Skip to main content
SpringerLink
Log in

A Comprehensive Comparative Review of Various Advanced Finite Elements to Alleviate Shear, Membrane and Volumetric Locking

  • Review article
  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Abstract

Finite element analysis (FEA) is an extensively exercised numerical procedure to address numerous problems in several engineering fields. However, the accuracy of conventional FEA solutions is significantly affected in specific circumstances where the problem demands near-incompressibility or incompressibility of domain or analysis of thin structural geometries. Over time, several advanced FE models are developed to improve the quality of solutions in stated situations. However, the extensive comparative aspects of these methods are spared limited attention. In the present paper, a comprehensive review and comparison of the selected FE models have been presented. The detailed implementation procedure, along with the relative efficacy of the methods, has been derived for selective reduced integration (SRI), enhanced assumed strain (EAS), assumed natural strain (ANS), and a specific class of hybrid stress elements alongside the conventional FE formulation. The quality of results is assessed by evaluating the relative error norms in displacement and stress on well-established benchmark numerical examples. Furthermore, the paper investigates the methods for several parameters that include the method’s best-suited environment, robustness, and efficiency. The findings in the paper provide an elaborate understanding of the optimal choice of the method in locking-dominated problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Algorithm 1
Fig. 2
Algorithm 2
Algorithm 3
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32

Similar content being viewed by others

References

  1. Love AEH (1888) The small free vibrations and deformation of a thin elastic shell. Philos Trans R Soc Lond (A) 179:491–546. https://doi.org/10.1098/rsta.1888.0016

    Article  Google Scholar 

  2. Adam C, Hughes TJR, Bouabdallah S et al (2015) Selective and reduced numerical integrations for NURBS-based isogeometric analysis. Comput Methods Appl Mech Eng 284:732–761. https://doi.org/10.1016/j.cma.2014.11.001

    Article  MathSciNet  Google Scholar 

  3. Agrawal M, Jog CS (2017) Monolithic formulation of electromechanical systems within the context of hybrid finite elements. Comput Mech 59(3):443–457. https://doi.org/10.1007/s00466-016-1356-1

    Article  MathSciNet  Google Scholar 

  4. Agrawal M, Nandy A, Jog CS (2019) A hybrid finite element formulation for large-deformation contact mechanics. Comput Methods Appl Mech Eng 356:407–434. https://doi.org/10.1016/j.cma.2019.07.017

    Article  MathSciNet  Google Scholar 

  5. Ahmad S, Irons BM, Zienkiewicz OC (1970) Analysis of thick and thin shell structures by curved finite elements. Int J Numer Methods Eng 2(3):419–451. https://doi.org/10.1002/nme.1620020310

    Article  Google Scholar 

  6. Ainsworth M, Parker C (2022) Unlocking the secrets of locking: Finite element analysis in planar linear elasticity. Comput Methods Appl Mech Eng 395(115):034. https://doi.org/10.1016/j.cma.2022.115034

    Article  MathSciNet  Google Scholar 

  7. Alves De Sousa RJ, Natal Jorge RM, Fontes Valente RA et al (2003) A new volumetric and shear locking-free 3D enhanced strain element. Eng Comput (Swansea, Wales) 20(7–8):896–925. https://doi.org/10.1108/02644400310502036

    Article  Google Scholar 

  8. Alves de Sousa RJ, Cardoso RP, Fontes Valente RA et al (2005) A new one-point quadrature enhanced assumed strain (EAS) solid-shell element with multiple integration points along thickness: Part I - Geometrically linear applications. Int J Numer Methods Eng 62(7):952–977. https://doi.org/10.1002/nme.1226

    Article  Google Scholar 

  9. Alves de Sousa RJ, Cardoso RP, Fontes Valente RA et al (2006) A new one-point quadrature enhanced assumed strain (EAS) solid-shell element with multiple integration points along thickness - Part II: Nonlinear applications. Int J Numer Methods Eng 67(2):160–188. https://doi.org/10.1002/nme.1609

    Article  Google Scholar 

  10. Ambati M, Kiendl J, De Lorenzis L (2018) Isogeometric Kirchhoff-Love shell formulation for elasto-plasticity. Comput Methods Appl Mech Eng 340:320–339. https://doi.org/10.1016/j.cma.2018.05.023

    Article  MathSciNet  Google Scholar 

  11. Andelfinger U, Ramm E (1993) EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements. Int J Numer Methods Eng 36(8):1311–1337. https://doi.org/10.1002/nme.1620360805

