平面细长梁基于无网格径向基点插值的绝对节点坐标法
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摘要
为了消除或减弱传统绝对节点坐标法(Absolute Nodal Coordinate Formulation,ANCF)中缩减梁单元的"失真现象",构造了一种适用于描述柔性梁绝对位形的无网格径向基点插值(Radial Point Interpolation Method,RPIM)形函数,提出了柔性梁基于无网格RPIM的ANCF法。传统ANCF梁单元在描述纯弯曲悬臂梁的位形(一段圆弧)时,即便获得精确的单元节点坐标,通过梁单元插值得到的位形与悬臂梁的实际位形存在差异,即失真现象,悬臂梁越弯曲该差异越明显,失真越大。失真导致伪应变的产生,极大地影响数值求解的精度。而RPIM法采用一组场节点离散问题域,通过计算点支持域内的场节点构造形函数,计算点一般位于支持域的中心区域,不同计算点之间的支持域有较多重合的部分,加强了节点之间的联系,能更合理、准确地描述绝对位形,能有效减小失真。研究表明:基于RPIM的ANCF法较传统ANCF法精度更高、计算效率更快、对不等距分布节点的适应性更强,在大变形柔性多体系统动力学领域内具有推广性。
In order to alleviate or eliminate the‘distortion phenomenon'of the deficient beam elements in the traditional absolute nodal coordinate formulation(ANCF),an ANCF based on radial point interpolation method(RPIM)for flexible beams is proposed in which a new RPIM shape functions are constructed to describe the absolute configuration of flexible beams.For a pure bending cantilever beam(an arc),there is always difference between configuration of the beams by using the gradient deficient beam elements and the actual configuration of the beam in the traditional ANCF.The difference,namely‘distortion phenomenon',becomes more obvious with the increase of the bending deformation of the beam,which may cause the pseudo strain and have serious influence on accuracy of numerical solution.In the present method,the RPIM is used to discretize the deformation field through a set of field nodes and the shape functions are generally established based on field nodes within a support domain of the calculating point.The calculating points are generally located in the central region of the support domain,and the support domain of different calculating points can have more coincident parts.Thus,the connection between field nodes is strengthened,which makes the method describe the configuration in more reasonable and effectively alleviate influence of the distortion and pseudo strain.The simulation results show that the proposed method has higher calculation accuracy and efficiency and is more adaptive for the non-equidistant nodes compared with the traditional ANCF,which can be further extended in the dynamic field of flexible multi-body system.
In order to alleviate or eliminate the‘distortion phenomenon'of the deficient beam elements in the traditional absolute nodal coordinate formulation(ANCF),an ANCF based on radial point interpolation method(RPIM)for flexible beams is proposed in which a new RPIM shape functions are constructed to describe the absolute configuration of flexible beams.For a pure bending cantilever beam(an arc),there is always difference between configuration of the beams by using the gradient deficient beam elements and the actual configuration of the beam in the traditional ANCF.The difference,namely‘distortion phenomenon',becomes more obvious with the increase of the bending deformation of the beam,which may cause the pseudo strain and have serious influence on accuracy of numerical solution.In the present method,the RPIM is used to discretize the deformation field through a set of field nodes and the shape functions are generally established based on field nodes within a support domain of the calculating point.The calculating points are generally located in the central region of the support domain,and the support domain of different calculating points can have more coincident parts.Thus,the connection between field nodes is strengthened,which makes the method describe the configuration in more reasonable and effectively alleviate influence of the distortion and pseudo strain.The simulation results show that the proposed method has higher calculation accuracy and efficiency and is more adaptive for the non-equidistant nodes compared with the traditional ANCF,which can be further extended in the dynamic field of flexible multi-body system.
引文
[1]Shabana A A.An absolute nodal coordinate formulation for the large rotation and deformation analysis of flexible bodies[R].Department of Mechanical and Industrial Engineering,University of Illinois at Chicago,1996.
