Finite Element Method

  • Meinhard KunaEmail author
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 201)


The finite element method (FEM) is currently one of the most efficient and universal methods of numerical calculation for solving partial differential equations from engineering and scientific fields. The basic mathematical concepts are based on the work of Ritz, Galerkin, Trefftz and others at the beginning of the twentieth century. With the advance of modern computer science in the 1960s, these approaches of numerical solution could be successfully implemented with FEM. This development was motivated to an enormous extent by tasks of structural analyses in aviation, construction and mechanical engineering. The formulation of the finite element method in its current standard was developed thanks to the pioneer work of (among others) Argyris, Zienkiewicz, Turner, and Wilson. Therein, the system of differential equations is converted into an equivalent variational problem (weak formulation), mostly utilizing mechanical principles or weighted residual methods.


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  1. 1.
    Washizu K (1975) Variational methods in elasticity and plasticity. Pergamon Press, OxfordGoogle Scholar
  2. 2.
    Pian THH, Tong P (1969) Basis of finite element methods for solid continua. Int J Numer Methods Eng 1:3–28zbMATHCrossRefGoogle Scholar
  3. 3.
    Atluri SN, Kobayashi AS, Nakagaki M (1975) An assumed displacement hybrid finite element model for linear fracture mechanics. Int J Fract 11:257–271CrossRefGoogle Scholar
  4. 4.
    Atluri SN, Kartiresan K (1979) 3D analysis of surface flaws in thick-walled reactor pressure vessels using displacement-hybrid finite element method. Nucl Eng Des 51:163–176CrossRefGoogle Scholar
  5. 5.
    Tong P, Pian THH, Lasry H (1973) A hybrid element approach to crack problems in plane elasticity. Int J Numer Methods Eng 7:297–308zbMATHCrossRefGoogle Scholar
  6. 6.
    Ueberhuber C (1995) Computer-Numerik. Springer, BerlinCrossRefGoogle Scholar
  7. 7.
    Bathe KJ (1986) Finite-Elemente-Methoden. Matrizen und lineare Algebra. Springer, BerlinzbMATHCrossRefGoogle Scholar
  8. 8.
    Schwarz HR (1991) Methoden der finiten Elemente. Teubner StudienbüchereiGoogle Scholar
  9. 9.
    Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite element method: its basis and fundamentals, vol 1. Elsevier, AmsterdamzbMATHGoogle Scholar
  10. 10.
    Stroud AH (1971) Approximate calculation of multiple integrals. Prentice Hall, Englewood CliffszbMATHGoogle Scholar
  11. 11.
    Wriggers P (2001) Nichtlineare finite-elemente-methoden. Springer, BerlinGoogle Scholar
  12. 12.
    Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New YorkzbMATHGoogle Scholar
  13. 13.
    Belytschko T, Liu WK, Moran B (2001) Nonlinear finite elements for continua and structures. Wiley, ChichesterGoogle Scholar
  14. 14.
    Hughes TJR (1987) The finite element method. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  15. 15.
    Belytschko T, Hughes TJR (1983) Computational methods for transient analysis. North-Holland, AmsterdamGoogle Scholar

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Inst für Mechanik und FluiddynamikTU Bergakademie FreibergFreibergGermany

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