On the equivalence of mixed element formulations and the concept of reduced integration in large deformation problems

International Journal of Nonlinear Sciences and Numerical Simulation

Volumn 3, Issue 1, 2002, Pages 1-33

  Reese, Stefanie ^a  

^a RUHR UNIVERSITY BOCHUM   (Germany)

Author keywords

Enhanced strain method; Finite elasticity; Finite element technology; Hourglassing; Stabilization

Indexed keywords

EID: 0346008409     PISSN: 15651339     EISSN: None     Source Type: Journal    
DOI: 10.1515/ijnsns.2002.3.1.1     Document Type: Article

References (80)

1. U. Andelfinger and E. Ramm (1993), EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements, International Journal for Numerical Methods in Engineering 36, 1311-1337.
2. U. Andelfinger, E. Ramm and D. Roehl (1992), 2D-and 3D-enhanced assumed strain elements and their application in plasticity, in Proc. 3rd Int. Conference on Computational Plasticity, Fundamentals and Applications (eds.: E. Onate, D. R. J. Owen), Pineridge Press, Swansea.
3. T. Belytschko, J. S.-J. Ong, W. K. Liu and J. M. Kennedy (1984), Hourglass control in linear and nonlinear problems, Computer Methods in Applied Mechanics and Engineering 43, 251-276.
4. P. Betsch, F. Gruttmann and E. Stein (1996), A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains, Computer Methods in Applied Mechanics and Engineering 130, 57-79.
5. P. Betsch and E. Stein (1999), Numerical implementation of multiplicative elastoplasticity into assumed strain elements with application to shells at large strains, Computer Methods in Applied Mechanics and Engineering 179, 215-245.
6. M. Bischoff, E. Ramm (1997), Shear deformable shell elements for large strains and rotations, International Journal for Numerical Methods in Engineering 40, 4427-4449.
7. M. Bischoff, E. Ramm and D. Braess (1999), A class of equivalent enhanced assumed strain and hybrid stress finite elements, Computational Mechanics 22, 443-449.
8. D. Braess, (1997), Finite Elements - Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, Cambridge.
9. N. Büchter, E. Ramm and D. Roehl (1994), Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept, International Journal for Numerical Methods in Engineering 37, 2551-2568.
10. J. M. A. Cesar de Sa and R. M. Jorge (1999), New enhanced strain elements for incompressible problems, International Journal for Numerical Methods in Engineering 44, 229-248.
11. R. D. Cook (1977a), Ways to improve the bending response of finite elements, International Journal for Numerical Methods in Engineering 11, 1029-1039.
12. R. D. Cook (1977b), More about artificial softening of finite elements, International Journal for Numerical Methods in Engineering 11, 1334-1339.
13. M. A. Crisfield, G. F. Moita, G. Jelenic and L. P. R. Lyons (1995), Enhanced lower-order element formulations for large strains, in Computational Plasticity - Fundamentals and Applications, Part I (D. R. J. Owen, E. Onate and E. Hinton, eds.), Pineridge Press, Swansea.
14. W. P. Doherty, E. L. Wilson and R. L. Taylor (1969), Stress analysis of axisymmetric solids utilizing higher order quadrilateral finite elements, SESM Report No. 69-3, Department of Civil Engineering, University of California, Berkeley.
15. S. Doll, R. Hauptmann, K. Schweizerhof and C. Freischlager (2000), On volumetric locking of low-order solid and solid-shell elements for finite elastoviscoplastic deformations and selective reduced integration, Engineering Computations 17, 874-902.
16. E. N. Dvorkin and A. P. Assanelli (2000), Implementation and stability analysis of the QMITC-TLH elaso-plastic finite strain (2D) element formulation, Computers and Structures 75, 305-312.
17. Eberlein, R. and P. Wriggers, Finite element formulations of five and six parameter shell theories accounting for finite plastic strain, in Computational Plasticity - Fundamentals and Applications, 1898-1903, D. R. J. Owen, E. Onate, E. Hinton (Ed.), Pineridge Press, Swansea.
