Abstract
Simulations of environmental processes are usually modelled by partial differential equations that are approximated with numerical methods, based on regular grids. An attractive alternative for simulating a fluid flow is the Free-Lagrange Method (FLM). In this paper, I discuss the use of the FLM—based on the Voronoi diagram (VD)—for the modelling of fluid flow in three dimensions (e.g. the movement of underground water or of pollution plumes in the ocean). Such a technique requires the kinetic three-dimensional VD, which is a VD for which the points are allowed to move freely in space. I present a new algorithm for the movement of points in a three-dimensional VD, and show that it can be relatively easy to implement as it is the extension of a simple two-dimensional algorithm.
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References
Albers G (1991) Three-dimensional dynamic Voronoi diagrams (in German). Ph.D. thesis, Universität Würzburg, Würzburg, Germany.
Albers G, Guibas LJ, Mitchell JSB, and Roos T (1998) Voronoi diagrams of moving points. International Journal of Computational Geometry and Applications, 8:365–380.
Albers G and Roos T (1992) Voronoi diagrams of moving points in higher dimensional spaces. In Proceedings 3rd Scandinavian Workshop On Algorithm Theory (SWAT’92), volume 621 of Lecture Notes in Computer Science, pages 399–409. Springler-Verlag, Helsinki, Finland.
Anselin L (1999) Interactive techniques and exploratory spatial data analysis. In PA Longley, MF Goodchild, DJ Maguire, and DW Rhind, editors, Geographical Information Systems, pages 253–266. John Wiley & Sons, second edition.
Augenbaum JM (1985) A Lagrangian method for the shallow water equations based on the Voronoi mesh-flows on a rotating sphere. In MJ Fritts, WP Crowley, and HE Trease, editors, Free-Lagrange method, volume 238, pages 54–87. Springer-Verlag, Berlin.
Aurenhammer F (1991) Voronoi diagrams: A survey of a fundamental geometric data structure. ACM Computing Surveys, 23(3):345–405.
Bajaj C and Bouma W (1990) Dynamic Voronoi diagrams and Delaunay triangulations. In Proceedings 2nd Annual Canadian Conference on Computational Geometry, pages 273–277. Ottawa, Canada.
Bivand R and Lucas A (2000) Integrating models and geographical information systems. In S Openshaw and RJ Abrahardt, editors, Geocomputation, pages 331–363. Taylor & Francis, London.
Bourke P (1992) Intersection of a line and a sphere (or circle). http://astronomy.swin.edu.au/~pbourke/geometry/sphereline/.
Burrough PA, van Deursen W, and Heuvelink G (1988) Linking spatial process models and GIS: A marriage of convenience or a blossoming partnership? In Proceedings GIS/LIS’ 88, volume 2, pages 598–607. San Antonio, Texas, USA.
Chen J, Zhao R, and Li Z (2004) Voronoi-based k-order neighbour relations for spatial analysis. ISPRS Journal of Photogrammetry & Remote Sensing, 59:60–72.
Cignoni P, Montani C, and Scopigno R (1998) DeWall: A fast divide & conquer Delaunay triangulation algorithm in E d. Computer-Aided Design, 30(5):333–341.
De Fabritiis G and Coveney PV (2003) Dynamical geometry for multiscale dissipative particle dynamics. Computer Physics Communications, 153:209–226.
Devillers O (2002) On deletion in Delaunay triangulations. International Journal of Computational Geometry and Applications, 12(3):193–205.
Edelsbrunner H and Shah NR (1996) Incremental topological flipping works for regular triangulations. Algorithmica, 15:223–241.
Erlebacher G (1985) Finite difference operators on unstructured triangular meshes. In MJ Fritts, WP Crowley, and HE Trease, editors, Free-Lagrange Method, volume 238, pages 21–53. Springer-Verlag, Berlin.
Ferrez JA (2001) Dynamic triangulations for efficient 3D simulation of granular materials. Ph.D. thesis, Département de Mathématiques, École Polytechnique Fédérale de Lausanne, Switzerland.
Field DA (1986) Implementing Watson’s algorithm in three dimensions. In Proceedings 2nd Annual Symposium on Computational Geometry, volume 246–259. ACM Press, Yorktown Heights, New York, USA.
Freda K (1993) GIS and environment modeling. In MF Goodchild, BO Parks, and LT Steyaert, editors, Environmental modeling with GIS, pages 35–50. Oxford University Press, New York.
Fritts MJ, Crowley WP, and Trease HE (1985) The Free-Langrange method, volume 238. Springler-Verlag, Berlin.
Gahegan M and Lee I (2000) Data structures and algorithms to support interactive spatial analysis using dynamic Voronoi diagrams. Computers, Environment and Urban Systems, 24(6):509–537.
Gavrilova ML and Rokne J (2003) Updating the topology of the dynamic Voronoi diagram for spheres in Euclidean d-dimensional space. Computer Aided Geometric Design, 20:231–242.
Gold CM (1990) Spatial data structures—the extension from one to two dimensions. In Mapping and Spatial Modelling for Navigation, volume 65, pages 11–39. Springer-Verlag, Berlin, Germany.
Gold CM (1991) Problems with handling spatial data—the Voronoi approach. CISM Journal, 45(1):65–80.
Gold CM (1996) An event-driven approach to spatio-temporal mapping. Geomatica, Journal of the Canadian Institute of Geomatics, 50(4):415–424.
