Skip to main content

Unstructured Methods

  • Chapter
  • First Online:
Grid Generation Methods

Part of the book series: Scientific Computation ((SCIENTCOMP))

  • 1542 Accesses

Abstract

Unstructured mesh techniques occupy an important niche in grid generation. The major feature of unstructured grids consists, in contrast to structured grids, of a nearly absolute absence of any restrictions on grid cells, grid organization, or grid structure.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
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

References

  • Anderson, W. K. (1994). A grid generation and flow solution method for the Euler equations in unstructured grids. Journal of Computational Physics, 110, 23–38.

    Article  MATH  ADS  Google Scholar 

  • Baker, T. J. (1987). Three-dimensional mesh generation by triangulation of arbitrary points sets. AIAA Paper 87-1124-CP.

    Google Scholar 

  • Baker, T. J. (1995). Prospects and expectations for unstructured methods. In Proceedings of the Surface Modeling, Grid Generation and Related Issues in Computational Fluid Dynamics Workshop. NASA Conference Publication 3291. (pp. 273–287). Cleveland, OH: NASA Lewis Research Center.

    Google Scholar 

  • Baker, T. J. (1989). Automatic mesh generation for complex three-dimensional region using a constrained Delaunay triangulation. Engineering with Computers, 5, 161–175.

    Article  Google Scholar 

  • Baker, T. J. (1994). Triangulations, mesh generation and point placement strategies. In D. Caughey (Ed.), Computing the Future (pp. 1–15). New York: Wiley.

    Google Scholar 

  • Blacker, T. D., & Stephenson, M. B. (1991). Paving a new approach to automated quadrilateral mesh generation. International Journal for Numerical Methods in Engineering, 32, 811–847.

    Article  MATH  ADS  Google Scholar 

  • Bowyer, A. (1981). Computing Dirichlet tessellations. The Computer Journal, 24(2), 162–166.

    Article  MathSciNet  Google Scholar 

  • Brostow, W., Dussault, J. P., & Fox, B. L. (1978). Construction of Voronoi polyhedra. Journal of Computational Physics, 29, 81–92.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Carey, G. F. (1997). Computational grids: Generation, adaptation, and solution strategies. London: Taylor and Francis.

    Google Scholar 

  • Cavendish, J. C., Field, D. A., & Frey, W. H. (1985). An approach to automatic three-dimensional finite element mesh generation. International Journal for Numerical Methods in Engineering, 21, 329–347.

    Article  MATH  ADS  Google Scholar 

  • Chew, P. (1993). Mesh generation, curved surfaces and guaranteed quality triangles. Technical report, IMA, Workshop on Modeling, Mesh Generation and Adaptive Numerical Methods for Partial Differential Equations, University of Minnesota, Minneapolis.

    Google Scholar 

  • Chew, L. P. (1989). Constrained Delaunay triangulations. Algorithmica, 4, 97–108.

    Article  MathSciNet  MATH  Google Scholar 

  • Chrisochoides, N. (2006). Chapter 7. Parallel Mesh Generation. In Numeriacal Solution of PDEs on Parallel Computers. Lecture Notes in Computational Science and Engineering (vol. 51, pp. 237–264).

    Google Scholar 

  • Cline, A. K., & Renka, R. L. (1990). A constrained two-dimensional triangulation and the solution of closest node problems in the presence of barriers. SIAM Journal on Numerical Analysis, 27, 1305–1321.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • DeFloriani, L. (1987). Surface representations on triangular grids. The Visual Computer, 3, 27–50.

    Article  Google Scholar 

  • Delaunay, B. (1947). Petersburg School of Number Theory. USSR, Moscow (Russian): Ak. Sci.

    Google Scholar 

  • Delaunay, B. (1934). Sur la sphere vide. Bull. Acad Sci. USSR VII: Class Sci. Mat. Nat., 6, 793–800.

    MATH  Google Scholar 

  • Dirichlet, G. L. (1850). Uber die Reduction der positiven quadratischen Formen mit drei underbestimmten ganzen Zahlen. Z. Reine Angew. Math., 40(3), 209–227.

    Article  Google Scholar 

  • Edelsbrunner, H. (1987). Algorithms in combinatorial geometry. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Du, D.-Z., & Hwang, F. (Eds.). (1992). Computing in euclidean geometry. Singapore: World Scientific.

