Flight load computations (FLC) are generally expensive and time-consuming. This paper studies deep learning (DL)-based surrogate models of FLC to provide a reliable basis for the strength design of aircraft structures. We mainly analyze the influence of Mach number, overload, angle of attack, elevator deflection, altitude, and other factors on the loads of key monitoring components, based on which input and output variables are set. The data used to train and validate the DL surrogate models are derived using aircraft flight load simulation results based on wind tunnel test data. According to the FLC features, a deep neural network (DNN) and a random forest (RF) are proposed to establish the surrogate models. The DNN meets the FLC accuracy requirement using rich data sources in the FLC; the RF can alleviate overfitting and evaluate the importance of flight parameters. Numerical experiments show that both the DNN-and RF-based surrogate models achieve high accuracy. The input variables importance analysis demonstrates that vertical overload and elevator deflection have a significant influence on the FLC. We believe that synthetic applications of these DL-based surrogate methods show a great promise in the field of FLC.

Flight loads are forces and moments borne by different components of an aircraft in actual flight states. The flight loads consist of aerodynamic loads, inertial loads, and elastic loads. Flight loads are influenced by complex factors, including working conditions (taking-off, climbing, cruising, maneuvering, landing), atmospheric environment (temperature, air density, gust), and aircraft features (configuration, weight, speed, acceleration).

Flight loads are preconditions of aircraft structural strength design. If the design strength is lower than the actual value, the structure may break down in extreme flight conditions. If the design results are too conservative, a large weight cost has to be paid. Accuracy and efficiency of flight load computation (FLC) directly affect the design quality, progress, and cost, which are of great significance in aircraft design [

Modern aircraft design involves large loads, large deformations, and multiple transmission paths. Conventional numerical simulation techniques, such as the finite element method, the panel method, and the CFD method require high resolutions and large discretization scales when applied to the FLC. Thus, the load computations are time-consuming, which significantly restricts aircraft research and development. To improve the efficiency of load computation, model order reduction (MOR) of conventional load computation models has attracted research interest [

In recent years, rapid developments in deep learning (DL) have attracted significant attention in the field of aircraft design [

DL technologies are widely used in the field of international aviation, and have achieved fruitful results in the field of aircraft design. Neural network models of aerodynamics are referred to [

This paper studies DL-based surrogate models of FLC to provide a reliable basis for aircraft structure strength design. The surrogate models are established using flight load simulation results from aircraft based on wind tunnel test data. The flight loads are affected by complex factors including body parameters, flight conditions, and control parameters. This paper is focused on symmetrical maneuvers to analyze and verify the effectiveness of the proposed method. In a situation of typical weight, the main flight conditions, altitude, Mach number, speed pressure, are considered in input variables. Furthermore, trim degrees of freedom and trim variables are crucial for the loads in the symmetrical maneuvers, including vertical overload, pitch angular acceleration, angle of attack, elevator deflection, pitch rate. These movement parameters and the flight conditions are set as the input variables of DL surrogate models. In choosing extreme loading situations, the loads at connecting joints of key components of an aircraft are the significant monitoring indicators [

The remainder of this paper is organized as follows. Conventional FLC methods are described in Section 2. In Section 3, we introduce DL-based surrogate models using the DNN and RF, and the computation procedure for establishing surrogate models to predict and analyze the flight loads. Numerical verification is presented in Section 4. Conclusions are presented in Section 5.

The purpose of flight load analysis is to obtain the maximum loads of main aircraft components and the corresponding flight conditions yielding these loads. The aircraft attitude is determined by solving a series of kinetic equations for aircraft, and obtaining the aerodynamic load distribution data, inertial loads, and elastic loads under equilibrium states for the entire aircraft. The maneuvers used in the flight load analysis mainly include symmetrical maneuver flight (pitch maneuver) and asymmetric maneuver flight (roll maneuver, yaw maneuver) [

First, an analysis axis system was designed. The origin

The selection of flight load conditions must cover all flight states within the flight envelope. Usually, standard specifications are chosen based on the type of aircraft; the flight dynamics equations are solved to simulate aircraft maneuvers under the constraints of the specifications. The maneuvers generally include combinations of flight situations, including weights, gravity centers, mass distributions, aerodynamic configurations, speeds, altitudes, engine thrusts, flight control systems, plug-in configurations, maneuvering actions, and control parameters. Based on the maneuvers, the main aircraft maneuver flight parameters are determined as the specific flight load conditions.

