^{1}

^{2}

^{1}

^{1}

^{1}

^{2}

It is not easy for the system identification-based reduced-order model (ROM) and even eigenmode based reduced-order model to predict the limit cycle oscillation generated by the nonlinear unsteady aerodynamics. Most of these traditional ROMs are sensitive to the flow parameter variation. In order to deal with this problem, a support vector machine- (SVM-) based ROM was investigated and the general construction framework was proposed. The two-DOF aeroelastic system for the NACA 64A010 airfoil in transonic flow was then demonstrated for the new SVM-based ROM. The simulation results show that the new ROM can capture the LCO behavior of the nonlinear aeroelastic system with good accuracy and high efficiency. The robustness and computational efficiency of the SVM-based ROM would provide a promising tool for real-time flight simulation including nonlinear aeroelastic effects.

Aeroelasticity is the science concerned with the fluid-structure interaction including the inertial, elastic, and aerodynamic forces. The prediction of aeroelastic instability in the transonic regime plays a very important role in the definition of the flight envelope for many high-performance aircraft. For example, flutter and limit cycle oscillation (LCO) are the major nonlinear aeroelastic unstable phenomena, which are very dangerous to the aircraft structure. With the development of computational aeroelasticity, the nonlinear aeroelastic response can be accurately predicted by the high-fidelity physics-based CFD/CSD couple solver. However, the use of multistep time domain calculations for each aircraft state is computationally expensive and provides limited insight into the dependence of the parameters on the type of response in the vicinity of the instability boundary. In order to reduce the expensive computational cost, a novel conception called reduced-order model (ROM) based on high-fidelity physics model has been put forward in recent years. ROM seeks to capture the dominant nonlinear behavior of the aeroelastic system by constructing a simple mathematical representative model, which is very convenient to be used in conceptual design, control, and data-driven systems [

Different approaches for reduced-order modeling of aerodynamic systems have been investigated, including linearization about a nonlinear steady-state flow data-driven model such as Volterra theory of nonlinear systems [

Nonlinear system-identification-based ROMs have been widely used to predict the transonic flutter boundary such as Volterra series [

In this study, we develop an SVM-based ROM for predicting the LCO induced by the nonlinear aerodynamics with high efficiency and good accuracy. Firstly, we gave a brief introduction about the regression SVM machine; secondly, we proposed a general construction framework of the SVM-based ROM for the aeroelastic system; finally, we demonstrated the ROM by the two-DOF NACA 64A010 airfoil aeroelastic model in detail.

The basic idea of SVM theory is to map the data to a higher-dimensional feature space via nonlinear mapping functions and then do the linear regression in this space. SVM-based nonlinear system identification approaches have taken advantage of both the kernel trick and the well-developed SVR algorithmic implementations [

Based on statistical learning theory, SVRs can be applicable especially to small-sample learning problems. Here, we give a brief summary of SVR. Given an input-output data set

After calculating the partial derivatives of

Equation (

In (

The full-coupled nonlinear semi-discrete aeroelastic equation is given as

CFD/CSD couple algorithm.

The typical two degrees of freedom (2DOF) aeroelastic model with plunge and pitch freedom has been widely used to validate the LCO prediction method [

Two-DOF aeroelastic model.

For CFD/CSD couple simulation, most of the computational cost is spent in solving the unsteady aerodynamics. The reduced-order model method is introduced for unsteady flow to improve the computational efficiency in the aeroelastic simulation. In the aerodynamic subsystem, the inputs are the structure state values

The flow chart of the construction procedure of the SVM-based aeroelastic ROM is illustrated in Figure

The workflow of SVM-based ROM.

As the initial demonstration of the SVM-based ROM, the NACA 64A010 airfoil model was selected. The structure parameters are

At the given Mach number 0.825, two white noise signals for pitch motion whose amplitudes are 0.01 and 0.2, respectively, were selected and federated into the CFD solver to compute the time series response of the unsteady aerodynamic coefficients. Selecting

Comparison of aerodynamic response at Mach = 0.825.

The accuracy of the SVM-based ROM has been verified for the unsteady aerodynamics. Now, we would like to couple the SVM-based ROM with the structure dynamics equation and investigate its effectiveness in aeroelastic response analysis. At the Mach number 0.825 and reduced velocity

Comparison of LCO response at Mach = 0.825, and

In order to check the capability of the ROM, we construct another SVM model at Mach = 0.8 to investigate the LCO behavior. We succeed to capture some LCO at

LCO response at different reduced velocity at Mach = 0.8.

As noted previously, the traditional ROMs such as POD/ROM and Volterra/ROM are very sensitive to the flow parameter variation, especially for the Mach numbers and Reynolds numbers [

Firstly, we prepare the unsteady responses

Figure

Comparison of LCO response at Mach = 0.85, and

In order to further validate the performance of the SVM-based ROM, we calculated the LCO behavior of the aeroelastic system in different Mach numbers. All of the reduced velocity is

The LCO amplitude with Mach number (ROM versus full-order model).

There is also an interesting thing that outside of the sampling Mach number set, such as at 0.7625, the development trend of LCO predicted by SVM does not agree with CFD solver, but the error is not so large. Maybe it is because of that SVR has excellent learning capability and generalization capability. We also investigated the LCO response above the Mach number 0.9, but the SVM model fail to capture the LCO. It seems that the aeroelastic system runs into divergence or convergence from the simulation of the full-order solver. It is obviously and reasonable that no data-driven model can do everything.

It takes about 1 hour for the CFD solver to capture the LCO response, but it is no more than one minutes for the SVM-based ROM to predict the same response. The computational efficiency of ROM is obvious, which is very important to the near real-time flight dynamic simulation and flight controller design. The fast prediction of the system response with good accuracy is one of the most important factors to realize these challenge applications successfully.

An SVM-based reduced-order model was developed for fast prediction of the response of the nonlinear aeroelastic system. We proposed a general construction framework for the SVM-based ROM. The two-dimensional aerofoil aeroelastic system was used to demonstrate the capability and performance of the ROM in detail. The simulation results show the capability, accuracy, and high efficiency of the SVM-based ROM for the LCO prediction. The SVM-based ROM can also fairly predict the LCO response of the 2DOF aeroelastic model with the Mach number variation. The robustness of the SVM-based ROM with varying flow parameters provides a useful tool for real-time flight simulation of flexible vehicle. Further research will focus on developing new training methods and improving the accuracy of the ROM, especially for the cases with parameters variation.

This work was supported by the National Natural Science Foundation of China (10902082, 91016008), New Faculty Research Foundation of XJTU, and the Fundamental Research Funds for the Central Universities (xjj20100126). The first author acknowledges W. T. Mao for the discussion of SVM algorithms. All the authors thank H. N. Agiza and the reviewers for their good comments.