r"""
Constitutive Relations
======================

When solving PDEs using the finite volume approach, inner products may
contain constitutive relations; examples include Ohm's law and Hooke's law.
For this class of inner products, you will learn how to:

    - Construct the inner-product matrix in the case of isotropic and anisotropic constitutive relations
    - Construct the inverse of the inner-product matrix
    - Work with constitutive relations defined by the reciprocal of a parameter

Let :math:`\vec{J}` and :math:`\vec{E}` be two physically related
quantities. If their relationship is isotropic (defined by a constant
:math:`\sigma`), then the constitutive relation is given by:

.. math::
    \vec{J} = \sigma \vec{E}

The inner product between a vector :math:`\vec{v}` and the right-hand side
of this expression is given by:

.. math::
    (\vec{v}, \sigma \vec{E} ) = \int_\Omega \vec{v} \cdot \sigma \vec{E} \, dv

Just like in the previous tutorial, we would like to approximate the inner
product numerically using an *inner-product matrix* such that:

.. math::
    (\vec{v}, \sigma \vec{E} ) \approx \mathbf{v^T M_\sigma e}

where the inner product matrix :math:`\mathbf{M_\sigma}` now depends on:

    1. the dimensions and discretization of the mesh
    2. where :math:`\mathbf{v}` and :math:`\mathbf{e}` live
    3. the spatial distribution of the property :math:`\sigma`

In the case of anisotropy, the constitutive relations are defined by a tensor
(:math:`\Sigma`). Here, the constitutive relation is of the form:

.. math::
    \vec{J} = \Sigma \vec{E}

where

.. math::
    \Sigma = \begin{bmatrix} \sigma_{1} & \sigma_{4} & \sigma_{5} \\
    \sigma_{4} & \sigma_{2} & \sigma_{6} \\
    \sigma_{5} & \sigma_{6} & \sigma_{3} \end{bmatrix}

Is symmetric and defined by 6 independent parameters. The inner product between
a vector :math:`\vec{v}` and the right-hand side of this expression is given
by:

.. math::
    (\vec{v}, \Sigma \vec{E} ) = \int_\Omega \vec{v} \cdot \Sigma \vec{E}  \, dv

Once again we would like to approximate the inner product numerically using an
*inner-product matrix* :math:`\mathbf{M_\Sigma}` such that:

.. math::
    (\vec{v}, \Sigma \vec{E} ) \approx \mathbf{v^T M_\Sigma e}

"""

####################################################
#
# Import Packages
# ---------------
#
# Here we import the packages required for this tutorial
#

from discretize import TensorMesh
import numpy as np
import matplotlib.pyplot as plt

rng = np.random.default_rng(87236)

# sphinx_gallery_thumbnail_number = 1

#####################################################
# Inner Product for a Single Cell
# -------------------------------
#
# Here we compare the inner product matricies for a single cell when the
# constitutive relationship is:
#
#     - **isotropic:** :math:`\sigma_1 = \sigma_2 = \sigma_3 = \sigma` and :math:`\sigma_4 = \sigma_5 = \sigma_6 = 0`; e.g. :math:`\vec{J} = \sigma \vec{E}`
#     - **diagonal anisotropic:** independent parameters :math:`\sigma_1, \sigma_2, \sigma_3` and :math:`\sigma_4 = \sigma_5 = \sigma_6 = 0`
#     - **fully anisotropic:** independent parameters :math:`\sigma_1, \sigma_2, \sigma_3, \sigma_4, \sigma_5, \sigma_6`
#
# When approximating the inner product according to the finite volume approach,
# the constitutive parameters are defined at cell centers; even if the
# fields/fluxes live at cell edges/faces. As we will see, inner-product
# matricies are generally diagonal; except for in the fully anisotropic case
# where the inner product matrix contains a significant number of non-diagonal
# entries.
#

# Create a single 3D cell
h = np.ones(1)
mesh = TensorMesh([h, h, h])

# Define 6 constitutive parameters for the cell
sig1, sig2, sig3, sig4, sig5, sig6 = 6, 5, 4, 3, 2, 1

# Isotropic case
sig = sig1 * np.ones((1, 1))
sig_tensor_1 = np.diag(sig1 * np.ones(3))
Me1 = mesh.get_edge_inner_product(sig)  # Edges inner product matrix
Mf1 = mesh.get_face_inner_product(sig)  # Faces inner product matrix

# Diagonal anisotropic
sig = np.c_[sig1, sig2, sig3]
sig_tensor_2 = np.diag(np.array([sig1, sig2, sig3]))
Me2 = mesh.get_edge_inner_product(sig)
Mf2 = mesh.get_face_inner_product(sig)

