新型协同转动四边形曲壳单元研究
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摘要
本文发展了一种新型9节点四边形曲壳单元。单元的局部坐标系采用了先进的协同转动框架,它能随单元的刚体转动和平动而运动,但不受单元变形的影响,从而在计算单元节点位移时可以排除其中的刚体位移成分,降低了建立单元切线刚度矩阵的复杂性。单元每个节点包含3个平动自由度和2个转动自由度,其中转动自由度为单元节点处中性面法向矢量的两个较小分量,在增量求解过程中它们是可以直接累加的。采用矢量型转动变量,通过简单的矢量变换即可建立起局部变量与整体变量之间的关系,从而非常方便地将单元局部切线刚度矩阵转换到整体坐标系下。此外,引进适用于浅壳的Green-Lagrange应变来计算单元应变能,然后对单元应变能进行一阶和二阶微分,即可得到单元内力矢量和对称的切线刚度矩阵。
采用降阶积分法可以克服闭锁现象,但有时也会使单元因缺秩而导致伪零能模式出现,从而使单元切线刚度矩阵产生病态现象。为了消除伪零能模式,引进Hellinger-Reissner函数,用假设应变部分代替协调应变。假设应变由低阶应变和高阶应变组成,其中低阶应变由协调应变采用2×2积分得到;而作为稳定应变的高阶应变由假设应变法构造而得。由Hellinger-Reissner变分原理得到的切线刚度矩阵仍是对称的。
最后,本文采用8个经典算例对单元的可靠性、计算效率和精度进行测试,它们由3个小变形算例和5个大变形算例(包括3个屈曲问题)组成。非线性增量方程组的求解分别采用了广义位移法和位移控制法,以越过结构屈曲后平衡路径中的极值点和反跳点,从而有效跟踪结构的屈曲后响应。分析结果表明:本文所发展的9节点四边形曲壳单元已有效地消除了闭锁现象和伪零能模式,单元的可靠性、收敛性和计算效率也是非常满意的。
An advanced 9-node curved quadrilateral shell element for large displacement and large rotation analysis is developed, where the local coordinate system is a co-rotational framework, it translates and rotates rigidly with the element, but does not deform with the element. Thus, the contribution of rigid-body motion to nodal displacements and rotations can be excluded in advance, making it simpler to obtain the element tangent stiffness matrix. There are 3 translational degrees of freedom and 2 rotational degrees of freedom at each node, and the latters are the two smallest components of the mid-surface normal vector at each node, and additive in an incremental solution procedure, taking advantage in updating the tangent stiffness matrix. Furthermore, the Green-Lagrange strains specialized for the shallow curved shell are employed, and the internal force vector and the element tangent stiffness matrix are calculated respectively as the first derivative and the second derivative of the strain energy with respect to local nodal variables. Considering the commutativity of all nodal variables, the achieved element tangent stiffness matrix is symmetric.
The uniformly reduced integration method can eliminate/alleviate locking problems, but sometimes it may lead to the occurrence of element rank deficiency and spurious singular modes. To overcome these problems, the Hellinger-Reissner functional is adopted in which part of conforming strains are replaced by assumed strains. The assumed strains consist of the lower-order strain and the higher-order strain, and the former is interpolated linearly by using the corresponding displacement-based strain, while the latter playing the role of stabilizing strain is formulated based on an assumed strain procedure. The element tangent stiffness matrix derived from Hellinger-Reissner variational principle is still symmetric.
Finally, 8 elastic flat plate and curved shell problems are solved to evaluate the performance of the proposed element. The generalized displacement control method is employed in solving these problems. It is shown that locking phenomena has been eliminated/alleviated in the present element, and the reliability, computational efficiency and convergence of the element are satisfying.
