Unsteady aerodynamic reduced-order modeling of an aeroelastic wing using arbitrary mode shapes

Computational fluid dynamics (CFD) based unsteady aerodynamic reduced-order model (ROM) can offer significant improvements to the efficiency of transonic aeroelastic analysis. To construct a ROM based on mode shapes, one run of CFD solver is needed to compute aerodynamic responses corresponding to mode excitations. When mode shapes change with structure, another run of the CFD solver is required to construct the new ROM. The typically large computational cost associated with repeated runs of the CFD solver impedes the application of existing unsteady aerodynamic reduced-order modeling methods to transonic aeroelastic design optimization and aeroelastic uncertainty analysis. This paper demonstrates a method that can replace the CFD solver used in the process of existing unsteady aerodynamic reduced-order modeling. It can produce aerodynamic responses corresponding to mode excitations for arbitrary mode shapes within a few seconds. Computational cost can be reduced by two orders of magnitude using the mode excitations and the corresponding aerodynamic responses computed by the method to construct the ROMs used for flutter analyses in aeroelastic design optimization or aeroelastic uncertainty analysis in transonic regime compared with the existing unsteady aerodynamic reduced-order modeling methods. Results show that the method can accurately produce the aerodynamic responses corresponding to the mode excitations and predict the flutter characteristics of AGARD 445.6 wings root-attached in three different ways.     

基于特征正交分解的非定常气动力建模技术

采用特征正交分解(proper orthogonal decomposition, POD)方法, 建立了基于状态空间的非定常气动力降阶模型, 并耦合结构方程, 建立了降阶的气动弹性系统, 开展了颤振分析的初步研究, 计算效率提高了2~3个数量级. 具体过程是:首先获取全阶系统的频域快照构成关联矩阵, 通过对关联矩阵进行奇异值分解提取流场模态(或流场基), 对低能量模态截断形成降阶子空间, 并将其映射到全阶系统, 从而形成基于状态空间的降阶非定常气动力模型. 对气动弹性标模AGARD445.6进行算例验证, 证明了降阶方法正确, 可以提供高效、高精度的气动弹性分析.      

Study of linear response identification techniques and reduced‐order model generation for a 2D CFD scheme

Reduced‐Order Models (ROMs) have been the focus of research in various engineering situations, but it is only relatively recently that such techniques have begun to be introduced into the CFD field. The purpose of generating such models is to capture the dominant dynamics of the full set of CFD equations, but at much lower cost. One method that has been successfully implemented in the field of fluid flows is based on the calculation of the linear pulse responses of the CFD scheme coupled with an Eigensystem Realization algorithm (ERA), resulting in a compact aerodynamic model. The key to the models is the identification of the linear responses of the non‐linear CFD code. Two different methods have been developed and reported in literature for linear response identification; the first method linearizes the CFD code and the second method uses Volterra theory and the non‐linear code. As these methods were developed independently they have not previously been brought together and compared. This paper first explains the subtle, but fundamental differences between the two methods. In addition, a series of test cases are shown to demonstrate the performance and drawbacks of the ROMs derived from the different linear responses. The conclusions of this study provide useful guidance for the implementation of either of the two approaches to obtain the linear responses of an existing CFD code. Copyright © 2006 John Wiley & Sons, Ltd.     

跨音速极限环型颤振的高效数值分析方法

事先建立一个低阶的非线性、非定常气动力模型是开展非线性流场中气动弹性问题研究的一个捷径. 基于CFD方法, 通过计算结构在流场中自激振动的响应来获得系统的训练数据. 采用带输出反馈的循环RBF神经网络, 建立时域非线性气动力降阶模型.耦合结构运动方程和非线性气动力降阶模型, 采用杂交的线性多步方法计算结构在不同速度(动压)下的响应历程, 从而获得模型极限环随速度(动压)变化的特性. 两个典型的跨音速极限环型颤振算例表明, 基于气动力降阶模型方法的计算结果与直接CFD仿真结果吻合很好, 与后者相比其将计算效率提高了1~2个数量级.      

