In this example, we will model a busbar simply modeled as rectangular parallelepiped.A differential electrical potential is applied to the entry/exit of the busbar. We note respectively \(V_0\) the electrical potential on the entry and and \(V_1\) on the exit.

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/busbar}" --case.config-file 2d.cfg

The command line to run this case in 3D is

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

2. Geometry

The busbar conductor consists in a rectangular cross section extruded along the x axis.+ In 2D, the geometry is as follow: 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 the Electric model, namely a poisson equation for the electrical potential \(V\) with Dirichlet boundary conditions on entry /exit.

  • 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







Lside, Rside, 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:

\[V(x) = V_0 + (\frac{x}{L}-1) (V_1-V_0)\]

Note that we may get an expression for the resistance \(R\) of the busbar from this equation. We recall that \(R\) is defined as \(R = V_D/I\) where \(I\) stands for the total current flowing in the busbar (\(V_D\) corresponds to the differential applied voltage).

By definition:

\[I = \int_{V0} \mathbf{j} \cdot \mathbf{n} \,d\Gamma\]

From Gauss law we have: \(\mathbf{j} = \sigma\,\mathbf{E} = -\sigma \nabla V\). It follows:

\[R = \frac{1}{\sigma} \frac{Lx}{S}\]

with \(S=Ly*Lz\).

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 \(\mathbf{j}\)