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 :
-
CFG file - Edit the file
-
JSON file - Edit the file
-
GEO file - Edit the file
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.
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.
![]() Geometry in Axisymmetrical cut
|
![]() Geometry in three dimensions
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The geometrical domains are :
-
Conductor
: the torus, it is composed of conductive materials -
Air
: the air surroundConductor
-
zAxis
: a bound ofAir
, correspond to zOzO axis ({(z,r),z=0}{(z,r),z=0}) -
infty
: the rest of bound ofAir
-
Symbol |
Description |
value |
unit |
rintrint |
interior radius of torus |
75e−375e−3 |
m |
rextrext |
exterior radius of torus |
100.2e−3100.2e−3 |
m |
z1z1 |
half-height of torus |
25e−325e−3 |
m |
rinftyrinfty |
radius of infty border |
5∗rext5∗rext |
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 :
"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 :
With ˜∇=(∂∂r∂∂z)~∇=(∂∂r∂∂z)
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.10−7 |
kgm/A2/S2 |
-
On
Air
:
Symbol |
Description |
Value |
Unit |
μ=μ0 |
magnetic permeability of vacuum |
4π.10−7 |
kgm/A2/S2 |
On JSON file, the parameters are writed :
"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 :
"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
-
![]() 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 :
So, with rotational in cylindric and axisymmetric condition :
We export from Φ :
-
Aθ=Φr
-
Br=−1r∂Φ∂z
-
Bz=1r∂Φ∂r
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.
13.3. 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(T) on Or axis
|
![]() Bz(T) on Or axis
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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(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 1r∂Φ∂r 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(T) on P0
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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