Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Mixed Variational Methods: Considerations on Numerical Applications

  • Massimo CuomoEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_181-1



Mixed variational principles are multi-field principles that involve all or some of the field variables of the physical model examined. The solution of the problem is characterized as an optimum point of the functional. Mixed methods are largely used in computational mechanics for obtaining accurate solutions and for removing locking and instability phenomena.


Irreducible formulations in solid mechanics are the most commonly employed, both for continuum and discrete models. However irreducible formulations require some of the field equations to be exactly satisfied, and this poses very strong conditions on the functional spaces where the field variables are defined. These are particularly severe when numerical solutions based on interpolations are used. The inability of the interpolating spaces to exactly fulfill some of the field equations gives rise to the phenomenon of locking. Mixed formulations turn out to...

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


  1. Adam C, Hughes T, Bouabdallah S, Zarroug M, Maitournam H (2015) Selective and reduced numerical integrations for nurbs-based isogeometric analysis. Comput Methods Appl Mech Eng 284:732–761MathSciNetCrossRefGoogle Scholar
  2. Amanatidou E, Aravas N (2002) Mixed finite element formulations of strain-gradient elasticity problems. Comput Methods Appl Mech Eng 191:1723–1751CrossRefGoogle Scholar
  3. Andelfinger E, 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 36:1311–1337CrossRefGoogle Scholar
  4. Antolin P, Bressan A, Buffa A, Sangalli G (2017) An isogeometric method for linear nearly-incompressible elasticity with local stress projection. Comput Methods Appl Mech Eng 316:694–719MathSciNetCrossRefGoogle Scholar
  5. Armero F, Valverde J (2012) Invariant Hermitian finite element for thin Kirchhoff rods. I: the linear plane case. Comput Methods Appl Mech Eng 213–216:427–457MathSciNetCrossRefGoogle Scholar
  6. Auricchio F, Brezzi F, Lovadina C (2004) Mixed finite elements method. In: Stein E, de Borst R, Hughes T (eds) Encyclopedia of computational mechanics. Wiley, ChichesterGoogle Scholar
  7. Babuska I, Gatica G (2003) On the mixed finite element method with lagrange multipliers. Numer Methods Partial Differential Equ 19(2):192–210MathSciNetCrossRefGoogle Scholar
  8. Babuska I, Suri M (1992) On locking and robustness in the finite element method. SIAM J Numer Anal 29:1261–1293MathSciNetCrossRefGoogle Scholar
  9. Bao W, Wang X, Bathe K (2001) On the Inf–Sup condition of mixed finite element formulations for acoustic fluids. Math Models Methods Appl Sci 11:883–901MathSciNetCrossRefGoogle Scholar
  10. Bathe K (1996) Finite element procedures. Prentice Hall, LondonzbMATHGoogle Scholar
  11. Bathe K (2001) The Inf-Sup condition and its evaluation for mixed finite element methods. Comput Struct 79:243–252MathSciNetCrossRefGoogle Scholar
  12. Bathe K, Dvorkin E (1985) A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. Int J Numer Methods Eng 21:367–383CrossRefGoogle Scholar
  13. Bathe K, Dvorkin E (1986) A formulation of general shell elements? The use of mixed interpolation of tensorial components. Int J Numer Methods Eng 22:697–722CrossRefGoogle Scholar
  14. Bauer A, Breitenberger M, Philipp B, Wüchner R, Bletzinger k (2016) Nonlinear isogeometric spatial bernoulli beam. Comput Methods Appl Mech Eng 198:101–127MathSciNetCrossRefGoogle Scholar
