Test of Maxwell Quasi Static with gauge condition on Two Torus

1. Introduction

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

2. Run the calculation

The command line to run this case is :

    mpirun -np 32 feelpp_toolbox_coefficientformpdes --config-file=mqs.cfg --cfpdes.gmsh.hsize=5e-3

3. Data Files

The case data files are available in Github here :

4. Equation

Thus Ω the domain, comprising the conductor (or superconductor) domain Ωc and non conducting materials Ωn (J=0) like air. Let Γ=Ω the bound of Ω, Γc=Ωc the bound of Ωc, ΓD, ΓDc the bound with Dirichlet boundary condition and ΓN, ΓNc the bound with Neumann boundary condition, such that Γ=ΓDΓN and Γc=ΓDcΓNc.

We introduce :

  • Magnetic potential field A : the magnetic field writes B=×A

  • Electric potential scalar : V=EdAdt

We have the conditions :

  • A=0

  • μ is independant of space

We want to resolve the electromagnetism problem ( with A and V the unknows) :

A-V Formulation with gauge condition
(AV+gauge){σdAdt+1μΔA=σV on Ω(AV+gauge-1)(σV)=(σdAdt) on Ωc(AV'-2)A×n=0 on ΓD(DA)(×A)×n=0 on ΓN(NA)V=Vd on ΓDc(DV)Vn=0 on ΓNc(NV)

This equation is obtain from the A-V Formulation with the gauge conditions. The detail calculus are in page MQS + gauge condition.

5. Geometry

The geometry is two tores of conductor surrounded by air.

2torus 3d
Geometry
2torus 3d(1)
Geometry loop on Conductors

The geometrical domains are :

  • Conductor_0 : the first torus, it is composed of conductive materials

    • V0_0 : enter of electrical potential

    • V1_0 : exit of electrical potential

  • Conductor_1 : the second torus, it is composed of conductive materials

    • V0_1 : enter of electrical potential

    • V1_1 : exit of electrical potential

( Conductor_0 Conductor_1 =Ωc and V0_0 V1_0 V0_1 V1_1 =ΓDc)

  • air (Ω/Ωc) : the air surround Conductor

    • OXOZ : OxOz plan

    • OYOZ : OyOz plan

    • Infty : the rest of bound of Air

Symbol

Description

value

unit

rint

interior radius of tores

75e3

m

rext

exterior radius of tores

100.2e3

m

z1

half-height of torus

25e3

m

rinfty

radius of infty border

5rext

6. Initial/Boundary Conditions

We impose the Dirichlet boundary conditions :

  • For V equation :

    • On V0_0 and V0_1 : V=0

    • On V1_0 : V=14{t0.1 on 0st<0.1s1 on 0.1st<0.5st on 0.5st

    • On V1_1 : V=14{t0.1 on 0st<0.1s1 on 0.1st<0.7st on 0.7st

  • For A :

    • On OXOZ, V0_0 and V0_1 : Ax=Az=0, we want A orthogonal to OXOZ and V0

    • On OYOZ, V1_0 and V1_1 : Ay=Az=0, we want A orthogonal to OYOZ and V1

    • Infty : We approximate the problem, Infty is the physical infty thus B=0 at Infty thus A=0.

We initialize at t=0s :

  • V=0

  • A=0

On JSON file, the boundary conditions are writed :

Boundary conditions on JSON file
    "BoundaryConditions":
    {
        "magnetic":
        {
            "Dirichlet":
            {
                "Infty":
                {
                    "expr":"{0,0,0}"
                }
            },
            "Dirichlet_x":
            {
                "mydir_x":
                {
                    "markers":["V0_0","V0_1","OXOZ"],
                    "expr":0
                }
            },
            "Dirichlet_y":
            {
                "mydir_y":
                {
                    "markers":["V1_0","V1_1","OYOZ"],
                    "expr":0
                }
            },
            "Dirichlet_z":
            {
                "mydir_z":
                {
                    "markers":["V0_0","V0_1","OXOZ","V1_0","V1_1","OYOZ"],
                    "expr":0
                }
            }
        },
        "electric":
        {
            "Dirichlet":
            {
                "V0_0":
                {
                    "expr":"materials_Conductor_0_V0:materials_Conductor_0_V0"
                },
                "V0_1":
                {
                    "expr":"materials_Conductor_1_V0:materials_Conductor_1_V0"
                },
                "V1_0":
                {
                    "expr":"materials_Conductor_0_V1:materials_Conductor_0_V1"
                },
                "V1_1":
                {
                    "expr":"materials_Conductor_1_V1:materials_Conductor_1_V1"
                }
            }
        }
    }

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

7. Weak Formulation

The weak formulation of A-V Formulation + gauge is :

\text{(AV+gauge Weak)} \left\{ \begin{eqnarray*} \int_{\Omega}{ \sigma \frac{d \mathbf{A}}{d t} \mathbf{ϕ} } + \int_{\Omega}{ \frac{1}{\mu} \nabla \mathbf{A} \cdot \nabla \mathbf{ϕ} } &=& - \int_{\Omega_c}{\sigma \nabla V \, \mathbf{ϕ}} \\ \int_{\Omega_c}{ \sigma \nabla V \cdot \nabla \psi } &=& - \int_{\Omega_c}{\sigma \frac{d \mathbf{A}}{d t} \cdot \nabla \psi} \end{eqnarray*} \right.

8. Parameters

The parameters of problem are :

  • On Conductor :

Symbol

Description

Value

Unit

V0

scalar electrical potential on V0

0

Volt

V1

scalar electrical potential on V1

\frac{1}{4} \left\{ \begin{matrix} t \ \text{on} \ 0s \leq t < 1s \\ 1 \ \text{on} \ 1s \leq t < 20s \\ 0 \ \text{on} \ 20s \leq t \end{matrix} \right.