    MATH  Google Scholar 

  • Field, D. A., Nehl, T. W. (1992). Stitching together tetrahedral meshes. In Field, D., Komkov, V. (Eds.), Geometric Aspects of Industrial Design. Philadelphia: SIAM, Chap. 3, 25–38

    Google Scholar 

  • Field, D. A. (1995). The legacy of automatic mesh generation from solid modeling. Computer Aided Geometric Design, 12, 651–673.

    Article  MathSciNet  MATH  Google Scholar 

  • Finney, J. L. (1979). A procedure for the construction of Voronoi polyhedra. Journal of Computational Physics, 32, 137–143.

    Article  MATH  ADS  Google Scholar 

  • Formaggia, L. (1991). An unstructured mesh generation algorithm for three-dimensional aeronautical configurations. In A. S. Arcilla, J. Hauser, P. R. Eiseman, & J. F. Thompson (Eds.), Numerical Grid Generation in Computational Fluid Dynamics and Related Fields (pp. 249–260). New York: North-Holland.

    Google Scholar 

  • Fortune, S. (1985). A sweepline algorithm for Voronoi diagrams. Murray Hill, NJ: AT&T Bell Laboratory Report.

    Google Scholar 

  • Frey, P. J., & George, P.-L. (2008). Mesh generation application to finite elements. ISTE Ltd and Wiley Inc.

    Google Scholar 

  • George, A. J. (1971). Computer Implementation of the Finite Element Method. Stanford University Department of Computer Science, STAN-CS-71-208.

    Google Scholar 

  • George, P. L., Hecht, F., & Saltel, E. (1990). Automatic 3d mesh generation with prescribed meshed boundaries. IEEE Transactions on Magnetics, 26(2), 771–774.

    Article  ADS  Google Scholar 

  • George, P. L., & Borouchaki, H. (1998a). Delaunay triangulation and meshing: application to finite elements. Paris: Editions Hermes.

    Google Scholar 

  • George, P. L., & Borouchaki, H. (1998b). Delaunay triangulation and meshing. Paris: Editions Hermes.

    MATH  Google Scholar 

  • George, P. L., & Hermeline, F. (1992). Delaunay’s mesh of convex polyhedron in dimension \(d\): application for arbitrary polyhedra. International Journal for Numerical Methods in Engineering, 33, 975–995.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Green, P. J., & Sibson, R. (1978). Computing Dirichlet tessellations in the plane. The Computer Journal, 21(2), 168–173.

    Article  MathSciNet  MATH  Google Scholar 

  • Guibas, L., & Stolfi, J. (1985). Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. ACM Transactions on Graphics, 4, 74–123.

    Article  MATH  Google Scholar 

  • Haman, B., Chen, J.-L., & Hong, G. (1994). Automatic generation of unstructured volume grids inside or outside closed surfaces. In N. P. Weatherill, P. R. Eiseman, J. Hauser, & J. F. Thompson (Eds.), Numerical Grid Generation in Computational Field Simulation and Related Fields (p. 187). Swansea: Pineridge.

    Google Scholar 

  • Hassan, O., Probert, E. J., Morgan, K., & Peraire, J. (1994). Unstructured mesh generation for viscous high speed flows. In N. P. Weatherill, P. R. Eiseman, J. Hauser, & J. F. Thompson (Eds.), Numerical Grid Generation in Computational Field Simulation and Related Fields (p. 779). Swansea p: Pineridge.

    Google Scholar 

  • Hazlewood, C. (1993). Approximating constrained tetrahedrizations. Computer Aided Geometric Design, 10, 67–87.

    Article  MathSciNet  MATH  Google Scholar 

  • Henle, M. (1979). A combinatorial introduction to topology. San Francisco: W.H Freeman.

    MATH  Google Scholar 

  • Holmes, D. G., & Lamson, S. H. (1986). Adaptive triangular meshes for compressible flow solutions. In J. Hauser & C. Taylor (Eds.), Numerical Grid Generation in Computational Fluid Dynamics (p. 413). Swansea: Pineridge.

    Google Scholar 

  • Holmes, D. G., & Snyder, D. D. (1988). The generation of unstructured triangular meshes using Delaunay triangulation. In S. Sengupta, J. Hauser, P. R. Eiseman, & J. F. Thompson (Eds.), Numerical Grid Generation in Computational Fluid Dynamics (pp. 643–652). Swansea: Pineridge.

