Distributed Octree Data Structures and Local Refinement Method for the Parallel Solution of Three-Dimensional Conservation Laws

  • J. E. Flaherty
  • R. M. Loy
  • M. S. Shephard
  • M. L. Simone
  • B. K. Szymanski
  • J. D. Teresco
  • L. H. Ziantz
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 113)


Conservation laws are solved by a local Galerkin finite element procedure with adaptive space-time mesh refinement and explicit time integration. A distributed octree structure representing a spatial decomposition of the domain is used for mesh generation, and later may be used to correct for processor load imbalances introduced at adaptive enrichment steps. A Courant stability condition is used to select smaller time steps on smaller elements of the mesh, thereby greatly increasing efficiency relative to methods having a single global time step. To accommodate the variable time steps, octree partitioning is extended to use weights derived from element size. Computational results are presented for the three-dimensional Euler equations of compressible flow solved on an IBM SP2 computer. The problem examined is the flow inside a perforated shock tube.


Mesh Generation Error Indicator Adaptive Finite Element Size Weighting Rejection Threshold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Adjerid, J.E. Flaherty, P. Moore, and Y. Wang. High-order adaptive methods for parabolic systems. Physica-D, 60: 94–111, 1992.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    P.L. Baehmann, S.L. Wittchen, M.S. Shephard, K.R. Grice, and M.A. Yerry. Robust, geometrically based, automatic two-dimensional mesh generation. Int. J. Numer. Meth. Engng., 24: 1043–1078, 1987.MATHCrossRefGoogle Scholar
  3. [3]
    M.W. Beall and M.S. Shephard. A general topology-based mesh data structure. Int. J. Numer. Meth. Engng., 40(9): 1573–1596, 1997.MathSciNetCrossRefGoogle Scholar
  4. [4]
    M.J. Berger. On conservation at grid interfaces. SI AM J. Numer. Anal, 24(5):967–984, 1987.MATHCrossRefGoogle Scholar
  5. [5]
    M.J. Berger and S.H. Bokhari. A partitioning strategy for nonuniform problems on multiprocessors. IEEE Trans. Computers, 36(5): 570–580, 1987.CrossRefGoogle Scholar
  6. [6]
    M.J. Berger and J. Oliger. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys., 53: 484–512, 1984.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    K.S. Bey, A. Patra, and J.T. Oden. hp-version discontinuous Galerkin methods for hyperbolic conservation laws: a parallel adaptive strategy. Int. J. Numer. Meth. Engng., 38(22): 3889–3907, 1995.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    R. Biswas, K.D. Devine, and J.E. Flaherty. Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math., 14: 255–283, 1994.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    C.L. Bottasso, H.L. de Cougny, M. Dindar, J.E. Flaherty, C. Özturan, Z. Rusak, and M.S. Shephard. Compressible aerodynamics using a parallel adaptive time-discontinuous Galerkin least-squares finite element method. In Proc. 12th AIAA Appl. Aero. Conf., number 94-1888, Colorado Springs, 1994.Google Scholar
  10. [10]
    B. Cockburn and P.-A. Gremaud. Error estimates for finite element methods for scalar conservation laws. SIAM J. Numer. Anal, 33: 522–554, 1996.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    B. Cockburn, S.-Y. Lin, and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-Dimensional systems. J. Comput. Phys., 84: 90–113, 1989.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    B. Cockburn and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework. Math. Comp., 52: 411–435, 1989.MathSciNetMATHGoogle Scholar
  13. [13]
    G. Cybenko. Dynamic load balancing for distributed memory multiprocessors. J. Parallel and Dist. Comput, 7: 279–301, 1989.CrossRefGoogle Scholar
  14. [14]
    H.L. de Cougny, K.D. Devine, J.E. Flaherty, R.M. Loy, C. Ozturan, and M.S. Shephard. Load balancing for the parallel adaptive solution of partial differential equations. Appl. Numer. Math., 16: 157–182, 1994.MathSciNetCrossRefGoogle Scholar
  15. [15]
    H.L. de Cougny, M.S. Shephard, and C. Özturan. Parallel three-dimensional mesh generation. Comp. Sys. Engng., 5: 311–323, 1994.CrossRefGoogle Scholar
  16. [16]
    H.L. de Cougny, M.S. Shephard, and C. Özturan. Parallel three-dimensional mesh generation on distributed memory MIMD computers. Engng. with Computers, 12(2): 94–106, 1996.CrossRefGoogle Scholar
  17. [17]
