ZHANG Can-hui, WANG Dong-dong, LI Tong-shan. Orthogonal Basic Deformation Mode Method for Zero-Energy Mode Suppression of Hybrid Stress Element[J]. Applied Mathematics and Mechanics, 2011, 32(1): 79-92. doi: 10.3879/j.issn.1000-0887.2011.01.009
Citation: ZHANG Can-hui, WANG Dong-dong, LI Tong-shan. Orthogonal Basic Deformation Mode Method for Zero-Energy Mode Suppression of Hybrid Stress Element[J]. Applied Mathematics and Mechanics, 2011, 32(1): 79-92. doi: 10.3879/j.issn.1000-0887.2011.01.009

Orthogonal Basic Deformation Mode Method for Zero-Energy Mode Suppression of Hybrid Stress Element

doi: 10.3879/j.issn.1000-0887.2011.01.009
  • Received Date: 2010-06-25
  • Rev Recd Date: 2010-11-30
  • Publish Date: 2011-01-15
  • A set of basic deformation modes for hybrid stress finite element were directly derived from the element displacement field.Subsequently by employing the so-called united orthogonal conditions a new orthogonalization method was also proposed.The resulting orthogonal basic deformation modes exhibit simple and clear physical meanings.In addition,they do not involve any material parameters and thus can be efficiently used to examine the element performance and serve as a unified tool to assess different hybrid elements.Therafter a convenient approach for identification of spurious zero-energy modes was presented through using the positive definiteness property of flexibility matrix.Moreover,based upon the orthogonality relationship between the given initial stress modes and the orthogonal basic deformation modes,an alternative method of assumed stress modes to formulate a hybrid element free of spurious modes was discussed.It was also found that the orthogonality of the basic deformation modes was the sufficient and necessary condition for suppression of spurious zero-energy modes.Numerical examples of 2D 4-node quadrilateral element and 3D 8-node hexahedral element were illustrated in details to demonstrate the efficacy of the proposed orthogonal basic deformation mode method.
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  • [1]
    Pian T H H. Derivation of element stiffness matrices[J], AIAA Journal, 1964, 2(3): 576-577.
    [2]
    Chen W J, Cheung Y K. Nonconforming element method and refined hybrid element method for axisymmetric solid[J]. International Journal for Numerical Methods in Engineering, 1996, 39(15): 2509-2529. doi: 10.1002/(SICI)1097-0207(19960815)39:15<2509::AID-NME963>3.0.CO;2-8
    [3]
    Sze K Y. Admissible matrix formulation-from orthogonal approach to explicit hybrid stabilization[J]. Finite Elements in Analysis and Design, 1996, 24(1): 1-30. doi: 10.1016/0168-874X(95)00026-P
    [4]
    张灿辉, 冯伟, 黄黔. 杂交应力元的应力子空间和柔度矩阵H对角化方法[J]. 应用数学和力学, 2002, 23(11): 1124- 1132.(ZHANG Can-hui, FENG Wei, HUANG Qian. The stress subspace of hybrid stress element and the diagonalization method for flexibility matrix H[J]. Applied Mathematics and Mechanics (English Edition), 2002, 23(11): 1263- 1273.)
