QuarterTurn

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:
In 3D, this is the same geometry, but extruded along the z axis.

3. Input parameters

Name	Description	Value	Unit
\(r_i\)	internal radius	1	\(m\)
\(r_e\)	external radius	2	\(m\)
\(\delta\)	angle	\(\pi/2\)	\(rad\)
\(V_D\)	electrical potential	9	\(V\)

Name

Description

Value

Unit

\(r_i\)

internal radius

\(m\)

\(r_e\)

external radius

\(m\)

\(\delta\)

angle

\(\pi/2\)

\(rad\)

\(V_D\)

electrical potential

\(V\)

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
\(\sigma\)	electric conductivity	omega	\(4.8e7\)	\(S.m^{-1}\)

Name

Description

Marker

Value

Unit

\(\sigma\)

electric conductivity

omega

\(4.8e7\)

\(S.m^{-1}\)

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
V0	Dirichlet	0
V1	Dirichlet	\(V_D\)
Rint, Rext, top, bottom	Neumann	0

Marker

Type

Value

Dirichlet

\(V_D\)

Rint, Rext, top*, bottom*

Neumann

*: 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

Table 2. Electric field 2D and 3D