Evaluation of linear,inviscid, viscous,and reduced-order modelling aeroelastic solutions of the AGARD 445.6 wing using root locus analysis

Reduced-order modelling (ROM) methods are applied to the Computational Fluid Dynamics (CFD)-based aeroelastic analysis of the AGARD 445.6 wing in order to gain insight regarding well-known discrepancies between the aeroelastic analyses and the experimental results. The results presented include aeroelastic solutions using the inviscid Computational Aeroelasticity Programme–Transonic Small Disturbance (CAP-TSD) code and the FUN3D code (Euler and Navier–Stokes). Full CFD aeroelastic solutions and ROM aeroelastic solutions, computed at several Mach numbers, are presented in the form of root locus plots in order to better reveal the aeroelastic root migrations with increasing dynamic pressure. Important conclusions are drawn from these results including the ability of the linear CAP-TSD code to accurately predict the entire experimental flutter boundary (repeat of analyses performed in the 1980s), that the Euler solutions at supersonic conditions indicate that the third mode is always unstable, and that the FUN3D Navier–Stokes solutions stabilize the unstable third mode seen in the Euler solutions.     

Stabilisation of reduced order models via restarting

This paper describes the development, implementation and validation of an explicit restarting scheme to stabilise reduced order models produced using the Eigensystem Realisation Algorithm. Here, the method is applied to a continuous time fluid dynamics model that is actually solved in discrete time space using a CFD approach. However, the methods described are generic and could be applied to any other similar dynamically linear system. The basic restarting approach successfully stabilises ROMs in the majority of cases. However, there are cases where this approach breaks down. The reasons for this breakdown in the basic restarting scheme are discussed, and the strategies for overcoming this problem are described. These include creating ROMs restricted to a range of time step sizes for a specific finite difference scheme and a more general approach using inexact shifts that overcome the problems associated with closely located eigenvalues. Copyright © 2013 John Wiley & Sons, Ltd.     

基于气动力降阶模型的跨音速气动弹性稳定性分析

基于离散型输入输出差分模型,运用非定常CFD方法训练信号,然后运用最小二乘方法进行参数辨识,得到降阶的非定常气动力模型,再将该离散差分模型转换为连续时间域内的状态方程。耦合气动状态方程和结构状态方程,得到耦合系统的气动弹性状态方程。求解不同动压下状态矩阵的特征值,根据根轨迹图分析系统的稳定性特性。分析结果与直接耦合CFD/CSD方法结果相吻合,可以计算跨音速非线性气动弹性问题。其计算效率比直接耦合CFD/CSD方法提高1~2个数量级。针对Isogai wing在跨音速出现的S型颤振边界进行了较为细致的分析,阐述了该现象是由于系统诱发颤振的分支随着速度(来流动压)的提高而发生转移所导致的。     

基于CFD的气动力建模及其在气动弹性中的应用

CFD技术为带有气动力非线性的气动弹性分析提供了一种研究途径,但是基于CFD的气动弹性直接数值模拟方法的计算量很大,不便开展定性分析和参数设计.基于CFD的非定常气动力模型的降阶技术为缓解计算效率与计算精度之间的矛盾以及系统的复杂性与易分析、易设计性之间的矛盾提供了行之有效的技术途径.综述了近年来发展的两类基于CFD技术的非定常气动力降阶技术和一种非线性气动力的谐波平衡方法,以及这些方法在非线性气动弹性研究中的运用.对比了各种方法的优越性并作了进一步的展望.      

Reduced order state‐space models from the pulse responses of a linearized CFD scheme

This paper describes a method for obtaining a time continuous reduced order model (ROM) from a system of time continuous linear differential equations. These equations are first put into a time discrete form using a finite difference approximation. The unit sample responses of the discrete system are calculated for each system input and these provide the Markov parameters of the system. An eigenvalue realization algorithm (ERA) is used to construct a discrete ROM. This ROM is then used to obtain a continuous ROM of the original continuous system. The focus of this paper is on the application of this method to the calculation of unsteady flows using the linearized Euler equations on moving meshes for aerofoils undergoing heave or linearized pitch motions. Applying a standard cell‐centre spatial discretization and taking account of mesh movement a continuous system of differential equations is obtained which are continuous in time. These are put into discrete time form using an implicit finite difference approximation. Results are presented demonstrating the efficiency of the system reduction method for this system. Copyright © 2003 John Wiley & Sons, Ltd.     

