Abstract:In order to improve solving efficiency of a structural nonlinear dynamic system under large geometric deformation condition and study its dynamic behavior in a specified frequency range, the proper orthogonal decomposition (POD) method in frequency domain was used to study the dynamic order reduction problem of a geometrically nonlinear structure with a cantilevered plate taken as the study object.The geometrically nonlinear stiffness of the plate was solved using the cooperative rotation (CR) method.POD base vectors were generated with snapshots computed in a specified frequency domain, and Galerkin method was used to realize the order reduction of dynamic system.The nonlinear stiffness was added to the external force term in the form of increment, and the nonlinear behavior of the system was reflectedin the form of generalized external force.The POD order reduction analysis in frequency domain and comparison were done for the cantilevered plate.Results showed that (1) for a linear system, the POD order reduction analysis in frequency-domain has high precision, the error is less than 1%, and its solving time is far less than that for the full order system, the solving time for 1 order POD is less than 50% of that for the full order system; (2) for a nonlinear system, the error of 1 order POD analysis is less than 1.5%, and the error of 3 order POD analysis is less than 0.5%, the solving time for the two cases is less than 75% of that for the full order analysis under sine and step loads; (3) for a geometrically nonlinear dynamic system under multi-point random loads, if the first 6 orders POD base vectors are kept after order reduction in frequency domain, the reduced order system’s analysis error is less than 0.5% and its solving time is just 79% of that for the full order system.
陈兵1,龚春林1,仇理宽2,谷良贤1. 基于频域本征正交分解的几何非线性动力学降阶[J]. 振动与冲击, 2020, 39(21): 163-172.
CHEN Bing1, GONG Chunlin1, QIU Likuan2, GU Liangxian1. Order reduction of geometrically nonlinear dynamic system based on POD in frequency domain. JOURNAL OF VIBRATION AND SHOCK, 2020, 39(21): 163-172.
[1] 王勖成. 有限单元法[M]. 北京:清华大学出版社,2003.
Wang Xu-cheng. Finite Element Method. Beijing: Tsinghua Press, 2003.
[2] C. A. Felippa, B. Haugen. A unified formulation of small-strain corotational finite elements: I. Theory [J]. Computer Methods in Applied Mechnics and Engneering, 2005, 194(21): 2285-2335.
[3] S. E. Stapleton, A. M. Waas. Co-rotational Formulation for Bounded Joint Finite Elements [C]. 53rd AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA 2012-1449.
[4] YANG JinSong, XIA PinQi. Finite element corotational formulation for geometric nonlinear analysis of thin shells with large rotation and small strain [J]. Science China, 2012, 55(11): 3142-3152.
[5] P. Bisegna, F. Caselli, S. Marfia, et al. A new SMA shell element based on the corotational formulation [J]. Computational Mechanics, 2014, 54(5): 1315-1329.
[6] S. K. Chimakurthi, B. K. Stanford, C. E. S. Cesinik, et al. Flapping Wing CFD/CSD Aeroelastic Formulation Based on a Co-rotational Shell Finite Element [C]. 51rd AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA 2009-2412.
[7] 周强,陈刚,李跃明.基于CFD降阶的非线性气动弹性稳定性分析[J]. 振动与冲击,2016, 35(16):17-23.
ZHOU Qiang, CHEN Gang, LI Yueming. Nonlinear Aeroelastic Stability analysis based on CFD reduced order model[J]. Journal of Vibration and Shock, 2016, 35(16): 17-23.
[8] 仲继泽,徐自力.基于动网格降阶算法的机翼颤振边界预测[J]. 振动与冲击,2017,36(4):185-191.
ZHONG Jize, XU Zili. Wing Flutter Prediction using a Reduced Dynamic Mesh Method[J]. Journal of Vibration and Shock, 2017, 36(4): 185-191.
[9] D. Bonomi, A. Manzoni, A. Quarteroni. A Matrix DEIM Technique for Model Reduction of Nonlinear Parametrized problems in Cardiac Mechanics. Comput. Methods Appl. Mech Engrg. 324 (2017) :300-326.
[10] S. Chaturantabut. Temporal Localized Nonlinear Model Reduction with a Priori error Estimate. Applied Numerical Mathematics, 119 (2017) 255-238.