[2]Yakoub R Y,Shabana A A.Three dimensional absolute nodal coordinate formulation for beam elements:implementation and applications[J].Journal of Mechanical Design,2001,123(4):614—621.
[3]Shabana A A,Yakoub R Y.Three dimensional absolute nodal coordinate formulation for beam elements:theory[J].Journal of Mechanical Design,2001,123(4):606—613.
[4]Omar M A,Shabana A A.A two-dimensional shear deformable beam for large rotation and deformation problems[J].Journal of Sound and Vibration,2001,243(3):565—576.
[5]Berzeri M,Shabana A A.Development of simple models for the elastic forces in the absolute nodal co-ordinate formulation[J].Journal of Sound and Vibration,2000,235(4):539—565.
[6]Olshevskiy A,Dmitrochenko O,Dai M D,et al.The simplest 3-,6-and 8-noded fully-parameterized ANCF plate elements using only transverse slopes[J].Multibody System Dynamics,2014,34(1):23—51.
[7]Yan D,Liu C,Tian Q,et al.A new curved gradient deficient shell element of absolute nodal coordinate formulation for modeling thin shell structures[J].Nonlinear Dynamics,2013,74(1-2):153—164.
[8]Dmitrochenko O,Mikkola A.Two simple triangular plate elements based on the absolute nodal coordinate formulation[J].Journal of Computational and Nonlinear Dynamics,2008,3(4):041012.
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[10]Dufva K E,Sopanen J T,Mikkola A M.A two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation[J].Journal of Sound and Vibration,2005,280(3-5):719—738.
[11]Dufva K,Shabana A A.Analysis of thin plate structures using the absolute nodal coordinate formulation[J].Proceedings of the Institution of Mechanical Engineers,Part K:Journal of Multi-body Dynamics,2005,219(4):345—355.
[12]Sanborn G G,Choi J,Choi J H.Curve-induced distortion of polynomial space curves,flat-mapped extension modeling,and their impact on ANCF thin-plate finite elements[J].Multibody System Dynamics,2011,26(2):191—211.
[13]张越,赵阳,谭春林,等.ANCF索梁单元应变耦合问题与模型解耦[J].力学学报,2016,48(6):1406—1415.Zhang Yue,Zhao Yan,Tan Chunlin,et al.The strain coupling problem and model decoupling of ANCF cable/beam element[J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1406—1415.
[14]Hyldahl P,Mikkola A M,Balling O,et al.Behavior of thin rectangular ANCF shell elements in various mesh configurations[J].Nonlinear Dynamics,2014,78(2):1277—1291.
[15]刘桂荣,顾元通.无网格法理论及程序设计[M].王建明,周学军,译.济南:山东大学出版社,2007:45—133.Liu G R,Gu Y T.An Introduction to Meshfree Methods and Their Programming[M].Jinan:Shandong University Press,2007:45—133.
[16]Chen J S,Pan C,Roque C M O L,et al.A lagrangian reproducing kernel particle method for metal forming analysis[J].Computational Mechanics,1998,22(3):289—307.
[17]Chen J S,Pan C,Wu C T,et al.Reproducing kernel particle methods for large deformation analysis of nonlinear structures[J].Computer Methods in Applied Mechanics&Engineering,1996,139(1-4):195—227.
[18]Liu W K,Jun S,Zhang Y F.Reproducing kernel particle methods.Int J Numer Meth Eng[J].International Journal for Numerical Methods in Fluids,1995,20(8-9):1081—1106.
[19]Xiong Y B,Long S Y.Local petrov-galerkin method for a thin plate[J].Applied Mathematics and Mechanics,2004,25(2):210—218.
[20]Wang J G,Liu G R.A point interpolation meshless method based on radial basis functions[J].International Journal for Numerical Methods in Engineering,2002,54(11):1623—1648.