The flight load analysis of elastic aircraft is based on numerically coupling the models of structural data, aerodynamic data, and mass distributions. The flight load data in complicated flight conditions is derived using static finite element analysis methods. The flight load analysis of elastic aircraft is mainly focused on the influence of aircraft structural deformations on aircraft loads. This includes the change in aircraft balance state caused by aerodynamic features and the redistribution of aerodynamic loads caused by structural elastic deformations. The model describing the flight load computations is dominated by a series of equilibrium equations that are based on principles of statics analysis and established by adding aerodynamic forces and considering inertia release theory. The finite element method is used in solving these equations to derive the flight loads. The major equation characterizing static aeroelastic responses is expressed as follows [

where _{d}

In this paper we mainly study effects of elastic deformations on steady aerodynamic loads. To this end, time independent simplification of _{x}

The flight parameters for different conditions of FLC are used as the input of

In this paper we use the SOL144 solver to conduct the flight load analysis of an aircraft. The FEM structural mesh model and aerodynamic panel mesh model are constructed in the FLC, as shown in

Thus, conventional methods of FLC depend heavily on aircraft shape, structural features, flight parameters, external conditions, and flow field information, and have a strong nonlinear relationship with them. These relationships are usually described by coupling a series of complex mathematical and physical equations. Solving these equations requires significant computational resources, which hinders aircraft design quality and schedules. Thus, development of surrogate models of FLC is required. In this paper, DL-based surrogate models of flight loads are developed [

This section establishes surrogate models for flight load analysis based on two deep learning algorithms, a deep neural network (DNN) and a random forest (RF).

Let

where

A surrogate model views

The surrogate model of

The

The flight loads are affected by complex factors including body parameters, flight parameters, and control parameters. The loads differ greatly in different flight stages such as take-off, climbing, cruising, gliding down, and landing. Flight parameters such as mass, speed, acceleration, flight attitude, and operation movements influence the flight loads. External flight factors such as temperature, pressure, wind gusts, and atmospheric turbulence also have great impact. In addition, the loads and severe load conditions differ for different parts of the aircraft. For example, the rib and beam of the wing and the frame of the fuselage have different severe load conditions; flight loads are complex and diverse. This paper is focused on symmetrical maneuvers to analyze and verify the effectiveness of the proposed method. In a situation of typical weight, the main flight conditions, altitude (H), Mach number (M), speed pressure (SP), are considered in input variables. The SP is incorporated to clearly identify its relationship with the flight load. Furthermore, the trim DOFs and trim variables are crucial for the loads in the symmetrical maneuvers, including vertical overload (OL), angle of attack (AoA), elevator deflection (ED), pitch rate (PR), and pitch angular acceleration (PAA). These movement parameters and the flight conditions are set as the input variables, i.e.,

To study the most extreme loading conditions, typical sections are selected as monitoring objects. The quantities of interests on these sections, including the bending moment, torque, and shear force, are the key indicators characterizing the flight loads during maneuvering. We choose the root and the middle of the wing and the root of the horizontal tail as the major objects because the most extreme loads generally occur in these sections [

We introduce two typical deep learning algorithms, a deep neural network and a random forest, to establish the surrogate model

A deep neural network (DNN) [^{(l)}, by an associated wight matrix ^{(l)}, and then add a bias term ^{(l)}. The summation is mapped by a nonlinear activation function

and so on.

