Archives of Computational Methods in Engineering

, Volume 26, Issue 4, pp 1117–1151 | Cite as

An Overview of Mixed Finite Elements for the Analysis of Inelastic Bidimensional Structures

  • Nicola A. NodargiEmail author
Original Paper


As inelastic structures are ubiquitous in many engineering fields, a central task in computational mechanics is to develop accurate, robust and efficient tools for their analysis. Motivated by the poor performances exhibited by standard displacement-based finite element formulations, attention is here focused on the use of mixed methods as approximation technique, within the small strain framework, for the mechanical problem of inelastic bidimensional structures. Despite a great flexibility characterizes mixed element formulations, several theoretical and numerical aspects have to be carefully taken into account in the design of a high-performance element. The present work aims at providing the basis for methodological analysis and comparison in such aspects, within the unified mathematical setting supplied by generalized standard material model and with special interest towards elastoplastic media. A critical review of the state-of-the-art computational methods is delivered in regard to variational formulations, selection of interpolation spaces, numerical solution strategies and numerical stability. Though those arguments are interrelated, a topic-oriented presentation is resorted to, for the very rich available literature to be properly examined. Finally, the performances of several significant mixed finite element formulations are investigated in numerical simulations.



The author expresses his sincere gratitude to Professor Paolo Bisegna for valuable comments and stimulating discussions on the present work.

Compliance with Ethical Standards

Conflict of interest

The author declares that he has no conflict of interest.


  1. 1.
    Allman DJ (1988) A quadrilateral finite element including vertex rotations for plane elasticity analysis. Int J Numer Methods Eng 26(3):717–730. CrossRefzbMATHGoogle Scholar
  2. 2.
    Aminpour MA (1992) An assumed-stress hybrid 4-node shell element with drilling degrees of freedom. Int J Numer Methods Eng 33(1):19–38. MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Armero F (2004) Elastoplastic and viscoplastic deformations in solids and structures. In: Stein E, de Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics, vol 2. Wiley, Chichester, pp 227–266. CrossRefGoogle Scholar
  4. 4.
    Arnold DN, Brezzi F, Fortin M (1984) A stable finite element for the stokes equations. Calcolo 21(4):337–344. MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arnold DN, Boffi D, Falk RS, Gastaldi L (2001) Finite element approximation on quadrilateral meshes. Commun Numer Methods Eng 17(11):805–812. MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Arnold DN, Boffi D, Falk RS (2002) Approximation by quadrilateral finite elements. Math Comput 71(239):909–922. MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bathe KJ (1996) Finite element procedures. Prentice Hall, Englewood CliffszbMATHGoogle Scholar
  8. 8.
    Bathe KJ (2001) The inf–sup condition and its evaluation for mixed finite element methods. Comput Struct 79(2):243–252. MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bergmann VL, Mukherjee S (1990) A hybrid strain finite element for plates and shells. Int J Numer Methods Eng 30(2):233–257. CrossRefzbMATHGoogle Scholar
  10. 10.
    Bilotta A, Casciaro R (2002) Assumed stress formulation of high order quadrilateral elements with an improved in-plane bending behaviour. Comput Methods Appl Mech Eng 191(15–16):1523–1540. CrossRefzbMATHGoogle Scholar
  11. 11.
    Bilotta A, Casciaro R (2007) A high-performance element for the analysis of 2D elastoplastic continua. Comput Methods Appl Mech Eng 196(4–6):818–828. CrossRefzbMATHGoogle Scholar
  12. 12.
    Bilotta A, Leonetti L, Garcea G (2011) Three field finite elements for the elastoplastic analysis of 2D continua. Finite Elem Anal Des 47(10):1119–1130. MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bilotta A, Leonetti L, Garcea G (2012) An algorithm for incremental elastoplastic analysis using equality constrained sequential quadratic programming. Comput Struct 102–103:97–107. CrossRefGoogle Scholar
  14. 14.