采用降阶积分法可以克服闭锁现象,但有时也会使单元因缺秩而导致伪零能模式出现,从而使单元切线刚度矩阵产生病态现象。为了消除伪零能模式,引进Hellinger-Reissner函数,用假设应变部分代替协调应变。假设应变由低阶应变和高阶应变组成,其中低阶应变由协调应变采用2×2积分得到;而作为稳定应变的高阶应变由假设应变法构造而得。由Hellinger-Reissner变分原理得到的切线刚度矩阵仍是对称的。
最后,本文采用8个经典算例对单元的可靠性、计算效率和精度进行测试,它们由3个小变形算例和5个大变形算例(包括3个屈曲问题)组成。非线性增量方程组的求解分别采用了广义位移法和位移控制法,以越过结构屈曲后平衡路径中的极值点和反跳点,从而有效跟踪结构的屈曲后响应。分析结果表明:本文所发展的9节点四边形曲壳单元已有效地消除了闭锁现象和伪零能模式,单元的可靠性、收敛性和计算效率也是非常满意的。
An advanced 9-node curved quadrilateral shell element for large displacement and large rotation analysis is developed, where the local coordinate system is a co-rotational framework, it translates and rotates rigidly with the element, but does not deform with the element. Thus, the contribution of rigid-body motion to nodal displacements and rotations can be excluded in advance, making it simpler to obtain the element tangent stiffness matrix. There are 3 translational degrees of freedom and 2 rotational degrees of freedom at each node, and the latters are the two smallest components of the mid-surface normal vector at each node, and additive in an incremental solution procedure, taking advantage in updating the tangent stiffness matrix. Furthermore, the Green-Lagrange strains specialized for the shallow curved shell are employed, and the internal force vector and the element tangent stiffness matrix are calculated respectively as the first derivative and the second derivative of the strain energy with respect to local nodal variables. Considering the commutativity of all nodal variables, the achieved element tangent stiffness matrix is symmetric.
The uniformly reduced integration method can eliminate/alleviate locking problems, but sometimes it may lead to the occurrence of element rank deficiency and spurious singular modes. To overcome these problems, the Hellinger-Reissner functional is adopted in which part of conforming strains are replaced by assumed strains. The assumed strains consist of the lower-order strain and the higher-order strain, and the former is interpolated linearly by using the corresponding displacement-based strain, while the latter playing the role of stabilizing strain is formulated based on an assumed strain procedure. The element tangent stiffness matrix derived from Hellinger-Reissner variational principle is still symmetric.
Finally, 8 elastic flat plate and curved shell problems are solved to evaluate the performance of the proposed element. The generalized displacement control method is employed in solving these problems. It is shown that locking phenomena has been eliminated/alleviated in the present element, and the reliability, computational efficiency and convergence of the element are satisfying.
引文
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[3]Bathe KJ,Iosilevich A,Chapelle D.An inf-sup test for shell finite elements.Computers and Structures,2000,75(5):439-456.
[4]Chapelle D,Oliveira DL,Bucalem ML.MITC elements for a classical shell model.Computers and Structures,2003,81:523-533.
[5]关玉璞,唐立民.结构分析中的退化壳有限元方法.力学进展.1994,24(1).
[6]Richard H,MacNeal.Perspective on finite elements for shell analysis.Finite Elements in Analysis and Design,1998,30:175-186.
[7]高光藩,丁信伟.薄壳结构分析有限元方法.石油化工设备.2002,31(5).
[8]Yang HTY,Masud SSA,Kapania RK.A survey of recent shell finite elements.International Journal for Numerical Methods in Engineering,2000,47:101—127.
[9]龙驭球,龙志飞,岑松.新型有限元论.清华大学出版社.北京.2004.
[10]Belytschko T,Liu WK,Moran B.Nonlinear Finite Elements for Continua and Structures.John Wiley & Sons.2000.
[11]Clough RW,FeEppa CA.A Refined Quadrilateral Element for Analysis of Plate Bending.Proc 2nd Conf Matrix Methods in Struct Mech.1968.
[12]Zienkiewitz OC,Cheung YK.The Finite Element Method for Analysis of Elastic Isotropic and Orthotropic Slabs.Proc Inst Civ Eng,1964,28.
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[16]李忠学.梁板壳有限单元中的各种闭锁现象及解决方法.浙江大学学报(工学版).2007,41(7):1119-1125,1167.
