A global mapping based reduced order method for transonic unsteady aerodynamic force

Abstract

The application provides a global mapping-based cross-sonic unsteady aerodynamic force reduction method, which comprises the following steps: using filtered Gaussian white noise random signals as input signals, the input signals need to cover a wide range of frequencies and amplitudes; using a CFD solver to perform numerical calculation on the designed input signals, and constructing a multiple-input-multiple-output mapping relationship between generalized displacement and aerodynamic force, which is used as training data of the model; using a data-driven Fourier neural operator model to predict the unsteady aerodynamic force, and training the Fourier neural operator model; and inputting generalized displacement data into the trained Fourier neural operator model to predict corresponding aerodynamic force coefficients. The application has reasonable concept, compared with previous reduction methods, the time spent for training the same data is shorter, the accuracy and efficiency of aerodynamic force coefficient prediction are higher, and linear characteristics and nonlinear characteristics can be captured simultaneously.

Description

Technical Field

[0001] This invention relates to the field of transonic unsteady aerodynamics technology, specifically to a method for reducing the order of transonic unsteady aerodynamics based on global mapping. Background Technology

[0002] Transonic flows are prevalent during the cruise and maneuvering of aircraft. Their aerodynamic characteristics are influenced by complex nonlinear phenomena such as shock wave-boundary layer interactions, turbulence, and flow separation, making accurate prediction and description difficult using traditional modeling methods. Under unsteady transonic conditions, aerodynamic responses typically exhibit significant time lag effects and nonlinear characteristics. Accurately capturing these dynamic features is crucial for high-precision modeling. Specifically, the time lag effect refers to the lag in aerodynamic response relative to flow field evolution or changes in flight state, while nonlinear dynamic characteristics arise not only from shock wave motion and flow separation but are also influenced by turbulence, unsteady vortex shedding, and other factors. These nonlinear and unsteady effects make traditional assumptions difficult to apply. Therefore, high-precision modeling methods need to comprehensively consider the historical dependence of flow evolution (such as the influence of previous aerodynamic states on the current response) and high-precision prediction capabilities under complex aerodynamic environments. These factors not only increase the difficulty of reduced-order modeling but also place higher demands on the stability and control performance of aircraft. Currently, the main methods for order reduction of transonic unsteady aerodynamic forces include the AutoRegressive with eXogenous input (ARX) model, the Nonli near AutoRegressive with eXogenous input (NARX) model, the Volterra sequence model, and neural network-based models.

[0003] The existing technology has the following problems:

[0004] (1) Existing models still need to be improved in terms of prediction accuracy, generalization ability and computational efficiency, while also facing challenges in robustness and computational cost control under complex flow field conditions.

[0005] (2) Existing methods are usually based on fixed input-output mapping for single-step prediction, which can only gradually deduce the temporal evolution of unsteady aerodynamic forces, and it is difficult to capture the global spatial-temporal distribution characteristics at the same time to achieve multi-input-multi-output overall prediction, thus limiting the ability to model complex unsteady phenomena.

[0006] (3) Existing methods do not have time discrete invariance. The data time step used during model training must be consistent with that used during the testing phase, otherwise it may lead to the accumulation of prediction errors.

[0007] (4) Existing methods are difficult to simultaneously take into account the dynamic linear characteristics under small disturbances and the nonlinear effects under large disturbances, resulting in limited applicability under different flow states.

[0008] In conclusion, it is necessary to further innovate existing technologies. Summary of the Invention

[0009] To address the technical problems existing in the background art, this invention proposes a transonic unsteady aerodynamic order reduction method based on global mapping. Its concept is reasonable, and compared with previous order reduction methods, it takes less time to train the same data, and the accuracy and efficiency of aerodynamic coefficient prediction are higher. It can simultaneously capture dynamic linear and nonlinear characteristics.

[0010] To address the aforementioned technical problems, this invention provides a transonic unsteady aerodynamic order reduction method based on global mapping, which specifically includes the following steps:

[0011] (1) Design the input signal

[0012] The filtered Gaussian white noise random signal is used as the input signal. The input signal needs to cover a wide range of frequencies and amplitudes to reflect the dynamic linear characteristics of the flow field under small disturbances and the nonlinear characteristics under large disturbances.

[0013] (2) Obtain training data

[0014] The input signal designed in step (1) above is numerically calculated using a CFD solver. Based on the results of the numerical calculation, a multi-input-multi-output mapping relationship between generalized displacement and aerodynamic force is constructed and used as the training data for the model.