    Bilotta A, Garcea G, Leonetti L (2016) A composite mixed finite element model for the elasto-plastic analysis of 3D structural problems. Finite Elem Anal Des 113:43–53. CrossRefGoogle Scholar
  15. 15.
    Boffi D, Brezzi F, Fortin M (2013) Mixed finite element methods and applications, Springer series in computational mathematics, vol 44. Springer, BerlinCrossRefGoogle Scholar
  16. 16.
    Bolzon G (2017) Complementarity problems in structural engineering: an overview. Arch Comput Methods Eng 24(1):23–36. MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Borja RI (2013) Plasticity: modeling and computation. Springer, Berlin. CrossRefzbMATHGoogle Scholar
  18. 18.
    Capsoni A, Corradi L (1997) A mixed finite element model for plane strain elastic–plastic analysis, part I. Formulation and assessment of the overall behaviour. Comput Methods Appl Mech Eng 141(1–2):67–79. CrossRefzbMATHGoogle Scholar
  19. 19.
    Caylak I, Mahnken R (2014) Stabilized mixed triangular elements with area bubble functions at small and large deformations. Comput Struct 138(1):172–182. CrossRefGoogle Scholar
  20. 20.
    Cen S, Fu XR, Zhou MJ (2011) 8- and 12-node plane hybrid stress-function elements immune to severely distorted mesh containing elements with concave shapes. Comput Methods Appl Mech Eng 200(29–32):2321–2336. MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cervera M, Chiumenti M, Codina R (2010) Mixed stabilized finite element methods in nonlinear solid mechanics: part I: formulation. Comput Methods Appl Mech Eng 199(37–40):2559–2570. MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Cervera M, Chiumenti M, Codina R (2015) Mixed stabilized finite element methods in nonlinear solid mechanics: part III: compressible and incompressible plasticity. Comput Methods Appl Mech Eng 285(1):752–775. MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Cervera M, Lafontaine N, Rossi R, Chiumenti M (2016) Explicit mixed strain–displacement finite elements for compressible and quasi-incompressible elasticity and plasticity. Comput Mech 58(3):511–532. MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Chapelle D, Bathe KJ (1993) The inf–sup test. Comput Struct 47(4–5):537–545. MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Choi N, Choo YS, Lee BC (2006) A hybrid Trefftz plane elasticity element with drilling degrees of freedom. Comput Methods Appl Mech Eng 195(33–36):4095–4105. CrossRefzbMATHGoogle Scholar
  26. 26.
    Choo YS, Choi N, Lee BC (2006) Quadrilateral and triangular plane elements with rotational degrees of freedom based on the hybrid Trefftz method. Finite Elem Anal Des 42(11):1002–1008. CrossRefGoogle Scholar
  27. 27.
    Comi C, Perego U (1995) A unified approach for variationally consistent finite elements in elastoplasticity. Comput Methods Appl Mech Eng 121(1–4):323–344. MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Contrafatto L, Ventura G (2004) Numerical analysis of augmented Lagrangian algorithms in complementary elastoplasticity. Int J Numer Methods Eng 60(14):2263–2287. CrossRefzbMATHGoogle Scholar
  29. 29.
    Cook RD, Malkus DS, Plesha ME, Witt RJ (2001) Concepts and applications of finite element analysis, 4th edn. Wiley, New YorkGoogle Scholar
  30. 30.
    Darilmaz K, Kumbasar N (2006) An 8-node assumed stress hybrid element for analysis of shells. Comput Struct 84:1990–2000. CrossRefGoogle Scholar
  31. 31.
    Eve RA, Reddy BD, Rockafellar RT (1990) An internal variable theory of plasticity based on the maximum plastic work inequality. Q Appl Math 48:59–83CrossRefGoogle Scholar
  32. 32.
    Felippa CA (2011) Introduction to finite element methods. University of Colorado at Boulder. d/IFEM.d
  33. 33.