    Article  Google Scholar 

  12. Arnold DN (1981) Discretization by finite elements of a model parameter dependent problem. Numer Math 37(3):405–421. https://doi.org/10.1007/BF01400318

    Article  MathSciNet  Google Scholar 

  13. Arnold DN, Brezzi F, Fortin M (1984) A stable finite element for the stokes equations. Calcolo 21(4):337–344. https://doi.org/10.1007/BF02576171

    Article  MathSciNet  Google Scholar 

  14. Babuška I, Suri M (1992) Locking effects in the finite element approximation of elasticity problems. Numer Math 62(1):439–463. https://doi.org/10.1007/BF01396238

    Article  MathSciNet  Google Scholar 

  15. Babuska I, Suri M (1992) On locking and robustness in the finite element method. SIAM J Numer Anal 29(5):1261–1293. https://doi.org/10.1137/0729075

    Article  MathSciNet  Google Scholar 

  16. Beirão Da Veiga L, Hughes TJR, Kiendl J et al (2015) A locking-free model for Reissner-Mindlin plates: analysis and isogeometric implementation via NURBS and triangular NURPS. Math Models Methods Appl Sci 25(8):1519–1551. https://doi.org/10.1142/S0218202515500402

    Article  MathSciNet  Google Scholar 

  17. Belytschko T, Leviathan I (1994) Physical stabilization of the 4-node shell element with one point quadrature. Comput Methods Appl Mech Eng 113(3–4):321–350. https://doi.org/10.1016/0045-7825(94)90052-3

    Article  Google Scholar 

  18. Belytschko T, Tsay CS (1983) A stabilization procedure for the quadrilateral plate element with one-point quadrature. Int J Numer Methods Eng 19(3):405–419. https://doi.org/10.1002/nme.1620190308

    Article  Google Scholar 

  19. Belytschko T, Ong JSJ, Liu Wing Kam et al (1984) Hourglass control in linear and nonlinear problems. Comput Methods Appl Mech Eng 43(3):251–276. https://doi.org/10.1016/0045-7825(84)90067-7

    Article  Google Scholar 

  20. Benson D, Bazilevs Y, Hsu M et al (2010) Isogeometric shell analysis: the Reissner-Mindlin shell. Comput Methods Appl Mech Eng 199(5–8):276–289. https://doi.org/10.1016/j.cma.2009.05.011

    Article  MathSciNet  Google Scholar 

  21. Betsch P, Gruttmann F, Stein E (1996) A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains. Comput Methods Appl Mech Eng 130(1–2):57–79. https://doi.org/10.1016/0045-7825(95)00920-5

    Article  MathSciNet  Google Scholar 

  22. Bischoff M, Ramm E (1997) Shear deformable shell elements for large strains and rotations. Int J Numer Methods Eng 40(23):4427–4449

    Article  Google Scholar 

  23. Bischoff M, Ramm E, Braess D (1999) A class of equivalent enhanced assumed strain and hybrid stress finite elements. Comput Mech 22(6):443–449. https://doi.org/10.1007/s004660050378

    Article  Google Scholar 

  24. Bischoff M, Ramm E, Irslinger J (2017) Models and Finite Elements for Thin-Walled Structures. 1859, https://doi.org/10.1002/9781119176817.ecm2026

  25. Bombarde DS, Nandy A, Gautam SS (2021) A two-field formulation in isogeometric analysis to alleviate locking. In: Joshi P, Gupta SS, Shukla AK et al (eds) Advances in engineering design. Springer Singapore, Singapore, pp 191–199

    Chapter  Google Scholar 

  26. Bombarde DS, Agrawal M, Gautam SS et al (2022) Hellinger-Reissner principle based stress-displacement formulation for three-dimensional isogeometric analysis in linear elasticity. Comput Methods Appl Mech Eng 394:114920. https://doi.org/10.1016/j.cma.2022.114920

    Article  MathSciNet  Google Scholar 

  27. Bombarde DS, Agrawal M, Gautam SS et al (2022) A locking-free formulation for three-dimensional isogeometric analysis. Mater Today: Proc 66:1710–1715. https://doi.org/10.1016/j.matpr.2022.05.266

    Article  Google Scholar 

  28. Bombarde DS, Gautam SS, Nandy A (2022) A novel hybrid isogeometric element based on two-field Hellinger-Reissner principle to alleviate different types of lockingy. Sadhana 47:148. https://doi.org/10.1007/s12046-022-01867-6

