From ae501a8a32c57aeba2e107193b5a4fc5a363960b Mon Sep 17 00:00:00 2001 From: dccowan Date: Tue, 17 Nov 2020 23:06:39 -0800 Subject: [PATCH 01/35] Update basic inner products theory --- docs/content/finite_volume.rst | 121 +++++++- docs/content/inner_products.rst | 306 +++++++++++-------- docs/content/inner_products_basic.rst | 282 +++++++++++++++++ docs/content/inner_products_constitutive.rst | 6 + docs/images/edge_discretization.PNG | Bin 0 -> 41008 bytes docs/images/face_discretization.PNG | Bin 0 -> 50289 bytes 6 files changed, 574 insertions(+), 141 deletions(-) create mode 100644 docs/content/inner_products_basic.rst create mode 100644 docs/content/inner_products_constitutive.rst create mode 100644 docs/images/edge_discretization.PNG create mode 100644 docs/images/face_discretization.PNG diff --git a/docs/content/finite_volume.rst b/docs/content/finite_volume.rst index 3703dd5cc..493bc8c99 100644 --- a/docs/content/finite_volume.rst +++ b/docs/content/finite_volume.rst @@ -1,19 +1,116 @@ .. _api_FiniteVolume: -Finite Volume -************* - -Any numerical implementation requires the discretization of continuous -functions into discrete approximations. These approximations are typically -organized in a mesh, which defines boundaries, locations, and connectivity. Of -specific interest to geophysical simulations, we require that averaging, -interpolation and differential operators be defined for any mesh. In SimPEG, -we have implemented a staggered mimetic finite volume approach (`Hyman and -Shashkov, 1999 `_). This -approach requires the definitions of variables at either cell-centers, nodes, -faces, or edges as seen in the figure below. +Finite Volume Method Concept +**************************** + +Introduction +------------ + +The finite volume method is a method for numerically solving partial differential +equations. Implementation of the finite volume method requires the discretization +of continuous functions, resulting in discrete approximations. The discrete +approximations are typically organized on a numerical grid (or mesh), +which defines boundaries, locations, and connectivity. The final product of the approach +is a linear system of equations :math:`\mathbf{Ax=b}` that can be solved to compute +the discrete approximation of a desired quantity. .. image:: ../images/finitevolrealestate.png :width: 400 px :alt: FiniteVolume :align: center + +In SimPEG, we have implemented a staggered mimetic finite volume approach (`Hyman and +Shashkov, 1999 `_). This +approach requires the definitions of variables at either cell-centers, nodes, +faces, or edges as seen in the figure above. For geophysical simulations, we also +require that the following operations be discretized: + + - Averaging between nodes, faces, edges and cell-centers + - Interpolation from nodes, faces, edges or cell-centers to other locations + - Differential operators such as the gradient, divergence and curl + + +Examples +-------- + +Below are several examples of the final linear system resulting from the finite volume approach. +The details of constructing the final solution are not discussed here. Details for constructing +the final linear system for a PDE are discussed in :ref:`inner products ` + +Direct Current Resistivity +^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The governing equation for the direct current resistivity problem is given by: + +.. math:: + \nabla \cdot \sigma \nabla \phi = -q_s + +where + + - :math:`\phi` is the electric potential + - :math:`\sigma` is the electrical conductivity within the domain + - :math:`q_s` is a general representation of the source term + - :math:`\nabla` is the gradient operator + - :math:`\nabla \cdot` is the divergence operator + +If we choose to define the discrete representation of the electric potential on the nodes, +the solution for the electric potentials after applying the finite volume approach is given by: + +.. math:: + \boldsymbol{[G^T M_{\sigma e} G ]} \boldsymbol{\phi} = \mathbf{q_s} + +where :math:`\boldsymbol{G^T M_{\sigma e} G }` is a sparse matrix and + + - :math:`\boldsymbol{\phi}` is the discrete approximation to the electric potentials on the nodes + - :math:`\boldsymbol{G}` is a discrete gradient operator + - :math:`\boldsymbol{G^T}` is the transpose of :math:`\boldsymbol{G}` but acts as a divergence operator + - :math:`\boldsymbol{M_{\sigma e}}` is the mass matrix + - :math:`\boldsymbol{q_s}` is the discrete representation of the source term on the nodes + + +Frequency Domain Electromagnetics +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The governing equations for the frequency domain electromagnetic problems, +for a source current, can be expressed using Maxwell's equations: + +.. math:: + \begin{align} + &\nabla \times \mu^{-1} \vec{B} - \sigma \vec{E} = \vec{J}_s \\ + &\nabla \times \vec{E} = - i\omega \vec{B} + \end{align} + +where + + - :math:`\vec{E}` is the electric field + - :math:`\vec{B}` is the magnetic flux density + - :math:`\vec{J}_s` is a general representation of the source term + - :math:`\sigma` is the electrical conductivity within the domain + - :math:`\mu` is the magnetic permeability within the domain + - :math:`\omega` is the angular frequency + - :math:`\nabla \times` is the curl operator + +Here we choose to define the discrete representation of the electric field on edges +and the discrete representation of the magnetic flux density on faces. +The solution for the electric potentials after applying the finite volume approach is given by: + +.. math:: + \begin{align} + \boldsymbol{C^T M_{\mu f} \, b } - \boldsymbol{M_{\sigma e} \, e} = \mathbf{j_s} \\ + \mathbf{C \, e} = -i \omega \mathbf{b} + \end{align} + +which can be combined to form a single linear system: + +.. math:: + \boldsymbol{[C^T M_{\mu f} C } + i\omega \boldsymbol{M_{\sigma e}]} \mathbf{e} = -i \omega \mathbf{j_s} + +where :math:`\boldsymbol{C^T M_{\mu f} C } + i\omega \boldsymbol{M_{\sigma e}}` is a sparse matrix and + + - :math:`\boldsymbol{e}` is the discrete approximation to the electric field on edges + - :math:`\boldsymbol{b}` is the discrete approximation to the magnetic flux density on faces + - :math:`\boldsymbol{C}` is a discrete curl operator from edges to faces + - :math:`\boldsymbol{M_{\sigma e}}` is the mass matrix for electrical conductivity + - :math:`\boldsymbol{M_{\mu f}}` is the mass matrix for inverse of the magnetic permeability + - :math:`\boldsymbol{j_s}` is the discrete representation of the source term on the edges + diff --git a/docs/content/inner_products.rst b/docs/content/inner_products.rst index 6c3690d65..f9c13856f 100644 --- a/docs/content/inner_products.rst +++ b/docs/content/inner_products.rst @@ -1,6 +1,49 @@ +.. _inner_products: + Inner Products ************** +Inner products provide the building blocks for discretizing and solving PDEs with the +finite volume method. For scalar quantities :math:`\psi` and :math:`\phi`, the inner +product is given by: + +.. math:: + (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv + +And for two vector quantities :math:`\vec{u}` and :math:`\vec{v}`, the inner product is +given by: + +.. math:: + (\vec{u}, \vec{v}) = \int_\Omega \vec{u} \cdot \vec{v} \, dv + +.. To implement the finite volume method, we: + +.. 1. take the inner product between each differential equation and a test function +.. 2. re-express the inner product using the weak formulation by applying the divergence theorem when necessary +.. 3. formulate a discrete approximation to each inner product which respects the boundary conditions on our variables + +When implementing the finite volume method, we construct discrete approximations to inner products. +The approximations to inner products are combined and simplified to form a linear system in terms +of an unknown variable, then solved. The approximation to each inner product depends on the quantities +and differential operators present in the inner product. Here, we demonstrate how to formulate the +discrete approximation for several classes of inner products. + +**Contents:** + +.. toctree:: + :maxdepth: 1 + + inner_products_basic + inner_products_constitutive + + + + +xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx + + + + By using the weak formulation of many of the PDEs in geophysical applications, we can rapidly develop discretizations. Much of this work, however, needs a good understanding of how to approximate inner products on our discretized @@ -18,8 +61,13 @@ where a and b are either scalars or vectors. for meshes and cannot run on its own. -Example problem for DC resistivity ----------------------------------- + + + + + + +**Example problem for DC resistivity** We will start with the formulation of the Direct Current (DC) resistivity problem in geophysics. @@ -151,143 +199,143 @@ matrices look a bit different because we have 12 edges in 3D instead of just 6 faces. The interface to the code is exactly the same. -Defining Tensor Properties --------------------------- - -**For 3D:** - -Depending on the number of columns (either 1, 3, or 6) of mu, the material -property is interpreted as follows: - -.. math:: - - \vec{\mu} = \left[\begin{matrix} \mu_{1} & 0 & 0 \\ 0 & \mu_{1} & 0 \\ 0 & 0 & \mu_{1} \end{matrix}\right] - - \vec{\mu} = \left[\begin{matrix} \mu_{1} & 0 & 0 \\ 0 & \mu_{2} & 0 \\ 0 & 0 & \mu_{3} \end{matrix}\right] - - \vec{\mu} = \left[\begin{matrix} \mu_{1} & \mu_{4} & \mu_{5} \\ \mu_{4} & \mu_{2} & \mu_{6} \\ \mu_{5} & \mu_{6} & \mu_{3} \end{matrix}\right] - -**For 2D:** - - Depending on the number of columns (either 1, 2, or 3) of mu, the material property is interpreted as follows: - -.. math:: - \vec{\mu} = \left[\begin{matrix} \mu_{1} & 0 \\ 0 & \mu_{1} \end{matrix}\right] - - \vec{\mu} = \left[\begin{matrix} \mu_{1} & 0 \\ 0 & \mu_{2} \end{matrix}\right] - - \vec{\mu} = \left[\begin{matrix} \mu_{1} & \mu_{3} \\ \mu_{3} & \mu_{2} \end{matrix}\right] - - -Structure of Matrices ---------------------- - -Both the isotropic, and anisotropic material properties result in a diagonal mass matrix. -Which is nice and easy to invert if necessary, however, in the fully anisotropic case which is not aligned with the grid, the matrix is not diagonal. This can be seen for a 3D mesh in the figure below. +.. Defining Tensor Properties +.. -------------------------- -.. plot:: +.. **For 3D:** - import discretize - import numpy as np - import matplotlib.pyplot as plt - mesh = discretize.TensorMesh([10,50,3]) - m1 = np.random.rand(mesh.nC) - m2 = np.random.rand(mesh.nC,3) - m3 = np.random.rand(mesh.nC,6) - M = list(range(3)) - M[0] = mesh.getFaceInnerProduct(m1) - M[1] = mesh.getFaceInnerProduct(m2) - M[2] = mesh.getFaceInnerProduct(m3) - plt.figure(figsize=(13,5)) - for i, lab in enumerate(['Isotropic','Anisotropic','Tensor']): - plt.subplot(131 + i) - plt.spy(M[i],ms=0.5,color='k') - plt.tick_params(axis='both',which='both',labeltop='off',labelleft='off') - plt.title(lab + ' Material Property') - plt.show() +.. Depending on the number of columns (either 1, 3, or 6) of mu, the material +.. property is interpreted as follows: +.. .. math:: -Taking Derivatives ------------------- +.. \vec{\mu} = \left[\begin{matrix} \mu_{1} & 0 & 0 \\ 0 & \mu_{1} & 0 \\ 0 & 0 & \mu_{1} \end{matrix}\right] -We will take the derivative of the fully anisotropic tensor for a 3D mesh, the -other cases are easier and will not be discussed here. Let us start with one -part of the sum which makes up :math:`\mathbf{M}^f_\Sigma` and take the -derivative when this is multiplied by some vector :math:`\mathbf{v}`: +.. \vec{\mu} = \left[\begin{matrix} \mu_{1} & 0 & 0 \\ 0 & \mu_{2} & 0 \\ 0 & 0 & \mu_{3} \end{matrix}\right] -.. math:: - - \mathbf{P}^\top \boldsymbol{\Sigma} \mathbf{Pv} +.. \vec{\mu} = \left[\begin{matrix} \mu_{1} & \mu_{4} & \mu_{5} \\ \mu_{4} & \mu_{2} & \mu_{6} \\ \mu_{5} & \mu_{6} & \mu_{3} \end{matrix}\right] -Here we will let :math:`\mathbf{Pv} = \mathbf{y}` and :math:`\mathbf{y}` will have the form: +.. **For 2D:** -.. math:: +.. Depending on the number of columns (either 1, 2, or 3) of mu, the material property is interpreted as follows: - \mathbf{y} = \mathbf{Pv} = - \left[ - \begin{matrix} - \mathbf{y}_1\\ - \mathbf{y}_2\\ - \mathbf{y}_3\\ - \end{matrix} - \right] +.. .. math:: +.. \vec{\mu} = \left[\begin{matrix} \mu_{1} & 0 \\ 0 & \mu_{1} \end{matrix}\right] -.. math:: +.. \vec{\mu} = \left[\begin{matrix} \mu_{1} & 0 \\ 0 & \mu_{2} \end{matrix}\right] - \mathbf{P}^\top\Sigma\mathbf{y} = - \mathbf{P}^\top - \left[\begin{matrix} - \boldsymbol{\sigma}_{1} & \boldsymbol{\sigma}_{4} & \boldsymbol{\sigma}_{5} \\ - \boldsymbol{\sigma}_{4} & \boldsymbol{\sigma}_{2} & \boldsymbol{\sigma}_{6} \\ - \boldsymbol{\sigma}_{5} & \boldsymbol{\sigma}_{6} & \boldsymbol{\sigma}_{3} - \end{matrix}\right] - \left[ - \begin{matrix} - \mathbf{y}_1\\ - \mathbf{y}_2\\ - \mathbf{y}_3\\ - \end{matrix} - \right] - = - \mathbf{P}^\top - \left[ - \begin{matrix} - \boldsymbol{\sigma}_{1}\circ \mathbf{y}_1 + \boldsymbol{\sigma}_{4}\circ \mathbf{y}_2 + \boldsymbol{\sigma}_{5}\circ \mathbf{y}_3\\ - \boldsymbol{\sigma}_{4}\circ \mathbf{y}_1 + \boldsymbol{\sigma}_{2}\circ \mathbf{y}_2 + \boldsymbol{\sigma}_{6}\circ \mathbf{y}_3\\ - \boldsymbol{\sigma}_{5}\circ \mathbf{y}_1 + \boldsymbol{\sigma}_{6}\circ \mathbf{y}_2 + \boldsymbol{\sigma}_{3}\circ \mathbf{y}_3\\ - \end{matrix} - \right] - -Now it is easy to take the derivative with respect to any one of the -parameters, for example, -:math:`\frac{\partial}{\partial\boldsymbol{\sigma}_1}` +.. \vec{\mu} = \left[\begin{matrix} \mu_{1} & \mu_{3} \\ \mu_{3} & \mu_{2} \end{matrix}\right] -.. math:: - \frac{\partial}{\partial \boldsymbol{\sigma}_1}\left(\mathbf{P}^\top\Sigma\mathbf{y}\right) - = - \mathbf{P}^\top - \left[ - \begin{matrix} - \text{diag}(\mathbf{y}_1)\\ - 0\\ - 0\\ - \end{matrix} - \right] - -Whereas :math:`\frac{\partial}{\partial\boldsymbol{\sigma}_4}`, for -example, is: -.. math:: - \frac{\partial}{\partial \boldsymbol{\sigma}_4}\left(\mathbf{P}^\top\Sigma\mathbf{y}\right) - = - \mathbf{P}^\top - \left[ - \begin{matrix} - \text{diag}(\mathbf{y}_2)\\ - \text{diag}(\mathbf{y}_1)\\ - 0\\ - \end{matrix} - \right] - -These are computed for each of the 8 projections, horizontally concatenated, -and returned. +.. Structure of Matrices +.. --------------------- + +.. Both the isotropic, and anisotropic material properties result in a diagonal mass matrix. +.. Which is nice and easy to invert if necessary, however, in the fully anisotropic case which is not aligned with the grid, the matrix is not diagonal. This can be seen for a 3D mesh in the figure below. + +.. .. plot:: + +.. import discretize +.. import numpy as np +.. import matplotlib.pyplot as plt +.. mesh = discretize.TensorMesh([10,50,3]) +.. m1 = np.random.rand(mesh.nC) +.. m2 = np.random.rand(mesh.nC,3) +.. m3 = np.random.rand(mesh.nC,6) +.. M = list(range(3)) +.. M[0] = mesh.getFaceInnerProduct(m1) +.. M[1] = mesh.getFaceInnerProduct(m2) +.. M[2] = mesh.getFaceInnerProduct(m3) +.. plt.figure(figsize=(13,5)) +.. for i, lab in enumerate(['Isotropic','Anisotropic','Tensor']): +.. plt.subplot(131 + i) +.. plt.spy(M[i],ms=0.5,color='k') +.. plt.tick_params(axis='both',which='both',labeltop='off',labelleft='off') +.. plt.title(lab + ' Material Property') +.. plt.show() + + +.. Taking Derivatives +.. ------------------ + +.. We will take the derivative of the fully anisotropic tensor for a 3D mesh, the +.. other cases are easier and will not be discussed here. Let us start with one +.. part of the sum which makes up :math:`\mathbf{M}^f_\Sigma` and take the +.. derivative when this is multiplied by some vector :math:`\mathbf{v}`: + +.. .. math:: + +.. \mathbf{P}^\top \boldsymbol{\Sigma} \mathbf{Pv} + +.. Here we will let :math:`\mathbf{Pv} = \mathbf{y}` and :math:`\mathbf{y}` will have the form: + +.. .. math:: + +.. \mathbf{y} = \mathbf{Pv} = +.. \left[ +.. \begin{matrix} +.. \mathbf{y}_1\\ +.. \mathbf{y}_2\\ +.. \mathbf{y}_3\\ +.. \end{matrix} +.. \right] + +.. .. math:: + +.. \mathbf{P}^\top\Sigma\mathbf{y} = +.. \mathbf{P}^\top +.. \left[\begin{matrix} +.. \boldsymbol{\sigma}_{1} & \boldsymbol{\sigma}_{4} & \boldsymbol{\sigma}_{5} \\ +.. \boldsymbol{\sigma}_{4} & \boldsymbol{\sigma}_{2} & \boldsymbol{\sigma}_{6} \\ +.. \boldsymbol{\sigma}_{5} & \boldsymbol{\sigma}_{6} & \boldsymbol{\sigma}_{3} +.. \end{matrix}\right] +.. \left[ +.. \begin{matrix} +.. \mathbf{y}_1\\ +.. \mathbf{y}_2\\ +.. \mathbf{y}_3\\ +.. \end{matrix} +.. \right] +.. = +.. \mathbf{P}^\top +.. \left[ +.. \begin{matrix} +.. \boldsymbol{\sigma}_{1}\circ \mathbf{y}_1 + \boldsymbol{\sigma}_{4}\circ \mathbf{y}_2 + \boldsymbol{\sigma}_{5}\circ \mathbf{y}_3\\ +.. \boldsymbol{\sigma}_{4}\circ \mathbf{y}_1 + \boldsymbol{\sigma}_{2}\circ \mathbf{y}_2 + \boldsymbol{\sigma}_{6}\circ \mathbf{y}_3\\ +.. \boldsymbol{\sigma}_{5}\circ \mathbf{y}_1 + \boldsymbol{\sigma}_{6}\circ \mathbf{y}_2 + \boldsymbol{\sigma}_{3}\circ \mathbf{y}_3\\ +.. \end{matrix} +.. \right] + +.. Now it is easy to take the derivative with respect to any one of the +.. parameters, for example, +.. :math:`\frac{\partial}{\partial\boldsymbol{\sigma}_1}` + +.. .. math:: +.. \frac{\partial}{\partial \boldsymbol{\sigma}_1}\left(\mathbf{P}^\top\Sigma\mathbf{y}\right) +.. = +.. \mathbf{P}^\top +.. \left[ +.. \begin{matrix} +.. \text{diag}(\mathbf{y}_1)\\ +.. 0\\ +.. 0\\ +.. \end{matrix} +.. \right] + +.. Whereas :math:`\frac{\partial}{\partial\boldsymbol{\sigma}_4}`, for +.. example, is: + +.. .. math:: +.. \frac{\partial}{\partial \boldsymbol{\sigma}_4}\left(\mathbf{P}^\top\Sigma\mathbf{y}\right) +.. = +.. \mathbf{P}^\top +.. \left[ +.. \begin{matrix} +.. \text{diag}(\mathbf{y}_2)\\ +.. \text{diag}(\mathbf{y}_1)\\ +.. 