Parallel mesh generation is a relatively new research area between the boundaries of two scientific computing disciplines: computational geometry and parallel computing. In this chapter we present a survey of parallel unstructured mesh generation methods. Parallel mesh generation methods decompose the original mesh generation problem into smaller sub-problems which are meshed in parallel. We organize the parallel mesh generation methods in terms of two basic attributes: (1) the sequential technique used for meshing the individual subproblems and (2) the degree of coupling between the subproblems. This survey shows that without compromising in the stability of parallel mesh generation methods it is possible to develop parallel meshing software using off-the-shelf sequential meshing codes. However, more research is required for the efficient use of the state-of-the-art codes which can scale from emerging chip multiprocessors (CMPs) to clusters built from CMPs.


Domain Decomposition Mesh Generation Delaunay Triangulation Medial Axis Dynamic Load Balance 
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  1. 1.
    S. Baden, N. Chrisochoides, D. Gannon, and M. Norman, editors. Structured Adaptive Mesh Refinement Grid Methods. Springer-Verlag, 1999.Google Scholar
  2. 2.
    K. Barker, A. Chernikov, N. Chrisochoides, and K. Pingali. A load balancing framework for adaptive and asynchronous applications. IEEE Transactions on Parallel and Distributed Systems, 15(2):183–192, Feb. 2004.CrossRefGoogle Scholar
  3. 3.
    A. Basermann, J. Clinckemaillie, T. Coupez, J. Fingberg, H. Digonnet, R. Ducloux, J. Gratien, U. Hartmann, G. Lonsdale, B. Maerten, D. Roose, and C. Walshaw. Dynamic load-balancing of finite element applications with the drama library. Applied Mathematical Modeling, 25:83–98, 2000.CrossRefzbMATHGoogle Scholar
  4. 4.
    A. Belguelin, J. Dongarra, A. Geist, R. Manchek, S. Otto, and J. Walpore. PVM: Experiences, current status, and future direction. In Supercomputing’ 93 Proceedings, pp. 765–766, 1993.Google Scholar
  5. 5.
    D. Bertsekas and J. Tsitsiklis. Parallel and Distributed Computation: Numerical Methods. Prentice Hall, 1989.Google Scholar
  6. 6.
    T. B.H.V. and B. Cheng. Parallel adaptive quadrilateral mesh generation. Computers and Structures, 73:519–536, 1999.CrossRefGoogle Scholar
  7. 7.
    G. E. Blelloch, J. Hardwick, G. L. Miller, and D. Talmor. Design and implementation of a practical parallel Delaunay algorithm. Algorithmica, 24:243–269, 1999.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    G. E. Blelloch, G. L. Miller, and D. Talmor. Developing a practical projection-based parallel Delaunay algorithm. In 12th Annual Symposium on Computational Geometry, pp. 186–195, 1996.Google Scholar
  9. 9.
    H. Blum. A transformation for extracting new descriptors of shape. In Models for the Perception of speech and Visual Form, pp. 362–380. MIT Press, 1967.Google Scholar
  10. 10.
    A. Bowyer. Computing Dirichlet tesselations. Computer Journal, 24:162–166, 1981.MathSciNetCrossRefGoogle Scholar
  11. 11.
    J. G. Castãnos and J. E. Savage. The dynamic adaptation of parallel mesh-based computation. In SIAM 7th Symposium on Parallel and Scientific Computation, 1997.Google Scholar
  12. 12.
    J. G. Castaños and J. E. Savage. Parallel refinement of unstructured meshes. In Proceedings of the IASTED, International Conference of Parallel and Distributed Computing and Systems, 1999.Google Scholar
  13. 13.
    J. G. Castaños and J. E. Savage. PARED: a framework for the adaptive solution of PDEs. In 8th IEEE Symposium on High Performance Distributed Computing, 1999.Google Scholar
  14. 14.
