Skip to main content
SpringerLink
Log in

Lloyd’s energy minimization in the L p norm for quadrilateral surface mesh generation

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

Abstract

Indirect methods recombine the elements of triangular meshes to produce quadrilaterals. The resulting quadrilaterals are usually randomly oriented, which is not desirable. However, by aligning the vertices of the initial triangular mesh, precisely oriented quads can be produced. Levy’s algorithm is a non-linear optimization procedure that can align points according to a locally defined orientation matrix. It minimizes an energy functional based on the L p distance. The triangulation of a set of vertices smoothed with Levy’s algorithm is mainly composed of right-angled triangles, which is ideal for quad recombination. An implementation of Levy’s algorithm based on numerical integration is presented. The implementation has the advantage of not modifying the edge meshes. It also features automatic calculation of the orientation angle. When used in combination with an indirect recombination algorithm, it can create quads of varying size and orientation. It has been tested on two-dimensional surfaces as well as globally parametrized three-dimensional surfaces. The results demonstrate an increase in the number of nodes having four neighbors and an improvement of the quads quality. The development took place in the framework of the Gmsh free software.

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
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25

Similar content being viewed by others

References

  1. Blundell BG (2008) An introduction to computer graphics and creative 3-D environments. Springer, London

    Book  MATH  Google Scholar 

  2. Bochkanov S, Bystritsky V (1999–2011) Alglib. http://www.alglib.net

  3. Du Q, Wang D (2003) Tetrahedral mesh generation and optimization based on centroidal voronoi tessellations. Int J Numer Meth Eng 56:1355–1373

    Article  MATH  MathSciNet  Google Scholar 

  4. Du Q, Faber V, Gunzburger M (1999) Centroidal voronoi tessellations: applications and algorithms. Siam Rev 41:637–676

    Article  MATH  MathSciNet  Google Scholar 

  5. Ferziger J, Peric M (1999) Computational methods for fluid dynamics. Springer, Berlin

    Book  MATH  Google Scholar 

  6. Hausner A (2001) Simulating decorative mosaics. In: Proceedings of the 28th annual conference on computer graphics and interactive techniques, New York

  7. Hughes T (2000) The finite element method: linear static and dynamic finite element analysis. Dover Publications, New York

    Google Scholar 

  8. Lai YK, Hu SM, Martin RR (2006) Surface mosaics. Visual Comput 22:604–611

    Article  Google Scholar 

  9. Lai YK, Kobbelt L, Hu SM (2010) Feature aligned quad dominant remeshing using iterative local updates. Comput Aided D 42:109–117

    Article  Google Scholar 

  10. Levy B, Liu Y (2010) l p centroidal voronoi tessellation and its applications. In: Hoppe H (ed) ACM Transactions on Graphics, University of California, Los Angeles

  11. Liu Y, Wang W, Levy B, Sun F, Yan DM, Lu L, Yang C (2009) On centroidal voronoi tessellation-energy smoothness and fast computation. ACM Trans Graphic 28:1–17

    Google Scholar 

  12. Liu Y, Xu W, Wang J, Zhu L, Guo B, Chen F, Wang G (2011) General planar quadrilateral mesh design using conjugate direction field. In: Bala K (ed) ACM Transactions on Graphics, Hong-Kong

  13. Liziér MAS, Siqueira MF, Daniels J, Silva CT, Nonato LG (2011) Template-based quadrilateral mesh generation from imaging data. Visual Comput 27:887–903

    Article  Google Scholar 

  14. Marchandise E, de Wiart CC, Vos WG, Geuzaine C, Remacle JF (2011) High-quality surface remeshing using harmonic maps-part II: surfaces with high genus and of large aspect ratio. Int J Numer Meth Eng 86:1303–1321

    Article  MATH  Google Scholar 

  15. Prakash S, Ethier CR (2001) Requirements for mesh resolution in 3d computational hemodynamics. J Biomech Eng 123:134–144

    Article  Google Scholar 

  16. Remacle JF, Henrotte F, Baudouin TC, Geuzaine C, Béchet E, Mouton T, Marchandise E (2011) A frontal delaunay quad mesh generator using the \(l^{\infty}\) norm. In: Proceedings of the 20th International Meshing Roundtable, Paris

  17. Remacle JF, Lambrechts J, Seny B, Marchandise E, Johnen A, Geuzaine C (2011) Blossom-quad: a non-uniform quadrilateral mesh generator using a minimum cost perfect matching algorithm. Int J Numer Meth Eng 89:1102–1119

    Google Scholar 

  18. Roache P (1992) Computational fluid dynamics. Hermosa, Albuquerque

    Google Scholar 

  19. Sherwin SJ, Peiró J (2002) Mesh generation in curvilinear domains using high-order elements. Int J Numer Meth Eng 53:207–223

    Article  MATH  Google Scholar 

  20. Shimada K, Liao JH, Itoh T (1998) Quadrilateral meshing with directionality control through the packing of square cells. In: Proceedings of the 7th International Meshing Roundtable, Dearborn

  21. Tarini M, Puppo E, Panozzo D, Pietroni N, Cignoni P (2011) Simple quad domains for field aligned mesh parametrization. In: Bala K (ed) ACM Transactions on Graphics, Hong-Kong

  22. Tchon KF, Camarero R (2006) Quad-dominant mesh adaptation using specialized simplicial optimization. In: Proceedings of the 15th International Meshing Roundtable, Birmingham

  23. Vinchurkar S, Longest PW (2007) Effect of mesh style and grid convergence on particle deposition in bifurcating airway models with comparisons to experimental data. Med Eng Phys 29:350–366

    Article  Google Scholar 

  24. Vinchurkar S, Longest PW (2008) Evaluation of hexahedral, prismatic and hybrid mesh styles for simulating respiratory aerosol dynamics. Comput Fluids 37:317–331

    Article  MATH  Google Scholar 

  25. Yamakawa S, Shimada K (2003) Fully-automated hex-dominant mesh generation with directionality control via packing rectangular solid cells. Int J Numer Meth Eng 57:2099–2129

    Article  MATH  MathSciNet  Google Scholar 

  26. Yan DM, Wang W, Levy B, Liu Y (2010) Efficient computation of 3d clipped voronoi diagram. In: Mourrain B, Schaefer S, Xu G (eds) GMP 2010 Conference Proceedings, Castro Urdiales

  27. Zienkiewicz OC, Taylor RL (2000) The finite element method: the basis, vol 1. Butterworth-Heinemann, Oxford

    Google Scholar 

  28. Zienkiewicz OC, Rojek J, Taylor RL, Pastor M (1998) Triangles and tetrahedra in explicit dynamic codes for solids. Int J Numer Meth Eng 43:565–583

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work has been partially supported by the Belgian Walloon Region under WIST grants ONELAB 1017086 and DOMHEX 1017074.

Author information

Authors and Affiliations

  1. Institute of Mechanics, Materials and Civil Engineering, Université catholique de Louvain, Avenue Georges-Lemaître 4, 1348, Louvain-la-Neuve, Belgium

    Tristan Carrier Baudouin, Jean-François Remacle, Emilie Marchandise, Jonathan Lambrechts & François Henrotte

Authors
  1. Tristan Carrier Baudouin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-François Remacle
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Emilie Marchandise
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jonathan Lambrechts
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. François Henrotte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tristan Carrier Baudouin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Baudouin, T.C., Remacle, JF., Marchandise, E. et al. Lloyd’s energy minimization in the L p norm for quadrilateral surface mesh generation. Engineering with Computers 30, 97–110 (2014). https://doi.org/10.1007/s00366-012-0290-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-012-0290-x

Keywords

Search

Navigation