Formulations in 3D DC Conduction

3D Electric field of the DC conduction currents in the media is described by the Laplace equation for the scalar electric potential U (E = –gradU, where E is the electric stress vector). This equation looks like:

Δ(γU) = 0,    (1)

where Δ — Laplace operator, U – scalar electric potential, γ - electric conductivity which is piece-wise constant for each body, comprising the model. As it is seen from the equation (1), electric field of DC currents has no volume sources. Currents, which generate the field, should be defined as boundary conditions.

3D vector of the current density J relates to the potential by the following expression:

J = −γ gradU        (2)

Following parameters may be defined for 3D DC Conduction problems:

1. Normal component of the surface current density on the faces;
2. Linear current density on the edges;
3. Singular sources and sinks of the electric current in the model vertices.

In all three cases the source current density may be defined by formula as a function of coordinates: j = j(x,y,z).

Boundary conditions for DC conduction problems include:

1. Dirichlet condition specifies a known value of electric potential U₀ on faces, edges or vertices of the model.
2. Neumann condition specifies a known value of normal component of the current density vector j_n= j on external faces and edges.
3. Constant potential boundary condition is used to describe surface of an isolated «floating» conductor that has constant but unknown potential value.

Non-zero Dirichlet and Neumann boundary conditions can be specified as a function of coordinates.

Calculated local physical values are electric potential U, electric field stress vector E, electric current density vector J. Total current I flowing through the defined surface may be calculated as an integral value.