Transonic unsteady aerodynamic force order reduction method based on global mapping

Abstract

The invention provides a transonic unsteady aerodynamic force order reduction method based on global mapping, and the method comprises the steps: employing a filtered Gaussian white noise random signal as an input signal which needs to cover a wide frequency and amplitude range; carrying out numerical calculation on the designed input signal by adopting a CFD solver, constructing a multi-input-multi-output mapping relation between generalized displacement and aerodynamic force, and taking the multi-input-multi-output mapping relation as training data of the model; a Fourier neural operator model based on data driving is adopted to predict the unsteady aerodynamic force, and the Fourier neural operator model is trained; and inputting generalized displacement data into the trained Fourier neural operator model to predict a corresponding aerodynamic coefficient. The method is reasonable in conception, compared with a previous order reduction method, the time spent on training the same data is shorter, the precision and efficiency of aerodynamic coefficient prediction are higher, and dynamic linear characteristics and nonlinear characteristics can be captured at the same time.

Description

Technical Field

[0001] The present invention relates to the technical field of transonic unsteady aerodynamics, and in particular to a transonic unsteady aerodynamics order reduction method based on global mapping. Background Art

[0002] Transonic flows are common during aircraft cruising and maneuvering. Their aerodynamic characteristics are influenced by complex nonlinear phenomena such as shock wave-boundary layer interactions, turbulence, and flow separation, making them difficult to accurately predict and describe using traditional modeling methods. Under unsteady transonic conditions, aerodynamic responses often exhibit significant time lags and nonlinear characteristics. Accurately capturing these dynamic characteristics is crucial for high-precision modeling. Specifically, time lag refers to the hysteresis of aerodynamic responses relative to flow field evolution or flight state changes. Nonlinear dynamic characteristics arise not only from shock wave motion and flow separation, but also from turbulence, unsteady vortex shedding, and other factors. These nonlinear and unsteady effects make traditional assumptions untenable. Therefore, high-precision modeling methods must comprehensively consider the historical dependence of flow evolution (such as the influence of previous aerodynamic states on the current response) and the ability to accurately predict in 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 the aircraft. At present, the order reduction methods for transonic unsteady aerodynamics mainly include AutoRegressive with eXogenous input (ARX) model, Non-linear AutoRegressive with eXogenous input (NARX) model, Volterra sequence model and neural network-based model.

[0003] The prior art has the following problems:

[0004] (1) Existing models still need to be improved in terms of prediction accuracy, generalization ability, and computational efficiency. At the same time, there are also challenges in terms of robustness under complex flow conditions and computational cost control.

[0005] (2) Existing methods usually perform single-step predictions based on fixed input-output mappings, which can only gradually deduce the temporal evolution of unsteady aerodynamic forces. However, it is difficult to simultaneously capture the global space-time distribution characteristics and achieve multi-input-multi-output overall predictions, which limits the modeling ability of complex unsteady phenomena.

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

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

[0008] In summary, it is necessary to make further innovations to the existing technologies. Summary of the Invention

[0009] In response to the technical problems existing in the above-mentioned background technology, the present invention proposes a transonic unsteady aerodynamic force reduction method based on global mapping. The method has a reasonable concept. Compared with previous reduction methods, it takes less time to train the same data, has higher accuracy and efficiency in predicting aerodynamic coefficients, and can capture both dynamic linear and nonlinear characteristics at the same time.