0\\ +.. \end{matrix} +.. \right] + +.. These are computed for each of the 8 projections, horizontally concatenated, +.. and returned. diff --git a/docs/content/inner_products_basic.rst b/docs/content/inner_products_basic.rst new file mode 100644 index 000000000..69196de02 --- /dev/null +++ b/docs/content/inner_products_basic.rst @@ -0,0 +1,282 @@ +.. _inner_products_basic: + +Basic Inner Products +******************** + +Summary +------- + +Inner products between two scalar or vector quantities represents the most +basic class of inner products. For scalar quantities :math:`\psi` and :math:`\phi`, +we will show how the inner product is approximation by: + +.. math:: + (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \mathbf{\psi^T \, M \, \phi} + :label: inner_product_basic_scalar + +And for vector quantities :math:`\vec{u}` and :math:`\vec{w}`, the +inner product is approximated by: + +.. math:: + (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + :label: inner_product_basic_vector + +where :math:`\mathbf{M}` in either equation represents an +*inner-product matrix*. :math:`\mathbf{\psi}`, :math:`\mathbf{\phi}`, +:math:`\mathbf{u}` and :math:`\mathbf{w}` are discrete variables that live +on the mesh. It is important to note a few things about the +inner-product matrix in this case: + + 1. It depends on the dimensions and discretization of the mesh + 2. It depends on where the discrete variables live; e.g. edges, faces, nodes, centers + +For this simple class of inner products, the form of the inner product matricies for +discrete quantities living on various parts of the mesh are shown below: + +.. math:: + \textrm{Centers:} \; \mathbf{M_c} &= \textrm{diag} (\mathbf{v} ) \\ + \textrm{Nodes:} \; \mathbf{M_n} &= \frac{1}{2^{2k}} \mathbf{P_n^T } \textrm{diag} (\mathbf{v} ) \mathbf{P_n} \\ + \textrm{Faces:} \; \mathbf{M_f} &= \frac{1}{4} \mathbf{P_f^T } \textrm{diag} (\mathbf{I_k \otimes v} ) \mathbf{P_f} \\ + \textrm{Edges:} \; \mathbf{M_e} &= \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} (\mathbf{I_k \otimes v}) \mathbf{P_e} + +where :math:`k = 1,2,3`, :math:`\mathbf{I_k}` is the identity matrix and +:math:`\otimes` is the kronecker product. :math:`\mathbf{P}` are projection +matricies that map quantities from one part of the cell (nodes, faces, edges) +to cell centers. + + +Scalars at Cell Centers +----------------------- + +We want to approximate the inner product of two scalar quantities :math:`\psi` and :math:`\phi` +where the discrete quantities :math:`\mathbf{\psi}` and :math:`\mathbf{\phi}` are defined +to live at cell centers. Assuming we know the values of :math:`\psi` and :math:`\phi` at cell centers, +our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: + +.. math:: + (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \mathbf{\psi^T \, M \, \phi} + +A simple approximation for the inner product is obtained by summing the product of +:math:`\psi` at cell centers, :math:`\phi` at cell centers and the cell volumes (:math:`\mathbf{v}`) +over all cells: + +.. math:: + (\psi , \phi ) \approx \sum_i^{nc} \psi_i \phi_i v_i + +Expressing the sum in terms of linear equations, we obtain: + +.. math:: + (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \mathbf{\psi^T \, M_c \, \phi} + +where the mass matrix for cell centered quantities is just a diagonal matrix containing +the cell volumes, i.e.: + +.. math:: + \mathbf{M_c} = diag(\mathbf{v}) + + +.. note:: To see a validation of the discrete approximation, see our tutorials section (link) + + +Scalars at Nodes +---------------- + +We want to approximate the inner product of two scalar quantities :math:`\psi` and :math:`\phi` +where the discrete quantities :math:`\mathbf{\psi}` and :math:`\mathbf{\phi}` are defined +to live on cell nodes. Assuming we know the values of :math:`\psi` and :math:`\phi` at the nodes, +our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: + +.. math:: + (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \mathbf{\psi^T \, M \, \phi} + :label: inner_product_basic_nodes + +Whereas :math:`\mathbf{\psi}` and :math:`\mathbf{\phi}` are defined +to live on cell nodes, it makes more sense for the cell volumes to be defined to live +at cell centers. This makes evaluating the inner product more complicated as +discrete quantities do not live at the same place. + +For each cell, the contribution towards the inner product is approximated by +taking the average of the quantity defined at each nodes and multiplying it +by the cell volume, i.e.: + +.. math:: + (\psi , \phi ) \approx \sum_i^{nc} \tilde{\psi}_i \tilde{\phi}_i v_i + :label: inner_product_basic_nodes_sum + +where :math:`\tilde{\psi}_i` and :math:`\tilde{\phi}_i` are just the average value of the quantities +at the nodes of the cell. Where each cell has *N=4* nodes for 2D problems and each cell has *N=8* nodes for +3D problems: + +.. math:: + \begin{align} + \tilde{\psi}_i = \frac{1}{N} \sum_k^N \tilde{\psi}_i^{(k)} \\ + \tilde{\phi}_i = \frac{1}{N} \sum_k^N \tilde{\phi}_i^{(k)} + \end{align} + +We want to express the sum in equation :eq:`inner_product_basic_nodes_sum` in the form described by +equation :eq:`inner_product_basic_nodes`. To accomlish this, we construct a sparse matrix +:math:`\mathbf{P_n}` which projects quantities at the nodes to their corresponding cell centers. + +Our final approximation for the inner product is therefore: + +.. math:: + (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \mathbf{\psi^T \, M_n \, \phi} + +where the mass matrix for nodal quantities has the form: + +.. math:: + \mathbf{M_n} = \frac{1}{2^{2k}} \mathbf{P_n^T } \textrm{diag} (\mathbf{v} ) \mathbf{P_n} + +and :math:`k=1,2,3` refers to the dimension (1D, 2D or 3D). The term containing :math:`k` is responsible for +averaging once nodal values have been projected to cell centers. + +.. note:: To see a validation of the discrete approximation, see our tutorials section (link) + + +Vectors on Cell Faces +--------------------- + +For the mimetic finite volume approach, fluxes are generally defined on cell faces; +as it allows cells to share faces while preserving natural boundary conditions. + +We want to approximate the inner product of two vector quantities :math:`\vec{u}` and :math:`\vec{w}` +where the discrete quantities :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined +to live on cell faces. Assuming we know the values of :math:`\vec{u}` and :math:`\vec{w}` on the faces, +our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: + +.. math:: + (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + :label: inner_product_basic_faces + +We must respect the dot product. For vectors defined on cell faces, we discretize such that the +x-component of the vectors live on the x-faces, the y-component lives y-faces and the z-component +lives on the z-faces. For a single cell, this is illustrated in 2D and 3D below. + +.. image:: ../images/face_discretization.png + :align: center + :width: 500 + + +As we can see there are 2 faces for each component. Therefore, we need to project each component of the +vector from its faces to the cell centers and take their averages separately. For a single cell with volume :math:`v_i`, +the contribution towards the inner product is: + +.. math:: + \begin{align} + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i}{4} \Big ( u_x^{(f1)} + u_x^{(f2)} \Big ) \Big ( w_x^{(f1)} + w_x^{(f2)} \Big ) \\ + & + \frac{v_i}{4} \Big ( u_y^{(f1)} + u_y^{(f2)} \Big ) \Big ( w_y^{(f1)} + w_y^{(f2)} \Big ) \\ + & \\ + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i}{4} \Big ( u_x^{(f1)} + u_x^{(f2)} \Big ) \Big ( w_x^{(f1)} + w_x^{(f2)} \Big ) \\ + & + \frac{v_i}{4} \Big ( u_y^{(f1)} + u_y^{(f2)} \Big ) \Big ( w_y^{(f1)} + w_y^{(f2)} \Big ) \\ + & + \frac{v_i}{4} \Big ( u_z^{(f1)} + u_z^{(f2)} \Big ) \Big ( w_z^{(f1)} + w_z^{(f2)} \Big ) + \end{align} + :label: inner_product_basic_faces_1 + +where superscripts :math:`(f1)` and :math:`(f2)` denote face 1 and face 2, respectively. +Using the contribution for each cell described in expression :eq:`inner_product_basic_faces_1`, +we want to approximate the inner product in the form described by +equation :eq:`inner_product_basic_faces`. To accomlish this, we construct a sparse matrix +:math:`\mathbf{P_f}` which projects quantities on the x, y and z faces separately to the +the cell centers. + +For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +are organized on cell faces as follows: + +.. math:: + \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \;\;\;\; \textrm{and} \;\;\;\; + \mathbf{w} = \begin{bmatrix} \mathbf{w_x} \\ \mathbf{w_y} \\ \mathbf{w_y} \\ \end{bmatrix} + +the approximation to the inner product is given by: + +.. math:: + (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \mathbf{\mathbf{u} \, M_f \, \mathbf{w}} + +where the mass matrix for face quantities has the form: + +.. math:: + \mathbf{M_f} = \frac{1}{4} \mathbf{P_f^T } \textrm{diag} (\mathbf{I_k \otimes v} ) \mathbf{P_f} + +and :math:`k=1,2,3` refers to the dimension (1D, 2D or 3D). :math:`\mathbf{I_k}` is the identity matrix and +:math:`\otimes` is the kronecker product. + +.. note:: To see a validation of the discrete approximation, see our tutorials section (link) + + + +Vectors on Cell Edges +--------------------- + +For the mimetic finite volume approach, fields are generally defined on cell edges; +as it allows cells to share edges while preserving natural boundary conditions. +We want to approximate the inner product of two vector quantities :math:`\vec{u}` and :math:`\vec{w}` +where the discrete quantities :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined +to live at cell edges. Assuming we know the values of :math:`\vec{u}` and :math:`\vec{w}` at the edges, +our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: + +.. math:: + (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + :label: inner_product_basic_edges + +We must respect the dot product. For vectors defined on cell edges, we discretize such that the +x-component of the vectors live on the x-edges, the y-component lives y-edges and the z-component +lives on the z-edges. This is illustrated in 2D and 3D below. + +.. image:: ../images/edge_discretization.png + :align: center + :width: 500 + + +As we can see there are 2 edges for each component in 2D and 4 edges for each component in 3D. +Therefore, we need to project each component of the +vector from its edges to the cell centers and take their averages separately. For a single cell with volume :math:`v_i`, +the contribution towards the inner product is: + +.. math:: + \begin{align} + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i}{4} \Big ( u_x^{(e1)} + u_x^{(e2)} \Big ) \Big ( w_x^{(e1)} + w_x^{(e2)} \Big ) \\ + & + \frac{v_i}{4} \Big ( u_y^{(e1)} + u_y^{(e2)} \Big ) \Big ( w_y^{(e1)} + w_y^{(e2)} \Big ) \\ + & \\ + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i}{16} \Bigg ( \sum_{ei=1}^4 u_x^{(ei)} \Bigg ) \Bigg ( \sum_{ei=1}^4 w_x^{(ei)} \Bigg ) \\ + & + \frac{v_i}{16} \Bigg ( \sum_{ei=1}^4 u_y^{(ei)} \Bigg ) \Bigg ( \sum_{ei=1}^4 w_y^{(ei)} \Bigg ) \\ + & + \frac{v_i}{16} \Bigg ( \sum_{ei=1}^4 u_z^{(ei)} \Bigg ) \Bigg ( \sum_{ei=1}^4 w_z^{(ei)} \Bigg ) + \end{align} + :label: inner_product_basic_edges_1 + +where superscripts :math:`(e1)` and :math:`(e2)` denote eges 1 and edges 2, respectively. +Using the contribution for each cell described in expression :eq:`inner_product_basic_edges_1`, +we want to approximate the inner product in the form described by +equation :eq:`inner_product_basic_edges`. To accomlish this, we construct a sparse matrix +:math:`\mathbf{P_e}` which projects quantities on the x, y and z edges separately to the +the cell centers. + +For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +are organized on cell edges as follows: + +.. math:: + \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \;\;\;\; \textrm{and} \;\;\;\; + \mathbf{w} = \begin{bmatrix} \mathbf{w_x} \\ \mathbf{w_y} \\ \mathbf{w_y} \\ \end{bmatrix} + +the approximation to the inner product is given by: + +.. math:: + (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \mathbf{\mathbf{u} \, M_e \, \mathbf{w}} + +where the mass matrix for face quantities has the form: + +.. math:: + \mathbf{M_e} = \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} (\mathbf{I_k \otimes v}) \mathbf{P_e} + +and :math:`k=1,2,3` refers to the dimension (1D, 2D or 3D). :math:`\mathbf{I_k}` is the identity matrix and +:math:`\otimes` is the kronecker product. + +.. note:: To see a validation of the discrete approximation, see our tutorials section (link) + + + + + + + + diff --git a/docs/content/inner_products_constitutive.rst b/docs/content/inner_products_constitutive.rst new file mode 100644 index 000000000..ad31a98a4 --- /dev/null +++ b/docs/content/inner_products_constitutive.rst @@ -0,0 +1,6 @@ +.. _inner_products_constitutive: + +Constitutive Relationships +************************** + + diff --git a/docs/images/edge_discretization.PNG b/docs/images/edge_discretization.PNG new file mode 100644 index 0000000000000000000000000000000000000000..4c801abfe0e0c987c94a9d44fdeeaf5b4479bb52 GIT binary patch literal 41008 zcmZ_0by$^Mw>=C97x_pr}W1bMh3rO3EDmUDU5sw|BCM4 zAIkl|ekj2`Sz-7&`C}LHaI=S-u{4!&nnw-O!vmrmOdt|*li%&~8gI{LY2QhOlN9}i?o}jC07xJnoAm8j-@t-b8_Uf#D<&PoYY!8o1)p5U#OM|PFI`L z#WE;Y415*v@+?rR%;mk>$f|I^w53z-%HVn!G>X70IvQ#F-RUvR=|bhgXIOkE5w}-e 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z2DyYO570GyG5i3&v+%Z(2TNB2vYo7kQHEz>0+X}zB-bf3-xdAxdR>Kdgb2e?lRMX>B{-R0{+#GPuj^U( z6WVe_aQvY~@~iVB8PMTFa;K28OvaQ#VS6|3JVWM5WRg4j#JvG^?bGLPORG5H5A4ip-?eMT%XUg2=hq0*6Ndmlq} z?k3YmK*5))r~Wof3VB67gm~{M-1f2T=I#0iAK_$#k{{4~2X&`2;LwTQ0tG-OO=+nC z7l|7CSV6Ps&#?FWl$bTTl2cIT4@|<$h{0nqo(f);_!@I3hK9O@%j~dBhuyLZm^`?~ zyG9+p8j9mPx)Tw@SD~oPwCm-NZxo5{`WxgMmUf7d{J#COUugfnef#R3Z)KG>9kboP zPa&}c{%_ZPVBfxUHHv-vVtDxX{hz;x`?O!WQM6BJWcDmPY@eF4=EZy^^T7WHFIWQ5 literal 0 HcmV?d00001 From 794db2b74a2596954e8af722b87bb27d80dbd4ff Mon Sep 17 00:00:00 2001 From: dccowan Date: Wed, 18 Nov 2020 14:29:34 -0800 Subject: [PATCH 02/35] Final draft of basic inner products theory --- docs/content/inner_products_basic.rst | 138 +++++++++++++++----------- docs/images/center_discretization.PNG | Bin 0 -> 26747 bytes docs/images/edge_discretization.PNG | Bin 41008 -> 43975 bytes docs/images/face_discretization.PNG | Bin 50289 -> 52889 bytes docs/images/node_discretization.PNG | Bin 0 -> 35649 bytes 5 files changed, 81 insertions(+), 57 deletions(-) create mode 100644 docs/images/center_discretization.PNG create mode 100644 docs/images/node_discretization.PNG diff --git a/docs/content/inner_products_basic.rst b/docs/content/inner_products_basic.rst index 69196de02..40665be58 100644 --- a/docs/content/inner_products_basic.rst +++ b/docs/content/inner_products_basic.rst @@ -8,21 +8,21 @@ Summary Inner products between two scalar or vector quantities represents the most basic class of inner products. For scalar quantities :math:`\psi` and :math:`\phi`, -we will show how the inner product is approximation by: +we will show that a disrete approximation to the inner product is given by: .. math:: - (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \mathbf{\psi^T \, M \, \phi} + (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \boldsymbol{\psi^T \, M \, \phi} :label: inner_product_basic_scalar -And for vector quantities :math:`\vec{u}` and :math:`\vec{w}`, the -inner product is approximated by: +And for vector quantities :math:`\vec{u}` and :math:`\vec{w}`, a discrete approximation +to the inner product is given by: .. math:: (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \mathbf{u^T \, M \, w} :label: inner_product_basic_vector where :math:`\mathbf{M}` in either equation represents an -*inner-product matrix*. :math:`\mathbf{\psi}`, :math:`\mathbf{\phi}`, +*inner-product matrix*, and :math:`\boldsymbol{\psi}`, :math:`\boldsymbol{\phi}`, :math:`\mathbf{u}` and :math:`\mathbf{w}` are discrete variables that live on the mesh. It is important to note a few things about the inner-product matrix in this case: @@ -30,38 +30,53 @@ inner-product matrix in this case: 1. It depends on the dimensions and discretization of the mesh 2. It depends on where the discrete variables live; e.g. edges, faces, nodes, centers -For this simple class of inner products, the form of the inner product matricies for +For this simple class of inner products, the corresponding form of the inner product matricies for discrete quantities living on various parts of the mesh are shown below: .. math:: - \textrm{Centers:} \; \mathbf{M_c} &= \textrm{diag} (\mathbf{v} ) \\ - \textrm{Nodes:} \; \mathbf{M_n} &= \frac{1}{2^{2k}} \mathbf{P_n^T } \textrm{diag} (\mathbf{v} ) \mathbf{P_n} \\ - \textrm{Faces:} \; \mathbf{M_f} &= \frac{1}{4} \mathbf{P_f^T } \textrm{diag} (\mathbf{I_k \otimes v} ) \mathbf{P_f} \\ - \textrm{Edges:} \; \mathbf{M_e} &= \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} (\mathbf{I_k \otimes v}) \mathbf{P_e} - -where :math:`k = 1,2,3`, :math:`\mathbf{I_k}` is the identity matrix and -:math:`\otimes` is the kronecker product. :math:`\mathbf{P}` are projection + \textrm{Scalars at centers:} \; \mathbf{M_c} &= \textrm{diag} (\mathbf{v} ) \\ + \textrm{Scalars at nodes:} \; \mathbf{M_n} &= \frac{1}{2^{2k}} \mathbf{P_n^T } \textrm{diag} (\mathbf{v} ) \mathbf{P_n} \\ + \textrm{Vectors on faces:} \; \mathbf{M_f} &= \frac{1}{4} \mathbf{P_f^T } \textrm{diag} (\mathbf{I_k \otimes v} ) \mathbf{P_f} \\ + \textrm{Vectors on edges:} \; \mathbf{M_e} &= \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} (\mathbf{I_k \otimes v}) \mathbf{P_e} + +where :math:`k = 1,2,3` represents the dimension (1D, 2D or 3D), +:math:`\mathbf{I_k}` is the identity matrix of dimension :math:`k\times k`, +and :math:`\otimes` is the kronecker product. :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) -to cell centers. +to cell centers. The volumes of all cells in the mesh are store in a vector :math:`\mathbf{v}`. +The term containing :math:`k` is responsible for averaging +once quantities are mapped to cell centers. Scalars at Cell Centers ----------------------- We want to approximate the inner product of two scalar quantities :math:`\psi` and :math:`\phi` -where the discrete quantities :math:`\mathbf{\psi}` and :math:`\mathbf{\phi}` are defined +where the discrete quantities :math:`\boldsymbol{\psi}` and :math:`\boldsymbol{\phi}` are defined to live at cell centers. Assuming we know the values of :math:`\psi` and :math:`\phi` at cell centers, -our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: +our goal is to construct the inner product matrix :math:`\mathbf{M}` such that: .. math:: (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \mathbf{\psi^T \, M \, \phi} -A simple approximation for the inner product is obtained by summing the product of -:math:`\psi` at cell centers, :math:`\phi` at cell centers and the cell volumes (:math:`\mathbf{v}`) -over all cells: + +.. image:: ../images/center_discretization.png + :align: center + :width: 600 + + +For a single cell (see above), the contribution towards the inner product is obtained by multiplying +:math:`\psi` and :math:`\phi` at the cell centers (represented by :math:`\psi_i and \phi_i`) +by the cell's volume (:math:`v_i`), i.e.: .. math:: - (\psi , \phi ) \approx \sum_i^{nc} \psi_i \phi_i v_i + \int_{\Omega_i} \psi \, \phi \, dv \approx \psi_i \phi_i v_i + +Therefore a simple approximation to the inner product is obtained by summing the above +approximation over all cells. Where :math:`nc` refers to the number of cells in the mesh: + +.. math:: + \int_\Omega \psi \, \phi \, dv = \approx \sum_i^{nc} \psi_i \phi_i v_i Expressing the sum in terms of linear equations, we obtain: @@ -69,7 +84,7 @@ Expressing the sum in terms of linear equations, we obtain: (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \mathbf{\psi^T \, M_c \, \phi} where the mass matrix for cell centered quantities is just a diagonal matrix containing -the cell volumes, i.e.: +the cell volumes (:math:`\mathbf{v}`), i.e.: .. math:: \mathbf{M_c} = diag(\mathbf{v}) @@ -82,40 +97,45 @@ Scalars at Nodes ---------------- We want to approximate the inner product of two scalar quantities :math:`\psi` and :math:`\phi` -where the discrete quantities :math:`\mathbf{\psi}` and :math:`\mathbf{\phi}` are defined +where the discrete quantities :math:`\boldsymbol{\psi}` and :math:`\boldsymbol{\phi}` are defined to live on cell nodes. Assuming we know the values of :math:`\psi` and :math:`\phi` at the nodes, -our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: +our goal is to construct the inner product matrix :math:`\mathbf{M}` such that: .. math:: (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \mathbf{\psi^T \, M \, \phi} :label: inner_product_basic_nodes -Whereas :math:`\mathbf{\psi}` and :math:`\mathbf{\phi}` are defined -to live on cell nodes, it makes more sense for the cell volumes to be defined to live -at cell centers. This makes evaluating the inner product more complicated as +Whereas :math:`\boldsymbol{\psi}` and :math:`\boldsymbol{\phi}` are defined +to live on cell nodes, it makes more sense for cell volumes to be considered a property +which lives at cell centers. This makes evaluating the inner product more complicated as discrete quantities do not live at the same place. -For each cell, the contribution towards the inner product is approximated by -taking the average of the quantity defined at each nodes and multiplying it -by the cell volume, i.e.: - -.. math:: - (\psi , \phi ) \approx \sum_i^{nc} \tilde{\psi}_i \tilde{\phi}_i v_i - :label: inner_product_basic_nodes_sum +.. image:: ../images/node_discretization.png + :align: center + :width: 600 -where :math:`\tilde{\psi}_i` and :math:`\tilde{\phi}_i` are just the average value of the quantities -at the nodes of the cell. Where each cell has *N=4* nodes for 2D problems and each cell has *N=8* nodes for -3D problems: +For a single