Skip to main content

Higher-Order Accurate Meshing of Nonsmooth Implicitly Defined Surfaces and Intersection Curves

Abstract

A higher-order accurate meshing algorithm for nonsmooth surfaces defined via Boolean set operations from smooth surfaces is presented. Input data are a set of level-set functions and a bounding box containing the domain of interest. This geometry definition allows the treatment of edges as intersection curves. Initially, the given bounding box is partitioned with an octree that is used to locate corners and points on the intersection curves. Once a point on an intersection curve is found, the edge is traced. Smooth surfaces are discretized using marching cubes and then merged together with the advancing-front method. The piecewise linear geometry is lifted by projecting the inner nodes of the Lagrangian elements onto the surface or intersection curve. To maintain an accurate mesh, special attention is paid to the accurate meshing of tangential intersection curves. Optimal convergence properties for approximation problems are confirmed in numerical studies.

This is a preview of subscription content, access via your institution.

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.

REFERENCES

  1. N. H. Abdel-All, S. A.-N. Badr, M. A. Soliman, and S. A. Hassan, “Intersection curves of two implicit surfaces in R3,” J. Math. Comput. Sci. 2 (2), 152–171 (2012).

    MathSciNet  Google Scholar 

  2. S. S. Abhyankar and C. J. Bajaj, “Automatic parameterization of rational curves and surfaces. IV: Algebraic space curves,” ACM Trans. Graphics 8 (4), 325–334 (1989).

    Article  Google Scholar 

  3. C. Asteasu, “Intersection of arbitrary surfaces,” Comput.-Aided Des. 20 (9), 533–538 (1988).

    Article  Google Scholar 

  4. C. L. Bajaj, C. M. Hoffmann, R. E. Lynch, and J. E. H. Hopcroft, “Tracing surface intersections,” Comput.-Aided Geom. Des. 5 (4), 285–307 (1988).

    MathSciNet  Article  Google Scholar 

  5. A. I. Belokrys-Fedotov, V. A. Garanzha, and L. N. Kudryavtseva, “Generation of Delaunay meshes in implicit domains with edge sharpening,” Comput. Math. Math. Phys. 56 (11), 1901–1918 (2016).

    MathSciNet  Article  Google Scholar 

  6. H. Borouchaki and P. L. George, Meshing, Geometric Modeling and Numerical Simulation: 1. Form Functions, Triangulations, and Geometric Modeling (Wiley-ISTE, 2017).

    Book  Google Scholar 

  7. J. P. Boyd, “Computing the zeros, maxima, and inflection points of Chebyshev, Legendre and Fourier series: Solving transcendental equations by spectral interpolation and polynomial rootfinding,” J. Eng. Math. 56 (3), 203–219 (2006).

    MathSciNet  Article  Google Scholar 

  8. S. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, 3rd ed. (Springer, Berlin, 2008).

    Book  Google Scholar 

  9. E. Burman, P. Hansbo, M. G. Larson, G. Mats, and A. Massing, “Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions,” ESAIM: Math. Model Numer. Anal. (2018).

  10. B. R. de Araújo, D. S. Lopes, P. Jepp, J. A. Jorge, and B. Wyvill, “A survey on implicit surface polygonization,” ACM Comput. Surv. 47 (4), Article 60 (2015).

    Article  Google Scholar 

  11. S. Dey, R. M. O’Bara, and M. S. Shephard, “Towards curvilinear meshing in 3D: The case of quadratic simplices,” Comput.-Aided Des. 33 (3), 199–209 (2001).

    Article  Google Scholar 

  12. S. Dey, M. S. Shephard, and E. J. Flaherty, “Geometry representation issues associated with p-version finite element computations,” Comput. Methods Appl. Mech. Eng. 150 (1), 39–55 (1997).

    MathSciNet  Article  Google Scholar 

  13. S. Dey, R. M. O’bara, and M. S. Shephard, “Curvilinear mesh generation in 3D,” Proceedings of the 8th International Meshing Roundtable (1999), pp. 407–417.

  14. B. U. Düldül and M. Düldül, “Can we find Willmore-like method for the tangential intersection problems?” J. Comput. Appl. Math. 302, 301–311 (2016).

    MathSciNet  Article  Google Scholar 

  15. G. Dziuk, “Finite elements for the Beltrami operator on arbitrary surfaces,” in Partial Differential Equations and Calculus of Variations, Ed. by S. Hildebrandt and R. Leis, Lecture Notes in Mathematics (Springer, Berlin, 1988), Vol. 1357, pp. 142–155.

  16. P. Frey and P.-L. George, Mesh Generation: Application to Finite Elements (Wiley, Chichester, 2008).

    Book  Google Scholar 

  17. T. P. Fries, “Higher-order conformal decomposition FEM (CDFEM),” Comput. Methods Appl. Mech. Eng. 328, 75–98 (2018).

    MathSciNet  Article  Google Scholar 

  18. T. P. Fries, S. Omerovic, D. Schöllhammer, and J. Steidl, “Higher-order meshing of implicit geometries: Part I. Integration and interpolation in cut elements,” Comput. Methods Appl. Mech. Eng. 313, 759–784 (2017).

    MathSciNet  Article  Google Scholar 

  19. T. P. Fries and S. Omerovic, “Higher-order accurate integration of implicit geometries,” Int. J. Numer. Methods Eng. 106 (5), 323–371 (2016).

