fix bug in leading surface form conductivity

[rtimms]

Description

Fixes a bug where a factor of electrode surface area to volume ratio is missing in the rhs of the LeadingOrderDifferential conductivity model.

In particular, this fixes the missing semi-circle when simulating EIS

import pybamm
import numpy as np
import matplotlib.pyplot as plt
import time as timer
from scipy.fft import fft

# Set up
model = pybamm.lithium_ion.SPM(options={"surface form": "differential"}, name="SPM")
V = model.variables["Terminal voltage [V]"]
I = model.variables["Current [A]"]
parameter_values = pybamm.ParameterValues("Marquis2019")
frequencies = np.logspace(-4, 2, 30)

# Time domain
I = 50 * 1e-3
number_of_periods = 20
samples_per_period = 16


def current_function(t):
    return I * pybamm.sin(2 * np.pi * pybamm.InputParameter("Frequency [Hz]") * t)


parameter_values["Current function [A]"] = current_function

start_time = timer.time()

sim = pybamm.Simulation(
    model, parameter_values=parameter_values, solver=pybamm.ScipySolver()
)

impedances_time = []
for frequency in frequencies:
    # Solve
    period = 1 / frequency
    dt = period / samples_per_period
    t_eval = np.array(range(0, 1 + samples_per_period * number_of_periods)) * dt
    sol = sim.solve(t_eval, inputs={"Frequency [Hz]": frequency})
    # Extract final two periods of the solution
    time = sol["Time [s]"].entries[-3 * samples_per_period - 1 :]
    current = sol["Current [A]"].entries[-3 * samples_per_period - 1 :]
    voltage = sol["Voltage [V]"].entries[-3 * samples_per_period - 1 :]
    # FFT
    current_fft = fft(current)
    voltage_fft = fft(voltage)
    # Get index of first harmonic
    idx = np.argmax(np.abs(current_fft))
    impedance = -voltage_fft[idx] / current_fft[idx]
    impedances_time.append(impedance)

end_time = timer.time()
time_elapsed = end_time - start_time
print("Time domain method: ", time_elapsed, "s")


fig, ax = plt.subplots()
data = np.array(impedances_time)
ax.plot(data.real, -data.imag, "o")
ax_max = max(data.real.max(), -data.imag.max()) * 1.1
plt.axis([0, ax_max, 0, ax_max])
plt.gca().set_aspect("equal", adjustable="box")
ax.set_xlabel(r"$Z_\mathrm{Re}$ [Ohm]")
ax.set_ylabel(r"$-Z_\mathrm{Im}$ [Ohm]")
plt.show()

Before fix:

After fix:

Fixes # (issue)

Type of change

Please add a line in the relevant section of CHANGELOG.md to document the change (include PR #) - note reverse order of PR #s. If necessary, also add to the list of breaking changes.

• New feature (non-breaking change which adds functionality)
• Optimization (back-end change that speeds up the code)
• Bug fix (non-breaking change which fixes an issue)

Key checklist:

• No style issues: $ pre-commit run (or $ nox -s pre-commit) (see CONTRIBUTING.md for how to set this up to run automatically when committing locally, in just two lines of code)
• All tests pass: $ python run-tests.py --all (or $ nox -s tests)
• The documentation builds: $ python run-tests.py --doctest (or $ nox -s doctests)

You can run integration tests, unit tests, and doctests together at once, using $ python run-tests.py --quick (or $ nox -s quick).

Further checks:

• Code is commented, particularly in hard-to-understand areas
• Tests added that prove fix is effective or that feature works

[brosaplanella]

Looks good, feel free to merge after adding a line to the README.