    Germain P, Nguyen QS, Suquet P (1983) Continuum thermodynamics. J Appl Mech-Trans ASME 50(4b):1010–1020. CrossRefzbMATHGoogle Scholar
  34. 34.
    Goldfarb D, Idnani A (1983) A numerically stable dual method for solving strictly convex quadratic programs. Math Program 27(1):1–33. MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Halphen B, Nguyen QS (1975) Sur les matériaux standards généralisés. J Méc 14:39–63zbMATHGoogle Scholar
  36. 36.
    Han W, Reddy BD (1999) Plasticity: mathematical theory and numerical analysis. Springer, New York. CrossRefzbMATHGoogle Scholar
  37. 37.
    Hill R (1950) The mathematical theory of plasticity. Oxford University Press, OxfordzbMATHGoogle Scholar
  38. 38.
    Hueck U, Reddy BD, Wriggers P (1994) On the stabilization of the rectangular 4-node quadrilateral element. Int J Numer Methods Biomed 10(7):555–563. CrossRefzbMATHGoogle Scholar
  39. 39.
    Hughes TJR (1980) Generalization of selective integration procedures to anisotropic and nonlinear media. Int J Numer Methods Eng 15(9):1413–1418. MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Hughes TJR, Brezzi F (1989) On drilling degrees of freedom. Comput Methods Appl Mech Eng 72(1):105–121. MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Hughes TJR, Franca LP (1987) A new finite element formulation for computational fluid dynamics: VII. The stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces. Comput Methods Appl Mech Eng 65(1):85–96. MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ibrahimbegovic A (1990) A novel membrane finite element with an enhanced displacement interpolation. Finite Elem Anal Des 7(2):167–179. MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Ibrahimbegovic A, Taylor RL, Wilson EL (1990) A robust quadrilateral membrane finite element with drilling degrees of freedom. Int J Numer Methods Eng 30(3):445–457. CrossRefzbMATHGoogle Scholar
  44. 44.
    Irons BM (1966) Engineering applications of numerical integration in stiffness methods. AIAA J 4(11):2035–2037. CrossRefzbMATHGoogle Scholar
  45. 45.
    Karaoulanis FE (2013) Implicit numerical integration of nonsmooth multisurface yield criteria in the principal stress space. Arch Comput Methods Eng 20(3):263–308. MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Kasper EP, Taylor RL (2000) A mixed-enhanced strain method: part I: geometrically linear problems. Comput Struct 75(3):237–250. CrossRefGoogle Scholar
  47. 47.
    Koiter WT (1960) General theorems for elastic–plastic solids, progress in solid mechanics, vol 6. North-Holland, AmsterdamGoogle Scholar
  48. 48.
    Krabbenhoft K, Lyamin AV, Sloan SW, Wriggers P (2007) An interior-point algorithm for elastoplasticity. Int J Numer Methods Eng 69(3):592–626. MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Leonetti L, Aristodemo M (2015) A composite mixed finite element model for plane structural problems. Finite Elem Anal Des 94:33–46. CrossRefGoogle Scholar
  50. 50.
    Madeo A, Zagari G, Casciaro R (2012) An isostatic quadrilateral membrane finite element with drilling rotations and no spurious modes. Finite Elem Anal Des 50:21–32. CrossRefGoogle Scholar
  51. 51.
    Madeo A, Casciaro R, Zagari G, Zinno R, Zucco G (2014) A mixed isostatic 16 dof quadrilateral membrane element with drilling rotations, based on airy stresses. Finite Elem Anal Des 89:52–66. MathSciNetCrossRefGoogle Scholar
  52. 52.
