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Deep neural network for unsteady aerodynamic and aeroelastic modeling across multiple Mach numbers

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Abstract

Aerodynamic reduced-order model (ROM) is a useful tool to predict nonlinear unsteady aerodynamics with reasonable accuracy and very low computational cost. The efficacy of this method has been validated by many recent studies. However, the generalization capability of aerodynamic ROMs with respect to different flow conditions and different aeroelastic parameters should be further improved. In order to enhance the predicting capability of ROM for varying operating conditions, this paper presents an unsteady aerodynamic model based on long short-term memory (LSTM) network from deep learning theory for large training dataset and sampling space. This type of network has attractive potential in modeling temporal sequence data, which is well suited for capturing the time-delayed effects of unsteady aerodynamics. Different from traditional reduced-order models, the current model based on LSTM network does not require the selection of delay orders. The performance of the proposed model is evaluated by a NACA 64A010 airfoil pitching and plunging in the transonic flow across multiple Mach numbers. It is demonstrated that the model can accurately capture the dynamic characteristics of aerodynamic and aeroelastic systems for varying flow and structural parameters.

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References

  1. Chen, G., Sun, J., Li, Y.: Active flutter suppression control law design method based on balanced proper orthogonal decomposition reduced order model. Nonlinear Dyn. 70, 1–12 (2012)

    Article  MathSciNet  Google Scholar 

  2. Da Ronch, A., Badcock, K.J., Wang, Y., Wynn, A., Palacios, R.: Nonlinear model reduction for flexible aircraft control design. In: AIAA Atmospheric Flight Mechanic Conference, AIAA- 2012-4404. American Institute of Aeronautics and Astronautics, Minneapolis, Minnesota (2012)

  3. Brunton, S., Noack, B.: Closed-loop turbulence control: progress and challenges. Appl. Mech. Rev. 67(5), 050801 (2015)

    Article  Google Scholar 

  4. Wang, Z., Zhang, W., Wu, X., Chen, K.: A novel unsteady aerodynamic reduced-order modeling method for transonic aeroelastic optimization. J. Fluid. Struct. 82, 308–328 (2018)

    Article  Google Scholar 

  5. Zhang, W., Li, X., Ye, Z., Jiang, Y.: Mechanism of frequency lock-in in vortex-induced vibrations at low Reynolds numbers. J. Fluid Mech. 783, 72–102 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gao, C., Zhang, W., Li, X., Liu, Y., Quan, J., Ye, Z., Jiang, Y.: Mechanism of frequency lock-in in transonic buffeting flow. J. Fluid Mech. 818, 528–561 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hesse, H., Palacios, R.: Reduced-order aeroelastic models of maneuvering flexible aircraft. AIAA J. 52(8), 1717–1732 (2014)

    Article  Google Scholar 

  8. Lucia, D.J., Beran, P.S., Silva, W.A.: Reduced-order modeling: new approaches for computational physics. Prog. Aerosp. Sci. 40(1–2), 51–117 (2004)

    Article  Google Scholar 

  9. Ghoreyshi, M., Jirasek, A., Cummings, R.M.: Reduced order unsteady aerodynamic modeling for stability and control analysis using computational fluid dynamics. Prog. Aerosp. Sci. 71, 167–217 (2014)

    Article  Google Scholar 

  10. Kou, J., Zhang, W.: A hybrid reduced-order framework for complex aeroelastic simulations. Aerosp. Sci. Technol. 84, 880–894 (2019)

    Article  Google Scholar 

  11. Willcox, K., Peraire, J.: Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40(11), 2323–2330 (2002)

    Article  Google Scholar 

  12. Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brewick, P.T., Masri, S.F.: An evaluation of data-driven identification strategies for complex nonlinear dynamic systems. Nonlinear Dyn. 85, 1297–1318 (2016)

    Article  MathSciNet  Google Scholar 

  14. He, S., Yang, Z., Gu, Y.: Nonlinear dynamics of an aeroelastic airfoil with free-play in transonic flow. Nonlinear Dyn. 87, 2099–2125 (2017)

    Article  Google Scholar 

  15. Zhang, W., Ye, Z.: Reduced-order-model-based flutter analysis at high angle of attack. J. Aircr. 44(6), 2086–2089 (2007)

    Article  MathSciNet  Google Scholar 

  16. Zhang, W., Wang, B., Ye, Z., Quan, J.: Efficient method for limit cycle flutter analysis based on nonlinear aerodynamic reduced-order models. AIAA J. 50(5), 1019–1028 (2012)

