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

$\vec{J} = \sigma \vec{E}$

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

$(\vec{v}, \sigma \vec{E} ) = \int_\Omega \vec{v} \cdot \sigma \vec{E} \, dv$

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

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

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

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

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

$\vec{J} = \Sigma \vec{E}$

where

\begin{align}\begin{aligned}\Sigma = \begin{bmatrix} \sigma_{1} & \sigma_{4} & \sigma_{5}\\\sigma_{4} & \sigma_{2} & \sigma_{6}\\\sigma_{5} & \sigma_{6} & \sigma_{3} \end{bmatrix}\end{aligned}\end{align}

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

$(\vec{v}, \Sigma \vec{E} ) = \int_\Omega \vec{v} \cdot \Sigma \vec{E} \, dv$

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

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

## Import Packages¶

Here we import the packages required for this tutorial

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

# 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: $$\sigma_1 = \sigma_2 = \sigma_3 = \sigma$$ and $$\sigma_4 = \sigma_5 = \sigma_6 = 0$$; e.g. $$\vec{J} = \sigma \vec{E}$$
• diagonal anisotropic: independent parameters $$\sigma_1, \sigma_2, \sigma_3$$ and $$\sigma_4 = \sigma_5 = \sigma_6 = 0$$
• fully anisotropic: independent parameters $$\sigma_1, \sigma_2, \sigma_3, \sigma_4, \sigma_5, \sigma_6$$

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.getEdgeInnerProduct(sig)  # Edges inner product matrix
Mf1 = mesh.getFaceInnerProduct(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.getEdgeInnerProduct(sig)
Mf2 = mesh.getFaceInnerProduct(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.getEdgeInnerProduct(sig)
Mf3 = mesh.getFaceInnerProduct(sig)

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

ax1.imshow(sig_tensor_1)
ax1.set_title('Property Tensor (isotropic)')

ax2.imshow(sig_tensor_2)
ax2.set_title('Property Tensor (diagonal anisotropic)')

ax3.imshow(sig_tensor_3)
ax3.set_title('Property Tensor (full anisotropic)')

ax4.imshow(Mf1.todense())
ax4.set_title('M-faces Matrix (isotropic)')

ax5.imshow(Mf2.todense())
ax5.set_title('M-faces Matrix (diagonal anisotropic)')

ax6.imshow(Mf3.todense())
ax6.set_title('M-faces Matrix (full anisotropic)')

ax7.imshow(Me1.todense())
ax7.set_title('M-edges Matrix (isotropic)')

ax8.imshow(Me2.todense())
ax8.set_title('M-edges Matrix (diagonal anisotropic)')

ax9.imshow(Me3.todense())
ax9.set_title('M-edges Matrix (full anisotropic)')


Out:

Text(0.5, 1.0, 'M-edges Matrix (full anisotropic)')


## Spatially Variant Parameters¶

In practice, the parameter $$\sigma$$ or tensor $$\Sigma$$ will vary spatially. In this case, we define the parameter $$\sigma$$ (or parameters $$\Sigma$$) 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 = np.random.rand(mesh.nC)        # sig for each cell
Me1 = mesh.getEdgeInnerProduct(sig)  # Edges inner product matrix
Mf1 = mesh.getFaceInnerProduct(sig)  # Faces inner product matrix

# Linear case: (nC, dim) numpy array
sig = np.random.rand(mesh.nC, mesh.dim)
Me2 = mesh.getEdgeInnerProduct(sig)
Mf2 = mesh.getFaceInnerProduct(sig)

# Anisotropic case: (nC, 3) for 2D and (nC, 6) for 3D
sig = np.random.rand(mesh.nC, 6)
Me3 = mesh.getEdgeInnerProduct(sig)
Mf3 = mesh.getFaceInnerProduct(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')


Out:

 FACE INNER PRODUCT MATRIX
- Number of faces              : 450
- Dimensions of operator       : 450 x 450
- Number non-zero (isotropic)  : 450
- Number non-zero (linear)     : 450
- Number non-zero (anisotropic): 3450