An evolve‐then‐filter regularized reduced order model for convection‐dominated flows

In this paper, we propose a new evolve‐then‐filter reduced order model (EF‐ROM). This is a regularized ROM (Reg‐ROM), which aims to add numerical stabilization to proper orthogonal decomposition (POD) ROMs for convection‐dominated flows. We also consider the Leray ROM (L‐ROM). These two Reg‐ROMs use explicit ROM spatial filtering to smooth (regularize) various terms in the ROMs. Two spatial filters are used: a POD projection onto a POD subspace (Proj) and a POD differential filter (DF). The four Reg‐ROM/filter combinations are tested in the numerical simulation of the three‐dimensional flow past a circular cylinder at a Reynolds number Re=1000. Overall, the most accurate Reg‐ROM/filter combination is EF‐ROM‐DF. Furthermore, the spatial filter has a higher impact on the Reg‐ROM than the regularization used. Indeed, the DF generally yields better results than Proj for both the EF‐ROM and L‐ROM. Finally, the CPU times of the four Reg‐ROM/filter combinations are orders of magnitude lower than the CPU time of the DNS. Copyright © 2017 John Wiley & Sons, Ltd.     

Identification of limit cycles for piecewise nonlinear aeroelastic systems

This paper describes two methods for the analysis of aeroelastic systems with complex piecewise nonlinear structural stiffness. These methods are tested and compared for low speed incompressible and transonic flows. The first technique employed in this paper uses a new application of the analytical solution of linear algebraic systems, the second technique utilises logarithmic and tanh functions to both represent discrete nonlinearities and to act as a switch between different nonlinear areas. The transonic aerodynamic models used are generated using an eigenvalue realisation algorithm (ERA) which produces reduced order models (ROMs) from the pulse responses of time linearised Euler simulations. It is shown that such aerodynamic models are well suited to use with continuation methods. Flutter boundaries and limit cycle oscillations can then be rapidly identified with good accuracy.     

弹性支撑圆柱绕流稳定性分析

基于CFD 技术,采用系统辨识方法,建立了亚临界雷诺数(Re < 47) 下绕圆柱流动的非定常气动力模型(reduced order model, ROM). 耦合结构运动方程和降阶气动力模型,建立了弹性支撑圆柱绕流的稳定性分析模型. 算例分析了亚临界雷诺数下,结构固有频率、质量比等参数以及支撑方式对弹性系统稳定性的影响. 对于单自由度横向支撑圆柱,当结构固有频率趋近流动最不稳定模态频率时,弹性系统会在一定频率范围内失稳,这种现象最低可在Re~20 时出现. 旋转自由度的释放能够进一步降低系统的稳定性,可将临界雷诺数进一步降低至18 左右. ROM 方法不仅具有很高的效率,而且清晰地指出了弹性系统失稳的根本原因:流动模态和结构模态耦合作用导致结构模态失稳所致. 因此,失稳状态下系统振荡频率锁定于结构固有频率. 基于ROM 技术预测的失稳边界与直接CFD/CSD 仿真结果吻合,证明了该方法的正确性和精度.      

基于POD和代理模型的静气动弹性分析方法

静气动弹性问题考虑弹性结构与定常气动力间的相互耦合作用,对飞行器的性能和安全具有显著的影响.在现代飞行器设计阶段,计算流体力学(CFD)/计算结构力学(CSD)直接耦合方法是精确考察静气动弹性影响的重要手段.然而,基于CFD技术的气动力仿真手段在耦合过程中计算量大且耗时长,难以满足设计阶段的需求.因此,为了兼顾计算精度与效率,文章采用本征正交分解(POD)和Kriging代理模型相结合的模型降阶方法,替代CFD求解过程并耦合有限元分析(FEA)方法,建立了高效、准确的静气动弹性分析框架.相较于传统的以模态法为主的静气动弹性分析方法,该方法能够解决更为复杂的静气动弹性问题以及提供静气动弹性变形过程中的气动分布载荷.针对典型三维跨声速HIRENASD机翼模型开展的马赫数、迎角变化的算例验证表明:由建立的静气动弹性分析方法与CFD/CSD直接耦合方法计算得到机翼翼梢处的静变形量间的相对误差在5%以内;同时该方法预测静平衡位置处的气动分布载荷的误差在5%以内,静气动弹性分析的计算效率至少提升了6倍.     

