Skip to main content
SpringerLink
Log in

Discontinuous Galerkin Based Isogeometric Analysis for Geometric Flows and Applications in Geometric Modeling

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We propose a method which combines isogeometric analysis with the discontinuous Galerkin (DG) method for second and fourth order geometric flows to generate fairing surfaces, which are composed of multiple patches. This technique can be used to tackle a challenging problem in geometric modeling–gluing multi-patches together smoothly to create complex models. Non-uniform rational B-splines (NURBS), the most popular representations of geometric models developed in Computer Aided Design, are employed to describe the geometry and represent the numerical solution. Since NURBS basis functions over two different patches are independent, DG methods can be appropriately applied to glue the multiple patches together to obtain smooth solutions. We present semi-discrete DG schemes to solve the problem, and \(\mathcal {L}^{2}\)-stability is proved for the proposed schemes. Our method enjoys the following advantages. Firstly, the geometric flexibility of NURBS basis functions, especially the use of multiple patches, enable us to construct surface models with complex geometry and topology. Secondly, the constructed geometry is fair. Thirdly, since only the control points of the NURBS patches evolve in accordance with the geometric flows, and their number (degrees of freedom) is very small, our algorithm is very efficient. Finally, this method can be easily formulated and implemented. We apply the method in mean curvature flows and in quasi surface diffusion flows to solve various geometric modeling problems, such as minimal surface generation, surface blending and hole filling, etc. Examples are provided to illustrate the effectiveness of our method.

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

Similar content being viewed by others

References

  1. Apostolatos, A., Schmidt, R., Wuchner, R., Bletzinger, K.U.: A Nitsche-type formulation and comparison of the most common domain decomposition methods in isogeometric analysis. Int. J. Numer. Methods Eng. 97, 473–504 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bazilevs, Y., Beirao de Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci. 16(07), 1031–1090 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bazilevs, Y., Calo, V.M., Cottrell, J.A., Hughes, T.J.R., Reali, A., Scovazzi, G.: Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput. Methods Appl. Mech. Eng. 197, 173–201 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bazilevs, Y., Calo, V.M., Hughes, T.J.R., Zhang, Y.: Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput. Mech. 43, 3–37 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bloor, M.I.G., Wilson, M.J.: Generating blend surfaces using partial differential equations. Comput. Aided Des. 21(3), 165–171 (1989)

    Article  MATH  Google Scholar 

  7. Bloor, M.I.G., Wilson, M.J.: Generating N-sided patches with partial differential equations. In: Advances in Computer Graphics, pp. 129–145. Springer, Berlin, (1989)

  8. Bloor, M.I.G., Wilson, M.J.: Using partial differential equations to generate free-form surfaces. Comput. Aided Des. 22(4), 221–234 (1990)

    Article  MATH  Google Scholar 

  9. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  10. Brivadis, E., Buffa, A., Wohlmuth, B., Wunderlich, L.: Isogeometric mortar methods. Comput. Methods Appl. Mech. Eng. 284, 292–319 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, C., Xu, G.: Construction of geometric partial differential equations for level sets. J. Comput. Math. 28(1), 105–121 (2010)

  12. Ciarlet, P.G., Raviart, P.A.: A mixed finite element method for the biharmonic equation. SIAM J. Numer. Anal. 24(4), 737–749 (1987)

    Article  MathSciNet  Google Scholar 

  13. Clarenz, U., Diewald, U., Dziuk, G., Rumpf, M., Rusu, R.: A finite element method for surface restoration with boundary conditions. Comput. Aided Geom. Des. 21(5), 427–445 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Clarenz, U., Diewald, U., Rumpf, M.: Anisotropic geometric diffusion in surface processing. In: Proceedings of Viz2000, IEEE Visualization, Salt Lake City, Utah, pp. 397–405 (2000)

  15. Cockburn, B., Karniadakis, G., Shu, C.W.: The development of discontinuous Galerkin methods. In: Cockburn, B., Karniadakis, G., Shu, C.W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering. Part I: Overview, vol. 11, pp. 3–50. Springer, Berlin (2000)