# Full anisotropic
sig = np.c_[sig1, sig2, sig3, sig4, sig5, sig6]
sig_tensor_3 = np.diag(np.array([sig1, sig2, sig3]))
sig_tensor_3[(0, 1), (1, 0)] = sig4
sig_tensor_3[(0, 2), (2, 0)] = sig5
sig_tensor_3[(1, 2), (2, 1)] = sig6
Me3 = mesh.get_edge_inner_product(sig)
Mf3 = mesh.get_face_inner_product(sig)

# Plotting matrix entries
fig = plt.figure(figsize=(12, 12))

ax1 = fig.add_subplot(331)
ax1.imshow(sig_tensor_1)
ax1.set_title("Property Tensor (isotropic)")

ax2 = fig.add_subplot(332)
ax2.imshow(sig_tensor_2)
ax2.set_title("Property Tensor (diagonal anisotropic)")

ax3 = fig.add_subplot(333)
ax3.imshow(sig_tensor_3)
ax3.set_title("Property Tensor (full anisotropic)")

ax4 = fig.add_subplot(334)
ax4.imshow(Mf1.todense())
ax4.set_title("M-faces Matrix (isotropic)")

ax5 = fig.add_subplot(335)
ax5.imshow(Mf2.todense())
ax5.set_title("M-faces Matrix (diagonal anisotropic)")

ax6 = fig.add_subplot(336)
ax6.imshow(Mf3.todense())
ax6.set_title("M-faces Matrix (full anisotropic)")

ax7 = fig.add_subplot(337)
ax7.imshow(Me1.todense())
ax7.set_title("M-edges Matrix (isotropic)")

ax8 = fig.add_subplot(338)
ax8.imshow(Me2.todense())
ax8.set_title("M-edges Matrix (diagonal anisotropic)")

ax9 = fig.add_subplot(339)
ax9.imshow(Me3.todense())
ax9.set_title("M-edges Matrix (full anisotropic)")


#############################################################
# Spatially Variant Parameters
# ----------------------------
#
# In practice, the parameter :math:`\sigma` or tensor :math:`\Sigma` will
# vary spatially. In this case, we define the parameter
# :math:`\sigma` (or parameters :math:`\Sigma`) for each cell. When
# creating the inner product matrix, we enter these parameters as
# a numpy array. This is demonstrated below. Properties of the resulting
# inner product matricies are discussed.
#

# Create a small 3D mesh
h = np.ones(5)
mesh = TensorMesh([h, h, h])

# Isotropic case: (nC, ) numpy array
sig = rng.random(mesh.nC)  # sig for each cell
Me1 = mesh.get_edge_inner_product(sig)  # Edges inner product matrix
Mf1 = mesh.get_face_inner_product(sig)  # Faces inner product matrix

# Linear case: (nC, dim) numpy array
sig = rng.random((mesh.nC, mesh.dim))
Me2 = mesh.get_edge_inner_product(sig)
Mf2 = mesh.get_face_inner_product(sig)

# Anisotropic case: (nC, 3) for 2D and (nC, 6) for 3D
sig = rng.random((mesh.nC, 6))
Me3 = mesh.get_edge_inner_product(sig)
Mf3 = mesh.get_face_inner_product(sig)

# Properties of inner product matricies
print("\n FACE INNER PRODUCT MATRIX")
print("- Number of faces              :", mesh.nF)
print("- Dimensions of operator       :", str(mesh.nF), "x", str(mesh.nF))
print("- Number non-zero (isotropic)  :", str(Mf1.nnz))
print("- Number non-zero (linear)     :", str(Mf2.nnz))
print("- Number non-zero (anisotropic):", str(Mf3.nnz), "\n")

print("\n EDGE INNER PRODUCT MATRIX")
print("- Number of faces              :", mesh.nE)
print("- Dimensions of operator       :", str(mesh.nE), "x", str(mesh.nE))
print("- Number non-zero (isotropic)  :", str(Me1.nnz))
print("- Number non-zero (linear)     :", str(Me2.nnz))
print("- Number non-zero (anisotropic):", str(Me3.nnz), "\n")


#############################################################
# Inverse
# -------
#
# The final discretized system using the finite volume method may contain
# the inverse of the inner-product matrix. Here we show how to call this
# using the *invert_matrix* keyword argument.
#
# For the isotropic and diagonally anisotropic cases, the inner product matrix
# is diagonal. As a result, its inverse can be easily formed. For the full
# anisotropic case however, we cannot expicitly form the inverse because the
# inner product matrix contains a significant number of off-diagonal elements.
#
# For the isotropic and diagonal anisotropic cases we can form
# :math:`\mathbf{M}^{-1}` then apply it to a vector using the :math:`*`
# operator. For the full anisotropic case, we must form the inner product
# matrix and do a numerical solve.
#

# Create a small 3D mesh
h = np.ones(5)
mesh = TensorMesh([h, h, h])

# Isotropic case: (nC, ) numpy array
sig = rng.random(mesh.nC)
Me1_inv = mesh.get_edge_inner_product(sig, invert_matrix=True)
Mf1_inv = mesh.get_face_inner_product(sig, invert_matrix=True)