Nonlinear dynamics of an aeroelastic airfoil with free-play in transonic flow

Nonlinear dynamic behaviors of an aeroelastic airfoil with free-play in transonic air flow are studied. The aeroelastic response is obtained by using time-marching approach with computational fluid dynamics (CFD) and reduced order model (ROM) techniques. Several standardized tests of transonic flutter are presented to validate numerical approaches. It is found that in time-marching approach with CFD technique, the time-step size has a significant effect on the calculated aeroelastic response, especially for cases considering both structural and aerodynamic nonlinearities. The nonlinear dynamic behavior for the present model in transonic air flow is greatly different from that in subsonic regime where only simple harmonic oscillations are observed. Major features of the responses in transonic air flow at different flow speeds can be summarized as follows. The aeroelastic responses with the amplitude near the free-play are dominated by single degree of freedom flutter mechanism, and snap-though phenomenon can be observed when the air speed is low. The bifurcation diagram can be captured by using ROM technique, and it is observed that the route to chaos for the present model is via period-doubling, which is essentially caused by the free-play nonlinearity. When the flow speed approaches the linear flutter speed, the aeroelastic system vibrates with large amplitude, which is dominated by the aerodynamic nonlinearity. Effects of boundary layer and airfoil profile on the nonlinear responses of the aeroelastic system are also discussed.     

基于ROM技术的阵风响应分析方法

阵风响应分析是大型飞机设计过程中必不可少的环节. 现有的阵风响应分析主要采用基于线化升力面理论的气动力模型,不能考虑到各种非线性效应,不适合于跨音速气动弹性的分析. 基于CFD技术,采用系统辨识方法,在状态空间内建立了降阶的非定常气动力模型(reduced order model, ROM). 耦合结构运动方程、非定常气动力模型(结构运动)、外激阵风的气动力模型,建立了基于CFD技术的阵风响应分析模型.算例研究了某一典型机翼在方波激励下的阵风响应问题,对比了各阶模态位移的响应以及翼根弯矩的响应. 基于ROM技术的计算结果与CFD/CSD直接耦合仿真结果吻合,证明了该方法的正确性和精度.      

一种高效的叶轮机叶片气动阻尼计算方法

运用叠加原理, 发展了一种可以运用于小振幅运动的叶轮机叶片非定常气动力降阶模型, 并将该模型与传统的能量法相结合, 提出了一种叶轮机叶片气动阻尼的高效求解方法. 运用该方法求解叶轮机叶片的气动阻尼系数, 对某个频率、某个模态只需要进行一次非定常计算, 就可以求出所有叶间振动相角下的气动阻尼系数, 提高了气动阻尼的求解效率. 在STCF4和NASA Rotor67两个算例上运用非定常雷诺平均N-S(RANS)方程和提出的降阶模型进行了对比计算.算例表明, 在小振幅下该方法的计算结果与RANS方程计算得到的气动阻尼系数能很好地吻合, 而计算效率相比多通道非定常RANS方程计算提升了近一个数量级, 并且该方法还可以运用于有失谐情况的颤振分析, 在工程上有较高的应用价值.      

