HPC MSU
    Cover
    Preface
  1. Introduction
  2. Boundary-Conforming Coordinate Systems
  3. Transformation Relations
  4. Numerical Implementation
  5. Truncation Error
  6. Elliptic Generation Systems
  7. Parabolic and Hyperbolic Generation Systems
  8. Algebraic Generation Systems
  9. Orthogonal Systems
  10. Conformal Mapping
  11. Adaptive Grids
  12. Appendix A
    Appendix B
    Appendix C
    References

    Downloadable Version (PDF)
Numerical Grid Generation
Foundations and Applications
By: Joe E. Thompson, Z.U.A. Warsi and C. Wayne Mastin

REFERENCES

1. Thompson, Joe F., Warsi, Z. U. A. and Mastin, C. W., "Boundary-Fitted Coordinate Systems for Numerical Solution of Partial Differential Equations -- A Review", Journal of Computational Physics, 47, 1, 1982.

2. Thompson, Joe F. (Ed.) Numerical Grid Generation, North-Holland 1982. (Also published as Vol. 10 11 of Applied Mathematics and Computation, 1982).

3. Smith, Robert E., (Ed), Numerical Grid Generation Techniques, NASA Conference Publication 2166, NASA Langley Research Center, 1980.

4. Ghia, K. N. and Ghia, U., (Ed), Advances in Grid Generation, FED-Vol. 5, ASME Applied Mechanics, Bioengineering, and Fluids Engineering Conference, Houston, 1983.

5. Thompson, Joe F., "Grid Generation Techniques in Computational Fluid Dynamics", AIAA Journal, 22, 1505, 1984.

6. Halsey, Douglas, "Conformal Grid Generation for Multi-element Airfoils", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 585, 1982.

7. Ives, David C., "Conformal Grid Generation", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 107, 1982.

8. Smith, Robert E., "Three-Dimensional Algebraic Grid Generation", AIAA-83-1904, AIAA 6th Computational Fluid Dynamics Conference, Danvers, Massachusetts, 1983.

9. Thompson, J. F. and Warsi, Z. U. A., "Three-Dimensional Grid Generation from Elliptic Systems", AIAA-83-1905, AIAA 6th Computational Fluid Dynamics Conference, Danvers, Massachussetts, 1983.

10. Coleman, Roderick M., "Generation of Boundary-Fitted Coordinate Systems Using Segmented Computational Regions", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 633, 1982.

11. Rubbert, P. E. and Lee, K. D., "Patched Coordinate Systems", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 235, 1982.

12. Thomas, P. D., "Numerical Generation of Composite Three-Dimensional by Quasilinear Elliptic Systems", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 667, 1982.

13. Miki, Kazuyoshi and Takagi, Toshiyuki, "A Domain Decomposition and Overlapping Method for the Generation of Three-Dimensional Boundary-Fitted Coordinate Systems", Journal of Computational Physics, 53, 319, 1984.

14. Thompson, Joe F., unpublished research, 1984.

15. Warsi, Z. U. A., "Tensors and Differential Geometry Applied to Analytic and Numerical Coordinate Generation", MSUU-EIRS-81-1, Mississippi State University, 1981.

16. Eiseman, P. R., "Geometric Methods in Computational Fluid Dynamics", ICASE 80-11, NASA Langley Research Center, 1980.

17. Thompson, Joe F. and Mastin, C. Wayne, "Order of Difference Expressions on Curvilinear Coordinate Systems", Advances in Grid Generation, FED-Vol. 5, Ed. K. N. Ghia and U. Ghia, ASME Applied Mechanics, Bioengineering, and Fluids Engineering Conference, Houston, 1983.

18. Vinokur, Marcel, "On One-Dimensional Stretching Functions for Finite-Difference Calculations", Journal of Computational Physics, 50, 215, 1983.

19. Warsi, Z. U. A., "Basic Differential Models for Coordinate Generation", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 41, 1982.

20. Mastin, C. Wayne and Thompson, Joe F., "Elliptic Systems and Numerical Transformations", Journal of Mathematical Analysis and Applications, 62, 52, 1978.

