Application of Kufarev method to problem of subsoil waters movement under hydraulic engineering constructions

S. Jambaa   ; T. V. Kasatkina   ; A. M. Bubenchikov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 47 (2017), pp. 15-21

Résumé

To solve plane filtration problems that are described by the classical Darcy motion law, it is proposed to apply the method of conformal mappings implemented in the form of Kufarev’s approach. This approach makes it convenient to find the constants entering into the Schwarz–Christoffel integral as the result of solving a system of ordinary differential equations. The system of differential equations for $b_k=a_k-\lambda$ (here $a_k$ are the prototypes of the polygon vertices, $\lambda$ is the prototype of the cut vertex) is solved with the use of matrix technologies in the MatLab system. In this case, the solution of the problem of constructing groundwater streamlines and lines of constant pressure is reduced to computing the matrix on a discrete set of its arguments and displaying the rows or columns of this matrix. Using the described solution construction technique, the motion of groundwater under a dam with a specific geometrical shape and depth in the ground at the existing difference between flood levels before and after the dam is considered.

Détail

Keywords: conformal mapping of rectilinear polygons, prototypes of polygon vertices, streamlines and lines of constant pressure.

Bibliographie

[1] Kufarev P. P., “A method for determining the parameters of the Schwarz–Christoffel integral”, Dokl. Akad. Nauk SSSR, 57:6 (1947), 535–537 | Zbl

[2] Kolesnikov I. A., “Determination of accessory parameters for mapping onto a numerable polygon”, Tomsk State University Journal of Mathematics and Mechanics, 2014, no. 2(28), 18–28

[3] Kopanev S. A., Kopaneva L. S., “The Formula type formula Christoffel–Schwarz for numerable polygon”, Tomsk State University Journal, 2003, no. 280, 52–54

[4] Chistyakov Yu. V., A numerical method for determining the function that conformally mapping a circle onto polygons, Physics and Mathematics Cand. Diss., Tomsk, 1953

[5] Baybarin B. G., On One Numerical method of determining the parameters of the Schwarz derivative for a function that conformally mapping the half-plane onto a circular area, Physics and Mathematics Cand. Diss., Tomsk, 1966, 97 pp.

[6] Nakipov N. N., Nasyrov S. R., “A parametric method of finding accessory parameters for the generalized Schwarz–Christoffel integrals”, Uchenye zapiski Kazanskogo universiteta. Seriya Fiziko-matematicheskie nauki, 158:2 (2016), 202–220

[7] Nasyrov S. R., Nizamieva L. Yu., “Finding of Accessory Parameters for Mixed Inverse Boundary Value Problem with Polygonal Known Part of Boundary”, Izvestiya Saratovskogo universiteta. Novaya seriya-Izvestiya of Saratov University. New Series, 11:4 (2011), 34–40 | Zbl

[8] Gutlyanskii V. Y., Zaidan A. O., “On conformal mapping of polygonal regions”, Ukrainian Mathematical Journal, 45:11 (1993), 1669–1680 | DOI | MR | Zbl

[9] Sobolev V. V., “The numeric method of the conformal mapping of the half-plane into self with the hydrodynamics normalization”, Tomsk State University Journal, 2003, no. 280, 81–85

[10] Jambaa S., Kasatkina T. V., Bubenchikov A. M., “On the determination of constants in the Schwarz–Christoffel integral by P. P. Kufarev's method”, Tomsk State University Journal of Mathematics and Mechanics, 2016, no. 5(43), 21–27 | DOI

[11] Fil'chakov P. F., Approximate methods of conformal mappings, Naukova Dumka, Kiev, 1964, 530 pp.

[12] Polubarinova-Kochina P. Ya., Theory of ground water movement, Nauka, M., 1977, 663 pp.

[13] Pavlovsky N. N., Coll. w., v. II, Theory of ground water movement under hydraulic structures and its main applications, Izd-vo Akad. Nauk SSSR, M.–L., 1956, 352 pp.