In this paper we use a so-called residual neural network [

where

The network updates the parameters using back propagation algorithm until the desired results are achieved. In the back propagation algorithm, parameters are updated based on gradients of loss function with respect to the parameters, and these gradients are computed using an adaptive moment method, Adam et al. [

The residual network better fits high-dimensional functions. The fitting ability is not affected by network width. The residual network can significantly increase training speed and pre-precision of deep networks, break the symmetry of networks, reduce network degradation, and improve network characterization ability.A residual DNN is accurate with sufficient sample data, and meets the accuracy requirements with rich data sources in FLC.

A random forest is a machine learning method that uses decision trees to train samples and predict objectives [

A random forest is established using the bagging (bootstrap aggregating) algorithm to vote the decision tree. In statistics, the bootstrap is a kind of ensemble technology that trains classifiers by selecting new data sets from the original data set through sampling with replacement. The number of selected objects will accounts for approximately 63% of the source samples; the remaining 37% of the samples are used to test the generalization ability of the constructed model. We randomly select ^{(t)} from ^{(t)}. At each node of this tree, we randomly select _{t}

In our computations, we use a RF regressor. The number of trees in the forest is 100, and the maximum depth of tree is 30. The number of samples is also 24619, from which 17233 samples are selected randomly for the bootstrap sample. 100 trees are built based on 100 bootstrap sample sets obtained from these 17233 samples. The quality of a split is measured by the mean square error (MSE), and the variance reduction serves as the feature selection criterion. The load regression prediction of an input is computed as the mean regression predictions of the trees in the forest.

An RF is not easily overfitted and has excellent generalization ability. Most important, an RF can evaluate the importance of input variables, which is critical in analyzing factors affecting flight loads.

The FLC procedure using DL-based surrogate models is described as follows:

analyze the factors affecting FLC and key monitoring components to set input and output variables;

compute the data used to train and validate the surrogate models using conventional flight load simulation algorithms based on wind tunnel test data;

train and validate the DNN and RF surrogate models;

compare the accuracy of the surrogate models;

identify the importance of input variables to determine the main factors affecting FLC;

adjust Steps (1) and (2) according to Steps (4) and (5) and repeat the procedure until a reasonable result is produced, comparable to results from conventional methods.

Using an example aircraft, we perform FLC using the proposed deep-learning surrogate models, DNN and RF. We test the accuracy of the two surrogate models and analyze their load prediction results through finite element analysis. The importance of the input variables is evaluated using the RF model to identify the main factors influencing loads.

We consider a high-speed and high-maneuverability aircraft with a conventional configurations. The wings have a double beam wing box structure. The specific parameters of the aircraft are shown in

As analyzed in Section 3.2, the input variables include the flight altitude (H), Mach number (M), speed pressure (SP), vertical overload (OL), angle of attack (AoA), elevator deflection (ED), pitch rate (PR), and pitch angular acceleration (PAA), which are the main factors affecting flight loads, see

Notation | Description | Values | Unit |
---|---|---|---|

m | Mass | 22000 | kg |

CG | Centroid | 31.4% | MAC |

b | Chord length | 4.8 | m |

l | Wing span | 15.2 | m |

S | Wing area | 70 | |

Moments of inertia | 50000 | ||

Moments of inertia | 300000 | ||

Moments of inertia | 300000 |

Input | H | M | SP | OL | AoA | ED | PR | PAA |
---|---|---|---|---|---|---|---|---|

Unit | m | 1 | Pa | 1 | degree | degree | degree/s | degree/ss |

Output | S-WR | BM-WR | T-WR | S-WM | S-HTR | BM-HTR | T-HTR | |

Unit | tf | tf | tf |

We present major parameters in the DNN and RF surrogate models, proposed in Sections 3.3 and 3.4. For the DNN, we use the residual neural network [

A total of 24619 data pairs are used to construct the DNN surrogate model, of which 17233 pairs are used to train the model. The model is adjusted according to the error computed from these pairs. A total of 3693 data pairs are used to judge when the training is finished. The training is stopped when these pairs are used to continue training but the network structure is not greatly improved. Finally, a total of 3693 data pairs are used to test the model errors of the surrogate model. The tanh function, whose range is