cell :math:`i`, the contribution towards the inner product is approximated by +mapping the values at the nodes to cell centers, taking the average, then multiplying +by the cell volume. For 2D cells there are 4 nodes. And for 3D cells there are 8 nodes +Thus: .. math:: \begin{align} - \tilde{\psi}_i = \frac{1}{N} \sum_k^N \tilde{\psi}_i^{(k)} \\ - \tilde{\phi}_i = \frac{1}{N} \sum_k^N \tilde{\phi}_i^{(k)} + \mathbf{In \; 2D:} \; \int_{\Omega_i} \psi \, \phi \, dv \approx & \;\; + \frac{v_i}{16} \Bigg ( \psi_i^{(1)} \! + \psi_i^{(2)} \! + \psi_i^{(3)} \! + \psi_i^{(4)} \Bigg ) + \Bigg ( \phi_i^{(n1)} \! + \phi_i^{(n2)} \! + \phi_i^{(n3)} \! + \phi_i^{(n4)} \Bigg ) \\ + & \\ + \mathbf{In \; 3D:} \; \int_{\Omega_i} \psi \, \phi \, dv \approx & \;\; + \frac{v_i}{16} \Bigg ( \sum_{n=1}^8 \psi_i^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^8 \psi_i^{(n)} \Bigg ) \end{align} + :label: inner_product_basic_nodes_1 -We want to express the sum in equation :eq:`inner_product_basic_nodes_sum` in the form described by +where the superscript :math:`(n)` is used to point to a specific node. +Using the contribution for each cell described in expression :eq:`inner_product_basic_nodes_1`, +we want to approximate the inner product in the form described by equation :eq:`inner_product_basic_nodes`. To accomlish this, we construct a sparse matrix -:math:`\mathbf{P_n}` which projects quantities at the nodes to their corresponding cell centers. +:math:`\mathbf{P_n}` which projects quantities on the nodes to the +the cell centers. Our final approximation for the inner product is therefore: @@ -128,7 +148,8 @@ where the mass matrix for nodal quantities has the form: \mathbf{M_n} = \frac{1}{2^{2k}} \mathbf{P_n^T } \textrm{diag} (\mathbf{v} ) \mathbf{P_n} and :math:`k=1,2,3` refers to the dimension (1D, 2D or 3D). The term containing :math:`k` is responsible for -averaging once nodal values have been projected to cell centers. +averaging once nodal values have been projected to cell centers. The cell volumes are stored in +a vector :math:`\mathbf{v}`. .. note:: To see a validation of the discrete approximation, see our tutorials section (link) @@ -154,7 +175,7 @@ lives on the z-faces. For a single cell, this is illustrated in 2D and 3D below. .. image:: ../images/face_discretization.png :align: center - :width: 500 + :width: 600 As we can see there are 2 faces for each component. Therefore, we need to project each component of the @@ -163,16 +184,16 @@ the contribution towards the inner product is: .. math:: \begin{align} - \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i}{4} \Big ( u_x^{(f1)} + u_x^{(f2)} \Big ) \Big ( w_x^{(f1)} + w_x^{(f2)} \Big ) \\ - & + \frac{v_i}{4} \Big ( u_y^{(f1)} + u_y^{(f2)} \Big ) \Big ( w_y^{(f1)} + w_y^{(f2)} \Big ) \\ + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ + & + \frac{v_i}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ & \\ - \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i}{4} \Big ( u_x^{(f1)} + u_x^{(f2)} \Big ) \Big ( w_x^{(f1)} + w_x^{(f2)} \Big ) \\ - & + \frac{v_i}{4} \Big ( u_y^{(f1)} + u_y^{(f2)} \Big ) \Big ( w_y^{(f1)} + w_y^{(f2)} \Big ) \\ - & + \frac{v_i}{4} \Big ( u_z^{(f1)} + u_z^{(f2)} \Big ) \Big ( w_z^{(f1)} + w_z^{(f2)} \Big ) + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ + & + \frac{v_i}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + & + \frac{v_i}{4} \Big ( u_z^{(1)} + u_z^{(2)} \Big ) \Big ( w_z^{(1)} + w_z^{(2)} \Big ) \end{align} :label: inner_product_basic_faces_1 -where superscripts :math:`(f1)` and :math:`(f2)` denote face 1 and face 2, respectively. +where superscripts :math:`(1)` and :math:`(2)` denote face 1 and face 2, respectively. Using the contribution for each cell described in expression :eq:`inner_product_basic_faces_1`, we want to approximate the inner product in the form described by equation :eq:`inner_product_basic_faces`. To accomlish this, we construct a sparse matrix @@ -198,7 +219,8 @@ where the mass matrix for face quantities has the form: \mathbf{M_f} = \frac{1}{4} \mathbf{P_f^T } \textrm{diag} (\mathbf{I_k \otimes v} ) \mathbf{P_f} and :math:`k=1,2,3` refers to the dimension (1D, 2D or 3D). :math:`\mathbf{I_k}` is the identity matrix and -:math:`\otimes` is the kronecker product. +:math:`\otimes` is the kronecker product. Cell volumes are store in a vector :math:`\mathbf{v}`. The term containing :math:`k` is responsible for +averaging once values have been projected to cell centers .. note:: To see a validation of the discrete approximation, see our tutorials section (link) @@ -224,7 +246,7 @@ lives on the z-edges. This is illustrated in 2D and 3D below. .. image:: ../images/edge_discretization.png :align: center - :width: 500 + :width: 600 As we can see there are 2 edges for each component in 2D and 4 edges for each component in 3D. @@ -234,16 +256,16 @@ the contribution towards the inner product is: .. math:: \begin{align} - \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i}{4} \Big ( u_x^{(e1)} + u_x^{(e2)} \Big ) \Big ( w_x^{(e1)} + w_x^{(e2)} \Big ) \\ - & + \frac{v_i}{4} \Big ( u_y^{(e1)} + u_y^{(e2)} \Big ) \Big ( w_y^{(e1)} + w_y^{(e2)} \Big ) \\ + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ + & + \frac{v_i}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ & \\ - \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i}{16} \Bigg ( \sum_{ei=1}^4 u_x^{(ei)} \Bigg ) \Bigg ( \sum_{ei=1}^4 w_x^{(ei)} \Bigg ) \\ - & + \frac{v_i}{16} \Bigg ( \sum_{ei=1}^4 u_y^{(ei)} \Bigg ) \Bigg ( \sum_{ei=1}^4 w_y^{(ei)} \Bigg ) \\ - & + \frac{v_i}{16} \Bigg ( \sum_{ei=1}^4 u_z^{(ei)} \Bigg ) \Bigg ( \sum_{ei=1}^4 w_z^{(ei)} \Bigg ) + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i}{16} \Bigg ( \sum_{n=1}^4 u_x^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_x^{(n)} \Bigg ) \\ + & + \frac{v_i}{16} \Bigg ( \sum_{n=1}^4 u_y^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_y^{(n)} \Bigg ) \\ + & + \frac{v_i}{16} \Bigg ( \sum_{n=1}^4 u_z^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_z^{(n)} \Bigg ) \end{align} :label: inner_product_basic_edges_1 -where superscripts :math:`(e1)` and :math:`(e2)` denote eges 1 and edges 2, respectively. +where the superscript :math:`(n)` denotes a particular edge. Using the contribution for each cell described in expression :eq:`inner_product_basic_edges_1`, we want to approximate the inner product in the form described by equation :eq:`inner_product_basic_edges`. To accomlish this, we construct a sparse matrix @@ -269,7 +291,9 @@ where the mass matrix for face quantities has the form: \mathbf{M_e} = \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} (\mathbf{I_k \otimes v}) \mathbf{P_e} and :math:`k=1,2,3` refers to the dimension (1D, 2D or 3D). :math:`\mathbf{I_k}` is the identity matrix and -:math:`\otimes` is the kronecker product. +:math:`\otimes` is the kronecker product. 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z0Wa8^i`|6@N|SWxi@E}r#-QFc!1WL^vZF`jRbelYsdV&pOTg3`2UzKeK3L>R1O^2f zLuIufxa)*!3vr_FY8SkZpUOSK>={5ul`TnHfCP^ZxCA&y-hE|{PQl&Ee}zrk4V0rE z5+S~`_g$+KJ~72Gp?=`709q{Uuvnng^n(v9CvY#BwGQ30Adj^eN)ih*ga52PKh%1l zW7f?9lB!WB`}ZMIPmf(0M@TV*zNKyAxoxuiHC?a-AdI<+E3H97U$GXsq2E6q&cli< zgCXlvzDc7icLD}wA~hzO$v5BhgHvmX(1mOOG~?W6(Ts1nm~wg2&mgox3Ywyf?Wz z>q^MMYDq5Mmyb@IFmRkgw@OEhz3|un%$I_VPfrZ%@c)I7>?8(Ngi=ou4_)|+o`IZr zXuw~}J9iNCDfml`o0wt1U*r@d|KGm!FZzvub$=_NmjNqp{(}FhpVmB8qHG=X{{ZJ} B`R)J! literal 0 HcmV?d00001 From d13ed2197206cf142800d40388ddd09ac2c6de2d Mon Sep 17 00:00:00 2001 From: dccowan Date: Wed, 18 Nov 2020 20:41:37 -0800 Subject: [PATCH 03/35] preliminary add of isotropic and anisotropic inner products. Fix some typos as well. --- docs/content/inner_products.rst | 3 +- docs/content/inner_products_anisotropic.rst | 459 +++++++++++++++++++ docs/content/inner_products_basic.rst | 51 ++- docs/content/inner_products_constitutive.rst | 6 - docs/content/inner_products_isotropic.rst | 202 ++++++++ 5 files changed, 693 insertions(+), 28 deletions(-) create mode 100644 docs/content/inner_products_anisotropic.rst delete mode 100644 docs/content/inner_products_constitutive.rst create mode 100644 docs/content/inner_products_isotropic.rst diff --git a/docs/content/inner_products.rst b/docs/content/inner_products.rst index f9c13856f..332b61551 100644 --- a/docs/content/inner_products.rst +++ b/docs/content/inner_products.rst @@ -34,7 +34,8 @@ discrete approximation for several classes of inner products. :maxdepth: 1 inner_products_basic - inner_products_constitutive + inner_products_isotropic + inner_products_anisotropic diff --git a/docs/content/inner_products_anisotropic.rst b/docs/content/inner_products_anisotropic.rst new file mode 100644 index 000000000..5ecbc71ce --- /dev/null +++ b/docs/content/inner_products_anisotropic.rst @@ -0,0 +1,459 @@ +.. _inner_products_anisotropic: + +Anisotropic Constitutive Relationships +************************************** + +Summary +------- + +A constitutive relationship quantifies the response of a material to an external stimulus. +Examples include Ohm's law and Hooke's law. For practical applications of the finite volume method, +we may need to take the inner product of expressions containing constitutive relationships. + +Let :math:`\vec{v}` and :math:`\vec{w}` be two physically related quantities. +If their relationship is anisotropic (defined by a tensor :math:`\Sigma`), the constitutive relation is of the form: + +.. math:: + \vec{v} = \Sigma \vec{w} \;\;\;\;\; \textrm{where} \;\;\;\;\; + \Sigma = \begin{bmatrix} \sigma_{1} & \sigma_{4} & \sigma_{5} \\ + \sigma_{4} & \sigma_{2} & \sigma_{6} \\ + \sigma_{5} & \sigma_{6} & \sigma_{3} \end{bmatrix} + :label: inner_product_anisotropic + +Here we show that for anisotropic constitutive relationships, the inner +product between a vector :math:`\vec{u}` and the right-hand side of +equation :eq:`inner_product_anisotropic` is approximated by: + +.. math:: + (\vec{u}, \Sigma \vec{w} ) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \boldsymbol{u^T M w} + +where :math:`\boldsymbol{M}` represents an *inner-product matrix*, and vectors +:math:`\mathbf{u}` and :math:`\mathbf{w}` are discrete variables that live +on the mesh. It is important to note a few things about the +inner-product matrix in this case: + + 1. It depends on the dimensions and discretization of the mesh + 2. It depends on where the discrete variables live; e.g. edges, faces, nodes, centers + 3. It depends on the spacial variation of the tensor property :math:`\Sigma` + +For this class of inner products, the corresponding form of the inner product matricies for +discrete quantities living on various parts of the mesh are shown below. We let: + + - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) + - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\odot` is the Hadamard product + - :math:`\otimes` is the kronecker product + - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers + - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + +**Diagonal Anisotropic:** + +For the diagonal anisotropic case, the tensor characterzing the material properties +has the form + +.. math:: + \Sigma = \begin{bmatrix} \sigma_{x} & 0 & 0 \\ + 0 & \sigma_{y} & 0 \\ + 0 & 0 & \sigma_{z} \end{bmatrix} + +The resulting inner products have form. + +.. math:: + \textrm{Vectors on faces:} \; \mathbf{M_{\Sigma f}} &= \frac{1}{4} \mathbf{P_f^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \mathbf{P_f} \\ + \textrm{Vectors on edges:} \; \mathbf{M_{\Sigma e}} &= \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \mathbf{P_e} + +where :math:`\boldsymbol{\sigma}` organizes :math:`\sigma_x`, :math:`\sigma_y` and :math:`\sigma_z` within a vector: + +.. math:: + \boldsymbol{\sigma} = \begin{bmatrix} \boldsymbol{\sigma_x} \\ \boldsymbol{\sigma_y} \\ \boldsymbol{\sigma_z} \\ \end{bmatrix} + + +**Fully Anisotropic:** + +For a fully anisotropic case, the tensor characterizing the material properties +has the form: + +.. math:: + \Sigma = \begin{bmatrix} \sigma_{1} & \sigma_{4} & \sigma_{5} \\ + \sigma_{4} & \sigma_{2} & \sigma_{6} \\ + \sigma_{5} & \sigma_{6} & \sigma_{3} \end{bmatrix} + + + + +Diagonally Anisotropic Case +--------------------------- + +Vectors on Cell Faces +^^^^^^^^^^^^^^^^^^^^^ + +We want to approximate the inner product between a vector quantity :math:`\vec{u}` and the product of +:math:`\Sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined +to live on cess faces. Our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: + +.. math:: + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \mathbf{u^T \, M \, e} + :label: inner_product_anisotropic_faces + +where + +.. math:: + \Sigma = \begin{bmatrix} \sigma_{x} & 0 & 0 \\ + 0 & \sigma_{y} & 0 \\ + 0 & 0 & \sigma_{z} \end{bmatrix} + +We must respect the dot product and the tensor. For vectors defined on cell faces, we discretize such that the +x-component of the vectors live on the x-faces, the y-component lives y-faces and the z-component +lives on the z-faces. For a single cell, this is illustrated in 2D and 3D below. By decomposing the +domain into a set of finite cells, we assume the tensor properties are spacial invariant within each cell. + +.. image:: ../images/face_discretization.png + :align: center + :width: 600 + +As we can see there are 2 faces for each component. Therefore, we need to project each component of the +vector from its faces to the cell centers and take their averages separately. We must also recognize that +x-components are only multiplied by :math:`\sigma_x`, y_components by :math:`\sigma_y` and z-components +by :math:`\sigma_z`. + +For a single cell with volume :math:`v^{(i)}` and tensor properties defined by +:math:`\sigma_x^{(i)}`, :math:`\sigma_y^{(i)}`, :math:`\sigma_z^{(i)}` +the contribution towards the inner product is: + +.. math:: + \begin{align} + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; + \frac{v^{(i)} \sigma_x^{(i)}}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ + & + \frac{v^{(i)} \sigma_y^{(i)}}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + & \\ + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; + \frac{v^{(i)} \sigma_x^{(i)}}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ + & + \frac{v^{(i)} \sigma_y^{(i)}}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + & + \frac{v^{(i)} \sigma_z^{(i)}}{4} \Big ( u_z^{(1)} + u_z^{(2)} \Big ) \Big ( w_z^{(1)} + w_z^{(2)} \Big ) + \end{align} + :label: inner_product_anisotropic_faces_1 + +where superscripts :math:`(1)` and :math:`(2)` denote face 1 and face 2, respectively. +Using the contribution for each cell described in expression :eq:`inner_product_anisotropic_faces_1`, +we want to approximate the inner product in the form described by +equation :eq:`inner_product_anisotropic_faces`. To accomlish this, we construct a sparse matrix +:math:`\mathbf{P_f}` which projects quantities on the x, y and z faces separately to the +the cell centers. + +For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +are organized on cell faces as follows: + +.. math:: + \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \;\;\;\; \textrm{and} \;\;\;\; + \mathbf{w} = \begin{bmatrix} \mathbf{w_x} \\ \mathbf{w_y} \\ \mathbf{w_y} \\ \end{bmatrix} + +the approximation to the inner product is given by: + +.. math:: + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \boldsymbol{\mathbf{u} \, M_{\sigma f}} \, \mathbf{w} + +where the mass matrix has the form: + +.. math:: + \mathbf{M_{\Sigma f}} = \frac{1}{4} \mathbf{P_f^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \mathbf{P_f} + +:math:`\boldsymbol{\sigma}` organizes :math:`\sigma_x`, :math:`\sigma_y` and :math:`\sigma_z` within a vector: + +.. math:: + \boldsymbol{\sigma} = \begin{bmatrix} \boldsymbol{\sigma_x} \\ \boldsymbol{\sigma_y} \\ \boldsymbol{\sigma_z} \\ \end{bmatrix} + +and + + - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) + - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\odot` is the Hadamard product + - :math:`\otimes` is the kronecker product + - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers + - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{\sigma}` is a vector that stores the tensor property values for the cells + + +.. note:: To see a validation of the discrete approximation, see our tutorials section (link) + + + + +Vectors on Cell Edges +^^^^^^^^^^^^^^^^^^^^^ + +We want to approximate the inner product between a vector quantity :math:`\vec{u}` and the product of +:math:`\Sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined +to live at cell edges. Our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: + +.. math:: + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + :label: inner_product_anisotropic_edges + +.. math:: + \Sigma = \begin{bmatrix} \sigma_{x} & 0 & 0 \\ + 0 & \sigma_{y} & 0 \\ + 0 & 0 & \sigma_{z} \end{bmatrix} + +We must respect the dot product and the tensor. For vectors defined on cell edges, we discretize such that the +x-component of the vectors live on the x-edges, the y-component lives y-edges and the z-component +lives on the z-edges. This is illustrated in 2D and 3D below. By decomposing the +domain into a set of finite cells, we assume the tensor properties are spacial invariant within each cell. + +.. image:: ../images/edge_discretization.png + :align: center + :width: 600 + +As we can see there are 2 edges for each component in 2D and 4 edges for each component in 3D. +Therefore, we need to project each component of the +vector from its edges to the cell centers and take their averages separately. +We must also recognize that +x-components are only multiplied by :math:`\sigma_x`, y_components by :math:`\sigma_y` and z-components +by :math:`\sigma_z`. + +For a single cell with volume :math:`v^{(i)}` and tensor properties defined by +:math:`\sigma_x^{(i)}`, :math:`\sigma_y^{(i)}`, :math:`\sigma_z^{(i)}` +the contribution towards the inner product is: + +.. math:: + \begin{align} + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; + \frac{v^{(i)} \sigma_x^{(i)}}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ + & + \frac{v^{(i)} \sigma_y^{(i)}}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + & \\ + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; + \frac{v^{(i)} \sigma_x^{(i)}}{16} \Bigg ( \sum_{n=1}^4 u_x^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_x^{(n)} \Bigg ) \\ + & + \frac{v^{(i)} \sigma_y^{(i)}}{16} \Bigg ( \sum_{n=1}^4 u_y^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_y^{(n)} \Bigg ) \\ + & + \frac{v^{(i)} \sigma_z^{(i)}}{16} \Bigg ( \sum_{n=1}^4 u_z^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_z^{(n)} \Bigg ) + \end{align} + :label: inner_product_anisotropic_edges_1 + +where the superscript :math:`(n)` denotes a particular edge. +Using the contribution for each cell described in expression :eq:`inner_product_anisotropic_edges_1`, +we want to approximate the inner product in the form described by +equation :eq:`inner_product_anisotropic_edges`. To accomlish this, we construct a sparse matrix +:math:`\mathbf{P_e}` which projects quantities on the x, y and z edges separately to the +the cell centers. + +For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +are organized on cell edges as follows: + +.. math:: + \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \;\;\;\; \textrm{and} \;\;\;\; + \mathbf{w} = \begin{bmatrix} \mathbf{w_x} \\ \mathbf{w_y} \\ \mathbf{w_y} \\ \end{bmatrix} + +the approximation to the inner product is given by: + +.. math:: + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \boldsymbol{\mathbf{u} \, M_{\sigma e} \, \mathbf{w}} + +where the mass matrix for face quantities has the form: + +.. math:: + \mathbf{M_{\Sigma e}} = \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \mathbf{P_e} + +where :math:`\boldsymbol{\sigma}` organizes :math:`\sigma_x`, :math:`\sigma_y` and :math:`\sigma_z` within a vector: + +.. math:: + \boldsymbol{\sigma} = \begin{bmatrix} \boldsymbol{\sigma_x} \\ \boldsymbol{\sigma_y} \\ \boldsymbol{\sigma_z} \\ \end{bmatrix} +and + + - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) + - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\odot` is the Hadamard product + - :math:`\otimes` is the kronecker product + - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers + - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{\sigma}` is a vector that stores the tensor property values for the cells + +.. note:: To see a validation of the discrete approximation, see our tutorials section (link) + + + +Fully Anisotropic Case +---------------------- + +Vectors on Cell Faces +^^^^^^^^^^^^^^^^^^^^^ + +We want to approximate the inner product between a vector quantity :math:`\vec{u}` and the product of +:math:`\Sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined +to live on cess faces. Our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: + +.. math:: + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \mathbf{u^T \, M \, e} + :label: inner_product_anisotropic_faces + +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} + +We must respect the dot product and the tensor. For vectors defined on cell faces, we discretize such that the +x-component of the vectors live on the x-faces, the y-component lives y-faces and the z-component +lives on the z-faces. For a single cell, this is illustrated in 2D and 3D below. By decomposing the +domain into a set of finite cells, we assume the tensor properties are spacial invariant within each cell. + +.. image:: ../images/face_discretization.png + :align: center + :width: 600 + +As we can see there are 2 faces for each component. Therefore, we need to project each component of the +vector from its faces to the cell centers and take their averages separately. We must also recognize that +x-components are only multiplied by :math:`\sigma_x`, y_components by :math:`\sigma_y` and z-components +by :math:`\sigma_z`. + +For a single cell with volume :math:`v^{(i)}` and tensor properties defined by +:math:`\sigma_x^{(i)}`, :math:`\sigma_y^{(i)}`, :math:`\sigma_z^{(i)}` +the contribution towards the inner product is: + +.. math:: + \begin{align} + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; + \frac{v^{(i)} \sigma_x^{(i)}}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ + & + \frac{v^{(i)} \sigma_y^{(i)}}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + & \\ + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; + \frac{v^{(i)} \sigma_x^{(i)}}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ + & + \frac{v^{(i)} \sigma_y^{(i)}}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + & + \frac{v^{(i)} \sigma_z^{(i)}}{4} \Big ( u_z^{(1)} + u_z^{(2)} \Big ) \Big ( w_z^{(1)} + w_z^{(2)} \Big ) + \end{align} + :label: inner_product_anisotropic_faces_1 + +where superscripts :math:`(1)` and :math:`(2)` denote face 1 and face 2, respectively. +Using the contribution for each cell described in expression :eq:`inner_product_anisotropic_faces_1`, +we want to approximate the inner product in the form described by +equation :eq:`inner_product_anisotropic_faces`. To accomlish this, we construct a sparse matrix +:math:`\mathbf{P_f}` which projects quantities on the x, y and z faces separately to the +the cell centers. + +For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +are organized on cell faces as follows: + +.. math:: + \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \;\;\;\; \textrm{and} \;\;\;\; + \mathbf{w} = \begin{bmatrix} \mathbf{w_x} \\ \mathbf{w_y} \\ \mathbf{w_y} \\ \end{bmatrix} + +the approximation to the inner product is given by: + +.. math:: + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \boldsymbol{\mathbf{u} \, M_{\sigma f}} \, \mathbf{w} + +where the mass matrix has the form: + +.. math:: + \mathbf{M_{\sigma f}} = \frac{1}{4} \mathbf{P_f^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \mathbf{P_f} + +:math:`\boldsymbol{\sigma}` organizes :math:`\sigma_x`, :math:`\sigma_y` and :math:`\sigma_z` within a vector: + +.. math:: + \boldsymbol{\sigma} = \begin{bmatrix} \boldsymbol{\sigma_x} \\ \boldsymbol{\sigma_y} \\ \boldsymbol{\sigma_z} \\ \end{bmatrix} + +and + + - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) + - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\odot` is the Hadamard product + - :math:`\otimes` is the kronecker product + - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers + - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{\sigma}` is a vector that stores the tensor property values for the cells + + +.. note:: To see a validation of the discrete approximation, see our tutorials section (link) + + + + +Vectors on Cell Edges +^^^^^^^^^^^^^^^^^^^^^ + +We want to approximate the inner product between a vector quantity :math:`\vec{u}` and the product of +:math:`\Sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined +to live at cell edges. Our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: + +.. math:: + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + :label: inner_product_anisotropic_edges + +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} + +We must respect the dot product and the tensor. For vectors defined on cell edges, we discretize such that the +x-component of the vectors live on the x-edges, the y-component lives y-edges and the z-component +lives on the z-edges. This is illustrated in 2D and 3D below. By decomposing the +domain into a set of finite cells, we assume the tensor properties are spacial invariant within each cell. + +.. image:: ../images/edge_discretization.png + :align: center + :width: 600 + +As we can see there are 2 edges for each component in 2D and 4 edges for each component in 3D. +Therefore, we need to project each component of the vector from its edges to the cell centers and take their averages separately. +Since the tensor is symmetric, it has 3 independent components in 2D and 6 independent components in 3D. +Using this, we can reduce the size of the computation. + +For a single cell with volume :math:`v^{(i)}` and tensor properties defined by +:math:`\sigma_n^{(i)}` for :math:`n=1,2,3,4,5,6` the contribution towards the inner product is: + +.. math:: + \begin{align} + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; + \frac{v^{(i)} \sigma_x^{(i)}}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ + & + \frac{v^{(i)} \sigma_y^{(i)}}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + & \\ + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; + \frac{v^{(i)} \sigma_x^{(i)}}{16} \Bigg ( \sum_{n=1}^4 u_x^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_x^{(n)} \Bigg ) \\ + & + \frac{v^{(i)} \sigma_y^{(i)}}{16} \Bigg ( \sum_{n=1}^4 u_y^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_y^{(n)} \Bigg ) \\ + & + \frac{v^{(i)} \sigma_z^{(i)}}{16} \Bigg ( \sum_{n=1}^4 u_z^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_z^{(n)} \Bigg ) + \end{align} + :label: inner_product_anisotropic_edges_1 + +where the superscript :math:`(n)` denotes a particular edge. +Using the contribution for each cell described in expression :eq:`inner_product_anisotropic_edges_1`, +we want to approximate the inner product in the form described by +equation :eq:`inner_product_anisotropic_edges`. To accomlish this, we construct a sparse matrix +:math:`\mathbf{P_e}` which projects quantities on the x, y and z edges separately to the +the cell centers. + +For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +are organized on cell edges as follows: + +.. math:: + \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \;\;\;\; \textrm{and} \;\;\;\; + \mathbf{w} = \begin{bmatrix} \mathbf{w_x} \\ \mathbf{w_y} \\ \mathbf{w_y} \\ \end{bmatrix} + +the approximation to the inner product is given by: + +.. math:: + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \boldsymbol{\mathbf{u} \, M_{\sigma e} \, \mathbf{w}} + +where the mass matrix for face quantities has the form: + +.. math:: + \mathbf{M_{\sigma e}} = \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \mathbf{P_e} + +where :math:`\boldsymbol{\sigma}` organizes :math:`\sigma_x`, :math:`\sigma_y` and :math:`\sigma_z` within a vector: + +.. math:: + \boldsymbol{\sigma} = \begin{bmatrix} \boldsymbol{\sigma_x} \\ \boldsymbol{\sigma_y} \\ \boldsymbol{\sigma_z} \\ \end{bmatrix} +and + + - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) + - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\odot` is the Hadamard product + - :math:`\otimes` is the kronecker product + - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers + - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{\sigma}` is a vector that stores the tensor property values for the cells + +.. note:: To see a validation of the discrete approximation, see our tutorials section (link) \ No newline at end of file diff --git a/docs/content/inner_products_basic.rst b/docs/content/inner_products_basic.rst index 40665be58..175665743 100644 --- a/docs/content/inner_products_basic.rst +++ b/docs/content/inner_products_basic.rst @@ -36,16 +36,16 @@ discrete quantities living on various parts of the mesh are shown below: .. math:: \textrm{Scalars at centers:} \; \mathbf{M_c} &= \textrm{diag} (\mathbf{v} ) \\ \textrm{Scalars at nodes:} \; \mathbf{M_n} &= \frac{1}{2^{2k}} \mathbf{P_n^T } \textrm{diag} (\mathbf{v} ) \mathbf{P_n} \\ - \textrm{Vectors on faces:} \; \mathbf{M_f} &= \frac{1}{4} \mathbf{P_f^T } \textrm{diag} (\mathbf{I_k \otimes v} ) \mathbf{P_f} \\ - \textrm{Vectors on edges:} \; \mathbf{M_e} &= \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} (\mathbf{I_k \otimes v}) \mathbf{P_e} + \textrm{Vectors on faces:} \; \mathbf{M_f} &= \frac{1}{4} \mathbf{P_f^T } \textrm{diag} (\mathbf{e_k \otimes v} ) \mathbf{P_f} \\ + \textrm{Vectors on edges:} \; \mathbf{M_e} &= \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} (\mathbf{e_k \otimes v}) \mathbf{P_e} -where :math:`k = 1,2,3` represents the dimension (1D, 2D or 3D), -:math:`\mathbf{I_k}` is the identity matrix of dimension :math:`k\times k`, -and :math:`\otimes` is the kronecker product. :math:`\mathbf{P}` are projection -matricies that map quantities from one part of the cell (nodes, faces, edges) -to cell centers. The volumes of all cells in the mesh are store in a vector :math:`\mathbf{v}`. -The term containing :math:`k` is responsible for averaging -once quantities are mapped to cell centers. +where + + - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) + - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\otimes` is the kronecker product + - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers + - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells Scalars at Cell Centers @@ -147,9 +147,11 @@ where the mass matrix for nodal quantities has the form: .. math:: \mathbf{M_n} = \frac{1}{2^{2k}} \mathbf{P_n^T } \textrm{diag} (\mathbf{v} ) \mathbf{P_n} -and :math:`k=1,2,3` refers to the dimension (1D, 2D or 3D). The term containing :math:`k` is responsible for -averaging once nodal values have been projected to cell centers. The cell volumes are stored in -a vector :math:`\mathbf{v}`. +where + + - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) + - :math:`\mathbf{P_n}` is a projection matrix that maps quantities from nodes to cell centers + - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells .. note:: To see a validation of the discrete approximation, see our tutorials section (link) @@ -216,11 +218,15 @@ the approximation to the inner product is given by: where the mass matrix for face quantities has the form: .. math:: - \mathbf{M_f} = \frac{1}{4} \mathbf{P_f^T } \textrm{diag} (\mathbf{I_k \otimes v} ) \mathbf{P_f} + \mathbf{M_f} = \frac{1}{4} \mathbf{P_f^T } \textrm{diag} (\mathbf{e_k \otimes v} ) \mathbf{P_f} -and :math:`k=1,2,3` refers to the dimension (1D, 2D or 3D). :math:`\mathbf{I_k}` is the identity matrix and -:math:`\otimes` is the kronecker product. Cell volumes are store in a vector :math:`\mathbf{v}`. The term containing :math:`k` is responsible for -averaging once values have been projected to cell centers +and + + - :math:`k = 1,2,3` represents the dimension (1D, 2D or 3D) + - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\otimes` is the kronecker product + - :math:`\mathbf{P_f}` is the projection matrix that maps quantities from faces to cell centers + - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells .. note:: To see a validation of the discrete approximation, see our tutorials section (link) @@ -288,12 +294,15 @@ the approximation to the inner product is given by: where the mass matrix for face quantities has the form: .. math:: - \mathbf{M_e} = \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} (\mathbf{I_k \otimes v}) \mathbf{P_e} + \mathbf{M_e} = \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} (\mathbf{e_k \otimes v}) \mathbf{P_e} + +and -and :math:`k=1,2,3` refers to the dimension (1D, 2D or 3D). :math:`\mathbf{I_k}` is the identity matrix and -:math:`\otimes` is the kronecker product. Cell volumes are store in a vector :math:`\mathbf{v}`. -The term containing :math:`k` is responsible for -averaging once values have been projected to cell centers + - :math:`k = 1,2,3` represents the dimension (1D, 2D or 3D) + - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\otimes` is the kronecker product + - :math:`\mathbf{P_e}` is the projection matrix that maps quantities from edges to cell centers + - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells .. note:: To see a validation of the discrete approximation, see our tutorials section (link) diff --git a/docs/content/inner_products_constitutive.rst b/docs/content/inner_products_constitutive.rst deleted file mode 100644 index ad31a98a4..000000000 --- a/docs/content/inner_products_constitutive.rst +++ /dev/null @@ -1,6 +0,0 @@ -.. _inner_products_constitutive: - -Constitutive Relationships -************************** - - diff --git a/docs/content/inner_products_isotropic.rst b/docs/content/inner_products_isotropic.rst new file mode 100644 index 000000000..4e83d4f54 --- /dev/null +++ b/docs/content/inner_products_isotropic.rst @@ -0,0 +1,202 @@ +.. _inner_products_isotropic: + +Isotropic Constitutive Relationships +************************************ + +Summary +------- + +A constitutive relationship quantifies the response of a material to an external stimulus. +Examples include Ohm's law and Hooke's law. For practical applications of the finite volume method, +we may need to take the inner product of expressions containing constitutive relationships. + +Let :math:`\vec{v}` and :math:`\vec{w}` 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{v} = \sigma \vec{w} + :label: inner_product_isotropic + +Here we show that for isotropic constitutive relationships, the inner +product between a vector :math:`\vec{u}` and the right-hand side of +equation :eq:`inner_product_isotropic` is approximated by: + +.. math:: + (\vec{u}, \sigma \vec{w} ) = \int_\Omega \vec{u} \cdot \sigma \vec{w} \, dv \approx \boldsymbol{u^T M w} + +where :math:`\boldsymbol{M}` represents an *inner-product matrix*, and vectors +:math:`\mathbf{u}` and :math:`\mathbf{w}` are discrete variables that live +on the mesh. It is important to note a few things about the +inner-product matrix in this case: + + 1. It depends on the dimensions and discretization of the mesh + 2. It depends on where the discrete variables live; e.g. edges, faces + 3. It depends on the spacial variation of the material property :math:`\sigma` + +For this class of inner products, the corresponding form of the inner product matricies for +discrete quantities living on various parts of the mesh are given by + +.. math:: + \textrm{Vectors on faces:} \; \mathbf{M_{\sigma f}} &= \frac{1}{4} \mathbf{P_f^T } \textrm{diag} \boldsymbol{\big ( e_k \otimes (v \odot \sigma ) \big )} \mathbf{P_f} \\ + \textrm{Vectors on edges:} \; \mathbf{M_{\sigma e}} &= \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} \boldsymbol{\big ( e_k \otimes (v \odot \sigma) \big )} \mathbf{P_e} + +where + + - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) + - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\odot` is the Hadamard product + - :math:`\otimes` is the kronecker product + - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (faces, edges) to cell centers + - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{\sigma}` is a vector containing the physical property values for the cells + + +Vectors on Cell Faces +--------------------- + +We want to approximate the inner product between a vector quantity :math:`\vec{u}` and the product of +:math:`\sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined +to live on cess faces. Our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: + +.. math:: + (\vec{u}, \sigma \vec{w}) = \int_\Omega \vec{u} \cdot \sigma \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + :label: inner_product_isotropic_faces + +We must respect the dot product. For vectors defined on cell faces, we discretize such that the +x-component of the vectors live on the x-faces, the y-component lives y-faces and the z-component +lives on the z-faces. For a single cell, this is illustrated in 2D and 3D below. By decomposing the +domain into a set of finite cells, we assume the physical properties are spacial invariant within each cell. + +.. image:: ../images/face_discretization.png + :align: center + :width: 600 + +As we can see there are 2 faces for each component. Therefore, we need to project each component of the +vector from its faces to the cell centers and take their averages separately. +For a single cell with volume :math:`v_i` and material property value :math:`\sigma_i`, +the contribution towards the inner product is: + +.. math:: + \begin{align} + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i \sigma_i}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ + & + \frac{v_i \sigma_i}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + & \\ + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i \sigma_i}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ + & + \frac{v_i \sigma_i}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + & + \frac{v_i \sigma_i}{4} \Big ( u_z^{(1)} + u_z^{(2)} \Big ) \Big ( w_z^{(1)} + w_z^{(2)} \Big ) + \end{align} + :label: inner_product_isotropic_faces_1 + +where superscripts :math:`(1)` and :math:`(2)` denote face 1 and face 2, respectively. +Using the contribution for each cell described in expression :eq:`inner_product_isotropic_faces_1`, +we want to approximate the inner product in the form described by +equation :eq:`inner_product_isotropic_faces`. To accomlish this, we construct a sparse matrix +:math:`\mathbf{P_f}` which projects quantities on the x, y and z faces separately to the +the cell centers. + +For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +are organized on cell faces as follows: + +.. math:: + \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \;\;\;\; \textrm{and} \;\;\;\; + \mathbf{w} = \begin{bmatrix} \mathbf{e_x} \\ \mathbf{e_y} \\ \mathbf{e_y} \\ \end{bmatrix} + +the approximation to the inner product is given by: + +.. math:: + (\vec{u}, \sigma \vec{w}) = \int_\Omega \vec{u} \cdot \sigma \vec{w} \, dv \approx \boldsymbol{\mathbf{u} \, M_{\sigma f}} \, \mathbf{w} + +where the mass matrix has the form: + +.. math:: + \mathbf{M_{\sigma f}} = \frac{1}{4} \mathbf{P_f^T } \textrm{diag} \boldsymbol{\big ( e_k \otimes (v \odot \sigma ) \big )} \mathbf{P_f} + +and + + - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) + - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\odot` is the Hadamard product + - :math:`\otimes` is the kronecker product + - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers + - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{\sigma}` is a vector containing the physical property values for the cells + + +.. note:: To see a validation of the discrete approximation, see our tutorials section (link) + + +Vectors on Cell Edges +--------------------- + +We want to approximate the inner product between a vector quantity :math:`\vec{u}` and the product of +:math:`\sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined +to live at cell edges. Our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: + +.. math:: + (\vec{u}, \sigma \vec{w}) = \int_\Omega \vec{u} \cdot \sigma \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + :label: inner_product_isotropic_edges + +We must respect the dot product. For vectors defined on cell edges, we discretize such that the +x-component of the vectors live on the x-edges, the y-component lives y-edges and the z-component +lives on the z-edges. This is illustrated in 2D and 3D below. By decomposing the +domain into a set of finite cells, we assume the material properties are spacial invariant within each cell. + +.. image:: ../images/edge_discretization.png + :align: center + :width: 600 + +As we can see there are 2 edges for each component in 2D and 4 edges for each component in 3D. +Therefore, we need to project each component of the +vector from its edges to the cell centers and take their averages separately. For a single cell with volume :math:`v_i` +and material property value :math:`\sigma_i`, the contribution towards the inner product is: + +.. math:: + \begin{align} + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i \sigma_i}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ + & + \frac{v_i \sigma_i}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + & \\ + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v_i \sigma_i}{16} \Bigg ( \sum_{n=1}^4 u_x^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_x^{(n)} \Bigg ) \\ + & + \frac{v_i \sigma_i}{16} \Bigg ( \sum_{n=1}^4 u_y^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_y^{(n)} \Bigg ) \\ + & + \frac{v_i \sigma_i}{16} \Bigg ( \sum_{n=1}^4 u_z^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_z^{(n)} \Bigg ) + \end{align} + :label: inner_product_isotropic_edges_1 + +where the superscript :math:`(n)` denotes a particular edge. +Using the contribution for each cell described in expression :eq:`inner_product_isotropic_edges_1`, +we want to approximate the inner product in the form described by +equation :eq:`inner_product_isotropic_edges`. To accomlish this, we construct a sparse matrix +:math:`\mathbf{P_e}` which projects quantities on the x, y and z edges separately to the +the cell centers. + +For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +are organized on cell edges as follows: + +.. math:: + \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \;\;\;\; \textrm{and} \;\;\;\; + \mathbf{w} = \begin{bmatrix} \mathbf{e_x} \\ \mathbf{e_y} \\ \mathbf{e_y} \\ \end{bmatrix} + +the approximation to the inner product is given by: + +.. math:: + (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \mathbf{\mathbf{u} \, M_e \, \mathbf{w}} + +where the mass matrix for face quantities has the form: + +.. math:: + \mathbf{M_{\sigma e}} = \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} \boldsymbol{\big ( e_k \otimes (v \odot \sigma) \big )} \mathbf{P_e} + +and + + - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) + - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\odot` is the Hadamard product + - :math:`\otimes` is the kronecker product + - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers + - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{\sigma}` is a vector containing the physical property values for the cells + +.. note:: To see a validation of the discrete approximation, see our tutorials section (link) + From 39a8b1caf6cee6cd67664da96dbfabcf51081aeb Mon Sep 17 00:00:00 2001 From: dccowan Date: Wed, 18 Nov 2020 22:24:44 -0800 Subject: [PATCH 04/35] Update tutorials and theory to have consistent vector notation --- docs/content/inner_products.rst | 1 + docs/content/inner_products_differential.rst | 111 ++++++++++++++++++ tutorials/inner_products/1_basic.py | 58 +++++---- .../inner_products/2_physical_properties.py | 40 +++---- tutorials/inner_products/3_calculus.py | 34 +++--- 5 files changed, 176 insertions(+), 68 deletions(-) create mode 100644 docs/content/inner_products_differential.rst diff --git a/docs/content/inner_products.rst b/docs/content/inner_products.rst index 332b61551..e5593ca9a 100644 --- a/docs/content/inner_products.rst +++ b/docs/content/inner_products.rst @@ -36,6 +36,7 @@ discrete approximation for several classes of inner products. inner_products_basic inner_products_isotropic inner_products_anisotropic + inner_products_differential diff --git a/docs/content/inner_products_differential.rst b/docs/content/inner_products_differential.rst new file mode 100644 index 000000000..ea7506f2d --- /dev/null +++ b/docs/content/inner_products_differential.rst @@ -0,0 +1,111 @@ +.. _inner_products_differential: + +Differential Operators +********************** + +Summary +------- + +For practical applications of the finite volume method, +we may need to take the inner product of expressions containing differential operators. +These operators include: + + - the gradient (:math:`\nabla`) + - the divergence (:math:`\nabla \cdot \;`) + - the curl (:math:`\nabla \times \;`) + +For scalar quantities :math:`\psi` and :math:`\phi` and for vector quantities :math:`\vec{u}` and :math:`\vec{w}` +we are interested in approximating the following inner products: + +.. math:: + \begin{align} + (\vec{u}, \nabla \phi ) &= \int_\Omega \vec{u} \cdot \nabla \phi \, dv\\ + (\psi, \nabla \cdot \vec{w} ) &= \int_\Omega \psi \; (\nabla \cdot \vec{w}) \, dv \\ + (\vec{u}, \nabla \times \vec{w} ) &= \int_\Omega \vec{u} \cdot (\nabla \times \vec{w} ) \, dv + \end{align} + +**Gradient:** + +For the inner product between a vector (:math:`\vec{u}`) and the gradient of a scalar (:math:`\phi`), +there are two options. Either: + + (1) :math:`\boldsymbol{\phi}` is defined on the nodes and :math:`\boldsymbol{u}` is defined on cell edges, or + (2) :math:`\boldsymbol{\phi}` is defined at cell centers and :math:`\boldsymbol{u}` is defined on cell faces + +For each case, the approximation to the inner product is given by: + +.. math:: + \begin{align} + \textrm{Case 1:} \; (\vec{u}, \nabla \phi ) &\approx \boldsymbol{u^T M_e G_n \, \phi + B.C.} \\ + \textrm{Case 2:} \; (\vec{u}, \nabla \phi ) &\approx - \boldsymbol{u^T D^T M_c \, \phi + B.C.} + \end{align} + +where + + - :math:`\mathbf{G_n}` is a discrete gradient operator that maps from nodes to edges + - :math:`\mathbf{D}` is a discrete divergence operator whose transpose acts as a gradient operator from faces to cell centers + - :math:`\mathbf{M_c}` is the :ref:`basic inner product matrix for vectors at cell centers ` + - :math:`\mathbf{M_e}` is the :ref:`basic inner product matrix for vectors at edges ` + - :math:`\mathbf{BC}` represents an additional contribution accounting for the boundary conditions + +**Divergence:** + +For the inner product between a scalar (:math:`\psi`) and the divergence of a vector (:math:`\vec{w}`), +there are two options. Either: + + (1) :math:`\boldsymbol{\psi}` is defined at cell centers :math:`\boldsymbol{w}` is defined on cell faces + (2) :math:`\boldsymbol{\psi}` is defined on the nodes and :math:`\boldsymbol{w}` is defined on cell edges, or + +For each case, the approximation to the inner product is given by: + +.. math:: + \begin{align} + \textrm{Case 1:} \; (\psi, \nabla \cdot \vec{w} ) &\approx \boldsymbol{\psi^T M_c D \, w + B.C.} \\ + \textrm{Case 2:} \; (\psi, \nabla \cdot \vec{w} ) &\approx - \boldsymbol{\psi^T G_n^T M_e \, w + B.C.} + \end{align} + +where + + - :math:`\mathbf{D}` is a discrete divergence operator from faces to cell centers + - :math:`\mathbf{G_n}` is a discrete gradient operator whose transpose acts as a divergence operator from edges to nodes + - :math:`\mathbf{M_c}` is the :ref:`basic inner product matrix for vectors at cell centers ` + - :math:`\mathbf{M_e}` is the :ref:`basic inner product matrix for vectors at edges ` + - :math:`\mathbf{BC}` represents an additional contribution accounting for the boundary conditions + +**Curl:** + +For the inner product between a vector (:math:`\vec{u}`) and the curl of another vector (:math:`\vec{w}`), +there are two options. Either: + + (1) :math:`\boldsymbol{u}` is defined on the faces and :math:`\boldsymbol{w}` is defined on cell edges, or + (2) :math:`\boldsymbol{u}` is defined on the edges :math:`\boldsymbol{w}` is defined on cell faces + +For each case, the approximation to the inner product is given by: + +.. math:: + \begin{align} + \textrm{Case 1:} \; (\vec{u}, \nabla \times \vec{w} ) &\approx \boldsymbol{u^T M_f C \, w + B.C.} \\ + \textrm{Case 2:} \; (\vec{u}, \nabla \times \vec{w} ) &\approx \boldsymbol{u^T C^T M_f \, w + B.C.} + \end{align} + +where + + - :math:`\mathbf{C}` is a discrete curl operator from edges to faces, whose transpose acts as a gradient operator from faces to edges + - :math:`\mathbf{M_f}` is the :ref:`basic inner product matrix for vectors on cell faces ` + - :math:`\mathbf{BC}` represents an additional contribution accounting for the boundary conditions + +Vector and the Gradient of a Scalar +----------------------------------- + + + + +Scalar and the Divergence of a Vector +------------------------------------- + + + + + +Vector and the Curl of a Vector +------------------------------- \ No newline at end of file diff --git a/tutorials/inner_products/1_basic.py b/tutorials/inner_products/1_basic.py index 426290631..c3d8db8c5 100644 --- a/tutorials/inner_products/1_basic.py +++ b/tutorials/inner_products/1_basic.py @@ -16,11 +16,11 @@ (\\psi , \\phi ) = \\int_\\Omega \\psi \\, \\phi \\, dv -And for vector quantities :math:`\\vec{u}` and :math:`\\vec{v}`, the +And for vector quantities :math:`\\vec{u}` and :math:`\\vec{w}`, the inner product is given by: .. math:: - (\\vec{u}, \\vec{v}) = \\int_\\Omega \\vec{u} \\cdot \\vec{v} \\, dv + (\\vec{u}, \\vec{w}) = \\int_\\Omega \\vec{u} \\cdot \\vec{w} \\, dv In discretized form, we can approximate the aforementioned inner-products as: @@ -32,12 +32,12 @@ and .. math:: - (\\vec{u}, \\vec{v}) \\approx \\mathbf{u^T \\, M \\, v} + (\\vec{u}, \\vec{w}) \\approx \\mathbf{u^T \\, M \\, w} where :math:`\\mathbf{M}` in either equation represents an *inner-product matrix*. :math:`\\mathbf{\\psi}`, :math:`\\mathbf{\\phi}`, -:math:`\\mathbf{u}` and :math:`\\mathbf{v}` are discrete variables that live +:math:`\\mathbf{u}` and :math:`\\mathbf{w}` are discrete variables that live on the mesh. It is important to note a few things about the inner-product matrix in this case: @@ -50,13 +50,16 @@ .. math:: \\textrm{Centers:} \\; \\mathbf{M_c} &= \\textrm{diag} (\\mathbf{v} ) \n \\textrm{Nodes:} \\; \\mathbf{M_n} &= \\frac{1}{2^{2k}} \\mathbf{P_n^T } \\textrm{diag} (\\mathbf{v} ) \\mathbf{P_n} \n - \\textrm{Faces:} \\; \\mathbf{M_f} &= \\frac{1}{4} \\mathbf{P_f^T } \\textrm{diag} (\\mathbf{I_k \\otimes v} ) \\mathbf{P_f} \n - \\textrm{Edges:} \\; \\mathbf{M_e} &= \\frac{1}{4^{k-1}} \\mathbf{P_e^T } \\textrm{diag} (\\mathbf{I_k \\otimes v}) \\mathbf{P_e} + \\textrm{Faces:} \\; \\mathbf{M_f} &= \\frac{1}{4} \\mathbf{P_f^T } \\textrm{diag} (\\mathbf{e_k \\otimes v} ) \\mathbf{P_f} \n + \\textrm{Edges:} \\; \\mathbf{M_e} &= \\frac{1}{4^{k-1}} \\mathbf{P_e^T } \\textrm{diag} (\\mathbf{e_k \\otimes v}) \\mathbf{P_e} -where :math:`k = 1,2,3`, :math:`\\mathbf{I_k}` is the identity matrix and -:math:`\\otimes` is the kronecker product. :math:`\\mathbf{P}` are projection -matricies that map quantities from one part of the cell (nodes, faces, edges) -to cell centers. +where + + - :math:`k = 1,2,3` is the dimension (1D, 2D or 3D) + - :math:`\\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\\otimes` is the kronecker product + - :math:`\\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers + - :math:`\\mathbf{v}` is a vector that contains the cell volumes """ @@ -149,13 +152,13 @@ def fcn_gaussian(x, mu, sig): # will define a 2D vector quantity: # # .. math:: -# \vec{v}(x,y) = \Bigg [ \frac{-y}{r} \hat{x} + \frac{x}{r} \hat{y} \Bigg ] +# \vec{u}(x,y) = \Bigg [ \frac{-y}{r} \hat{x} + \frac{x}{r} \hat{y} \Bigg ] # \, e^{-\frac{x^2+y^2}{2\sigma^2}} # # We will then evaluate the following inner product: # # .. math:: -# (\vec{v}, \vec{v}) = \int_\Omega \vec{v} \cdot \vec{v} \, da +# (\vec{u}, \vec{u}) = \int_\Omega \vec{u} \cdot \vec{u} \, da # = 2 \pi \sigma^2 # # using inner-product matricies. Next we compare the numerical evaluation @@ -185,33 +188,33 @@ def fcn_y(xy, sig): sig = 1.5 # Evaluate inner-product using edge-defined discrete variables -vx = fcn_x(mesh.gridEx, sig) -vy = fcn_y(mesh.gridEy, sig) -v = np.r_[vx, vy] +ux = fcn_x(mesh.gridEx, sig) +uy = fcn_y(mesh.gridEy, sig) +u = np.r_[ux, uy] Me = mesh.getEdgeInnerProduct() # Edge inner product matrix -ipe = np.dot(v, Me * v) +ipe = np.dot(u, Me * u) # Evaluate inner-product using face-defined discrete variables -vx = fcn_x(mesh.gridFx, sig) -vy = fcn_y(mesh.gridFy, sig) -v = np.r_[vx, vy] +ux = fcn_x(mesh.gridFx, sig) +uy = fcn_y(mesh.gridFy, sig) +u = np.r_[ux, uy] Mf = mesh.getFaceInnerProduct() # Edge inner product matrix -ipf = np.dot(v, Mf * v) +ipf = np.dot(u, Mf * u) -# The analytic solution of (v, v) +# The analytic solution of (u, u) ipt = np.pi * sig ** 2 # Plot the vector function fig = plt.figure(figsize=(5, 5)) ax = fig.add_subplot(111) mesh.plotImage( - v, ax=ax, v_type="F", view="vec", stream_opts={"color": "w", "density": 1.0} + u, ax=ax, v_type="F", view="vec", stream_opts={"color": "w", "density": 1.0} ) -ax.set_title("v at cell faces") +ax.set_title("u at cell faces") fig.show() @@ -232,7 +235,7 @@ def fcn_y(xy, sig): # L2-norm for each: # # .. math:: -# \| \mathbf{v - M^{-1} M v} \|^2 +# \| \mathbf{u - M^{-1} M u} \|^2 # # @@ -245,10 +248,6 @@ def fcn_y(xy, sig): Mc = sdiag(mesh.vol) Mc_inv = sdiag(1 / mesh.vol) -# Nodes for scalar quantities (*functionality pending*) -# Mn = mesh.getNodalInnerProduct() -# Mn_inv = mesh.getNodalInnerProduct(invMat=True) - # Edges for vector quantities Me = mesh.getEdgeInnerProduct() Me_inv = mesh.getEdgeInnerProduct(invMat=True) @@ -259,19 +258,16 @@ def fcn_y(xy, sig): # Generate some random vectors phi_c = np.random.rand(mesh.nC) -phi_n = np.random.rand(mesh.nN) vec_e = np.random.rand(mesh.nE) vec_f = np.random.rand(mesh.nF) # Generate some random vectors norm_c = np.linalg.norm(phi_c - Mc_inv.dot(Mc.dot(phi_c))) -# norm_n = np.linalg.norm(phi_n - Mn_inv*Mn*phi_n) norm_e = np.linalg.norm(vec_e - Me_inv * Me * vec_e) norm_f = np.linalg.norm(vec_f - Mf_inv * Mf * vec_f) # Verify accuracy print("ACCURACY") print("Norm for centers:", norm_c) -# print('Norm for nodes: ', norm_n) print("Norm for edges: ", norm_e) print("Norm for faces: ", norm_f) diff --git a/tutorials/inner_products/2_physical_properties.py b/tutorials/inner_products/2_physical_properties.py index 0b5d9e71e..2b2a3b1ad 100644 --- a/tutorials/inner_products/2_physical_properties.py +++ b/tutorials/inner_products/2_physical_properties.py @@ -10,36 +10,36 @@ - 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 +Let :math:`\\vec{v}` and :math:`\\vec{w}` 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} + \\vec{v} = \\sigma \\vec{w} -The inner product between a vector :math:`\\vec{v}` and the right-hand side +The inner product between a vector :math:`\\vec{u}` 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 + (\\vec{u}, \\sigma \\vec{w} ) = \\int_\\Omega \\vec{v} \\cdot \\sigma \\vec{w} \\, 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} + (\\vec{u}, \\sigma \\vec{w} ) \\approx \\mathbf{u^T M w} -where the inner product matrix :math:`\\mathbf{M_\\sigma}` now depends on: +where the inner product matrix :math:`\\mathbf{M}` now depends on: 1. the dimensions and discretization of the mesh - 2. where :math:`\\mathbf{v}` and :math:`\\mathbf{e}` live + 2. where :math:`\\mathbf{u}` and :math:`\\mathbf{w}` 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} + \\vec{v} = \\Sigma \\vec{w} where @@ -49,17 +49,17 @@ \\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 +a vector :math:`\\vec{u}` 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 + (\\vec{u}, \\Sigma \\vec{w} ) = \\int_\\Omega \\vec{u} \\cdot \\Sigma \\vec{w} \\, dv Once again we would like to approximate the inner product numerically using an -*inner-product matrix* :math:`\\mathbf{M_\\Sigma}` such that: +*inner-product matrix* :math:`\\mathbf{M}` such that: .. math:: - (\\vec{v}, \\Sigma \\vec{E} ) \\approx \\mathbf{v^T M_\\Sigma e} + (\\vec{u}, \\Sigma \\vec{w} ) \\approx \\mathbf{u^T M w} @@ -265,25 +265,25 @@ # In this case, the constitutive relation is given by: # # .. math:: -# \vec{J} = \frac{1}{\rho} \vec{E} +# \vec{v} = \frac{1}{\rho} \vec{w} # -# The inner product between a vector :math:`\\vec{v}` and the right-hand side +# The inner product between a vector :math:`\\vec{u}` 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 +# (\vec{u}, \rho^{-1} \vec{w} ) = \int_\Omega \vec{u} \cdot \rho^{-1} \vec{w} \, 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} +# (\vec{u}, \rho^{-1} \vec{w} ) \approx \mathbf{u^T M_{\rho^{-1}} w} # # In the case that the constitutive relation is defined by a # tensor :math:`P`, e.g.: # # .. math:: -# \vec{J} = P \vec{E} +# \vec{v} = P \vec{w} # # where # @@ -292,17 +292,17 @@ # \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 +# The inner product between a vector :math:`\vec{u}` 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 +# (\vec{u}, P \vec{w} ) = \int_\Omega \vec{u} \cdot P \vec{w} \, 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} +# (\vec{u}, P \vec{w} ) \approx \mathbf{u^T M_P w} # # Here we demonstrate how to form the inner-product matricies # :math:`\mathbf{M_{\rho^{-1}}}` and :math:`\mathbf{M_P}`. diff --git a/tutorials/inner_products/3_calculus.py b/tutorials/inner_products/3_calculus.py index 81fda6834..d3aa3d2a2 100644 --- a/tutorials/inner_products/3_calculus.py +++ b/tutorials/inner_products/3_calculus.py @@ -5,13 +5,13 @@ When solving PDEs using the finite volume approach, inner products may contain differential operators. Where :math:`\\psi` and :math:`\\phi` are -scalar quantities, and :math:`\\vec{u}` and :math:`\\vec{v}` are vector +scalar quantities, and :math:`\\vec{u}` and :math:`\\vec{w}` are vector quantities, we may need to derive a discrete approximation for the following inner products: 1. :math:`(\\vec{u} , \\nabla \\phi)` - 2. :math:`(\\psi , \\nabla \\cdot \\vec{v})` - 3. :math:`(\\vec{u} , \\nabla \\times \\vec{v})` + 2. :math:`(\\psi , \\nabla \\cdot \\vec{w})` + 3. :math:`(\\vec{u} , \\nabla \\times \\vec{w})` 4. :math:`(\\psi, \\Delta^2 \\phi)` In this section, we demonstrate how to go from the inner product to the @@ -102,21 +102,21 @@ # Divergence # ---------- # -# Where :math:`\psi` is a scalar quantity and :math:`\vec{v}` is a vector +# Where :math:`\psi` is a scalar quantity and :math:`\vec{w}` is a vector # quantity, we would like to evaluate the following inner product: # # .. math:: -# (\psi , \nabla \cdot \vec{v}) = \int_\Omega \psi \nabla \cdot \vec{v} \, dv +# (\psi , \nabla \cdot \vec{w}) = \int_\Omega \psi \nabla \cdot \vec{w} \, dv # # The divergence defines a measure of the flux leaving/entering a volume. As a -# result, it is natural for :math:`\vec{v}` to be a flux defined on cell faces. +# result, it is natural for :math:`\vec{w}` to be a flux defined on cell faces. # The face divergence operator (:math:`\mathbf{D}`) maps from cell faces to -# cell centers, therefore # we should define :math:`\psi` at cell centers. The +# cell centers, therefore we should define :math:`\psi` at cell centers. The # inner product is ultimately computed using an inner product matrix -# (:math:`\mathbf{M_f}`) for quantities living on cell faces, e.g.: +# (:math:`\mathbf{M_c}`) for quantities living on cell faces, thus: # # .. math:: -# (\psi , \nabla \cdot \vec{v}) \approx \mathbf{\psi^T} \textrm{diag} (\mathbf{vol} ) \mathbf{D v} +# (\psi , \nabla \cdot \vec{w}) \approx \boldsymbol{\psi^T M_c D w} # # Make basic mesh @@ -138,30 +138,30 @@ # Curl # ---- # -# Where :math:`\vec{u}` and :math:`\vec{v}` are vector quantities, we would +# Where :math:`\vec{u}` and :math:`\vec{w}` are vector quantities, we would # like to evaluate the following inner product: # # .. math:: -# (\vec{u} , \nabla \times \vec{v}) = \int_\Omega \vec{u} \nabla \times \vec{v} \, dv +# (\vec{u} , \nabla \times \vec{w}) = \int_\Omega \vec{u} \nabla \times \vec{w} \, dv # # **Inner Product at Faces:** # -# Let :math:`\vec{u}` denote a flux and let :math:`\vec{v}` denote a field. +# Let :math:`\vec{u}` denote a flux and let :math:`\vec{w}` denote a field. # In this case, it is natural for the flux :math:`\vec{u}` to live on cell -# faces and for the field :math:`\vec{v}` to live on cell edges. The discrete +# faces and for the field :math:`\vec{w}` to live on cell edges. The discrete # curl operator (:math:`\mathbf{C_e}`) in this case naturally maps from cell # edges to cell faces without the need to define boundary conditions. The # inner product can be approxiated using an inner product matrix # (:math:`\mathbf{M_f}`) for quantities living on cell faces, e.g.: # # .. math:: -# (\vec{u} , \nabla \times \vec{v}) \approx \mathbf{u^T M_f C_e v} +# (\vec{u} , \nabla \times \vec{w}) \approx \mathbf{u^T M_f C_e w} # # **Inner Product at Edges:** # -# Now let :math:`\vec{u}` denote a field and let :math:`\vec{v}` denote a flux. +# Now let :math:`\vec{u}` denote a field and let :math:`\vec{w}` denote a flux. # Now it is natural for the :math:`\vec{u}` to live on cell edges -# and for :math:`\vec{v}` to live on cell faces. We would like to compute the +# and for :math:`\vec{w}` to live on cell faces. We would like to compute the # inner product using an inner product matrix (:math:`\mathbf{M_e}`) for # quantities living on cell edges. However, this requires a discrete curl # operator (:math:`\mathbf{C_f}`) that maps from cell faces @@ -169,7 +169,7 @@ # If done successfully: # # .. math:: -# (\vec{u} , \nabla \times \vec{v}) \approx \mathbf{u^T M_e C_f v} +# (\vec{u} , \nabla \times \vec{w}) \approx \mathbf{u^T M_e C_f v} # # Make basic mesh From 05ba1f6271ffdaba1726b709ad8c96da8d2ed2aa Mon Sep 17 00:00:00 2001 From: dccowan Date: Wed, 18 Nov 2020 22:50:21 -0800 Subject: [PATCH 05/35] Add link to theory and update tutorials for latest difinition of origin property --- docs/content/inner_products_anisotropic.rst | 14 ++++++-------- docs/content/inner_products_basic.rst | 11 ++++++----- docs/content/inner_products_differential.rst | 2 ++ docs/content/inner_products_isotropic.rst | 6 ++++-- tutorials/mesh_generation/4_tree_mesh.py | 18 +++++++++--------- 5 files changed, 27 insertions(+), 24 deletions(-) diff --git a/docs/content/inner_products_anisotropic.rst b/docs/content/inner_products_anisotropic.rst index 5ecbc71ce..64b95278e 100644 --- a/docs/content/inner_products_anisotropic.rst +++ b/docs/content/inner_products_anisotropic.rst @@ -46,6 +46,8 @@ discrete quantities living on various parts of the mesh are shown below. We let: - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells +.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` + **Diagonal Anisotropic:** For the diagonal anisotropic case, the tensor characterzing the material properties @@ -79,8 +81,6 @@ has the form: \sigma_{5} & \sigma_{6} & \sigma_{3} \end{bmatrix} - - Diagonally Anisotropic Case --------------------------- @@ -113,7 +113,7 @@ domain into a set of finite cells, we assume the tensor properties are spacial i As we can see there are 2 faces for each component. Therefore, we need to project each component of the vector from its faces to the cell centers and take their averages separately. We must also recognize that -x-components are only multiplied by :math:`\sigma_x`, y_components by :math:`\sigma_y` and z-components +x-components are only multiplied by :math:`\sigma_x`, y-components by :math:`\sigma_y` and z-components by :math:`\sigma_z`. For a single cell with volume :math:`v^{(i)}` and tensor properties defined by @@ -208,7 +208,7 @@ As we can see there are 2 edges for each component in 2D and 4 edges for each co Therefore, we need to project each component of the vector from its edges to the cell centers and take their averages separately. We must also recognize that -x-components are only multiplied by :math:`\sigma_x`, y_components by :math:`\sigma_y` and z-components +x-components are only multiplied by :math:`\sigma_x`, y-components by :math:`\sigma_y` and z-components by :math:`\sigma_z`. For a single cell with volume :math:`v^{(i)}` and tensor properties defined by @@ -267,9 +267,7 @@ and - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells - :math:`\boldsymbol{\sigma}` is a vector that stores the tensor property values for the cells -.. note:: To see a validation of the discrete approximation, see our tutorials section (link) - - +.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` Fully Anisotropic Case ---------------------- @@ -303,7 +301,7 @@ domain into a set of finite cells, we assume the tensor properties are spacial i As we can see there are 2 faces for each component. Therefore, we need to project each component of the vector from its faces to the cell centers and take their averages separately. We must also recognize that -x-components are only multiplied by :math:`\sigma_x`, y_components by :math:`\sigma_y` and z-components +x-components are only multiplied by :math:`\sigma_x`, y-components by :math:`\sigma_y` and z-components by :math:`\sigma_z`. For a single cell with volume :math:`v^{(i)}` and tensor properties defined by diff --git a/docs/content/inner_products_basic.rst b/docs/content/inner_products_basic.rst index 175665743..d1efffe94 100644 --- a/docs/content/inner_products_basic.rst +++ b/docs/content/inner_products_basic.rst @@ -47,6 +47,8 @@ where - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells +.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` + Scalars at Cell Centers ----------------------- @@ -90,7 +92,7 @@ the cell volumes (:math:`\mathbf{v}`), i.e.: \mathbf{M_c} = diag(\mathbf{v}) -.. note:: To see a validation of the discrete approximation, see our tutorials section (link) +.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` Scalars at Nodes @@ -153,7 +155,7 @@ where - :math:`\mathbf{P_n}` is a projection matrix that maps quantities from nodes to cell centers - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells -.. note:: To see a validation of the discrete approximation, see our tutorials section (link) +.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` Vectors on Cell Faces @@ -228,8 +230,7 @@ and - :math:`\mathbf{P_f}` is the projection matrix that maps quantities from faces to cell centers - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells -.. note:: To see a validation of the discrete approximation, see our tutorials section (link) - +.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` Vectors on Cell Edges @@ -304,7 +305,7 @@ and - :math:`\mathbf{P_e}` is the projection matrix that maps quantities from edges to cell centers - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells -.. note:: To see a validation of the discrete approximation, see our tutorials section (link) +.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` diff --git a/docs/content/inner_products_differential.rst b/docs/content/inner_products_differential.rst index ea7506f2d..759dcc1f3 100644 --- a/docs/content/inner_products_differential.rst +++ b/docs/content/inner_products_differential.rst @@ -24,6 +24,8 @@ we are interested in approximating the following inner products: (\vec{u}, \nabla \times \vec{w} ) &= \int_\Omega \vec{u} \cdot (\nabla \times \vec{w} ) \, dv \end{align} +.. note:: To construct differential operators and/or approximate inner products of this type, see the :ref:`tutorial on inner products with differential operators ` + **Gradient:** For the inner product between a vector (:math:`\vec{u}`) and the gradient of a scalar (:math:`\phi`), diff --git a/docs/content/inner_products_isotropic.rst b/docs/content/inner_products_isotropic.rst index 4e83d4f54..b21b1b069 100644 --- a/docs/content/inner_products_isotropic.rst +++ b/docs/content/inner_products_isotropic.rst @@ -51,6 +51,8 @@ where - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells - :math:`\boldsymbol{\sigma}` is a vector containing the physical property values for the cells +.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` + Vectors on Cell Faces --------------------- @@ -124,7 +126,7 @@ and - :math:`\boldsymbol{\sigma}` is a vector containing the physical property values for the cells -.. note:: To see a validation of the discrete approximation, see our tutorials section (link) +.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` Vectors on Cell Edges @@ -198,5 +200,5 @@ and - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells - :math:`\boldsymbol{\sigma}` is a vector containing the physical property values for the cells -.. note:: To see a validation of the discrete approximation, see our tutorials section (link) +.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` diff --git a/tutorials/mesh_generation/4_tree_mesh.py b/tutorials/mesh_generation/4_tree_mesh.py index a857a9de9..e252eddf3 100644 --- a/tutorials/mesh_generation/4_tree_mesh.py +++ b/tutorials/mesh_generation/4_tree_mesh.py @@ -95,7 +95,7 @@ # Define the base mesh hx = [(dx, nbcx)] hy = [(dy, nbcy)] -mesh = TreeMesh([hx, hy], x0="CC") +mesh = TreeMesh([hx, hy], origin="CC") # Refine surface topography xx = mesh.vectorNx @@ -117,8 +117,8 @@ fig = plt.figure(figsize=(6, 6)) ax = fig.add_subplot(111) mesh.plotGrid(ax=ax) -ax.set_xbound(mesh.x0[0], mesh.x0[0] + np.sum(mesh.hx)) -ax.set_ybound(mesh.x0[1], mesh.x0[1] + np.sum(mesh.hy)) +ax.set_xbound(mesh.origin[0], mesh.origin[0] + np.sum(mesh.hx)) +ax.set_ybound(mesh.origin[1], mesh.origin[1] + np.sum(mesh.hy)) ax.set_title("QuadTree Mesh") #################################################### @@ -142,7 +142,7 @@ # Define the base mesh hx = [(dx, nbcx)] hy = [(dy, nbcy)] -mesh = TreeMesh([hx, hy], x0="CC") +mesh = TreeMesh([hx, hy], origin="CC") # Refine surface topography xx = mesh.vectorNx @@ -161,7 +161,7 @@ mesh.finalize() # The bottom west corner -x0 = mesh.x0 +origin = mesh.origin # The total number of cells nC = mesh.nC @@ -178,8 +178,8 @@ fig = plt.figure(figsize=(6, 6)) ax = fig.add_subplot(111) mesh.plotImage(np.log10(s), grid=True, ax=ax) -ax.set_xbound(mesh.x0[0], mesh.x0[0] + np.sum(mesh.hx)) -ax.set_ybound(mesh.x0[1], mesh.x0[1] + np.sum(mesh.hy)) +ax.set_xbound(mesh.origin[0], mesh.origin[0] + np.sum(mesh.hx)) +ax.set_ybound(mesh.origin[1], mesh.origin[1] + np.sum(mesh.hy)) ax.set_title("Log of Cell Areas") ############################################### @@ -207,7 +207,7 @@ hx = [(dx, nbcx)] hy = [(dy, nbcy)] hz = [(dz, nbcz)] -mesh = TreeMesh([hx, hy, hz], x0="CCC") +mesh = TreeMesh([hx, hy, hz], origin="CCC") # Refine surface topography [xx, yy] = np.meshgrid(mesh.vectorNx, mesh.vectorNy) @@ -226,7 +226,7 @@ mesh.finalize() # The bottom west corner -x0 = mesh.x0 +origin = mesh.origin # The total number of cells nC = mesh.nC From 306bef5f4727fff7e650b83629a671cab75159b9 Mon Sep 17 00:00:00 2001 From: dccowan Date: Thu, 19 Nov 2020 17:32:40 -0800 Subject: [PATCH 06/35] Clean up some typos and improvements to formulations --- docs/content/inner_products_anisotropic.rst | 341 ++++++++++--------- docs/content/inner_products_basic.rst | 98 +++--- docs/content/inner_products_differential.rst | 68 ++-- docs/content/inner_products_isotropic.rst | 64 ++-- 4 files changed, 292 insertions(+), 279 deletions(-) diff --git a/docs/content/inner_products_anisotropic.rst b/docs/content/inner_products_anisotropic.rst index 64b95278e..56aa6ccb7 100644 --- a/docs/content/inner_products_anisotropic.rst +++ b/docs/content/inner_products_anisotropic.rst @@ -3,6 +3,8 @@ Anisotropic Constitutive Relationships ************************************** +.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` + Summary ------- @@ -14,21 +16,32 @@ Let :math:`\vec{v}` and :math:`\vec{w}` be two physically related quantities. If their relationship is anisotropic (defined by a tensor :math:`\Sigma`), the constitutive relation is of the form: .. math:: - \vec{v} = \Sigma \vec{w} \;\;\;\;\; \textrm{where} \;\;\;\;\; - \Sigma = \begin{bmatrix} \sigma_{1} & \sigma_{4} & \sigma_{5} \\ - \sigma_{4} & \sigma_{2} & \sigma_{6} \\ - \sigma_{5} & \sigma_{6} & \sigma_{3} \end{bmatrix} + \vec{v} = \Sigma \vec{w} :label: inner_product_anisotropic +where + +.. math:: + \mathbf{In \; 2D:} \; + \Sigma = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} \\ + \sigma_{yx} & \sigma_{yy} \end{bmatrix} + \;\;\;\;\;\;\;\; \mathbf{In \; 3D:} \; + \Sigma = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ + \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ + \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix} + +Note that for real materials, the tensor is symmetric and has 6 independent variables +(i.e. :math:`\sigma_{pq}=\sigma_{qp}` for :math:`p,q=x,y,z`). Here we show that for anisotropic constitutive relationships, the inner product between a vector :math:`\vec{u}` and the right-hand side of equation :eq:`inner_product_anisotropic` is approximated by: .. math:: (\vec{u}, \Sigma \vec{w} ) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \boldsymbol{u^T M w} + :label: inner_product_anisotropic_general where :math:`\boldsymbol{M}` represents an *inner-product matrix*, and vectors -:math:`\mathbf{u}` and :math:`\mathbf{w}` are discrete variables that live +:math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` are discrete variables that live on the mesh. It is important to note a few things about the inner-product matrix in this case: @@ -37,48 +50,84 @@ inner-product matrix in this case: 3. It depends on the spacial variation of the tensor property :math:`\Sigma` For this class of inner products, the corresponding form of the inner product matricies for -discrete quantities living on various parts of the mesh are shown below. We let: - - - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) - - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` - - :math:`\odot` is the Hadamard product - - :math:`\otimes` is the kronecker product - - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers - - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells - -.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` +discrete quantities living on various parts of the mesh are shown below. **Diagonal Anisotropic:** For the diagonal anisotropic case, the tensor characterzing the material properties -has the form +has the form: .. math:: + \mathbf{In \; 2D:} \; + \Sigma = \begin{bmatrix} \sigma_{x} & 0 \\ + 0 & \sigma_{y} \end{bmatrix} + \;\;\;\;\;\;\;\; \mathbf{In \; 3D:} \; \Sigma = \begin{bmatrix} \sigma_{x} & 0 & 0 \\ 0 & \sigma_{y} & 0 \\ 0 & 0 & \sigma_{z} \end{bmatrix} + :label: inner_product_tensor_diagonal -The resulting inner products have form. + +The inner product matrix defined in expression :eq:`inner_product_anisotropic_general` is given by: .. math:: - \textrm{Vectors on faces:} \; \mathbf{M_{\Sigma f}} &= \frac{1}{4} \mathbf{P_f^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \mathbf{P_f} \\ - \textrm{Vectors on edges:} \; \mathbf{M_{\Sigma e}} &= \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \mathbf{P_e} + \textrm{Vectors on faces:} \; \boldsymbol{M_{\Sigma f}} &= \frac{1}{4} \boldsymbol{P_f^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \boldsymbol{P_f} \\ + \textrm{Vectors on edges:} \; \boldsymbol{M_{\Sigma e}} &= \frac{1}{4^{k-1}} \boldsymbol{P_e^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \boldsymbol{P_e} -where :math:`\boldsymbol{\sigma}` organizes :math:`\sigma_x`, :math:`\sigma_y` and :math:`\sigma_z` within a vector: +where :math:`\boldsymbol{\sigma}` organizes vectors :math:`\boldsymbol{\sigma_x}`, +:math:`\boldsymbol{\sigma_y}` and :math:`\boldsymbol{\sigma_z}` as: .. math:: \boldsymbol{\sigma} = \begin{bmatrix} \boldsymbol{\sigma_x} \\ \boldsymbol{\sigma_y} \\ \boldsymbol{\sigma_z} \\ \end{bmatrix} +and + + - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) + - :math:`\boldsymbol{e_k}` is a vector of 1s of length :math:`k` + - :math:`\odot` is the Hadamard product + - :math:`\otimes` is the kronecker product + - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{P_f}` and :math:`\boldsymbol{P_e}` are projection matricies that map quantities from faces and edges to cell centers, respectively **Fully Anisotropic:** For a fully anisotropic case, the tensor characterizing the material properties -has the form: +has the form is given by: + +.. math:: + \mathbf{In \; 2D:} \; + \Sigma = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} \\ + \sigma_{yx} & \sigma_{yy} \end{bmatrix} + \;\;\;\;\;\;\;\; \mathbf{In \; 3D:} \; + \Sigma = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ + \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ + \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix} + :label: inner_product_tensor + +The inner product matrix defined in expression :eq:`inner_product_anisotropic_general` is given by: + +.. math:: + \textrm{Vectors on faces:} \; \boldsymbol{M_{\Sigma f}} &= \frac{1}{4} \boldsymbol{P_f^T Q_u^T} \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \boldsymbol{Q_w P_f} \\ + \textrm{Vectors on edges:} \; \boldsymbol{M_{\Sigma e}} &= \frac{1}{4^{k-1}} \boldsymbol{P_e^T Q_u^T} \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \boldsymbol{Q_w P_e} + +where :math:`\boldsymbol{\sigma}` is a large vector that organizes vectors :math:`\boldsymbol{\sigma_{pq}}` for :math:`p,q=x,y,z` as: .. math:: - \Sigma = \begin{bmatrix} \sigma_{1} & \sigma_{4} & \sigma_{5} \\ - \sigma_{4} & \sigma_{2} & \sigma_{6} \\ - \sigma_{5} & \sigma_{6} & \sigma_{3} \end{bmatrix} + \boldsymbol{\sigma} = \begin{bmatrix} + \boldsymbol{\sigma_{xx}} , \; \boldsymbol{\sigma_{xy}} , \; \boldsymbol{\sigma_{xz}} , \; + \boldsymbol{\sigma_{yx}} , \; \boldsymbol{\sigma_{yy}} , \; \boldsymbol{\sigma_{yz}} , \; + \boldsymbol{\sigma_{zx}} , \; \boldsymbol{\sigma_{zy}} , \; \boldsymbol{\sigma_{zz}} \end{bmatrix}^T + +and + + - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) + - :math:`\boldsymbol{e_k}` is a vector of 1s of length :math:`k` + - :math:`\odot` is the Hadamard product + - :math:`\otimes` is the kronecker product + - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{P_f}` and :math:`\boldsymbol{P_e}` are projection matricies that map quantities from faces and edges to cell centers, respectively + - :math:`\boldsymbol{Q_u}` is a sparse replication matrix that augments a vector of the form :math:`[\boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}]^T` to create a vector of the form :math:`[\boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}, \; \boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}, \; \boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z} ]^T` + - :math:`\boldsymbol{Q_w}` is a sparse replication matrix that augments a vector of the form :math:`[\boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}]^T` to create a vector of the form :math:`[\boldsymbol{u_x}, \; \boldsymbol{u_x}, \; \boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_y}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}, \; \boldsymbol{u_z} \; \boldsymbol{u_z} ]^T` Diagonally