A noniterative method for the generation of orthogonal coordinates in doubly-connected regions

Authors:
Z. U. A. Warsi and J. F. Thompson

Journal:
Math. Comp. **38** (1982), 501-516

MSC:
Primary 65N50; Secondary 30C60, 31A05

DOI:
https://doi.org/10.1090/S0025-5718-1982-0645666-3

MathSciNet review:
645666

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Abstract: In this paper a noniterative method for the numerical generation of orthogonal curvilinear coordinates for plane annular regions between two arbitrary smooth closed curves has been developed. The basic generating equation is the Gaussian equation for a Euclidean space under the constraint of orthogonality. The resulting equation has been solved analytically for the case of isothermic coordinates. A coordinate transformation then yields nonisothermic coordinates in which the outgoing coordinate can be distributed in any desired manner. The method has been applied in many cases and these test results demonstrate that the proposed method can be readily applied in a variety of problems.

**[1]**A. A. Amsden & C. W. Hirt, "A simple scheme for generating general curvilinear grids,"*J. Comput. Phys.*, v. 11, 1973, pp. 348-359.**[2]**W. D. Barfield,*An optimal mesh generator for Lagrangian hydrodynamic calculations in two space dimensions*, J. Computational Phys.**6**(1970), 417–429. MR**0273829****[3]**J. Burbea,*A numerical determination of the modulus of doubly connected domains by using the Bergman curvature*, Math. Comp.**25**(1971), 743–756. MR**0289758**, https://doi.org/10.1090/S0025-5718-1971-0289758-2**[4]**W. H. Chu, "Development of a general finite difference approximation for a general domain, Part I: Machine transformation,"*J. Comput. Phys.*, v. 8, 1971, pp. 392-408.**[5]**A. Cohen,*An Introduction to the Lie Theory of One-Parameter Groups*, Stechert, New York, 1931.**[6]**R. T. Davis,*Numerical Methods for Coordinate Generation Based on Schwarz-Christoffel Transformation*, AIAA Computational Fluid Dynamics Conference, Paper No. 79-1463, 1979.**[7]**P. R. Eiseman, "A coordinate system for a viscous transonic cascade analysis,"*J. Comput. Phys.*, v. 26, 1978, pp. 307-338.**[8]**Luther Pfahler Eisenhart,*Riemannian Geometry*, Princeton University Press, Princeton, N. J., 1949. 2d printing. MR**0035081****[9]**Dieter Gaier,*Determination of conformal modules of ring domains and quadrilaterals*, Functional analysis and its applications (Internat. Conf., Eleventh Anniversary of Matscience, Madras, 1973; dedicated to Alladi Ramakrishnan), Springer, Berlin, 1974, pp. 180–188. Lecture Notes in Math., Vol. 399. MR**0393446****[10]**S. K. Godunov and G. P. Prokopov,*The utilization of movable grids in gas dynamic calculations*, Ž. Vyčisl. Mat. i Mat. Fiz.**12**(1972), 429–440 (Russian). MR**0314360****[11]**G. M. Goluzin,*Geometric theory of functions of a complex variable*, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR**0247039****[12]**F. B. Hildebrand,*Introduction to numerical analysis*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956. MR**0075670****[13]**H. Kober,*Dictionary of conformal representations*, Dover Publications, Inc., New York, N. Y., 1952. MR**0049326****[14]**P. D. Thomas and J. F. Middlecoff,*Direct control of the grid point distribution in meshes generated by elliptic equations*, AIAA J.**18**(1980), no. 6, 652–656. MR**571548**, https://doi.org/10.2514/3.50801**[15]**C. D. Mobley & R. J. Stewart, "On the numerical generation of boundary-fitted orthogonal curvilinear coordinate systems,"*J. Comput. Phys.*, v. 34, 1980, pp. 124-135.**[16]**S. B. Pope, "The calculation of turbulent recirculating flows in general orthogonal coordinates,"*J. Comput. Phys.*, v. 26, 1978, pp. 197-217.**[17]**D. E. Potter & G. H. Tuttle, "The construction of discrete orthogonal coordinates,"*J. Comput. Phys.*, v. 13, 1973, p. 483.**[18]**M. K. Richardson & H. B. Wilson, "A numerical method for the conformal mapping of finite doubly connected regions,"*Developments in Theoretical and Applied Mechanics*, Vol. 3, Ed., W. A. Shaw, Plenum Press, 1967.**[19]**Göran Starius,*Constructing orthogonal curvilinear meshes by solving initial value problems*, Numer. Math.**28**(1977), no. 1, 25–48. MR**0440968**, https://doi.org/10.1007/BF01403855**[20]**J. F. Thompson, F. C. Thames & C. W. Mastin, "Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies,"*J. Comput. Phys.*, v. 15, 1974, pp. 299-319.**[21]**J. F. Thompson, F. C. Thames & C. W. Mastin, "'TOMCAT'--A code for numerical generation of boundary-fitted curvilinear coordinate system on fields containing any number of arbitrary two-dimensional bodies,"*J. Comput. Phys.*, v. 24, 1977, pp. 274-302.**[22]**Z. U. A. Warsi, R. A. Weed & J. F. Thompson,*Numerical Generation of Two-Dimensional Orthogonal Coordinates in an Euclidean Space*, Engineering and Experimental Research Station, Mississippi State University, Rep. No. MSSU-EIRS-ASE-80-3, June 1980.**[23]**Z. U. A. Warsi & J. F. Thompson,*Machine Solutions of Partial Differential Equations in the Numerically Generated Coordinate Systems*, Engineering and Experimental Research Station, Mississippi State University, Rep. No. MSSU-EIRS-ASE-77-1, August 1976.**[24]**A. J. Winslow, "Numerical solution of the quasi-linear Poisson equation in a non-uniform triangular mesh,"*J. Comput. Phys.*, v. 2, 1966, pp. 149-172.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1982-0645666-3

Keywords:
Grid generation,
mappings,
potential theory

Article copyright:
© Copyright 1982
American Mathematical Society