    Article  Google Scholar 

  29. Bucalem ML, Bathe KJ (1993) Higher-order MITC general shell elements. Int J Numer Methods Eng 36(21):3729–3754. https://doi.org/10.1002/nme.1620362109

    Article  MathSciNet  Google Scholar 

  30. Buechter N, Ramm E (1992) Shell theory versus degeneration–a comparison in large rotation finite element analysis. Int J Numer Methods Eng 34(1):39–59. https://doi.org/10.1002/nme.1620340105

    Article  Google Scholar 

  31. Cardoso RPR, Cesar de Sa JMA (2012) The enhanced assumed strain method for the isogeometric analysis of nearly incompressible deformation of solids. Int J Numer Methods Eng 92(1):56–78. https://doi.org/10.1002/nme.4328

    Article  MathSciNet  Google Scholar 

  32. Cardoso RPR, Yoon JW, Mahardika M et al (2008) Enhanced assumed strain (EAS) and assumed natural strain (ANS) methods for one-point quadrature solid-shell elements. Int J Numer Methods Eng 75(2):156–187. https://doi.org/10.1002/nme.2250

    Article  MathSciNet  Google Scholar 

  33. Caseiro JF, Valente RAF, Reali A et al (2014) On the Assumed Natural Strain method to alleviate locking in solid-shell NURBS-based finite elements. Comput Mech 53(6):1341–1353. https://doi.org/10.1007/s00466-014-0978-4

    Article  MathSciNet  Google Scholar 

  34. Caseiro JF, Valente RA, Reali A et al (2015) Assumed natural strain NURBS-based solid-shell element for the analysis of large deformation elasto-plastic thin-shell structures. Comput Methods Appl Mech Eng 284:861–880. https://doi.org/10.1016/j.cma.2014.10.037

    Article  MathSciNet  Google Scholar 

  35. Chiumenti M, Valverde Q, Agelet De Saracibar C et al (2002) A stabilized formulation for incompressible elasticity using linear displacement and pressure interpolations. Comput Methods Appl Mech Eng 191(46):5253–5264. https://doi.org/10.1016/S0045-7825(02)00443-7

    Article  MathSciNet  Google Scholar 

  36. Cook RD (1974) Improved two-dimensional finite element. J Struct Div 100(9):1851–1863. https://doi.org/10.1061/JSDEAG.0003877

    Article  Google Scholar 

  37. Courant R (1943) Variational methods for the solution of problems of equilibrium and vibrations. Bull Am Math Soc 49(1):1–23. https://doi.org/10.1090/S0002-9904-1943-07818-4

    Article  MathSciNet  Google Scholar 

  38. De César Sá JM, Natal Jorge RM, Fontes Valente RA et al (2002) Development of shear locking-free shell elements using an enhanced assumed strain formulation. Int J Numer Methods Eng 53(7):1721–1750. https://doi.org/10.1002/nme.360

  39. De Souza Neto EA, Perić D, Dutko M et al (1996) Design of simple low order finite elements for large strain analysis of nearly incompressible solids. Int J Solids Struct 33(20–22):3277–3296. https://doi.org/10.1016/0020-7683(95)00259-6

    Article  MathSciNet  Google Scholar 

  40. de Souza Neto EA, Andrade Pires FM, Owen DR (2005) F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. Part I: Formulation and benchmarking. Int J Numer Methods Eng 62(3):353–383. https://doi.org/10.1002/nme.1187

    Article  Google Scholar 

  41. Dolbow J, Belytschko T (1999) Volumetric locking in the element free Galerkin method. Int J Numer Methods Eng 46(6):925–942

    Article  MathSciNet  Google Scholar 

  42. Dornisch W, Klinkel S, Simeon B (2013) Isogeometric Reissner-Mindlin shell analysis with exactly calculated director vectors. Comput Methods Appl Mech Eng 253:491–504. https://doi.org/10.1016/j.cma.2012.09.010

    Article  MathSciNet  Google Scholar 

  43. Duong TX, Roohbakhshan F, Sauer RA (2017) A new rotation-free isogeometric thin shell formulation and a corresponding continuity constraint for patch boundaries. Comput Methods Appl Mech Eng 316:43–83. https://doi.org/10.1016/j.cma.2016.04.008

    Article  MathSciNet  Google Scholar 

  44. Dvorkin EN, Bathe KJ (1984) A continuum mechanics based four-node shell element for general nonlinear analysis. Eng Comput 1(1):77–88. https://doi.org/10.1108/eb023562

