In this example, we will estimate the current distribution in a stranded conductor. An electrical potential \(V_0\) is applied to the entry/exit of the conductor.

1. Running the case

The command line to run this case in 2D is

mpirun -np 4 feelpp_toolbox_electric --case "github:{path:toolboxes/electric/cvg}" --case.config-file 2d.cfg

The command line to run this case in 3D is

mpirun -np 4 feelpp_toolbox_electric --case "github:{path:toolboxes/electric/cvg}" --case.config-file 3d.cfg

2. Geometry

The conductor consists in a rectangular cross section torus which is somehow "cut" to allow for applying electrical potential.+ In 2D, the geometry is as follow: geometry
In 3D, this is the same geometry, but extruded along the z axis.

3. Input parameters

Name Description Value Unit


internal radius




external radius








electrical potential



3.1. Model & Toolbox

  • This problem is fully described by an Electric model, namely a poisson equation for the electrical potential \(V\).

  • toolbox: electric

3.2. Materials

Name Description Marker Value Unit


electric conductivity




3.3. Boundary conditions

The boundary conditions for the electrical probleme are introduced as simple Dirichlet boundary conditions for the electric potential on the entry/exit of the conductor. For the remaining faces, as no current is flowing througth these faces, we add Homogeneous Neumann conditions.

Marker Type Value







Rint, Rext, top*, bottom*



*: only in 3D

4. Outputs

The main fields of concern are the electric potential \(V\), the current density \(\mathbf{j}\) and the electric field \(\mathbf{E}\). // presented in the following figure.

5. Verification Benchmark

The analytical solution is given by:

\[\begin{align*} V&=\frac{V_D}{\delta}\theta=\frac{V_D}{\delta}\operatorname{atan2}(y,x)\\ \mathbf{E}&=\left( -\frac{V_D}{\delta}\frac{y}{x^2+y^2}, \frac{V_D}{\delta}\frac{x}{x^2+y^2}\right)\\ \end{align*}\]

We will check if the approximations converge at the appropriate rate:

  • k+1 for the \(L_2\) norm for \(V\)

  • k for the \(H_1\) norm for \(V\)

  • k for the \(L_2\) norm for \(\mathbf{E}\) and \(\mathbf{j}\)

  • k-1 for the \(H_1\) norm for \(\mathbf{E}\) and \(\mathbf{j}\)

Table 1. Electric potential 2D and 3D

potential 2D

potential 3D

Table 2. Electric field 2D and 3D

electric field 2D

electric field 3D