Skip to main content

The Finite Element (FE) Method

  • Chapter
  • First Online:
  • 2523 Accesses

Abstract

The finite element (FE) method has become the standard numerical calculation scheme for the computer simulation of physical systems [13].

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. K.J. Bathe, Finite Element Procedures (Prentice Hall, New Jersey, 1996)

    Google Scholar 

  2. N. Ida, Engineering Electromagnetics (Springer, 2004)

    Google Scholar 

  3. O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, vol. 1 (Butterworth-Heinemann, UK, 2003)

    Google Scholar 

  4. T.J.R. Hughes, The Finite Element Method, 1st edn. (Prentice-Hall, New Jersey, 1987)

    MATH  Google Scholar 

  5. F.X. Zgainski, J.L. Coulomb, Y. Marechal, A new family of finite elements: the pyramidal element. IEEE Trans. Magn. 32, 1393–1396 (1996)

    Article  Google Scholar 

  6. M. Jung, U. Langer, Methode der Finiten Elemente für Ingenieure, 2nd edn. (Springer, 2013)

    Google Scholar 

  7. K. Meyberg, P. Vachenauer, Höhere Mathematik 1 (Springer, 1993)

    Google Scholar 

  8. H. Whitney, Geometric Integration Theory (Princeton University Press, Princeton, 1957)

    Google Scholar 

  9. J.C. Nédélec, Mixed finite elements in \(R^3\). Numer. Math. 35, 315–341 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  10. M.L. Barton, Z.J. Cendes, New vector finite elements for three-dimensional magnetic field computation. J. Appl. Phys. 61(8), 3919–3921 (1987)

    Article  Google Scholar 

  11. A. Bossavit, J.C. Verite, A mixed FEM-BIEM method to solve 3-D eddy current problems. IEEE Trans. Magn. 18, 431–435 (1982)

    Article  Google Scholar 

  12. A. Kameari, Three dimensional eddy current calculation using edge elements for magnetic vector potential. Appl. Electromagn. Mater. 225–236 (1986)

    Google Scholar 

  13. G. Mur, A.T. Hoop, A finite-element method for computing three-dimensional electromagnetic fields in inhomogeneous media. IEEE Trans. Magn. 21, 2188–2191 (1985)

    Article  Google Scholar 

  14. J.S. Welij, Calculation of eddy current in terms of H on hexahedra. IEEE Trans. Magn. 21, 2239–2241 (1985)

    Article  Google Scholar 

  15. P. Silvester, R. Ferrari, Finite Elements for Electrical Engineers (Cambridge, 1996)

    Google Scholar 

  16. M. Ainsworth, J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis, (Wiley, 2000)

    Google Scholar 

  17. Ch. Großmann, H.G. Roos, Numerik Partieller Differentialgleichungen (Teubner, 1994)

    Google Scholar 

  18. G. Strang, G. Fix, An Analysis of the Finite Element Method (Cambridge Press, Wellesley, 2008)

    Google Scholar 

  19. I. Babuska, B. Szabo, I. Katz, The p-version of the finite element method. SIAM J. Numer. Anal. 18(3), 515–545 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Szabó, I. Babuška, Finite Element Analysis, 1st edn. (Wiley, 1991)

    Google Scholar 

  21. O.C. Zienkiewicz, J.P. De, S.R. Gago, D.W. Kelly, The hierarchical concept in finite element analysis. Comput. Struct. 16, 53–65 (1983)

    Article  MATH  Google Scholar 

  22. G. Szegö, Orthogonal polynomials. American Mathematical Society Colloquium Publications, no. Bd. 23. American Mathematical Society (1959)

    Google Scholar 

  23. S. Zaglmayr, High order finite element methods for electromagnetic field computation. Ph.D. thesis, Johannes Kepler University, Linz (2006)

    Google Scholar 

  24. A. Düster, Lecture notes: high order FEM. 132 (2005)

    Google Scholar 

  25. G. Karniadakis, S.J. Sherwin, Spectral/HP Element Methods for CFD (Oxford University Press, 1999)

    Google Scholar 

  26. D.A. Kopriva, Implementing Spectral Methods for Partial Differential Equations (Springer, Dordrecht, 2009)

    Google Scholar 

  27. A.T. Patera, A spectral element method for fluid dynamics—laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488 (1984)

    Article  MATH  Google Scholar 

  28. G.C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations (Springer, New York, 2002)

    Google Scholar 

  29. William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, Numerical Recipes 3rd edition: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, New York, 2007)

    Google Scholar 

  30. R. Courant, K. Friedrichs, H. Lewy, On the partial difference equations of mathematical physics. IBM J. Res. Dev. 11, 215–234 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  31. C. Bernardi, Y. Maday, F. Rapetti, Basics and some applications of the mortar element method. GAMM-Mitt. 28(2), 97–123 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. B. Flemisch, Non-matching triangulations of curvilinear interfaces applied to electro-mechanics and elasto-acoustics, Ph.D. thesis, University of Stuttgart, (2006)

    Google Scholar 

  33. J. Danek, H. Kutakova, The mortar finite element method in 2D: implementation in MATLAB, in 16th Annual Conference Proceedings of Technical Computing (Prague, Czech Republic, 2008)

    Google Scholar 

  34. B.I. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38(3), 989–1012 (2000). MRMR1781212 (2001h:65132)

    Google Scholar 

  35. S. Triebenbacher, Nonmatching grids for the numerical simulation of problems from aeroacoustics and vibroacoustics. Ph.D. thesis, Alpen-Adria-Universität Klagenfurt, Austria (2012)

    Google Scholar 

  36. S. Triebenbacher, M. Kaltenbacher, B. Flemisch, B. Wohlmuth, Applications of the mortar finite element method in vibroacoustics and flow induced noise computations. Acta Acust. United Acust. 96, 536–553 (2010)

    Article  Google Scholar 

  37. Ivan E. Sutherland, Gary W. Hodgman, Reentrant polygon clipping. Commun. ACM 17, 32–42 (1974)

    Article  MATH  Google Scholar 

  38. J. Grabinger, Mechanical-acoustic coupling on non-matching finite element grids. Master’s thesis, University Erlangen-Nuremberg, June 2007

    Google Scholar 

  39. J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Tech. Report 36, Abh. Math. Univ. Hamburg, (1971)

    Google Scholar 

  40. A. Hansbo, P. Hansbo, M.G. Larson, A finite element method on composite grids based on nitsche’s method. ESAIM Math. Model. Numer. Anal. 37(3), 495–514 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  41. A. Fritz, S. Hüeber, B. Wohlmuth, A comparison of mortar and Nitsche techniques for linear elasticity. CALCOLO (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Manfred Kaltenbacher .

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Kaltenbacher, M. (2015). The Finite Element (FE) Method. In: Numerical Simulation of Mechatronic Sensors and Actuators. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40170-1_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-40170-1_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40169-5

  • Online ISBN: 978-3-642-40170-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics