Test Case of Maxwell Quasi Static with Saddle-Point Formulation on One Torus
1. Introduction
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.
2. Run the Calculation
The command line to run this case is :
feelpp_toolbox_coefficientformpdes --config-file=mqs_saddle.cfg --cfpdes.gmsh.hsize=5e-3
This case is run with the latest version 109 of Feelpp.
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
The domain \(\Omega\) is composed of the conductor \(\Omega_c\) and non conducting materials \(\Omega/\Omega_c\) where \(\mathbf{J} = 0\), like the air for example. \(\Gamma = \partial \Omega\) is the bound of \(\Omega\), decomposed in \(\Gamma_D\) with Dirichlet boundary condition and \(\Gamma_N\) with Neumann boundary condition, such that \(\Gamma = \Gamma_D \cup \Gamma_N\).
We introduce :
-
Magnetic potential field \(\mathbf{A}\) : the magnetic field is \(\textbf{B} = \nabla \times \textbf{A}\)
-
Electric potential scalar : \(\nabla V = - \textbf{E}\)
We have the conditions :
-
\(\nabla \cdot \mathbf{A} = 0\)
We want to resolve the electromagnetic problem ( with \(\mathbf{A}\), \(p\) and \(V\) as the unknowns) :
With :
-
\(\sigma\) : electric conductivity \(S/m\)
-
\(\mu\) : electric permeability \(kg/A^2/S^2\)
(See A-V Saddle-Point Formulation).
5. Geometry
The geometry is a quarter of a conducting torus, surrounded by air.
Geometry
|
The geometrical domains are :
-
Conductor
: the torus, composed by a conductor-
V0
: entrance of electrical potential -
V1
: exit of electrical potential
-
-
Air
: the air surroundingConductor
-
OXOZ
: \(OxOz\) plan -
OYOZ
: \(OyOz\) plan -
Infty
: the rest ofAir
's bound
-
Symbol |
Description |
value |
unit |
\(r_{int}\) |
interior radius of torus |
\(75e-3\) |
m |
\(r_{ext}\) |
exterior radius of torus |
\(100.2e-3\) |
m |
\(z_1\) |
half-height of torus |
\(25e-3\) |
m |
\(r_{infty}\) |
radius of infty border |
\(5*r_{ext}\) |
m |
6. Boundary Conditions
The Dirichlet boundary conditions imposed are :
-
For \(V\) equation :
-
On
V0
: \(V=0\) -
On
V1
: \(V = \begin{cases}\frac{t}{4}&\quad\text{if }t<1\\ \frac{1}{4}&\quad\text{if }1<t<20\\ 0&\quad\text{if }t>20\end{cases}\)
-
-
For \(\mathbf{A}\) :
-
On
OXOZ
andV0
: \(A_x = A_z = 0\), we want \(\mathbf{A}\) orthogonal toOXOZ
andV0
-
On
OYOZ
andV1
: \(A_y = A_z = 0\), we want \(\mathbf{A}\) orthogonal toOYOZ
andV1
-
Infty
: We approximate the problem,Infty
is the physical infty so \(\mathbf{B}=0\) atInfty
thus \(\mathbf{A} = 0\)
-
-
For \(p\) :
-
On
OXOZ
andV0
: \(p = 0\) -
On
OYOZ
andV1
: \(p = 0\) -
Infty
: \(p = 0\)
-
On JSON file, the boundary conditions are written :
"BoundaryConditions": { "magnetic": { "Dirichlet": { "boundary": { "markers":["Infty"], "expr":"{0,0,0}" }, "mydir_x": { "markers":["V0","OXOZ"], "expr":"{0,0,0}" }, "mydir_y": { "markers":["V1","OYOZ"], "expr":"{0,0,0}" }, "mydir_z": { "markers":["V0","OXOZ","V1","OYOZ"], "expr":"{0,0,0}" } } }, "constraint": { "Dirichlet": { "boundary": { "markers":["Infty"], "expr":"0" }, "mydir_x": { "markers":["V0","OXOZ"], "expr":"0" }, "mydir_y": { "markers":["V1","OYOZ"], "expr":"0" }, "mydir_z": { "markers":["V0","OXOZ","V1","OYOZ"], "expr":"0" } } }, "electric": { "Dirichlet": { "V0": { "markers":["V0"], "expr":"V0:V0" }, "V1": { "markers":["V1"], "expr":"V1:V1" } } } },