    Article  Google Scholar 

  45. Echter R (2013) Isogeometric Analysis of Shells. PhD thesis, Universitat Stuttgart

  46. Echter R, Oesterle B, Bischoff M (2013) A hierarchic family of isogeometric shell finite elements. Comput Methods Appl Mech Eng 254:170–180. https://doi.org/10.1016/j.cma.2012.10.018

    Article  MathSciNet  Google Scholar 

  47. Elguedj T, Bazilevs Y, Calo VM et al (2008) B and F bar projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements. Comput Methods Appl Mech Eng 197(33–40):2732–2762. https://doi.org/10.1016/j.cma.2008.01.012

    Article  Google Scholar 

  48. Flanagan DP, Belytschko T (1981) A uniform strain hexahedron and quadrilateral with orthogonal hourglass control. Int J Numer Methods Eng 17(5):679–706. https://doi.org/10.1002/nme.1620170504

    Article  Google Scholar 

  49. Fontes Valente RA, Natal Jorge RM, Cardoso RP et al (2003) On the use of an enhanced transverse shear strain shell element for problems involving large rotations. Comput Mech 30(4):286–296. https://doi.org/10.1007/s00466-002-0388-x

    Article  Google Scholar 

  50. Franco Brezzi and Klaus-jurgen Bathe (1986) Studies of Finite Element Procedures The Inf-Sup Condition Equivalent, Forms and Applications. In: Reliability of Methods for Engineering Analysis

  51. Gmür TC, Schorderet AM (1993) A set of three-dimensional solid to shell transition elements for structural dynamics. Comput Struct 46(4):583–591. https://doi.org/10.1016/0045-7949(93)90387-S

    Article  Google Scholar 

  52. Greco L, Cuomo M (2016) An isogeometric implicit G1 mixed finite element for Kirchhoff space rods. Comput Methods Appl Mech Eng 298:325–349. https://doi.org/10.1016/j.cma.2015.06.014

    Article  Google Scholar 

  53. Griffiths DV, Mustoe GGW (1995) Selective reduced integration of four-node plane element in closed form. J Eng Mech 121(6):725–729. https://doi.org/10.1061/(ASCE)0733-9399(1995)121:6(725)

    Article  Google Scholar 

  54. Hauptmann R, Schweizerhof K (1998) A systematic development of ‘solid-shell’ element formulations for linear and non-linear analyses employing only displacement degrees of freedom. Int J Numer Methods Eng 42(1):49–69

    Article  Google Scholar 

  55. Herrmann LR (1965) Elasticity equations for incompressible and nearly incompressible materials by a variational theorem. AIAA J 3(10):1896–1900. https://doi.org/10.2514/3.3277

    Article  MathSciNet  Google Scholar 

  56. Hrennikoff A (1941) Solution of problems of elasticity by the framework method. J Appl Mech 8(4):A169–A175

    Article  MathSciNet  Google Scholar 

  57. Huerta A, Fernández-Méndez S (2001) Locking in the incompressible limit for the element-free Galerkin method. Int J Numer Methods Eng 51(11):1361–1383. https://doi.org/10.1002/nme.213

  58. Hughes TJR (1977) Equivalence of finite elements for nearly incompressible elasticity. J Appl Mech 44(1):181–183. https://doi.org/10.1115/1.3423994

    Article  Google Scholar 

  59. Hughes TJR (1980) Generalization of selective integration procedures to anisotropic and nonlinear media. Int J Numer Methods Eng 15(9):1413–1418. https://doi.org/10.1002/nme.1620150914

    Article  MathSciNet  Google Scholar 

  60. Hughes TJR (1987) The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, Englewood Cliffs, New Jersey

    Google Scholar 

  61. Hughes TJR, Liu WK (1981) Nonlinear finite element analysis of shells: Part I. three-dimensional shells. Comput Methods Appl Mech Eng 26(3):331–362. https://doi.org/10.1016/0045-7825(81)90121-3

    Article  Google Scholar 

  62. Hughes TJR, Tezduyar TE (1981) Finite elements based upon mindlin plate theory with particular reference to the four-node bilinear isoparametric element. J Appl Mech 48(3):587–596. https://doi.org/10.1115/1.3157679

    Article  Google Scholar 

  63. Hughes TJR, Tezduyar TE (1981) Finite elements based upon mindlin plate theory with particular reference to the four-node bilinear isoparametric element. J Appl Mech Trans ASME 48(3):587–596. https://doi.org/10.1115/1.3157679