    Google Scholar 

  • Ivanov, E. (2008). Parallel Tetrahedral meshing based on a-priori domain decomposition: from scratch to results by utilizing off-the-shelf sequential software. Saarbrucken: VDM Verlag Dr. Muller.

    Google Scholar 

  • Jameson, A., Baker, T. J., Weatherill, N. P. (1986). Calculation of inviscid transonic flow over a complete aircraft. AIAA Paper 86-0103.

    Google Scholar 

  • Klee, V. (1964). The number of vertices of a convex polytope. Can. Math., 16, 37.

    MathSciNet  MATH  Google Scholar 

  • Lawson, C. L. (1986). Properties of \(n\)-dimensional triangulations. Computer Aided Geometric Design, 3, 231–246.

    Article  MathSciNet  MATH  Google Scholar 

  • Lee, K. D. (1976). On finding \(k\)-nearest neighbours in the plane. Technical Report 76–2216. University of Illinois, Urbana, IL.

    Google Scholar 

  • Lee, D. T. (1978). Proximity and reachibility in the plane. Techical Report R-831, University of Illinois, Urbana, IL.

    Google Scholar 

  • Lee, D. T., & Lin, A. K. (1986). Generalized Delaunay triangulation for planar graphs. Discrete & Computational Geometry, 1, 201–217.

    Article  MathSciNet  MATH  Google Scholar 

  • Lee, C. K., & Lo, S. H. (1994). A New Scheme for the Generation of a Graded Quadrilateral Mesh. Computers and Structures., 52, 847–857.

    Article  MATH  Google Scholar 

  • Lee, D. T., & Schachter, B. J. (1980). Two algorithms for constructing a Delaunay triangulation. International Journal of Computer & Information Science, 9(3), 219–241.

    Article  MathSciNet  MATH  Google Scholar 

  • Lo, S. H. (1985). A new mesh generation scheme for arbitrary planar domains. International Journal for Numerical Methods in Engineering, 21, 1403–1426.

    Article  MATH  ADS  Google Scholar 

  • Lo, S. H. (2015). Finite element mesh generation. Boca Raton: CRC Press Taylor and Francis Group.

    Book  MATH  Google Scholar 

  • Lohner, R. (1988a). Generation of three-dimensional unstructured grids by the advancing-front method. AIAA Paper 88-0515.

    Google Scholar 

  • Lohner, R. (1993). Matching semi-structured and unstructured grids for Navier–Stokes calculations. AIAA Paper 933348-CP.

    Google Scholar 

  • Lohner, R. (1988b). Some useful data structures for the generation of unstructured grids. Communications in Applied Numerical Methods, 4, 123–135.

    Article  MathSciNet  MATH  Google Scholar 

  • Lohner, R., & Parikh, P. (1988). \(3\)-dimensional grid generation by the advancing front method. Intertnational Journal for Numerical Methods in Fluids, 8, 1135–1149.

    Article  MATH  ADS  Google Scholar 

  • Marchant, M. J., & Weatherill, N. P. (1994). Unstructured grid generation for viscous flow simulations. In N. P. Weatherill, P. R. Eiseman, J. Hauser, & J. F. Thompson (Eds.), Numerical Grid Generation in Computational Field Simulations and Related Fields (p. 151). Swansea, UK: Pineridge.

    Google Scholar 

  • Marcum, D. L., & Weatherill, N. P. (1995). Unstructured grid generation using iterative point insertion and local reconnection. AIAA Journal, 33(9), 1619–1625.

    Article  MATH  ADS  Google Scholar 

  • Mavriplis, D. J. (1993). An advancing front Delaunay triangulation algorithm designed for robustness. AIAA Paper 93-0671.

    Google Scholar 

  • Mavriplis, D. J. (1990). Adaptive mesh generation for viscous flows using Delaunay triangulation. Journal of Computational Physics, 90, 271–291.

    Article  MATH  ADS  Google Scholar 

  • Mavriplis, D. J. (1991). Unstructured and adaptive mesh generation for high Reynolds number viscous flows. In A. S. Arcilla, J. Hauser, P. R. Eiseman, & J. F. Thompson (Eds.), Numerical Grid Generation in Computational Fluid Dynamics and Related Fields (pp. 79–92). Amsterdam: North-Holland.