18. D. P. Flanagan and T. Belytschko (1981), A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, International Journal for Numerical Methods in Engineering 17, 679-706.
19. C. Freischläger and K. Schweizerhof (1996), On a systematic development of trilinear three-dimensional solid elements based on Simos enhanced strain formulation, International Journal of Solids and Structures 33, 2993-3017.
20. S. Glaser and F. Armero (1997), On the formulation of enhanced strain finite elements in finite deformations, Engineering Computations 14, 759-791.
21. R. Hauptmann, K. Schweizerhof and S. Doll (2000), Extension of the "solid-shell" concept for application to large elastic and large elastoplastic deformations, International Journal for Numerical Methods in Engineering 49, 1121-1141.
22. U. Hueck and P. Wriggers (1995), A formulation for the 4-node quadrilateral element, International Journal for Numerical Methods in Engineering 38, 3007-3037.
23. T. J. R. Hughes (1977), Equivalence of finite elements for nearly incompressible elasticity, Journal of Applied Mechanics, 181-183.
24. T. J. R. Hughes (1980), Generalization of selctive integration procedures to anisotropic and nonlinear media, International Journal for Numerical Methods in Engineering 15, 1413-1418.
25. B. Irons and S. Ahmad (1980), Techniques of Finite Elements, Ellis Horwood, Chichester, England.
26. O.-P. Jacquotte and J. T. Oden (1986), An accurate and efficient a posteriori control of hourglass instabilities in underintegrated linear and nonlinear elasticity, Computer Methods in Applied Mechanics and Engineering 55, 105-128.
27. E. P. Kasper and R. L. Taylor (2000a), A mixed-enhanced strain method, Part I: Geometrically linear problems, Computers and Structures 75, 237-250.
28. E. P. Kasper and R. L. Taylor (2000b), A mixed-enhanced strain method, Part II: Geometrically non-linear problems, Computers and Structures 75, 251-260.
29. S. W. Key (1974), The large deformation dynamic response of axisymmetric solids, Computer Methods in Applied Mechanics and Engineering 4, 195-218.
30. O. Klaas, A. Maniatty and M. S. Shephard (1999), A stabilized mixed finite element method for finite elasticity. Formulation for linear displacement and pressure interpolation, Computer Methods in Applied Mechanics and Engineering 180, 65-79.
31. B. C. Koh and N. Kikuchi (1987), New improved hourglass control for bilinear and trilinear elements in anisotropic linear elasticity, Computer Methods in Applied Mechanics and Engineering 65, 1-46.
32. J. Korelc and P. Wriggers (1996a), An efficient 3D enhanced strain element with Taylor expansion of the shape functions, Computational Mechanics 19, 30-40.
33. J. Korelc and P. Wriggers (1996b), Consistent gradient formulation for a stable enhanced strain method for large deformations, Engineering Computations 13, 103-123.
34. J. Korelc and P. Wriggers (1997), Improved enhanced strain four-node element with Taylor expansion of the shape functions, International Journal for Numerical Methods in Engineering 40, 407-421.
35. D. Kosloff and G. A. Frazier (1978), Treatment of hourglass pattern in low order finite element codes, International Journal for Numerical and Analytical Methods in Geomechanics 2, 57-72.
36. M. Küssner and B. D. Reddy (1998), The equivalent parallelogram and its application in 2D and 3D finite element analysis, Proceedings of SACAM 98, Cape Town, South Africa.
37. X. Li, A. J. L. Crook and L. P. R. Lyons (1993), Mixed strain elements for non-linear analysis, Engineering Computations 10, 223-242.
38. W. K. Liu, Y. Guo, S. Tang and T. Belytschko (1998), A multiple quadrature eight-node hexahedral finite element for large deformation elastoplastic analysis, Computer Methods in Applied Mechanics and Engineering 154, 69-132.
