Test Case : One Torus

This is the test case of Maxwell Quasi Static Problem with the A-V Formulation and Gauge Condition on a torus geometry surrounded by air for the unstationary case in axisymmetric coordinates.

1. Run the Calculation

The command line to run this case is :

    mpirun -np 16 feelpp_toolbox_coefficientformpdes --config-file=mqs_axis.cfg --cfpdes.gmsh.hsize=1e-3

This case is run with the latest version 109 of Feelpp.

2. Data Files

The case data files are available in Github here :

CFG file - Edit the file
JSON file - Edit the file
GEO file - Edit the file

3. Equation

Assuming that \(V\) is known, the A-V Formulation in axisymmetric coordinates is :

A-V Formulation in axisymmetric coordinates

\[\text{(AV Axis)} \left\{ \begin{matrix} -\frac{1}{\mu}\Delta A_\theta + \frac{1}{\mu r^2}A_\theta + \sigma \frac{\partial A_\theta}{\partial t} + \frac{\sigma}{r} \frac{\partial V}{ \partial\theta}=0 & \text{on } \Omega^{axis} & \text{(AV-1 Axis)} \\ V = \frac{U}{2\pi} \, \theta & \text{ on } \Omega_c^{axis} & \text{(AV-2 Axis)}\\ A_{\theta} = 0 & \text{ on } \Gamma_D^{axis} & \text{(D Axis)} \\ \frac{\partial A_{\theta}}{\partial \mathbf{n}^{axis}} = 0 & \text{ on } \Gamma_N^{axis} & \text{(N Axis)} \end{matrix} \right.\]

With :

\(A_{\theta}\) : \(\theta\) component of potential magnetic field
\(\sigma\) : electric conductivity \(S/m\)
\(\mu\) : electric permeability \(kg/A^2/S^2\)
\(U\) : tension \(Volt\)

4. Geometry

The geometry is a rectangle in axisymmetric coordinates \((r,z)\) representing a conducting torus, surrounded by air.

Geometry in Axisymmetrical cut

The geometrical domains are :

Conductor : the torus, composed by a conductor
Air : the air surrounding Conductor
- zAxis : Air 's bound, correspond to \(Oz\) axis (\(\{(z,r), \, z=0 \}\))
- infty : the rest of the Air 's bound

Symbol

Description

value

unit

\(r_{int}\)

interior radius of torus

\(75e-3\)

\(r_{ext}\)

exterior radius of torus

\(100.2e-3\)

\(z_1\)

half-height of torus

\(25e-3\)

\(r_{infty}\)

radius of infty border

\(5*r_{ext}\)

5. Boundary Conditions

The Dirichlet boundary conditions imposed are :

On zAxis : \(A_{\theta} = 0\)
On infty : \(A_{\theta} = 0\)

On JSON file, the boundary conditions are written :

Boundary conditions on JSON file

    "BoundaryConditions":
    {
        "magnetic":
        {
            "Dirichlet":
            {
                "ZAxis":
                {
                    "expr":"0"
                },
                "Infty":
                {
                    "expr":"0"
                }
            }
        }
    }

6. Weak Formulation

Weak Formulation

\[\scriptsize{ \int_{\Omega^{axis}}{ \frac{r}{\mu} \tilde\nabla A_\theta \cdot \tilde\nabla \phi \ drdz } + \int_{\Omega^{axis}}{\frac{1}{\mu r} A_\theta \cdot \phi \ drdz} +\int_{\Omega^{axis}}{ \sigma r \frac{\partial A_\theta}{\partial t} \cdot \phi \ drdz} = \int_{\Omega^{axis}_c}{ \sigma \frac{\partial V}{\partial \theta} \cdot\phi \ rdrdz } }\]

With \(\tilde{\nabla} = \begin{pmatrix} \partial r \\ \partial z \end{pmatrix}\)

7. Parameters

The parameters of the problem are :

On Conductor :

Symbol

Description

Value

Unit

\(V\)

scalar electrical potential

\( U \, \frac{\theta}{2\pi}\)

\(Volt\)

\(U\)

electrical potential

\(\begin{cases}t\quad \text{if } t<1\\ 1 \quad \text{if } 1<t<20\\ 0 \quad \text{if } t>20 \end{cases}\)

\(Volt / rad\)

\(\sigma\)

electrical conductivity

\(58e6\)

\(S/m\)

\(\mu=\mu_0\)

magnetic permeability of vacuum

\(4\pi.10^{-7}\)

\(kg \, m / A^2 / S^2\)

On Air :

Symbol

Description

Value

Unit

\(\mu=\mu_0\)

magnetic permeability of vacuum

\(4\pi.10^{-7}\)

\(kg \, m / A^2 / S^2\)

On JSON file, the parameters are written :

Parameters on JSON file

    "Parameters":
    {
	    "sigma":58e6,                           //S/m
	    "mu":"4*pi*1e-7",                       //kg m / A^2 / S^2
	    "U":"t/1.*(t<1.)+(t<20.)*(t>(1.)):t"    //Volt
    }

8. Coefficient Form PDEs

The Feelpp toolboxe Coefficient Form PDEs is used here. The coefficients associated to the Weak Formulation are :

On Conductor :

Coefficient

Description

Expression

\(d\)

damping or mass coefficient

\(\sigma r\)

\(c\)

diffusion coefficient

\(\frac{r}{\mu}\)

\(a\)

absorption or reaction coefficient

\(\frac{1}{\mu r}\)

\(f\)

source term

\(- \sigma \frac{U}{2\pi}\)

On Air :

Coefficient

Description

Expression

\(c\)

diffusion coefficient

\(\frac{r}{\mu}\)

\(a\)

absorption or reaction coefficient

\(\frac{1}{\mu r}\)

On JSON file, the coefficients are written :

CFPDEs coefficients on JSON file

    "Materials":
    {
        "Conductor":
        {
	        "magnetic_c":"x/mu:x:mu",
            "magnetic_a":"1/mu/x:mu:x",
            "magnetic_f":"-sigma*U/2/pi:sigma:U",
            "magnetic_d":"sigma*x:sigma:x"
        },
        "Air":
        {
            "magnetic_c":"x/mu:x:mu",
            "magnetic_a":"1/mu/x:mu:x"
        }
    }

9. Numeric Parameters

Time
- Initial Time : \(0s\)
- Final Time : \(240s\)
- Time Step : \(10s\)
Mesh size :
- Interior of torus : \(0.001 m\)
- Far of torus : \(0.004 m\)

Mesh of Geometry

10. Results

The results are calculated using the method AIR_OUT.

10.1. Magnetic Potential Field

The magnetic potential field \(\mathbf{A}\) defined by :

\[ \mathbf{A} = \begin{pmatrix} 0 \\ A_{\theta} \\ 0 \end{pmatrix}_{cyl}\]

\(\theta\) component of Magnetic potential field \(A_\theta (A/m^2)\)

The behavior of \(A_\theta\) on the \(O_r\) axis at \(t=15s\) is as follows :

Table 1. L2 Error Norm for \(A_\theta\) on \(O_r\) at \(t=15s\)
h	L2 Error Norm	L2 Relative Error Norm
\(9e-3\)	\(1.85301e-3\)	\(1.38\%\)
\(5e-3\)	\(1.48345e-3\)	\(1.10\%\)
\(1e-3\)	\(4.00114e-5\)	\(0.03\%\)
\(5e-4\)	\(2.46014e-5\)	\(0.02\%\)

10.2. Magnetic Field

The magnetic field \(\mathbf{B}\) is defined by :

\[ \mathbf{B} = \nabla \times \mathbf{A} = \begin{pmatrix} -\partial_z A_\theta \\ 0 \\ \frac{1}{r}\partial_r (rA_\theta) \end{pmatrix}_{cyl}\]

\(r\) component of Magnetic field \(B_r (T)\)

\(z\) component of Magnetic field \(B_z (T)\)

The behavior of \(\mathbf{B}_z\) on the \(O_z\) axis at \(t=15s\) :

Table 2. L2 Error Norm for \(B_z\) on \(O_z\) at \(t=15s\)
h	L2 Error Norm	L2 Relative Error Norm
\(9e-3\)	\(4.235206e-2\)	\(1.84\%\)
\(5e-3\)	\(2.294347e-2\)	\(1.00\%\)
\(1e-3\)	\(3.908771e-3\)	\(0.17\%\)
\(5e-4\)	\(1.292564e-3\)	\(0.06\%\)