Introduction

In this example, we check the correctness of SFEMaNS for a magnetic problem involving a conducting domain. We consider Dirichlet and Neumann boundary conditions. We use P1 finite elements for the magnetic field.

We solve the Maxwell equations:

\begin{align} \begin{cases} \partial_t (\mu^c \mathbf{H}) + \nabla \times \left(\frac{1}{\Rm \sigma }\nabla \times \mathbf{H} \right) = \nabla\times (\bu^\text{given} \times \mu^c \mathbf{H}) + \nabla \times \left(\frac{1}{\Rm \sigma} \nabla\times \mathbf{j} \right) & \text{in } \Omega_1, \\ \text{div} (\mu^c \mathbf {H}) = 0 & \text{in } \Omega_1 ,\\ \bH \times \bn = \bH_{\text{bdy}} \times \bn & \text{on } \Gamma_1,\\ - \left( \frac{1}{\Rm \sigma} \left( \ROT (\mathbf{H}) - \mathbf{j} \right) - \bu^\text{given} \times \mu \mathbf{H} \right) = \textbf{a} \times \bn & \text{on } \Gamma_2,\\ \bH_{|t=0}= \bH_0, \end{cases} \end{align}

in the domain $\Omega_1= \{ (r,\theta,z) \in {R}^3 : (r,\theta,z) \in [0,0.5] \times [0,2\pi) \times [0,1]\} $. We also set $\Gamma_1= \partial \Omega_1\cap \{ z\in \{0,1\}\} $ and $\Gamma_2= \partial \Omega_1 \cap \{ r=1\} $. The data are the source term $\mathbf{j}$, the given velocity $\bu^\text{given}$, the boundary datas $\bH_{\text{bdy}}$ $\textbf{a}$, the initial data $\bH_0$. The parameter $\Rm$ is the magnetic Reynolds number. The parameter $\mu^c$ is the magnetic permeability of the conducting region $\Omega_1$. The parameter $\sigma$ is the conductivity in the conducting region $\Omega_1$.

Manufactured solutions

We approximate the following analytical solutions:

\begin{align*} H_r(r,\theta,z,t) &= 0, \\ H_{\theta}(r,\theta,z,t) &= r - r^2 \sin(\theta), \\ H_z(r,\theta,z,t) &= r z \cos(\theta). \end{align*}

We also set the given velocity field as follows:

\begin{align*} u^\text{given}_r(r,\theta,z,t) &= 0, \\ u^\text{given}_{\theta}(r,\theta,z,t) &= 0, \\ u^\text{given}_z(r,\theta,z,t) &= 0. \end{align*}

The source term $ \mathbf{j} $ and the boundary datas $\bH_{\text{bdy}}, \textbf{a}$ are computed accordingly.

Generation of the mesh

The finite element mesh used for this test is named Mesh_10_form.FEM and has a mesh size of $0.1$ for the P1 approximation. You can generate this mesh with the files in the following directory: ($SFEMaNS_MESH_GEN_DIR)/EXAMPLES/EXAMPLES_MANUFACTURED_SOLUTIONS/Mesh_10_form. The following images show the mesh for P1 finite elements.

Finite element mesh (P1).

Information on the file `condlim.f90`

The initial conditions, boundary conditions and the forcing term $\textbf{j}$ in the Maxwell equations are set in the file condlim_test_12.f90. Here is a description of the subroutines and functions of interest.

The subroutine init_maxwell initializes the magnetic field and the scalar potential at the time $-dt$ and $0$ with $dt$ being the time step. It is done by using the functions Hexact, Phiexact as follows:
time = -dt

DO k=1,6

DO i=1, SIZE(list_mode)

Hn1(:,k,i) = Hexact(H_mesh,k, H_mesh%rr, list_mode(i), mu_H_field, time)

IF (inputs%nb_dom_phi>0) THEN

IF (k<3) THEN

phin1(:,k,i) = Phiexact(k, phi_mesh%rr, list_mode(i) , mu_phi, time)

ENDIF

END IF

ENDDO

ENDDO

time = time + dt

DO k=1,6

DO i=1, SIZE(list_mode)