    M.-B. Chen, T. R. Chuang, and J.-J. Wu. Efficient parallel implementations of near Delaunay triangulation with high performance Fortran. Concurrency: Practice and Experience, 16(12), 2004.Google Scholar
  15. 15.
    A. N. Chernikov and N. P. Chrisochoides. Parallel guaranteed quality planar Delaunay mesh generation by concurrent point insertion. In 14th Annual Fall Workshop on Computational Geometry, pp. 55–56. MIT, Nov. 2004.Google Scholar
  16. 16.
    A. N. Chernikov and N. P. Chrisochoides. Practical and efficient point insertion scheduling method for parallel guaranteed quality Delaunay refinement. In Proceedings of the 18th annual international conference on Supercomputing, pp. 48–57. ACM Press, 2004.Google Scholar
  17. 17.
    A. N. Chernikov, N. P. Chrisochoides, and L. P. Chew. Design of a parallel constrained Delaunay meshing algorithm, 2005.Google Scholar
  18. 18.
    L. P. Chew. Constrained Delaunay triangulations. Algorithmica, 4(1):97–108, 1989.zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    L. P. Chew, N. Chrisochoides, and F. Sukup. Parallel constrained Delaunay meshing. In ASME/ASCE/SES Summer Meeting, Special Symposium on Trends in Unstructured Mesh Generation, pp. 89–96, Northwestern University, Evanston, IL, 1997.Google Scholar
  20. 20.
    N. Chrisochoides. An alternative to data-mapping for parallel iterative PDE solvers: Parallel grid generation. In Scalable Parallel Libraries Conference, pp. 36–44. IEEE, 1993.Google Scholar
  21. 21.
    N. Chrisochoides. Multithreaded model for load balancing parallel adaptive computations. Applied Numerical Mathematics, 6:1–17, 1996.Google Scholar
  22. 22.
    N. Chrisochoides, C. Houstis, E.N. Houstis, P. Papachiou, S. Kortesis, and J. Rice. Domain decomposer: A software tool for partitioning and allocation of PDE computations based on geometry decomposition strategies. In 4th International Symposium on Domain Decomposition Methods, pp. 341–357. SIAM, 1991.Google Scholar
  23. 23.
    N. Chrisochoides, E. Houstis, and J. Rice. Mapping algorithms and software environment for data parallel PDE iterative solvers. Special issue of the Journal of Parallel and Distributed Computing on Data-Parallel Algorithms and Programming, 21(1):75–95, 1994.Google Scholar
  24. 24.
    N. Chrisochoides and D. Nave. Parallel Delaunay mesh generation kernel. Int. J. Numer. Meth. Engng., 58:161–176, 2003.CrossRefzbMATHGoogle Scholar
  25. 25.
    N. Chrisochoides and F. Sukup. Task parallel implementation of the Bowyer-Watson algorithm. In Proceedings of Fifth International Conference on Numerical Grid Generation in Computational Fluid Dynamics and Related Fields, 1996.Google Scholar
  26. 26.
    N. P. Chrisochoides. A new approach to parallel mesh generation and partitioning problems. Computational Science, Mathematics and Software, pp. 335–359, 2002.Google Scholar
  27. 27.
    P. Cignoni, D. Laforenza, C. Montani, R. Perego, and R. Scopigno. Evaluation of parallelization strategies for an incremental Delaunay triangulator in E3. Concurrency: Practice and Experience, 7(1):61–80, 1995.Google Scholar
  28. 28.
    H. D. Cougny and M. Shephard. Parallel refinement and coarsening of tetrahedral meshes. Int. J. Meth. Eng., 46(1101–1125), 1999.Google Scholar
  29. 29.
    T. Culver. Computing the Medial Axis of a Polyhedron Reliably and Efficiently. PhD thesis, The University of North Carolina at Chapel Hill, 2000.Google Scholar
  30. 30.