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

9. Coefficient Form PDEs

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

In the json file, we note magnetic the equation (AV'-1) and electric the equation (AV'-2).

  • On Conductor_0 and Conductor_1 :

Coefficient

Description

Expression

magnetic_d

damping or mass coefficient

\sigma

magnetic_c

diffusion coefficient

\frac{1}{\mu}

magnetic_f

source term

- \sigma \nabla V

electric_c

damping or mass coefficient

\sigma

electric_gamma

source term

\sigma \, \frac{d \mathbf{A}}{d t}

  • On Air (the lectrical potential isn’t computed on Air) :

Coefficient

Description

Expression

magnetic_c

diffusion coefficient

\frac{1}{\mu}

On JSON file, the coefficients are writed :

CFPDEs coefficients on JSON file
    "Materials":
    {
        "Conductor_0":
        {
            "V0":0,
            "V1":"1/4*(t/(0.1)*(t<(0.1))+(t<(0.5))*(t>(0.1))):t",
            "sigma":58e6,       // S.m-1
            "mu":"4*pi*1e-7",   // kg.m/A2/S2

            "magnetic_d":"sigma:sigma",
            "magnetic_c":"1/mu:mu",
            "magnetic_f":"{-sigma*electric_grad_V_0,-sigma*electric_grad_V_1,-sigma*electric_grad_V_2}:sigma:electric_grad_V_0:electric_grad_V_1:electric_grad_V_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"
        },
        "Conductor_1":
        {
            "V0":0,
            "V1":"1/4*(t/(0.1)*(t<(0.1))+(t<(0.7))*(t>(0.1))):t",
            "sigma":58e6,       // S.m-1
            "mu":"4*pi*1e-7",   // kg.m/A2/S2

            "magnetic_d":"sigma:sigma",
            "magnetic_c":"1/mu:mu",
            "magnetic_f":"{-sigma*electric_grad_V_0,-sigma*electric_grad_V_1,-sigma*electric_grad_V_2}:sigma:electric_grad_V_0:electric_grad_V_1:electric_grad_V_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"
        },
        "Air":
        {
            "physics":"magnetic",
            "mu":"4*pi*1e-7",   // kg.m/A2/S2
            "magnetic_c":"1/mu:mu"
        }
    }

10. Numeric Parameters

This section show 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

  • Element type : P1

  • Solver : automatic

  • Number of CPU core : 32

2torus 3d mesh
Mesh of Geometry

11. Results

11.1. Magnetic Fields

I plot the horizontal magnetic field by time on point (0,0,0).

Bz P0 t
Horizontal Magnetic Field B_z (T) by time

We can see the four step :

  • the ramp for the two tensions

  • the tray for the two tensions

  • the tray for the tension U_1 and the tray at 0 for the tension U_0

  • the tray for the tray at 0 for the tensions U_0 and U_1

11.2. Intensities

The analytical solve of intensities is computed by python3’s script : Edit the file, with the formula :

\small{ \mathbf{I}(t) = \left\{ \begin{matrix} \left( t \, \left( \mathbf{R} \mathbf{L}^{-1} \right)^{-1} - \left( \mathbf{R} \mathbf{L}^{-1} \right)^{-2} \left( Id - e^{-\mathbf{R} \mathbf{L}^{-1} t} \right) \right) \ \mathbf{L}^{-1} \, \begin{pmatrix} 1/0.1 \\ 1/0.1 \end{pmatrix} & 0s \leq t \leq 0.1s \\ e^{-\mathbf{R} \mathbf{L}^{-1} t} \, \mathbf{C}_1 + \mathbf{R}^{-1} \begin{pmatrix} 1 \\ 1 \end{pmatrix} & 0.1s \leq t \leq 0.5s \\ e^{-\mathbf{R} \mathbf{L}^{-1} t} \, \mathbf{C}_2 + \mathbf{R}^{-1} \begin{pmatrix} 0 \\ 1 \end{pmatrix} & 0.5s \leq t \leq 0.7s \\ e^{-\mathbf{R} \mathbf{L}^{-1} t} \, \mathbf{C}_3 & 0.7s \leq t \leq 1s \\ \end{matrix} \right. }

And the values :

Symbol

Description

Value

Unit

R_0

resistance of Conductor_0

7.479339e-06

Ohm

R_1

resistance of Conductor_0

7.479339e-06

Ohm

L_0

auto-inductance of Conductor_0

1.920400e-07

Henry

L_0

auto-inductance of Conductor_0

1.920400e-07

Henry

M_{01}

inductance from Conductor_0 to Conductor_1

1.705000e-08

Henry

M_{10}

inductance from Conductor_1 to Conductor_0

1.705000e-08

Henry

I t
Intensities : Comparison between numerical and analytical results
elec
Figure 1. Intensities I_i (A) and Tensions U_i(V), caption=

The intensities follow the tensions like for 1-Torus problem with the four step (ramp, trays and cuts). But, we can see little peaks. This peak on one intensity appears when the tension of other conductor is cut. This phenomena is called Electromagnetism Induction.

The Electromagnetism Induction is the production of an electromotive force across an electrical conductor in a changing magnetic field.

With the equation of Maxwell-Faraday :

\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}

We can see, with time variation of magnetic field \mathbf{B}, we have creation of electrical field. For us case, when we cut the tension for one conductor, we have great time variation of magnetic field thus the other conductor answer by Maxwell-Faraday equation with creation of electrical field and current. When the magnetic field is at zeros, is phenomena disapears.