    Article  Google Scholar 

  17. Da Ronch, A., Ghoreyshi, M., Badcock, K.J.: On the generation of flight dynamics aerodynamic tables by computational fluid dynamics. Prog. Aerosp. Sci. 47(8), 597–620 (2011)

    Article  Google Scholar 

  18. Kou, J., Zhang, W., Yin, M.: Novel Wiener models with a time-delayed nonlinear block and their identification. Nonlinear Dyn. 85, 2389–2404 (2016)

    Article  Google Scholar 

  19. Duriez, Thomas, Brunton, Steven L., Noack, Bernd R.: Machine Learning Control-Taming Nonlinear Dynamics and Turbulence. Springer, Switzerland (2017)

    Book  MATH  Google Scholar 

  20. Tracey, B.D., Duraisamy, K., Alonso, J.J.: A machine learning strategy to assist turbulence model development. In: 53rd AIAA Aerospace Sciences Meeting, AIAA SciTech, Kissimmee, FL (2015)

  21. Mannarino, A., Mantegazza, P.: Nonlinear aeroelastic reduced order modeling by recurrent neural networks. J. Fluid. Struct. 48, 103–121 (2014)

    Article  Google Scholar 

  22. Amsallem, D., Farhat, C.: Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 46(7), 1803–1813 (2008)

    Article  Google Scholar 

  23. Skujins, T., Cesnik, C.E.S.: Reduced-order modeling of unsteady aerodynamics across multiple mach regimes. J. Aircr. 51(6), 1681–1704 (2014)

    Article  Google Scholar 

  24. Winter, M., Breitsamter, C.: Neurofuzzy-model-based unsteady aerodynamic computations across varying freestream conditions. AIAA J. 54(9), 2705–2720 (2016)

    Article  Google Scholar 

  25. Liu, H., Huang, R., Zhao, Y., Hu, H.: Reduced-order modeling of unsteady aerodynamics for an elastic wing with control surfaces. J. Aerosp. Eng. 30(3), 04016083 (2016)

    Article  Google Scholar 

  26. Kou, J., Zhang, W.: Reduced-order modeling for nonlinear aeroelasticity with varying Mach numbers. J. Aerosp. Eng. 31(6), 04018105 (2018)

    Article  Google Scholar 

  27. Liu, H., Hu, H., Zhao, Y., et al.: Efficient reduced-order modeling of unsteady aerodynamics robust to flight parameter variations. J. Fluids Struct. 49(8), 728–741 (2014)

    Article  Google Scholar 

  28. Zhang, W., Kou, J., Wang, Z.: Nonlinear aerodynamic reduced-order model for limit-cycle oscillation and flutter. AIAA J. 54(10), 3302–3310 (2016)

    Google Scholar 

  29. Schmidhuber, J.: Deep learning in neural networks: an overview. Neural Netw. 61, 85–117 (2015)

    Article  Google Scholar 

  30. Sainath, T.N., Mohamed, A.R., Kingsbury, B., Ramabhadran, B.: Deep convolutional neural networks for LVCSR. In: 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 8614–8618 (2013)

  31. Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classification with deep convolutional neural networks. In: 2012 International Conference on Neural Information Processing Systems, pp. 1097–1105 (2012)

  32. Xiong, H., Alipanahi, B., et al.: The human splicing code reveals new insights into the genetic determinants of disease. Science 347(6218), 1254806 (2015)

    Article  Google Scholar 

  33. Jin, X., Shao, J., Zhang, X., An, W., Malekian, R.: Modeling of nonlinear system based on deep learning framework. Nonlinear Dyn. 84, 1327–1340 (2016)

    Article  MathSciNet  Google Scholar 

  34. Rosa, E.D.L., Yu, W.: Randomized algorithms for nonlinear system identification with deep learning modification. Inf. Sci. 364–365, 197–212 (2016)

    Article  Google Scholar 

  35. Ling, J., Kurzawski, A., Templeton, J.: Reynolds averaged turbulence modeling using deep learning neural networks with embedded invariance. J. Fluid Mech. 807, 155–166 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Efrati, A.: Apples machines can learn too. The Information. https://www.theinformation.com/articles/apples-machines-can-learn-too (2016). Accessed 13 June 2016

  37. Vogels, W.: Bringing the magic of amazon ai and alexa to apps on aws. All Things Distributed. https://www.allthingsdistributed.com (2016). Accessed 30 Nov 2016

  38. Zhao, Z., Chen, W., Wu, X., et al.: LSTM network: a deep learning approach for short-term traffic forecast. IET Intell. Transp. Syst. 11(2), 68–75 (2017)

    Article  Google Scholar 

  39. Fu, R., Zhang, Z., Li L.: Using LSTM and GRU neural network methods for traffic flow prediction. In: 2016 31st Youth Academic Annual Conference of Chinese Association Automation, pp. 324–328 (2016)

