Repository: pybamm-team/PyBaMM (python)

Issue:
Harmonic mean for node to edge in fluxes
Use the harmonic mean when doing `node_to_edge` in binary operators so that fluxes have the correct behavior 


Pull request description:
#782 add harmonic mean
# Description
Adds the harmonic mean to the finite element method. This is then used in `process_binary_operators` so that fluxes are computed using the harmonic mean of the diffusivity (or conductivity etc.) 

Fixes #782 

## Type of change

Please add a line in the relevant section of [CHANGELOG.md](https://github.com/pybamm-team/PyBaMM/blob/master/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: `$ flake8`
- [ ] All tests pass: `$ python run-tests.py --unit`
- [ ] The documentation builds: `$ cd docs` and then `$ make clean; make html`

You can run all three at once, using `$ python run-tests.py --quick`.

## Further checks:

- [ ] Code is commented, particularly in hard-to-understand areas
- [ ] Tests added that prove fix is effective or that feature works


Patch:
diff --git a/CHANGELOG.md b/CHANGELOG.md
index ee7535c19..35202d66e 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2,6 +2,7 @@
 
 ## Features
 
+-   Added the harmonic mean to the Finite Volume method, which is now used when computing fluxes ([#783](https://github.com/pybamm-team/PyBaMM/pull/783))
 -   Added notebook to explain broadcasts ([#776](https://github.com/pybamm-team/PyBaMM/pull/776))
 -   Added a step to discretisation that automatically compute the inverse of the mass matrix of the differential part of the problem so that the underlying DAEs can be provided in semi-explicit form, as required by the CasADi solver ([#769](https://github.com/pybamm-team/PyBaMM/pull/769))
 -   Added the gradient operation for the Finite Element Method ([#767](https://github.com/pybamm-team/PyBaMM/pull/767))
diff --git a/pybamm/spatial_methods/finite_volume.py b/pybamm/spatial_methods/finite_volume.py
index eba8ab4bd..3533b1f86 100644
--- a/pybamm/spatial_methods/finite_volume.py
+++ b/pybamm/spatial_methods/finite_volume.py
@@ -967,7 +967,11 @@ class FiniteVolume(pybamm.SpatialMethod):
     def process_binary_operators(self, bin_op, left, right, disc_left, disc_right):
         """Discretise binary operators in model equations.  Performs appropriate
         averaging of diffusivities if one of the children is a gradient operator, so
-        that discretised sizes match up.
+        that discretised sizes match up. For this averaging we use the harmonic
+        mean [1].
+
+        [1] Recktenwald, Gerald. "The control-volume finite-difference approximation to
+        the diffusion equation." (2012).
 
         Parameters
         ----------
@@ -1003,11 +1007,23 @@ class FiniteVolume(pybamm.SpatialMethod):
         elif left_evaluates_on_edges == right_evaluates_on_edges:
             pass
         # If only left child evaluates on edges, map right child onto edges
+        # using the harmonic mean if the left child is a gradient (i.e. this
+        # binary operator represents a flux)
         elif left_evaluates_on_edges and not right_evaluates_on_edges:
-            disc_right = self.node_to_edge(disc_right)
+            if isinstance(left, pybamm.Gradient):
+                method = "harmonic"
+            else:
+                method = "arithmetic"
+            disc_right = self.node_to_edge(disc_right, method=method)
         # If only right child evaluates on edges, map left child onto edges
+        # using the harmonic mean if the right child is a gradient (i.e. this
+        # binary operator represents a flux)
         elif right_evaluates_on_edges and not left_evaluates_on_edges:
-            disc_left = self.node_to_edge(disc_left)
+            if isinstance(right, pybamm.Gradient):
+                method = "harmonic"
+            else:
+                method = "arithmetic"
+            disc_left = self.node_to_edge(disc_left, method=method)
         # Return new binary operator with appropriate class
         out = bin_op.__class__(disc_left, disc_right)
         return out
@@ -1039,27 +1055,28 @@ class FiniteVolume(pybamm.SpatialMethod):
                     )
         return pybamm.DomainConcatenation(disc_children, self.mesh)
 
-    def edge_to_node(self, discretised_symbol):
+    def edge_to_node(self, discretised_symbol, method="arithmetic"):
         """
         Convert a discretised symbol evaluated on the cell edges to a discretised symbol
         evaluated on the cell nodes.
         See :meth:`pybamm.FiniteVolume.shift`
         """
-        return self.shift(discretised_symbol, "edge to node")
+        return self.shift(discretised_symbol, "edge to node", method)
 
-    def node_to_edge(self, discretised_symbol):
+    def node_to_edge(self, discretised_symbol, method="arithmetic"):
         """
         Convert a discretised symbol evaluated on the cell nodes to a discretised symbol
         evaluated on the cell edges.
         See :meth:`pybamm.FiniteVolume.shift`
         """
-        return self.shift(discretised_symbol, "node to edge")
+        return self.shift(discretised_symbol, "node to edge", method)
 
-    def shift(self, discretised_symbol, shift_key):
+    def shift(self, discretised_symbol, shift_key, method):
         """
         Convert a discretised symbol evaluated at edges/nodes, to a discretised symbol
-        evaluated at nodes/edges.
-        For now we just take the arithemtic mean, though it may be better to take the
+        evaluated at nodes/edges. Can be the arithmetic mean or the harmonic mean.
+
+        Note: when computing fluxes at cell edges it is better to take the
         harmonic mean based on [1].
 