    Mahnken R, Caylak I, Laschet G (2008) Two mixed finite element formulations with area bubble functions for tetrahedral elements. Comput Methods Appl Mech Eng 197(9–12):1147–1165. MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Maier G (1968) Quadratic programming and theory of elastic-perfectly plastic structures. Meccanica 3(4):265–273. MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Maier G (1969) Shakedown theory in perfect elastoplasticity with associated and nonassociated flow-laws: a finite element, linear programming approach. Meccanica 4(3):250–260. CrossRefzbMATHGoogle Scholar
  55. 55.
    Melosh RJ (1963) Basis for derivation of matrices for the direct stiffness method. AIAA J 1(7):1631–1637. CrossRefGoogle Scholar
  56. 56.
    Mendes LAM, Castro LMSS (2009) Hybrid-mixed stress finite element models in elastoplastic analysis. Finite Elem Anal Des 45(12):863–875. MathSciNetCrossRefGoogle Scholar
  57. 57.
    Michel JC, Suquet P (2003) Nonuniform transformation field analysis. Int J Solids Struct 40(25):6937–6955. MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Miehe C, Schotte J, Lambrecht M (2002) Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to texture analysis of polycrystals. J Mech Phys Solids 50(10):2123–2167. MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Mielke A (2004) Existence of minimizers in incremental elasto-plasticity with finite strains. SIAM J Math Anal 36(2):384–404. MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Moharrami H, Mahini MR, Cocchetti G (2015) Elastoplastic analysis of plane stress/strain structures via restricted basis linear programming. Comput Struct 146:1–11. CrossRefGoogle Scholar
  61. 61.
    Mosler J (2010) Variationally consistent modeling of finite strain plasticity theory with non-linear kinematic hardening. Comput Methods Appl Mech Eng 199(45–48):2753–2764. MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Neuenhofer A, Filippou FC (1997) Evaluation of nonlinear frame finite-element models. J Struct Eng 123(7):958–966. CrossRefGoogle Scholar
  63. 63.
    Nocedal J, Wright S (2006) Numerical optimization. Springer, New York. CrossRefzbMATHGoogle Scholar
  64. 64.
    Nodargi NA, Bisegna P (2015a) Mixed tetrahedral elements for the analysis of structures with material and geometric nonlinearities. Proc Appl Math Mech 15(1):219–220. CrossRefGoogle Scholar
  65. 65.
    Nodargi NA, Bisegna P (2015b) State update algorithm for isotropic elastoplasticity by incremental energy minimization. Int J Numer Methods Eng 105(3):163–196. MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Nodargi NA, Bisegna P (2017) A novel high-performance mixed membrane finite element for the analysis of inelastic structures. Comput Struct 182:337–353. CrossRefGoogle Scholar
  67. 67.
    Nodargi NA, Artioli E, Caselli F, Bisegna P (2014) State update algorithm for associative elastic–plastic pressure-insensitive materials by incremental energy minimization. Fract Struct Integr 29:111–127. CrossRefGoogle Scholar
  68. 68.
    Nodargi NA, Caselli F, Artioli E, Bisegna P (2016) A mixed tetrahedral element with nodal rotations for large-displacement analysis of inelastic structures. Int J Numer Methods Eng. MathSciNetCrossRefGoogle Scholar
  69. 69.
    Petryk H (2003) Incremental energy minimization in dissipative solids. C R Mec 331(7):469–474. CrossRefzbMATHGoogle Scholar
  70. 70.
    Pian THH (1964) Derivation of element stiffness matrices by assumed stress distributions derivation of element stiffness matrices by assumed stress distributions. AIAA J 2(7):1333–1336. CrossRefGoogle Scholar
  71. 71.
    Pian THH (1995) State-of-the-art development of hybrid/mixed finite element method. Finite Elem Anal Des 21(1–2):5–20. MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Pian THH (2000) Some notes on the early history of hybrid stress finite element method. Int J Numer Methods Eng 47(1–3):419–425.<419::AID-NME778>3.0.CO;2-#Google Scholar
  73. 73.