  15. Bischoff M, Ramm E (1997) Shear deformable shell elements for large strains and rotations. Int J Numer Methods Eng 40:4427–4449CrossRefGoogle Scholar
  16. Bletzinger K, Bischoff M, Ramm E (2000) A unified approach for shear-locking-free triangular and rectangular shell finite elements. Comput Struct 75(3):321–334CrossRefGoogle Scholar
  17. Borja R (2008) Assumed enhanced strain and the extended finite element methods: a unification of concepts. Comput Methods Appl Mech Eng 197:2789—2803MathSciNetCrossRefGoogle Scholar
  18. Bouclier R, Elguedj T, Coumbescure A (2012) Locking free isogeometric formulations of curved thick beams. Comput Methods Appl Mech Eng 245–246: 144–162MathSciNetCrossRefGoogle Scholar
  19. Bouclier R, Elguedj T, Coumbescure A (2013) Efficient isogeometric NURBS-based solid-shell elements: mixed formulation and \(\bar {B}\)-method. Comput Methods Appl Mech Eng 267:86–110MathSciNetCrossRefGoogle Scholar
  20. Bouclier R, Elguedj T, Coumbescure A (2015) An isogeometric locking-free nurbs-based solid-shell element for geometrically nonlinear analysis. Int J Numer Methods Eng 101:774–808MathSciNetCrossRefGoogle Scholar
  21. Boutin C, dell’Isola F, Giorgio I, Placidi L (2017) Linear pantographic sheets: asymptotic micro-macro models identification. Math Mech Complex Syst 5:127–162MathSciNetCrossRefGoogle Scholar
  22. Brezzi F, Bathe K (1990) A discourse on the stability conditions for mixed finite element formulations. Comput Methods Appl Mech Eng 82:27–57MathSciNetCrossRefGoogle Scholar
  23. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, New YorkCrossRefGoogle Scholar
  24. Caseiro J, Valente R, Reali A, Kiendl J, Auricchio F, Alves de Sousa R (2014) On the assumed natural strain method to alleviate locking in solid-shell NURBS-based finite elements. Comput Mech 53(6):1341–1353MathSciNetCrossRefGoogle Scholar
  25. Chapelle D, Bathe K (1993) The inf–sup test. Comput Struct 47:537–545MathSciNetCrossRefGoogle Scholar
  26. Chapelle D, Bathe K (2003) The finite element analysis of shells? Fundamentals. Springer, Berlin/Heidelberg/NewYorkCrossRefGoogle Scholar
  27. Contrafatto L, Cuomo M (2002) A new thermodynamically consistent continuum model for hardening plasticity coupled with damage. Int J Solids Struct 39:6241–6271CrossRefGoogle Scholar
  28. Cuomo M (2017) Forms of the dissipation function for a class of viscoplastic models. Math Mech Complex Syst 5(3–4):217–237MathSciNetCrossRefGoogle Scholar
  29. Cuomo M, Contrafatto L (2000) Stress rate formulation for elastoplastic models with internal variables based on augmented Lagrangian regularisation. Int J Solids Struct 37:3935–3964CrossRefGoogle Scholar
  30. Cuomo M, Ventura G (1998) Complementary energy approach to contact problems based on augmented lagrangian regularization. Math Comput Model 28(4–8):185–204CrossRefGoogle Scholar
  31. dell’Isola F, Giorgio I, Pawlikowski M, Rizzi N (2016) Large deformations of planar extensible beams and pantographic lattices: heuristic homogenisation, experimental and numerical examples of equilibrium. Proc R Soc Lond A 472.  https://doi.org/10.1098/rspa.2015.0790 CrossRefGoogle Scholar
  32. Echter R, Bischoff M (2010) Numerical efficiency, locking and unlocking of NURBS finite elements. Comput Methods Appl Mech Eng 199(5–8):374–382CrossRefGoogle Scholar
  33. Elguedj T, Bazilevs Y, Calo V, Hughes T (2007) B-bar an F-bar projection methods for nearly incompressible linear and nonlinear elasticity and plasticity using higher order NURBS element. Comput Methods Appl Mech Eng 197:5257–5296Google Scholar