    MathSciNet  Article  Google Scholar 

  20. T. P. Fries and D. Schöllhammer, “Higher-order meshing of implicit geometries: Part II. Approximations on manifolds,” Comput. Methods Appl. Mech. Eng. 326, 270–297 (2017).

    MathSciNet  Article  Google Scholar 

  21. A. Gomes, Implicit Curves and Surfaces: Mathematics, Data Structures, and Algorithms (Springer, Dordrecht, 2009).

    Book  Google Scholar 

  22. W. J. Gordon and C. A. Hall, “Construction of curvilinear coordinate systems and applications to mesh generation,” Int. J. Numer. Methods Eng. 7 (4), 461–477 (1973).

    Article  Google Scholar 

  23. W. J. Gordon and C. A. Hall, “Transfinite element methods: Blending-function interpolation over arbitrary curved element domains,” Numer. Math. 21 (2), 109–129 (1973).

    MathSciNet  Article  Google Scholar 

  24. E. Hartmann, “A marching method for the triangulation of surfaces,” Visual Comput. 14 (3), 95–108 (1998).

    Article  Google Scholar 

  25. C.-Y. Hu, T. Maekawa, N. M. Patrikalakis, M. Nicholas, and X. Ye, “Robust interval algorithm for surface intersections,” Comput.-Aided Des. 29 (9), 617–627 (1997).

    Article  Google Scholar 

  26. P. M. Knupp, “Algebraic mesh quality metrics,” SIAM J. Sci. Comput. 23 (1), 193–218 (2001).

    MathSciNet  Article  Google Scholar 

  27. D. S. H. Lo, Finite Element Mesh Generation (CRC Press, Boca Raton, FL, 2014).

    Book  Google Scholar 

  28. S. H. Lo, “A new mesh generation scheme for arbitrary planar domains,” Int. J. Numer. Methods Eng. 21 (8), 1403–1426 (1985).

    Article  Google Scholar 

  29. R. Löhner and P. Parikh, “Generation of three-dimensional unstructured grids by the advancing-front method,” Int. J. Numer. Methods Fluids 8 (10), 1135–1149 (1988).

    Article  Google Scholar 

  30. W. E. Lorensen and H. E. Cline, “Marching cubes: A high resolution 3D surface construction algorithm,” ACM SIGGRAPH Comput. Graphics 21 (4), 163–169 (1987).

    Article  Google Scholar 

  31. B. Marussig and T. J. R. Hughes, “A review of trimming in isogeometric analysis: Challenges, data exchange, and simulation aspects,” Arch. Comput. Methods Eng. 25 (4), 1059–1127 (2018).

    MathSciNet  Article  Google Scholar 

  32. K. Nakahashi and D. Sharov, “Direct surface triangulation using the advancing front method,” AIAA Paper 95-1686-CP, 442–451 (1995).

  33. Y. Ohtake and A. G. Belyaev, “Dual/primal mesh optimization for polygonised implicit surfaces,” Proceedings of the Seventh ACM Symposium on Solid Modeling and Applications (ACM, 2002), pp. 171–178.

  34. A. Pasko, V. Adzhiev, A. Sourin, and V. Savchenko, “Function representation in geometric modeling: Concepts, implementation, and applications,” Visual Comput. 11 (8), 429–446 (1995).

    Article  Google Scholar 

  35. N. M. Patrikalakis, “Interrogation of surface intersections,” Geometry Processing for Design and Manufacturing (SIAM, 1992), pp. 161–185.

    Google Scholar 

  36. N. M. Patrikalakis, “Surface-to-surface intersections,” IEEE Comput. Graphics Appl. 13 (1), 89–95 (1993).

    Article  Google Scholar 

  37. P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,” SIAM Rev. 46 (2), 329–345 (2004).

    MathSciNet  Article  Google Scholar 

  38. A. A. G. Requicha, “Representations of rigid solid objects,” Computer Aided Design Modeling, Systems Engineering, CAD-Systems (Springer, Berlin, 1980), pp. 1–78.

    Google Scholar 

  39. X. Roca, A. Gargallo-Peiró, and J. Sarrate, “Defining quality measures for high-order planar triangles and curved mesh generation,” Proceedings of the 20th International Meshing Roundtable (2011), pp. 365–383.

  40. P. Solin, K. Segeth, and I. Dolezel, Higher-Order Finite Element Methods (Chapman and Hall/CRC, London, 2003).

    Book  Google Scholar 

  41. J. W. Stanford and T. P. Fries, “Higher-order accurate meshing of implicitly defined tangential and transversal intersection curves,” Lecture Notes in Computer Science and Engineering (2018) (accepted).

  42. J. W. Stanford and T. P. Fries, “A higher-order conformal decomposition finite element method for plane B-rep geometries,” Comput. Struct. 214, 15–27 (2019).

    Article  Google Scholar 

  43. M. Turner, High-Order Mesh Generation for CFD Solvers, PhD Thesis (2017).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. W. Stanford.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Stanford, J.W., Fries, T.P. Higher-Order Accurate Meshing of Nonsmooth Implicitly Defined Surfaces and Intersection Curves. Comput. Math. and Math. Phys. 59, 2093–2107 (2019). https://doi.org/10.1134/S0965542519120169

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542519120169

Keywords:

  • higher-order finite elements
  • meshing
  • higher-order
  • implicit surface
  • intersection problems