    Article  Google Scholar 

  64. Hughes TJR, Taylor RL, Kanoknukulchai W (1977) A simple and efficient finite element for plate bending. Int J Numer Methods Eng 11(10):1529–1543. https://doi.org/10.1002/nme.1620111005

    Article  Google Scholar 

  65. Hughes TJR, Cohen M, Haroun M (1978) Reduced and selective integration techniques in the finite element analysis of plates. Nucl Eng Des 46(1):203–222. https://doi.org/10.1016/0029-5493(78)90184-X

    Article  Google Scholar 

  66. Jog C (2010) Improved hybrid elements for structural analysis. J Mech Mater Struct 5(3):507–528. https://doi.org/10.2140/jomms.2010.5.507

    Article  Google Scholar 

  67. Jog CS (2005) A 27-node hybrid brick and a 21-node hybrid wedge element for structural analysis. Finite Elem Anal Des 41(11–12):1209–1232. https://doi.org/10.1016/j.finel.2004.11.007

    Article  Google Scholar 

  68. Jog CS, Nandy A (2014) Mixed finite elements for electromagnetic analysis. Comput Math Appl 68(8):887–902. https://doi.org/10.1016/j.camwa.2014.08.006

    Article  MathSciNet  Google Scholar 

  69. Jog CS, Nandy A (2015) Conservation properties of the trapezoidal rule in linear time domain analysis of acoustics and structures. J Vib Acoust Trans ASME 137(2):021,010. https://doi.org/10.1115/1.4029075

    Article  Google Scholar 

  70. Kadapa C (2014) Mixed Galerkin and Least-Squares formulations for Isogeometric analysis. PhD thesis, Swansea University, https://doi.org/10.13140/2.1.1546.1442

  71. Kasper EP, Taylor RL (2000) Mixed-enhanced strain method. Part I: geometrically linear problems. Comput Struct 75(3):237–250. https://doi.org/10.1016/S0045-7949(99)00134-0

    Article  Google Scholar 

  72. Kiendl J, Bletzinger KU, Linhard J et al (2009) Isogeometric shell analysis with Kirchhoff-Love elements. Comput Methods Appl Mech Eng 198(49–52):3902–3914. https://doi.org/10.1016/j.cma.2009.08.013

    Article  MathSciNet  Google Scholar 

  73. Kiendl J, Hsu MC, Wu MC et al (2015) Isogeometric Kirchhoff-Love shell formulations for general hyperelastic materials. Comput Methods Appl Mech Eng 291:280–303. https://doi.org/10.1016/j.cma.2015.03.010

    Article  MathSciNet  Google Scholar 

  74. Kim JG, Kim YY (1998) A new higher-order hybrid-mixed curved beam element. Int J Numer Methods Eng 43(5):925–940

    Article  Google Scholar 

  75. Kim KD, Liu GZ, Han SC (2005) A resultant 8-node solid-shell element for geometrically nonlinear analysis. Comput Mech 35(5):315–331. https://doi.org/10.1007/s00466-004-0606-9

    Article  CAS  Google Scholar 

  76. Korelc J, Wriggers P (1996) An efficient 3D enhanced strain element with Taylor expansion of the shape functions. Comput Mech 19(1):30–40

    Article  Google Scholar 

  77. Korelc J, Šolinc U, Wriggers P (2010) An improved EAS brick element for finite deformation. Comput Mech 46(4):641–659. https://doi.org/10.1007/s00466-010-0506-0

    Article  Google Scholar 

  78. Lee Y, Yoon K, Lee PS (2012) Improving the MITC3 shell finite element by using the Hellinger-Reissner principle. Comput Struct 110–111:93–106. https://doi.org/10.1016/j.compstruc.2012.07.004

    Article  Google Scholar 

  79. Leonetti L, Magisano D, Madeo A et al (2019) A simplified Kirchhoff-Love large deformation model for elastic shells and its effective isogeometric formulation. Comput Methods Appl Mech Eng 354:369–396. https://doi.org/10.1016/j.cma.2019.05.025

    Article  MathSciNet  Google Scholar 

  80. Li Q, Liu Y, Zhang Z et al (2015) A new reduced integration solid-shell element based on EAS and ANS with hourglass stabilization. Int J Numer Methods Eng 104(8):805–826. https://doi.org/10.1002/nme.4958

    Article  MathSciNet  Google Scholar 

  81. Liu WK, Hu YK, Belytschko T (1994) Multiple quadrature underintegrated finite elements. Int J Numer Methods Eng 37(19):3263–3289. https://doi.org/10.1002/nme.1620371905

