Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2014, Article ID 723832, 11 pages
http://dx.doi.org/10.1155/2014/723832
Review Article

Summary on Several Key Techniques in 3D Geological Modeling

Institute of Earth and Environmental Science, University of Freiburg, Albertstr. 23B, 79104 Freiburg im Breisgau, Germany

Received 23 August 2013; Accepted 23 December 2013; Published 16 February 2014

Academic Editors: G. Bordogna and H. Xu

Copyright © 2014 Gang Mei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. “Geomodelling,” 2013, http://en.wikipedia.org/wiki/Geologic_modelling.
  2. J.-L. Mallet, Geomodeling, Applied Geostatistics Series, Oxford University Press, New York, NY, USA, 2002.
  3. S. W. Houlding, “3D geoscience modeling: computer techniques for geological characterization,” in 3D Geoscience Modeling: Computer Techniques for Geological Characterization, Springer, New York, NY, USA, 1994. View at Google Scholar
  4. T. Fisher and R. Wales, “Three dimensional solid modeling of geo-objects using Non- Uniform Rational B-Splines (NURBS),” in Three-Dimensional Modeling with Geoscientific Information Systems, A. Turner, Ed., vol. 354 of NATO ASI Series, pp. 85–105, Springer, Amsterdam, The Netherlands, 1992. View at Google Scholar
  5. D.-H. Zhong, M.-C. Li, L.-G. Song, and G. Wang, “Enhanced NURBS modeling and visualization for large 3D geoengineering applications: an example from the Jinping first-level hydropower engineering project, China,” Computers and Geosciences, vol. 32, no. 9, pp. 1270–1282, 2006. View at Publisher · View at Google Scholar · View at Scopus
  6. E. A. De Kemp, “Visualization of complex geological structures using 3-D Bezier construction tools,” Computers and Geosciences, vol. 25, no. 5, pp. 581–5597, 1999. View at Google Scholar
  7. R. R. Moore and S. E. Johnson, “Three-dimensional reconstruction and modelling of complexly folded surfaces using mathematica,” Computers and Geosciences, vol. 27, no. 4, pp. 401–418, 2001. View at Publisher · View at Google Scholar · View at Scopus
  8. N. Xu and H. Tian, “Wire frame: a reliable approach to build sealed engineering geological models,” Computers and Geosciences, vol. 35, no. 8, pp. 1582–1591, 2009. View at Publisher · View at Google Scholar · View at Scopus
  9. J. L. Mallet, “Discrete modeling for natural objects,” Mathematical Geology, vol. 29, no. 2, pp. 199–219, 1997. View at Google Scholar · View at Scopus
  10. J. M. V. Pea, “A program in pascal to simulate the superposition of two or three fold systems,” Computers and Geosciences, vol. 26, no. 3, pp. 341–3349, 2000. View at Google Scholar
  11. T. Frank, A.-L. Tertois, and J.-L. Mallet, “3D-reconstruction of complex geological interfaces from irregularly distributed and noisy point data,” Computers and Geosciences, vol. 33, no. 7, pp. 932–943, 2007. View at Publisher · View at Google Scholar · View at Scopus
  12. B. Delaunay, “Sur la sphere vide,” IzvestIa AkademII Nauk SSSR, vol. 6, pp. 793–800, 1934. View at Google Scholar
  13. S. H. Lo, “A new mesh generation scheme for arbitrary planar domains,” International Journal for Numerical Methods in Engineering, vol. 21, no. 8, pp. 1403–1426, 1985. View at Google Scholar · View at Scopus
  14. G. H. Meisters, “Polygons have ears,” American Mathematical Monthly, vol. 82, 6, pp. 648–651, 1975. View at Google Scholar
  15. M. T. Dickerson, R. L. Scot Drysdale, S. A. McElfresh, and E. Welzl, “Fast greedy triangulation algorithms,” Computational Geometry, vol. 8, no. 2, pp. 67–86, 1997. View at Google Scholar · View at Scopus
  16. K. Ho-Le, “Finite element mesh generation methods: a review and classification,” Computer-Aided Design, vol. 20, no. 1, pp. 27–38, 1988. View at Google Scholar · View at Scopus
  17. S. J. Owen, “A survey of unstructured mesh generation technology,” in Proceedings of the 7th International Meshing Roundtable, vol. 3, pp. 26–28, 1998.
