Module 2 - Forward

module2_forward.current_method(L, l, method=1, value=1)

Create a numpy array (or a list of arrays) that represents the current pattern in the electrodes.

Parameters:

  • L (int) – Number of electrodes.

  • l (int) – Number of measurements.

  • method (int) – Current pattern. Possible values are 1, 2, 3, or 4 (default=1).

  • value (int or float) – Current density value (default=1).

Returns:

list of arrays or numpy array – Return list with current density in each electrode for each measurement.

Method Values:

  1. 1 and -1 in opposite electrodes.

  2. 1 and -1 in adjacent electrodes.

  3. 1 in one electrode and -1/(L-1) for the rest.

  4. For measurement k, we have: (sin(k*2*pi/16) sin(2*k*2*pi/16) … sin(16*k*2*pi/16)).

Example:

Create current pattern 1 with 4 measurements and 4 electrodes:

>>> I_all = current_method(L=4, l=4, method=1)
>>> print(I_all)
    [array([ 1.,  0., -1.,  0.]),
    array([ 0.,  1.,  0., -1.]),
    array([-1.,  0.,  1.,  0.]),
    array([ 0., -1.,  0.,  1.])]

Create current pattern 2 with 4 measurements and 4 electrodes:

>>> I_all = current_method(L=4, l=4, method=2)
>>> print(I_all)
    [array([ 1., -1.,  0.,  0.]),
    array([ 0.,  1., -1.,  0.]),
    array([0.,  0.,  1., -1.]),
    array([ 1.,  0.,  0., -1.])]
module2_forward.GammaCircle(mesh, in_v, out_v, radius, centerx, centery)

Function to create a circle in the mesh with specified properties.

Parameters:

  • mesh (dolfin.cpp.mesh.Mesh) – Mesh.

  • in_v (float) – Value inside the circle.

  • out_v (float) – Value outside the circle.

  • radius (float) – Circle radius.

  • centerx (float) – Circle center position x.

  • centery (float) – Circle center position y.

Returns:

numpy.array – Return a vector where each position corresponds to the value of the function in that element.

Example:

>>> ValuesCells0 = GammaCircle(mesh=mesh_refined, in_v=3.0, out_v=1.0, radius=0.50, centerx=0.25, centery=0.25)
>>> print(ValuesCells0)
    array([1., 1., 1., ..., 1., 1., 1.])
>>> Q = FunctionSpace(mesh, "DG", 0) #Define Function space with basis Descontinuous Galerkin
>>> gamma = Function(Q)
>>> gamma.vector()[:]=ValuesCells0
>>> plot_figure(gamma, name="", map="jet");
../_images/gamma13.png
class module2_forward.ForwardProblem(mesh, z)

Object representing the Forward Problem in 2D EIT.

Parameters:

  • mesh (MyMesh()) – Mesh.

  • z (array-like) – Vector of impedances in electrodes.

Example:

>>> # Basic Definitions
>>> L = 16
>>> l = int(L)  # Measurements number.
>>> z = np.ones(L) * 0.025  # Impedance
>>> I_all = current_method(L, l, method=1)  # Current pattern

>>> # Solver
>>> VD = FiniteElement('CG', mesh_refined.ufl_cell(), 1)  # Space Solution
>>> DirectProblem = ForwardProblem(mesh_refined, z)
>>> list_u0, list_U0 = DirectProblem.solve_forward(VD, gamma0, I_all)
>>> list_U0 = DirectProblem.sol_asarray()
>>> print(list_U0[0:L])
    [1.0842557  0.32826713 0.19591977 0.13158264 0.06214628 -0.03412964
    -0.17331413 -0.40308837 -1.18449889 -0.42369776 -0.21120216 -0.08218106
    0.01735219 0.10789938 0.20976791 0.37492101]
solve_forward(V, I_all, gamma)

Solver for the Forward Problem of 2D Electrical Impedance Tomography (EIT).

Parameters:

  • V (dolfin.cpp.fem.FiniteElement) – FiniteElement FEniCS object.

  • gamma (dolfin.function.Function) – Finite Element Function representing the electrical conductivity distribution.

  • I_all (current_method() or list of arrays) – Current density in each electrode for each measurement.

Returns:

tuple – A tuple containing two FEniCS objects, representing the potential distribution in the domain and the potentials at the electrodes, respectively.

Example:

>>> DirectProblem = ForwardProblem(mesh_refined, z)
>>> list_u0, list_U0 = DirectProblem.solve_forward(VD, I_all, gamma0)
sol_asarray()

Convert electrode potential results into an array and concatenate them.

Returns:

array – A vector with concatenated potential values for all electrodes and measurements.

Example:

>>> list_U0 = DirectProblem.sol_asarray()
add_noise(noise_level=0, noise_type='uniform', seed=42)

Add noise to the potential values.

Parameters:

  • noise_level (float) – Noise level in percentage (between 0 and 1).

  • noise_type (str.) – Type of noise to add (‘uniform’ or ‘cauchy’).

  • seed (int) – Seed for the random number generator.

Returns:

array – A vector with noised potential values for all electrodes and measurements.

Example:

>>> list_U0_noised = DirectProblem.add_noise(noise_level=0.01, noise_type='uniform')
verify_solution_graphs(gamma0, sol_index=0, method=1)

Plot boundary information to verify the solution.

Parameters:

  • gamma0 (dolfin.function.Function) – Finite Element Function representing the electrical conductivity distribution.

  • sol_index (int) – Index for the solution, ranging from 0 to l (number of measurements).

  • method (int) – Method for verification. 1: u+zi.gama.n.grad(u)=Ui, 2: boundary gamma.n.grad(u), 3: boundary gamma.n.grad(u) (only gaps).

Returns:

array – Plot boundary data.

Example:

>>> data = DirectProblem.verify_solution_graphs(gamma0, sol_index=0, method=2)
verify_solution_values(I_all, gamma0, sol_index=0, method=1)

Verify the solution values by comparing with the expected values.

Parameters:

  • I_all (array or list of arrays) – Current density in each electrode for each measurement.

  • gamma0 (dolfin.function.Function) – Finite Element Function representing the electrical conductivity distribution.

  • sol_index (int) – Index for the solution, ranging from 0 to l (number of measurements).

  • method (int) – Method for verification. 1: Current values, 2: Average potential on electrodes.

Returns:

None

Example:

>>> DirectProblem.verify_solution_values(I_all, gamma0, sol_index=0, method=2)