A noniterative method for the generation of orthogonal coordinates in doublyconnected regions
Authors:
Z. U. A. Warsi and J. F. Thompson
Journal:
Math. Comp. 38 (1982), 501516
MSC:
Primary 65N50; Secondary 30C60, 31A05
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.
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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. MSSUEIRSASE771, August 1976.
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A. J. Winslow, "Numerical solution of the quasilinear Poisson equation in a nonuniform triangular mesh," J. Comput. Phys., v. 2, 1966, pp. 149172.
 [1]
 A. A. Amsden & C. W. Hirt, "A simple scheme for generating general curvilinear grids," J. Comput. Phys., v. 11, 1973, pp. 348359.
 [2]
 W. D. Barfield, "An optimal mesh generator for Lagrangian hydrodynamic calculations in two space dimensions," J. Comput. Phys., v. 6, 1970, pp. 417429. MR 0273829 (42:8705)
 [3]
 J. Burbea, "A numerical determination of the modulus of doubly connected domains by using the Bergman curvature," Math. Comp., v. 25, 1971, pp. 743756. MR 0289758 (44:6946)
 [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. 392408.
 [5]
 A. Cohen, An Introduction to the Lie Theory of OneParameter Groups, Stechert, New York, 1931.
 [6]
 R. T. Davis, Numerical Methods for Coordinate Generation Based on SchwarzChristoffel Transformation, AIAA Computational Fluid Dynamics Conference, Paper No. 791463, 1979.
 [7]
 P. R. Eiseman, "A coordinate system for a viscous transonic cascade analysis," J. Comput. Phys., v. 26, 1978, pp. 307338.
 [8]
 L. P. Eisenhart, Riemannian Geometry, Princeton Univ. Press, Princeton, N. J., 1926. MR 0035081 (11:687g)
 [9]
 D. Gaier, Determination of Conformal Modulus of Ring Domains and Quadrilaterals, Lecture Notes in Math., vol. 399, SpringerVerlag, Berlin, 1974. MR 0393446 (52:14256)
 [10]
 S. K. Godunov & G. P. Prokopov, "The use of moving meshes in gas dynamics computations," USSR Comput. Math. and Math. Phys., v. 12, 1972, pp. 182195. MR 0314360 (47:2912)
 [11]
 G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Transl. Math. Monos., vol. 26, Amer. Math. Soc., Providence, R. I., 1969. MR 0247039 (40:308)
 [12]
 F. B. Hildebrand, Introduction to Numerical Analysis, McGrawHill, New York, 1956. MR 0075670 (17:788d)
 [13]
 H. Kober, Dictionary of Conformal Representations, Dover, New York, 1952. MR 0049326 (14:156d)
 [14]
 J. F. Middlecoff & P. D. Thomas, Direct Control of the Grid Point Distribution in Meshes Generated by Elliptic Equations, AIAA Computational Fluid Dynamics Conference, Paper No. 791462, 1979. MR 571548 (81f:65071)
 [15]
 C. D. Mobley & R. J. Stewart, "On the numerical generation of boundaryfitted orthogonal curvilinear coordinate systems," J. Comput. Phys., v. 34, 1980, pp. 124135.
 [16]
 S. B. Pope, "The calculation of turbulent recirculating flows in general orthogonal coordinates," J. Comput. Phys., v. 26, 1978, pp. 197217.
 [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]
 G. Starius, "Constructing orthogonal curvilinear meshes by solving initialvalue problems," Numer. Math., v. 28, 1977, pp. 2548. MR 0440968 (55:13836)
 [20]
 J. F. Thompson, F. C. Thames & C. W. Mastin, "Automatic numerical generation of bodyfitted curvilinear coordinate system for field containing any number of arbitrary twodimensional bodies," J. Comput. Phys., v. 15, 1974, pp. 299319.
 [21]
 J. F. Thompson, F. C. Thames & C. W. Mastin, "'TOMCAT'A code for numerical generation of boundaryfitted curvilinear coordinate system on fields containing any number of arbitrary twodimensional bodies," J. Comput. Phys., v. 24, 1977, pp. 274302.
 [22]
 Z. U. A. Warsi, R. A. Weed & J. F. Thompson, Numerical Generation of TwoDimensional Orthogonal Coordinates in an Euclidean Space, Engineering and Experimental Research Station, Mississippi State University, Rep. No. MSSUEIRSASE803, 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. MSSUEIRSASE771, August 1976.
 [24]
 A. J. Winslow, "Numerical solution of the quasilinear Poisson equation in a nonuniform triangular mesh," J. Comput. Phys., v. 2, 1966, pp. 149172.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206456663
PII:
S 00255718(1982)06456663
Keywords:
Grid generation,
mappings,
potential theory
Article copyright:
© Copyright 1982
American Mathematical Society