    Pian THH, Sumihara K (1984) Rational approach for assumed stress finite elements. Int J Numer Methods Eng 20(9):1685–1695. CrossRefzbMATHGoogle Scholar
  74. 74.
    Piltner R (2000) An alternative version of the pian-sumihara element with a simple extension to non-linear problems. Comput Mech 26(5):483–489. CrossRefzbMATHGoogle Scholar
  75. 75.
    Piltner R, Taylor RL (1995) A quadrilateral mixed finite element with two enhanced strain modes. Int J Numer Methods Eng 38(11):1783–1808. MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Piltner R, Taylor RL (1999) A systematic construction of B-bar functions for linear and non-linear mixed-enhanced finite elements for plane elasticity problems. Int J Numer Methods Eng 44(5):615–639.<615::AID-NME518>3.0.CO;2-UMathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Pimpinelli G (2004) An assumed strain quadrilateral element with drilling degrees of freedom. Finite Elem Anal Des 41(3):267–283. CrossRefGoogle Scholar
  78. 78.
    Pinsky PM (1987) A finite element formulation for elastoplasticity based on a three-field variational equation. Comput Methods Appl Mech Eng 61(1):41–60. CrossRefzbMATHGoogle Scholar
  79. 79.
    Rebiai C, Belounarb L (2014) An effective quadrilateral membrane finite element based on the strain approach. Measurement 50:263–269. CrossRefGoogle Scholar
  80. 80.
    Reddy BD, Martin JB (1991) Algorithms for the solution of internal variable problems in plasticity. Comput Methods Appl Mech Eng 93(2):253–273. MathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Rezaiee-Pajand M, Karkon M (2013) An effective membrane element based on analytical solution. Eur J Mech A Solids 39:268–279. MathSciNetCrossRefzbMATHGoogle Scholar
  82. 82.
    Rockafellar RT (1970) Convex analysis. Princeton University Press, PrincetonCrossRefGoogle Scholar
  83. 83.
    Saritas A, Soydas O (2012) Variational base and solution strategies for non-linear force-based beam finite elements. Int J Non-Linear Mech 47(3):54–64. CrossRefGoogle Scholar
  84. 84.
    Scalet G, Auricchio F (2017) Computational methods for elastoplasticity: an overview of conventional and less-conventional approaches. Arch Comput Methods Eng. CrossRefzbMATHGoogle Scholar
  85. 85.
    Schröder J, Klaas O, Stein E, Miehe C (1997) A physically nonlinear dual mixed finite element formulation. Comput Methods Appl Mech Eng 144(1–2):77–92. CrossRefzbMATHGoogle Scholar
  86. 86.
    Schröder J, Igelbüscher M, Schwarz A, Starke G (2017) A Prange–Hellinger–Reissner type finite element formulation for small strain elasto-plasticity. Comput Methods Appl Mech Eng 317:400–418. MathSciNetCrossRefGoogle Scholar
  87. 87.
    Simo JC, Hughes TJR (1986) On the variational foundations of assumed strain methods. J Appl Mech-Trans ASME 53(1):51–54. MathSciNetCrossRefzbMATHGoogle Scholar
  88. 88.
    Simo JC, Hughes TJR (1998) Computation inelasticity. Springer, New YorkzbMATHGoogle Scholar
  89. 89.
    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. MathSciNetCrossRefzbMATHGoogle Scholar
  90. 90.
    Simo JC, Kennedy JG, Govindjee S (1988) Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms. Int J Numer Methods Eng 26(10):2161–2185. CrossRefzbMATHGoogle Scholar
  91. 91.
    Simo JC, Kennedy JG, Taylor RL (1989) Complementary mixed finite element formulations for elastoplasticity. Comput Methods Appl Mech Eng 74(2):177–206. MathSciNetCrossRefzbMATHGoogle Scholar
  92. 92.
    de Souza Neto EA, Perić D, Owen DRJ (2008) Computational methods for plasticity: theory and applications. Wiley, Chichester. CrossRefGoogle Scholar
  93. 93.
