Abstract
In this paper, the Maple computer algebra system is used to construct a symbolic-numeric implementation of the method for calculating normal modes of square closed waveguides in a vector formulation. The method earlier proposed by Malykh et al. [M.D. Malykh, L.A. Sevastianov, A.A. Tiutiunnik, N.E. Nikolaev. On the representation of electromagnetic fields in closed waveguides using four scalar potentials // Journal of Electromagnetic Waves and Applications, 32 (7), 886–898 (2018)] will be referred to as the method of four potentials. The Maple system is used at all stages of treating the system of differential equations for four potentials: the generation of the Galerkin basis, the substitution of approximate solution into the system under study, the formulation of a computational problem, and its approximate solution.
Thanks to the symbolic-numeric implementation of the method, it is possible to carry out calculations for a large number of basis functions of the Galerkin decomposition with reasonable computation time and then to investigate the convergence of the method and verify it, which is done in the present paper, too.
The publication has been prepared with the support of the “RUDN University Program 5-100” and funded by RFBR according to the research projects Nos. 18-07-00567 and 18-51-18005.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Malykh, M.D., Sevastianov, L.A., Tiutiunnik, A.A., Nikolaev, N.E.: On the representation of electromagnetic fields in closed waveguides using four scalar potentials. J. Electromagn. Waves Appl. 32(7), 886–898 (2018)
Divakov, D.V., Lovetskiy, K.P., Malykh, M.D., Tiutiunnik, A.A.: The application of Helmholtz decomposition method to investigation of multicore fibers and their application in next-generation communications systems. Commun. Comput. Inf. Sci. 919, 469–480 (2018)
Gusev, A.A., et al.: Symbolic-numerical algorithms for solving the parametric self-adjoint 2D elliptic boundary-value problem using high-accuracy finite element method. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2017. LNCS, vol. 10490, pp. 151–166. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66320-3_12
Gusev, A.A., et al.: Symbolic-numerical algorithm for generating interpolation multivariate hermite polynomials of high-accuracy finite element method. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2017. LNCS, vol. 10490, pp. 134–150. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66320-3_11
Shapeev, V.P., Vorozhtsov, E.V.: The method of collocations and least residuals combining the integral form of collocation equations and the matching differential relations at the solution of pdes. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2017. LNCS, vol. 10490, pp. 346–361. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66320-3_25
Sevastyanov, L.A., Sevastyanov, A.L., Tyutyunnik, A.A.: Analytical calculations in maple to implement the method of adiabatic modes for modelling smoothly irregular integrated optical waveguide structures. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2014. LNCS, vol. 8660, pp. 419–431. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10515-4_30
Bertolazzi, E., Biral, F., Da Lio, M.: Symbolic-numeric efficient solution of optimal control problems for multibody systems. J. Comput. Appl. Math. 185(2), 404–421 (2006)
Gutnik, S.A., Sarychev, V.A.: Symbolic-numerical methods of studying equilibrium positions of a gyrostat satellite. Program. Comput. Softw. 40(3), 143–150 (2014)
Budzko, D.A., Prokopenya, A.N.: Symbolic-numerical analysis of Equilibrium solutions in a restricted four-body problem. Program. Comput. Softw. 36(2), 68–74 (2010)
Shapeev, V.P., Vorozhtsov, E.V.: Symbolic-numerical optimization and realization of the method of collocations and least residuals for solving the Navier-stokes equations. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2016. LNCS, vol. 9890, pp. 473–488. Springer, Cham (2016)
Semin, L., Shapeev, V.: Constructing the numerical method for Navier-stokes equations using computer algebra system. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 367–378. Springer, Berlin (2005)
Shapeev, V.P., Vorozhtsov, E.V.: CAS application to the construction of the collocations and least residuals method for the solution of 3D Navier-stokes equations. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 381–392. Springer, Cham (2013)
Shapeev, V.P., Vorozhtsov, E.V.: Symbolic-numeric implementation of the method of collocations and least squares for 3D Navier-stokes equations. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2012. LNCS, vol. 7442, pp. 321–333. Springer, Heidelberg (2012)
Kantorovich, L.V., Krylov, V.I.: Approximate Methods of Higher Analysis. Wiley, New York (1964)
Fletcher, C.A.J.: Computational Galerkin Methods. Springer-Verlag, Heidelberg (1984)
Adams, M.J.: An Introduction to Optical Waveguides. Wiley, New York (1981)
Marcuse, D.: Light Transmission Optics. Van Nostrand, New York (1974)
Tamir, T.: Guided-Wave Optoelectronics. Springer-Verlag, Berlin (1990)
Ladyzhenskaya, O.A.: The Boundary Value Problems of Mathematical Physics. Springer, Heidelberg (1985)
Hellwig, G.: Differential Operators of Mathematical Physics. Addison-Wesley, MA (1967)
Bathe, K.J.: Finite Element Procedures in Engineering Analysis. Prentice Hall, Englewood Cliffs (1982)
Ciarlet, P.: The Finite Element Method for Elliptic Problems. North Holland Publishing Company, Amsterdam (1978)
Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)
Bogolyubov, A.N., Mukhartova, Yu.V., Gao, J., Bogolyubov, N.A.: Mathematical modeling of plane chiral waveguide using mixed finite elements. In: Progress in Electromagnetics Research Symposium, pp. 1216–1219 (2012)
Bogolyubov, A.N., Mukhartova, Y.V., Gao, T.: Calculation of a parallel-plate waveguide with a chiral insert by the mixed finite element method. Math. Models Compu. Simul. 5(5), 416–428 (2013)
Mukhartova, Y.V., Mongush, O.O., Bogolyubov, A.N.: Application of the finite-element method for solving a spectral problem in a waveguide with piecewise constant bi-isotropic filling. J. Commun. Technol. Electron. 62(1), 1–13 (2017)
Sveshnikov, A.G.: The basis for a method of calculating irregular waveguides. Comput. Math. Math. Phys. 3(1), 170–179 (1963)
Sveshnikov, A.G.: A substantiation of a method for computing the propagation of electromagnetic oscillations in irregular waveguides. Comput. Math. Math. Phys. 3(2), 314–326 (1963)
Mathematics-based software and services for education, engineering, and research. https://www.maplesoft.com/
Anderson, E., et al.: LAPACK Users’ Guide, 3rd edn. SIAM, Philadelphia (1999). http://www.netlib.org/lapack/lug
LAPACK Users’ Guide Release. http://www.netlib.org/lapack/lug/node93.html
Bellman, R.: Introduction to Matrix Analysis. McGraw-Hill, New York (1960)
Kressner, D.: Numerical Methods for General and Structured Eigenvalue Problems. Springer, Berlin (2006)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
Van Loan, C.: On estimating the condition of eigenvalues and eigenvectors. Linear Algebra Appl. 88–89, 715–732 (1987)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Tiutiunnik, A.A., Divakov, D.V., Malykh, M.D., Sevastianov, L.A. (2019). Symbolic-Numeric Implementation of the Four Potential Method for Calculating Normal Modes: An Example of Square Electromagnetic Waveguide with Rectangular Insert. In: England, M., Koepf, W., Sadykov, T., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2019. Lecture Notes in Computer Science(), vol 11661. Springer, Cham. https://doi.org/10.1007/978-3-030-26831-2_27
Download citation
DOI: https://doi.org/10.1007/978-3-030-26831-2_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-26830-5
Online ISBN: 978-3-030-26831-2
eBook Packages: Computer ScienceComputer Science (R0)