    K.D. Devine and J.E. Flaherty. Parallel adaptive hp-refinement techniques for conservation laws. Appl. Numer. Math., 20: 367–386, 1996.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    K.D. Devine, J.E. Flaherty, R. Loy, and S. Wheat. Parallel partitioning strategies for the adaptive solution of conservation laws. In, I. Babuška, J.E. Flaherty, W.D. Henshaw, J.E. Hopcroft, J.E. Oliger, and T. Tezduyar, editors, Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations, volume 75, pages 215–242, Berlin-Heidelberg, 1995. Springer-Verlag.Google Scholar
  19. [19]
    R.E. Dillon Jr. A parametric study of perforated muzzle brakes. ARDC Tech. Report ARLCB-TR-84015, Benét Weapons Laboratory, Watervliet, 1984.Google Scholar
  20. [20]
    C. Farhat and M. Lesoinne. Automatic partitioning of unstructured meshes for the parallel solution of problems in computational mechanics. Int. J. Numer. Meth. Engng., 36: 745–764, 1993.MATHCrossRefGoogle Scholar
  21. [21]
    J.E. Flaherty, M. Dindar, R.M. Loy, M.S. Shephard, B.K. Szymanski, J.D. Teresco, and L.H. Ziantz. An adaptive and parallel framework for partial differential equations. Numerical Analysis 1997 (Proc. 17th Dundee Biennial Conf.). In, D.F. Griffiths and D.J. Higham and G.A. Watson, editors, Pitman Research Notes in Mathematics Series, volume 380, pages 74–90, Addison Wesley Longman, 19Google Scholar
  22. [22]
    J.E. Flaherty, R.M. Loy, C. Özturan, M.S. Shephard, B.K. Szymanski, J.D. Teresco, and L.H. Ziantz. Parallel structures and dynamic load balancing for adaptive finite element computation. Appl. Numer. Math., 26: 241–263, 1998.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    J.E. Flaherty, R.M. Loy, P.C. Scully, M.S. Shephard, B.K. Szymanski, J.D. Teresco, and L.H. Ziantz. Load balancing and communication optimization for parallel adaptive finite element computation. Proc. XVII Int. Conf. Chilean Comp. Sci. Soc, 246–255, IEEE, Los Alamitos, CA, 1997.CrossRefGoogle Scholar
  24. [24]
    J.E. Flaherty, R.M. Loy, M.S. Shephard, B.K. Szymanski, J.D. Teresco, and L.H. Ziantz. Adaptive local refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws. IMA Preprint Series 1483, Institute for Mathematics and its Applications, University of Minnesota, 1997. To appear, J. Parallel and Dist. Comput.Google Scholar
  25. [25]
    J.E. Flaherty, R.M. Loy, M.S. Shephard, B.K. Szymanski, J.D. Teresco, and L.H. Ziantz. Predictive load balancing for parallel adaptive finite element computation. In H.R. Arabnia, editor, Proc. PDPTA ′97, volume I, 460–469, 1997.Google Scholar
  26. [26]
    P.L. George. Automatic Mesh Generation. John Wiley and Sons, Ltd., Chichester, 1991.MATHGoogle Scholar
  27. [27]
    G. Karypis and V. Kumar. Metis: Unstructured graph partitioning and sparse matrix ordering system. Tech. Report, University of Minnesota, Department of Computer Science, Minneapolis, MN, 1995.Google Scholar
  28. [28]
    A. Kela. Hierarchical octree approximations for boundary representation-based geometric models. Computer Aided Design, 21: 355–362, 1989.CrossRefGoogle Scholar
  29. [29]
    W.L. Kleb and J.T. Batina. Temporal adaptive Euler/Navier-Stokes algorithm involving unstructured dynamic meshes. AIAA J., 30(8): 1980–1985, 1992.MATHCrossRefGoogle Scholar
  30. [30]
    E. Leiss and H. Reddy. Distributed load balancing: design and performance analysis. W. M. Kuck Research Computation Laboratory, 5: 205–270, 1989.Google Scholar
  31. [31]
    R.A. Ludwig, J.E. Flaherty, F. Guerinoni, P.L. Baehmann, and M.S. Shephard. Adaptive solutions of the Euler equations using finite quadtree and octree grids. Computers and Structures, 30: 327–336, 1988.MATHCrossRefGoogle Scholar
  32. [32]
    H.T. Nagamatsu, K.Y. Choi, R.E. Duffy, and G.C. Carofano. An experimental and numerical study of the flow through a vent hole in a perforated muzzle brake. ARDEC Tech. Report ARCCB-TR-87016, Benet Weapons Laboratory, Watervliet, 1987.Google Scholar
  33. [33]
    L. Oliker and R. Biswas. Efficient load balancing and data remapping for adaptive grid calculations. In Proc. 9th ACM Symposium on Parallel Algorithms and Architectures (SPAA), 33–42, Newport, 1997.Google Scholar
  34. [34]
    L. Oliker, R. Biswas, and R.C. Strawn. Parallel implementation of an adaptive scheme for 3D unstructured grids on the SP2. In Proc. 3rd International Workshop on Parallel Algorithms for Irregularly Structured Problems, Santa Barbara, 1996.Google Scholar