    H. de Cougny and M. Shephard. CRC Handbook of Grid Generation, chapter Parallel unstructured grid generation, pp. 24.1–24.18. CRC Press, Inc., 1999.Google Scholar
  31. 31.
    H. de Cougny and M. Shephard. Parallel volume meshing using face removals and hierarchical repartitioning. Comp. Meth. Appl. Mech. Engng., 174(3–4):275–298, 1999.zbMATHGoogle Scholar
  32. 32.
    H. L. de Cougny, M. S. Shephard, and C. Ozturan. Parallel three-dimensional mesh generation on distributed memory mimd computers. Engineering with Computers, 12:94–106, 1995.CrossRefGoogle Scholar
  33. 33.
    K. Devine, B. Hendrickson, E. Boman, M. S. John, and C. Vaughan. Design of dynamic load-balancing tools for parallel applications. In Proc. of the Int. Conf. on Supercomputing, Santa Fe, May 2000.Google Scholar
  34. 34.
    E. W. Dijkstra and C. Sholten. Termination detection for diffusing computations. Inf. Proc. Lettres, 11, 1980.Google Scholar
  35. 35.
    H. Edelsbrunner and D. Guoy. Sink-insertion for mesh improvement. In Proceedings of the Seventeenth Annual Symposium on Computational Geometry, pp. 115–123. ACM Press, 2001.Google Scholar
  36. 36.
    P. J. Frey and P. L. George. Mesh Generation: Applications to Finite Element. Hermis; Paris, 2000.Google Scholar
  37. 37.
    J. Gaither, D. Marcum, and B. Mitchell. Solidmesh: A solid modeling approach to unstructured grid generation. In 7th International Conference on Numerical Grid Generation in Computational Field Simulations, 2000.Google Scholar
  38. 38.
    J. Galtier and P. L. George. Prepartitioning as a way to mesh subdomains in parallel. In Special Symposium on Trends in Unstructured Mesh Generation, pp. 107–122. ASME/ ASCE/SES, 1997.Google Scholar
  39. 39.
    P. L. George and H. Borouchaki. Delaunay Triangulation and Meshing: Applications to Finite Element. Hermis; Paris, 1998.zbMATHGoogle Scholar
  40. 40.
    H. N. Gürsoy. Shape interrogation by medial axis transform for automated analysis. PhD thesis, Massachusetts Institute of Technology, 1989.Google Scholar
  41. 41.
    J. C. Hardwick. Implementation and evaluation of an efficient 2D parallel Delaunay triangulation algorithm. In Proceedings of the 9th Annual ACM Symposium on Parallel Algorithms and Architectures, 1997.Google Scholar
  42. 42.
    B. Hendrickson and R. Leland. The Chaco user’s guide, version 2.0. Technical Report SAND94-2692, Sandia National Laboratories., 1994.Google Scholar
  43. 43.
    A. M. S. Ito, Yasushi and B. K. Soni. Reliable isotropic tetrahedral mesh generation based on an advancing front method. In Proceedings 13th International Meshing Roundtable, Williamsburg, VA, Sandia National Laboratories, pp. 95–106, 2004.Google Scholar
  44. 44.
    D. A. Jefferson. Virtual time. In ACM Transactions on Programming Languages and Systems, volume 7, pp. 404–425, July 1985.CrossRefGoogle Scholar
  45. 45.
    M. T. Jones and P. E. Plassmann. Parallel algorithms for the adaptive refinement and partitioning of unstructured meshes. In Proceedings of the Scalable High-Performance Computing Conference, 1994.Google Scholar
  46. 46.
    C. Kadow. Adaptive dynamic projection-based partitioning for parallel Delaunay mesh generation algorithms. In SIAM Workshop on Combinatorial Scientific Computing, Feb. 2004.Google Scholar
  47. 47.
    C. Kadow and N. Walkington. Design of a projection-based parallel Delaunay mesh generation and refinement algorithm. In Fourth Symposium on Trends in Unstructured Mesh Generation, July 2003. Scholar
  48. 48.
