An Efficient Algorithm Based on Dynamic System Properties and Group Theory for Transient Responses of 1D Periodic Structures

LIANG Xiqiang; GAO Qiang; YAO Weian

doi:10.21656/1000-0887.380129

Volume 39 Issue 2

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LIANG Xiqiang, GAO Qiang, YAO Weian. An Efficient Algorithm Based on Dynamic System Properties and Group Theory for Transient Responses of 1D Periodic Structures[J]. Applied Mathematics and Mechanics, 2018, 39(2): 170-182. doi: 10.21656/1000-0887.380129

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An Efficient Algorithm Based on Dynamic System Properties and Group Theory for Transient Responses of 1D Periodic Structures

doi: 10.21656/1000-0887.380129

Department of Engineering Mechanics, Dalian University of Technology; State Key Laboratory of Structural Analysis for Industrial Equipment(Dalian University of Technology), Dalian, Liaoning 116024, P.R.China

Funds: The National Natural Science Foundation of China（11572076）；The National Basic Research Program of China(973 Program)（2014CB049000）

Received Date: 2017-05-09
Rev Recd Date: 2017-11-01
Publish Date: 2018-02-15

Abstract

Abstract

Based on the condensation technology, the dynamic periodic structure properties and the group theory, an efficient numerical method for computing the transient responses of 1D periodic structures was proposed. Efficiently solving linear equations is an issue for computing the dynamic responses. Based on the periodic properties of the structure and with the condensation technology, the scale of the linear equation corresponding to the structure was reduced. By means of the properties of linear equations for dynamic periodic systems, it was proved that the force on any chosen unit cell can only influence a finite number of adjacent unit cells within a time step. Then, the dynamic response computation of 1D periodic structures was converted into the computation of a series of small-scale substructures. Subsequently, the dynamic response computation of the substructures can be converted into the computation of the cyclic-periodic structures. Then, the cyclic-periodic structures were solved efficiently in light of the group theory. Numerical examples illustrate the high efficiency and memory saving of the proposed method.
- periodic structure,
- group theory,
- Newmark-beta method,
- dynamic response

