Engineering with Computers

, Volume 20, Issue 3, pp 273–285 | Cite as

Automatic p-version mesh generation for curved domains

  • Xiao-Juan LuoEmail author
  • Mark S. Shephard
  • Robert M. O’Bara
  • Rocco Nastasia
  • Mark W. Beall
Original article


To achieve the exponential rates of convergence possible with the p-version finite element method requires properly constructed meshes. In the case of piecewise smooth domains, these meshes are characterized by having large curved elements over smooth portions of the domain and geometrically graded curved elements to isolate the edge and vertex singularities that are of interest. This paper presents a procedure under development for the automatic generation of such meshes for general three-dimensional domains defined in solid modeling systems. Two key steps in the procedure are the determination of the singular model edges and vertices, and the creation of geometrically graded elements around those entities. The other key step is the use of general curved element mesh modification procedures to correct any invalid elements created by the curving of mesh entities on the model boundary, which is required to ensure a properly geometric approximation of the domain. Example meshes are included to demonstrate the features of the procedure.


p-version method Curved meshes Graded meshes 



This work was supported by the National Science Foundation through SBIR grant number DMI-0132742.


  1. 1.
    Anderson B, Falk U, Babuska I, Petersdorff TV (1995) Reliable stress and fracture mechanics analysis of complex components using a hp version of FEM. Int J Numer Methods Eng 38:2135–2163Google Scholar
  2. 2.
    Babuska I, Suri M (1994) The p and h-p versions of the finite element method, basic principles and properties. SIAM Rev 36(4):578–632MathSciNetzbMATHGoogle Scholar
  3. 3.
    Babuska I, Petersdorff TV, Anderson B (1994) Numerical treatment of vertex singularities and intensity factors for mixed boundary value problems for the Laplace equation. SIAM J Numer Anal 31(5):1265–1288MathSciNetzbMATHGoogle Scholar
  4. 4.
    Babuska I, Anderson B, Guo B, Melenk JM, Oh HS (1996) Finite element method for solving problems with singular solutions. J Comp Appl Math 74:51–70CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Dey S, O’Bara RM, Shephard MS (2001) Towards curvilinear meshing in 3D: the case of quadratic simplices. CAD Comput Aided Des 33(3):199–209CrossRefGoogle Scholar
  6. 6.
    Dey S, Shephard MS, Flaherty JE (1997) Geometry representation issues associated with p-version finite element computations. Comput Methods Appl Mech Eng 150(1–4):39–55Google Scholar
  7. 7.
    Dorr MR (1986) The approximation of solutions of elliptic boundary-value problems via the p-version of the finite element method. SIAM J Numer Anal 23(1):58–77MathSciNetzbMATHGoogle Scholar
  8. 8.
    Farin GE (1993) Curves and surfaces for computer aided geometric design: a practical guide, 3rd edn. Academic Press, BostonzbMATHGoogle Scholar
  9. 9.
    Farouki RT, Rajan VT (1988) Algorithms for polynomials in Bernstein. Comput Aided Geom Des 5:1–26CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Garimella R, Shephard MS (2000) Boundary layer mesh generation for viscous flow simulations in complex geometric domains. Int J Numer Methods Eng 49(1–2):193–218Google Scholar
  11. 11.
    Harber R, Shephard MS, Abel JF, Gallagher RH, Greenberg DP (1981) A general two-dimensional, graphical finite element preprocessor utilizing discrete transfinite mappings. Int J Numer Methods Eng 17:1015–1044Google Scholar
  12. 12.
    Li X, Shephard MS, Beall MW (2003) 3-D anisotropic mesh adaptation by mesh modifications. Comput Methods Appl Mech Eng (submitted)Google Scholar
  13. 13.
    Luo XJ, Shephard MS, Remacle JF, O’Bara RM, Beall MW, Szabo BA, Actis R (2002) p-version mesh generation issues. In: Proceedings of the 11th international meshing roundtable, Ithaca, New York, September 2002. Sandia National Laboratories, pp 343–354Google Scholar
  14. 14.
    Sherwin SJ, Peiro J (2002) Mesh generation in curvilinear domains using high order elements. Int J Numer Methods Eng 53:207–223CrossRefzbMATHGoogle Scholar
  15. 15.
    Szabo BA (1986) Mesh design for the p-version of the finite element method. Comput Methods Appl Mech Eng 55:181–197CrossRefzbMATHGoogle Scholar
  16. 16.
    Szabo BA, Babuska I (1991) Finite element analysis. Wiley, New YorkGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2004

Authors and Affiliations

  • Xiao-Juan Luo
    • 1
    Email author
  • Mark S. Shephard
    • 1
  • Robert M. O’Bara
    • 2
  • Rocco Nastasia
    • 2
  • Mark W. Beall
    • 2
  1. 1.Scientific Computation Research CenterRensselaer Polytechnic InstituteTroyUSA
  2. 2.Simmetrix Inc.Clifton ParkUSA

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