21. Mastin, C. Wayne and Thompson, Joe F., "Transformationof Three-Dimensional Regions onto Rectangular Regions by Elliptic Systems", Numerische Mathematik, 29, 397, 1978.

22. Mastin, C. Wayne and Thompson, Joe F., "Discrete Quasiconformal Mappings", Journal of Applied Mathematics and Physics (ZAMP), 29, 1978.

23. Mastin, C. W. and Thompson, J. F., "Quasiconformal Mappings and Grid Generation", SIAM Journal On Scientific and Statistical Computing, 5, 305, 1984.

24. Sorenson, R. L., "A Computer Program to Generate Two-Dimensional Grids About Airfoils and Other Shapes by the Use of Poisson's Equations", NASA Ames Research Center, NASA TM 81198, 1980.

25. Sorenson, Reese L., "Grid Generation by Elliptic Partial Differential Equations for a Tri-Element Augmentor-Wing Airfoil", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 653, 1982.

26. Sorenson, R. L. and Steger, J. L., "Grid Generation in Three Dimensions by Poisson Equations with Control of Cell Size and Skewness at Boundary Surfaces", Advances in Grid Generation, FED-Vol. 5, Ed. K. N. Ghia and U. Ghia, ASME Applied Mechanics, Bioengineering, and Fluids Engineering Conference, Houston, 1983.

27. Warsi, Z. U. A., "A Note on the Mathematical Formulation of the Problem of Numerical Coordinate Generation", Quarterly of Applied Mathematics, 41, 221, 1983.

28. Steger, J. L. and Chaussee, D. S., "Generation of Body Fitted Coordinates Using Hyperbolic Partial Differential Equations", SIAM J. Sci. Stat. Comput. 1, 431, 1980.

29. Steger, J. L. and Sorenson, R. L., "Use of Hyperbolic Partial Differential Equations to Generate Body Fitted Coordinates", Numerical Grid Generation Techniques, Ed. Robert E. Smith, NASA-CP-2166, 463, 1980.

30. Nakamura, S., "Marching Grid Generation Using Parabolic Partial Differential Equations", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 775, 1982.

31. Smith, Robert E., "Algebraic Grid Generation", Numerical Grid Generation, Ed. Joe F. Thompson, North Holland, 137, 1982.

32. Eiseman, P. R., "A Multi-Surface Method of Coordinate Generation", Journal of Computational Physics, 33, 118, 1979.

33. Eiseman, Peter R. and Smith, Robert, "Mesh Generation Using Algebraic Techniques", Numerical Grid Generation Technigues, Ed. Robert E. Smith, NASA CP-2166, 1980.

34. Eiseman, Peter R., "Automatic Algebraic Coordinate Generation", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 447, 1982.

35. Eiseman, Peter R., "Coordinate Generation with Precise Controls Over Mesh Properties", Journal of Computational Physics, 47, 331, 1982.

36. Eiseman, Peter R., "High Level Continuity for Coordinate Generation with Precise Controls", Journal of Computational Physics, 47, 352, 1982.

37. Eiseman, Peter R. "Grid Generation for Fluid Mechanics Computations", Annual Review of Fluid Mechanics, Vol. 17, 1985.

38. Eiseman, Peter R. unpublished result referred to in Ref. [33].

39. Roberts, A., "Automatic Topology Generation and Generalized B-Spline Mapping", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 465, 1982.

40. Gordon, William J. and Thiel, Linda C., "Transfinite Mappings and Their Application to Grid Generation", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 171, 1982.

41. Gordon, W. J. "Blending Function Methods of Bivariate and Multivariate Interpolation", SIAM J. of Numerical Analysis, 8, 158, 1971.

42. Eiseman, Peter R., "Orthogonal Grid Generation", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 193, 1982.

43. Ives, D. C. and Siddons, W. D., "Orthogonal Grid Generation", AIAA-84-1248, AIAA/SAE/ASME 20th Joint Propulsion Conference, Cincinnati, 1984.