EDGE INNER PRODUCT MATRIX
- Number of faces              : 540
- Dimensions of operator       : 540 x 540
- Number non-zero (isotropic)  : 540
- Number non-zero (linear)     : 540
- Number non-zero (anisotropic): 4140


## 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 invMat 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 $$\mathbf{M}^{-1}$$ then apply it to a vector using the $$*$$ 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 = np.random.rand(mesh.nC)
Me1_inv = mesh.getEdgeInnerProduct(sig, invMat=True)
Mf1_inv = mesh.getFaceInnerProduct(sig, invMat=True)

# Diagonal anisotropic: (nC, dim) numpy array
sig = np.random.rand(mesh.nC, mesh.dim)
Me2_inv = mesh.getEdgeInnerProduct(sig, invMat=True)
Mf2_inv = mesh.getFaceInnerProduct(sig, invMat=True)

# Full anisotropic: (nC, 3) for 2D and (nC, 6) for 3D
sig = np.random.rand(mesh.nC, 6)
Me3 = mesh.getEdgeInnerProduct(sig)
Mf3 = mesh.getFaceInnerProduct(sig)


## Reciprocal Properties¶

At times, the constitutive relation may be defined by the reciprocal of a parameter ($$\rho$$). Here we demonstrate how inner product matricies can be formed using the keyword argument invProp. 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:

$\vec{J} = \frac{1}{\rho} \vec{E}$

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

$(\vec{v}, \rho^{-1} \vec{E} ) = \int_\Omega \vec{v} \cdot \rho^{-1} \vec{E} \, dv$

where the inner product is approximated using an inner product matrix $$\mathbf{M_{\rho^{-1}}}$$ as follows:

$(\vec{v}, \rho^{-1} \vec{E} ) \approx \mathbf{v^T M_{\rho^{-1}} e}$

In the case that the constitutive relation is defined by a tensor $$P$$, e.g.:

$\vec{J} = P \vec{E}$

where

$\begin{split}P = \begin{bmatrix} \rho_{1}^{-1} & \rho_{4}^{-1} & \rho_{5}^{-1} \\ \rho_{4}^{-1} & \rho_{2}^{-1} & \rho_{6}^{-1} \\ \rho_{5}^{-1} & \rho_{6}^{-1} & \rho_{3}^{-1} \end{bmatrix}\end{split}$

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

$(\vec{v}, P \vec{E} ) = \int_\Omega \vec{v} \cdot P \vec{E} \, dv$

Once again we would like to approximate the inner product numerically using an inner-product matrix $$\mathbf{M_P}$$ such that:

$(\vec{v}, P \vec{E} ) \approx \mathbf{v^T M_P e}$

Here we demonstrate how to form the inner-product matricies $$\mathbf{M_{\rho^{-1}}}$$ and $$\mathbf{M_P}$$.

# 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./6., 1./5., 1./4., 1./3., 1./2., 1

# Isotropic case
rho = rho1*np.ones((1, 1))
Me1 = mesh.getEdgeInnerProduct(rho, invProp=True)  # Edges inner product matrix
Mf1 = mesh.getFaceInnerProduct(rho, invProp=True)  # Faces inner product matrix

# Diagonal anisotropic case
rho = np.c_[rho1, rho2, rho3]
Me2 = mesh.getEdgeInnerProduct(rho, invProp=True)
Mf2 = mesh.getFaceInnerProduct(rho, invProp=True)

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

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

ax1.imshow(Mf1.todense())
ax1.set_title('Isotropic (Faces)')

ax2.imshow(Mf2.todense())
ax2.set_title('Diagonal Anisotropic (Faces)')

ax3.imshow(Mf3.todense())
ax3.set_title('Full Anisotropic (Faces)')

ax4.imshow(Me1.todense())
ax4.set_title('Isotropic (Edges)')

ax5.imshow(Me2.todense())
ax5.set_title('Diagonal Anisotropic (Edges)')


Text(0.5, 1.0, 'Full Anisotropic (Edges)')