    Article  MathSciNet  Google Scholar 

  82. Liu WK, Li S, Park H (2021) Eighty Years of the Finite Element Method: Birth, Evolution, and Future, arXiv:2107.04960

  83. MacNeal RH (1993) Finite elements their design and performance. Marcel Dekke, New York

    Google Scholar 

  84. Magisano D, Leonetti L, Garcea G (2021) Isogeometric analysis of 3D beams for arbitrarily large rotations: locking-free and path-independent solution without displacement DOFs inside the patch. Comput Methods Appl Mech Eng 373(113):437. https://doi.org/10.1016/j.cma.2020.113437

    Article  MathSciNet  Google Scholar 

  85. Malkus DS, Hughes TJR (1978) Mixed finite element methods - reduced and selective integration techniques: a unification of concepts. Comput Methods Appl Mech Eng 15:63–81

    Article  Google Scholar 

  86. Marinković D, Köppe H, Gabbert U (2008) Degenerated shell element for geometrically nonlinear analysis of thin-walled piezoelectric active structures. Smart Mater Struct. https://doi.org/10.1088/0964-1726/17/01/015030

    Article  Google Scholar 

  87. McHenry D (1943) A lattice analogy for the solution of stress problem. J Inst Civil Eng 21(2):59–82. https://doi.org/10.1680/ijoti.1943.13967

    Article  Google Scholar 

  88. Militello C, Felippa CA (1990) A variational justification of the assumed natural strain formulation of finite elements-I. Variational principles. Comput Struct 34(3):431–438. https://doi.org/10.1016/0045-7949(90)90267-6

    Article  Google Scholar 

  89. Militello C, Felippa CA (1990) A variational justification of the assumed natural strain formulation of finite elements-II. The C0 four-node plate element. Comput Struct 34(3):439–444. https://doi.org/10.1016/0045-7949(90)90268-7

    Article  Google Scholar 

  90. Mindlin RD (1951) Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 18(1):31–38. https://doi.org/10.1115/1.4010217

    Article  Google Scholar 

  91. Nakshatrala KB, Masud A, Hjelmstad KD (2008) On finite element formulations for nearly incompressible linear elasticity. Comput Mech 41(4):547–561. https://doi.org/10.1007/s00466-007-0212-8

    Article  MathSciNet  Google Scholar 

  92. Narayan SL, Bombarde DS, Gautam SS et al (2023) Comparison of selective reduced integration, enhanced assumed strain and assumed natural strain formulation in alleviating locking. In: Manik G, Kalia S, Verma OP et al (eds) Recent advances in mechanical engineering. Springer Nature Singapore, Singapore, pp 643–654

    Chapter  Google Scholar 

  93. Naylor DJ (1974) Stresses in nearly incompressible materials by finite elements with application to the calculation of excess pore pressures. Int J Numer Methods Eng 8(3):443–460. https://doi.org/10.1002/nme.1620080302

    Article  Google Scholar 

  94. Nguyen NH (2009) Development of solid-shell elements for large deformation simulation and springback prediction. PhD thesis, University of Liège, Belgium

  95. Pastor M, Quecedo M, Zienkiewicz OC (1997) A mixed displacement-pressure formulation for numerical analysis of plastic failure. Comput Struct 62(1):13–23. https://doi.org/10.1016/S0045-7949(96)00208-8

    Article  Google Scholar 

  96. Pian THH, Sumihara K (1984) Rational approach for assumed stress finite elements. Int J Numer Methods Eng 20(9):1685–1695. https://doi.org/10.1002/nme.1620200911

    Article  Google Scholar 

  97. Piltner R (2000) An implementation of mixed enhanced finite elements with strains assumed in Cartesian and natural element coordinates using sparse B (overline)-matrices. Eng Comput 17(8):933–949. https://doi.org/10.1108/02644400010379776

    Article  Google Scholar 

  98. Prager W, Synge JL (1947) Approximations in elasticity based on the concept of function space. Q Appl Math 5(3):241–269

    Article  MathSciNet  Google Scholar 

  99. Prathap G (1993) The Finite Element Method in Structural Mechanics, Solid Mechanics and Its Applications,. Springer, Netherlands, Dordrecht. https://doi.org/10.1007/978-94-017-3319-9

    Book  Google Scholar 

  100. Prathap G, Bhashyam GR (1982) Reduced integration and the shear-flexible beam element. Int J Numer Methods Eng 18(2):195–210. https://doi.org/10.1002/nme.1620180205