         [1] Recktenwald, Gerald. "The control-volume finite-difference approximation to
@@ -1074,6 +1091,8 @@ class FiniteVolume(pybamm.SpatialMethod):
         shift_key : str
             Whether to shift from nodes to edges ("node to edge"), or from edges to
             nodes ("edge to node")
+        method : str
+            Whether to use the "arithmetic" or "harmonic" mean
 
         Returns
         -------
@@ -1121,10 +1140,141 @@ class FiniteVolume(pybamm.SpatialMethod):
 
             return pybamm.Matrix(matrix) @ array
 
+        def harmonic_mean(array):
+            """
+            Calculate the harmonic mean of an array using matrix multiplication.
+            The harmonic mean is computed as
+
+            .. math::
+                D_{eff} = \\frac{D_1  D_2}{\\beta D_2 + (1 - \\beta) D_1},
+
+            where
+
+            .. math::
+                \\beta = \\frac{\\Delta x_1}{\\Delta x_2 + \\Delta x_1}
+
+            accounts for the difference in the control volume widths. This is the
+            definiton from [1], which is the same as that in [2] but with slightly
+            different notation.
+
+            [1] Torchio, M et al. "LIONSIMBA: A Matlab Framework Based on a Finite
+            Volume Model Suitable for Li-Ion Battery Design, Simulation, and Control."
+            (2016).
+            [2] Recktenwald, Gerald. "The control-volume finite-difference
+            approximation to the diffusion equation." (2012).
+            """
+            # Create appropriate submesh by combining submeshes in domain
+            submesh_list = self.mesh.combine_submeshes(*array.domain)
+            submesh = submesh_list[0]
+
+            # Get second dimension length for use later
+            second_dim_len = len(submesh_list)
+
+            # Create 1D matrix using submesh
+            n = submesh.npts
+
+            if shift_key == "node to edge":
+                # Matrix to compute values at the exterior edges
+                edges_sub_matrix_left = csr_matrix(
+                    ([1.5, -0.5], ([0, 0], [0, 1])), shape=(1, n)
+                )
+                edges_sub_matrix_center = csr_matrix((n - 1, n))
+                edges_sub_matrix_right = csr_matrix(
+                    ([-0.5, 1.5], ([0, 0], [n - 2, n - 1])), shape=(1, n)
+                )
+                edges_sub_matrix = vstack(
+                    [
+                        edges_sub_matrix_left,
+                        edges_sub_matrix_center,
+                        edges_sub_matrix_right,
+                    ]
+                )
+
+                # Generate full matrix from the submatrix
+                # Convert to csr_matrix so that we can take the index (row-slicing),
+                # which is not supported by the default kron format
+                # Note that this makes column-slicing inefficient, but this should
+                # not be an issue
+                edges_matrix = csr_matrix(kron(eye(second_dim_len), edges_sub_matrix))
+
+                # Matrix to extract the node values running from the first node
+                # to the penultimate node in the primary dimension (D_1 in the
+                # definiton of the harmonic mean)
+                sub_matrix_D1 = hstack([eye(n - 1), csr_matrix((n - 1, 1))])
+                matrix_D1 = csr_matrix(kron(eye(second_dim_len), sub_matrix_D1))
+                D1 = pybamm.Matrix(matrix_D1) @ array
+
+                # Matrix to extract the node values running from the second node
+                # to the final node in the primary dimension  (D_2 in the
+                # definiton of the harmonic mean)
+                sub_matrix_D2 = hstack([csr_matrix((n - 1, 1)), eye(n - 1)])
+                matrix_D2 = csr_matrix(kron(eye(second_dim_len), sub_matrix_D2))
+                D2 = pybamm.Matrix(matrix_D2) @ array
+
+                # Compute weight beta
+                dx = submesh.d_edges
+                sub_beta = (dx[:-1] / (dx[1:] + dx[:-1]))[:, np.newaxis]
+                beta = pybamm.Array(np.kron(np.ones((second_dim_len, 1)), sub_beta))
+
+                # Compute harmonic mean on internal edges
+                # Note: add small number to denominator to regularise D_eff
+                D_eff = D1 * D2 / (D2 * beta + D1 * (1 - beta) + 1e-16)
+
+                # Matrix to pad zeros at the beginning and end of the array where
+                # the exterior edge values will be added
+                sub_matrix = vstack(
+                    [csr_matrix((1, n - 1)), eye(n - 1), csr_matrix((1, n - 1))]
+                )
+
+                # Generate full matrix from the submatrix
+                # Convert to csr_matrix so that we can take the index (row-slicing),
+                # which is not supported by the default kron format
+                # Note that this makes column-slicing inefficient, but this should
+                # not be an issue
+                matrix = csr_matrix(kron(eye(second_dim_len), sub_matrix))
+
+                return (
+                    pybamm.Matrix(edges_matrix) @ array + pybamm.Matrix(matrix) @ D_eff
+                )
+
+            elif shift_key == "edge to node":
+                # Matrix to extract the edge values running from the first edge
+                # to the penultimate edge in the primary dimension (D_1 in the
+                # definiton of the harmonic mean)
+                sub_matrix_D1 = hstack([eye(n), csr_matrix((n, 1))])
+                matrix_D1 = csr_matrix(kron(eye(second_dim_len), sub_matrix_D1))
+                D1 = pybamm.Matrix(matrix_D1) @ array
+
+                # Matrix to extract the edge values running from the second edge
+                # to the final edge in the primary dimension  (D_2 in the
+                # definiton of the harmonic mean)
+                sub_matrix_D2 = hstack([csr_matrix((n, 1)), eye(n)])
+                matrix_D2 = csr_matrix(kron(eye(second_dim_len), sub_matrix_D2))
+                D2 = pybamm.Matrix(matrix_D2) @ array
+
+                # Compute weight beta
+                dx0 = submesh.nodes[0] - submesh.edges[0]  # first edge to node
+                dxN = submesh.edges[-1] - submesh.nodes[-1]  # last node to edge
+                dx = np.concatenate(([dx0], submesh.d_nodes, [dxN]))
+                sub_beta = (dx[:-1] / (dx[1:] + dx[:-1]))[:, np.newaxis]
+                beta = pybamm.Array(np.kron(np.ones((second_dim_len, 1)), sub_beta))
+
+                # Compute harmonic mean on nodes
+                # Note: add small number to denominator to regularise D_eff
+                D_eff = D1 * D2 / (D2 * beta + D1 * (1 - beta) + 1e-16)
+
+                return D_eff
+
+            else:
+                raise ValueError("shift key '{}' not recognised".format(shift_key))
+
         # If discretised_symbol evaluates to number there is no need to average
         if discretised_symbol.evaluates_to_number():
             out = discretised_symbol
-        else:
+        elif method == "arithmetic":
             out = arithmetic_mean(discretised_symbol)
-
+        elif method == "harmonic":
+            out = harmonic_mean(discretised_symbol)
+        else:
+            raise ValueError("method '{}' not recognised".format(method))
         return out