    Spacone E, Filippou FC, Taucer FF (1996) Fibre beam-column model for non-linear analysis of R/C frames: part I. Formulation. Earthq Eng Struct Dyn 25(7):711–725.<711::AID-EQE576>3.0.CO;2-9CrossRefGoogle Scholar
  94. 94.
    Taylor RL (2000) A mixed-enhanced formulation tetrahedral finite elements. Int J Numer Methods Eng 47(1–3):205–227.<205::AID-NME768>3.0.CO;2-JMathSciNetCrossRefzbMATHGoogle Scholar
  95. 95.
    Taylor RL, Beresford PJ, Wilson EL (1976) A non-conforming element for stress analysis. Int J Numer Methods Eng 10(6):1211–1219. CrossRefzbMATHGoogle Scholar
  96. 96.
    Taylor RL, Simo JC, Zienkiewicz OC, Chan ACH (1986) The patch test—a condition for assessing FEM convergence. Int J Numer Methods Eng 22(1):39–62. MathSciNetCrossRefzbMATHGoogle Scholar
  97. 97.
    Taylor RL, Filippou FC, Saritas A, Auricchio F (2003) A mixed finite element method for beam and frame problems. Comput Mech 31(1):192–203. CrossRefzbMATHGoogle Scholar
  98. 98.
    Turner MJ, Clough RW, Martin HC, Topp LJ (1956) Stiffness and deflection analysis of complex structures. J Aerosp Sci 23(9):805–823. CrossRefzbMATHGoogle Scholar
  99. 99.
    Washizu K (1982) Variational methods in elasticity and plasticity, 3rd edn. Pergamon Press, OxfordzbMATHGoogle Scholar
  100. 100.
    Weissman SL, Jamjian M (1993) Two-dimensional elastoplasticity: approximation by mixed finite elements. Int J Numer Methods Eng 36(21):3703–3727. CrossRefzbMATHGoogle Scholar
  101. 101.
    Wilkins ML (1964) Calculation of elastic–plastic flow. In: Alder B, Fernbach S, Rotenberg M (eds) Methods in computational physics, vol 3. Academic Press, New York, pp 211–263Google Scholar
  102. 102.
    Wilson EL (1963) Finite element analysis of two-dimensional structures. PhD thesis, Department of Civil Engineering, University of California at BerkeleyGoogle Scholar
  103. 103.
    Wilson EL, Taylor RL, Doherty WP, Ghaboussi J (1973) Incompatible displacement models. In: Fenves SJ (ed) Numerical and computer methods in structural mechanics. Academic Press, New York, p 43Google Scholar
  104. 104.
    Wisniewski K, Turska E (2009) Improved 4-node Hu–Washizu elements based on skew coordinates. Comput Struct 87(7–8):407–424. CrossRefGoogle Scholar
  105. 105.
    Xie X, Zhou T (2006) Accurate 4-node quadrilateral elements with a new version of energy-compatible stress mode. Int J Numer Methods Biomed 24(2):125–139. MathSciNetCrossRefzbMATHGoogle Scholar
  106. 106.
    Yunus SM, Saigal S, Cook RD (1989) On improved hybrid finite elements with rotational degrees of freedom. Int J Numer Methods Eng 28(4):785–800. CrossRefGoogle Scholar
  107. 107.
    Zienkiewicz OC, Irons BM, Ergatoudis J, Ahmad S, Scott FC (1969) Iso-parametric and associate element families for two- and three-dimensional analysis. In: Holland I, Bell K (eds) Finite element methods for stress analysis. Tapir, TrondheimGoogle Scholar
  108. 108.
    Zienkiewicz OC, Taylor RL, Zhu JZ (2013) The finite element method. Its basis and fundamentals, 7th edn. Butterworth-Heinemann, OxfordzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Civil Engineering and Computer ScienceUniversity of Rome Tor VergataRomeItaly

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