The Boundary Value Problem for Elliptic Equation in the Corner Domain in the Numerical Simulation of Magnetic Systems

Abstract

Russian FederationModern accelerator systems and detectors contain magnetic systems of complex geometrical configuration. Design and optimization of the magnetic systems demands solving a nonlinear boundary-value problem of magnetostatic.The region in which the boundary-value problem is solved, consists of two sub-domains: a domain of vacuum and a domain of ferromagnetic. In view of the complex geometrical configuration of magnetic systems, the ferromagnetic/vacuum boundary can be nonsmooth, i.e. it contains a corner point near of which the boundary is formed by two smooth curves crossed in a corner point at some angle.Thereby, the solution of such a problem has to be found by numerical methods, a question arises about the behavior of the boundary value problem solution around the angular point of the ferromagnetic. This work shows that if the magnetic permeability function meets certain requirements, the corresponding solution of the boundary value problem will have a limited gradient. In this paper, an upper estimate of maximum possible growth of the magnetic field in the corner domain is given.In terms of this estimate, a method of condensing the differential mesh near the corner domain is proposed. This work represents an algorithm of constructing an adaptive mesh in the domain with a boundary corner point of ferromagnetic taking into account the character of behavior of the solution of the boundary value problem. An example of calculating a model problem in the domain containing a corner point is given.Key words and phrases: magnet systems, mathematical modeling, boundary value problem, elliptic equations, the behavior of solutions in the corner domain

Estimation of Magnetic Field Growth and Construction of Adaptive Mesh in Corner Domain for the Magnetostatic Problem in Three-Dimensional Space

“…The following theorem characterizes the magnetic field growth in the 3D space (the estimation for the 2D space was obtained in [1]):…”

Optimization of the Magnetic Field Homogeneity Area for Solenoid Type Magnets

“…In this work we discuss two basic solenoid magnets models, ways of optimizing of these models and our new model within the designated problem. For comparing the models we selected parameters which provide the same magnetic field at the center of each model B center = 0.5 T. Computations were performed (using TOSCA software) by the finite element method on tetrahedral mesh, which had been specially generated taking into account the structural features of the detectors [1].…”

The Boundary Value Problem for Elliptic Equation in the Corner Domain in the Numerical Simulation of Magnetic Systems

“…It is therefore necessary to use special methods for solving this problem. One of these methods is considered in [2,3] for the solution of the equation…”