Skip to main content
SpringerLink
Log in

The simplex subdivision of a complex region: a positive and negative finite element superposition principle

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

In the case of very complex geometric morphology in three-dimensional (3-d) calculation regions, the subdivision of finite element meshes tends to be very difficult or even impossible. This problem was especially dominant in the element subdivision of rock blocks cut by arbitrary fracture networks. This study proposed a directed simplex subdivision method of 2- and 3-d regions which can readily subdivide a complex region and automatically adapt to arbitrary variations of region boundaries. The directed simplex was employed as finite elements for the first time. When the nodes of 2-d simplex (triangle) were connected anticlockwise and the nodes of 3-d simplex (tetrahedron) were linked by the right hand screw rule, the finite element is treated as a positive element. Otherwise, it is treated as a negative element. However, some generated elements presented superposition, which was unallowable in the normal finite element method (FEM). Therefore, the superposition principle of positive and negative finite elements was proposed. Such superposition method can transform a normal FEM mesh into a positive–negative FEM mesh with maintaining its completeness and compatibility. Finally, 1-d, 2-d, and 3-d cases were used as examples to verify the validity of the proposed method. The completeness and compatibility of the positive–negative FEM and the normal FEM are consistent. The accuracy of these two methods differs due to the difference of nodal connection relationships, node numbers, and element sizes. However, as the mesh is refined, the calculated results tend to be consistent.

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
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. Baker TJ (1989) Automatic mesh generation for complex three-dimensional regions using a constrained Delaunay triangulation. Eng Comput 5:161–175

    Article  Google Scholar 

  2. Boltcheva D, Yvinec M, Boissonnat J-D (2009) Feature preserving Delaunay mesh generation from 3D multi-material images. Comput Graph Forum 28(5):1455–1464

    Article  Google Scholar 

  3. Borouchaki H, Lo SH (1995) Fast Dealunay triangulation in three dimensions. Comput Methods Appl Mech Eng 128:153–167

    Article  MATH  Google Scholar 

  4. Cavalcante Neto JB, Wawrzynek PA, Carvalho MTM, Martha LF, Ingraffea AR (2001) An algorithm for three-dimensional mesh generation for arbitrary regions with cracks. Eng Comput 17:75–91

    Article  MATH  Google Scholar 

  5. Chen J, Zhao D, Huang Z, Zheng Y, Gao S (2011) Three-dimensional constrained boundary recovery with an enhanced Steiner point suppression procedure. Comput Struct 89:455–466

    Article  Google Scholar 

  6. Du Q, Wang D (2004) Boundary recovery for three dimensional conforming Delaunay triangulation. Comput Methods Appl Mech Eng 193:2547–2563

    Article  MathSciNet  MATH  Google Scholar 

  7. Du Q, Wang D (2005) The optimal centroidal voronoi tessellations and the Gersho’s conjecture in the three-dimensional space. Comput Math Appl 49:1355–1373

    Article  MathSciNet  MATH  Google Scholar 

  8. Du Q, Wang D (2006) Recent progress in robust and quality Delaunay mesh generation. J Comput Appl Math 195:8–23

    Article  MathSciNet  MATH  Google Scholar 

  9. Frey PJ, Borouchaki H, George P-L (1998) 3D Delaunay mesh generation coupled with an advancing-front approach. Comput Methods Appl Mech Eng 157:115–131

    Article  MathSciNet  MATH  Google Scholar 

  10. Ghadyani H, Sullivan J, Wu Z (2010) Boundary recovery for Delaunay tetrahedral meshes using local topological transformations. Finite Elem Anal Des 46:74–83

    Article  MathSciNet  Google Scholar 

  11. Jing L (2000) Block system construction for three-dimensional discrete element models of fractured rocks. Int J Rock Mech Min Sci 37(4):645–659

    Article  Google Scholar 

  12. Jin L, Stephansson O (1994) Topological identification of block assemblages for jointed rock masses. Int J Rock Mech Min Sci 31(2):163–172

    Article  Google Scholar 

  13. Karamete BK, Beall M, Shephard M (2000) Triangulation of arbitrary polyhedra to support automatic mesh generators. Int J Numer Methods Eng 49:167–191

    Article  MATH  Google Scholar 

  14. Lederman C, Joshi A, Dinov I, Vese L, Toga A, Van Horn JD (2011) The generation of tetrahedral mesh models for neuroanatomical MRI. NeuroImage 55(1):153–164