    [5]
    张灿辉, 冯伟, 黄黔. 用单元柔性矩阵H对角化方法建立杂交应力有限单元[J]. 计算力学学报, 2002, 19(4): 409-413.(ZHANG Can-hui, FENG Wei, HUANG Qian. A method of flexibility matrix H diagonalization for constructing hybrid stress finite elements[J]. Chinese Journal of Computational Mechanics, 2002, 19(4): 409-413.(in Chinese))
    [6]
    Tian Z, Zhao F, Yang Q. Straight free-edge effects in laminated composites[J]. Finite Elements in Analysis and Design, 2004, 41(1): 1-14. doi: 10.1016/j.finel.2004.03.004
    [7]
    Zhang C, Wang D, Zhang J, Feng W, Huang Q. On the equivalence of various hybrid finite elements and a new orthogonalization method for explicit element stiffness formulation[J]. Finite Elements in Analysis and Design, 2007, 43(4): 321-332. doi: 10.1016/j.finel.2006.11.002
    [8]
    张灿辉, 王东东, 张建霖. 三维杂交应力元性能分析的基本变形模式方法[J]. 工程力学, 2009, 26(8): 44-49.(ZHANG Can-hui, WANG Dong-dong, ZHANG Jian-lin. Performance analysis of 3D hybrid stress elements with a basic deformation-based approach[J]. Engineering Mechanics, 2009, 26(8): 44-49.(in Chinese))
    [9]
    Pian T H H, Wu C C. Hybrid and Incompatible Finite Element Methods[M]. Boca Raton: Chapman & Hall/CRC Press, 2006.
    [10]
    Babuska I, Oden J T, Lee J K. Mixed-hybrid finite element approximation of second-order elliptic boundary-value problems[J]. Computer Methods in Applied Mechanics and Engineering, 1977, 11(2): 175-206. doi: 10.1016/0045-7825(77)90058-5
    [11]
    Pian T H H, Chen D P. On the suppression of zero-energy deformation modes[J]. International Journal for Numerical Methods in Engineering, 1983, 19(12): 1741-1752. doi: 10.1002/nme.1620191202
    [12]
    Pian T H H, Sumihara K. Rational approach for assumed stress finite elements[J]. International Journal for Numerical Methods in Engineering, 1984, 20(9): 1685-1965. doi: 10.1002/nme.1620200911
    [13]
    Pian T H H, Wu C C. A rational approach for choosing stress terms of hybrid finite element formulations[J]. International Journal for Numerical Methods in Engineering, 1988, 26(10): 2331-2343. doi: 10.1002/nme.1620261014
    [14]
    HUANG Qian. Modal analysis of deformable bodies with finite degree of deformation freedom-an approach to determination of natural stress modes in hybrid finite elements[C]Chien Wei-zang, FU Zi-zhi. Advances in Applied Mathematics & Mechanics in China. Beijing: IAP (International Academic Publishers), 1991, 3: 283-303.
    [15]
    Feng W , Hoa S V, Huang Q. Classification of stress modes in assumed stress fields of hybrid finite elements[J]. International Journal for Numerical Methods in Engineering, 1997, 40(23): 4313-4339. doi: 10.1002/(SICI)1097-0207(19971215)40:23<4313::AID-NME259>3.0.CO;2-N
    [16]
    张灿辉, 王东东. 一种抑制杂交元零能模式的假设应力场方法[J]. 固体力学学报, 2010, 31(1): 40-47.(ZHANG Can-hui, WANG Dong-dong. An assumed stress method for zero-energy mode suppression in hybrid finite elements[J]. Chinese Journal of Solid Mechanics, 2010, 31(1): 40-47.(in Chinese))
    [17]
    张灿辉, 冯伟, 黄黔. 杂交元假设应力模式的变形刚度分析[J]. 应用数学和力学, 2006, 27(7): 757-764.(ZHANG Can-hui, FENG Wei, HUANG Qian. Deformation rigidity of assumed stress modes in hybrid elements[J]. Applied Mathematics and Mechanics(English Edition), 2006, 27(7): 861-869.)
    [18]
    Han J, Hoa S V. A three-dimensional multilayer composite finite element for stress analysis of composite laminates[J]. International Journal for Numerical Methods in Engineering, 1993,36(22): 3903-3914. doi: 10.1002/nme.1620362209
    [19]
    Rubinstein R, Punch E F, Atluri S N. An analysis of, and remedies for, kinematic modes in hybrid-stress finite elements: selection of stable, invariant stress fields[J]. Computer Methods in Applied Mechanics and Engineering, 1983, 38(1): 63-92. doi: 10.1016/0045-7825(83)90030-0
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