  34. Felippa C (1994) A survey of parametrized variational principles and applications to computational mechanics. Comput Methods Appl Mech Eng 113:109–139MathSciNetCrossRefGoogle Scholar
  35. Gao D (1999) General analytic solutions and complementary variational principles for large deformation nonsmooth mechanics. Meccanica 34:169–198MathSciNetzbMATHGoogle Scholar
  36. Giorgio I, Della Corte A, dell’Isola F, Steigmann D (2016) Buckling modes in pantographic lattices. C R Meécanique 344:487–501CrossRefGoogle Scholar
  37. Giorgio I, Andreaus U, dell’Isola F, Lekszycki T (2017) Viscous second gradient porous materials for bones reconstructed with bio-resorbable grafts. Extreme Mech Lett 13:141–147CrossRefGoogle Scholar
  38. Greco L, Cuomo M (2016) An isogeometric implicit G 1 mixed finite element for Kirchhoff space rods. Comput Methods Appl Mech Eng 298:325–349MathSciNetCrossRefGoogle Scholar
  39. Greco L, Cuomo M, Contrafatto L, Gazzo S (2017) An efficient blended mixed b-spline formulation for removing membrane locking in plane curved Kirchhoff rods. Comput Methods Appl Mech Eng 324:476–511MathSciNetCrossRefGoogle Scholar
  40. Greco L, Cuomo M, Contrafatto L (2018) A reconstructed local \(\bar {B}\) formulation for isogeometric Kirchhoff–Love shells. Comput Methods Appl Mech Eng 332: 462–467MathSciNetCrossRefGoogle Scholar
  41. Hu P, Hu Q, Xia Y (2016) Order reduction method for locking free isogeometric analysis of Timoshenko beams. Comput Methods Appl Mech Eng 308:1–22MathSciNetCrossRefGoogle Scholar
  42. Hughes T, Cohen M, Haroun M (1978) Reduced and selective integration techniques in the finite element analysis of plates. Nucl Eng Des 46(1):203–222CrossRefGoogle Scholar
  43. Hughes T, Cottrell A, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195MathSciNetCrossRefGoogle Scholar
  44. Iosilevic A, Bathe K, Brezzi F (1997) The inf–sup condition and its evaluation for mixed finite element methods. Int J Numer Methods Eng 40:3639–3663CrossRefGoogle Scholar
  45. Ishaquddin M, Raveendranath P, Reddy J (2012) Flexure and torsion locking phenomena in out-of-plane deformation of timoshenko curved beam element. Finite Elem Anal Des 51:22–30MathSciNetCrossRefGoogle Scholar
  46. Koschnick F, Bischoff M, Camprubí N, K-U B (2005) The discrete strain gap method and membrane locking. Comput Methods Appl Mech Eng 194(21):2444–2463CrossRefGoogle Scholar
  47. Madeo A, Neff P, Ghiba I, Placidi L, Rosi G (2015) Band gaps in the relaxed linear micromorphic continuum. Zeitschrift für Angewandte Mathematik und Mechanik 95(9):880–887MathSciNetCrossRefGoogle Scholar
  48. Malkus D, Hughes T (1978) Mixed finite element methods – reduced and selective integration tecniques: a unification of the concepts. Comput Methods Appl Mech Eng 15:63–81CrossRefGoogle Scholar
  49. Marino E (2016) Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams. Comput Methods Appl Mech Eng 307:383–410MathSciNetCrossRefGoogle Scholar
  50. Meier C, Popp A, Wall W (2014) An objective 3D large deformation finite element formulation for geometrically exact curved Kirchhoff rods. Comput Methods Appl Mech Eng 278:445–478MathSciNetCrossRefGoogle Scholar
  51. Meier C, Popp A, Wall W (2015) A locking-free finite element formulation and reduced models for geometrically exact Kirchhoff rods. Comput Methods Appl Mech Eng 290:314–341MathSciNetCrossRefGoogle Scholar
  52. Militello C, Felippa C (1990) A variational justification of the assumed natural strain formulation of finite elements – I. Variational principles. Comput Struct 34:431–438CrossRefGoogle Scholar
  53. Neff P, Ghiba I, Madeo A, Placidi L, Rosi G (2014) A unifying perspective: the relaxed linear micromorphic continuum. Contin Mech Thermodyn 26(5):639–681MathSciNetCrossRefGoogle Scholar