    C. M. Kadow. Parallel Delaunay Refinement Mesh Generation. PhD thesis, Carnegie Mellon University, May 2004.Google Scholar
  49. 49.
    S. Kohn and S. Baden. Parallel software abstractions for structured adaptive mesh methods. Journal of Par. and Dist. Comp., 61(6):713–736, 2001.CrossRefzbMATHGoogle Scholar
  50. 50.
    R. Konuru, J. Casas, R. Prouty, S. Oto, and J. Walpore. A user level process package for pvm. In Proceedings of Scalable High-Performance Computing Conferene, pp. 48–55. IEEE, 1997.Google Scholar
  51. 51.
    L. Linardakis and N. Chrisochoides. Parallel Delaunay domain decoupling method for non-uniform mesh generation. SIAM Journal on Scientific Computing, 2005.Google Scholar
  52. 52.
    L. Linardakis and N. Chrisochoides. Parallel domain decoupling Delaunay method. SIAM Journal on Scientific Computing, in print, accepted Nov. 2004.Google Scholar
  53. 53.
    L. Linardakis and N. Chrisochoides. Medial axis domain decomposition method. ACM Trans. Math. Software, To be submited, 2005.Google Scholar
  54. 54.
    R. Lober, T. Tautges, and R. Cairncross. The parallelization of an advancing-front, all-quadrilateral meshing algorithm for adaptive analysis. In 4th International Meshing Roundtable, pp. 59–70, October 1995.Google Scholar
  55. 55.
    R. Löhner, J. Camberos, and M. Marshal. Unstructured Scientific Computation on Scalable Multiprocessors (Eds. Piyush Mehrotra and Joel Saltz), chapter Parallel Unstructured Grid Generation, pp. 31–64. MIT Press, 1990.Google Scholar
  56. 56.
    R. Löhner and J. R. Cebral. Parallel advancing front grid generation. In Proceedings of the Eighth International Meshing Roundtable, pp. 67–74, 1999.Google Scholar
  57. 57.
    F. Lori, M. Jones, and P. Plassmann. An efficient parallel algorithm for mesh smoothing. In Proceedings 4th International Meshing Roundtable, pp. 47–58, 1995.Google Scholar
  58. 58.
    R. P. M. Saxena. Parallel FEM algorithm based on recursive spatial decomposition. Computers and Structures, 45(9–6):817–831, 1992.zbMATHCrossRefGoogle Scholar
  59. 59.
    S. N. Muthukrishnan, P. S. Shiakolos, R. V. Nambiar, and K. L. Lawrence. Simple algorithm for adaptative refinement of three-dimensionalfinite element tetrahedral meshes. AIAA Journal, 33:928–932, 1995.CrossRefzbMATHGoogle Scholar
  60. 60.
    D. Nave, N. Chrisochoides, and L. P. Chew. Guaranteed: quality parallel Delaunay refinement for restricted polyhedral domains. In SCG’ 02: Proceedings of the eighteenth annual symposium on Computational geometry, pp. 135–144. ACM Press, 2002.Google Scholar
  61. 61.
    D. Nave, N. Chrisochoides, and L. P. Chew. Guaranteed-quality parallel Delaunay refinement for restricted polyhedral domains. Computational Geometry: Theory and Applications, 28:191–215, 2004.MathSciNetzbMATHGoogle Scholar
  62. 62.
    T. Okusanya and J. Peraire. Parallel unstructured mesh generation. In 5th International Conference on Numerical Grid Generation on Computational Field Simmulations, pp. 719–729, April 1996.Google Scholar
  63. 63.
    T. Okusanya and J. Peraire. 3D parallel unstructured mesh generation. In S. A. Canann and S. Saigal, editors, Trends in Unstructured Mesh Generation, pp. 109–116, 1997.Google Scholar
  64. 64.
