stage, when the model is asked to supply the structural state of the wing solicited by loads external from the selected n, the GAPPY algorithm starts to work on the gappy snapshot s * 0 made of an initial guess of strains, ϵ 0 and a known SFBM (from DLL book), q: the 'Gappyness level' correspond to the ratio between the sub-vector q and ϵ 0 , hence, in general, the 'gappy' region is the whole sub-array strains, [ϵ], unknown at querying stage.At the end of the computation, it is obtained a vector s * with predicted strains, ϵ ROM and a reconstruction of SFBM, q ROM .

Adaptive Surrogate Model
This iterative approach aims to enhance the reliability of the reduced order model by adding new samples, i.e., new FEM analyses to update training DB.To realize this, we have identified infill criteria based on 'Empirical norm': metric based on Load approximations, q, and relative features of the Strain-State ('POD energy' spectra).The Input Norm is defined as ∥ q rom −q ∥ 2 , Empirical norm as ⟨∥ q rom − q ∥ 2 ⟩ • S rom , ϕ N  Here are briefly shown the results of applying the adaptive surrogate model working on FL with the DFEM model and performing 3 Iterations.In Figure 7 there is evidence of Input and Empirical norms reduction: circles indicate Selected 3-ples of ID-Loads per iteration.

Results
Since the values of the interface forces calculated at the attacks between wing and fuselage for the coarser model or Global FEM (GFEM), it was possible to make an effective comparison in order to evaluate the reliability of the surrogate model.The wing structure object of the present work is connected to the fuselage via eight attachment points.In particular, we report the forces calculated at the fuselage wing interface elements and their POD prediction for all loading conditions.In order to make a comparison preliminary among the results, the trends of the forces have been reported on the same diagram.In particular, the distribution of POD Based on the previously shown results, some sizing load conditions have been subjected to related verification for the POD methodology regarding axial forces that solicit the wing-fuselage interface attachments.The predicted strength values with the surrogate model (F POD) were compared with those calculated by the GFEM of the wing(F Gfem).It is worth noting that from a first comparison of the results obtained, some differences are found within a maximum percentage error that oscillates in a range (0.05% -16%): it is necessary to insert further FEM investigations to train the surrogate model, but in any case, is in a trend of error reduction.Figure 9 shows diagrams for flight and ground load, where small deviations between FEM and ROM are noticed. .

Conclusions
The structural safety assessment aimed at obtaining the first flight permit may require, in accordance with current regulations, a very high number of structural analyses.In this work, we have developed and explored the potential of an interactive strategy aimed at minimizing the number of DLL analyses required for aeronautical structures.The proposed approach is based on a surrogate model built with the Gappy-POD technique aimed at identifying the most critical set of loads that must have priority in the structural analysis sequence.The proposed Gappy-POD methodology has proven capable of reconstructing the structural stress state, discerning the loading conditions, and obtaining the structural response for all the thousands of loading conditions prescribed by the DLL.The metrics introduced, namely input norm and empirical norm, allow us to have a good level of confidence in the quality of the approximation obtained and potentially allow us to reduce significantly the number of FEM analyses necessary for airworthiness certification.The two proposed rules only allow a probabilistic estimate of the error introduced by the surrogate model.Therefore, although the results obtained show an acceptable degree of approximation, it is necessary to continue to implement and refine the model to improve its reliability and accuracy.However, we can state that in less critical applications than certification, such as design, the careful use of this technique, perhaps in a multi-fidelity optimization context, can already lead to significant increases in the efficiency and effectiveness of optimization and design procedures.

Figure 6 .
Figure 6.Torsional Moment distributions on wing (left) and Strains on control points (right)

7 Figure 7 .Figure 8 .
Figure 7. Evolution of input and empirical norms after three iterations

Figure 9 .Figure 10 .
Figure 9. Differences are found within a maximum percentage error that oscillates between a minimum of 0.05% and a maximum of 16%, after 3 SM update