基于混合近似法的纤维梁单元非线性求解方法
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摘要
环境荷载作用下土木工程结构往往产生较大的非线性变形,包括材料非线性和几何大变形两部分,而两者非线性问题的数值求解往往需要花费高昂的计算代价。隔离非线性有限元法是将材料应变分解为弹性与塑性两部分,在控制方程中实现刚度矩阵的弹塑性分离,使用Woodbury公式进行非线性求解。目前,该方法仅用于求解局部材料非线性问题。文章基于隔离非线性有限元法,推导同时考虑材料和几何非线性(简称混合非线性)的纤维梁单元控制方程;依据Woodbury公式和组合近似法的基本理论,提出混合近似法的高效非线性求解方法,并利用时间复杂度理论,对该方法与传统有限元法进行计算效率对比分析;将提出的方法应用于某钢框架结构的地震非线性反应分析,计算结果表明:文中方法可以在求解材料非线性时考虑几何大变形问题,在保证计算精度的同时使计算效率得到明显提升。
Nonlinearity problems in civil engineering structures mainly involve the material nonlinearity and geometric nonlinearity(i.e., hybrid nonlinearities) under environmental loads. Because the computational cost is high, great attentions have been paid to the numerical solution of hybrid nonlinearity problem. The inelasticity-separated finite element method(IS FEM) is an efficient nonlinear solution method, and based on the concept, the strain of a nonlinear material can be decomposed into the linear-elastic and inelastic components.Consequently, the elastic and plastic components in the stiffness matrices from the governing equation can be separated, such that the Woodbury formula can be used to solve local material nonlinearity problems. In this study, the governing equation of the fiber beam element considering hybrid nonlinear behaviors is derived using the IS FEM. In order to efficiently solve the governing equation, the efficient hybrid approximations method is proposed based on the Woodbury formula and the combined approximations(CA) method. In addition, the time complexity theory is used to evaluate the calculation efficiency of the present method, and the results show that it is obviously superior to the conventional finite element method. Finally, based on a numerical example, it is proved that the proposed method can produce accurate results efficiently, and thus, appear to be appealing in solving hybrid nonlinear problems of the fiber beam element.
Nonlinearity problems in civil engineering structures mainly involve the material nonlinearity and geometric nonlinearity(i.e., hybrid nonlinearities) under environmental loads. Because the computational cost is high, great attentions have been paid to the numerical solution of hybrid nonlinearity problem. The inelasticity-separated finite element method(IS FEM) is an efficient nonlinear solution method, and based on the concept, the strain of a nonlinear material can be decomposed into the linear-elastic and inelastic components.Consequently, the elastic and plastic components in the stiffness matrices from the governing equation can be separated, such that the Woodbury formula can be used to solve local material nonlinearity problems. In this study, the governing equation of the fiber beam element considering hybrid nonlinear behaviors is derived using the IS FEM. In order to efficiently solve the governing equation, the efficient hybrid approximations method is proposed based on the Woodbury formula and the combined approximations(CA) method. In addition, the time complexity theory is used to evaluate the calculation efficiency of the present method, and the results show that it is obviously superior to the conventional finite element method. Finally, based on a numerical example, it is proved that the proposed method can produce accurate results efficiently, and thus, appear to be appealing in solving hybrid nonlinear problems of the fiber beam element.
引文
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[20]Kirsch U.Design-oriented analysis of structures-unified approach[J].Journal of engineering mechanics,2003,129(3):264-272
[21]Kirsch U,Bogomolni M.Error evaluation in approximate reanalysis of structures[J].Structural&Multidisciplinary Optimization,2004,28(2/3):77-86
[22]Kirsch U,Papalambros P Y.Exact and accurate solutions in the approximate reanalysis of structures[J].Aiaa Journal,2001,39(11):2198-2205
[23]Kirsch U.A unified reanalysis approach for structural analysis,design,and optimization[J].Structural&Multidisciplinary Optimization,2003,25(2):67-85
[24]Gao G,Wang H,Li G.An adaptive time-based global method for dynamic reanalysis[J].Structural&Multidisciplinary Optimization,2013,48(2):355-365
[25]Bazaraa M S,Jarvis J J.Linear programming and network flows[M].John Wiley,1977
[26]李钢,贾硕,李宏男.基于算法复杂度理论的拟力法计算效率评价[J].计算力学学报,2018,35(2):129-137(Li Gang,Jia Shuo,Li Hongnan.The efficiency evalution of force analogy method base on the algorithm complexity theory[J].Chinese Journal of Computational Mechanics,2018,35(2):129-137(in Chinese))
[27]Li G,Jia S,Yu D H.Woodbury approximation method for structural nonlinear analysis[J].Journal of Engineering Mechanics,2018,144(7):04018052
[28]Golub G H,Van Loan C F.Matrix computations[M].4th edition.Beijing:Posts&Telecom Press,2014