Gold CM, Remmele PR, and Roos T (1995) Voronoi diagrams of line segments made easy. In Proceedings 7th Canadian Conference on Computational Geometry, pages 223–228. Quebec City, Canada.
Gold CM, Remmele PR, and Roos T (1997) Voronoi methods in GIS. In M van Kreveld, J Nievergelt, T Roos, and P Widmayer, editors, Algorithmic Foundations of Geographic Information Systems, volume 1340 of Lecture Notes in Computer Science, pages 21–35. Springer-Verlag.
Guibas L, Karaveles M, and Russel D (2004) A Computational Framework for Handling Motion. In Proceedings 6th Workshop on Algorithm Engineering and Experiments, pages 129–141. New Orleans, USA.
Guibas L and Russel D (2004) An empirical comparison of techniques for updating Delaunay triangulations. In Proceedings 20th Annual Symposium on Computational Geometry, pages 170–179. ACM Press, Brooklyn, New York, USA.
Guibas LJ and Stolfi J (1985) Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. ACM Transactions on Graphics, 4:74–123.
Imai K, Sumino S, and Imai H (1989) Geometric fitting of two corresponding sets of points. In Proceedings 5th Annual Symposium on Computational Geometry, pages 266–275. ACM Press, Saarbrücken, West Germany.
Joe B (1989) Three-dimensional triangulations from local transformations. SIAM Journal on Scientific and Statistical Computing, 10(4):718–741.
Karavelas MI (2004) A robust and efficient implementation for the segment Voronoi diagram. In International Symposium on Voronoi Diagrams in Science and Engineering, pages 51–62. Tokyo, Japan.
Lawson CL (1986) Properties of n-dimensional triangulations. Computer Aided Geometric Design, 3:231–246.
Ledoux H (2006) Modelling three-dimensional fields in geoscience with the Voronoi diagram and its dual. Ph.D. thesis, School of Computing, University of Glamorgan, Pontypridd, Wales, UK.
Ledoux H, Gold CM, and Baciu G (2005) Flipping to robustly delete a vertex in a Delaunay tetrahedralization. In Proceedings International Conference on Computational Science and its Applications—ICCSA 2005, volume 3480 of Lecture Notes in Computer Science, pages 737–747. Springer-Verlag, Singapore.
Mioc D (2002) The Voronoi spatio-temporal data structure. Ph.D. thesis, Département des Sciences Géomatiques, Université Laval, Québec, Canada.
Mostafavi MA (2001) Development of a global dynamic data structure. Ph.D. thesis, Département des Sciences Géomatiques, Université Laval, Québec City, Canada.
Mostafavi MA (2006) Personal communication.
Mostafavi MA and Gold CM (2004) A global kinetic spatial data structure for a marine simulation. International Journal of Geographical Information Science, 18(3):211–228.
Nativi S, Blumenthal MB, Caron J, Domenico B, Habermann T, Hertzmann D, Ho Y, Raskin R, and Weber J (2004) Differences among the data models used by the geographic information systems and atmospheric science communities. In Proceedings 84th Annual Meeting of the American Meteorological Society. Seattle, USA.
Okabe A, Boots B, Sugihara K, and Chiu SN (2000) Spatial tessellations: Concepts and applications of Voronoi diagrams. John Wiley and Sons, second edition.
Parks BO (1993) The need for integration. In MF Goodchild, BO Parks, and LT Steyaert, editors, Environmental Modeling with GIS, pages 31–34. Oxford University Press, New York.
Raper J (2000) Multidimensional geographic information science.Taylor & Francis, London.
Roos T (1991) Dynamic Voronoi diagrams. Ph.D. thesis, Universität Würzburg, Germany.
Schaller G and Meyer-Hermann M (2004) Kinetic and dynamic Delaunay tetrahedralizations in three dimensions. Computer Physics Communications, 162(1):9–23.
Schirra S (1997) Precision and robustness in geometric computations. In M van Kreveld, J Nievergelt, T Roos, and P Widmayer, editors, Algorithmic Foundations of Geographic Information Systems, volume 1340 of Lecture Notes in Computer Science, pages 255–287. Springer-Verlag, Berlin.
Strang WG and Fix GJ (1973) An analysis of the finite element method. Prentice-Hall, Englewood Cliffs, USA.
Sugihara K and Inagaki H (1995) Why is the 3D Delaunay triangulation difficult to construct? Information Processing Letters, 54:275–280.
Sui DZ and Maggio RC (1999) Integrating GIS with hydrological modeling: Practices, problems, and prospects. Computers, Environment and Urban Systems, 23:33–51.
Watson DF (1981) Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes. Computer Journal, 24(2):167–172.
Zlatanova S, Abdul Rahman A, and Pilouk M (2002) 3D GIS: Current status and perspectives. In Proceedings Joint Conference on Geo-spatial Theory, Processing and Applications. Ottawa, Canada.
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Ledoux, H. (2008). The Kinetic 3D Voronoi Diagram: A Tool for Simulating Environmental Processes. In: van Oosterom, P., Zlatanova, S., Penninga, F., Fendel, E.M. (eds) Advances in 3D Geoinformation Systems. Lecture Notes in Geoinformation and Cartography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72135-2_20
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DOI: https://doi.org/10.1007/978-3-540-72135-2_20
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