Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia8383Research ArticleThe Volume Integral Equations Method in Magnetostatics ProblemsAkishinP GLaboratory of Information Technologiesakishin@jinr.ruSapozhnikovA ALaboratory of Information Technologiesasap@jinr.ruJoint Institute for Nuclear Research15022014231031508092016Copyright © 2014,2014In this article, use of volume integral equations method for calculations of magnetic systems is considered. GFUN program package based on integral approach to magnetostatics applies the method of collocations and piecewise constant approximations of unknown variables within the elements to discretization of equations. The limitation of this approach is related to singularity of the integral equations kernel. An alternative to collocation method, replacing observation point by integration over discretization elements, is considered in this article. This approach enables one to use higher order approximations for unknown variables. In the context of the finite element method, the piecewise constant and piecewise linear approximations of unknown variables are considered. The problem of computing matrix elements for discretized systems of equations can be reduced to evaluation of sixth-order integrals, singular ones in the general case, over two different elements of the computational region. Possible methods are proposed for calculating this kind of integrals. Iterative processes for solving the arising nonlinear systems of discretized equation are discussed. The proposed approach enables one to build discretizations with higher precision of approximation for the initial volume integral equations of magnetostatics. The proposed method was used for 3D modeling of a dipole magnet. Comparison of results obtained for simulation of the dipole magnet using different versions of integral magnetostatics problem discretization are given.magnetostaticsvolume integral equations methodthe finite element methoddiscretizationnonlinear problemiterative processмагнитостатикаметод объёмных интегральных уравненийметод конечных элементовдискретизациянелинейная проблемаитерационный процесс[Armstrong A. G. et al., 1976. - GFUN3D User Guide. - RL-76-029/A.][Акишин П. Г. Метод интегральных уравнений в задачах магнитостатики: Автореф. дис. канд. физ.-мат. наук. - Дубна: ОИЯИ, 1983. - 11-83-558.][Pasciak J. E. An Iterative Algorithm for the Volume Integral Method for Magnetostatics Problems // Comp. Maths. with Appls. - 1982. - Vol. 8, No 8. - Pp. 283-290.][Meijerink J. A., van der Vorst H. A. An Iterative Solution Method for Linear Systems, of Which the Coefficient Matrix is a Symmetric M-matrix // Math. Comput. - 1977. - Vol. 31, No 137. - Pp. 148-162.]