  18. J. O'Rourke, Computational Geometry in C, Cambridge University Press, New York, NY, USA, 1998.
  19. M. d. Berg, O. Cheong, M. v. Kreveld, and M. Overmars, Computational Geometry: Algorithms and Applications, Springer, New York, NY, USA, 2008.
  20. L. P. Chew, “Constrained delaunay triangulations,” Algorithmica, vol. 4, no. 1–4, pp. 97–108, 1989. View at Publisher · View at Google Scholar · View at Scopus
  21. G. L. Dirichlet, “Uber die Reduktion der positiven quadratischen formen mit drei unbes-timmten ganzen Zahlen,” Journal für die Reine und Angewandte Mathematik, vol. 40, pp. 209–227, 1850. View at Google Scholar
  22. G. Voronoi, “Nouvelles applications des parametres continus a la theorie des formes quadratiques. Deuxieme memoire. Recherches sur les parallelloedres primitifs,” Journal für die Reine und Angewandte Mathematik, vol. 1908, no. 134, pp. 198–287, 1908. View at Google Scholar
  23. A. Okabe, B. Boots, K. Sugihara, and S. N. Chiu, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, Wiley Series in Probability and Statistics, John Wiley & Sons, 2nd edition, 2000.
  24. J. C. Tipper, “A straightforward iterative algorithm for the planar Voronoi diagram,” Information Processing Letters, vol. 34, no. 3, pp. 155–160, 1990. View at Publisher · View at Google Scholar · View at Scopus
  25. J. C. Tipper, “FORTRAN programs to construct the planar Voronoi diagram,” Computers and Geosciences, vol. 17, no. 5, pp. 597–632, 1991. View at Google Scholar · View at Scopus
  26. F. Aurenhammer, “Voronoi diagrams a survey of a fundamental geometric data structure,” ACM Computing Surveys, vol. 23, no. 3, pp. 345–405, 1991. View at Google Scholar
  27. S. Fortune, “Voronoi diagrams and Delaunay triangulations,” in Handbook of Discrete and Computational Geometry, J. E. Goodman and J. O'Rourke, Eds., Chapter 23, pp. 513–528, Chapman and Hall/CRC, Boca Raton, Fla, USA, 2004. View at Google Scholar
  28. S. Fortune, “A sweepline algorithm for Voronoi diagrams,” Algorithmica, vol. 2, no. 1–4, pp. 153–174, 1987. View at Publisher · View at Google Scholar · View at Scopus
  29. D. T. Lee and B. J. Schachter, “Two algorithms for constructing a Delaunay triangulation,” International Journal of Computer & Information Sciences, vol. 9, no. 3, pp. 219–242, 1980. View at Publisher · View at Google Scholar · View at Scopus
  30. C. L. Lawson, “Software for C1 surface interpolation,” in Mathematical Software III, Academic Press, New York, NY, USA, 1977. View at Google Scholar
  31. A. Bowyer, “Computing dirichlet tessellations,” The Computer Journal, vol. 24, no. 2, pp. 162–166, 1981. View at Google Scholar · View at Scopus
  32. D. F. Watson, “Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes,” The Computer Journal, vol. 24, no. 2, pp. 167–172, 1981. View at Google Scholar · View at Scopus
  33. D.-T. Lee and A. K. Lin, “Generalized delaunay triangulation for planar graphs,” Discrete & Computational Geometry, vol. 1, no. 1, pp. 201–217, 1986. View at Publisher · View at Google Scholar · View at Scopus
  34. V. Domiter and B. Žalik, “Sweep-line algorithm for constrained Delaunay triangulation,” International Journal of Geographical Information Science, vol. 22, no. 4, pp. 449–462, 2008. View at Publisher · View at Google Scholar · View at Scopus
  35. J. R. Shewchuk, “Triangle: engineering a 2D quality mesh generator and delaunay triangulator,” in Proceedings of the Workshop on Applied Computational Geometry, Towards Geometric Engineering (FCRC/WACG'96), pp. 203–222, Springer, 1996.