[11] M. Alotaibi, E. Chung. Global-loacal Model Reduction for Heterogeneous Forchheimer Flow. Journal of Computational and Applied Mathematics, 321 (2017) 160-184.
[12] S. Ilbeigi, D. Chelidze. Persistent Model Order Reduction for Complex Dynamical Systems using Sooth Orthogonal Decomposition. Mechanical Systems and Signal Processing, 96 (2017) 125-138.
[13] M. Mordhorst, T. Strecker, D. Wirtz, et al. POD-DEIM Reduction of Computational EMG Models. Journal of Computational Science, 19 (2017) 86-96.
[14] D. F. C. Silva, A. L. G. A. Coutinho. Practical implementation aspects of Galerkin reduced order models based on proper orthogonal decomposition for computational fluid dynamics [J]. J Braz. Soc. Mech. Sci. Eng., 2015, 37(4): 1309-1327.
[15] 罗佳奇,段焰辉,夏振华. 基于自适应本征正交分解混合模型的跨音速流场分析. 物理学报,2016,65(2):1-9
Luo Jiaqi, Duan Yanhui, Xia Zhenhua. Transonic Flow Reconstruction by an Adaptive Proper Orthogonal Decomposition Hybrid Model. Acta Phys. Sin, 2016, 65(12):1-9.
[16] 张立章,尹泽勇,米栋,等. 基于本征正交分解的离心压气机多学科设计优化. 推进技术,2017,02(38):323-333.
Zhang Lizhang, Yin Zeyong, Mi Dong, et al. Multidisciplinary Design Optimization for Centrifugal Compressor based on Proper Orthogonal Decomposition. Journal of Propulsion Technology, 2017, 02(38):323-333.
[17] 梅冠华,康灿,张家忠.二维壁板颤振的本征正交分解降阶模型研究[J].振动与冲击,2017,36(23):144-151.
MEI Guanhua, KANG Can, ZHANG Jiazhong. Reduced Order Model based on Proper Orthogonal Decomposition for Two-dimensional Pannel Flutter[J]. Journal of Vibration and Shock, 2017, 36(23): 144-151.
[18] CHEN Gang, LI YueMing and YAN GuiRong. A nonlinear POD reduced order model for limit cycle oscillation prediction [J]. SCIENCE CHINA Physics, Mechanics & Astronomy, 2010, 53(7): 1325-1332.
[19] K. Willcox. Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition [J]. Computers & Fluids , 2006, 35(2): 208-226.
[20] Cusumano J., Sharkady M., Kimble B. . Spatial coherence measurements of a chaotic flexible-beam impact oscillator [J]. Am. Soc. Mech. Eng. Aerospace Div. (Publication) AD, 1993, 33(1): 13-22.
[21] T. Kim. Frequency-Domain Karhunen-Loeve Method and Its Application to Linear Dynamic Systems [J]. AIAA Journal, 1998, 36(11): 2117-2123.
[22] T. Kim. Efficient Reduced-Order System Identification for Linear Systems with Multiple Inputs [J]. AIAA Journal, 2005, 43(7): 1455-1464.
[23] T. Kim. Surrogate reduction for linear dynamic systems based on a frequency domain modal analysis [J]. Computational Mechanics, 2015, 56(4): 709-723.
[24] T. Kim, K. S. Nagaraja, K. G. Bhatia. Order Reduction of State-Space Aeroelastic Models Using Optimal Modal Analysis [J]. Journal of Aircraft, 2004, 41(6): 1440-1448.
[25] D. Amsallem, Charbel Farhat. Interpolation Method for Adapting Reducer-Order Models and Application to Aeroelasticity. AIAA Journal, 2008, 46(7): 1803-1813.
[26] G. Weickum, M. S. Eldred, K. Maute. A multi-point reduced-order modeling approach of transient structural dynamics with application to robust design optimization [J]. Struc Multidisc Optim, 2009, 38(6): 599-611.
[27] F. Bamer, C. Bucher. Application of the proper orthogonal decomposition for linear and nonlinear structures under transient excitations [J]. Acta Mechnica, 2012, 223(12): 2549-2563.
[28] K. Calberg, C. Bou-Mosleh, C. Farhat. Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations [J]. International Journal for Numerical Methods in Engneering, 2011, 86(2): 155-181.