# Diagonal anisotropic: (nC, dim) numpy array
sig = rng.random((mesh.nC, mesh.dim))
Me2_inv = mesh.get_edge_inner_product(sig, invert_matrix=True)
Mf2_inv = mesh.get_face_inner_product(sig, invert_matrix=True)

# Full anisotropic: (nC, 3) for 2D and (nC, 6) for 3D
sig = rng.random((mesh.nC, 6))
Me3 = mesh.get_edge_inner_product(sig)
Mf3 = mesh.get_face_inner_product(sig)


###########################################################################
# Reciprocal Properties
# ---------------------
#
# At times, the constitutive relation may be defined by the reciprocal of
# a parameter (:math:`\rho`). Here we demonstrate how inner product matricies
# can be formed using the keyword argument *invert_model*. We will do this for a
# single cell and plot the matrix elements. We can easily extend this to
# a mesh comprised of many cells.
#
# In this case, the constitutive relation is given by:
#
# .. math::
#     \vec{J} = \frac{1}{\rho} \vec{E}
#
# The inner product between a vector :math:`\vec{v}` and the right-hand side
# of the expression is given by:
#
# .. math::
#     (\vec{v}, \rho^{-1} \vec{E} ) = \int_\Omega \vec{v} \cdot \rho^{-1} \vec{E} \, dv
#
# where the inner product is approximated using an inner product matrix
# :math:`\mathbf{M_{\rho^{-1}}}` as follows:
#
# .. math::
#     (\vec{v}, \rho^{-1} \vec{E} ) \approx \mathbf{v^T M_{\rho^{-1}} e}
#
# In the case that the constitutive relation is defined by a
# tensor :math:`P`, e.g.:
#
# .. math::
#     \vec{J} = P \vec{E}
#
# where
#
# .. math::
#     P = \begin{bmatrix} \rho_{1}^{-1} & \rho_{4}^{-1} & \rho_{5}^{-1} \\
#     \rho_{4}^{-1} & \rho_{2}^{-1} & \rho_{6}^{-1} \\
#     \rho_{5}^{-1} & \rho_{6}^{-1} & \rho_{3}^{-1} \end{bmatrix}
#
# The inner product between a vector :math:`\vec{v}` and the right-hand side of
# this expression is given by:
#
# .. math::
#     (\vec{v}, P \vec{E} ) = \int_\Omega \vec{v} \cdot P \vec{E}  \, dv
#
# Once again we would like to approximate the inner product numerically using an
# *inner-product matrix* :math:`\mathbf{M_P}` such that:
#
# .. math::
#     (\vec{v}, P \vec{E} ) \approx \mathbf{v^T M_P e}
#
# Here we demonstrate how to form the inner-product matricies
# :math:`\mathbf{M_{\rho^{-1}}}` and :math:`\mathbf{M_P}`.
#

# Create a small 3D mesh
h = np.ones(1)
mesh = TensorMesh([h, h, h])

# Define 6 constitutive parameters for the cell
rho1, rho2, rho3, rho4, rho5, rho6 = (
    1.0 / 6.0,
    1.0 / 5.0,
    1.0 / 4.0,
    1.0 / 3.0,
    1.0 / 2.0,
    1,
)

# Isotropic case
rho = rho1 * np.ones((1, 1))
Me1 = mesh.get_edge_inner_product(rho, invert_model=True)  # Edges inner product matrix
Mf1 = mesh.get_face_inner_product(rho, invert_model=True)  # Faces inner product matrix

# Diagonal anisotropic case
rho = np.c_[rho1, rho2, rho3]
Me2 = mesh.get_edge_inner_product(rho, invert_model=True)
Mf2 = mesh.get_face_inner_product(rho, invert_model=True)

# Full anisotropic case
rho = np.c_[rho1, rho2, rho3, rho4, rho5, rho6]
Me3 = mesh.get_edge_inner_product(rho, invert_model=True)
Mf3 = mesh.get_face_inner_product(rho, invert_model=True)

# Plotting matrix entries
fig = plt.figure(figsize=(14, 9))

ax1 = fig.add_subplot(231)
ax1.imshow(Mf1.todense())
ax1.set_title("Isotropic (Faces)")

ax2 = fig.add_subplot(232)
ax2.imshow(Mf2.todense())
ax2.set_title("Diagonal Anisotropic (Faces)")

ax3 = fig.add_subplot(233)
ax3.imshow(Mf3.todense())
ax3.set_title("Full Anisotropic (Faces)")

ax4 = fig.add_subplot(234)
ax4.imshow(Me1.todense())
ax4.set_title("Isotropic (Edges)")

ax5 = fig.add_subplot(235)
ax5.imshow(Me2.todense())
ax5.set_title("Diagonal Anisotropic (Edges)")

ax6 = fig.add_subplot(236)
ax6.imshow(Me3.todense())
ax6.set_title("Full Anisotropic (Edges)")