    Google Scholar 

  16. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester (2009)

    Book  Google Scholar 

  17. Cottrell, J.A., Hughes, T.J.R., Reali, A.: Studies of refinement and continuity in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 196, 4160–4183 (2007)

    Article  MATH  Google Scholar 

  18. Cottrell, J.A., Reali, A., Bazilevs, Y., Hughes, T.J.R.: Isogeometric analysis of structural vibrations. Comput. Methods Appl. Mech. Eng. 195, 5257–5296 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Deckelnick, K., Dziuk, G.: A fully descrite numerical scheme for weighted mean curvature flow. Numer. Math. 91, 423–452 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14, 139–232 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dedner, A., Madhavan, P., Stinner, B.: Analysis of the discontinuous Galerkin method for elliptic problems on surfaces. IMA J. Numer. Anal. 33(3), 952–973 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: SIGGRAPH99, Los Angeles, USA, pp. 317C324 (2002)

  23. Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  24. Farin, G.: Curves and Surfaces for CAGD, 5th edn. Morgan Kaufmann Publishers, Burlington (2002)

    Google Scholar 

  25. Gudi, T., Nataraj, N., Pani, A.K.: Mixed discontinuous Galerkin finite element method for the biharmonic equation. J. Sci. Comput. 37(2), 139–161 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Langer, U., Moore, S.E.: Discontinuous Galerkin Isogeometric Analysis of Elliptic PDEs on Surfaces, arXiv:1402.1185 (2014)

  28. Li, M., Xu, G.: \(G^{1}\) B-spline surface construction by geometric partial differential equations. J. Comput. Aided Des. Comput. Graphics 22(7), 1087–1093 (2010)

    Google Scholar 

  29. Ohtake, Y., Belyaev, A.G., Bogaevski, I.A.: Polyhedral surface smoothing with simultaneous mesh regularization. In: Geometric Modeling and Processing Proceedings, pp. 229–237 (2000)

  30. Riviere, B.: Discontinuous Galerkin methods for Solving Elliptic and Parabolic Equations. Society for Industrial and Applied Mathematics, SIAM, Philadelphia (2008)

    Book  MATH  Google Scholar 

  31. Schneider, R., Kobbelt, L.: Generating fair meshes with \(G^{1}\) boundary conditions. In: Geometric Modeling and Processing, Hong Kong, China, pp. 251–261 (2000)

  32. Schneider, R., Kobbelt, L.: Geometric fairing of irregular meshes for free-form surface design. Comput. Aided Geom. Des. 18(4), 359–379 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Xu, G., Zhang, Q.: A general framework for surface modeling using geometric partial differential equations. Comput. Aided Geom. Des. 18(4), 359–379 (2001)

    Article  Google Scholar 

  34. Xu, G., Zhang, Q.: Geometric Partial Differential Equation Methods in Computational Geometry. Science Press, Beijing (2013)

    Google Scholar 

  35. Yoshizawa, S., Belyaev, A.G.: Fair triangle mesh generation with discrete elastica. In: Geometric Modeling and Processing, Saitama, Japan, pp. 119–123 (2002)

  36. Zhang, F., Xu, Y., Chen, F.: Discontinuous Galerkin methods for isogeometric analysis for elliptic equations on surfaces. Commun. Math. Stat. 2, 431–461 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhao, H., Osher, S., Merriman, B., Kang, M.: Implicit and nonparametric shape reconstruction from unorganized data using a variational level set method. Comput. Vis. Image Underst. 80, 295–314 (2000)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Futao Zhang, Yan Xu & Falai Chen

Authors
  1. Futao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Falai Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Falai Chen.

Additional information

The Research of Yan Xu is supported by NSFC Grant Nos. 11371342 and 11526212. The Research of Falai Chen is supported by NSFC Grant No.11571138.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, F., Xu, Y. & Chen, F. Discontinuous Galerkin Based Isogeometric Analysis for Geometric Flows and Applications in Geometric Modeling. J Sci Comput 71, 525–546 (2017). https://doi.org/10.1007/s10915-016-0307-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-016-0307-5

Keywords

Search

Navigation