44. Kober, H., Dictionary of Conformal Representations, Dover, New York, 1952.

45. Thompson, Joe F., "A Survey of Dynamically-Adaptive Grids in the Numerical Solution of Partial Differential Equations", to appear in Journal of Numerical Mathematics, 1984. (also AIAA-84-1606, AIAA Fluid and Plasma Dynamics Conference, Snowmass, Colorado, 1984).

46. Dwyer, H. A., Smooke, Mitchell, D. and Kee, Robert J., "Adaptive Gridding for Finite Difference Solutions to Heat and Mass Transfer Problems", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 339, 1982.

47. Brackbill, J. U. and Saltzman, J. S., "Adaptive Zoning for Singular Problems in Two Dimensions", Journal of Computational Physics, 46, 342, 1982.

48. Anderson, Dale, A., and Rai, M. M., "The Use of Solution Adaptive Grids in Solving Partial Differential Equations", Numerical Grid Generation, ed. Joe F. Thompson, North-Holland, 317. 1982.

49. Miller, Keith and Miller, Robert N., "Moving Finite Elements. I", SIAM Journal of Numerical Analysis, 18, 1019, 1981.

50. Miller, Keith, "Moving Finite Elements. II", SIAM Journal of Numerical Analysis, 18, 1033, 1981.

51. Weatherill, N. C. and Forsey, C. R. "Grid Generation and Flow Calculations for Complex Aircraft Geometries Using a Multi-Block Scheme", AIAA-84-1665, AIAA 17th Fluid Dynamics, Plasma Dynamics, and Lasers Conference, Snowmass, CO, 1984.

52. Rai, N. M. "A Conservative Treatment of Zonal Boundaries for Euler Equations Calculations", AIAA-84-0164, AIAA 22nd Aerospace Sciences Meeting, Reno, NV, 1984.

53. Hessenius, K. A. and Rai, M. M., "Applications of a Conservative Zonal Scheme to Transient and Geometrically Complex Problems", AIAA-S4-1532, AIAA 17th Fluid Dynamics, Plasma Dynamics, and Lasers Conference, Snowmass, CO, 1984.

GRID ILLUSTRATION REFERENCES

The grids used for illustration are drawn from those in the works cited here:

Page

20 Anderson, 0. L., Davis, R. T., Hankins, G. B., and Ewards, D. E., "Solution of Viscous Internal Flows on Curvilinear Grids Generated by the Schwarz-Christoffel Transformation." in Ref. [2].

21 Kumar, D., Hester, L. R., and Thompson, J. F., "Development of Partial Channel Flow for Arbitrary Input Velocity Distribution Using Boundary-Fitted Coordinate Systems", in Nonsteady Fluid Dynamics, ASME Winter Annual Meeting, San Francisco, 53, 1978.

22, McWhorter, John C., "Solid Mechanics Applications of Boundary Fitted Coordinate Systems", in Ref. [2].
23

24, Lee, K. D., Huang, Yu, N. J., and Rubbert, P. E., "Grid Generation for General Three-Dimensional Configurations", in Ref. [3].
25

27, Coleman, Roderick M., "Generation of Boundary-Fitted Coordinate Systems Using Segmented Computational Regions", in Ref. [2].
28

29 Thompson, J. F., Thames, F. C., and Mastin, C. W., "'TOMCAT' -- A Code for Numerical Generation of Boundary- Fitted Curvilinear Coordinate Systems on Fields Containing any Number of Arbitrary Two-Dimensional Bodies", Journal of Comptational Physics, 245, 1977.

30 Reddy, R. N. and Thompson, Joe F., "Numerical Solution of Incompressible Navier-Stokes Equations in the Integro-Differential Formulation Using Boundary-Fitted Coordinate Systems", Proceedings of the AIAA 3rd Computational Dynamics Conference, Albuquerque, 1977.

33 Thompson, J. F., "A Boundary-Fitted Coordinate Code for General Two-Dimensional Regions with Obstacles and Boundary Intrusions", Technical Report E-83-8, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Mississippi, 1983.

