Skip to main content
SpringerLink
Log in

Least-squares approximation of affine mappings for sweep mesh generation: functional analysis and applications

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

Sweep methods are one of the most robust techniques to generate hexahedral meshes in extrusion volumes. The main issue in sweep algorithms is the projection of cap surface meshes along the sweep path. The most competitive technique to determine this projection is to find a least-squares approximation of an affine mapping. Several functional formulations have been defined to carry out this least-squares approximation. However, these functionals generate unacceptable meshes for several common geometries in CAD models. In this paper we present a new comparative analysis between these classical functional formulations and a new functional presented by the authors. In particular, we prove under which conditions the minimization of the analyzed functionals leads to a full rank linear system. Moreover, we also prove the equivalences between these formulations. These allow us to point out the advantages of the proposed functional. Finally, from this analysis we outline an automatic algorithm to compute the nodes location in the inner layers.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Thompson JF, Soni B, Weatherill N (1999) Handbook of grid generation. CRC Press, Boca Raton

    MATH  Google Scholar 

  2. Frey PJ, George PL (2000) Mesh generation. Application to finite elements. Hermes Science Publishing, Oxford

    MATH  Google Scholar 

  3. Tautges TJ (2001) The generation of hexahedral meshes for assembly geometry: survey and progress. Int J Numer Meth Eng 50:2617–2642

    Article  MATH  Google Scholar 

  4. Cook WA, Oakes WR (1982) Mapping methods for generating three-dimensional meshes. Comput Mech Eng 1:67–72

    Google Scholar 

  5. Whiteley M, White D, Benzley S, Blacker T (1996) Two and three-quarter dimensional meshing facilitators. Eng Comput 12:144–154

    Article  Google Scholar 

  6. Knupp PM (1998) Next-generation sweep tool: a method for generating all-hex meshes on two-and-one-half dimensional geometries. In: Proceedings of 7th International Meshing Roundtable, Sandia National Laboratory, pp 505–513

  7. Knupp PM (1999) Applications of mesh smoothing: copy, morph, and sweep on unstructured quadrilateral meshes. Int J Numer Meth Eng 45:37–45

    Article  MathSciNet  MATH  Google Scholar 

  8. Blacker TD (1996) The cooper tool. In: Proceedings of 5th International Meshing Roundtable, Sandia National Laboratory, pp 13–30

  9. Mingwu L, Benzley SE (1996) A multiple source and target sweeping method for generating all-hexahedral finite element meshes. In: Proceedings of 5th International Meshing Roundtable, Sandia National Laboratory, pp 217–225

  10. Staten ML, Canann SA, Owen SJ (1999) BMSweep: locating interior nodes during sweeping. Eng Comput 15:212–218

    Article  MATH  Google Scholar 

  11. Scott MA, Earp MN, Benzley SE, Stephenson MB (2005) Adaptive sweeping techniques. In: Proceedings of 14th International Meshing Roundtable, Sandia National Laboratory, pp 417–432

  12. Roca X, Sarrate J, Huert, A (2005) A new least-squares approximation of affine mappings for sweep algorithms. In: Proceedings of 14th International Meshing Roundtable, Sandia National Laboratory, pp 433–448

  13. Roca X, Sarrate J, Huerta A (2006) Mesh projection between parametric surfaces. Commun Numer Meth Eng 22:591–603

    Article  MathSciNet  MATH  Google Scholar 

  14. Roca X, Sarrate J (2006) An automatic and general least-squares projection procedure for sweep meshing. In: Proceedings of 15th International Meshing Roundtable, Sandia National Laboratory, pp 437–506

  15. White DR, Tautges TJ (2000) Automatic scheme selection for toolkit hex meshing. Int J Numer Meth Eng 49:127–144

    Article  MATH  Google Scholar 

  16. Cass RJ, Benzley SE, Meyers RJ, Blacker TD (1996) Generalized 3-D paving: an automated quadrilateral surface mesh generation algorithm. Int J Numer Meth Eng 39:1475–1489

    Article  MATH  Google Scholar 

  17. Goodrich D (1997) Generation of all-quadrilateral surface meshes by mesh morphing. MSc Thesis, Brigham Young University

  18. Sarrate J, Huerta A (2000) Efficient unstructured quadrilateral mesh generation. Int J Numer Meth Eng 49:1327–1350

    Article  MATH  Google Scholar 

  19. Sarrate J, Huerta A (2000) Automatic mesh generation of nonstructured quadrilateral meshes over curved surfaces in \({{\mathbb{R}}^3}\). In Proceedings of 3th European congress on computational methods in applied sciences and engineering, ECCOMAS, Barcelona, Spain

  20. Gill PE, Murray W, Wright MH (1991) Numerical linear algebra and optimization. Addison-Wesley, Redwood City

    MATH  Google Scholar 

Download references

Acknowledgments

This work was partially sponsored by the Spanish Ministerio de Ciencia e Innovación under grants DPI2007-62395, BIA2007-66965 and CGL2008-06003-C03-02/CLI.

Author information

Authors and Affiliations

  1. Laboratori de Càlcul Numèric (LaCàN), Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Jordi Girona 1-3, 08034, Barcelona, Spain

    Xevi Roca & Josep Sarrate

Authors
  1. Xevi Roca
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Josep Sarrate
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xevi Roca.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Roca, X., Sarrate, J. Least-squares approximation of affine mappings for sweep mesh generation: functional analysis and applications. Engineering with Computers 29, 1–15 (2013). https://doi.org/10.1007/s00366-012-0260-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-012-0260-3

Keywords

Search

Navigation