[0015] (3) Training the model

[0016] A data-driven Fourier neural operator model is used to predict unsteady aerodynamic forces, and the Fourier neural operator model is trained.

[0017] (4) Prediction of aerodynamic coefficients

[0018] By inputting generalized displacement data into the Fourier neural operator model trained in step (3) above, the corresponding aerodynamic coefficients can be predicted quickly and accurately.

[0019] The transonic unsteady aerodynamic order reduction method based on global mapping is described in the following steps: the signal amplitude of the input signal in step (1) is small in the initial stage and gradually increases to a larger amplitude; by designing a mixture of small amplitude and large amplitude signals, the input signal can simulate the dynamic response of the aerodynamic system under different disturbance intensities.

[0020] The transonic unsteady aerodynamic order reduction method based on global mapping, wherein the input signal in step (2) includes heave motion and pitch motion, which correspond to the bending mode and torsional mode in the two-dimensional airfoil aeroelastic analysis, respectively;

[0021] In step (2), during the CFD calculation of the input signal, the values ​​of both the heave displacement and the pitch angle of attack are simultaneously used as inputs in the CFD solver for calculation. Furthermore, by calculating the mesh deformation, the rotation center of the pitch motion changes with the heave displacement, thereby obtaining the corresponding aerodynamic coefficients.

[0022] During mesh deformation, the coordinates of each node are updated using the following formula:

[0023] x′=rcos(θ+Δθ);

[0024] y′=rsin(θ+Δθ)+Δy;

[0025] Where (x′, y′) are the new coordinates of the nodes after deformation. is the radial distance of the node in the polar coordinate system, and (x, y) are the coordinates of the node before deformation. Δθ is the polar angle before deformation, Δθ is the increment of pitch angle of attack, and Δy is the increment of buoyancy displacement.

[0026] The transonic unsteady aerodynamic order reduction method based on global mapping, wherein the specific process of training the Fourier neural operator model in step (3) is as follows:

[0027] First, the input data is transformed into a high-dimensional representation through a dimensionality boosting operation, which is implemented by a shallow fully connected neural network.

[0028] ω0=P(a(x));

[0029] Where ω0 is the initial state, and P represents the local transformation implemented by the shallow fully connected neural network.

[0030] a(x) represents the input;

[0031] Next, the hidden layers (Fourier layers) in the Fourier neural operator model are used as part of an iterative architecture and mapped to the next layer iteratively. The mapping between different function spaces is gradually approximated using convolution operators in Fourier space. The specific iterative process is as follows:

[0032]

[0033] Where, ω t Represents an intermediate state. θ represents a nonlinear mapping, and θ represents the parameter set of the Fourier neural operator;

[0034]

[0035] Among them, the operator Let φ represent the mapping of the bounded linear operator space, W represent the weight matrix, and σ represent the local nonlinear activation function.

[0036] For Fourier neural operators, the mapping process of Fourier layers depends on the Fourier transform and its inverse; applying the convolution theorem, the kernel integral operator can be replaced by convolution operations in Fourier space; the replacement form of the kernel integral operator in Fourier space is:

[0037]

[0038] in, Represents Fourier transform, Represents the inverse Fourier transform;

[0039] Finally, the output of the Fourier layer is locally transformed and projected back into the original output space to obtain the final prediction result.

[0040] u(x)=Q(ω T (x));

[0041] Where, ω T Let Q represent the final state, where Q represents the local transformation and u(x) represents the output.

[0042] The transonic unsteady aerodynamic order reduction method based on global mapping is described in which: the Fourier neural operator model trained in step (3) extracts the inherent nonlinear characteristics of the aerodynamic system through a hierarchical combination of spectrum and point-by-point operation, and effectively captures the time delay characteristics by utilizing frequency domain transformation.

[0043] The transonic unsteady aerodynamic order reduction method based on global mapping, wherein the specific process of effectively capturing time delay characteristics using frequency domain transformation is as follows:

[0044] Represent the input function in the frequency domain using Fourier transform:

[0045]

[0046] Convert the frequency domain data back to the original space:

[0047]

[0048] Where, x j k represents the discrete coordinate index in physical space. jHere, f(x) represents the Fourier frequency index, f(k) represents the signal in physical space, and f(k) represents the signal in the frequency domain, i.e., the Fourier transform result. j The exponent term represents the resolution of the discrete grid. Control the change of direction.