    Google Scholar 

  • Merriam, M. (1991). An efficient advancing front algorithm for Delaunay triangulation. Technical report, AIAA Paper 91-0792.

    Google Scholar 

  • Muller, J. D., Roe, P. L., & Deconinck, H. (1993). A frontal approach for internal node generation in Delaunay triangulations. International Journal for Numerical Methods in Fluids, 17(3), 241–256.

    Article  MATH  ADS  Google Scholar 

  • Muller, J. D. (1994). Quality estimates and stretched meshes based on Delaunay triangulation. AIAA Journal, 32, 2372–2379.

    Article  MATH  ADS  Google Scholar 

  • Okabe, A., Boots, B., & Sugihara, K. (1992). Spatial tessellations concepts and applications of voronoi diagrams. New York: Wiley.

    MATH  Google Scholar 

  • Owen, S. (1998). A survey of unstructured mesh generation technology. In 7th International Roundtable

    Google Scholar 

  • Parthasarathy, V., & Kallinderis, Y. (1995). Directional viscous multigrid using adaptive prismatic meshes. AIAA Journal, 33(1), 69–78.

    Article  MATH  ADS  Google Scholar 

  • Peraire, J. (1986). A Finite Element Method for Convection Dominated Flows. Ph.D. dissertation, University of Wales.

    Google Scholar 

  • Peraire, J., Peiro, J., Formaggia, L., Morgan, K., Zienkiewicz, O. C. (1988). Finite element Euler computations in three dimensions. AIAA Paper 88-0032.

    Google Scholar 

  • Peraire, J., Vahdati, M., Morgan, H., & Zienkiewicz, O. C. (1987). Adaptive remeshing for compressible flow computations. Journal of Computational Physics, 72, 449–466.

    Article  MATH  ADS  Google Scholar 

  • Perronet, A. (1988). A generator of tetrahedral finite elements for multimaterial objects or fluids. In S. Sengupta, J. Hauser, P. R. Eiseman, & J. F. Thompson (Eds.), Numerical Grid Generation in Computational Fluid Mechanics (pp. 719–728). Swansea: Pineridge.

    Google Scholar 

  • Pirzadeh, S. (1992). Recent progress in unstructured grid generation. AIAA Paper 92-0445.

    Google Scholar 

  • Pirzadeh, S. (1994). Viscous unstructured three-dimensional grids by the advancing-layers method. AIAA Paper 94-0417.

    Google Scholar 

  • Pirzadeh, S. (1993). Structured background grids for generation of unstructured grids by advancing front method. AIAA Journal, 31(2), 257–265.

    Article  ADS  Google Scholar 

  • Powell, K. G., Roe, P. L., & Quirk, J. J. (1992). Adaptive-mesh algorithms for computational fluid dynamics. In M. Y. Hussaini, A. Kumar, & M. D. Salas (Eds.), Algorithmic Trends in Computational Fluid Dynamics (pp. 301–337). New York: Springer.

    Google Scholar 

  • Preparata, F. P., & Shamos, M. I. (1985). Computational geometry: an introduction. New York: Springer.

    Book  MATH  Google Scholar 

  • Rebay, S. (1993). Efficient unstructured mesh generation by means of Delaunay triangulation and Bowyer–Watson algorithm. Journal of Computational Physics, 106, 125–138.

    Article  MATH  ADS  Google Scholar 

  • Ruppert, J. (1992). Results on Triangulation and High Quality Mesh Generation. Ph.D. thesis, University of California, Berkeley.

    Google Scholar 

  • Schneiders, R., & Bunten, R. (1995). Automatic generation of hexahedral finite element meshes. Computers Aided Geometry Design, 12(7), 693.

    Article  MathSciNet  MATH  Google Scholar 

  • Shenton, D. N., Cendes, Z. J. (1985). Three-dimensional finite element mesh generation using Delaunay tessellation. IEEE Transactions on Magnetics MAG-21, 2535–2538.

    Google Scholar 

  • Shephard, M. S., Guerinoni, F., Flaherty, J. E., Ludwig, R. A., & Bachmann, P. L. (1988). Finite octree mesh generation for automated adaptive three-dimensional flow analysis. In S. Sengupta, J. Hauser, P. R. Eiseman, & J. F. Thompson (Eds.), Numerical grid generation in computational fluid mechanics (pp. 709–718). Swansea: Pineridge.