    L. Oliker and R. Biswas. Plum: Parallel load balancing for adaptive unstructured meshes. Journal of Par. and Dist. Comp., 52(2):150–177, 1998.CrossRefzbMATHGoogle Scholar
  65. 65.
    L. Oliker, R. Biswas, and H. Gabow. Parallel tetrahedral mesh adaptation with dynamic load balancing. Parallel Computing Journal, pp. 1583–1608, 2000.Google Scholar
  66. 66.
    S. Owen. A survey of unstructured mesh generation. Technical report, ANSYS Inc., 2000.Google Scholar
  67. 67.
    M. Parashar and J. Browne. Dagh: A data-management infrastructure for parallel adaptive mesh refinement techniques. Technical report, Dept. of Comp. Sci., Univ. of Texas at Austin, 1995.Google Scholar
  68. 68.
    N. Patrikalakis and H. Gürsoy. Shape interrogation by medial axis transform. In Design Automation Conference (ASME), pp. 77–88, 1990.Google Scholar
  69. 69.
    P. Pebay and D. Thompson. Parallel mesh refinement without communication. In Proceedings of International Meshing Roundtable, pp. 437–443, 2004.Google Scholar
  70. 70.
    S. Prassidis and N. Chrisochoides. A categorical approach for parallel Delaunay mesh generation, July 2004.Google Scholar
  71. 71.
    M. C. Rivara. Algorithms for refining triangular grids suitable for adaptive and multigrid techniques. International Journal for Numerical Methods in Engineering, 20:745–756, 1984.zbMATHMathSciNetCrossRefGoogle Scholar
  72. 72.
    M. C. Rivara. Selective refinement/derefinement algorithms for sequences of nested triangulations. International Journal for Numerical Methods in Engineering, 28:2889–2906, 1989.zbMATHMathSciNetCrossRefGoogle Scholar
  73. 73.
    M. C. Rivara. New longest-edge algorithms for the refinement and/or improvement of unstructured triangulations. International Journal for Numerical Methods in Engineering, 40:3313–3324, 1997.zbMATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    M.-C. Rivara, C. Calderon, D. Pizarro, A. Fedorov, and N. Chrisochoides. Parallel decoupled terminal-edge bisection algorithm for 3D meshes. (Invited) Engineering with Computers, 2005.Google Scholar
  75. 75.
    M. C. Rivara, N. Hitschfeld, and R. B. Simpson. Terminal edges Delaunay (small angle based) algorithm for the quality triangulation problem. Computer-Aided Design, 33:263–277, 2001.CrossRefGoogle Scholar
  76. 76.
    M. C. Rivara and C. Levin. A 3D refinement algorithm for adaptive and multigrid techniques. Communications in Applied Numerical Methods, 8:281–290, 1992.CrossRefzbMATHGoogle Scholar
  77. 77.
    M. C. Rivara and M. Palma. New lepp algorithms for quality polygon and volume triangulation: Implementation issues and practical behavior. In In Trends unstructured mesh generationi Eds: S. A. Cannan. Saigal, AMD, volume 220, pp. 1–8, 1997.Google Scholar
  78. 78.
    M.-C. Rivara, D. Pizarro, and N. Chrisochoides. Parallel refinement of tetrahedral meshes using terminal-edge bisection algorithm. In 13th International Meshing Roundtable, Sept. 2004.Google Scholar
  79. 79.
    R. Said, N. Weatherill, K. Morgan, and N. Verhoeven. Distributed parallel Delaunay mesh generation. Computer Methods in Applied Mechanics and Engineering, (177):109–125, 1999.CrossRefzbMATHGoogle Scholar
  80. 80.
    K. Schloegel, G. Karypis, and V. Kumar. Parallel multilevel diffusion schemes for repartitioning of adaptive meshes. Technical Report 97-014, Univ. of Minnesota, 1997.Google Scholar
  81. 81.
    Sciclone cluster project. Last accessed, March 2005. Scholar
  82. 82.