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[20]Chinosi C,Lovadina C.Remarks on partial selective reduced integration method for Reissner-Mindlin plate problem.Computers and Structures,1999,73:73-78.
[21]章向明,王安稳,何汉林.无锁定的退化的等参壳元.应用数学和力学.2001,22(5).
[22]陈大鹏,潘亦苏.壳单元的自锁问题.西南交大学报.1993,1:4-8.
[23]Xenophontos C,Kurtz J,Fulton S.A p-version MITC finite element method for Reissner-Mindlin plates with curved boundaries.Journal of Computational and Applied Mathematics,2006,192:374-395.
[24]Dvorkin EN,Bathe KJ.A continuum mechanics based four-node shell element for general nonlinear analysis.Eng Comput.1984,1:77-88.
[25]Bathe K J,Dvorkin EN.A formulation of general shell elements—the use of mixed interpolation of tensorial components,lnt JNumer Methods Eng,1986,22:697-722.
[26]Bucalem ML,Bathe KJ.On higher-order general shell elements.Int J Numer Meth Eng 1993,36:3729-54.
[27]Bathe KJ,Iosilevich A,Chapelle D.Dominique Chapelle.An evaluation of the MITC shell elements.Computers and Structures,2000,75(1):1-30
[28]Guzey S,Stolarski HK,Cockburn B,Tamma KK.Design and development of a discontinuous Galerkin method for shells.Computer methods in applied mechanics and engineering,2006,195:3582-3548.
[29]Belytschko T,Ong JSJ,Liu WK,Kennedy JM.Hourglass control in linear and nonlinear problems.Comp.MethodsAppl.Mech.Eng,1984,43:251-276.
[30]Jacquotte OP,Oden JT.Analysis of hourglass instabilities and control in underintegrated finite element methods.Computer Methods in Applied Mechanics and Engineering,1984,44(3):339-363.
[31]Belytschko T,Liu WK,Ong JSJ.Mixed variational principles and stabilization of spurious modes in the 9-node element.Computer Methods in Applied Mechanics and Engineering,1987,62(3):275-292.
[32]Briassoulis D.The zero energy modes problem of the nine-node lagrangian degenerated shell element.Computers and Structures,1988,30(6):1389-1402.
[33]Vu-Quoc L,Mora JA.A class of simple and efficient degenerated shell elements—Analysis of global spurious-mode filtering.Computer Methods in Applied Mechanics and Engineering,1989,74(2):117-175.
[34]Vu-Quoc L.A perturbation method for dynamic analyses using under-integrated shell elements.Computer Methods in Applied Mechanics and Engineering,1990,79(2):129-172.
[35]Wempner G.Finite elements,finite rotations and small strains of flexible shells.International Journal of Solids and Structures,1969,5(2):117-153.
[36]Belytschko T,Hseih BJ.Non-linear transient finite element analysis with convected co-ordinates.International Journal for Numerical Methods in Engineering,1973,7:255-271
[37]Belytschko T,Glaum LW.Application of higher order corotational stretch theories to nonlinear finite element analysis.Computers and structures,1979,10:175-182.
[38]Li ZX,Izzuddin BA,Vu-Quoc L.A 9-node co-rotational quadrilateral shell element. Computational Mechanics.DOI10.1007/s00466-008-0289-8.
[39]Teh LH,Clarke MJ.Co-rotational and Lagrangian formulations for elastic three-dimensional beam finite elements.Journal of Constructional Steel Research,1998,48:123-144.
[40]Jetteur PH,Cescotto S.(1991).A mixed finite element for the analysis of large inelastic strains.International Journal for Numerical Methods in Engineering,31:229-239.
[41]Crisfield MA,Moita GF,Jelenic G,Lyons,LPR.(1995).Enhanced lower-order element formulations for large strains.In Computational Plasticity,Fundamentals & Applications,Part Ⅰ(Eds D.R.J.Owen et al.).Pineridge Press,Swansea,pp.293-320.
[42]Crisfield MA,Moita GF.A unified co-rotational framework for solids,shells and beams.International Journal of Solids and Structures,1996,33(20-22):2969-2992.
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