×

Electromagnetic problems solving by conformal mapping: a mathematical operator for optimization. (English) Zbl 1213.78033

Summary: Having the property to modify only the geometry of a polygonal structure, preserving its physical magnitudes, the Conformal Mapping is an exceptional tool to solve electromagnetism problems with known boundary conditions. This work aims to introduce a new developed mathematical operator, based on polynomial extrapolation. This operator has the capacity to accelerate an optimization method applied in conformal mappings, to determinate the equipotential lines, the field lines, the capacitance, and the permeance of some polygonal geometry electrical devices with an inner dielectric of permittivity \(\epsilon\) . The results obtained in this work are compared with other simulations performed by the software of finite elements method, Flux \(2D\).

MSC:

78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] P. Henrici, Applied and Computational Complex Analysis, vol. 3 of Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1986. · Zbl 0578.30001
[2] M. R. Spiegel, Complex Variable, McGraw-Hill, New York, NY, USA, 1967.
[3] J. D. Kraus and K. R. Carver, Electromagnetics, McGraw-Hill, New York, NY, USA, 1973.
[4] W. P. Calixto, E. G. Marra, L. C. Brito, and B. P. Alvarenga, “A new methodology to calculate carter factor using geneticalgorithms,” International Journal of Numerical Modelling: Electronic Networks, Devices and Fields. In press. · Zbl 1220.78075
[5] Z. Nehari, Conformal Mapping, McGraw-Hill, New York, NY, USA, 1952. · Zbl 0049.17603
[6] H. Cohn, Conformal Mapping on Riemann Surfaces, Dover, New York, NY, USA, 1967. · Zbl 0498.16002
[7] W. J. Gibbs, Conformal Transformations in Electrical Engineering, Chapman and Hall, London, UK, 1958. · Zbl 0084.07006
[8] G. C. Wen, Conformal Mappings and Boundary Value Problems, vol. 106 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1992. · Zbl 1020.37549
[9] L. N. Trefethen, “Numerical computation of the Schwarz-Christoffel transformation,” Society for Industrial and Applied Mathematics, vol. 1, no. 1, pp. 82-102, 1980. · Zbl 0451.30004
[10] T. A. Driscoll and S. A. Vavasis, “Numerical conformal mapping using cross-ratios and Delaunay triangulation,” SIAM Journal on Scientific Computing, vol. 19, no. 6, pp. 1783-1803, 1998. · Zbl 0915.30006
[11] E. Costamagna, “On the numerical inversion of the schwarz-christoffel conformal transformation,” IEEE Transactions on Microwave Theory and Techniques, vol. 35, no. 1, pp. 35-40, 1987.
[12] T. A. Driscoll and L. N. Trefethen, Schwarz-Christoffel Mapping, vol. 8 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2002. · Zbl 1003.30005
[13] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Dover, New York, NY, USA, 1968. · Zbl 0171.38503
[14] S. C. Milne, Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions, vol. 5 of Developments in Mathematics, Kluwer Academic Publishers, Boston, Mass, USA, 2002. · Zbl 1125.11315
[15] W. P. Calixto, Application of conformal mapping to the calculus of Carter’s factor, M.S. thesis, Electrical & Computer Engineering School, Federal University of Goias, Goiania, Brazil, 2008.
[16] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Artificial Intelligence, Springer, Berlin, Germany, 1992. · Zbl 0763.68054
[17] Z. Michalewicz and D. B. Fogel, How to Solve it: Modern Heuristics, Springer, Berlin, Germany, 2000. · Zbl 0943.90002
[18] J. H. Holland, Adaptation in Natural and Artificial Systems: an Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, University of Michigan Press, Ann Arbor, Mich, USA, 1975. · Zbl 0317.68006
[19] F. Herrera, M. Lozano, and J. L. Verdegay, “Crossover operators and offspring selection for real coded genetic algorithms,” Tech. Rep., Department of Intelligence of the Computation and Artificial intelligence, University of Granada, Granada, Spain, 1994. · Zbl 0833.68054
[20] Cedrat, Flux 2D User’s Guide, Cedrat, Grenoble, France, 2000.
[21] R. E. Collins, Mathematical Methods for Physicists and Engineers, Dover, New York, NY, USA, 2nd edition, 1999. · Zbl 0176.00401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.