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Numerical Algorithms

, Volume 77, Issue 4, pp 1029–1068 | Cite as

Robust intersection of structured hexahedral meshes and degenerate triangle meshes with volume fraction applications

  • Frida Svelander
  • Gustav Kettil
  • Tomas Johnson
  • Andreas Mark
  • Anders Logg
  • Fredrik Edelvik
Open Access
Original Paper
First Online:
Received:
Accepted:

Abstract

Two methods for calculating the volume and surface area of the intersection between a triangle mesh and a rectangular hexahedron are presented. The main result is an exact method that calculates the polyhedron of intersection and thereafter the volume and surface area of the fraction of the hexahedral cell inside the mesh. The second method is approximate, and estimates the intersection by a least squares plane. While most previous publications focus on non-degenerate triangle meshes, we here extend the methods to handle geometric degeneracies. In particular, we focus on large-scale triangle overlaps, or double surfaces. It is a geometric degeneracy that can be hard to solve with existing mesh repair algorithms. There could also be situations in which it is desirable to keep the original triangle mesh unmodified. Alternative methods that solve the problem without altering the mesh are therefore presented. This is a step towards a method that calculates the solid area and volume fractions of a degenerate triangle mesh including overlapping triangles, overlapping meshes, hanging nodes, and gaps. Such triangle meshes are common in industrial applications. The methods are validated against three industrial test cases. The validation shows that the exact method handles all addressed geometric degeneracies, including double surfaces, small self-intersections, and split hexahedra.

Keywords

Cut-cell Volume fraction Mesh repair Overlapping triangles Split hexahedra 
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Notes

Acknowledgments

This work was supported in part by the Swedish Governmental Agency for Innovation Systems, VINNOVA, through the FFI Sustainable Production Technology program, and in part by the Sustainable Production Initiative and the Production Area of Advance at Chalmers University of Technology. The support is gratefully acknowledged.

References

  1. 1.
    Aftosmis, M.J., Berger, M.J., Melton, J.E.: Robust and efficient Cartesian mesh generation for component-based geometry. AIAA J. 36(6), 952–960 (1998)CrossRefGoogle Scholar
  2. 2.
    Aftosmis, M.J., Berger, M.J., Melton, J.E.: Adaptive Cartesian mesh generation. In: Weatherill, N. P., Soni, B. K., Thompson, J. (eds.) Handbook of Grid Generation, pp 22–1–22-21. CRC Press, Boca Raton (1998)Google Scholar
  3. 3.
    Akenine-Möller, T.: Fast 3D triangle-box overlapping test. J. Graph. Tools 6(1), 29–33 (2001)CrossRefGoogle Scholar
  4. 4.
    Attene, M., Campen, M., Kobbelt, L.: Polygon mesh repairing: an application perspective. ACM Comput. Surv. 45(2), Article No. 15 (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    Attene, M.: Direct repair of self-intersecting meshes. Graph. Model. 76(14), 658–668 (2014)CrossRefGoogle Scholar
  6. 6.
    Brentzen, J.A., Aans, H.: Signed distance computation using the angle weighted pseudonormal. IEEE Trans. Vis. Comput. Graph. 11(3), 243–253 (2005)CrossRefGoogle Scholar
  7. 7.
    Barequet, G., Sharir, M.: Filling gaps in the boundary of a polyhedron. Comput.-Aided Geom. Des. 12(2), 207–229 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bashein, G., Detmer, P.R.: Centroid of a polygon. In: Heckbert, P. S. (ed.) Graphics Gems IV, pp 3–6. Academic, New York (1994)CrossRefGoogle Scholar
  9. 9.
    Bischoff, S., Pavic, D., Kobbelt, L.: Automatic restoration of polygon models. Trans. Graph. 24(4), 1332–1352 (2005)CrossRefGoogle Scholar
  10. 10.
    Botsch, M., Pauly, M., Kobbelt, L., Alliez, P., Levy, B.: Geometric modeling based on polygonal meshes http://lgg.epfl.ch/publications/2008/botsch_2008_GMPeg.pdf. Accessed 27 March 2016 (2007)
  11. 11.
    Cieslak, S., Ben Khelil, S., Choquet, I., Merlen, A.: Cut cell strategy for 3-D blast waves numerical simulations. Shock Waves 10, 421–429 (2001)CrossRefzbMATHGoogle Scholar
  12. 12.
    Ericson, C.: Real-Time Collision Detection. Morgan Kaufmann, Burlington (2004)Google Scholar
  13. 13.
    Gander, W., Hrebicek, J.: Solving Problems in Scientific Computing Using MAPLE and MATLAB. Springer Verlag, Heidelberg (1993)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gouraud, H.: Continuous shading of curved surfaces. IEEE Trans. Comput. 20(6), 124–133 (1971)zbMATHGoogle Scholar
  15. 15.
    Guéziec, A., Taubin, G., Lazarus, F., Horn, B.: Cutting and stitching: converting sets of polygons to manifold surfaces. IEEE Trans. Vis. Comput. Graph. 7(2), 136–151 (2001)CrossRefGoogle Scholar
  16. 16.
    IBOFlow. http://www.iboflow.com. Accessed 27 March 2016
  17. 17.
    Ju, T.: Robust repair of polygonal models. ACM Trans. Graph. (TOG) 23(3), 888–895 (2004)CrossRefGoogle Scholar
  18. 18.
    Liepa, P.: Filling holes in meshes. In: Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on geometry processing, pp 200–205 (2003)Google Scholar
  19. 19.
    Mark, A., Rundqvist, R., Edelvik, F.: Comparison between different immersed boundary conditions for simulation of complex fluid flows. Fluid Dyn. Mater. Process. 7(3), 241–258 (2011)Google Scholar
  20. 20.
    Mark, A., Svenning, E., Edelvik, F.: An immersed boundary method for simulation of flow with heat transfer. Int. J. Heat Mass Transf. 56, 424–435 (2013)CrossRefGoogle Scholar
  21. 21.
    Nooruddin, F.S., Turk, G.: Simplification and repair of polygonal models using volumetric techniques. IEEE Trans. Vis. Comput. Graph. 9(2), 191–205 (2003)CrossRefGoogle Scholar
  22. 22.
    Peskin, C.S.: Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220–252 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Quirk, J.J.: An alternative to unstructured grids for computing gas dynamic flows around arbitrarile complex two-dimensional bodies. Comput. Fluids 23, 125–142 (1994)CrossRefzbMATHGoogle Scholar
  24. 24.
    Schirra, S.: Robustness and precision issues in geometric computation. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp 597–632. Elsevier (1997)Google Scholar
  25. 25.
    Sutherland, I.E., Hodgman, G.W.: Reentrant polygon clipping. Comm. ACM 17(1), 32–42 (1974)CrossRefzbMATHGoogle Scholar
  26. 26.
    Yang, G., Causon, D.M., Ingram, D.M.: Calculation of compressible flows about complex moving geometries using a three-dimensional Cartesian cut cell method. Int. J. Numer. Meth. Fluids 33, 1121–1151 (2000)CrossRefzbMATHGoogle Scholar
  27. 27.
    Zhou, Q., Grinspun, E., Zorin, D., Jacobson, A.: Mesh arrangements for solid geometry. ACM Trans. Graph. 35(4), Article No. 39 (2016)CrossRefGoogle Scholar

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Fraunhofer-Chalmers Research Centre for Industrial MathematicsGöteborgSweden
  2. 2.Department of Mathematical SciencesChalmers University of Technology and University of GothenburgGothenburgSweden

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