In this paper, a fuzzy scalar radial basis function neural network is proposed, in order to overcome the limitation of traditional aerodynamic reduced-order models having difficulty in adapting to input variables with different orders of magnitude. This network is a combination of fuzzy rules and standard radial basis function neural network, and all the basis functions are defined as scalar basis functions. The use of scalar basis function will increase the flexibility of the model, thus enhancing the generalization capability on complex dynamic behaviors. Particle swarm optimization algorithm is used to find the optimal width of the scalar basis function. The constructed reduced-order models are used to model the unsteady aerodynamics of an airfoil in transonic flow. Results indicate that the proposed reduced-order models can capture the dynamic characteristics of lift coefficients at different reduced frequencies and amplitudes very accurately. Compared with the conventional reduced-order model based on recursive radial basis function neural network, the fuzzy scalar radial basis function neural network shows better generalization capability for different test cases with multiple normalization methods.
ZhaoHGaoSHeZ, et al.Identification of nonlinear dynamic system using a novel recurrent wavelet neural network based on the pipelined architecture. IEEE Trans Ind Electron2014; 61: 4171–4182.
2.
AdhikariR. A neural network based linear ensemble framework for time series forecasting. Neurocomputing2015; 157: 231–242.
3.
LouYFBrunnP. An offset error compensation method for improving ANN accuracy when used for position control of precision machinery. Neural Comput Appl1998; 7: 90–95.
4.
LiKPengJXBaiEW. A two-stage algorithm for identification of nonlinear dynamic systems. Automatica2006; 42: 1189–1197.
5.
WuGDZhuZW. An enhanced discriminability recurrent fuzzy neural network for temporal classification problems. Fuzzy Sets Syst2014; 237: 47–62.
6.
WangXMaLWangB, et al.A hybrid optimization-based recurrent neural network for real-time data prediction. Neurocomputing2013; 120: 547–559.
7.
KurtulusDF. Ability to forecast unsteady aerodynamic forces of flapping airfoils by artificial neural network. Neural Comput Appl2009; 18: 359–359.
8.
MannarinoAMantegazzaP. Nonlinear aeroelastic reduced order modeling by recurrent neural networks. J Fluids Struct2014; 48: 103–121.
9.
ZhangWWangBYeZ, et al.Efficient method for limit cycle flutter analysis based on nonlinear aerodynamic reduced-order models. AIAA J2012; 50: 1019–1028.
10.
KouJZhangW. A hybrid reduced-order framework for complex aeroelastic simulations. Aerosp Sci Technol2019; 84: 880–894.
11.
ZhangWYeZ. On unsteady aerodynamic modeling based on CFD technique and its applications on aeroelastic analysis. Adv Mech2008; 38: 77–86.
12.
ZhangWLiXYeZ, et al.Mechanism of frequency lock-in in vortex-induced vibrations at low Reynolds numbers. J Fluid Mech2015; 783: 72–102.
13.
GaoCZhangWLiX, et al.Mechanism of frequency lock-in in transonic buffeting flow. Journal of Fluid Mechanics2017; 818: 528–561.
14.
HesseHPalaciosR. Reduced-order aeroelastic models for dynamics of maneuvering flexible aircraft. AIAA Journal2014; 52(8): 1717–1732.
15.
ZhangWChenKYeZ. Unsteady aerodynamic reduced-order modeling of an aeroelastic wing using arbitrary mode shapes. Journal of Fluids and Structures2015; 58: 254–270.
WillcoxKPeraireJ. Balanced model reduction via the proper orthogonal decomposition. AIAA Journal2002; 40(11): 2323–2330.
18.
ZhangWYeZZhangC. Aeroservoelastic analysis for transonic missile based on computational fluid dynamics. Journal of Aircraft2009; 46(6): 2178–2183.
19.
LiXLiuYKouJ, et al.Reduced-order thrust modeling for an efficiently flapping airfoil using system identification method. Journal of Fluids and Structures2017; 69: 137–153.
20.
JuangJNPappaRS. An eigensystem realization algorithm for modal parameter identification and model reduction. Journal of guidance, control, and dynamics1985; 8(5): 620–627.
21.
Griffiths L M, Gaitonde A L, Jones D P, et al. Updating of aerodynamic reduced order models generated using computational fluid dynamics. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2018, 232(9): 1739-1763.
22.
SilvaW. Identification of nonlinear aeroelastic systems based on the Volterra theory: progress and opportunities. Nonlinear Dynamics2005; 39(1-2): 25–62.
23.
BalajewiczMDowellE. Reduced-order modeling of flutter and limit-cycle oscillations using the sparse Volterra series. Journal of Aircraft2012; 49(6): 1803–1812.
TobakM. On the use of the indicial-function concept in the analysis of unsteady motions of wings and wing-tail combinations1954.
26.