[21]Cui X Y,Feng H,Li G Y,et al.A cell-based smoothed radial point interpolation method(CSRPIM)for three-dimensional solids[J].Engineering Analysis with Boundary Elements,2015,50:474—485.
[22]Pilafkan R,Folkow P D,Darvizeh M,et al.Three dimensional frequency analysis of bidirectional functionally graded thick cylindrical shells using a radial point interpolation method(RPIM)[J].European Journal of Mechanics/A Solids,2013,39(5):26—34.
[23]Gu Y T,Wang W L,Fu Q.An enriched radial point interpolation method(e-RPIM)for the analysis of crack tip[J].Engineering Fracture Mechanics,2010,78(1):175—190.
[24]Chen S L,Li Y X.An efficient RPIM for simulating wave motions in saturated porous media[J].International Journal of Solids&Structures,2008,45(25):6316—6332.
[25]Dai K Y,Liu G R,Han X,et al.Inelastic analysis of2Dsolids using a weak-form RPIM based on deformation theory[J].Computer Methods in Applied Mechanics&Engineering,2006,195(33):4179—4193.
[26]杜超凡,章定国,洪嘉振.径向基点插值法在旋转柔性梁动力学中的应用[J].力学学报,2015,47(2):279—288.Du Chaofan,Zhang Dingguo,Hong Jiazhen.A meshfree method based on radial point interpolation method for the dynamic analysis of rotating flexible beams[J].Chinese Journal of Theoretical and Applied Mechanics,2015,47(2):279—288.
[27]杜超凡,章定国.光滑节点插值法:计算固有频率下界值的新方法[J].力学学报,2015,47(5):839—847.Du Chaofan,Zhang Dingguo.Node-based smoothed point interpolation method:a new method for computing lower bound of natural frequency[J].Chinese Journal of Theoretical and Applied Mechanics,2015,47(5):839—847.
[28]杜超凡,章定国.基于无网格点插值法的旋转悬臂梁的动力学分析[J].物理学报,2015,64(3):396—405.Du Chaofan,Zhang Dingguo.A meshfree method based on point interpolation for dynamic analysis of rotating cantilever beams[J].Acta Phys.Sin.,2015,64(3):396—405.
[29]章孝顺,章定国,陈思佳,等.基于绝对节点坐标法的大变形柔性梁几种动力学模型研究[J].物理学报,2016,65(9):148—157.Zhang Xiaoshun,Zhang Dingguo,Chen Sijia,et al.Several dynamic models of a large deformation flexible beam based on the absolute nodal coordinate formulation[J].Acta Phys.Sin.,2016,65(9):148—157.
[30]陈思佳,章定国,洪嘉振.大变形旋转柔性梁的一种高次刚柔耦合动力学模型[J].力学学报,2013,45(2):251—256.Chen Sijia,Zhang Dingguo,Hong Jiazhen.A high-order rigid-flexible coupling model of a rotating flexible beam under large deformation[J].Chinese Journal of Theoretical and Applied Mechanics,2013,45(2):251—256.
[31]李彬,刘锦阳.大变形柔性梁系统的绝对坐标方法[J].上海交通大学学报,2005,39(5):827—831.Li Bin,Liu Jinyang.Application of absolute nodal coordination formulation in flexible beams with large deformation[J].Journal of Shanghai Jiaotong University,2005,39(5):827—831.
[32]章孝顺,章定国,洪嘉振.考虑曲率纵向变形效应的大变形柔性梁刚柔耦合动力学建模与仿真[J].力学学报,2016,48(3):692—701.Zhang X H,Zhang D G,Hong J Z.Rigid-flexible couple dynamic modeling and simulation with the longitudinal deformation induced curvature effect for a rotating flexible beam under larger deformation[J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):692—701.
[33]Amold M,Brüls O.Convergence of the generalized-αscheme for constrained mechanical systems[J].Multibody System Dynamics,2007,18(2):185—202.