A random forest is a meta estimator that fits a number of classifying decision trees on different sub-samples of the data set and uses averaging to improve the predictive accuracy and control overfitting [

We employ mean square error (MSE) and coefficients of determination ^{2} to examine the accuracy of models, defined as follows:
_{i}^{2} is to 1, the higher the accuracy of models is. The MSE and ^{2} for the training, validation, and test sets for the DNN and RF surrogate models are presented in

According to the results in ^{2} values for both models are both close to 1, indicating that the surrogate models demonstrate the high fitting accuracy. The MSE of the RF model is 0.03267 for the training set, which is better than 0.05963 in the DNN model. However, the MSE of the RF in the test set is 0.08412, which is larger than 0.05995 in the DNN model. The MSEs in the DNN model for the training, validation, and test sets are similar; the DNN model is more stable. The DNN model is easier to train because residual terms are introduced. However, the RF model does not need to normalize the data in training, and can identify the importance of input variables.

The predicted shear force (S), bending moment (BM), and torque (T) of the DNN and RF surrogate models are presented in

To demonstrate the efficiency of the DNN and RF surrogate models, we compare them with the conventional neural network method. The neural network has been applied to aeronautical areas quite early, as reviewed in the Introduction, see [^{2} for the training, validation, and test sets for the neural network model are presented in

Set | Number of data pairs | MSE | ^{2} |
---|---|---|---|

Training | 17233 | 0.05963 | 0.9972 |

Validation | 3693 | 0.05641 | 0.9980 |

Test | 3693 | 0.05995 | 0.9961 |

Set | Number of data pairs | MSE | ^{2} |
---|---|---|---|

Training | 17233 | 0.03267 | 0.9999 |

Test | 7386 | 0.08412 | 0.9999 |

Set | Number of data pairs | MSE | ^{2} |
---|---|---|---|

Training | 17233 | 0.1264 | 0.9991 |

Validation | 3693 | 0.1454 | 0.9991 |

Test | 3693 | 0.1531 | 0.9991 |

We introduce many variables to train the surrogate models; their influences on flight loads are different. Thus, identifying the importance of different variables is critical in the analysis of loads. The main factors are instructive in developing more efficient load computation approaches. We apply the RF to identify the importance of input variables; this is an advantages of RF over other deep learning techniques, including the DNN. In the RF, the importance of a variable is computed as the (normalized) total reduction of the criterion brought by that variable. It is also known as the Gini importance. The importance of a variable is calculated as follows. A baseline metric, defined by scoring, is evaluated on a (potentially different) dataset defined by

Input | H | M | SP | OL | AoA | ED | PR | PAA |
---|---|---|---|---|---|---|---|---|

Importance | 0.0055 | 0.0609 | 0.0122 | 0.5979 | 0.0719 | 0.2413 | 0.0012 | 0.0096 |

This paper studied deep learning-(DL) based surrogate models of flight load computations (FLC). A deep neural network (DNN) and a random forest (RF) were proposed to establish the surrogate models according to the features of FLC. The DNN meets the accuracy requirement of FLC with rich data sources in FLC; the RF can alleviate overfitting and evaluate the importance of flight parameters. The data used to train and validate the DL surrogate models were derived using aircraft flight load simulation results based on wind tunnel test data. Numerical experiments showed that both the DNN-and RF-based surrogate models achieve high accuracy. The input variable importance analysis was conducted to identify the main factors in FLC. This paper was focused on typical symmetric flight conditions, steady pitch and steep pitch, to test the surrogate models. Additional flight conditions, such as roll maneuvers, yaw maneuvers, and severe load conditions within the flight envelope will be investigated in future research.

The authors wish to express their appreciation to the anonymous reviewers for their helpful suggestions which greatly improved the presentation of this paper.