基于桁架单元的能量一致积分方法
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摘要
基于能量平衡理论,提出针对桁架单元的能量一致积分方法。该方法具有非线性无条件稳定性,2阶精度。利用中值定理证明算法参数的存在性,并给出参数的求解形式。对离散后的动力方程线性化得到用于迭代的等效刚度矩阵。实现新算法在非线性有限元程序中的嵌入,并以此为基础完成单摆、输电塔体结构的非线性动力分析。数值结果表明,经典的平均加速度方法与隐式中点方法均会表现出能量不一致现象,甚至会产生发散结果;相比而言,该文方法在不同的时间步长情况下都表现出良好的数值稳定性。
Based on the energy equilibrium theory, an energy consistent integration method for truss elements is proposed in this paper. The method is unconditionally stable in nonlinear systems, and its accuracy is second order. The existence of algorithm parameters is proved by mean value theorem, and the solution form of the parameters is also provided. The discrete dynamic equations are linearized to obtain the equivalent stiffness matrices for iteration. The new algorithm is embedded in a nonlinear finite element program. On the basis of this program, the nonlinear dynamic analysis of a single pendulum and a transmission tower structure is completed. The numerical results show that the classic average acceleration method and implicit midpoint method are both energy inconsistent and may even produce divergent results. In contrast, the proposed method has good stability within different time steps.
Based on the energy equilibrium theory, an energy consistent integration method for truss elements is proposed in this paper. The method is unconditionally stable in nonlinear systems, and its accuracy is second order. The existence of algorithm parameters is proved by mean value theorem, and the solution form of the parameters is also provided. The discrete dynamic equations are linearized to obtain the equivalent stiffness matrices for iteration. The new algorithm is embedded in a nonlinear finite element program. On the basis of this program, the nonlinear dynamic analysis of a single pendulum and a transmission tower structure is completed. The numerical results show that the classic average acceleration method and implicit midpoint method are both energy inconsistent and may even produce divergent results. In contrast, the proposed method has good stability within different time steps.
引文
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[30]潘天林.能量一致积分方法及其在混合实验中的应用[D].哈尔滨:哈尔滨工业大学, 2016:20―27.Pan Tianlin. Energy consistent integration method and its applicationstohybridtesting[D].Harbin:Harbin Institute of Technology, 2016:20―27.(in Chinese)
[31]潘天林,吴斌,郭丽娜,等.能量守恒逐步积分方法在工程结构动力分析中的应用[J].工程力学,2014,31(9):21―27.PanTianlin,WuBin,GuoLina,etal.Applicationof energyconservingstep-by-stepintegrationalgorithmin dynamicanalysisofengineeringstructures[J].EngineeringMechanics,2014,31(9):21―27.(in Chinese)
[32]BatheKJ,BaigMMI.Onacompositeimplicittime integrationprocedurefornonlineardynamics[J].Computers&Structures, 2005, 83(31):2513―2524.
[2]Chen C, Ricles J M. Stability analysis of SDOF real-time hybridtestingsystemswithexplicitintegration algorithmsandactuatordelay[J].Earthquake Engineering&Structural Dynamics. 2008, 37(4):597―613.
[3]Chen C, Ricles J M, Marullo T M, et al. Real-time hybrid testingusingtheunconditionallystableexplicitCR integrationalgorithm[J].EarthquakeEngineering&Structural Dynamics. 2009, 38(1):3―44.
[4]HughesTJR.Thefiniteelementmethod:linearstatic and dynamic finite element analysis[M]. Massachusetts:Courier Corporation, 2000:551―553.
[5]CombescureD,PegonP.α-Operatorsplittingtime integrationtechniqueforpseudodynamictestingerror propagationanalysis[J].SoilDynamicsandEarthquake Engineering, 1997, 16(7):427―443.
[6]NewmarkNM.Amethodofcomputationforstructural dynamics[J].JournaloftheEngineeringMechanics Division, 1959, 85(3):67―94.
[7]周惠蒙,吴斌,王涛,等.基于速度的显式等效力控制方法的研究[J].工程力学, 2016, 33(6):15―22.ZhouHuimeng,WuBin,WangTao,etal.Explicit equivalentforcecontrolmethodbasedonvelocity[J].EngineeringMechanics,2016,33(6):15―22.(in Chinese)
[8]HilberHM,HughesTJR,TaylorRL.Improved numericaldissipationfortimeintegrationalgorithmsin structuraldynamics[J].EarthquakeEngineering&Structural Dynamics, 1977, 5(3):283―292.
[9]Hilber H M, Hughes T J R. Collocation, dissipation and overshootfortimeintegrationschemesinstructural dynamics[J].EarthquakeEngineering&Structural Dynamics, 1978, 6(1):99―117.
[10]WoodWL,BossakM,ZienkiewiczOC.Analpha modificationofnewmark’smethod[J].International JournalforNumericalMethodsinEngineering,1980,15(10):1562―1566.