    M. Shephard and M. Georges. Automatic three-dimensional mesh generation by the finite octree technique. International Journal for Numerical Methods in Engineering, 32:709–749, 1991.CrossRefzbMATHGoogle Scholar
  83. 83.
    E. C. Sherbrooke. 3-D shape interrogation by medial axial transform. PhD thesis, Massachusetts Institute of Technology, 1995.Google Scholar
  84. 84.
    J. Shewchuk. Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator. In Proceedings of the First workshop on Applied Computational Geometry, pp. 123–133, Philadelphia, PA, 1996.Google Scholar
  85. 85.
    J. R. Shewchuk. Delaunay Refinement Mesh Generation. PhD thesis, Carnegie Mellon University, 1997.Google Scholar
  86. 86.
    M. Snir, S. Otto, S. Huss-Lederman, and D. Walker. MPI the complete reference. MIT Press, 1996.Google Scholar
  87. 87.
    A. Sohn and H. Simon. Jove: A dynamic load balancing framework for adaptive computations on an SP-2 distributed memory multiprocessor, 1994. Technical Report 94-60, Dept. of Comp. and Inf. Sci., New Jersey Institute of Technology, 1994.Google Scholar
  88. 88.
    D. A. Spielman, S.-H. Teng, and A. Üngör. Parallel Delaunay refinement: Algorithms and analyses. In Proceedings of the Eleventh International Meshing Roundtable, pp. 205–217, 2001.Google Scholar
  89. 89.
    D. A. Spielman, S.-H. Teng, and A. Üngör. Time complexity of practical parallel Steiner point insertion algorithms. In Proceedings of the sixteenth annual ACM symposium on Parallelism in algorithms and architectures, pp. 267–268. ACM Press, 2004.Google Scholar
  90. 90.
    M. Stuti and A. Moitra. Considerations of computational optimality in parallel algorithms for grid generation. In 5th International Conference on Numerical Grid Generation in Computational Field Simulations, pp. 753–762, 1996.Google Scholar
  91. 91.
    T. Tam, M. Price, C. Armstrong, and R. McKeag. Computing the critical points on the medial axis of a planar object using a Delaunay point triangulation algorithm.Google Scholar
  92. 92.
    Y. A. Teng, F. Sullivan, I. Beichl, and E. Puppo. A data-parallel algorithm for three-dimensional Delaunay triangulation and its implementation. In SuperComputing, pp. 112–121. ACM, 1993.Google Scholar
  93. 93.
    J. F. Thompson, B. K. Soni, and N. P. Weatherill. Handbook of Grid Generation. CRC Press, 1999.Google Scholar
  94. 94.
    T. von Eicken, D. Culler, S. Goldstein, and K. Schauser. Active messages: A mechanism for integrated communication and computation. In Proceedings of the 19th Int. Symp. on Comp. Arch., pp. 256–266. ACM Press, May 1992.Google Scholar
  95. 95.
    C. Walshaw, M. Cross, and M. Everett. Parallel dynamic graph partitioning for adaptive unstructured meshes. Journal of Par. and Dist. Comp., 47:102–108, 1997.CrossRefGoogle Scholar
  96. 96.
    D. F. Watson. Computing the n-dimensional Delaunay tesselation with application to Voronoi polytopes. Computer Journal, 24:167–172, 1981.MathSciNetCrossRefGoogle Scholar
  97. 97.
    R. Williams. Adaptive parallel meshes with complex geometry. Numerical Grid Generation in Computational Fluid Dynamics and Related Fields, 1991.Google Scholar
  98. 98.
    X. Yuan, C. Salisbury, D. Balsara, and R. Melhem. Load balancing package on distributed memory systems and its application particle-particle and particle-mesh (P3M) methods. Parallel Computing, 23(10):1525–1544, 1997.CrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Nikos Chrisochoides
    • 1
    • 2
  1. 1.Computer Science DepartmentCollege of William and MaryWilliamsburgUSA
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

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