  40. Wang, Z., Xiao, D., Fang, F., Govindan, R., Pain, C.C., Guo, Y.: Model identification of reduced order fluid dynamics systems using deep learning. Int. J. Numer. Meth. Fluids. 86(4), 255–268 (2018)

    Article  MathSciNet  Google Scholar 

  41. Mohan, A.T., Gaitonde, D.V.: A Deep Learning based Approach to Reduced Order Modeling for Turbulent Flow Control using LSTM Neural Networks (2018). arXiv:1804.09269v1 [physics.comp-ph]

  42. LeCun, Y., Bengio, Y., Hinton, G.E.: Deep learning. Nature 521(7553), 436–444 (2015)

    Article  Google Scholar 

  43. Hochreiter, S., Schmidhuber, J.: Long short-term memory. Neural Comput. 9(8), 1735–1780 (1997)

    Article  Google Scholar 

  44. Pascanu. R., Mikolov, T., Bengio, Y.: On the difficulty of training recurrent neural networks. In: International Conference on International Conference on Machine Learning. JMLR.org, III-1310 (2013)

  45. Chollet, F.: Keras (2015). https://github.com/fchollet/keras

  46. Kou, J., Zhang, W.: Multi-kernel neural networks for nonlinear unsteady aerodynamic reduced-order modeling. Aerosp. Sci. Technol. 67, 309–326 (2017)

    Article  Google Scholar 

  47. Kou, J., Zhang, W.: An approach to enhance the generalization capability of nonlinear aerodynamic reduced-order models. Aerosp. Sci. Technol. 49(1), 197–208 (2016)

    Article  Google Scholar 

  48. Goodfellow, Ian, Bengio, Yoshua, Courville, Aaron: Deep Learning, pp. 246–252. MIT Press, Cambridge (2016)

    MATH  Google Scholar 

  49. Jiang, Y.: Numerical Solution of Navier–Stokes Equations on Generalized Mesh and Its Applications. Ph.D. Dissertation, Northwestern Polytechnical Univ., Xi’an, China (2013)

  50. Boer, A., Schoot, M.S., Faculty, H.B.: Mesh deformation based on radial basis function interpolation. Comput. Struct. 85(11–14), 784–795 (2007)

    Article  Google Scholar 

  51. Zhang, W., Gao, C., Liu, Y., Ye, Z., Jiang, Y.: The interaction between flutter and buffet in transonic flows. Nonlinear Dyn. 82(4), 1851–1865 (2015)

    Article  Google Scholar 

  52. Gao, C., Zhang, W., Liu, Y., Ye, Z., Jiang, Y.: Numerical study on the correlation of transonic single-degree-of-freedom flutter and buffet. Sci. China Phys. Mech. Astron. 58(8), 084701 (2015)

    Article  Google Scholar 

  53. Zhang, W., Jiang, Y., Ye, Z.: Two better loosely coupled solution algorithms of CFD based aeroelastic simulation. Eng. Appl. Comput. Fluid Mech. 1(4), 253–262 (2007)

    Google Scholar 

  54. Graves, A., Mohamed, A.R., Hinton G.E.: Speech recognition with deep recurrent neural networks. In: 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 6645–6649 (2013)

  55. Pascanu, R., Gulcehre, C., Cho, K., Bengio, Y.: How to construct deep recurrrent neural networks (2013). arXiv: 1312.6026 [cs.NE]

  56. Kingma, D., Adam B.J.: A method for stochastic optimization (2014). arXiv: 1412.6980 [cs.LG]

  57. Thomas, J.P., Dowell, E.H., Hall, K.C.: Nonlinear inviscid aerodynamic effects on transonic divergence, flutter and limit cycle oscillations. AIAA J. 40(4), 638–646 (2002)

    Article  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11572252), the National Science Fund for Excellent Young Scholars (No. 11622220) and “111” project of China (No. B17037) and ATCFD project (2015-F-016). We thank Professor Bernd Noack of LIMSI-CNRS for his valuable discussions and suggestions on this work.

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  1. School of Aeronautics, Northwestern Polytechnical University, Xi’an, 710072, China

    Kai Li, Jiaqing Kou & Weiwei Zhang

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  1. Kai Li
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Correspondence to Jiaqing Kou.

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Li, K., Kou, J. & Zhang, W. Deep neural network for unsteady aerodynamic and aeroelastic modeling across multiple Mach numbers. Nonlinear Dyn 96, 2157–2177 (2019). https://doi.org/10.1007/s11071-019-04915-9

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