34 Thompson, J. F., Thames, F. C., and Mastin, C. W., "'TOMCAT' -- A code for Numerical Generation of Boundary-Fitted Curvilinear Coordinate Systems on Fields Containing any Number of Arbitrary Two-Dimensional Bodies", Journal of Comptuational Physics, 245, 1977.

35, Thompson, J. F., Thames, F. C., and Mastin, C. W., "'TOMCAT' -- A Code for Numerical Generation of Boundary-Fitted Curvilinear Coordinate Systems on Fields Containing any Number of Arbitrary Two-Dimensional Bodies", Journal of Computational Physics, 245, 1977.
37

38 Halsey, Douglas, "Conformal Grid Generation for Multi-Element Airfoils", in Ref. [2].

39 Long, W. S., "Two-Body Coordinate System Generation Using Body-Fitted Coordinate System and Complex Variable Transformation", M.S. thesis, Mississippi State University, 1977.

40 Coleman, R. M. "NUMESH: A Computer Program to Generate Finite Difference Meshes for Arbitrary Double-Connected Two-Dimensional Regions," CMLD-77-05, David W. Taylor Naval Ship Research and Development Center, 1977.

45 Haussling, Henry J., "Solution of Nonlinear Water Wave Problems Using Boundary-Fitted Coordinate Systems," in Ref. [2].

48 Sorenson, Reese L., "Grid Generation by Elliptic Partial Differential Equations for a Tri-Element Augmentor-Wing Airfoil," in Ref. [2].

52 Thompson, Joe F., General Curvilinear Coordinate Systems," in Ref. [2].

53a Chen, Brian C-J, Sha, W. F., Doria, M. L., Schmidt, R. C., and Thompson, J. F., "BODYFIG-IFE: A Computer Code for the Three-Dimensional Steady-State/Transient Single-Phase Rod-Bundle Thermal-Hydraulic Analysis, NUREG/CR-1874, ANL-80-127, Argonne National Laboratory, 198O.

54 Dulikravich, Djordje S., "Fast Generation of Three-Dimensional Computational Boundary-Conforming Periodic Grids of C-Type," in Ref. [2].

54 Rai Man Mohan, "An Implicit, Conservative, Zoned-Boundary Scheme for Euler Equation Calculations", AIAA-85-0488, AIAA 23rd Aerospace Sciences Meeting, Reno, 1985.

55 Ives, David D., "Conformal Grid Generation," in Ref. [2].

55 Rai, Man Mohan, "A Relaxation Approach to Patched-Grid Calculations with the Euler equations". AIAA-85-0295, AIAA 23rd Aerospace Sciences Meeting, Reno, 1985.

56 Eiseman, P. R., "Alternating Direction Adaptive Grid Generation", AIAA-83-1937,AIAA 6th Computational Fluid Dynamics Conference, Danvers, Mass. 1983.

57 Nakahashi, Kazuhiro and Deiwert, George S., "A Practical Adaptive Grid Method for Complex Fluid-Flow Problems", NASA TM 85989, NASA Ames Research Center, 1984.

57a Jain, Sunil K., "Embedded-Grid Generation with Complete Continuity Across Interfaces for Multi-Element Airfoils." ASE 84-270, Ph-D Dissertation, Mississippi State University, 1984.

57b Sorenson, Reese L., "Grid Generation by Elliptic Partial Differential Equations for a Tri-Element Augmentor-Wing Airfoil," in Ref. [2].

57c Rubbert, P. E. and Lee, K. D., "Patched Coordinate Systems," in Ref. [21.

58 Halsey, Douglas, "Conformal Grid Generation for Multi-element Air foils," in Ref. [2].

65a Jain, Sunil K., "Embedded-Grid Generation With Complete Contintuity Across Interfaces for Multi-Element Airfoils." ASE 84-270, Ph-D Dissertation, Mississippi State University, 1984.

65b Sorenson, Reese L., "Grid Generation by Elliptic Partial Differential Equations for a Tri-Element Augmentor-Wing Airfoil," in Ref. [2].