[2]Yakoub R Y,Shabana A A.Three dimensional absolute nodal coordinate formulation for beam elements:implementation and applications[J].Journal of Mechanical Design,2001,123(4):614—621.
[3]Shabana A A,Yakoub R Y.Three dimensional absolute nodal coordinate formulation for beam elements:theory[J].Journal of Mechanical Design,2001,123(4):606—613.
[4]Omar M A,Shabana A A.A two-dimensional shear deformable beam for large rotation and deformation problems[J].Journal of Sound and Vibration,2001,243(3):565—576.
[5]Berzeri M,Shabana A A.Development of simple models for the elastic forces in the absolute nodal co-ordinate formulation[J].Journal of Sound and Vibration,2000,235(4):539—565.
[6]Olshevskiy A,Dmitrochenko O,Dai M D,et al.The simplest 3-,6-and 8-noded fully-parameterized ANCF plate elements using only transverse slopes[J].Multibody System Dynamics,2014,34(1):23—51.
[7]Yan D,Liu C,Tian Q,et al.A new curved gradient deficient shell element of absolute nodal coordinate formulation for modeling thin shell structures[J].Nonlinear Dynamics,2013,74(1-2):153—164.
[8]Dmitrochenko O,Mikkola A.Two simple triangular plate elements based on the absolute nodal coordinate formulation[J].Journal of Computational and Nonlinear Dynamics,2008,3(4):041012.
[9]Gerstmayr J,Shabana A A.Analysis of thin beams and cables using the absolute nodal coordinate formulation[J].Nonlinear Dynamics,2006,45(1-2):109—130.
[10]Dufva K E,Sopanen J T,Mikkola A M.A two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation[J].Journal of Sound and Vibration,2005,280(3-5):719—738.
[11]Dufva K,Shabana A A.Analysis of thin plate structures using the absolute nodal coordinate formulation[J].Proceedings of the Institution of Mechanical Engineers,Part K:Journal of Multi-body Dynamics,2005,219(4):345—355.
[12]Sanborn G G,Choi J,Choi J H.Curve-induced distortion of polynomial space curves,flat-mapped extension modeling,and their impact on ANCF thin-plate finite elements[J].Multibody System Dynamics,2011,26(2):191—211.
[13]张越,赵阳,谭春林,等.ANCF索梁单元应变耦合问题与模型解耦[J].力学学报,2016,48(6):1406—1415.Zhang Yue,Zhao Yan,Tan Chunlin,et al.The strain coupling problem and model decoupling of ANCF cable/beam element[J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1406—1415.
[14]Hyldahl P,Mikkola A M,Balling O,et al.Behavior of thin rectangular ANCF shell elements in various mesh configurations[J].Nonlinear Dynamics,2014,78(2):1277—1291.
[15]刘桂荣,顾元通.无网格法理论及程序设计[M].王建明,周学军,译.济南:山东大学出版社,2007:45—133.Liu G R,Gu Y T.An Introduction to Meshfree Methods and Their Programming[M].Jinan:Shandong University Press,2007:45—133.
[16]Chen J S,Pan C,Roque C M O L,et al.A lagrangian reproducing kernel particle method for metal forming analysis[J].Computational Mechanics,1998,22(3):289—307.
[17]Chen J S,Pan C,Wu C T,et al.Reproducing kernel particle methods for large deformation analysis of nonlinear structures[J].Computer Methods in Applied Mechanics&Engineering,1996,139(1-4):195—227.
[18]Liu W K,Jun S,Zhang Y F.Reproducing kernel particle methods.Int J Numer Meth Eng[J].International Journal for Numerical Methods in Fluids,1995,20(8-9):1081—1106.
[19]Xiong Y B,Long S Y.Local petrov-galerkin method for a thin plate[J].Applied Mathematics and Mechanics,2004,25(2):210—218.
[20]Wang J G,Liu G R.A point interpolation meshless method based on radial basis functions[J].International Journal for Numerical Methods in Engineering,2002,54(11):1623—1648.
