Engineering with Computers

, Volume 30, Issue 3, pp 375–382

Geometrical validity of high-order triangular finite elements

Original Article

Abstract

This paper presents a method to compute accurate bounds on Jacobian determinants of high-order (curvilinear) triangular finite elements. This method can be used to guarantee that a curvilinear triangle is geometrically valid, i.e., its Jacobian determinant is strictly positive everywhere in its reference domain. It also provides an efficient way to measure the quality of triangles. The key feature of the method is to expand the Jacobian determinant using a polynomial basis, built using Bézier functions, that has both properties of boundedness and positivity. Numerical results show the sharpness of our estimates.

Keywords

Finite element method High-order methods Mesh generation Bézier functions 

References

  1. 1.
    Dey S, O’Bara RM, Shephard MS (2001) Curvilinear mesh generation in 3D. Comput Aided Geom Des 33:199–209CrossRefGoogle Scholar
  2. 2.
    Shephard MS, Flaherty JE, Jansen KE, Li X, Luo X, Chevaugeon N, Remacle J-F, Beall MW, O’Bara RM (2005) Adaptive mesh generation for curved domains. Appl Numer Math 52:251–271CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Sherwin SJ, Peiró J (2002) Mesh generation in curvilinear domains using high-order elements. Int J Numer Meth Eng 53:207–223CrossRefMATHGoogle Scholar
  4. 4.
    Luo X-J, Shephard MS, O’Bara RM, Nastasia R, Beall MW (2004) Automatic p-version mesh generation for curved domains. Eng Comput 20:273–285CrossRefGoogle Scholar
  5. 5.
    Sahni O, Luo XJ, Jansen KE, Shephard MS (2010) Curved boundary layer meshing for adaptive viscous flow simulations. Finite Elem Anal Des 46:132–139CrossRefMathSciNetGoogle Scholar
  6. 6.
    Geuzaine C, Remacle J-F (2009) Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Meth Eng 79(11):1309–1331CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Hughes T (2003) The finite element method. Dover, MineolaGoogle Scholar
  8. 8.
    Babuška I, Szabò B, Actis RL (1992) Hierarchic models for laminated composites. Int J Numer Methods Eng 33:503–535CrossRefMATHGoogle Scholar
  9. 9.
    Farin GE (2002) Curves and surfaces for CAGD: a practicle guide. Morgan-Kaufmann,Google Scholar
  10. 10.
    Lane JM, Riesenfeld RF (1980) A theoretical development for the computer generation and display of piecewise polynomial surfaces. IEEE Trans Pattern Anal Mach Intell 2(1):35–46CrossRefMATHGoogle Scholar
  11. 11.
    Cohen E, Schumacker LL (1985) Rates of convergence of control polygons. Comput Aided Geom Des 2:229–235CrossRefMATHGoogle Scholar
  12. 12.
    Lambrechts J, Comblen R, Legat V, Geuzaine C, Remacle J-F (2008) Multiscale mesh generation on the sphere. Ocean Dyn 58:461–473CrossRefGoogle Scholar
  13. 13.
    Remacle J-F, Geuzaine C, Compère G, Marchandise E (2010) High-quality surface remeshing using harmonic maps. Int J Numer Methods Eng 83(4):403–425MATHGoogle Scholar
  14. 14.
    Marchandise E, Cartonde Wiart C, Vos WG, Geuzaine C, Remacle J-F (2011) High quality surface remeshing using harmonic maps. Part II: Surfaces with high genus and of large aspect ratio. Int J Numer Methods Eng 86(11):1303–1321CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  1. 1.Department of Electrical Engineering and Computer Science, Montefiore Institute B28Université de LiègeLiègeBelgium
  2. 2.Institute of Mechanics, Materials and Civil EngineeringUniversité catholique de LouvainLouvain-la-NeuveBelgium

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