  36. J. Peraire, M. Vahdati, K. Morgan, and O. C. Zienkiewicz, “Adaptive remeshing for compressible flow computations,” Journal of Computational Physics, vol. 72, no. 2, pp. 449–466, 1987. View at Google Scholar · View at Scopus
  37. R. Löhner and P. Parikh, “Generation of three-dimensional unstructured grids by the advancing-front method,” International Journal for Numerical Methods in Fluids, vol. 8, no. 10, pp. 1135–1149, 1988. View at Google Scholar · View at Scopus
  38. D. J. Mavriplis, “An advancing front Delaunay triangulation algorithm designed for robustness,” Journal of Computational Physics, vol. 117, no. 1, pp. 90–101, 1995. View at Publisher · View at Google Scholar · View at Scopus
  39. J. Schöberl, “Netgen an advancing front 2D/3D-mesh generator based on abstract rules,” Computing and Visualization in Science, vol. 1, no. 1, pp. 41–52, 1997. View at Google Scholar
  40. A. Shostko and R. Löhner, “Three-dimensional parallel unstructured grid generation,” International Journal for Numerical Methods in Engineering, vol. 38, no. 6, pp. 905–925, 1995. View at Google Scholar · View at Scopus
  41. T. D. Blacker and M. B. Stephenson, “Paving: a new approach to automated quadrilateral mesh generation,” International Journal for Numerical Methods in Engineering, vol. 32, no. 4, pp. 811–847, 1991. View at Google Scholar · View at Scopus
  42. T. D. Blacker and R. J. Meyers, “Seams and wedges in plastering: a 3-D hexahedral mesh generation algorithm,” Engineering with Computers, vol. 9, no. 2, pp. 83–93, 1993. View at Publisher · View at Google Scholar · View at Scopus
  43. M. Staten, S. Owen, and T. Blacker, “Unconstrained paving and plastering: a new idea for all hexahedral mesh generation,” in Proceedings of the 14th International Meshing Roundtable, pp. 399–416, Springer, 2005.
  44. R. Löhner, “Progress in grid generation via the advancing front technique,” Engineering with Computers, vol. 12, no. 3-4, pp. 186–210, 1996. View at Google Scholar · View at Scopus
  45. B. Chazelle, “A theorem on polygon cutting with applications,” in Proceedings of the 23rd Annual Symposium on Foundations of Computer Science (SFCS'08), pp. 339–349, IEEE, 1982.
  46. B. Chazelle, “Triangulating a simple polygon in linear time,” Discrete & Computational Geometry, vol. 6, no. 1, pp. 485–524, 1991. View at Publisher · View at Google Scholar · View at Scopus
  47. R. Seidel, “A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons,” Computational Geometry, vol. 1, no. 1, pp. 51–64, 1991. View at Google Scholar · View at Scopus
  48. H.-Y. Feng and T. Pavlidis, “Decomposition of polygons into simpler components: feature generation for syntactic pattern recognition,” IEEE Transactions on Computers, vol. 24, no. 6, pp. 636–650, 1975. View at Google Scholar · View at Scopus
  49. H. ElGindy, H. Everett, and G. Toussaint, “Slicing an ear using prune-and-search,” Pattern Recognition Letters, vol. 14, no. 9, pp. 719–722, 1993. View at Google Scholar · View at Scopus
  50. M. Held, “FIST: Fast Industrial-Strength Triangulation of polygons,” Algorithmica, vol. 30, no. 4, pp. 563–596, 2001. View at Google Scholar · View at Scopus
  51. G. Mei, J. C. Tipper, and N. Xu, “Ear-clipping based algorithms of generating high-quality polygon triangulation,” in Proceedings of the International Conference on Information Technology and Software Engineering, vol. 212, pp. 979–988, Springer, 2013.