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References

[1]	汪越胜, 张传增, 陈阿丽, 等. 声子晶体能带结构计算数值方法评述[C]//中国力学大会. 中国，陕西，西安, 2013.（WANG Yuesheng, ZHANG Chuanzeng, CHEN Ali, et al. The recitation of numerical methods of the phononic crystal band structure[C]// Proceedings of the Chinese Congress of Theoretical and Applied Mechanics . Xi’an, Shaanxi, China, 2013.（in Chinese））
[2]	温激鸿, 郁殿龙, 王刚, 等. 周期结构细直梁弯曲振动中的振动带隙[J]. 机械工程学报, 2005,41(4): 1-6.（WEN Jihong, YU Dianlong, WANG Gang, et al. Elastic wave band gaps in flexural vibrations of straight beams[J]. Chinese Journal of Mechanical Engineering,2005,41(4): 1-6.（in Chinese））
[3]	郑玲, 李以农. 基于多重散射理论的二维声子晶体带结构计算[J]. 重庆大学学报(自然科学版), 2007,30(6): 6-9.（ZHENG Ling, LI Yinong. Elastic band gaps in two-dimensional phononic crystals based on multiple scattering theory[J]. Journal of Chongqing University (Natural Science Edition),2007,30(6): 6-9.（in Chinese））
[4]	陈启勇, 胡少伟, 张子明. 基于声子晶体理论的弹性地基梁的振动特性研究[J]. 应用数学和力学, 2014,35(1): 29-38.（CHEN Qiyong, HU Shaowei, ZHANG Ziming. Research on the vibration property of the beam on elastic foundation based on the PCs theory[J]. Applied Mathematics and Mechanics,2014,35(1): 29-38.（in Chinese））
[5]	AXMANN W, KUCHMENT P. An efficient finite element method for computing spectra of photonic and acoustic band-gap materials—I： scalar case[J]. Journal of Computational Physics,1999,150(2): 468-481.
[6]	郁殿龙, 刘耀宗, 王刚, 等. 一维杆状结构声子晶体扭转振动带隙研究[J]. 振动与冲击, 2006,25(1): 104-106, 169-170.（YU Dianlong, LIU Yaozong, WANG Gang, et al. Research on torsional vibration band gaps of one dimensional phononic crystals composed of rod structures[J]. Journal of Vibration and Shock,2006,25(1): 104-106, 169-170.（in Chinese））
[7]	CAO Y J, HOU Z L, LIU Y Y. Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals[J]. Solid State Communications,2004,132(8): 539-543.
[8]	贠昊, 邓子辰, 朱志韦. 弹性波在星形节点周期结构蜂窝材料中的传播特性研究[J]. 应用数学和力学, 2015,36(8): 814-820.（YUN Hao, DENG Zichen, ZHU Zhiwei. Bandgap properties of periodic 4-point star-shaped honeycomb materials with negative Poisson’s ratios[J]. Applied Mathematics and Mechanics,2015,36(8): 814-820.（in Chinese））
[9]	刘维宁, 张昀青. 轨道结构在移动荷载作用下的周期解析解[J]. 工程力学, 2004,21(5): 100-102, 93.（LIU Weining, ZHANG Yunqing. A periodic analytical solution of railway track structure under moving loads[J]. Engineering Mechanics,2004,21(5): 100-102, 93.（in Chinese））
[10]	柴维斯, 刘锋. 周期加筋板动力响应的解析解[J]. 暨南大学学报(自然科学版), 2005,26(1): 46-49.（CHAI Weisi, LIU Feng. Exact solutions of vibration analysis for stiffened plates[J]. Journal of Jinan University (Natural Science),2005,〖STHZ〗 26(1): 46-49.（in Chinese））
[11]	HANS S, BOUTIN C, CHESNAIS C. A typical dynamic behavior of periodic frame structures with local resonance[J]. Journal of the Acoustical Society of America,2014,136(4): 2077.
[12]	RYCHLEWSKA J, SZYMCZYK J, WOZNIAK C. On the modelling of dynamic behavior of periodic lattice structures[J]. Acta Mechanica,2004,170(1/2): 57-67.
[13]	MENCIK J M, DUHAMEL D. A wave-based model reduction technique for the description of the dynamic behavior of periodic structures involving arbitrary-shaped substructures and large-sized finite element models[J]. Finite Elements in Analysis and Design,2015,101: 1-14.
[14]	高强, 姚伟岸, 吴锋, 等. 周期结构动力响应的高效数值方法[J]. 力学学报, 2011,43(6): 1181-1185.（GAO Qiang, YAO Weian, WU Feng, et al. An efficient algorithm for dynamic responses of periodic structures[J]. Chinese of Journal of Theoretical and Applied Mechanics,2011,43(6): 1181-1185.（in Chinese））
[15]	GAO Q, ZHANG H W, ZHONG W X, et al. An accurate and efficient method for dynamic analysis of two-dimensional periodic structures[J]. International Journal of Applied Mechanics,2016,8(2): 1650013.
[16]	NEWMARK N M. A method of computation for structural dynamics[J]. Journal of the Engineering Mechanics,1959,85(3): 67-94.
[17]	BURRAGE K, BUTCHER J C. Stability-criteria for implicit Runge-Kutta methods[J]. SIAM Journal on Numerical Analysis,1979,16(1): 46-57.
[18]	PARK K C, UNDERWOOD P G. A variable-step central difference method for structureal dynamics analysis—part 1: theoretical aspects[J]. Computer Methods in Applied Mechanics and Engineering,1980,22(2): 241-258.
[19]	同济大学应用数学系. 高等数学[M]. 北京: 高等教育出版社, 2002.（Department of Applied Mathematics of Tongji University. Advanced Mathematics [M]. Beijing: Higher Education Press, 2002.（in Chinese））
[20]	ZHONG W X, QIU C H. Analysis of symmetric or partially symmetric structures[J]. Computer Methods in Applied Mechanics and Engineering,1983,38(1): 1-18.