Hn(:,k,i) = Hexact(H_mesh, k, H_mesh%rr, list_mode(i), mu_H_field, time)

IF (inputs%nb_dom_phi>0) THEN

IF (k<3) THEN

phin(:,k,i) = Phiexact(k, phi_mesh%rr, list_mode(i), mu_phi, time)

ENDIF

END IF

ENDDO

ENDDO
The function Hexact contains the analytical magnetic field. It is used to initialize the magnetic field and to impose Dirichlet boundary conditions on $\textbf{H}\times\textbf{n}$ with $\textbf{n}$ the outter normal vector.
1. For the Fourier modes m∈{0,1}, we define the magnetic field depending of its TYPE (1 and 2 for the component radial cosine and sine, 3 and 4 for the component azimuthal cosine and sine, 5 and 6 for the component vertical cosine and sine) as follows:
  IF (m==1) THEN
  
  IF (TYPE==4) THEN
  
  vv = -rr(1,:)**2
  
  ELSE IF (TYPE==5) THEN
  
  vv = rr(1,:)*rr(2,:)
  
  ELSE
  
  vv = 0.d0
  
  END IF
  
  ELSE IF (m==0) THEN
  
  IF (TYPE==3) THEN
  
  vv = rr(1,:)
  
  ELSE IF (TYPE==5) THEN
  
  vv = 0.d0
  
  ELSE
  
  vv = 0.d0
  
  END IF
2. For the other Fourier modes, the magnetic field is set to zero.
  ELSE
  
  vv = 0.d0
  
  END IF
  
  RETURN
The function Vexact is used to define the given velocity field $\bu^\text{given}$. It is set to zero.
vv = 0.d0
The function Jexact_gauss is used to define the source term $\textbf{j}$. Since $\partial_t (\mu \textbf{H})=0$ and $ \bu^\text{given} \times \textbf{H}=0 $, we define this source term so it satisfies the relation $ \ROT\textbf{j}=\ROT(\ROT\textbf{H} )$. As this term only depends of the Fourier mode $m=1$, we define it as follows:
IF (m==1) THEN

IF (TYPE==2) THEN

vv = -rr(2)

ELSE IF (TYPE==3) THEN

vv = -rr(2)

ELSE IF (TYPE==6) THEN

vv = -3*rr(1)

ELSE

vv = 0.d0

END IF

ELSE

vv = 0.d0

END IF

RETURN
The function Eexact_gauss is used to define the electric field $\textbf{E}$. We remind that $ - \textbf{E}=\textbf{a} \times \textbf{n}$. In a conducting area, the electric field also satisfies the relation $\textbf{E}= \frac{1}{\Rm \sigma} \left( \ROT (\mathbf{H}) - \mathbf{j} \right) - \bu^\text{given} \times (\mu \textbf{H}) $. It is defined as follows.
IF (m/=0) THEN

vv = 0

ELSE

IF (TYPE==5) THEN

vv = 2.d0

ELSE

vv = 0.d0

END IF

END IF

RETURN

All the other subroutines present in the file condlim_test_12.f90 are not used in this test. We refer to the section Fortran file condlim.f90 for a description of all the subroutines of the condlim file.

Setting in the data file

We describe the data file of this test. It is called debug_data_test_12 and can be found in the following directory: ($SFEMaNS_DIR)/MHD_DATA_TEST_CONV_PETSC.

We use a formatted mesh by setting:
===Is mesh file formatted (true/false)?

.t.
The path and the name of the mesh are specified with the two following lines:
===Directory and name of mesh file

'.' 'Mesh_10_form.FEM'

where '.' refers to the directory where the data file is, meaning ($SFEMaNS_DIR)/MHD_DATA_TEST_CONV_PETSC.
We use two processors in the meridian section. It means the finite element mesh is subdivised in two.
===Number of processors in meridian section

2
We solve the problem for $2$ Fourier modes.
===Number of Fourier modes

2
We use $2$ processors in Fourier space.
===Number of processors in Fourier space

2

It means that each processors is solving the problem for $2/2=1$ Fourier modes.
We select specific Fourier modes to solve.
===Select Fourier modes? (true/false)

.t.
We give the list of the Fourier modes to solve.
===List of Fourier modes (if select_mode=.TRUE.)