    Article  Google Scholar 

  101. Pugh EDL, Hinton E, Zienkiewicz OC (1978) A study of quadrilateral plate bending elements with ‘reduced’ integration. Int J Numer Methods Eng 12(7):1059–1079. https://doi.org/10.1002/nme.1620120702

    Article  Google Scholar 

  102. Recio DP, Natal Jorge RM, Dinis LM (2006) Locking and hourglass phenomena in an element-free Galerkin context: the B-bar method with stabilization and an enhanced strain method. Int J Numer Methods Eng 68(13):1329–1357. https://doi.org/10.1002/nme.1741

    Article  Google Scholar 

  103. Reese S (2002) On the equivalent of mixed element formulations and the concept of reduced integration in large deformation problems. Int J Nonlinear Sci Numer Simul 3(1):1–34. https://doi.org/10.1515/IJNSNS.2002.3.1.1

    Article  MathSciNet  Google Scholar 

  104. Reese S (2006) A large deformation solid-shell concept based on reduced integration with hourglass stabilization. Int J Numer Methods Eng 69(July):1671–1716. https://doi.org/10.1002/nme. arXiv:1010.1724

    Article  MathSciNet  Google Scholar 

  105. Reese S (2007) A large deformation solid-shell concept based on reduced integration with hourglass stabilization. Int J Numer Methods Eng 69(8):1671–1716. https://doi.org/10.1002/nme.1827

    Article  MathSciNet  Google Scholar 

  106. Reissner E (1944) On the theory of bending of elastic plates. J Math Phys 23(1–4):184–191. https://doi.org/10.1002/sapm1944231184

    Article  MathSciNet  Google Scholar 

  107. Reissner E (1945) The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 12(2):A69–A77. https://doi.org/10.1115/1.4009435

    Article  MathSciNet  Google Scholar 

  108. Roehl D, Ramm E (1996) Large elasto-plastic finite element analysis of solids and shells with the enhanced assumed strain concept. Int J Solids Struct 33(20–22):3215–3237. https://doi.org/10.1016/0020-7683(95)00246-4

    Article  MathSciNet  Google Scholar 

  109. Roychowdhury A, Nandy A, Jog CS et al (2014) Hybrid elements for modelling squeeze film effects coupled with structural interactions in vibratory mems devices. Comput Model Eng Sci 103(2):91–110. https://doi.org/10.3970/cmes.2014.103.091

    Article  MathSciNet  Google Scholar 

  110. Schwarze M, Reese S (2009) A reduced integration solid-shell finite element based on the EAS and the ANS concept-Geometrically linear problems. Int J Numer Methods Eng 80(10):1322–1355. https://doi.org/10.1002/nme.2653

    Article  MathSciNet  Google Scholar 

  111. Schwarze M, Reese S (2011) A reduced integration solid-shell finite element based on the EAS and the ANS concept-Large deformation problems. Int J Numer Methods Eng 85(3):289–329. https://doi.org/10.1002/nme.2966

    Article  MathSciNet  Google Scholar 

  112. Scordelis C, Lo KS (1964) Computer analysis of cylindrical shells. ACI J Proc 61(5):539–562. https://doi.org/10.14359/7796

    Article  Google Scholar 

  113. Simo JC, Armero F (1992) Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int J Numer Methods Eng 33(7):1413–1449. https://doi.org/10.1002/nme.1620330705

    Article  Google Scholar 

  114. Simo JC, Hughes TJR (1986) On the variational foundations of assumed strain methods. J Appl Mech 53(1):51–54. https://doi.org/10.1115/1.3171737

    Article  MathSciNet  Google Scholar 

  115. Simo JC, Rifai MS (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29(8):1595–1638. https://doi.org/10.1002/nme.1620290802

    Article  MathSciNet  Google Scholar 

  116. Stolarski H, Belytschko T (1982) Membrane locking and reduced integration for curved elements. J Appl Mech 49(1):172–176

    Article  Google Scholar 

  117. Stolarski H, Belytschko T (1986) On the equivalence of mode decomposition and mixed finite elements based on the hellinger-reissner principle. part I: Theory. Comput Methods Appl Mech Eng 58(3):249–263. https://doi.org/10.1016/0045-7825(86)90149-0

    Article  Google Scholar 

  118. Sussman T, Bathe KJ (1987) A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Comput Struct 26(1–2):357–409. https://doi.org/10.1016/0045-7949(87)90265-3

    Article  Google Scholar 

  119. Sze KY (2000) On immunizing five-beta hybrid-stress element models from “trapezoidal locking” in practical analyses. Int J Numer Methods Eng 47(4):907–920

    Article  Google Scholar 

  120. Sze KY, Yao LQ (2000) A hybrid stress ANS solid-shell element and its generalization for smart structure modelling. Part I - Solid-shell element formulation. Int J Numer Methods Eng 48(4):545–564.