39. W. K. Liu, J. S.-J. Ong and R. A. Uras (1985), Finite element stabilization matrices: a unification approach, Computer Methods in Applied Mechanics and Engineering 53, 13-46.
40. D. Malkus and T. J. R Hughes (1978), Mixed finite element methods - reduced and selective integration techniques: a unification of concepts. Computer Methods in Applied Mechanics and Engineering 15, 68-81.
41. J. C. Nagtegaal and D. D. Fox (1996), Using assumed enhanced strain elements for large compressive deformation, International Journal of Solids and Structures 33, 3151-3159.
42. J. C. Nagtegaal, D. M. Parks and J. R. Rice (1974), On numerically accurate finite element solutions in the fully plastic range, Computer Methods in Applied Mechanics and Engineering 4, 153-177.
43. J. T. Oden and J. E. Key (1970), Numerical analysis of finite axisymmetric deformations of incompressible elastic solids of revolution, International Journal of Solids and Structures 6, 497-518.
44. D. Pantuso and K.-J. Bathe (1997), On the stability of mixed finite elements in large strain analysis of incompressible solids, Finite Elements in Analysis and Design 28, 83-104.
45. S. F. Pawsey and R. W. Clough (1971), Improved numerical integration of thick shell finite elements, International Journal for Numerical Methods in Engineering 3, 575-586.
46. U. Perego (2000), A variationally consistent generalized variable formulation for enhanced strain finite elements, Communications in Numerical Methods in Engineering 16, 151-163.
47. T. H. H. Pian (1964), Derivation of element stiffness matrices by assumed stress distribution, AIAA Journal 2, 1333-1336.
48. T. H. H. Pian (1995), State-of-the-art development of hybrid/mixed finite element method, Finite Elements in Analysis and Design 21, 5-20.
49. T. H. H. Pian and K. Sumihara (1984), Rational approach for assumed stress finite elements, International Journal for Numerical Methods in Engineering 20, 1685-1695.
50. R. Piltner and R. L. Taylor (1995), A quadrilateral mixed finite element with two enhanced strain modes, International Journal for Numerical Methods in Engineering 38, 1783-1808.
51. R. Piltner and R. L. Taylor (1999), A systematic construction of B-bar functions for linear and non-linear mixed-enhanced finite elements for plane elasticity problems, International Journal for Numerical Methods in Engineering 44, 615-639.
52. R. Piltner and R. L. Taylor (2000), Triangular finite elements with rotational degrees of freedom and enhanced strian modes, Computers & Structures 75, 361-368.
53. M. A. Puso (2000), A highly efficient enhanced assumed strain physically stabilized hexahedral element, International Journal for Numerical Methods in Engineering 49, 1029-1064.
54. E. Rank, M. Schweingruber and M. Sommer (1993), Adaptive mesh generation and transformation of triangular to quadrilateral meshes, Communications in Numerical Methods in Engineering 9, 121-129.
55. B. D. Reddy and J. C. Simo (1995), Stability and convergence of a class of enhanced strain methods, SIAM J. Numer. Anal. 32, 1705-1728.
56. S. Reese, M. K. Küssner, B. D. Reddy (1999), A new stabilization technique for finite elements in nonlinear elasticity, International Journal for Numerical Methods in Engineering 44, 1617-1652.
57. S. Reese and P. Wriggers (2000), A stabilization technique to avoid hourglassing in finite elasticity, International Journal for Numerical Methods in Engineering 48, 79-109.
58. S. Reese, P. Wriggers and B. D. Reddy (2000), A new locking-free brick element technique for large deformation problems in finite elasticity, Computers and Structures 75, 291-304.
59. D. Roehl and E. Ramm (1996), Large elastoplastic finite element analysis of solids and shells with the enhanced assumed strain concept, International Journal of Solids and Structures 33, 3215-3237.