69a Miki, Kazuyoshi and Takagi, Toshiyuki, "A Domain Decomposition and Overlapping Method for the Generation of Three-Dimensional Boundary-Fitted Coordinate Systems", Journal of Computational Physics 53, 319, 1984.

69b Steger, J. L., Dougherty, F. C., and Benek, J. A., "A Chimera Grid Scheme," Advances in Grid Generation, ASME Fluids Engineering Conference, Houston, June 1983.

190a Johnson, Billy H. and Thompson, Joe F., "A Discussion of Boundary-Fitted Coordinate Systems and Their Applicability to the Numerical Modeling of Hydraulio Problems," Miscellaneous Paper H-78-9, U. S. Army Engineer Waterways Experiment Station, Vicksburg, Mississippi, 1978.

190b Chae, Yeon Seok, "An Investigation of a Navier-Stokes Solution for Quasi-Three-Dimensional flow". ASE 84-282, M.S. thesis, Mississippi State University, 1984.

193 Thompson, J. F., Thames, F. C., and Mastin, C. W., "Automatic Numerical Generation of Body-Fitted Curvilinear Coordinate System for Fields Containing any Number of Arbitrary Two-Dimensional Bodies," Journal of Computatonal Physics, 15, 299, 1974.

194 Thompson, J. F., Thames, F. C., and Mastin, C. W., "TOMCAT" -- A Code for Numerical Generation of Boundary-Fitted Curvilinear Coordinate Systems on Fields Containing any Number of Arbitrary Two- Dimensional Bodies," Journal of Computational Physics, 245, 1977.

196 Shanks, S. P. and Thompson, J. F., "Numerical Solution of the Navier-Stokes Equation for 2D Hydrofoil in or Below a Free Surface," Proceedings of the 2nd International Conference on Numerical Ship Hydrodynamics, Berkeley, 1977.

208 Thompson, Joe F., General Curvilinear Coordinate Systems," in Ref. [2].

232a Sorenson, Reese L., "Grid Generation by Elliptic Partial Differential Equations for a Tri-Element Augmentor-Wing Airfoil," in Ref. [2].

232b Sorenson, R. L., "A Computer Program to Generate Two-Dimensional Grids About Airfoils and Other Shapes by the Use of Poisson's Equations," NASA TM 81198, 1980.

350a Ives, David C., "Conformal Grid Generation," in Ref. [2].

350b Dulikravich, Djordje S., "Fast Generation of Three-Dimensional Computational Boundary-Conforming Periodic Grids of C-Type," in Ref. [2].

383 Dywer, H. A., Smooke, Mitchell, D. and Kee, Robert J., "Adaptive Gridding for Finite Difference Solutions to Heat and Mass Transfer Problems", Numerical Grid Generation, Ed. Joe F. Thompson, North-Holland, 339, 1982.

388, Dwyer, H. A., Smooke, Mitchell, D. and Kee, Robert J., "Adaptive Gridding for Finite Difference Solutions to Heat and Mass Transfer Problems," in Ref. [2].
389a

389b Gnoffo, Peter A., "A Vectorized Finite-Volume, Adaptive Grid Algorithm for Navier-Stokes Calculations," in Ref. [2].

390 Nakamura, S., "Adaptive Grid Relocation Algorithm for Transonic Full Potential Calculators Using One-Dimensional or Two-Dimensional Diffusion Equations," Advances in Grid Generation, ASME Fluids Engineering Conference, Houston, 1983.

411, Saltzman, Jeffery and Brackbill, Jeremiah, "Applications and Generalizations of Variational Methods for Generating Adaptive Meshes," in Ref. [2].
412
       350a Ives, David C., "Conformal Grid Generation," in Ref. [2].
       350b Dulikravich, Dgordge S., "Fast Generation of Three-Dimensional Computational Boundary-Conforming Periodic Grids of C-Type," in Ref. [2].

413 Bell, J. B. and Shulin, G. R., "An Adaptive Grid Finite Difference Method for Conservation Laws", Journal of Computational Physics, 52, 569, 1983.