7. Weak Formulation
8. Parameters
The parameters of the problem are :
-
On
Conductor
:
Symbol |
Description |
Value |
Unit |
\(V0\) |
scalar electrical potential on |
\(0\) |
\(Volt\) |
\(V1\) |
scalar electrical potential on |
\(\begin{cases}\frac{t}{4}&\quad\text{if }t<1\\ \frac{1}{4}&\quad\text{if }1<t<20\\ 0&\quad\text{if }t>20\end{cases}\) |
\(Volt\) |
\(\sigma\) |
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": { "V0":0, "V1":"1/4*(t/(0.1*10)*(t<(0.1*10))+((t<(0.5*40))+0*(t>(0.5*40)))*(t>(0.1*10))):t", "sigma":58e6, "mu":"4*pi*1e-7" },
9. 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 |
|
damping or mass coefficient |
\(\sigma\) |
|
curlcurl coefficient |
\(\frac{1}{\mu}\) |
|
source term |
\(-\nabla p - \sigma \nabla V \) |
|
damping or mass coefficient |
\(\sigma\) |
|
source term |
\(\sigma \, \frac{d \mathbf{A}}{d t}\) |
|
conservative flux source term |
\(\mathbf{A}\) |
-
On
Air
:
Coefficient |
Description |
Expression |
|
curlcurl coefficient |
\(\frac{1}{\mu}\) |
|
source term |
\(-\nabla p\) |
|
conservative flux source term |
\(\mathbf{A}\) |
On JSON file, the coefficients are written :
"Materials": { "Conductor": { "magnetic_d":"sigma:sigma", "magnetic_zeta":"1/mu:mu", "magnetic_f":"{-constraint_grad_p_0-sigma*electric_grad_V_0,-constraint_grad_p_1-sigma*electric_grad_V_1,-constraint_grad_p_2-sigma*electric_grad_V_2}:sigma:electric_grad_V_0:electric_grad_V_1:electric_grad_V_2:constraint_grad_p_0:constraint_grad_p_1:constraint_grad_p_2", "electric_c":"sigma:sigma", "electric_gamma":"{sigma*magnetic_dA_dt_0,sigma*magnetic_dA_dt_1,sigma*magnetic_dA_dt_2}:sigma:magnetic_dA_dt_0:magnetic_dA_dt_1:magnetic_dA_dt_2", "constraint_gamma":"{magnetic_A_0,magnetic_A_1,magnetic_A_2}:magnetic_A_0:magnetic_A_1:magnetic_A_2" }, "Air": { "physics":["magnetic","constraint"], "magnetic_zeta":"1/mu:mu", "magnetic_f":"{-constraint_grad_p_0,-constraint_grad_p_1,-constraint_grad_p_2}:constraint_grad_p_0:constraint_grad_p_1:constraint_grad_p_2", "constraint_gamma":"{magnetic_A_0,magnetic_A_1,magnetic_A_2}:magnetic_A_0:magnetic_A_1:magnetic_A_2" } },
10. Numeric Parameters
This section shows the parameters used to compute the simulation.
-
Size of mesh :
-
On
Conductor
: \(0.5 \, mm\) -
On
Infty
: \(100 \, mm\)
-
-
Time Parameters :
-
Time step : \(0.1 \, s\)
-
Initial Time : \(0 \, s\)
-
Final Time : \(22 \, s\)
-
Mesh of Geometry
|
11. Results
12. References
-
Cecile Daversin - Catty. Reduced basis method applied to large non-linear multi-physics problems: application to high field magnets design. Electromagnetism. Université de Strasbourg, 2016. p56-64 PDF