Anisotropic Case @@ -88,20 +137,14 @@ Vectors on Cell Faces ^^^^^^^^^^^^^^^^^^^^^ We want to approximate the inner product between a vector quantity :math:`\vec{u}` and the product of -:math:`\Sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined -to live on cess faces. Our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: +:math:`\Sigma` and :math:`\vec{w}`, where :math:`\Sigma` given in expression :eq:`inner_product_tensor_diagonal`. +Here, we discretize such that :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` are defined +to live on cell faces. Our goal is to construct the inner product matrix :math:`\boldsymbol{M}` such that: .. math:: - (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \mathbf{u^T \, M \, e} + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \boldsymbol{u^T \, M \, w} :label: inner_product_anisotropic_faces -where - -.. math:: - \Sigma = \begin{bmatrix} \sigma_{x} & 0 & 0 \\ - 0 & \sigma_{y} & 0 \\ - 0 & 0 & \sigma_{z} \end{bmatrix} - We must respect the dot product and the tensor. For vectors defined on cell faces, we discretize such that the x-component of the vectors live on the x-faces, the y-component lives y-faces and the z-component lives on the z-faces. For a single cell, this is illustrated in 2D and 3D below. By decomposing the @@ -116,20 +159,17 @@ vector from its faces to the cell centers and take their averages separately. We x-components are only multiplied by :math:`\sigma_x`, y-components by :math:`\sigma_y` and z-components by :math:`\sigma_z`. -For a single cell with volume :math:`v^{(i)}` and tensor properties defined by -:math:`\sigma_x^{(i)}`, :math:`\sigma_y^{(i)}`, :math:`\sigma_z^{(i)}` +For a single cell :math:`i` with volume :math:`v` and tensor properties defined by +:math:`\sigma_x`, :math:`\sigma_y`, :math:`\sigma_z` the contribution towards the inner product is: .. math:: \begin{align} - \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; - \frac{v^{(i)} \sigma_x^{(i)}}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ - & + \frac{v^{(i)} \sigma_y^{(i)}}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v}{4} + \sum_{p=x,y} \sigma_{p} \Big ( u_p^{(1)} + u_p^{(2)} \Big ) \Big ( w_p^{(1)} + w_p^{(2)} \Big ) \\ & \\ - \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; - \frac{v^{(i)} \sigma_x^{(i)}}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ - & + \frac{v^{(i)} \sigma_y^{(i)}}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ - & + \frac{v^{(i)} \sigma_z^{(i)}}{4} \Big ( u_z^{(1)} + u_z^{(2)} \Big ) \Big ( w_z^{(1)} + w_z^{(2)} \Big ) + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v}{4} + \sum_{p=x,y,z} \sigma_{p} \Big ( u_p^{(1)} + u_p^{(2)} \Big ) \Big ( w_p^{(1)} + w_p^{(2)} \Big ) \end{align} :label: inner_product_anisotropic_faces_1 @@ -137,28 +177,29 @@ where superscripts :math:`(1)` and :math:`(2)` denote face 1 and face 2, respect Using the contribution for each cell described in expression :eq:`inner_product_anisotropic_faces_1`, we want to approximate the inner product in the form described by equation :eq:`inner_product_anisotropic_faces`. To accomlish this, we construct a sparse matrix -:math:`\mathbf{P_f}` which projects quantities on the x, y and z faces separately to the +:math:`\boldsymbol{P_f}` which projects quantities on the x, y and z faces separately to the the cell centers. -For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +For discretize vectors :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` whose x, y (and z) components are organized on cell faces as follows: .. math:: - \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_y} \\ \end{bmatrix} \;\;\;\; \textrm{and} \;\;\;\; - \mathbf{w} = \begin{bmatrix} \mathbf{w_x} \\ \mathbf{w_y} \\ \mathbf{w_y} \\ \end{bmatrix} + \boldsymbol{w} = \begin{bmatrix} \boldsymbol{w_x} \\ \boldsymbol{w_y} \\ \boldsymbol{w_y} \\ \end{bmatrix} the approximation to the inner product is given by: .. math:: - (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \boldsymbol{\mathbf{u} \, M_{\sigma f}} \, \mathbf{w} + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \boldsymbol{\boldsymbol{u} \, M_{\Sigma f}} \, \boldsymbol{w} -where the mass matrix has the form: +The inner product matrix defined in the previous expression is given by: .. math:: - \mathbf{M_{\Sigma f}} = \frac{1}{4} \mathbf{P_f^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \mathbf{P_f} + \boldsymbol{M_{\Sigma f}} = \frac{1}{4} \boldsymbol{P_f^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \boldsymbol{P_f} -:math:`\boldsymbol{\sigma}` organizes :math:`\sigma_x`, :math:`\sigma_y` and :math:`\sigma_z` within a vector: +where :math:`\boldsymbol{\sigma}` organizes vectors :math:`\boldsymbol{\sigma_x}`, +:math:`\boldsymbol{\sigma_y}` and :math:`\boldsymbol{\sigma_z}` as: .. math:: \boldsymbol{\sigma} = \begin{bmatrix} \boldsymbol{\sigma_x} \\ \boldsymbol{\sigma_y} \\ \boldsymbol{\sigma_z} \\ \end{bmatrix} @@ -166,35 +207,28 @@ where the mass matrix has the form: and - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) - - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\boldsymbol{e_k}` is a vector of 1s of length :math:`k` - :math:`\odot` is the Hadamard product - :math:`\otimes` is the kronecker product - - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers - - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells - - :math:`\boldsymbol{\sigma}` is a vector that stores the tensor property values for the cells + - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{P_f}` is a projection matrix that maps quantities from faces to cell centers .. note:: To see a validation of the discrete approximation, see our tutorials section (link) - - Vectors on Cell Edges ^^^^^^^^^^^^^^^^^^^^^ We want to approximate the inner product between a vector quantity :math:`\vec{u}` and the product of -:math:`\Sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined -to live at cell edges. Our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: +:math:`\Sigma` and :math:`\vec{w}`, where :math:`\Sigma` given in expression :eq:`inner_product_tensor_diagonal`. +Here, we discretize such that :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` are defined +to live on cell edges. Our goal is to construct the inner product matrix :math:`\boldsymbol{M}` such that: .. math:: - (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \boldsymbol{u^T \, M \, w} :label: inner_product_anisotropic_edges -.. math:: - \Sigma = \begin{bmatrix} \sigma_{x} & 0 & 0 \\ - 0 & \sigma_{y} & 0 \\ - 0 & 0 & \sigma_{z} \end{bmatrix} - We must respect the dot product and the tensor. For vectors defined on cell edges, we discretize such that the x-component of the vectors live on the x-edges, the y-component lives y-edges and the z-component lives on the z-edges. This is illustrated in 2D and 3D below. By decomposing the @@ -211,61 +245,59 @@ We must also recognize that x-components are only multiplied by :math:`\sigma_x`, y-components by :math:`\sigma_y` and z-components by :math:`\sigma_z`. -For a single cell with volume :math:`v^{(i)}` and tensor properties defined by -:math:`\sigma_x^{(i)}`, :math:`\sigma_y^{(i)}`, :math:`\sigma_z^{(i)}` +For a single cell :math:`i` with volume :math:`v` and tensor properties defined by +:math:`\sigma_x`, :math:`\sigma_y`, :math:`\sigma_z` the contribution towards the inner product is: .. math:: \begin{align} - \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; - \frac{v^{(i)} \sigma_x^{(i)}}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ - & + \frac{v^{(i)} \sigma_y^{(i)}}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v}{4} + \sum_{p=x,y} \sigma_{p} \Big ( u_p^{(1)} + u_p^{(2)} \Big ) \Big ( w_p^{(1)} + w_p^{(2)} \Big ) \\ & \\ - \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; - \frac{v^{(i)} \sigma_x^{(i)}}{16} \Bigg ( \sum_{n=1}^4 u_x^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_x^{(n)} \Bigg ) \\ - & + \frac{v^{(i)} \sigma_y^{(i)}}{16} \Bigg ( \sum_{n=1}^4 u_y^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_y^{(n)} \Bigg ) \\ - & + \frac{v^{(i)} \sigma_z^{(i)}}{16} \Bigg ( \sum_{n=1}^4 u_z^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_z^{(n)} \Bigg ) + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v}{16} + \sum_{p=x,y,z} \sigma_{p} \Big ( u_p^{(1)} + u_p^{(2)} + u_p^{(3)} + u_p^{(4)} \Big ) + \Big ( w_p^{(1)} + w_p^{(2)} + w_p^{(3)} + w_p^{(4)} \Big ) \end{align} :label: inner_product_anisotropic_edges_1 -where the superscript :math:`(n)` denotes a particular edge. +where the superscripts :math:`(1)` to :math:`(4)` denote a particular edges. Using the contribution for each cell described in expression :eq:`inner_product_anisotropic_edges_1`, we want to approximate the inner product in the form described by equation :eq:`inner_product_anisotropic_edges`. To accomlish this, we construct a sparse matrix -:math:`\mathbf{P_e}` which projects quantities on the x, y and z edges separately to the +:math:`\boldsymbol{P_e}` which projects quantities on the x, y and z edges separately to the the cell centers. -For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +For discretize vectors :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` whose x, y (and z) components are organized on cell edges as follows: .. math:: - \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_y} \\ \end{bmatrix} \;\;\;\; \textrm{and} \;\;\;\; - \mathbf{w} = \begin{bmatrix} \mathbf{w_x} \\ \mathbf{w_y} \\ \mathbf{w_y} \\ \end{bmatrix} + \boldsymbol{w} = \begin{bmatrix} \boldsymbol{w_x} \\ \boldsymbol{w_y} \\ \boldsymbol{w_y} \\ \end{bmatrix} the approximation to the inner product is given by: .. math:: - (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \boldsymbol{\mathbf{u} \, M_{\sigma e} \, \mathbf{w}} + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \boldsymbol{\boldsymbol{u} \, M_{\Sigma e} \, \boldsymbol{w}} -where the mass matrix for face quantities has the form: +The inner product matrix defined in the previous expression is given by: .. math:: - \mathbf{M_{\Sigma e}} = \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \mathbf{P_e} + \boldsymbol{M_{\Sigma e}} = \frac{1}{4^{k-1}} \boldsymbol{P_e^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \boldsymbol{P_e} -where :math:`\boldsymbol{\sigma}` organizes :math:`\sigma_x`, :math:`\sigma_y` and :math:`\sigma_z` within a vector: +where :math:`\boldsymbol{\sigma}` organizes vectors :math:`\boldsymbol{\sigma_x}`, +:math:`\boldsymbol{\sigma_y}` and :math:`\boldsymbol{\sigma_z}` as: .. math:: \boldsymbol{\sigma} = \begin{bmatrix} \boldsymbol{\sigma_x} \\ \boldsymbol{\sigma_y} \\ \boldsymbol{\sigma_z} \\ \end{bmatrix} and - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) - - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\boldsymbol{e_k}` is a vector of 1s of length :math:`k` - :math:`\odot` is the Hadamard product - :math:`\otimes` is the kronecker product - - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers - - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells - - :math:`\boldsymbol{\sigma}` is a vector that stores the tensor property values for the cells + - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{P_e}` is a projection matrix that maps quantities from edges to cell centers .. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` @@ -276,20 +308,14 @@ Vectors on Cell Faces ^^^^^^^^^^^^^^^^^^^^^ We want to approximate the inner product between a vector quantity :math:`\vec{u}` and the product of -:math:`\Sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined -to live on cess faces. Our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: +:math:`\Sigma` and :math:`\vec{w}`, where :math:`\Sigma` given in expression :eq:`inner_product_tensor`. +Here, we discretize such that :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` are defined +to live on cell faces. Our goal is to construct the inner product matrix :math:`\boldsymbol{M}` such that: .. math:: - (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \mathbf{u^T \, M \, e} + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \boldsymbol{u^T \, M \, e} :label: inner_product_anisotropic_faces -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} - We must respect the dot product and the tensor. For vectors defined on cell faces, we discretize such that the x-component of the vectors live on the x-faces, the y-component lives y-faces and the z-component lives on the z-faces. For a single cell, this is illustrated in 2D and 3D below. By decomposing the @@ -301,23 +327,19 @@ domain into a set of finite cells, we assume the tensor properties are spacial i As we can see there are 2 faces for each component. Therefore, we need to project each component of the vector from its faces to the cell centers and take their averages separately. We must also recognize that -x-components are only multiplied by :math:`\sigma_x`, y-components by :math:`\sigma_y` and z-components -by :math:`\sigma_z`. +different parameters :math:`\sigma_{pq}` for :math:`p,q=x,y,z` multiply different components of the vectors. -For a single cell with volume :math:`v^{(i)}` and tensor properties defined by -:math:`\sigma_x^{(i)}`, :math:`\sigma_y^{(i)}`, :math:`\sigma_z^{(i)}` +For a single cell :math:`i` with volume :math:`v` and tensor properties defined by +:math:`\sigma_{pq}` for :math:`p,q=x,y,z`, the contribution towards the inner product is: .. math:: \begin{align} - \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; - \frac{v^{(i)} \sigma_x^{(i)}}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ - & + \frac{v^{(i)} \sigma_y^{(i)}}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v}{4} + \sum_{p,q=x,y} \sigma_{pq} \Big ( u_p^{(1)} + u_p^{(2)} \Big ) \Big ( w_q^{(1)} + w_q^{(2)} \Big ) \\ & \\ - \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; - \frac{v^{(i)} \sigma_x^{(i)}}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ - & + \frac{v^{(i)} \sigma_y^{(i)}}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ - & + \frac{v^{(i)} \sigma_z^{(i)}}{4} \Big ( u_z^{(1)} + u_z^{(2)} \Big ) \Big ( w_z^{(1)} + w_z^{(2)} \Big ) + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v}{4} + \sum_{p,q=x,y,z} \sigma_{pq} \Big ( u_p^{(1)} + u_p^{(2)} \Big ) \Big ( w_q^{(1)} + w_q^{(2)} \Big ) \end{align} :label: inner_product_anisotropic_faces_1 @@ -325,66 +347,60 @@ where superscripts :math:`(1)` and :math:`(2)` denote face 1 and face 2, respect Using the contribution for each cell described in expression :eq:`inner_product_anisotropic_faces_1`, we want to approximate the inner product in the form described by equation :eq:`inner_product_anisotropic_faces`. To accomlish this, we construct a sparse matrix -:math:`\mathbf{P_f}` which projects quantities on the x, y and z faces separately to the +:math:`\boldsymbol{P_f}` which projects quantities on the x, y and z faces separately to the the cell centers. -For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +For discretize vectors :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` whose x, y (and z) components are organized on cell faces as follows: .. math:: - \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_z} \\ \end{bmatrix} \;\;\;\; \textrm{and} \;\;\;\; - \mathbf{w} = \begin{bmatrix} \mathbf{w_x} \\ \mathbf{w_y} \\ \mathbf{w_y} \\ \end{bmatrix} + \boldsymbol{w} = \begin{bmatrix} \boldsymbol{w_x} \\ \boldsymbol{w_y} \\ \boldsymbol{w_z} \\ \end{bmatrix} the approximation to the inner product is given by: .. math:: - (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \boldsymbol{\mathbf{u} \, M_{\sigma f}} \, \mathbf{w} + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \boldsymbol{\boldsymbol{u} \, M_{\Sigma f}} \, \boldsymbol{w} -where the mass matrix has the form: +The inner product matrix defined in the previous expression is given by: .. math:: - \mathbf{M_{\sigma f}} = \frac{1}{4} \mathbf{P_f^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \mathbf{P_f} + \boldsymbol{M_{\Sigma f}} = \frac{1}{4} \boldsymbol{P_f^T Q_u^T} \textrm{diag} \boldsymbol{\big ( (e_k \otimes e_k \otimes v) \odot \sigma \big )} \boldsymbol{Q_w P_f} -:math:`\boldsymbol{\sigma}` organizes :math:`\sigma_x`, :math:`\sigma_y` and :math:`\sigma_z` within a vector: +where :math:`\boldsymbol{\sigma}` is a large vector that organizes vectors :math:`\boldsymbol{\sigma_{pq}}` for :math:`p,q=x,y,z` as: .. math:: - \boldsymbol{\sigma} = \begin{bmatrix} \boldsymbol{\sigma_x} \\ \boldsymbol{\sigma_y} \\ \boldsymbol{\sigma_z} \\ \end{bmatrix} + \boldsymbol{\sigma} = \begin{bmatrix} + \boldsymbol{\sigma_{xx}} , \; \boldsymbol{\sigma_{xy}} , \; \boldsymbol{\sigma_{xz}} , \; + \boldsymbol{\sigma_{yx}} , \; \boldsymbol{\sigma_{yy}} , \; \boldsymbol{\sigma_{yz}} , \; + \boldsymbol{\sigma_{zx}} , \; \boldsymbol{\sigma_{zy}} , \; \boldsymbol{\sigma_{zz}} \end{bmatrix}^T and - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) - - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\boldsymbol{e_k}` is now a vector of 1s of length :math:`k` - :math:`\odot` is the Hadamard product - :math:`\otimes` is the kronecker product - - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers - - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells - - :math:`\boldsymbol{\sigma}` is a vector that stores the tensor property values for the cells - - -.. note:: To see a validation of the discrete approximation, see our tutorials section (link) - - + - :math:`\boldsymbol{P_f}` is a projection matrix that maps quantities from faces to cell centers + - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{Q_u}` is a sparse replication matrix that augments a vector of the form :math:`[\boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}]^T` to create a vector of the form :math:`[\boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}, \; \boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}, \; \boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z} ]^T` + - :math:`\boldsymbol{Q_w}` is a sparse replication matrix that augments a vector of the form :math:`[\boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}]^T` to create a vector of the form :math:`[\boldsymbol{u_x}, \; \boldsymbol{u_x}, \; \boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_y}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}, \; \boldsymbol{u_z} \; \boldsymbol{u_z} ]^T` Vectors on Cell Edges ^^^^^^^^^^^^^^^^^^^^^ We want to approximate the inner product between a vector quantity :math:`\vec{u}` and the product of -:math:`\Sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined -to live at cell edges. Our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: +:math:`\Sigma` and :math:`\vec{w}`, where :math:`\Sigma` given in expression :eq:`inner_product_tensor`. +Here, we discretize such that :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` are defined +to live on cell edges. Our goal is to construct the inner product matrix :math:`\boldsymbol{M}` such that: .. math:: - (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \Sigma \vec{w} \, dv \approx \boldsymbol{u^T \, M \, w} :label: inner_product_anisotropic_edges -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} - +where :math:`\Sigma` is defined in expression :eq:`inner_product_tensor`. We must respect the dot product and the tensor. For vectors defined on cell edges, we discretize such that the x-component of the vectors live on the x-edges, the y-component lives y-edges and the z-component lives on the z-edges. This is illustrated in 2D and 3D below. By decomposing the @@ -399,59 +415,62 @@ Therefore, we need to project each component of the vector from its edges to the Since the tensor is symmetric, it has 3 independent components in 2D and 6 independent components in 3D. Using