Test case : One Torus

1. Introduction

In this page, I present the test case of Maxwell Quasi Static Problem in A-V Formulation with gauge condition for a geometry of torus surrounded by air for instationnary case in Axisymmetrical case.

2. Run the calculation

The command line to run this case is :

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

3. Data Files

The case data files are available in Github here :

4. Equation

We solve the A-V Formulation in axisymmetric and we assume that VV is known.

The unknow of equation is Φ=rAθΦ=rAθ but we want Magnetic Potential field AθAθ.

Differential Formulation
(AV Axis){σΦt1μΔΦ+2μrΦr+σVθ=0 on Ωaxis(AV-1 Axis)V=U2πθ on Ωaxisc(AV-2 Axis)Aθ=0 on ΓaxisD(D Axis)Aθnaxis=0 on ΓaxisN(N Axis)(AV Axis)⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪σΦt1μΔΦ+2μrΦr+σVθ=0 on Ωaxis(AV-1 Axis)V=U2πθ on Ωaxisc(AV-2 Axis)Aθ=0 on ΓaxisD(D Axis)Aθnaxis=0 on ΓaxisN(N Axis)

With :

  • Φ=rAθΦ=rAθ : AθAθ component θθ of potential magnetic field

  • σσ : electric conductivity S/mS/m

  • μμ : electric permeability kg/A2/S2kg/A2/S2

  • UU : tension VoltVolt

5. Geometry

The geometry is a torus of conductor in cartesian coordinates (x,y,z)(x,y,z) or rectangle in axisymmetric coordinates (r,z)(r,z), surrounded by air.

1torus axis
Geometry in Axisymmetrical cut
1torus 3d
Geometry in three dimensions

The geometrical domains are :

  • Conductor : the torus, it is composed of conductive materials

  • Air : the air surround Conductor

    • zAxis : a bound of Air, correspond to zOzO axis ({(z,r),z=0}{(z,r),z=0})

    • infty : the rest of bound of Air

Symbol

Description

value

unit

rintrint

interior radius of torus

75e375e3

m

rextrext

exterior radius of torus

100.2e3100.2e3

m

z1z1

half-height of torus

25e325e3

m

rinftyrinfty

radius of infty border

5rext5rext

m

6. Initial/Boundary Conditions

We impose the Dirichlet boundary conditions :

  • On zAxis : Φ=0Φ=0 (Aθ=0Aθ=0 by symetric argument)

  • On infty : Φ=0Φ=0 (Aθ=0Aθ=0 we consider the bound of resolution like infty for magnetic field)

We initialize :

  • On Conductor : Φ(t=0,r,z)=0Φ(t=0,r,z)=0 (Aθ(t=0,r,z)=0Aθ(t=0,r,z)=0)

  • On Air : Φ(t=0,r,z)=0Φ(t=0,r,z)=0 (Aθ(t=0,r,z)=0Aθ(t=0,r,z)=0)

On JSON file, the boundary conditions are writed :

Boundary conditions on JSON file
    "BoundaryConditions":
    {
        "magnetic":
        {
            "Dirichlet":
            {
                "ZAxis":
                {
                    "expr":"0"
                },
                "Infty":
                {
                    "expr":"0"
                }
            }
        }
    }

For initial condition, we write nothing in JSON, by default the application put zeros on initialization.

7. Weak Formulation

We obtain :

Weak Formulation
ΩaxisσΦtϕ r drdz+Ωaxis1μ˜Φ˜ϕ r drdz+Ωaxis2μrΦrϕ r drdz=ΩaxiscσVθϕ r drdzΩaxisσΦtϕ r drdz+Ωaxis1μ~Φ~ϕ r drdz+Ωaxis2μrΦrϕ r drdz=ΩaxiscσVθϕ r drdz

With ˜=(rz)~=(rz)

8. Parameters

The parameters of problem are :

  • On Conductor :

Symbol

Description

Value

Unit

V

scalar electrical potential

Uθ2π

Volt

U

electrical potential

1

Volt/rad

σ

electrical conductivity

58e6

S/m

μ=μ0

magnetic permeability of vacuum

4π.107

kgm/A2/S2

  • On Air :

Symbol

Description

Value

Unit

μ=μ0

magnetic permeability of vacuum

4π.107

kgm/A2/S2

On JSON file, the parameters are writed :

Parameters on JSON file
    "Parameters":
    {
	    "U":"t/(0.1*10)*(t<0.1*10)+(t<0.5*40)*(t>(0.1*10)):t" // Volt
    }

9. Coefficient Form PDEs

We use the application Coefficient Form PDEs. The coefficient associate to Weak Formulation are :

  • On Conductor :

Coefficient

Description

Expression

d

damping or mass coefficient

σ

c

diffusion coefficient

rμ

β

convection coefficient

(2μ0)

f

source term

σU2πr

  • On Air :

Coefficient

Description

Expression

c

diffusion coefficient

rμ

β

convection coefficient

(2μ0)

On JSON file, the coefficients are writed :

CFPDEs coefficients on JSON file
    "Materials":
    {
        "Conductor":
        {
            "sigma":58e+6,      // S.m-1
            "mu":"4*pi*1e-7",   // kg.m/A2/S2
	        "magnetic_c":"x/mu:x:mu",
            "magnetic_beta":"{2/mu,0}:mu",
            "magnetic_f":"-sigma*U/2/pi*x:sigma:U:x:mu",
            "magnetic_d":"sigma*x:x:sigma"
        },
        "Air":
        {
            "mu":"4*pi*1e-7",   // kg.m/A2/s2
            "magnetic_c":"x/mu:x:mu",
            "magnetic_beta":"{2/mu,0}:mu"
        }
    }

10. Numeric Parameters

  • Time

    • Initial Time : 0s

    • Final Time : 240s

    • Time Step : 10s

  • Mesh size :

    • Interior of torus : 0.001m

    • Far of torus : 0.004m

1torus axis mesh
Mesh of Geometry

11. Export

We solve an equation (A-V Formulation in axisymmetric) of unknow Φ=rAθ but we want the magnetic field B.

The potential magntic vector A is define :

B=×A

So, with rotational in cylindric and axisymmetric condition :

B=(BrBθBz)=(Aθz01r(rAθ)r)

We export from Φ :

  • Aθ=Φr

  • Br=1rΦz

  • Bz=1rΦr

12. Run the calculation

12.1. CFPDEs

13. Result

The analytical solve of potential magnetic field and magnetic field (the cyan curve on plots) is computed by python3’s module MagnetTools.MagnetTools. The module based on paper : spire.

The analytical solve of intensity (and magnetic field) is computed on python3’s script Edit the file.

13.1. Intensity

I plot the intensity of current with analytical solve.

I t
Intensity I(A)

13.2. Magnetic Potential Field

Orthoraidal Magnetic Potential Field Aθ(T.m)

13.2.1. On Or axis at t=20s

I export the magnetic potential field at t=20s because we consider at this time, the system is stationary. Thus, we can compare with the analytical solve of stationnary problem.

a th Or t20s
Aθ(A/m2) on Or axis

13.3. Magnetic Field

Horizontal Magnetic Field Bz(T)

13.3.1. On Or axis at t=20s

I export the magnetic field at t=20s because we consider at this time, the system is stationary. Thus, we can compare with the analytical solve of stationnary problem.

Br Or t20s
Br(T) on Or axis
Bz Or t20s
Bz(T) on Or axis

Around the Or axis, the direction of Magnetic field B is horizontal, in particulary at P0=(r=0,z=0).

We can observe a great difference between the resolution with P1 and P2 elements at r=0 :

Bz Or t20s(1)
Bz(T) on Or axis

The result in P1 elements have instability near of r=0 but not with P2. This instability can be explain by the method of export of Magnetic Field : to have Bz, we export 1rΦr and the division 1r near of r=0 can be a problem.

13.3.2. On P0=(r=0,z=0) in terms of time

The value of magnetic field on P0=(r=0,z=0) is important because the research put theire experiments near of this point.

Bz P0 t
Bz(T) on P0

We can see a difference between the result Bz with P1 and P2 element. The curve of z-magnetic field Bz in P1 element is far of analytical solve and in P2, it seems good. The phenomena is explain by the remark above.

14. References

  • Calcul du champ magnétique pour les géométries axisymétriques simples, Christophe Trophime, 2002, unpublished, Download the PDF