[11]Chung J, Hulbert G M. A time integration algorithm for structural dynamics with improved numerical dissipation:thegeneralized-αmethod[J].Journalofapplied mechanics, 1993, 60(2):371―375.
[12]梁轩,杜建镔.采用减震榫桥梁非线性动力学分析计算方法[J].工程力学, 2016, 33(4):136―143.LiangXuan,DuJianbin.Anapproachtononlinear dynamicanalysisofbridgewithaseismicabsorbers[J].EngineeringMechanics,2016,33(4):136―143.(in Chinese)
[13]Kadapa C, Dettmer W G, Peri?D. On the advantages of usingthefirst-ordergeneralised-alphaschemefor structuraldynamicproblems[J].Computers&Structures, 2017, 193:226―238.
[14]RossiS,AbboudN,ScovazziG.Implicitfinite incompressibleelastodynamicswithlinearfinite elements:A stabilized method in rate form[J]. Computer MethodsinAppliedMechanicsandEngineering,2016,311:208―249.
[15]Butcher J C. Numerical methods for ordinary differential equations, second edition[M]. Chichester:John Wiley&Sons Ltd. 2008:248―252.
[16]Simo J C, Hughes T J R. Computational inelasticity[M].NewYork:SpringerScience&BusinessMedia,1998:53―57.
[17]LiY,WuB,OuJ.Stabilityofaverageacceleration methodforstructureswithnonlineardamping[J].EarthquakeEngineeringandEngineeringVibration,2006, 5(1):87―92.
[18]潘天林,吴斌.隐式中点法对于非线性阻尼结构的稳定性[J].振动与冲击, 2013, 32(23):38―42.PanTianlin,WuBin.Stabilityofimplicitmidpoint algorithmappliedtononlineardampingstructure[J].JournalofVibrationandShock,2013,32(23):38―42.(in Chinese)
[19]LabuddeRA,GreenspanD.Discretemechanics-A generaltreatment[J].Journalof Computational Physics,1974, 15(2):134―167.
[20]HughesTJR,CaugheyTK,LiuWK.Finite-element methodsfornonlinearelastodynamicswhichconserve energy[J].JournalofAppliedMechanics,1978,45(2):366―370.
[21]KuhlD,RammE.Generalizedenergy–momentum methodfornon-linearadaptiveshelldynamics[J].ComputerMethodsinAppliedMechanicsand Engineering, 1999, 178(3):343―366.
[22]Kuhl D, Crisfield M A. Energy-conserving and decaying algorithmsinnon-linearstructuraldynamics[J].InternationalJournalforNumericalMethodsin Engineering, 1999, 45(5):569―599.
[23]SimoJC,TarnowN.Thediscreteenergy-momentum method.Conservingalgorithmsfornonlinear elastodynamics[J].Zeitschriftfürangewandte Mathematik und Physik ZAMP, 1992, 43(5):757―792.
[24]RomeroI.Ananalysisofthestressformulafor energy-momentummethodsinnonlinearelastodynamics[J]. Computational Mechanics, 2012, 50(5):603―610.
[25]NoelsL,StainierL,PonthotJP.Afirst-order energy-dissipativemomentumconservingschemefor elasto-plasticity using the variational updates formulation[J].ComputerMethodsinAppliedMechanicsand Engineering, 2008, 197(6):706―726.
[26]MeierC,Popp A,WallWA.Geometrically exactfinite elementformulationsforslenderbeams:Kirchhoff-love theoryversussimo-reissnertheory[J].Archivesof Computational Methods in Engineering, 2017, 24:1―81.
[27]Nguyen T L, Sansour C, Hjiaj M. Long-term stable time integrationschemefordynamicanalysisofplanar geometricallyexactTimoshenkobeams[J].Journalof Sound and Vibration, 2017, 396:144―171.
[28]CrisfieldMA,ShiJ.ACo-rotationalelement time-integrationstrategyfornon-lineardynamics[J].InternationalJournalforNumericalMethodsin Engineering, 1994, 37(11):1897―1913.
[29]Crisfield M A, Shi J. An energy conserving co-rotational procedurefornon-lineardynamicswithfiniteelements[J]. Nonlinear Dynamics, 1996, 9(1/2):37―52.
[30]潘天林.能量一致积分方法及其在混合实验中的应用[D].哈尔滨:哈尔滨工业大学, 2016:20―27.Pan Tianlin. Energy consistent integration method and its applicationstohybridtesting[D].Harbin:Harbin Institute of Technology, 2016:20―27.(in Chinese)
[31]潘天林,吴斌,郭丽娜,等.能量守恒逐步积分方法在工程结构动力分析中的应用[J].工程力学,2014,31(9):21―27.PanTianlin,WuBin,GuoLina,etal.Applicationof energyconservingstep-by-stepintegrationalgorithmin dynamicanalysisofengineeringstructures[J].EngineeringMechanics,2014,31(9):21―27.(in Chinese)
[32]BatheKJ,BaigMMI.Onacompositeimplicittime integrationprocedurefornonlineardynamics[J].Computers&Structures, 2005, 83(31):2513―2524.
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