  54. Pian T (1964) Derivation od element stiffness matrix by assumed stress disrtibutions. AIAA J 2:1333–1336CrossRefGoogle Scholar
  55. Reiher J, Giorgio I, Bertram A (2017) Finite-element analysis of polyhedra under point and line forces in second-strain gradient elasticity. J Eng Mech 143.  https://doi.org/10.1061/(ASCE)EM.1943--7889.0001184
  56. Romano G, Rosati L, Marotti de Sciarra F (1993) Variational formulations of non-linear and non-smooth structural problems. Int J Non Linear Mech 28:195–208CrossRefGoogle Scholar
  57. Romano G, Marotti de Sciarra F, Diaco M (2001) Well-posedness and numerical performances of the strain gap method. Int J Numer Methods Eng 51:109–139MathSciNetCrossRefGoogle Scholar
  58. Scerrato D, Giorgio I, Rizzi N (2016) Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Zeitschrift für angewandte Mathematik und Physik 67:19MathSciNetCrossRefGoogle Scholar
  59. Scerrato D, Zhurba Eremeeva IA, Lekszycki T, Rizzi NL (2017) On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. Zeitschrift fuür Angewandte Mathematik und Mechanik 96:1268–1279MathSciNetCrossRefGoogle Scholar
  60. Schillinger D, Hossain S, Hughes TJR (2014) Reduced bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis. Comput Methods Appl Mech Eng 277:1–45CrossRefGoogle Scholar
  61. 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 85:289–329MathSciNetCrossRefGoogle Scholar
  62. Simo J, Hughes T (1986) On the variational foundations of assumed strain methods. J Appl Mech 53:51–54MathSciNetCrossRefGoogle Scholar
  63. Simo J, Rifai M (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29:1595–1638MathSciNetCrossRefGoogle Scholar
  64. Stolarski H, Belytschko T (1982) Membrane locking and reduced integration for curved elements. J Appl Mech 46:172–176CrossRefGoogle Scholar
  65. Thomas D, Scott M, Evans J, Tew K, Evans E (2015) Bézier projection: a unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis. Comput Methods Appl Mech Eng 284:55–105CrossRefGoogle Scholar
  66. Tonti E (1972) On the mathematical structure of a large class of physical theories. Rendiconti Accaademia Nazionale dei Lincei LII:48–56Google Scholar
  67. Tonti E (1976) The reason for analogies between physical theories. Appl Math Model 1:37–50MathSciNetCrossRefGoogle Scholar
  68. Turco E, Giorgio I, Misra A, dell’Isola F (2017a) King post truss as a motif for internal structure of (meta)material with controlled elastic properties. R Soc Open Sci 4(171153):141–147CrossRefGoogle Scholar
  69. Turco E, Golaszewski M, Giorgio I, D’Annibale F (2017b) Pantographic lattices with non-orthogonal fibres: experiments and their numerical simulations. Compos B Eng 118:1–14CrossRefGoogle Scholar
  70. Yeo S, Lee B (1996) Equivalence between enhanced assumed strain method and assumed stress hybrid method based on the Hellinger-Reissner Principle. Int J Numer Methods Eng 39:3083–3099MathSciNetCrossRefGoogle Scholar
  71. Zervos A (2008) Finite elements for elasticity with microstructure and gradient elasticity. Int J Numer Methods Eng 73:564–595MathSciNetCrossRefGoogle Scholar
  72. Zienkiewicz O, Taylor R, Too J (1971) Reduced integration technique in general analysis of plates and shells. Int J Numer Methods Eng 43:275–290CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil Engineering and ArchitectureUniversity of CataniaCataniaItaly

Section editors and affiliations

  • Francesco dell’Isola
    • 1
    • 2
  1. 1.DISGUniversity of Rome La SapienzaRomeItaly
  2. 2.International Research Center M&MoCSUniversity of L’AquilaL’AquilaItaly