Test patch:
diff --git a/tests/unit/test_spatial_methods/test_finite_volume/test_finite_volume.py b/tests/unit/test_spatial_methods/test_finite_volume/test_finite_volume.py
index 6c0100138..96de8693e 100644
--- a/tests/unit/test_spatial_methods/test_finite_volume/test_finite_volume.py
+++ b/tests/unit/test_spatial_methods/test_finite_volume/test_finite_volume.py
@@ -23,20 +23,33 @@ class TestFiniteVolume(unittest.TestCase):
 
         # node to edge
         c = pybamm.Vector(np.ones(n), domain=["negative electrode"])
-        diffusivity_c = fin_vol.node_to_edge(c)
-        np.testing.assert_array_equal(diffusivity_c.evaluate(), np.ones((n + 1, 1)))
+        diffusivity_c_ari = fin_vol.node_to_edge(c, method="arithmetic")
+        np.testing.assert_array_equal(diffusivity_c_ari.evaluate(), np.ones((n + 1, 1)))
+        diffusivity_c_har = fin_vol.node_to_edge(c, method="harmonic")
+        np.testing.assert_array_equal(diffusivity_c_har.evaluate(), np.ones((n + 1, 1)))
 
         # edge to node
         d = pybamm.StateVector(slice(0, n + 1), domain=["negative electrode"])
         y_test = np.ones(n + 1)
-        diffusivity_d = fin_vol.edge_to_node(d)
+        diffusivity_d_ari = fin_vol.edge_to_node(d, method="arithmetic")
         np.testing.assert_array_equal(
-            diffusivity_d.evaluate(None, y_test), np.ones((n, 1))
+            diffusivity_d_ari.evaluate(None, y_test), np.ones((n, 1))
+        )
+        diffusivity_d_har = fin_vol.edge_to_node(d, method="harmonic")
+        np.testing.assert_array_equal(
+            diffusivity_d_har.evaluate(None, y_test), np.ones((n, 1))
         )
 
         # bad shift key
         with self.assertRaisesRegex(ValueError, "shift key"):
-            fin_vol.shift(c, "bad shift key")
+            fin_vol.shift(c, "bad shift key", "arithmetic")
+
+        with self.assertRaisesRegex(ValueError, "shift key"):
+            fin_vol.shift(c, "bad shift key", "harmonic")
+
+        # bad method
+        with self.assertRaisesRegex(ValueError, "method"):
+            fin_vol.shift(c, "shift key", "bad method")
 
     def test_concatenation(self):
         mesh = get_mesh_for_testing()