[0049] The transonic unsteady aerodynamic order reduction method based on global mapping is described in which: the Fourier neural operator model in step (3) has time discrete invariance and can use different time steps in the training and testing phases, thereby achieving unsteady aerodynamic prediction more efficiently.

[0050] By adopting the above technical solution, the present invention has the following beneficial effects:

[0051] Compared with traditional CFD calculations, this invention improves the speed of predicting a working condition by three orders of magnitude.

[0052] Compared with previous reduced-order models, this invention is based on the Fourier neural operator model and uses a global mapping method, so it takes less time to train the same data; the training time of this invention is only about 25 minutes, while the training time of the previous model is about 4 hours.

[0053] Compared with the previous reduced-order model, the present invention has higher prediction accuracy; the average relative error of the prediction results of the present invention is about 2.4%, while the average relative error of the prediction results of the previous model is about 5%.

[0054] This invention can simultaneously capture dynamic linearity and nonlinearity.

[0055] Because the Fourier neural operator model has discrete invariance, different time steps can be used in the training and prediction phases, thus enabling more efficient prediction. Attached Figure Description

[0056] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0057] Figure 1 The flowchart shows the transonic unsteady aerodynamic order reduction method based on global mapping of the present invention.

[0058] Figure 2 This is a schematic diagram of the input signal designed in the transonic unsteady aerodynamic order reduction method based on global mapping of the present invention.

[0059] Figure 3This is a structural diagram of the Fourier neural operator model based on global mapping used in the transonic unsteady aerodynamic order reduction method based on global mapping of the present invention.

[0060] Figure 4 This is an internal unfolded diagram of the Fourier layer in the order reduction model structure based on global mapping used in the transonic unsteady aerodynamic order reduction method based on global mapping of this invention. Detailed Implementation

[0061] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0062] The present invention will be further explained below with reference to specific embodiments.

[0063] like Figure 1 As shown in the figure, this embodiment provides a transonic unsteady aerodynamic order reduction method based on global mapping, which specifically includes the following steps:

[0064] S100, Design Input Signal

[0065] The design of the input signal is crucial to the performance of the aerodynamic reduced-order model, especially in improving its generalization ability. To enable the model to effectively capture the dynamic characteristics of the aerodynamic system under different perturbation amplitudes, the input signal needs to cover a wide frequency and amplitude range to reflect the dynamic linear characteristics of the flow field under small perturbations and the nonlinear characteristics under large perturbations. Therefore, a filtered white Gaussian noise (FWGN) random signal is used as the input, where the signal amplitude is small in the initial stage and gradually increases to a larger amplitude, such as... Figure 2 As shown, by designing a hybrid of small-amplitude and large-amplitude signals, the input signal can effectively simulate the dynamic response of the aerodynamic system under different disturbance intensities, thereby enhancing the model's adaptability to complex dynamic behaviors.

[0066] S200, Obtain Training Data

[0067] A CFD solver is used to numerically calculate the input signal designed in step S100. This input signal includes heave and pitch motions, corresponding to the bending and torsional modes in the aeroelastic analysis of a two-dimensional airfoil, respectively. During the CFD calculation of the input signal, both the heave displacement and the pitch angle of attack are simultaneously used as inputs in the CFD solver. Furthermore, by calculating mesh deformation, the rotation center of the pitch motion is shown to change with the heave displacement, thereby obtaining the corresponding aerodynamic coefficients.

[0068] During mesh deformation, the coordinates of each node are updated using the following formula:

[0069] x′=rcos(θ+Δθ);

[0070] y′=rsin(θ+Δθ)+Δy;

[0071] Where (x′, y′) are the new coordinates of the nodes after deformation. is the radial distance of the node in the polar coordinate system, and (x, y) are the coordinates of the node before deformation. Δθ is the polar angle before deformation, Δθ is the increment of pitch angle of attack, and Δy is the increment of heave displacement.

[0072] Based on the numerical calculation results above, a multi-input / multi-output (MIMO) mapping relationship between generalized displacement and aerodynamic force is constructed and used as the training data for the model.

[0073] S300, training model

[0074] In this invention, a data-driven Fourier neural operator model is used to predict unsteady aerodynamic forces, and this model is trained. Unlike traditional neural networks that map finite-dimensional vectors to each other, the Fourier neural operator achieves global mapping by learning the low-level mappings between function spaces, thus providing greater flexibility and scalability, and offering significant advantages in characterizing complex physical phenomena. Furthermore, the Fourier neural operator utilizes integral kernel operators as basic units, rather than neurons, where each integral kernel operator is a function space analogous to the weight matrix in a standard feedforward neural network.