Marzocca P, Librescu L, Kim D H, et al. Development of an indicial function approach for the two-dimensional incompressible/compressible aerodynamic load modelling. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2007, 221(3): 453–463.
27.
GlazBLiuLFriedmannPP. Reduced-order nonlinear unsteady aerodynamic modeling using a surrogate-based recurrence framework. AIAA Journal2010; 48(10): 2418–2429.
28.
Li X, Guo Z, Dana G, et al. Efficient reduced-order modeling of unsteady aerodynamics under light dynamic stall conditions. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2018: 0954410018773628.
29.
Da RonchAGhoreyshiMBadcockKJ. On the generation of flight dynamics aerodynamic tables by computational fluid dynamics. Progress in Aerospace Sciences2011; 47(8): 597–620.
30.
WangBobinZhangWeiweiYeZhengyin. Unsteady nonlinear aerodynamics identification based on neural network model. Acta Aeronautica et Astronautica Sinica 31.72010, pp. 1379–1388. (in Chinese).
31.
KouJZhangWYinM. Novel Wiener models with a time-delayed nonlinear block and their identification. Nonlinear Dynamics2016; 85(4): 2389–2404.
GhoreyshiMCummingsRMRonchAD, et al.Transonic aerodynamic load modeling of X-31 aircraft pitching motions. AIAA journal2013; 51(10): 2447–2464.
34.
WinterMBreitsamterC. Neurofuzzy-model-based unsteady aerodynamic computations across varying freestream conditions. AIAA Journal2016, pp. 2705–2720.
35.
DeyPSarkarADasAK. Development of GEP and ANN model to predict the unsteady forced convection over a cylinder. Neural Computing and Applications2016; 27(8): 2537–2549.
36.
KouJZhangW. An approach to enhance the generalization capability of nonlinear aerodynamic reduced-order models. Aerospace Science and Technology2016; 49: 197–208.
Zhang W, Guo X, Wang C, et al. A POD-based center selection for RBF neural network in time series prediction problems//International Conference on Adaptive and Natural Computing Algorithms. Springer, Berlin, Heidelberg, 2007: 189-198.
39.
Fernández-NavarroFHervás-MartínezCSanchez-MonederoJ, et al.MELM-GRBF: a modified version of the extreme learning machine for generalized radial basis function neural networks. Neurocomputing2011; 74(16): 2502–2510.
40.
KouJZhangW. Multi-kernel neural networks for nonlinear unsteady aerodynamic reduced-order modeling. Aerospace Science and Technology2017; 67: 309–326.
41.
WinterMBreitsamterC. Nonlinear identification via connected neural networks for unsteady aerodynamic analysis. Aerospace Science and Technology2018, pp. 77.802–818.
42.
HanHGZhangSQiaoJF. An adaptive growing and pruning algorithm for designing recurrent neural network. Neurocomputing2017; 242: 51–62.
43.
ChenSBillingsSACowanCFN, et al.Non-linear systems identification using radial basis functions. International Journal of Systems Science1990; 21(12): 2513–2539.
44.
JangJSRSunCT. Functional equivalence between radial basis function networks and fuzzy inference systems. IEEE Transactions on Neural Networks1993; 4(1): 156–159.
45.
Boely N, Botez R M, Kouba G. Identification of a non-linear F/A-18 model by the use of fuzzy logic and neural network methods. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2011, 225(5): 559-574.
46.
ChoKBWangBH. Radial basis function based adaptive fuzzy systems and their applications to system identification and prediction. Fuzzy Sets and Systems1996; 83(3): 325–339.
47.
El-SousyFFM. Adaptive hybrid control system using a recurrent RBFN-based self-evolving fuzzy-neural-network for PMSM servo drives. Applied Soft Computing2014; 21: 509–532.
48.
Eberhart R, Kennedy J. A new optimizer using particle swarm theory//Micro Machine and Human Science, 1995. MHS’95., Proceedings of the Sixth International Symposium on. IEEE, 1995: 39-43.
49.
SpalartPRAAllmarasS. A one-equation turbulence model for aerodynamic flows//30th aerospace sciences meeting and exhibit1992439.
50.
De BoerAVan der SchootMSBijlH. Mesh deformation based on radial basis function interpolation. Computers & Structures2007; 85(11-14): 784–795.
51.
Jiang Y. Numerical solution of Navier–Stokes equations on generalized mesh and its applications. NWPU, Xi’an, China (Ph. D. thesis), 2013.
52.
ZhuLZhangWKouJ, et al.Machine learning methods for turbulence modeling in subsonic flows around airfoils. Physics of Fluids2019; 31(1): 015105–015105.
53.
Landon R H. NACA 0012 oscillatory and transient pitching[R]. AIRCRAFT RESEARCH ASSOCIATION LTD BEDFORD (UNITED KINGDOM), 2000.
54.
DowellEHIlgamovM. Studies in nonlinear aeroelasticity, Springer Science & Business Media, 2012.
