Boundary conditions in DC conduction

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The following boundary conditions can be specified at outward and inner boundaries of the region.

Dirichlet condition specifies a known value of electric potential U₀ at the vertex or at the edge of the model. U₀ value at the edge can be specified as a linear function of coordinates. The function parameters can vary from one edge to another, but have to be adjusted to avoid discontinuities at edges' junction points.

Note. For problem to be defined correctly the Dirichlet condition has to be specified at least at one point. If the region consists of two or more disjoint subregions, the Dirichlet conditions have to be specified at least at one point of every part.

Neumann condition is defined by the following equations:

where j_n is a normal component of the current density vector, "+" and "–" superscripts denote quantities to the left and to the right side of the boundary, and j at right hand side is a density of the external current. If j value is zero, the boundary condition is called homogeneous. This kind of boundary condition is used at an outward boundary of the region that is formed by the symmetry plane of the problem. The homogeneous Neumann condition is the natural one, it is defaulted at all outward boundary parts where no explicit boundary condition is specified.

If the surface-bound current density is to be specified at the plane of problem symmetry and this plane is the outward boundary of the region, the surface current density has to be halved.

Constant potential boundary condition is used to describe surface of a conductor having much greater conductivity than the surrounding medium. This conductor is assumed to have constant but unknown potential value.

Note. The edge described as possessing constant potential should not have common points with any Dirichlet edge. In that case the constant potential edge has to be described by the Dirichlet condition with an appropriate potential value.