Advertisement

On implementation aspects of finite element method and its application

  • Petr SváčekEmail author
Article
  • 38 Downloads

Abstract

This paper describes the usage of the finite element library CFEM for solution of boundary value problems for partial differential equations. The application of the finite element method is shown based on the weak formulation of a boundary value problem. A unified approach for solution of linear scalar, linear vector, and nonlinear vector problems is presented. A direct link between the mathematical formulation and the design of the computer code is shown. Several examples and results are shown.

Keywords

Finite element method Nonlinear problems 

Mathematics Subject Classification (2010)

65N30 76D05 35Q30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors acknowledge the support from the EU Operational Programme Research, Development and Education, and from the Center of Advanced Aerospace Technology (CZ.02.1.01/0.0/0.0/16_019/0000826), Faculty of Mechanical Engineering, Czech Technical University in Prague.

References

  1. 1.
  2. 2.
    FEM4FSI/CFEM (Finite element method for fluid structure interactions - core library). http://marian.fsik.cvut.cz/svacek/cfem
  3. 3.
    COMSOL multiphysics modeling software. www.comsol.com/
  4. 4.
    deal.II (an open source finite element library). http://www.dealii.org/
  5. 5.
  6. 6.
    Hermes (Hermes hp-FEM & hp-DG Library). http://hpfem.org/hermes
  7. 7.
  8. 8.
    MFEM: Modular finite element methods library. mfem.org.  https://doi.org/10.11578/dc.20171025.1248  https://doi.org/10.11578/dc.20171025.1248
  9. 9.
    MSC Nastran (Multidisciplinary Structural Analysis). www.mscsoftware.com/product/msc-nastran
  10. 10.
    OpenFOAM. The OpenFOAM Foundation. http://openfoam.org/
  11. 11.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  12. 12.
    Alns, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M., Wells, G.: The fenics project version 1.5. Arch. Numer. Softw 3(100) (2015)Google Scholar
  13. 13.
    Armaly, B.F., Durst, F., Pereira, J.C.F., Schoenung, B.: Experimental and theoretical investigation of backward-facing step flow. J. Fluid Mech. 127, 473–496 (1983)CrossRefGoogle Scholar
  14. 14.
    Bangerth, W., Davydov, D., Heister, T., Heltai, L., Kanschat, G., Kronbichler, M., Maier, M., Turcksin, B., Wells, D.: The deal.ii library, version 8.4. J. Numer. Math. 24(3), 135–141 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Brauer, J. (ed.): What Every Engineer Should Know about Finite Element Analysis, 2nd edn. Marcel Dekker Inc., New York (1993)Google Scholar
  16. 16.
    Ciarlet, P.G.: The Finite Element Methods for Elliptic Problems. North-Holland Publishing (1979)Google Scholar
  17. 17.
    Donald, B.J.M.: Practical Stress Analysis with Finite Elements. Glasnevin Publishing, Dublin (2007)Google Scholar
  18. 18.
    Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Springer, Berlin (1974)zbMATHGoogle Scholar
  19. 19.
    Geuzaine, C., Remacle, J. -F.: GMSH: A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009)CrossRefzbMATHGoogle Scholar
  20. 20.
    Holub, A.I.: C+ C++: Programming with Objects in C and C++. McGraw-Hill Companies, New York (1991)Google Scholar
  21. 21.
    John, V., Matthies, G.: MooNMD - a program package based on mapped finite element methods. Comput. Vis. Sci. 6(2-3), 163–169 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Koudelka, T., Krejcí, T., Kruis, J.: Moderate use of object oriented programming for scientific computing. In: Proceedings of the Seventh International Conference on Engineering Computational Technology. Paper 68. Civil-Comp Press, Stirlingshire (2010),  https://doi.org/10.4203/ccp.94.68
  23. 23.
    Kozel, K., Louda, P., Sváček, P., Příhoda, J.: Finite volume and finite element methods applied to backward facing step flows. In: 1st International Conference ”From Scientific Computing to Computational Engineering”, p. 10 pp. University of Patras, Patras. CD ROM Proceedings (2004)Google Scholar
  24. 24.
    Schmidt, A., Siebert, K.G.: Design of Adaptive Finite Element Software. The Finite Element Toolbox ALBERTA. Springer, Berlin (2005)zbMATHGoogle Scholar
  25. 25.
    Schreiner, A.: Object-Oriented Programming with ANSI-C (2011)Google Scholar
  26. 26.
    Solin, P., Andrs, D., Cerveny, J., Simko, M.: Pde-independent adaptive hp-FEM based on hierarchic extension of finite element spaces. J. Comput. Appl. Math. 233, 3086–3094 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Solin, P., Dubcova, L., Kruis, J.: Adaptive hp-FEM with dynamical meshes for transient heat and moisture transfer problems. J. Comput. Appl. Math. 233, 3103–3112 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sváček, P.: Finite element method application for turbulent and transitional flows. In: Proceedings of the Conference on Modelling Fluid Flow, p. 7 pp. Budapest University of Technology and Economics, Department of Fluid Mechanics, Budapest (2015). Art. no. 83Google Scholar
  29. 29.
    Sváček, P.: On the finite element method application for approximation of free-surface flows with surface tension and contact angles. Appl. Mech. Mater. 821, 129–137 (2016)CrossRefGoogle Scholar
  30. 30.
    Sváček, P., Horáček, J.: On application of finite element method for approximation of 3d flow problems. In: Simurda, D., Bodnar T. (eds.) Topical Problems of Fluid Mechanics 2015, pp. 175–182 (2015)Google Scholar
  31. 31.
    Sváček, P., Horáček, J.: On mathematical modeling of fluid–structure interactions with nonlinear effects: Finite element approximations of gust response. J. Comput. Appl. Math. 273(0), 394–403 (2015)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Wilbrandt, U., Bartsch, C., Ahmed, N., Alia, N., Anker, F., Blank, L., Caiazzo, A., Ganesan, S., Giere, S., Matthies, G., Meesala, R., Shamim, A., Venkatesan, J., John, V.: ParMooN - A modernized program package based on mapped finite elements. Comput. Math. Appl. 74, 74–88 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Technical Mathematics, Faculty of Mechanical Engineering, Center of Advanced Aerospace TechnologyCzech Technical University in PraguePrague 6Czech Republic

Personalised recommendations