Skip to main content
SpringerLink
Log in

Parametrization of Plane Irregular Regions: A Semi-automatic Approach I

  • Conference paper
  • First Online:
Numerical Geometry, Grid Generation and Scientific Computing

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 143))

  • 653 Accesses

Abstract

In some problems, the solutions of partial differential equations use parametrizations of plane regions. However, it is difficult to get suitable parametrizations of irregular regions. In this paper we introduce a method for finding a parametrization of a polygonal region Ω. Our method decomposes Ω into a finite collection of admissible subregions. We use compatible parametrizations of these subregions to construct the parametrization of Ω as a block structured mesh.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abello, I.A., Hernández, V., Barrera, P., González, G.F.: Parametrización B-spline de regiones planas con frontera irregular. Ci. Mat. 31:2, 95–107. Cuba (2017)

    Google Scholar 

  2. Abello, I.A., Hernández, V., Barrera, P., González, G. F.: Injectivity of B-spline biquadratic maps. Comput. Methods Appl. Mech. Eng. 341, 586–608. Elsevier (2018)

    Google Scholar 

  3. Barrera, P., González, G.F., Domínguez, F.J.: Robust discrete grid generation on plane irregular regions. USSR Comput. Math. Math. Phys. 43(6), 884–892 (2003)

    MathSciNet  Google Scholar 

  4. Barrera, P., Cortés, J.J., González, G.F., Domínguez, F.J., Tinoco, J.G.: Smoothness and convex area functionals revisited. SIAM J. Sci. Comput. 32(4), 1913–1928 (2010)

    Article  MathSciNet  Google Scholar 

  5. Bezanson, J., Edelman, A., Karpinski, S., Shah V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59(1), 65–98 (2017)

    Article  MathSciNet  Google Scholar 

  6. Bommes, D., Campen, M., Ebke, H.C., Alliez, P., Kobbelt, L.: Integer-grid maps for reliable quad meshing. ACM Trans. Graph. 32(4) (2013). https://doi.org/10.1145/2461912.2462014

  7. Forrest, J., Ralph, T., Vigerske, S.: coin-or/Cbc: Version 2.9.9. https://projects.coin-or.org/Cbc. Cited 19 July 2018

  8. Garanzha, V.A.: Barrier method for quasi-isometric grid generation. Comput. Math. Math. Phys. 40(11), 1617–1637 (2000)

    MathSciNet  MATH  Google Scholar 

  9. González G. F.: Generación Numérica de Mallas Estructuradas de Calidad y Adaptativas en Regiones Planas Irregulares. PhD. Thesis. Universidad Nacional Autónoma de México. México (2018)

    Google Scholar 

  10. Gravesen, J., Evgrafov, A., Nguyen, M., Nørtoft, P.: Planar parametrization in isogeometric analysis. In: Floater, M., et al. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2012. Lecture Notes in Computer Science, vol. 8177. Springer, New York (2014). https://doi.org/10.1007/978-3-642-54382-1_11

    MATH  Google Scholar 

  11. Ivanenko, S.A.: Control of cells shapes in the course of grid generation. Comput. Math. Math. Phys. 40(11), 1596–1616 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Lien, J.: Approximate Convex Decomposition and its Applications. PhD. Thesis. Texas A&M University, USA (2006)

    Google Scholar 

  13. Liu, G., Xi, Z., Lien, J.: Dual-Space Decomposition of 2D Complex Shapes. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 4154–4161 (2014)

    Google Scholar 

  14. Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1998)

    MATH  Google Scholar 

  15. Sheffer. A., Praun, E., Rose, K.: Mesh Parameterization Methods and Their Applications. Found. Trends Comput. Graphics Vision 2(2), 105–171 (2006)

    Google Scholar 

  16. UNAMalla Workgroup: UNAMalla: Version 5.0. http://www.mathmoo.unam.mx/unamalla. Cited 23 May 2013

  17. Xiao, Z., He, S., Xu, G., Chen, J., Wu, Q.: A boundary element-based automatic domain partitioning approach for semi-structured quad mesh generation. Engrg. Anal. Boundary Elem. 11, 133–144. Elsevier (2020)

    Google Scholar 

  18. Xu, G., Mourrain, B., Duvigneau, R., Galligo, A.: Parametrization of computational domain in isogeometric analysis: methods and comparison. Comput. Methods Appl. Mech. Eng. 200, 23–24, 2021–2031. Elsevier (2011). https://doi.org/10.1016/j.cma.2011.03.005

  19. Xu, G., Li, M., Mourrain, B., Rabczuk, T., Xu, J., Bordas, S.P.A.: Constructing IGA-suitable planar parametrization from complex CAD boundary by domain partition and global/local optimization. Comput. Methods Appl. Mech. Eng. 328, 175–200 (2018)

    Article  Google Scholar 

  20. Zhang, Y., Jia, Y.: 2D automatic body-fitted structured mesh generation using advancing extraction method. J. Comput. Phys. 353, 316–335. Elsevier (2018). https://doi.org/10.1016/j.jcp.2017.10.018

  21. Zint, D., Grosso, R., Aizinger, V., Köstler, H.: Generation of block structured grids on complex domains for high performance simulation. In: Garanzha, V.A., et al. (eds.) Numerical Geometry, Grid Generation and Scientific Computing. Lecture Notes in Computational Science and Engineering, vol. 131, pp. 87–99. Springer, New York (2019). https://doi.org/10.1007/978-3-030-23436-2_6

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Sciences, National Autonomous University of Mexico, Mexico City, Mexico

    Pablo Barrera & Iván Méndez

Authors
  1. Pablo Barrera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Iván Méndez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Iván Méndez .

Editor information

Editors and Affiliations

  1. Dorodnicyn Computing Centre, Federal Research Center of Informatics and Control, Russian Academy of Sciences Moscow, Moscow, Russia

    Vladimir A. Garanzha

  2. Berlin, Germany

    Lennard Kamenski

  3. Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany

    Hang Si

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Barrera, P., Méndez, I. (2021). Parametrization of Plane Irregular Regions: A Semi-automatic Approach I. In: Garanzha, V.A., Kamenski, L., Si, H. (eds) Numerical Geometry, Grid Generation and Scientific Computing. Lecture Notes in Computational Science and Engineering, vol 143. Springer, Cham. https://doi.org/10.1007/978-3-030-76798-3_17

Download citation

Publish with us

Policies and ethics

Search

Navigation