    Article  Google Scholar 

  121. Elguedj T, Bazilevs Y, V.M. Calo TJRH, (2008) F-bar projection method for finite deformation elasticity and plasticity using NURBS based isogeometric analysis T. Int J Mater Form 1:1091–1094. https://doi.org/10.1007/s12289-008-0

  122. Tan XG, Vu-Quoc L (2005) Efficient and accurate multilayer solid-shell element: Non-linear materials at finite strain. Int J Numer Methods Eng 63(15):2124–2170. https://doi.org/10.1002/nme.1360

    Article  Google Scholar 

  123. Taylor RL, Beresford PJ, Wilson EL (1976) A non-conforming element for stress analysis. Int J Numer Methods Eng 10(6):1211–1219. https://doi.org/10.1002/nme.1620100602

    Article  Google Scholar 

  124. Belytschko Ted, Wong Bak Leong, Chiang Huai-Yang (1992) Advances in one-point quadrature shell elements. Comput Methods Appl Mech Eng 96(1):93–107. https://doi.org/10.1016/0045-7825(92)90100-X

    Article  Google Scholar 

  125. Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates and shells, 2nd edn. McGraw-Hill, New York

    Google Scholar 

  126. Timoshenko SP, Goodier JN (2010) Theory of elasticity. Engineering societies monographs, McGraw-Hill Education (India) Pvt Limited

  127. Valente RA, De Sousa RJ, Jorge RM (2004) An enhanced strain 3D element for large deformation elastoplastic thin-shell applications. Comput Mech 34(1):38–52. https://doi.org/10.1007/s00466-004-0551-7

    Article  Google Scholar 

  128. Wriggers P, Korelc J (1996) On enhanced strain methods for small and finite deformations of solids. Comput Mech 18(6):413–428. https://doi.org/10.1007/BF00350250

    Article  Google Scholar 

  129. Yang HT, Saigal S, Liaw DG (1990) Advances of thin shell finite elements and some applications-version I. Comput Struct 35(4):481–504. https://doi.org/10.1016/0045-7949(90)90071-9

    Article  Google Scholar 

  130. Yang HT, Saigal S, Masud A et al (2000) A survey of recent shell finite elements. Int J Numer Methods Eng 47(1–3):101–127

    Article  MathSciNet  Google Scholar 

  131. Zienkiewicz OC (2001) Displacement and equilibrium models in the finite element method by B. Fraeijs de Veubeke, Chapter 9, Pages 145-197 of Stress Analysis, Edited by O. C. Zienkiewicz and G. S. Holister, Published by John Wiley & Sons, 1965. International Journal for Numerical Methods in Engineering 52(3):287–342. https://doi.org/10.1002/nme.339

  132. Zienkiewicz OC, Taylor RL (2000) Finite Element Method: Volume 1 - The Basis, 5th edn. Butterworth-Heinemann, Oxford, United Kingdom

  133. Zienkiewicz OC, Taylor RL, Too JM (1971) Reduced integration technique in general analysis of plates and shells. Int J Numer Methods Eng 3(2):275–290. https://doi.org/10.1002/nme.1620030211

    Article  Google Scholar 

  134. Zou Z, Scott MA, Miao D et al (2020) An isogeometric Reissner-Mindlin shell element based on Bézier dual basis functions: Overcoming locking and improved coarse mesh accuracy. Comput Methods Appl Mech Eng 370(113):283. https://doi.org/10.1016/j.cma.2020.113283

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the support from SERB, DST under the project IMP/2019/000276 and VSSC, ISRO through MoU No.: ISRO:2020:MOU:NO:480.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India

    Dhiraj S. Bombarde, Lakshmi Narayan Silla, Sachin S. Gautam & Arup Nandy

Authors
  1. Dhiraj S. Bombarde
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lakshmi Narayan Silla
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sachin S. Gautam
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Arup Nandy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dhiraj S. Bombarde.

Ethics declarations

Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bombarde, D.S., Silla, L., Gautam, S.S. et al. A Comprehensive Comparative Review of Various Advanced Finite Elements to Alleviate Shear, Membrane and Volumetric Locking. Arch Computat Methods Eng (2024). https://doi.org/10.1007/s11831-023-10050-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11831-023-10050-x

Search

Navigation