Energy Functional for a Nonlinear Magnetostatics Problem

The original boundary value problem of magnetostatics is a system of the first-order quasi-linear equations for an unknown vector function. The case is considered when an ideal conductor is located beyond the boundary of the domain under consideration, which corresponds to the vanishing of the normal component of magnetic induction at the boundary. This problem has been transformed into a problem for a single quasi-linear elliptic equation for a scalar potential by constructing a solution to an auxiliary linear problem. Only unambiguous dependencies of magnetic induction on the magnetic field strength are considered, that is, hysteresis is excluded. Within the framework of the energy method, the existence and uniqueness of a generalized solution to the potential problem is proved. This solution makes it possible to construct a finite-energy magnetic field that is a solution to the original problem. It is advisable to use the proposed and substantiated principle of the minimum of the energy functional in the numerical methods for magnetostatics problems, since it allows us to build finite element schemes, using standard approximation functions, for example, piecewise linear ones.