60. J. C. Simo and F. Armero (1992), Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes. Computer Methods in Applied Mechanics and Engineering 33, 1413-1449.
61. J. C. Simo, F. Armero and R. L. Taylor (1993) Improved version of assumed enhanced strain tri-linear elements for 3D finite deformation problems, Computer Methods in Applied Mechanics and Engineering 110, 359-366.
62. J. C. Simo and S. Rifai (1990), A class of mixed assumed strain methods and the method of incompatible modes, International Journal for Numerical Methods in Engineering 29, 1595-1638.
63. J. C. Simo, R. L. Taylor and K. S. Pister (1985), Variational and projection methods for the volume constraint in finite deformation elasto-plasticity, Computer Methods in Applied Mechanics and Engineering 51, 177-208.
64. E. A. de Souza Neto, D. Peric, G. C. Huang and D. R. J. Owen (1995), Remarks on the stability of enhanced strain elements in finite elasticity and elastoplasticity, in Computational Plasticity - Fundamentals and Applications, Part I (D. R. J. Owen, E. Onate and E. Hinton, eds.), Pineridge Press, Swansea.
65. L. Stainier and J. Ph. Ponthot (1994), An improved one-point integration method for large strain elastoplastic analysis, Computer Methods in Applied Mechanics and Engineering 118, 163-177.
66. H. K. Stolarski and Y.-I. Chen (1995), Extrapolated fields in the formulation of the assumed strain elements, Part 1: Two-dimensional problems, Computer Methods in Applied Mechanics and Engineering 123, 247-262.
67. T. Sussman and K.-J. Bathe (1987), A finite element formulation for nonlinear incompressible elastic and inelastic analysis, Computers and Structures 26, 357-409.
68. I. C. Taig (1961), Structural analysis by the matrix displacement method, English Electric Aviation Limited Report S.O.17.
69. I. C. Taig and R. I. Kerr (1964), Some problems in the discrete element representation of aircraft structures, in B. F. Veubeke (ed.): Matrix Methods of Structural Analysis, Pergamon Press, Oxford, 267-315.
70. R. L. Taylor (2000), A mixed-enhanced formulation for tetrahedral finite elements, International Journal for Numerical Methods in Engineering 47, 205-227.
71. R. L. Taylor, P. J. Beresford and E. L. Wilson (1976), A nonconforming element for stress analysis, International Journal for Numerical Methods in Engineering 10, 1211-1219.
72. E. L. Wachspress (1978), Incompatible quadrilateral basis functions, International Journal for Numerical Methods in Engineering 12, 589-595.
73. W. A. Wall, M. Bischoff and E. Ramm (2000), A deformation dependent stabilization technique, exemplified by EAS elements at large strain, Computer Methods in Applied Mechanics and Engineering 188, 859-871.
74. E. L. Wilson, R. L. Taylor, W. P. Doherty and J. Ghaboussi (1973), Incompatible displacement models, in Numerical and Computer Models in Structural Mechanics, S. J. Fenves et al., Academic Press, New York.
75. P. Wriggers and U. Hueck (1996), A formulation of the QS6 element for large elastic deformations, International Journal for Numerical Methods in Engineering 39, 1437-1454.
76. P. Wriggers and J. Korelc (1996), On enhanced strain methods for small and finite deformations of solids, Computational Mechanics 18, 413-428.
77. P. Wriggers and S. Reese (1996), A note on enhanced strain methods for large deformations, Computer Methods in applied mechanics and engineering 135, 201-209.
78. O. C. Zienkiewicz, R. L. Taylor (1989), The Finite Element Method, VoI I: Basis Formulation and Linear Problems, Mc Graw-Hill, Maidenhead.
79. O. C. Zienkiewicz, R. L. Taylor (1991), The Finite Element Method, Vol II: Solid and Fluid Mechanics, Dynamics and Nonlinearity, Mc Graw-Hill, Maidenhead.
80. O. C. Zienkiewicz, R. L. Taylor and J. M. Too (1971), Reduced integration technique in general analysis of plates and shells, International Journal for Numerical Methods in Engineering 3, 275-290.