  35. [35]
    S. Osher and S. Chakravarthy. Upwind schemes and boundary conditions with applications to the euler equations in general coordinates. J. Comput. Phys., 50, 1983.Google Scholar
  36. [36]
    A. Patra and J.T. Oden. Problem decomposition for adaptive hp finite element methods. Comp. Sys. Engng., 6(2): 97, 1995.CrossRefGoogle Scholar
  37. [37]
    R. Perucchio, M. Saxena, and A. Kela. Automatic mesh generation from solid modelsbased on recursive spatial decompositions. Int. J. Numer. Meth. Engng., 28: 2469–2501, 1989.MATHCrossRefGoogle Scholar
  38. [38]
    A. Pothen, H. Simon, and K.-P. Liou. Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Mat. Anal Appl, 11(3): 430–452, 1990.MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    P.L. Roe. Approximate riemann solvers, parametric vectors and difference schemes. J. Comput. Phys., 43, 1981.Google Scholar
  40. [40]
    P.L. Roe. Characteristic based schemes for the euler equations. Annual Review of Fluid Mechanics, 18, 1986.Google Scholar
  41. [41]
    K. Schloegel, G. Karypis, and V. Kumar. Parallel multilevel diffusion algorithms for repartitioning of adaptive meshes. Tech. Report 97-014, University of Minnesota, Department of Computer Science and Army HPC Center, Minneapolis, MN, 1997.Google Scholar
  42. [42]
    W.J. Schroeder and M.S. Shephard. Geometry-based fully automatic mesh generati on and the delaunay triangulation. Int. J. Numer. Meth. Engng., 26: 2503–2515, 1988.MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    M.S. Shephard. Approaches to the automatic generation and control of finite element meshes. Applied Mechanics Review, 41(4): 169–185, 1988.CrossRefGoogle Scholar
  44. [44]
    M.S. Shephard. Update to: Approaches to the automatic generation and control of finite element meshes. Applied Mechanics Reviews, 49(10, part 2): S5–S14, 1996.CrossRefGoogle Scholar
  45. [45]
    M.S. Shephard, J.E. Flaherty, H.L. de Cougny, C. Özturan, C.L. Bottasso, and M.W. Beall. Parallel automated adaptive procedures for unstructured meshes. In Parallel Comput. in CFD, number R-807, pages 6.1–6.49. Agard, Neuilly-Sur-Seine, 1995.Google Scholar
  46. [46]
    M.S. Shephard and M.K. Georges. Automatic three-dimensional mesh generation by the Finite Octree technique. Int. J. Numer. Meth. Engng., 32(4): 709–749, 1991.MATHCrossRefGoogle Scholar
  47. [47]
    C.-W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys., 27: 1–31, 1978.CrossRefGoogle Scholar
  48. [48]
    M.L. Simone, M.S. Shephard, J.E. Flaherty, and R.M. Loy. A distributed octree and neighbor-finding algorithms for parallel mesh generation. Tech. Report 23-1996, Rensselaer Polytechnic Institute, Scientific Computation Research Center, Troy, 1996.Google Scholar
  49. [49]
    P.K. Sweby. High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal, 21: 995–1011, 1984.MathSciNetMATHCrossRefGoogle Scholar
  50. [50]
    B. Van Leer. Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection. J. Comput. Phys., 23: 276–299, 1977.MATHCrossRefGoogle Scholar
  51. [51]
    B. Van Leer. Flux vector splitting for the Euler equations. ICASE Report 82-30, ICASE, NASA Langley Research Center, Hampton, 1982.Google Scholar
  52. [52]
    V. Vidwans, Y. Kallinderis, and V. Venkatakrishnan. Parallel dynamic load-balancing algorithm for three-dimensional adaptive unstructured grids. AIAA J., 32(3): 497–505, 1994.MATHCrossRefGoogle Scholar
  53. [53]
    C.H. Walshaw, M. Cross, and M. Everett. Mesh partitioning and load-balancing for distributed memory parallel systems. In Proc. Par. Dist. Comput. for Comput. Mech., Lochinver, Scotland, 1997.Google Scholar
  54. [54]
    S. Wheat, K. Devine, and A. MacCabe. Experience with automatic, dynamic load balancing and adaptive finite element computation. In H. El-Rewini and B. Shriver, editors, Proc. 27th Hawaii International Conference on System Sciences, 463–472, Kihei, 1994.Google Scholar
  55. [55]
    M.A. Yerry and M.S. Shephard. Automatic three-dimensional mesh generation by the modified octree technique. Int. J. Numer. Meth. Engng., 20: 1965–1990,1984.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • J. E. Flaherty
    • 1
  • R. M. Loy
    • 1
  • M. S. Shephard
    • 1
  • M. L. Simone
    • 1
  • B. K. Szymanski
    • 1
  • J. D. Teresco
    • 1
  • L. H. Ziantz
    • 1
  1. 1.Scientific Computation Research CenterRensselaer Polytechnic InstituteTroyUSA

Personalised recommendations