[0010] To solve the above technical problems, the present invention provides a transonic unsteady aerodynamic force reduction method based on global mapping, which specifically includes the following steps:

[0011] (1) Design input signal

[0012] A 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) Obtaining training data

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

[0015] (3) Training 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) Aerodynamic coefficient prediction

[0018] By inputting the 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 force reduction method based on global mapping, wherein: the signal amplitude of the input signal in the 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 force reduction method based on global mapping, wherein the input signal in step (2) includes heaving motion and pitching motion, which correspond to bending mode and torsional mode in the aeroelastic analysis of two-dimensional airfoil, 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 for calculation in the CFD solver, and the rotation center of the pitch motion is realized to change with the change of the heave displacement by calculating the grid deformation method, thereby obtaining the corresponding aerodynamic coefficient;

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

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

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

[0028] ω0=P(a(x));

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

[0030] a(x) represents the input;

[0031] Next, the hidden layer in the Fourier neural operator model, i.e., the Fourier layer, is used as part of the iterative architecture and is mapped to the next layer in an iterative manner. Through the convolution operator in the Fourier space, the mapping between different function spaces is gradually approximated. The specific iterative process is as follows:

[0032]

[0033] Among them, ω t Represents the intermediate state, represents the nonlinear mapping, θ represents the parameter set of the Fourier neural operator;

[0034]

[0035] Among them, the operator represents the mapping of bounded linear operator space, φ represents the parameter, W represents the weight matrix, and σ represents the local nonlinear activation function;

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

[0037]

[0038] in, stands for Fourier transform, stands for inverse Fourier transform;

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

[0040] u(x)=Q(ω T (x));

[0041] Among them, ω T is the final state, Q represents the local transformation, and u(x) represents the output.

[0042] The transonic unsteady aerodynamic force reduction method based on global mapping, wherein: 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-lag characteristics by using frequency domain transformation.

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

[0044] The input function is represented in the frequency domain by Fourier transform:

[0045]

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

[0047]

[0048] Among them, x j Represents the discrete coordinate index in physical space, k jRepresents the Fourier frequency index, f(x) represents the signal in the physical space, f(k) represents the signal in the frequency domain, that is, the Fourier transform result, s j Represents the resolution of the discrete grid, the exponential term Controls the direction of the transformation.

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

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

[0051] Compared with traditional CFD calculations, the speed of predicting a working condition is increased by 3 orders of magnitude.

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

[0053] The present invention has higher prediction accuracy than the previous reduced-order model; 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] The present invention can capture dynamic linear characteristics and nonlinear characteristics at the same time.

[0055] Since the Fourier neural operator model is discrete invariant, different time steps can be used in the training and prediction stages, thereby achieving more efficient prediction. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the specific embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0057] Figure 1 Flowchart of the transonic unsteady aerodynamic force reduction method based on global mapping of the present invention;

[0058] Figure 2 Schematic diagram of the input signal designed for the transonic unsteady aerodynamic force reduction method based on global mapping of the present invention;

[0059] Figure 34 is a structural diagram of a Fourier neural operator model based on global mapping adopted in the transonic unsteady aerodynamic force reduction method based on global mapping of the present invention;

[0060] Figure 4 This is an internal expansion diagram of the Fourier layer in the reduced-order model structure based on global mapping adopted in the transonic unsteady aerodynamic force reduction method based on global mapping of the present invention. DETAILED DESCRIPTION

[0061] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the embodiments described are only some embodiments of the present invention, not all embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall 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, this embodiment provides a transonic unsteady aerodynamic force 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 reduction model, especially in improving the generalization ability of the model. In order for the model to effectively capture the dynamic characteristics of the aerodynamic system under different disturbance amplitudes, 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. Therefore, a filtered white Gaussian noise (FWGN) random signal is used as the input, in which the signal amplitude is small in the initial stage and gradually increases to a larger amplitude, such as Figure 2 By mixing 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 perform numerical calculations on the input signal designed in step S100. The input signal includes heaving and pitching 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 heaving displacement and the pitch angle of attack are simultaneously input into the CFD solver. The center of rotation of the pitching motion is calculated to change with the heaving displacement by deforming the computational grid, 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] Among them, (x′, y′) is the new coordinate of the node after deformation, is the radial distance of the node in the polar coordinate system, (x,y) is the coordinate of the node before deformation, 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 results of the above numerical calculations, a multi-input-multi-output (MIMO) mapping relationship between generalized displacement and aerodynamic force is constructed and used as the training data of the model.