[2]Chen Z Q,Agar T J A.Geometric nonlinear analysis of flexible spatial beam structures[J].Computer&Structures,1993,49(6):1083-1093
[3]陈军明,吴代华.三维梁单元的弹塑性切线刚度矩阵[J].华中科技大学学报:自然科学版,2000,28(2):111-113(Chen Junming Wu Daihua.Elasto plastic tangent stiffness matrix of three dimensional beam elements[J].Journal of Huazhong University of Science and Technology,2000,28(2):111-113(in Chinese))
[4]Bathe K J,Ramm E,Wilson E L.Finite element formulations for large deformation dynamic analysis[J].International Journal for Numerical Methods in Engineering,1975,9(2):353-386
[5]Bathe K J,Bolourchi S.Large displacement analysis of three-dimensional beam structures[J].International Journal for Numerical Methods in Engineering,1979,14(7):961-986
[6]陈政清,曾庆元,颜全胜.空间杆系结构大挠度问题内力分析的UL列式法[J].土木工程学报,1992,25(5):34-44(Chen Zhengqing,Zeng Qingyuan,Yan Quansheng.A UL formulation for internal force analysis of spacial frame structures with large displacement[J].China Civil Engineering Journal,1992,25(5):34-44(in Chinese))
[7]吴庆雄,陈宝春,韦建刚.三维杆系结构的几何非线性有限元分析[J].工程力学,2007,24(12):19-24,42(Wu Qingxiong,Chen Baochun,Wei Jiangang.Ageometric nonlinear finite element analysis for 3Dframed structures[J].Journal of Engineering Mechanics,2007,24(12):19-24,42(in Chinese))
[8]古雅琦,王海龙,杨怀宇.一种大变形几何非线性Euler-Bernoulli梁单元[J].工程力学,2013,30(6):11-15(Gu Yaqi,Wang Hailong,Yang Huaiyu.A large deformation geometric nonlinear euler-bernoulli beam element[J].Journal of Engineering Mechanics,2013,30(6):11-15(in Chinese))
[9]Karkon M,Rezaiee-Pajand M.Geometrical nonlinear analysis of plane problems by corotational formulation[J].Journal of Engineering Mechanics,2016,142(10):04016073
[10]邓继华.基于共旋坐标法的结构非线性计算理论研究[D].长沙:湖南大学,2013
[11]Kirsch U.Reanalysis of structures:a unified approach for linear,nonlinear,static,and dynamic systems[M].New York:Springer,2010
[12]Amir O,Kirsch U,Sheinman I.Efficient non-linear reanalysis of skeletal structures using combined approximations[J].International Journal for Numerical Methods in Engineering,2008,73(9):1328-1346
[13]Akgün M A,Garcelon J H,Haftka R T.Fast exact linear and non-linear structural reanalysis and the Sherman-Morrison-Woodbury formulas[J].International Journal for Numerical Methods in Engineering,2001,50(7):1587-1606
[14]Li G,Yu D H.Efficient inelasticity-separated finite element method for material nonlinearity analysis[J].Journal of Engineering Mechanics,2018,144(1):04018008
[15]李钢,余丁浩,李宏男.基于拟力法的纤维梁有限元非线性分析方法[J].建筑结构学报,2016,37(9):61-68(Li Gang,Yu Dinghao,Li Hongnan.Nonlinear fiber beam element analysis based on force analogy method[J].Journal of Building Structures,2016,37(9):61-68(in Chinese))
[16]Li G,Yu D H,Li H N.Seismic response analysis of reinforced concrete frames using inelasticity-separated fiber beam-column model[J].Earthquake Engineering&Structural Dynamics,2018,47(5):1291-1308
[17]刘光栋,王解君,何放龙.空间梁单元的几何非线性刚度矩阵的分解形式[J].湖南大学学报,1992,19(1):60-71(Liu Guangdong,Wang Jiejun,He Fanglong.Resolved formulation of geometrical nonlinear stiffness matrix for three-dimensional beam element[J].Journal of Hunan University,1992,19(1):60-71(in Chinese))
[18]王勖成.有限单元法[M].北京:清华大学出版社,2003(Wang Xucheng.Finite element method[M].Beijing:Tsinghua University Press,2003(in Chinese))
[19]Leu L J,Huang C W.A reduced basis method for geometric nonlinear analysis of structures[J].Journal of the International Association for Shell&Spatial Structures,1998,39:71-76
[20]Kirsch U.Design-oriented analysis of structures-unified approach[J].Journal of engineering mechanics,2003,129(3):264-272
[21]Kirsch U,Bogomolni M.Error evaluation in approximate reanalysis of structures[J].Structural&Multidisciplinary Optimization,2004,28(2/3):77-86
[22]Kirsch U,Papalambros P Y.Exact and accurate solutions in the approximate reanalysis of structures[J].Aiaa Journal,2001,39(11):2198-2205
[23]Kirsch U.A unified reanalysis approach for structural analysis,design,and optimization[J].Structural&Multidisciplinary Optimization,2003,25(2):67-85
[24]Gao G,Wang H,Li G.An adaptive time-based global method for dynamic reanalysis[J].Structural&Multidisciplinary Optimization,2013,48(2):355-365
[25]Bazaraa M S,Jarvis J J.Linear programming and network flows[M].John Wiley,1977
[26]李钢,贾硕,李宏男.基于算法复杂度理论的拟力法计算效率评价[J].计算力学学报,2018,35(2):129-137(Li Gang,Jia Shuo,Li Hongnan.The efficiency evalution of force analogy method base on the algorithm complexity theory[J].Chinese Journal of Computational Mechanics,2018,35(2):129-137(in Chinese))
[27]Li G,Jia S,Yu D H.Woodbury approximation method for structural nonlinear analysis[J].Journal of Engineering Mechanics,2018,144(7):04018052
[28]Golub G H,Van Loan C F.Matrix computations[M].4th edition.Beijing:Posts&Telecom Press,2014