Skip to main content
SpringerLink
Log in

High Quality Mesh Morphing Using Triharmonic Radial Basis Functions

  • Conference paper
Proceedings of the 21st International Meshing Roundtable

Summary

The adaptation of an existing volumetric simulation mesh to updated parameters of the underlying CAD geometry is a crucial component within automatic design optimization. By avoiding costly automatic or even manual (re-)meshing it enables the automatic generation and evaluation of new design variations, e.g., through FEM or CFD simulations. This is particularly important for stochastic global optimization techniques—such as evolutionary algorithms—which typically require a large number of design variations to be created and evaluated. In this paper we present a simple yet versatile method for high quality mesh morphing. Building upon triharmonic radial basis functions, our shape deformations minimize distortion and thereby implicitly preserve shape quality. Moreover, the same unified code can be used to morph tetrahedral, hexahedral, or arbitrary polyhedral volume meshes. We compare our method to several other recently proposed techniques and show that ours yields superior results in most cases.

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 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia (1999)

    Book  Google Scholar 

  2. Baker, T.J.: Mesh movement and metamorphosis. In: Proceedings of the 10th International Meshing Roundtable, pp. 387–396 (2001)

    Google Scholar 

  3. Bechmann, D.: Space deformation models survey. Computers & Graphics 18(4), 571 (1994)

    Article  Google Scholar 

  4. de Boer, A., van der Schoot, M., Bijl, H.: Mesh deformation based on radial basis function interpolation. Computers & Structures 85, 784–795 (2007)

    Article  Google Scholar 

  5. Botsch, M., Kobbelt, L.: Real-time shape editing using radial basis functions. Computer Graphics Forum (Proc. Eurographics) 24(3), 611–621 (2005)

    Article  Google Scholar 

  6. Brewer, M., Diachin, L.F., Knupp, P., Leurent, T., Melander, D.: The Mesquite mesh quality improvement toolkit. In: Proceedings of the 12th International Meshing Roundtable, pp. 239–250 (2003)

    Google Scholar 

  7. Carr, J.C., Beatson, R.K., Cherrie, J.B., Mitchell, T.J., Fright, W.R., McCallum, B.C., Evans, T.R.: Reconstruction and representation of 3D objects with radial basis functions. In: Proc. of ACM SIGGRAPH, pp. 67–76. ACM, New York (2001)

    Google Scholar 

  8. Floater, M.S., Kos, G., Reimers, M.: Mean value coordinates in 3D. Computer Aided Geometric Design 22, 623–631 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. von Funck, W., Theisel, H., Seidel, H.P.: Vector field-based shape deformations. ACM Transactions on Graphics (Proc. SIGGRAPH) 25(3), 1118–1125 (2006)

    Article  Google Scholar 

  10. Gain, J., Bechmann, D.: A survey of spatial deformation from a user-centered perspective. ACM Transaction on Graphics 27, 107:1–107:21 (2008)

    Google Scholar 

  11. Helenbrook, B.T.: Mesh deformation using the biharmonic operator. International Journal for Numerical Methods in Engineering 56, 1007–1021 (2003)

    Article  MATH  Google Scholar 

  12. Hormann, K., Sukumar, N.: Maximum entropy coordinates for arbitrary polytopes. Computer Graphics Forum (Proc. Symp. Geometry Processing) 27(5), 1513–1520 (2008)

    Article  Google Scholar 

  13. Jakobsson, S., Amoignon, O.: Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization. Computers & Fluids 36(6), 1119–1136 (2007)

    Article  MATH  Google Scholar 

  14. Joshi, P., Meyer, M., DeRose, T., Green, B., Sanocki, T.: Harmonic coordinates for character articulation. ACM Transactions on Graphics (Proc. SIGGRAPH) 26(3) (2007)

    Google Scholar 

  15. Knupp, P.: Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part I. International Journal for Numerical Methods in Engineering 48(3), 401–420 (2000)

    Article  MATH  Google Scholar 

  16. Knupp, P.: Updating meshes on deforming domains: An application of the target-matrix paradigm. Commun. Num. Meth. Engr. 24(6), 467–476 (2007)

    Article  MathSciNet  Google Scholar 

  17. Martinez Esturo, J., Rössl, C., Fröhlich, S., Botsch, M., Theisel, H.: Pose correction by space-time integration. In: Proc. of Vision, Modeling, Visualization, pp. 33–40 (2011)

    Google Scholar 

  18. Michler, A.K.: Aircraft control surface deflection using RBF-based mesh deformation. International Journal for Numerical Methods in Engineering 88(10), 986–1007 (2011)

    Article  MATH  Google Scholar 

  19. OpenCASCADE: Open CASCADE Technology, 3D modeling & numerical simulation (2012), http://www.opencascade.org/

  20. Samareh, J.A.: A survey of shape parameterization techniques. Tech. Rep. NASA/CP-1999-209136/PT1, NASA Langley Research Center (1999)

    Google Scholar 

  21. Samet, H.: The Design and Analysis of Spatial Data Structures. Addison Wesley, Reading (1994)

    Google Scholar 

  22. Sederberg, T.W., Parry, S.R.: Free-form deformation of solid geometric models. In: Proc. of ACM SIGGRAPH, pp. 151–159. ACM, New York (1986)

    Google Scholar 

  23. Shontz, S.M., Vavasis, S.A.: A mesh warping algorithm based on weighted Laplacian smoothing. In: Proceedings of the 12th International Meshing Roundtable, pp. 147–158 (2003)

    Google Scholar 

  24. Sibson, R.: A brief description of natural neighbor interpolation. In: Interpreting Multivariate Data, vol. 21, John Wiley & Sons (1981)

    Google Scholar 

  25. Staten, M.L., Canann, S.A., Owen, S.J.: BMSWEEP: Locating interior nodes during sweeping. Eng. Comput. 15(3), 212–218 (1999)

    Article  MATH  Google Scholar 

  26. Staten, M.L., Owen, S.J., Shontz, S.M., Salinger, A.G., Coffey, T.S.: A Comparison of Mesh Morphing Methods for 3D Shape Optimization. In: Quadros, W.R. (ed.) Proceedings of the 20th International Meshing Roundtable, vol. 90, pp. 293–311. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  27. Sukumar, N.: Construction of polygonal interpolants: A maximum entropy approach. International Journal for Numerical Methods in Engineering 61(12), 2159–2181 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sukumar, N., Malsch, E.A.: Recent advances in the construction of polygonal finite element interpolants. Archives of Computational Methods in Engineering 13(1), 129–163 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wachspress, E.L.: A Rational Finite Element Basis. Academic Press (1975)

    Google Scholar 

  30. Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bielefeld University, Bielefeld, Germany

    Daniel Sieger & Mario Botsch

  2. Honda Research Institute Europe, Offenbach, Germany

    Stefan Menzel

Authors
  1. Daniel Sieger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Menzel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mario Botsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Sieger .

Editor information

Editors and Affiliations

  1. , Department of Applied Mathematics &, Stony Brook University, Stony Brook, 11794-3600, New York, USA

    Xiangmin Jiao

  2. , Bruyeres le Chatel, CEA, Arpajon Cedex, 91297, France

    Jean-Christophe Weill

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Sieger, D., Menzel, S., Botsch, M. (2013). High Quality Mesh Morphing Using Triharmonic Radial Basis Functions. In: Jiao, X., Weill, JC. (eds) Proceedings of the 21st International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33573-0_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-33573-0_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33572-3

  • Online ISBN: 978-3-642-33573-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation