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APPLIED SYMMETRICAL PRINCIPLE TO SOLVE SCHWARZ-CHRISTOFFEL PARAMETER PROBLEM

  • YUAN, MEI (SCHOOL OF RESOURCES AND ENVIRONMENTAL ENGINEERING, WUHAN UNIVERSITY OF TECHNOLOGY) ;
  • PENG, HUIQING (SCHOOL OF RESOURCES AND ENVIRONMENTAL ENGINEERING, WUHAN UNIVERSITY OF TECHNOLOGY) ;
  • LEI, YUN (SCHOOL OF RESOURCES AND ENVIRONMENTAL ENGINEERING, WUHAN UNIVERSITY OF TECHNOLOGY)
  • Published : 2018.10.31

Abstract

This paper adopted symmetry theory to solve the Schwarz-Christoffel parameter problem for axisymmetric polygons. Numerical conformal mappings were performed to shift the upper half-plane onto polygonal domains. Once the constraint conditions of the problem were treated in a special way such as added or deleted a little area, it turns to be a solution of a singular integral.In this paper, an auxiliary point was suggested to attach to the polygon that obeyed the principle of symmetry, which can accelerate the solving process of the singular integral. After that, several numerical examples, along with an application related to electrostatics, are provided to verify its feasibility and simplification. When the distance from the auxiliary point to polygon is controlled under 1E-08, the accuracy can be controlled within 1E-09, accuracy and consequences of the calculation basically meet the ordinary requirement.

References

  1. V. N. Dubinin, M. Vuorinen, On conformal moduli of polygonal quadrilaterals, Israel J Math, 171(2009), 111-125. https://doi.org/10.1007/s11856-009-0043-8
  2. R. Khnau, The conformal module of quadrilaterals and of rings, Handbook of Complex Analysis: Geometric Function Theory (Volumn 2). North Holland, Amsterdam, 2005.
  3. G. Polya, G. Szeg, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, Princeton University Press, Princeton, 1951.
  4. H. Hakula, R. Antti, V. Matti, On moduli of rings and quadrilaterals: algorithms and experiments, SIAM J Sci Comput, 33(1) (2011), 279-302 https://doi.org/10.1137/090763603
  5. T. A.Driscoll, L. N.Trefethen, Schwarz-Christoffel Mapping, Cambridge University Press, Cambridge, 2002.
  6. T. A.Driscoll, The Schwarz-Christoffel tool-box for MATLAB, Available via DIALOG, http://www.math.udel.edu/driscoll/SC/, Cite 12 Dec 2017.
  7. W. P.Calixto, J. C. da Mota, B. P. de Alvarenga, Methodology for the Reduction of Parameters in the Inverse Transformation of Schwarz-Christoffel Applied to Electromagnetic Devices with Axial Geometry, Int J Numer Model El, 24(6) (2011), 568-582. https://doi.org/10.1002/jnm.804
  8. D. M.Hough, Asymptotic Gauss-Jacobi quadrature error estimation for Schwarz-Christoffel integrals, J Approx Theory, 146 (2007), 157-173. https://doi.org/10.1016/j.jat.2006.10.004
  9. V. A.Koptsik, Symmetry principle in physics, J Phys C: Solid State Phys, 16 (1983), 23-34. https://doi.org/10.1088/0022-3719/16/1/007
  10. J. Merker, F. Meylan, On the Schwarz symmetry principle in a model case. P Am Math Soc, 127(4) (1999), 1097-1102. https://doi.org/10.1090/S0002-9939-99-04688-2
  11. D. Babbitt, V. Chari, R. Fioresi, Symmetry in Mathematics and Physics, American Mathematical Society, New York, 2009.
  12. L. Brickman, The symmetry principle for Mobius transformations, Am Math Mon. 100(8) (1993), 781-782. https://doi.org/10.2307/2324786
  13. Y. Q. Zhong, Theory of Functions of a Complex Variable (Third Edition), China Higher Education Press, Beijing, 2004.
  14. E. B. Saff, A. D. Snider Fundamentals of Complex Analysis with Applications to Engineering and Science (Third Edition), Prentice Hall, Upper Saddle River, 2003.
  15. J. K. Lu, Boundary Value Problem of Analytic Function, Wuhan University Press, Wuhan, 2004.
  16. C. H. Liang, Sketches of Complex Function, Science Press, Beijing, 2011.
  17. Z.C.Huang, X.J.Chen, Numerical Analysis, Science Press, Beijing, 2010.
  18. G.Wang, H.Z.Xu, Numerical SchwarzChristoffel Transformation and Numerical Gauss-Jacobi Quadrature, Academic Journal of Naval Engineering College, 67(2) (1994), 25-33.