    Google Scholar 

  • Sibson, R. (1978). Locally equiangular triangulations. The Computer Journal, 21(3), 243–245.

    Article  MathSciNet  Google Scholar 

  • Sloan, S. W., & Houlsby, G. T. (1984). An implementation of Watson’s algorithm for computing 2D Delaunay triangulations. Advances in Engineering Software, 6(4), 192–197.

    Article  Google Scholar 

  • Steinitz, E. (1922). Polyeder and Raumeintailungen. Enzykl. Mathematischen Wiss., 3, 163.

    Google Scholar 

  • Tam, T. K. H., & Armstrong, C. G. (1991). 2D finite element mesh generation by medial axis subdivision. Advance in Engineering Software and Workstations, 13, 313–344.

    Article  MATH  Google Scholar 

  • Tanemura, M., Ogawa, T., & Ogita, N. (1983). A new algorithm for three-dimensional Voronoi tessellation. Journal of Computational Physics, 51, 191–207.

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Thompson, J. F., Soni, B. K., & Weatherill, N. P. (Eds.). (1999). Handbook of grid generation. Boca Raton: CRC Press.

    MATH  Google Scholar 

  • Venkatakrishan, V. (1996). Perspective on unstructured grid flow solvers. AIAA Journal, 34(3), 533–547.

    Article  ADS  Google Scholar 

  • Voronoi, G. F. (1908). Nouvelles applications des parameters countinus a la theorie des formes quadratiques. Deuxieme Memoire: Recherches sur la parallelloedres primitifs. J. Reine Angew. Math., 134, 198–287.

    MathSciNet  MATH  Google Scholar 

  • Watson, D. (1981). Computing the \(n\)-dimensional Delaunay tessellation with application to Voronoi polytopes. The Computer Journal, 24(2), 167–172.

    Article  MathSciNet  Google Scholar 

  • Weatherill, N. P., Marchant, M. F., Hassan, O., Marcum, D. L (1993). Adaptive inviscid flow solutions for aerospace geometries on efficiently generated unstructured tetrahedral meshes. AIAA Paper 93-3390

    Google Scholar 

  • Weatherill, N. P. (1988). A method for generating irregular computational grids in multiply connected planar domains. International Journal for Numerical Methods Fluids, 8, 181–197.

    Article  MATH  ADS  Google Scholar 

  • Weatherill, N. P. (1990). The integrity of geometrical boundaries in the two-dimensional Delaunay triangulation. Communications in Applied Numerical Methods, 6, 101–109.

    Article  MATH  Google Scholar 

  • Weatherill, N. P., & Hassan, O. (1994). Efficient three-dimensional Delaunay triangulation with automatic point creation and imposed boundary constraints. International Journal for Numerical Methods Engineering, 37, 2005–2039.

    Article  MATH  ADS  Google Scholar 

  • Wordenweber, B. (1981). Automatic mesh generation of \(2\)- and \(3\)-dimensional curvilinear manifolds. Ph.D. Dissertation (available as Computer Laboratory Report N18), University of Cambridge).

    Google Scholar 

  • Wordenweber, B. (1983). Finite-element analysis from geometric models. The International Journal for Computational and Mathematics in Electrical and Electronics Engineering, 2, 23–33.

    Article  MATH  Google Scholar 

  • Yerry, M. A., & Shephard, M. S. (1985). Automatic three-dimensional mesh generation for three-dimensional solids. Computers & Structures, 20, 31–39.

    Article  Google Scholar 

  • Yerry, M. A., & Shephard, M. S. (1990). Automatic three-dimensional mesh generation by the modified-octree technique. International Journal for Numerical Methods in Engineering, 20, 1965–1990.

    Article  MATH  ADS  Google Scholar 

  • Zhou, J. M., Ke-Ran, S., Ke-Ding, Z., & Quing-Hua, Z. (1990). Computing constrained triangulation and Delaunay triangulation: a new algorithm. IEEE Transactions on Magnetics, 26(2), 692–694.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vladimir D. Liseikin .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Liseikin, V.D. (2017). Unstructured Methods. In: Grid Generation Methods. Scientific Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-57846-0_12

Download citation

Publish with us

Policies and ethics