[0073] S300, training model

[0074] In the present invention, a data-driven Fourier neural operator model is used to predict unsteady aerodynamic forces and is trained. Unlike traditional neural networks, which map finite-dimensional vectors to finite-dimensional vectors, the Fourier neural operator achieves global mapping by learning the underlying mapping between function spaces, thereby 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 the function space analog of the weight matrix in a standard feedforward neural network.

[0075] The Fourier neural operator model trained in the present invention 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-lag characteristics by using frequency domain transformation. In addition, the Fourier neural operator model also has time discrete invariance and can use different time steps in the training and testing phases, thereby more efficiently realizing the prediction of unsteady aerodynamic forces. The specific structure of the Fourier neural operator model is as follows: Figure 3 shown.

[0076] The Fourier neural operator model can be divided into three key parts: first, the input data is converted into a high-dimensional representation through a dimensionality lifting operation, which is implemented by a shallow fully connected neural network;

[0077] ω0=P(a(x));

[0078] Here, ω0 is the initial state and P represents the local transformation implemented by a 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 the features in the input data;

[0081] Next, the hidden layer in the Fourier neural operator model, namely the Fourier layer, is used as part of the iterative architecture and is mapped to the next layer in an iterative manner. The function of each Fourier layer is similar to that of a standard feedforward neural network, using a weight matrix to simulate the mapping relationship between function spaces. Specifically, the convolution operator in Fourier space is used to gradually approximate the mapping between different function spaces. The specific iterative process is as follows:

[0082]

[0083] Among them, ω t Represents the intermediate state, represents the nonlinear mapping, θ represents the parameter set of the Fourier neural operator;

[0084]

[0085] Among them, the operator represents the mapping of bounded linear operator space, φ represents the parameter, W represents the weight matrix, and σ represents the local nonlinear activation function;

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

[0087]

[0088] in, stands for Fourier transform, stands for inverse Fourier transform; this formula expresses the equivalent form of the kernel integral operator in Fourier space, that is, through Fourier transform, the kernel integral operation can be simplified to a point multiplication operation in Fourier space, and then returned to the original space through inverse transform.

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

[0090] u(x)=Q(ω T (x));

[0091] Among them, ω Tis the final state, Q represents the local transformation, and u(x) represents the output.

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

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

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

[0095] The specific process of effectively capturing the time-lag characteristics by using frequency domain transformation is as follows:

[0096] The input function is represented in the frequency domain by Fourier transform:

[0097]

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

[0099]

[0100] Among them, x j Represents the discrete coordinate index in physical space, k j Represents the Fourier frequency index, f(x) represents the signal in the physical space, f(k) represents the signal in the frequency domain, that is, the Fourier transform result, s j Represents the resolution of the discrete grid, the exponential term Controls the direction of the transformation.

[0101] During training, a custom Adam optimizer and a relative L2 loss function are used to gradually update model parameters through backpropagation and optimization steps. A learning rate scheduler (StepLR) is used to adjust the learning rate. Model performance on the test set is regularly evaluated, and the mean square error (MSE), relative mean square error (RMSE), and R2 are calculated to measure prediction effectiveness. After training, hyperparameters are further adjusted and model performance is evaluated by visually comparing predicted values to true values to ensure accuracy and robustness in aerodynamic force prediction tasks. Ultimately, a trained model is obtained.

[0102] S400, aerodynamic coefficient prediction

[0103] The Fourier neural operator model trained in step S300 is applied to the prediction of the aerodynamic coefficients. The present invention only needs to input the 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, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements 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 force reduction method based on global mapping, characterized in that: The order reduction method specifically comprises the following steps: (1) Design input signal A 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) Obtaining training data The CFD solver is used to perform numerical calculations on the input signals designed in the above step (1). Based on the results of the numerical calculations, a multi-input-multi-output mapping relationship between generalized displacement and aerodynamic force is constructed and used as training data for the model. (3) Training model A data-driven Fourier neural operator model is used to predict unsteady aerodynamic forces, and the Fourier neural operator model is trained; (4) Aerodynamic coefficient prediction By inputting the generalized displacement data into the Fourier neural operator model trained in step (3) above, the corresponding aerodynamic coefficients can be predicted quickly and accurately.