  52. G. Mei, J. C. Tipper, and N. Xu, “The modified direct method: an approach for smoothing planar and surface meshes,” http://arxiv.org/abs/1212.3133.
  53. L. R. Herrmann, “Laplacian-isoparametric grid generation scheme,” Journal of the Engineering Mechanics Division, vol. 102, no. 5, pp. 749–756, 1976. View at Google Scholar · View at Scopus
  54. D. A. Field, “Laplacian smoothing and Delaunay triangulations,” Communications in Applied Numerical Methods, vol. 4, no. 6, pp. 709–712, 1988. View at Google Scholar · View at Scopus
  55. P. Hansbo, “Generalized Laplacian smoothing of unstructured grids,” Communications in Numerical Methods in Engineering, vol. 11, no. 5, pp. 455–464, 1995. View at Google Scholar
  56. Z. Mao, L. Ma, M. Zhao, and Z. Li, “A modified Laplacian smoothing approach with mesh saliency,” in Smart Graphics, vol. 4073 of Lecture Notes in Computer Science, pp. 105–113, Springer, New York, NY, USA, 2006. View at Google Scholar
  57. T. Zhou and K. Shimada, “An angle-based approach to two-dimensional mesh smoothing,” in Proceedings of the 9th International Meshing Roundtable, pp. 373–384, 2000.
  58. D. Vartziotis and J. Wipper, “The geometric element transformation method for mixed mesh smoothing,” Engineering with Computers, vol. 25, no. 3, pp. 287–301, 2009. View at Publisher · View at Google Scholar · View at Scopus
  59. J. M. Escobar, R. Montenegro, E. Rodríguez, and G. Montero, “Simultaneous aligning and smoothing of surface triangulations,” Engineering with Computers, vol. 27, no. 1, pp. 17–29, 2011. View at Publisher · View at Google Scholar · View at Scopus
  60. J. M. Escobar, G. Montero, R. Montenegro, and E. Rodríguez, “An algebraic method for smoothing surface triangulations on a local parametric space,” International Journal for Numerical Methods in Engineering, vol. 66, no. 4, pp. 740–760, 2006. View at Publisher · View at Google Scholar · View at Scopus
  61. P. Knupp, “Introducing the target-matrix paradigm for mesh optimization via node-movement,” in in Proceedings of the 19th International Meshing Roundtable, pp. 67–83, 2010.
  62. X. Jiao, D. Wang, and H. Zha, “Simple and effective variational optimization of surface and volume triangulations,” Engineering with Computers, vol. 27, no. 1, pp. 81–94, 2011. View at Publisher · View at Google Scholar · View at Scopus
  63. J. Wang and Z. Yu, “A novel method for surface mesh smoothing: applications in biomedical modeling,” in Proceedings of the 18th International Meshing Roundtable, pp. 195–210, 2009.
  64. L. Freitag, M. Jones, and P. Plassmann, “An efficient parallel algorithm for mesh smoothing,” in Proceedings of 4th International Meshing Roundtable, pp. 47–58, 1995.
  65. P. M. Knupp, “Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part I—a framework for surface mesh optimization,” International Journal for Numerical Methods in Engineering, vol. 48, no. 3, pp. 401–420, 2000. View at Google Scholar · View at Scopus
  66. K. Shivanna, N. Grosland, and V. Magnotta, “An analytical framework for quadrilateral surface mesh improvement with an underlying triangulated surface definition,” in Proceedings of the 19th International Meshing Roundtable, pp. 85–102, 2010.
  67. R. Lohner, K. Morgan, and O. Zienkiewicz, “Adaptive grid refinement for the compressible Eeuler equations,” in Accurency Estimates and Adaptive Refinements in Finite Element Computations, Wiley Series in Numerical Methods in Engineering, pp. 281–297, John Wiley & Sons, New York, NY, USA, 1986. View at Google Scholar
  68. K. Shimada, A. Yamada, and T. Itoh, “Anisotropic triangular meshing of parametric surfaces via close packing of ellipsoidal bubbles,” in Proceedings of the 6th International Meshing Roundtable, pp. 375–390, 1997.