Analyzing mistuned multi-stage turbomachinery rotors with aerodynamic effects

A great deal of research has been conducted on accurately modeling large cyclic structures such as turbomachinery rotors. Accurate modeling of realistic industrial turbomachinery requires overcoming several challenges. The first is the excessively large size of the finite element models (FEMs) needed, which can contain millions of degrees of freedom per stage of the rotor. The second challenge is the presence of small random variations in the structural properties known as mistuning, which arise from operational wear and/or manufacturing tolerances, and destroy the cyclic symmetry of the FEMs. The third is the complexity of turbomachinery models, which often include multiple stages that often have a mismatched computational grid at the interface between stages. The fourth challenge is associated with modeling the aerodynamic loads on the turbomachinery rotor. Much research has been conducted to overcome the first two challenges. By combining cyclic symmetry analysis and component mode mistuning (CMM), compact single-stage reduced order models (ROMs) can be created to accurately capture the free and forced response of these systems. These highly efficient ROMs can be developed from single sector calculations and can be of the order of the number of sectors in the stage. Recently, the third challenge associated with the complexity of modeling multiple stages has been addressed by the authors. Their method uses cyclic symmetry and CMM to form single-stage ROMs (using only single sector models and single sector calculations), and then combines these single-stage ROMs by projecting the motion along the interface between stages along a set of harmonic shape functions. This method allows for the creation of compact ROMs of multi-stage systems with mistuning using sector only calculations. The fourth challenge has been addressed only for single-stage systems by computing a complex aerodynamic matrix (which contains stiffness and damping terms) using an iterative approach. In this work, some of the effects of the aerodynamics on multi-stage systems are explored. The methodology consists of first creating efficient structural ROMs of a multi-stage rotor using the method previously developed, and then iteratively calculating the complex aerodynamic matrices for each stage. A new way to account for the effects of a shift in frequency due to mistuning on the complex aerodynamic matrix is also proposed. Additionally, a new classification of complex multi-stage aeroelastic modes is introduced. The presented results focus on exploring the influences of the aerodynamics and mistuning on the multi-stage response. A variety of numerical results are analyzed for two stages of an industrial rotor.     

Efficient unsteady aerodynamic loads prediction based on nonlinear system identification and proper orthogonal decomposition

In the present work, an efficient surrogate-based framework is developed for the prediction of motion-induced surface pressure fluctuations and integral force and moment coefficients. The model construction is realized by performing forced-motion computational fluid dynamics (CFD) simulations, while the result is processed via the proper orthogonal decomposition (POD) to obtain the predominant flow modes. Subsequently, a nonlinear system identification is carried out with respect to the applied excitation and the resulting POD coefficients. For the input/output model identification task, a recurrent local linear neuro-fuzzy approach is employed in order to capture the linear and nonlinear characteristics of the dynamic system. Once the reduced-order model (ROM) is trained, it can substitute the flow solver within unsteady aerodynamic or aeroelastic simulation frameworks for a given configuration at fixed freestream conditions. For demonstration purposes, the ROM approach is applied to the LANN wing in high subsonic and transonic flow. Due to the characteristic lambda-shock system, the unsteady aerodynamic surface pressure distribution is dominated by nonlinear effects. Numerical investigations show a good correlation between the results obtained by the ROM methodology in comparison to the full-order CFD solution. In addition, the surrogate approach yields a significant speed-up regarding unsteady aerodynamic calculations, which is beneficial for multidisciplinary computations.     

Active flutter suppression control law design method based on balanced proper orthogonal decomposition reduced order model

Active control for nonlinear aeroelastic structures is an attractive innovative technology. The design of classic active flutter controllers has often been based on low-fidelity and low-accuracy linear aerodynamic models. Multi-physics high-fidelity reduced order model (ROM) was used to design active control laws. In order to provide a lower-order model for controllers design, a balanced proper orthogonal decomposition ROM (POD-BT/ROM) was investigated. A state-space aeroservoelastic model and the active flutter suppression control law design method based on POD-BT/ROM were proposed. The effectiveness of the proposed method was then demonstrated by NACA 0012 airfoil, AGARD 445.6 wing and the Goland wing+ aeroelastic model.     

基于SCI/ERA方法的高效气动力降阶模型

由正交Walsh函数构造Walsh-单信号-复合-输入,对其作用下的计算流体力学响应采用单信号-复合-输入/特征系统实现算法SCI/ERA(Single-Composite-Input/Eigensystem Realization Algorithm)辨识得到离散时间非定常气动力状态空间降阶模型。通过对Isogai机翼剖面气动弹性算例的计算证明该方法具有和非定常计算流体力学方法相当的精度同时模型维数降低2个数量级;在模型构造时间上,SCI/ERA方法比脉冲/ERA方法计算效率提高24%,同时内存占用减小34%;由理论分析可知当耦合结构模态数目增加时,SCI/ERA方法所需的计算开销增幅远小于脉冲/ERA方法;采用频域平衡特征正交分解BPOD(Balanced Proper Orthogonal Decomposition)方法可以准确地从降阶模型中提取出一个低频二次降阶模型,同时保持与原模型相当的精度。二次降阶后模型维数进一步减小88%。