[WIP] Allow 3D process variables

[rtimms]

Description

Allow 3D (in space) processed variables. There is a lot of code duplication in the processed variables code, but we can fix that in a separate PR.

This currently covers the case of variables that are 3D and have a tertiary domain (e.g. particle -> particle size -> electrode) but doesn't work for the special case of 3D variables that only have a primary and secondary domain (e.g. electrode -> current collector, when using the 2D current collector mesh).

Still needs tests.

Fixes #4776 and #2394.

Also, see #1549

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 -m pytest (or $ nox -s tests)
• The documentation builds: $ python -m pytest --doctest-plus src (or $ nox -s doctests)

You can run integration tests, unit tests, and doctests together at once, using $ 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

[rtimms]

All these cases work now

import pybamm

# Process variable in 3D
model = pybamm.lithium_ion.DFN(
    options={"current collector": "potential pair", "dimensionality": 1}
)
sim = pybamm.Simulation(model)
sol = sim.solve([0, 100])
sol["Negative particle concentration [mol.m-3]"].entries

# Experiment with 3D vars (particle size distribution)
model = pybamm.lithium_ion.DFN(
    {
        "working electrode": "positive",
        "surface form": "differential",
        "contact resistance": "true",
        "particle size": "distribution",
    }
)

parameter_values = model.default_parameter_values


def f_a_dist_p_dim(R):
    return pybamm.lognormal(R, 8e-6, 1e-6)


distribution_params = {
    "Positive minimum particle radius [m]": 0,
    "Positive maximum particle radius [m]": 3e-5,
    "Positive area-weighted " + "particle-size distribution [m-1]": f_a_dist_p_dim,
}
parameter_values.update(distribution_params, check_already_exists=False)


def D_p(sto, T):
    return 8e-15 + 3e-15 * sto


parameter_values.update(
    {"Positive particle diffusivity [m2.s-1]": D_p}, check_already_exists=False
)

experiment = pybamm.Experiment(["Charge at C/100 for 20 hours", "Rest for 2 hours"])

sim = pybamm.Simulation(
    model,
    parameter_values=parameter_values,
    experiment=experiment,
    solver=pybamm.IDAKLUSolver(),
)
sim.solve(initial_soc=0)

# Experiment with 3D vars (current collector)
options = {"cell geometry": "pouch", "thermal": "x-lumped", "dimensionality": 2}
model = pybamm.lithium_ion.SPMe(options=options)
experiment = pybamm.Experiment(
    [
        "Discharge at 5 C for 10 seconds",
        "Charge at 1 C for 10 seconds",
    ]
)
sim = pybamm.Simulation(model, experiment=experiment, solver=pybamm.IDAKLUSolver())
sol = sim.solve(calc_esoh=False)
sol["Negative electrode potential [V]"].entries