**MSC2010** 97-04; 90C99; 82D55

### Numerical modeling of the process of penetration of an external magnetic field into a thick disk-shaped of a high-temperature superconductors on the basis of the random walk algorithm

#### N. D. Kuzmichev^{1}, I. V. Buryanov^{2}, M. A. Vasyutin^{3}, A. Yu. Shitov^{4}

Annotation | In this paper a mathematical modeling of the process of penetration of an external magnetic field into a thick disk-shaped sample of high-temperature superconductor (HTSC) in a critical state is performed. The problem is reduced to finding the minimum of the objective function that includes integral equations of the first kind. At the same time, volume of the disk is occupied by the shielding superconducting current (overcurrent). Disk is separated by a curved conical surface with shape that is determined by a random walk in the disk ${(r, z)}$ plane. The result of the development of the program in language C\# is given; this program calculates the optimal configuration of the overcurrent volume in HTSC using the random walk algorithm. The results of the computational experiment based on Bean's model for cases when an external magnetic field in the $z=0$ plane penetrates in a superconducting disk to the depth of 20, 50 and 80 \% of the disk radius, are given. The results of the program's work for the grid of 50x50 in the plane ${(r, z)}$ of the quarter of the disk section are presented. Processing and visualization of obtained numerical data were carried out using OriginLab and MS Excel. |
---|---|

Keywords | type-II superconductor, high-temperature superconductor, critical state, shielding overcurrent, Bean's model,integral equations of the first kind, objective function, random walk algorithm |

^{1}**Nikolay D. Kuzmichev**, Professor of Department of Computer Science and CAD-technology, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russia), ORCID: http://orcid.org/0000-0001-6707-4950, kuzmichevnd@yandex.ru^{2}**Igor V. Buryanov**, Postgraduate student of the Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russia), ORCID: http://orcid.org/0000-0003-1033-0487, i.v.buryanov@gmail.com^{3}**Mikhael A. Vasyutin**, Docent of Department of Computer Science and CAD-technology, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russia), ORCID: http://orcid.org/0000-0002-4856-7407, vasyutinm@mail.ru^{4}**Almir Yu. Shitov**, Postgraduate student of the Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68 Bolshevistskaya St., Saransk 430005, Russia), ORCID: http://orcid.org/0000-0002-2029-8479, shishkin92@mail.ru

**Citation**: N. D. Kuzmichev, I. V. Buryanov, M. A. Vasyutin, A. Yu. Shitov, "[Numerical modeling of the process of penetration of an external magnetic field into a thick disk-shaped of a high-temperature superconductors on the basis of the random walk algorithm]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,20:1 (2018) 88–95 (In Russian)

**DOI** 10.15507/2079-6900.20.201801.88-95