2. The transonic unsteady aerodynamic force reduction method based on global mapping according to claim 1, characterized in that: In the step (1), the signal amplitude of the input signal 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.

3. The transonic unsteady aerodynamic force reduction method based on global mapping according to claim 1, characterized in that: The input signal in step (2) includes heaving motion and pitching motion, which correspond to bending mode and torsional mode in the aeroelastic analysis of two-dimensional airfoil, respectively; In step (2), during the CFD calculation of the input signal, the displacement generated by the heaving motion and the angle of attack generated by the pitching motion are simultaneously used as inputs for calculation in the CFD solver, and the rotation center of the pitching motion is realized to change with the change of the heaving displacement by calculating the grid deformation method, thereby obtaining the corresponding aerodynamic coefficient; During mesh deformation, the coordinates of each node are updated using the following formula: x ′ =rcos(θ+Δθ); y ′ =rsin(θ+Δθ)+Δy; Among them, (x ′ ,y ′ ) is the new coordinate of the node after deformation, is the radial distance of the node in the polar coordinate system, (x,y) is the coordinate of the node before deformation, is the polar angle before deformation, Δθ is the increment of pitch angle of attack, and Δy is the increment of heave displacement.

4. The transonic unsteady aerodynamic force reduction method based on global mapping according to 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 converted into a high-dimensional representation through a dimensionality lifting operation, which is implemented by a shallow fully connected neural network; ω0=P(a(x)); Where ω0 is the initial state, P represents the local transformation implemented by the shallow fully connected neural network, and a(x) represents the input; Next, the hidden layer in the Fourier neural operator model, i.e., the Fourier layer, is used as part of the iterative architecture and is mapped to the next layer in an iterative manner. Through the convolution operator in the Fourier space, the mapping between different function spaces is gradually approximated. The specific iterative process is as follows: Among them, ω t Represents the intermediate state, represents the nonlinear mapping, θ represents the parameter set of the Fourier neural operator; Among them, the operator represents the mapping of bounded linear operator space, φ represents the parameter, W represents the weight matrix, and σ represents the local nonlinear activation function; For the Fourier neural operator, the mapping process of the Fourier layer depends on the Fourier transform and its inverse; applying the convolution theorem, the kernel integral operator can be replaced by the convolution operation in the Fourier space; the replacement form of the kernel integral operator in the Fourier space is: in, stands for Fourier transform, stands for inverse Fourier transform; Finally, the output of the Fourier layer is locally transformed and projected back to the original output space to obtain the final prediction result; u(x)=Q(ω T (x)); Among them, ω T is the final state, Q represents the local transformation, and u(x) represents the output.

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

6. The transonic unsteady aerodynamic force reduction method based on global mapping according to claim 5 is characterized in that ,The specific process of effectively capturing the time lag characteristics by using the frequency domain transformation is as follows: The input function is represented in the frequency domain by Fourier transform: Convert the frequency domain data back to the original space: Among them, x j Represents the discrete coordinate index in physical space, k j Represents the Fourier frequency index, f(x) represents the signal in the physical space, f(k) represents the signal in the frequency domain, that is, the Fourier transform result, s j Represents the resolution of the discrete grid, the exponential term Controls the direction of the transformation.

7. The transonic unsteady aerodynamic force reduction method based on global mapping according to claim 1, characterized in that: The Fourier neural operator model in step (3) has time discrete invariance and can use different time steps in the training and testing stages, thereby more efficiently realizing unsteady aerodynamic force prediction.

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