  69. F. Bossen and P. S. Heckbert, “A pliant method for anisotropic mesh generation,” in Proceedings of the 5th International Meshing Roundtable, pp. 63–74, 1996.
  70. S. A. Canann, J. R. Tristano, and M. L. Staten, “An approach to combined Laplacian and optimization-based smoothing for triangular, quadrilateral, and quad-dominant meshes,” in Proceedings of 7th International Meshing Roundtable, pp. 479–494, 1998.
  71. Z. Chen, J. R. Tristano, and W. Kwok, “Combined Laplacian and optimization-based smoothing for quadratic mixed surface meshes,” in Proceedings of the 12th International Meshing Roundtable, pp. 201–213, 2003.
  72. L. A. Freitag, “On combining Laplacian and optimization-based mesh smoothing techniques,” in Trends in Unstructured Mesh Generation, pp. 37–43, 1997. View at Google Scholar
  73. B. Balendran, “A direct smoothing method for surface meshes,” in Proceedings of the 8th International MeshIng Roundtable, pp. 189–193, 1999. View at Google Scholar
  74. G. Mei, J. C. Tipper, and N. Xu, “The modified direct method: an iterative approach for smoothing planar meshes,” Procedia Computer Science, vol. 18, pp. 2436–2439, 2013. View at Google Scholar
  75. I. Bronshtein, K. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics, Springer, New York, NY, USA, 2007.
  76. M. A. Oliver and R. Webster, “Kriging: a method of interpolation for geographical information systems,” International Journal of Geographical Information Systems, vol. 4, no. 3, pp. 313–332, 1990. View at Google Scholar · View at Scopus
  77. J.-L. Mallet, “Discrete smooth interpolation,” ACM Transactions on Graphics, vol. 8, no. 2, pp. 121–144, 1989. View at Google Scholar
  78. J.-L. Mallet, “Discrete smooth interpolation in geometric modelling,” Computer-Aided Design, vol. 24, no. 4, pp. 178–191, 1992. View at Google Scholar · View at Scopus
  79. D. Shepard, “A two-dimensional interpolation function for irregularly-spaced data,” in Proceedings of the 23rd ACM National Conference, pp. 517–524.
  80. D. G. Krige, A statistical approach to some mine valuation and allied problems on the witwatersrand [M.S. thesis], University of the Witwatersrand, 1951.
  81. D. G. Krige, “A statistical approach to some basic mine valuation problems on the witwatersrand,” Journal of the Chemical, Metallurgical and Mining Society of South Africa, vol. 52, no. 6, pp. 119–139, 1951. View at Google Scholar
  82. G. Matheron, “Principles of geostatistics,” Economic Geology, vol. 58, no. 8, pp. 1246–1266, 1963. View at Google Scholar