    Article  Google Scholar 

  15. Lewis RW, Zheng Y, Usmani AS (1995) Aspects of adaptive mesh generation based on domain decomposition and Delaunay triangulation. Finite Elem Anal Des 20(1):47–70

    Article  MATH  Google Scholar 

  16. Lewis RW, Zheng Y, Gethin DT (1996) Three-dimensional unstructured mesh generation: Part 3. Volume meshes. Comput Methods Appl Mech Eng 134:285–310

    Article  MATH  Google Scholar 

  17. Lin D, Fairhurst C, Starfield AM (1987) Geometrical identification of three-dimensional rock block systems using topological techniques. Int J Rock Mech Min Sci 24(6):331–338

    Article  Google Scholar 

  18. Liu Y, Lo S, Guan Z, Zhang H (2014) Boundary recovery for 3D Delaunay triangulation. Finite Elem Anal Des 84:32–43

    Article  MathSciNet  Google Scholar 

  19. Lo SH, Wang WX (2005) Finite element mesh generation over intersecting curved surfaces by tracing of neighbours. Finite Elem Anal Des 41:351–370

    Article  Google Scholar 

  20. Löhner R (1996) Progress in grid generation via the advancing front technique. Eng Comput 12:186–199

    Article  Google Scholar 

  21. Radovitzky R, Ortiz M (2000) Tetrahedral mesh generation based on node insertion in crystal lattice arrangements and advancing-front-Delaunay triangulation. Comput Methods Appl Mech Eng 187:543–569

    Article  MathSciNet  MATH  Google Scholar 

  22. Rassineux A (1997) 3D mesh adaptation. Optimization of tetrahedral meshes by advancing front technique. Comput Methods Appl Mech Eng 141:335–354

    Article  MATH  Google Scholar 

  23. Schroeder WJ, Shephard MS (1990) A combined Octree/Delaunay method for fully automatic 3-D mesh generation. Int J Numer Methods Eng 29(1):37–55

    Article  MATH  Google Scholar 

  24. Secchi S, Simoni L (2003) An improved procedure for 2D unstructured Delaunay mesh generation. Adv Eng Softw 34:217–234

    Article  MATH  Google Scholar 

  25. Shi G, Goodman RE (1989) The key blocks of unrolled joint traces in developed maps of tunnel walls. Int J Numer Anal Methods Geomech 13:131–158

    Article  Google Scholar 

  26. Shi G (1995) Simplex integration for manifold method and discontinuous deformation analysis. In: Proceedings of the First International Conference on Analysis of Discontinuous Deformation. Chungli, Taiwan, China, pp 1–25

  27. Shi G (2006) Producing joint polygons, cutting joint blocks and finding key blocks from general free surfaces. Chin J Rock Mech Eng 25(11):2161–2170

    Google Scholar 

  28. Tabarraei A, Sukumar N (2005) Adaptive computations on conforming quadtree meshes. Finite Elem Anal Des 41(7–8):686–702

    Article  Google Scholar 

  29. Tian J, Jiang W, Luo T, Cai K, Peng J, Wang W (2012) Adaptive coding of generic 3D triangular meshes based on octree decomposition. Vis Comput 28:819–827

    Article  Google Scholar 

  30. Xing H, Liu Y, 2014. Mesh Generation for 3D geological reservoirs with arbitrary stratigraphic surface constraints. In: 14th international conference on computational science, pp 897–909

  31. Zhang Q-H, Yin J-M (2014) Solution of two key issues in arbitrary three-dimensional discrete fracture network flow models. J Hydrol 514:281–296

    Article  Google Scholar 

  32. Zhang Q-H (2015) Advances of study on three-dimensional block cutting analysis and its application. Comput Geotech 63:26–32

    Article  Google Scholar 

Download references

Acknowledgements

This research was sponsored by the General Program of National Natural Science Foundation of China (Grant No. 51679012) and the Key Program of National Natural Science Foundation of China (Grant No. 51539002).

Author information

Authors and Affiliations

  1. Yangtze River Scientific Research Institute, Wuhan, 430010, Hubei, People’s Republic of China

    Qi-Hua Zhang, Hai-Dong Su & Shao-Zhong Lin

Authors
  1. Qi-Hua Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hai-Dong Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shao-Zhong Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qi-Hua Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, QH., Su, HD. & Lin, SZ. The simplex subdivision of a complex region: a positive and negative finite element superposition principle. Engineering with Computers 34, 155–173 (2018). https://doi.org/10.1007/s00366-017-0527-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-017-0527-9

Keywords

Search

Navigation