[0075] The Fourier neural operator model trained in this invention extracts the inherent nonlinear characteristics of aerodynamic systems through a hierarchical combination of spectrum and pointwise operations, and effectively captures time-delay characteristics using frequency domain transformation. Furthermore, this Fourier neural operator model possesses time discrete invariance, allowing the use of different time steps during training and testing phases, thereby achieving more efficient prediction of unsteady aerodynamic forces. The specific structure of the Fourier neural operator model is as follows: Figure 3 As shown.

[0076] The Fourier neural operator model can be divided into three key parts: First, the input data undergoes a dimension-up operation to be converted into a high-dimensional representation, which is implemented by a shallow fully connected neural network.

[0077] ω0=P(a(x));

[0078] Where ω0 is the initial state, and P represents the local transformation implemented by the shallow fully connected neural network.

[0079] a(x) represents the input;

[0080] This significantly enhances the expressive power of the Fourier neural operator model, helping to better capture features in the input data;

[0081] Next, the hidden layers (Fourier layers) in the Fourier neural operator model are used as part of an iterative architecture and mapped to the next layer iteratively. Each Fourier layer functions similarly to a standard feedforward neural network, using a weight matrix to simulate the mapping relationship in the function space. Specifically, it uses convolution operators in Fourier space to gradually approximate the mapping between different function spaces. The specific iterative process is as follows:

[0082]

[0083] Where, ω t Represents an intermediate state. θ represents a nonlinear mapping, and θ represents the parameter set of the Fourier neural operator;

[0084]

[0085] Among them, the operator Let φ represent the mapping of the bounded linear operator space, W represent the weight matrix, and σ represent the local nonlinear activation function.

[0086] For Fourier neural operators, the mapping process of Fourier layers depends on the Fourier transform and its inverse; applying the convolution theorem, the kernel integral operator can be replaced by convolution operations in Fourier space; the replacement form of the kernel integral operator in Fourier space is:

[0087]

[0088] in, Represents Fourier transform, This represents the inverse Fourier transform; the formula expresses the equivalent form of the kernel integral operator in Fourier space, that is, through the Fourier transform, the kernel integral operation can be simplified to the dot product operation in Fourier space, and then the inverse transform returns to the original space.

[0089] Finally, the output of the Fourier layer is locally transformed and projected back into the original output space to obtain the final prediction result.

[0090] u(x)=Q(ω T (x));

[0091] Where, ω TLet Q represent the final state, where Q represents the local transformation and u(x) represents the output.

[0092] Since the input data of this invention contains two modes, namely bending and torsion, this invention also innovatively proposes a multi-branch structure, performs Fourier transforms on the two different modes respectively and extracts different Fourier mode numbers, thereby improving the aerodynamic prediction accuracy of the Fourier neural operator model under multimodal coupled motion.

[0093] The internal structure of the above Fourier layer is as follows Figure 4 As shown.

[0094] Where F1 and F2 represent Fourier transforms of different modes, respectively; R represents a linear transform; F1 -1 and F2 -1 , respectively, represent the inverse Fourier transform of different modes; b represents the bias matrix; W represents the weight matrix.

[0095] The specific process by which the time delay characteristics are effectively captured using frequency domain transformation is as follows:

[0096] Represent the input function in the frequency domain using Fourier transform:

[0097]

[0098] Convert the frequency domain data back to the original space:

[0099]

[0100] Where, x j k represents the discrete coordinate index in physical space. j Here, f(x) represents the Fourier frequency index, f(k) represents the signal in physical space, and f(k) represents the signal in the frequency domain, i.e., the Fourier transform result. j The exponent term represents the resolution of the discrete grid. Control the change of direction.

[0101] During training, a custom Adam optimizer and relative L2 loss function were used to progressively update model parameters through backpropagation and optimization steps. A learning rate scheduler (StepLR) was used to adjust the learning rate, and the model's performance on the test set was periodically evaluated. MSE, RMSE, and R2 were calculated to measure prediction effectiveness. After training, the predicted values ​​were compared with the true values ​​visually to further adjust the hyperparameters and evaluate the model's performance, ensuring its accuracy and robustness in aerodynamic prediction tasks, ultimately resulting in a well-trained model.

[0102] S400, Aerodynamic Coefficient Prediction

[0103] By applying the Fourier neural operator model trained in step S300 to the prediction of aerodynamic coefficients, this invention only needs to input generalized displacement data into the trained model to quickly and accurately predict the corresponding aerodynamic coefficients.