0 1

We note that setting select Fourier modes to false would give the same result as we select the first two Fourier modes.
We approximate the Maxwell equations by setting:
===Problem type: (nst, mxw, mhd, fhd)

'mxw'
We do not restart the computations from previous results.
===Restart on velocity (true/false)

.f.

===Restart on magnetic field (true/false)

.f.

It means the computation starts from the time $t=0$. We note that we need to say if we use a restart for the velocity. If yes, the $\bu^\text{given}$ comes from a suite file and not the function Vexact.
We use a time step of $1$ and solve the problem over $1$ time iterations.
===Time step and number of time iterations

1.d0, 1
We set the number of domains and their label, see the files associated to the generation of the mesh, where the code approximates the magnetic field.
===Number of subdomains in magnetic field (H) mesh

1

===List of subdomains for magnetic field (H) mesh

1
We set the number of interface in H_mesh.
===Number of interfaces in H mesh

0

Such interfaces represent interfaces with discontinuities in magnetic permeability or interfaces between the magnetic field mesh and the temperature or the velocity field meshes.
We set the number of boundaries with Dirichlet conditions on the magnetic field and give their respective labels.
===Number of Dirichlet sides for Hxn

2

===List of Dirichlet sides for Hxn

2 4

We note that we did not select the boundary of index 3 that correspond to the face $r=0.5$. As consequence, Neumann boundary condition are apply to this face.
We set the magnetic permeability in each domains where the magnetic field is approximated.
===Permeability in the conductive part (1:nb_dom_H)

1.d0
We set the conductivity in each domains where the magnetic field is approximated.
===Conductivity in the conductive part (1:nb_dom_H)

1.d0
We set the type of finite element used to approximate the magnetic field.
===Type of finite element for magnetic field

1
We set the magnetic Reynolds number $\Rm$.
===Magnetic Reynolds number

1.d0
We set stabilization coefficient for the divergence of the magnetic field and the penalty of the Dirichlet and interface terms.
===Stabilization coefficient (divergence)

1.d0

===Stabilization coefficient for Dirichlet H and/or interface H/H

1.d0

We note these coefficients are usually set to $1$.
We give information on how to solve the matrix associated to the time marching of the Maxwell equations.
1. ===Maximum number of iterations for Maxwell solver
  
  100
2. ===Relative tolerance for Maxwell solver
  
  1.d-6
  
  ===Absolute tolerance for Maxwell solver
  
  1.d-10
3. ===Solver type for Maxwell (FGMRES, CG, ...)
  
  GMRES
  
  ===Preconditionner type for Maxwell solver (HYPRE, JACOBI, MUMPS...)
  
  MUMPS

Outputs and value of reference

The outputs of this test are computed with the file post_processing_debug.f90 that can be found in the following: ($SFEMaNS_DIR)/MHD_DATA_TEST_CONV_PETSC.

To check the well behavior of the code, we compute three quantities:

The L2 norm of the error on the magnetic field $\textbf{H}$.
The L2 norm of the error on the curl of the magnetic field $\textbf{H}$.
The L2 norm of the divergence of the magnetic field $\textbf{B}=\mu\bH$.

They are compared to reference values to attest of the correctness of the code. These values of reference are in the last lines of the file debug_data_test_12 in the directory ($SFEMaNS_DIR)/MHD_DATA_TEST_CONV_PETSC. They are equal to:

============================================
Mesh_10_form.FEM, P1
===Reference results
4.823924970280468E-003 L2 error on  Hn     
1.130509829567467E-002 L2 error on Curl(Hn) 
4.821424313259546E-002 L2 norm of error on Div(mu Hn)
1.                      Dummy ref

To conclude this test, we show the profile of the approximated magnetic field $\textbf{H}$ at the final time. This figure is done in the plane $y=0$ which is the union of the half plane $\theta=0$ and $\theta=\pi$.