[21]Cui X Y,Feng H,Li G Y,et al.A cell-based smoothed radial point interpolation method(CSRPIM)for three-dimensional solids[J].Engineering Analysis with Boundary Elements,2015,50:474—485.
[22]Pilafkan R,Folkow P D,Darvizeh M,et al.Three dimensional frequency analysis of bidirectional functionally graded thick cylindrical shells using a radial point interpolation method(RPIM)[J].European Journal of Mechanics/A Solids,2013,39(5):26—34.
[23]Gu Y T,Wang W L,Fu Q.An enriched radial point interpolation method(e-RPIM)for the analysis of crack tip[J].Engineering Fracture Mechanics,2010,78(1):175—190.
[24]Chen S L,Li Y X.An efficient RPIM for simulating wave motions in saturated porous media[J].International Journal of Solids&Structures,2008,45(25):6316—6332.
[25]Dai K Y,Liu G R,Han X,et al.Inelastic analysis of2Dsolids using a weak-form RPIM based on deformation theory[J].Computer Methods in Applied Mechanics&Engineering,2006,195(33):4179—4193.
[26]杜超凡,章定国,洪嘉振.径向基点插值法在旋转柔性梁动力学中的应用[J].力学学报,2015,47(2):279—288.Du Chaofan,Zhang Dingguo,Hong Jiazhen.A meshfree method based on radial point interpolation method for the dynamic analysis of rotating flexible beams[J].Chinese Journal of Theoretical and Applied Mechanics,2015,47(2):279—288.
[27]杜超凡,章定国.光滑节点插值法:计算固有频率下界值的新方法[J].力学学报,2015,47(5):839—847.Du Chaofan,Zhang Dingguo.Node-based smoothed point interpolation method:a new method for computing lower bound of natural frequency[J].Chinese Journal of Theoretical and Applied Mechanics,2015,47(5):839—847.
[28]杜超凡,章定国.基于无网格点插值法的旋转悬臂梁的动力学分析[J].物理学报,2015,64(3):396—405.Du Chaofan,Zhang Dingguo.A meshfree method based on point interpolation for dynamic analysis of rotating cantilever beams[J].Acta Phys.Sin.,2015,64(3):396—405.
[29]章孝顺,章定国,陈思佳,等.基于绝对节点坐标法的大变形柔性梁几种动力学模型研究[J].物理学报,2016,65(9):148—157.Zhang Xiaoshun,Zhang Dingguo,Chen Sijia,et al.Several dynamic models of a large deformation flexible beam based on the absolute nodal coordinate formulation[J].Acta Phys.Sin.,2016,65(9):148—157.
[30]陈思佳,章定国,洪嘉振.大变形旋转柔性梁的一种高次刚柔耦合动力学模型[J].力学学报,2013,45(2):251—256.Chen Sijia,Zhang Dingguo,Hong Jiazhen.A high-order rigid-flexible coupling model of a rotating flexible beam under large deformation[J].Chinese Journal of Theoretical and Applied Mechanics,2013,45(2):251—256.
[31]李彬,刘锦阳.大变形柔性梁系统的绝对坐标方法[J].上海交通大学学报,2005,39(5):827—831.Li Bin,Liu Jinyang.Application of absolute nodal coordination formulation in flexible beams with large deformation[J].Journal of Shanghai Jiaotong University,2005,39(5):827—831.
[32]章孝顺,章定国,洪嘉振.考虑曲率纵向变形效应的大变形柔性梁刚柔耦合动力学建模与仿真[J].力学学报,2016,48(3):692—701.Zhang X H,Zhang D G,Hong J Z.Rigid-flexible couple dynamic modeling and simulation with the longitudinal deformation induced curvature effect for a rotating flexible beam under larger deformation[J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):692—701.
[33]Amold M,Brüls O.Convergence of the generalized-αscheme for constrained mechanical systems[J].Multibody System Dynamics,2007,18(2):185—202.