  83. M. L. Stein, Interpolation of Spatial Data: Some Theory for Kriging, Springer, New York, NY, USA, 1999.
  84. GOCAD, 2013, http://www.gocad.org.
  85. C. H. Lindenbeck, H. D. Ebert, H. Ulmer, L. P. Lavorante, and R. Pflug, “TRICUT: a program to clip triangle meshes using the rapid and triangle libraries and the visualization toolkit,” Computers and Geosciences, vol. 28, no. 7, pp. 841–850, 2002. View at Publisher · View at Google Scholar · View at Scopus
  86. S. Gottschalk, M. C. Lin, and D. Manocha, “OBBTree: a hierarchical structure for rapid interference detection,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, pp. 171–180, August 1996. View at Scopus
  87. A. A. Shostko, R. Löhner, and W. C. Sandberg, “Surface triangulation over intersecting geometries,” International Journal for Numerical Methods in Engineering, vol. 44, no. 9, pp. 1359–1376, 1999. View at Google Scholar · View at Scopus
  88. S. H. Lo and W. X. Wang, “A fast robust algorithm for the intersection of triangulated surfaces,” Engineering with Computers, vol. 20, no. 1, pp. 11–21, 2004. View at Publisher · View at Google Scholar · View at Scopus
  89. K.-B. Guo, L.-C. Zhang, C.-J. Wang, and S.-H. Huang, “Boolean operations of STL models based on loop detection,” The International Journal of Advanced Manufacturing Technology, vol. 33, no. 5-6, pp. 627–633, 2007. View at Publisher · View at Google Scholar · View at Scopus
  90. D. Pavić, M. Campen, and L. Kobbelt, “Hybrid booleans,” Computer Graphics forum, vol. 29, no. 1, pp. 75–87, 2010. View at Publisher · View at Google Scholar · View at Scopus
  91. C. C. L. Wang, “Approximate Boolean operations on large polyhedral solids with partial mesh reconstruction,” IEEE Transactions on Visualization and Computer Graphics, vol. 17, no. 6, pp. 836–849, 2011. View at Publisher · View at Google Scholar · View at Scopus
  92. H. Zhao, C. C. L. Wang, Y. Chen, and X. Jin, “Parallel and efficient Boolean on polygonal solids,” Visual Computer, vol. 27, no. 6-8, pp. 507–517, 2011. View at Publisher · View at Google Scholar · View at Scopus
  93. A. H. Elsheikh and M. Elsheikh, “A reliable triangular mesh intersection algorithm and its application in geological modelling,” Engineering with Computers, vol. 30, no. 1, pp. 143–157, 2014. View at Publisher · View at Google Scholar
  94. B. K. Karamete, S. Dey, E. L. Mestreau, R. Aubry, and F. A. Bulat-Jara, “An algorithm for discrete booleans with applications to finite element modeling of complex systems,” Finite Elements in Analysis and Design, vol. 68, pp. 10–27, 2013. View at Google Scholar
  95. G. Mei and J. C. Tipper, “Simple and robust Boolean operations for triangulated surfaces,” http://arxiv.org/abs/1308.4434.
  96. T. Möller, “A fast triangle-triangle intersection test,” Journal of Graphics Tools, vol. 2, no. 2, pp. 25–30, 1997. View at Google Scholar
  97. M. Held, “ERIT—a collection of efficient and reliable intersection tests,” Journal of Graphics Tools, vol. 2, no. 4, pp. 25–44, 1997. View at Google Scholar
  98. O. Devillers and R. Guigue, “Faster triangle-triangle intersection tests,” Tech. Rep. RR-4488, INRIA, 2002. View at Google Scholar
  99. R. Guigue and O. Devillers, “Fast and robust triangle-triangle overlap test using orientation predicates,” Journal of Graphics Tools, vol. 8, no. 1, pp. 25–32, 2003. View at Google Scholar
  100. O. Tropp, A. Tal, and I. Shimshoni, “A fast triangle to triangle intersection test for collision detection,” Computer Animation and Virtual Worlds, vol. 17, no. 5, pp. 527–535, 2006. View at Publisher · View at Google Scholar · View at Scopus
  101. H. Shen, R. A. Heng, and Z. Tang, “A fast triangle-triangle overlap test using signed distances,” Journal of Graphics Tools, vol. 8, no. 1, pp. 17–23, 2003. View at Google Scholar
  102. J. Suter, “Introduction to Octrees,” 1999, http://www.flipcode.com/archives/Introduction_To_Octrees.shtml.
  103. M. Campen and L. Kobbelt, “Exact and robust (self-)intersections for polygonal meshes,” Computer Graphics forum, vol. 29, no. 2, pp. 397–406, 2010. View at Publisher · View at Google Scholar · View at Scopus
  104. G. Van Den Bergen, “Efficient collision detection of complex deformable models using aabb trees,” Journal of Graphics Tools, vol. 2, no. 4, pp. 1–13, 1997. View at Google Scholar
  105. C. Ericson, Real-Time Collision Detection, Morgan Kaufmann, Boston, Mass, USA, 2004.
  106. OpenMP, 2013, http://www.openmp.org.
  107. MPI, 2013, http://www.mcs.anl.gov/research/projects/mpi/.
  108. CUDA, 2013, http://www.nvidia.com/object/cuda_home_new.html.
  109. OpenCL, 2013, http://www.khronos.org/opencl/.