[0104] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A transonic unsteady aerodynamic order reduction method based on global mapping, characterized in that, The order reduction method specifically includes the following steps: (1) Design the input signal The filtered Gaussian white noise random signal is used as the input signal. The input signal needs to cover a wide range of frequencies and amplitudes to reflect the dynamic linear characteristics of the flow field under small disturbances and the nonlinear characteristics under large disturbances. (2) Obtain training data The input signal designed in step (1) above is numerically calculated using a CFD solver. Based on the results of the numerical calculation, a multi-input-multi-output mapping relationship between generalized displacement and aerodynamic force is constructed and used as the training data for the model. (3) Training the model A data-driven Fourier neural operator model is used to predict unsteady aerodynamic forces, and the Fourier neural operator model is trained. (4) Prediction of aerodynamic coefficients By inputting generalized displacement data into the Fourier neural operator model trained in step (3) above, the corresponding aerodynamic coefficients can be predicted quickly and accurately. In step (1), the amplitude of the input signal gradually increases from small to large in the initial stage; by designing a mixture of small and large amplitude signals, the input signal can simulate the dynamic response of the aerodynamic system under different disturbance intensities. The input signal in step (2) includes heave motion and pitch motion, which correspond to the bending mode and torsional mode in the two-dimensional airfoil aeroelastic analysis, respectively; In step (2), during the CFD calculation of the input signal, the values ​​of the displacement generated by the heave motion and the angle of attack generated by the pitch motion are simultaneously used as inputs in the CFD solver for calculation. Furthermore, by calculating the mesh deformation, the rotation center of the pitch motion changes with the change of the heave displacement, thereby obtaining the corresponding aerodynamic coefficients. During mesh deformation, the coordinates of each node are updated using the following formula: ； ； in,( , () represents the new coordinates of the node after deformation. It is the radial distance of the node in the polar coordinate system. , () represents the coordinates of the node before deformation. It is the polar angle before deformation. It is the increment of pitch angle of attack. It is the increment of the floating and sinking displacement; The Fourier neural operator model in step (3) has time discrete invariance and can use different time steps during the training and testing phases, thereby achieving more efficient prediction of unsteady aerodynamic forces.

2. The transonic unsteady aerodynamic order reduction method based on global mapping as described in claim 1, characterized in that, The specific process of training the Fourier neural operator model in step (3) is as follows: First, the input data is transformed into a high-dimensional representation through a dimensionality boosting operation, which is implemented by a shallow fully connected neural network. ； in, This is the initial state. This represents a local transformation implemented by a shallow, fully connected neural network; Represents input; Next, the hidden layers (Fourier layers) in the Fourier neural operator model are used as part of an iterative architecture and mapped to the next layer iteratively. The mapping between different function spaces is gradually approximated using convolution operators in Fourier space. The specific iterative process is as follows: ； ； in, Represents an intermediate state. Represents a nonlinear mapping. The set of parameters representing Fourier neural operators; ； Among them, the operator Represents a mapping of a bounded linear operator space. Indicates parameters, Represents the weight matrix. Represents a local nonlinear activation function; For Fourier neural operators, the mapping process of Fourier layers depends on the Fourier transform and its inverse; applying the convolution theorem, the kernel integral operator can be replaced by convolution operations in Fourier space; the replacement form of the kernel integral operator in Fourier space is: ； in, Represents Fourier transform, Represents the inverse Fourier transform; Finally, the output of the Fourier layer is locally transformed and projected back into the original output space to obtain the final prediction result. ； in, For the final state, Represents a local transformation. Represents output.

3. The transonic unsteady aerodynamic order reduction method based on global mapping as described in claim 1, characterized in that: The Fourier neural operator model trained in step (3) extracts the inherent nonlinear characteristics of the aerodynamic system through a hierarchical combination of spectrum and point-by-point operations, and effectively captures the time delay characteristics by utilizing frequency domain transformation.

4. The transonic unsteady aerodynamic order reduction method based on global mapping as described in claim 3, characterized in that... The specific process of effectively capturing time delay characteristics using frequency domain transformation is as follows: Represent the input function in the frequency domain using Fourier transform: ； Convert the frequency domain data back to the original space: ； in, Represents discrete coordinate indices in physical space. Represents the Fourier frequency index. Represents signals in physical space. The signal in the frequency domain is represented by the Fourier transform result. The exponential term represents the resolution of the discrete grid. Control the change of direction.

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