this, we can reduce the size of the computation. -For a single cell with volume :math:`v^{(i)}` and tensor properties defined by -:math:`\sigma_n^{(i)}` for :math:`n=1,2,3,4,5,6` the contribution towards the inner product is: +For a single cell :math:`i` with volume :math:`v` and tensor properties defined by +:math:`\sigma_{pq}` for :math:`p,q=x,y,z`, +the contribution towards the inner product is: .. math:: \begin{align} - \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; - \frac{v^{(i)} \sigma_x^{(i)}}{4} \Big ( u_x^{(1)} + u_x^{(2)} \Big ) \Big ( w_x^{(1)} + w_x^{(2)} \Big ) \\ - & + \frac{v^{(i)} \sigma_y^{(i)}}{4} \Big ( u_y^{(1)} + u_y^{(2)} \Big ) \Big ( w_y^{(1)} + w_y^{(2)} \Big ) \\ + \mathbf{In \; 2D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v}{4} + \sum_{p,q=x,y} \sigma_{pq} \Big ( u_p^{(1)} + u_p^{(2)} \Big ) \Big ( w_q^{(1)} + w_q^{(2)} \Big ) \\ & \\ - \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; - \frac{v^{(i)} \sigma_x^{(i)}}{16} \Bigg ( \sum_{n=1}^4 u_x^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_x^{(n)} \Bigg ) \\ - & + \frac{v^{(i)} \sigma_y^{(i)}}{16} \Bigg ( \sum_{n=1}^4 u_y^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_y^{(n)} \Bigg ) \\ - & + \frac{v^{(i)} \sigma_z^{(i)}}{16} \Bigg ( \sum_{n=1}^4 u_z^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^4 w_z^{(n)} \Bigg ) + \mathbf{In \; 3D:} \; \int_{\Omega_i} \vec{u} \cdot \vec{w} \, dv \approx & \;\; \frac{v}{16} + \sum_{p,q=x,y,z} \sigma_{pq} \Big ( u_p^{(1)} + u_p^{(2)} + u_p^{(3)} + u_p^{(4)} \Big ) + \Big ( w_q^{(1)} + w_q^{(2)} + w_q^{(3)} + w_q^{(4)} \Big ) \end{align} :label: inner_product_anisotropic_edges_1 -where the superscript :math:`(n)` denotes a particular edge. +where the superscripts :math:`(1)` to :math:`(4)` denote a particular edges. Using the contribution for each cell described in expression :eq:`inner_product_anisotropic_edges_1`, we want to approximate the inner product in the form described by equation :eq:`inner_product_anisotropic_edges`. To accomlish this, we construct a sparse matrix -:math:`\mathbf{P_e}` which projects quantities on the x, y and z edges separately to the +:math:`\boldsymbol{P_e}` which projects quantities on the x, y and z edges separately to the the cell centers. -For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +For discretize vectors :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` whose x, y (and z) components are organized on cell edges as follows: .. math:: - \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_y} \\ \end{bmatrix} \;\;\;\; \textrm{and} \;\;\;\; - \mathbf{w} = \begin{bmatrix} \mathbf{w_x} \\ \mathbf{w_y} \\ \mathbf{w_y} \\ \end{bmatrix} + \boldsymbol{w} = \begin{bmatrix} \boldsymbol{w_x} \\ \boldsymbol{w_y} \\ \boldsymbol{w_y} \\ \end{bmatrix} the approximation to the inner product is given by: .. math:: - (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \boldsymbol{\mathbf{u} \, M_{\sigma e} \, \mathbf{w}} + (\vec{u}, \Sigma \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \boldsymbol{\boldsymbol{u} \, M_{\Sigma e} \, \boldsymbol{w}} -where the mass matrix for face quantities has the form: +The inner product matrix defined in the previous expression is given by: .. math:: - \mathbf{M_{\sigma e}} = \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} \boldsymbol{\big ( (e_k \otimes v) \odot \sigma \big )} \mathbf{P_e} + \boldsymbol{M_{\Sigma e}} = \frac{1}{4^{k-1}} \boldsymbol{P_e^T Q_u^T} \textrm{diag} \boldsymbol{\big ( (e_k \otimes e_k \otimes v) \odot \sigma \big )} \boldsymbol{Q_w P_e} -where :math:`\boldsymbol{\sigma}` organizes :math:`\sigma_x`, :math:`\sigma_y` and :math:`\sigma_z` within a vector: +where :math:`\boldsymbol{\sigma}` is a large vector that organizes vectors :math:`\boldsymbol{\sigma_{pq}}` for :math:`p,q=x,y,z` as: .. math:: - \boldsymbol{\sigma} = \begin{bmatrix} \boldsymbol{\sigma_x} \\ \boldsymbol{\sigma_y} \\ \boldsymbol{\sigma_z} \\ \end{bmatrix} + \boldsymbol{\sigma} = \begin{bmatrix} + \boldsymbol{\sigma_{xx}} , \; \boldsymbol{\sigma_{xy}} , \; \boldsymbol{\sigma_{xz}} , \; + \boldsymbol{\sigma_{yx}} , \; \boldsymbol{\sigma_{yy}} , \; \boldsymbol{\sigma_{yz}} , \; + \boldsymbol{\sigma_{zx}} , \; \boldsymbol{\sigma_{zy}} , \; \boldsymbol{\sigma_{zz}} \end{bmatrix}^T + and - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) - - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\boldsymbol{e_k}` is a vector of 1s of length :math:`k` - :math:`\odot` is the Hadamard product - :math:`\otimes` is the kronecker product - - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers - - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells - - :math:`\boldsymbol{\sigma}` is a vector that stores the tensor property values for the cells - -.. note:: To see a validation of the discrete approximation, see our tutorials section (link) \ No newline at end of file + - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{P_e}` is a projection matrix that maps quantities from edges to cell centers + - :math:`\boldsymbol{Q_u}` is a sparse replication matrix that augments a vector of the form :math:`[\boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}]^T` to create a vector of the form :math:`[\boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}, \; \boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}, \; \boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z} ]^T` + - :math:`\boldsymbol{Q_w}` is a sparse replication matrix that augments a vector of the form :math:`[\boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}]^T` to create a vector of the form :math:`[\boldsymbol{u_x}, \; \boldsymbol{u_x}, \; \boldsymbol{u_x}, \; \boldsymbol{u_y}, \; \boldsymbol{u_y}, \; \boldsymbol{u_y}, \; \boldsymbol{u_z}, \; \boldsymbol{u_z} \; \boldsymbol{u_z} ]^T` + \ No newline at end of file diff --git a/docs/content/inner_products_basic.rst b/docs/content/inner_products_basic.rst index d1efffe94..bbbf4aba3 100644 --- a/docs/content/inner_products_basic.rst +++ b/docs/content/inner_products_basic.rst @@ -11,19 +11,19 @@ basic class of inner products. For scalar quantities :math:`\psi` and :math:`\ph we will show that a disrete approximation to the inner product is given by: .. math:: - (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \boldsymbol{\psi^T \, M \, \phi} + (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \boldsymbol{\psi^T M \, \phi} :label: inner_product_basic_scalar And for vector quantities :math:`\vec{u}` and :math:`\vec{w}`, a discrete approximation to the inner product is given by: .. math:: - (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \boldsymbol{u^T M \, w} :label: inner_product_basic_vector -where :math:`\mathbf{M}` in either equation represents an +where :math:`\boldsymbol{M}` in either equation represents an *inner-product matrix*, and :math:`\boldsymbol{\psi}`, :math:`\boldsymbol{\phi}`, -:math:`\mathbf{u}` and :math:`\mathbf{w}` are discrete variables that live +:math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` are discrete variables that live on the mesh. It is important to note a few things about the inner-product matrix in this case: @@ -34,18 +34,18 @@ For this simple class of inner products, the corresponding form of the inner pro discrete quantities living on various parts of the mesh are shown below: .. math:: - \textrm{Scalars at centers:} \; \mathbf{M_c} &= \textrm{diag} (\mathbf{v} ) \\ - \textrm{Scalars at nodes:} \; \mathbf{M_n} &= \frac{1}{2^{2k}} \mathbf{P_n^T } \textrm{diag} (\mathbf{v} ) \mathbf{P_n} \\ - \textrm{Vectors on faces:} \; \mathbf{M_f} &= \frac{1}{4} \mathbf{P_f^T } \textrm{diag} (\mathbf{e_k \otimes v} ) \mathbf{P_f} \\ - \textrm{Vectors on edges:} \; \mathbf{M_e} &= \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} (\mathbf{e_k \otimes v}) \mathbf{P_e} + \textrm{Scalars at centers:} \; \boldsymbol{M_c} &= \textrm{diag} (\boldsymbol{v} ) \\ + \textrm{Scalars at nodes:} \; \boldsymbol{M_n} &= \frac{1}{2^{2k}} \boldsymbol{P_n^T } \textrm{diag} (\boldsymbol{v} ) \boldsymbol{P_n} \\ + \textrm{Vectors on faces:} \; \boldsymbol{M_f} &= \frac{1}{4} \boldsymbol{P_f^T } \textrm{diag} (\boldsymbol{e_k \otimes v} ) \boldsymbol{P_f} \\ + \textrm{Vectors on edges:} \; \boldsymbol{M_e} &= \frac{1}{4^{k-1}} \boldsymbol{P_e^T } \textrm{diag} (\boldsymbol{e_k \otimes v}) \boldsymbol{P_e} where - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) - - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\boldsymbol{e_k}` is a vector of 1s of length :math:`k` - :math:`\otimes` is the kronecker product - - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers - - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{P_n}`, :math:`\boldsymbol{P_f}` and :math:`\boldsymbol{P_e}` are projection matricies that map quantities from nodes, faces and edges to cell centers, respectively + - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells .. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` @@ -56,10 +56,10 @@ Scalars at Cell Centers We want to approximate the inner product of two scalar quantities :math:`\psi` and :math:`\phi` where the discrete quantities :math:`\boldsymbol{\psi}` and :math:`\boldsymbol{\phi}` are defined to live at cell centers. Assuming we know the values of :math:`\psi` and :math:`\phi` at cell centers, -our goal is to construct the inner product matrix :math:`\mathbf{M}` such that: +our goal is to construct the inner product matrix :math:`\boldsymbol{M}` such that: .. math:: - (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \mathbf{\psi^T \, M \, \phi} + (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \boldsymbol{\psi^T M \, \phi} .. image:: ../images/center_discretization.png @@ -68,7 +68,7 @@ our goal is to construct the inner product matrix :math:`\mathbf{M}` such that: For a single cell (see above), the contribution towards the inner product is obtained by multiplying -:math:`\psi` and :math:`\phi` at the cell centers (represented by :math:`\psi_i and \phi_i`) +:math:`\psi` and :math:`\phi` at the cell centers (represented by :math:`\psi_i` and :math:`\phi_i`) by the cell's volume (:math:`v_i`), i.e.: .. math:: @@ -83,13 +83,13 @@ approximation over all cells. Where :math:`nc` refers to the number of cells in Expressing the sum in terms of linear equations, we obtain: .. math:: - (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \mathbf{\psi^T \, M_c \, \phi} + (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \boldsymbol{\psi^T \, M_c \, \phi} where the mass matrix for cell centered quantities is just a diagonal matrix containing -the cell volumes (:math:`\mathbf{v}`), i.e.: +the cell volumes (:math:`\boldsymbol{v}`), i.e.: .. math:: - \mathbf{M_c} = diag(\mathbf{v}) + \boldsymbol{M_c} = diag(\boldsymbol{v}) .. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` @@ -101,10 +101,10 @@ Scalars at Nodes We want to approximate the inner product of two scalar quantities :math:`\psi` and :math:`\phi` where the discrete quantities :math:`\boldsymbol{\psi}` and :math:`\boldsymbol{\phi}` are defined to live on cell nodes. Assuming we know the values of :math:`\psi` and :math:`\phi` at the nodes, -our goal is to construct the inner product matrix :math:`\mathbf{M}` such that: +our goal is to construct the inner product matrix :math:`\boldsymbol{M}` such that: .. math:: - (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \mathbf{\psi^T \, M \, \phi} + (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \boldsymbol{\psi^T M \, \phi} :label: inner_product_basic_nodes Whereas :math:`\boldsymbol{\psi}` and :math:`\boldsymbol{\phi}` are defined @@ -128,7 +128,7 @@ Thus: \Bigg ( \phi_i^{(n1)} \! + \phi_i^{(n2)} \! + \phi_i^{(n3)} \! + \phi_i^{(n4)} \Bigg ) \\ & \\ \mathbf{In \; 3D:} \; \int_{\Omega_i} \psi \, \phi \, dv \approx & \;\; - \frac{v_i}{16} \Bigg ( \sum_{n=1}^8 \psi_i^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^8 \psi_i^{(n)} \Bigg ) + \frac{v_i}{64} \Bigg ( \sum_{n=1}^8 \psi_i^{(n)} \Bigg ) \Bigg ( \sum_{n=1}^8 \psi_i^{(n)} \Bigg ) \end{align} :label: inner_product_basic_nodes_1 @@ -136,24 +136,24 @@ where the superscript :math:`(n)` is used to point to a specific node. Using the contribution for each cell described in expression :eq:`inner_product_basic_nodes_1`, we want to approximate the inner product in the form described by equation :eq:`inner_product_basic_nodes`. To accomlish this, we construct a sparse matrix -:math:`\mathbf{P_n}` which projects quantities on the nodes to the +:math:`\boldsymbol{P_n}` which projects quantities on the nodes to the the cell centers. Our final approximation for the inner product is therefore: .. math:: - (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \mathbf{\psi^T \, M_n \, \phi} + (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \boldsymbol{\psi^T \, M_n \, \phi} where the mass matrix for nodal quantities has the form: .. math:: - \mathbf{M_n} = \frac{1}{2^{2k}} \mathbf{P_n^T } \textrm{diag} (\mathbf{v} ) \mathbf{P_n} + \boldsymbol{M_n} = \frac{1}{2^{2k}} \boldsymbol{P_n^T } \textrm{diag} (\boldsymbol{v} ) \boldsymbol{P_n} where - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) - - :math:`\mathbf{P_n}` is a projection matrix that maps quantities from nodes to cell centers - - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{P_n}` is a projection matrix that maps quantities from nodes to cell centers + - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells .. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` @@ -165,12 +165,12 @@ For the mimetic finite volume approach, fluxes are generally defined on cell fac as it allows cells to share faces while preserving natural boundary conditions. We want to approximate the inner product of two vector quantities :math:`\vec{u}` and :math:`\vec{w}` -where the discrete quantities :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined +where the discrete quantities :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` are defined to live on cell faces. Assuming we know the values of :math:`\vec{u}` and :math:`\vec{w}` on the faces, -our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: +our goal is to construct the inner product matrix :math:`\boldsymbol{M}` in the expression below: .. math:: - (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \boldsymbol{u^T M \, w} :label: inner_product_basic_faces We must respect the dot product. For vectors defined on cell faces, we discretize such that the @@ -201,34 +201,34 @@ where superscripts :math:`(1)` and :math:`(2)` denote face 1 and face 2, respect Using the contribution for each cell described in expression :eq:`inner_product_basic_faces_1`, we want to approximate the inner product in the form described by equation :eq:`inner_product_basic_faces`. To accomlish this, we construct a sparse matrix -:math:`\mathbf{P_f}` which projects quantities on the x, y and z faces separately to the +:math:`\boldsymbol{P_f}` which projects quantities on the x, y and z faces separately to the the cell centers. -For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +For discretize vectors :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` whose x, y (and z) components are organized on cell faces as follows: .. math:: - \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_y} \\ \end{bmatrix} \;\;\;\; \textrm{and} \;\;\;\; - \mathbf{w} = \begin{bmatrix} \mathbf{w_x} \\ \mathbf{w_y} \\ \mathbf{w_y} \\ \end{bmatrix} + \boldsymbol{w} = \begin{bmatrix} \boldsymbol{w_x} \\ \boldsymbol{w_y} \\ \boldsymbol{w_y} \\ \end{bmatrix} the approximation to the inner product is given by: .. math:: - (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \mathbf{\mathbf{u} \, M_f \, \mathbf{w}} + (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \boldsymbol{\boldsymbol{u} M_f \, \boldsymbol{w}} where the mass matrix for face quantities has the form: .. math:: - \mathbf{M_f} = \frac{1}{4} \mathbf{P_f^T } \textrm{diag} (\mathbf{e_k \otimes v} ) \mathbf{P_f} + \boldsymbol{M_f} = \frac{1}{4} \boldsymbol{P_f^T } \textrm{diag} (\boldsymbol{e_k \otimes v} ) \boldsymbol{P_f} and - :math:`k = 1,2,3` represents the dimension (1D, 2D or 3D) - - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\boldsymbol{e_k}` is a vector of 1s of length :math:`k` - :math:`\otimes` is the kronecker product - - :math:`\mathbf{P_f}` is the projection matrix that maps quantities from faces to cell centers - - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{P_f}` is the projection matrix that maps quantities from faces to cell centers + - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells .. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` @@ -239,12 +239,12 @@ Vectors on Cell Edges For the mimetic finite volume approach, fields are generally defined on cell edges; as it allows cells to share edges while preserving natural boundary conditions. We want to approximate the inner product of two vector quantities :math:`\vec{u}` and :math:`\vec{w}` -where the discrete quantities :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined +where the discrete quantities :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` are defined to live at cell edges. Assuming we know the values of :math:`\vec{u}` and :math:`\vec{w}` at the edges, -our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: +our goal is to construct the inner product matrix :math:`\boldsymbol{M}` in the expression below: .. math:: - (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \boldsymbol{u^T \, M \, w} :label: inner_product_basic_edges We must respect the dot product. For vectors defined on cell edges, we discretize such that the @@ -276,34 +276,34 @@ where the superscript :math:`(n)` denotes a particular edge. Using the contribution for each cell described in expression :eq:`inner_product_basic_edges_1`, we want to approximate the inner product in the form described by equation :eq:`inner_product_basic_edges`. To accomlish this, we construct a sparse matrix -:math:`\mathbf{P_e}` which projects quantities on the x, y and z edges separately to the +:math:`\boldsymbol{P_e}` which projects quantities on the x, y and z edges separately to the the cell centers. -For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +For discretize vectors :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` whose x, y (and z) components are organized on cell edges as follows: .. math:: - \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_y} \\ \end{bmatrix} \;\;\;\; \textrm{and} \;\;\;\; - \mathbf{w} = \begin{bmatrix} \mathbf{w_x} \\ \mathbf{w_y} \\ \mathbf{w_y} \\ \end{bmatrix} + \boldsymbol{w} = \begin{bmatrix} \boldsymbol{w_x} \\ \boldsymbol{w_y} \\ \boldsymbol{w_y} \\ \end{bmatrix} the approximation to the inner product is given by: .. math:: - (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \mathbf{\mathbf{u} \, M_e \, \mathbf{w}} + (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \boldsymbol{\boldsymbol{u} \, M_e \, \boldsymbol{w}} where the mass matrix for face quantities has the form: .. math:: - \mathbf{M_e} = \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} (\mathbf{e_k \otimes v}) \mathbf{P_e} + \boldsymbol{M_e} = \frac{1}{4^{k-1}} \boldsymbol{P_e^T } \textrm{diag} (\boldsymbol{e_k \otimes v}) \boldsymbol{P_e} and - :math:`k = 1,2,3` represents the dimension (1D, 2D or 3D) - - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\boldsymbol{e_k}` is a vector of 1s of length :math:`k` - :math:`\otimes` is the kronecker product - - :math:`\mathbf{P_e}` is the projection matrix that maps quantities from edges to cell centers - - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{P_e}` is the projection matrix that maps quantities from edges to cell centers + - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells .. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` diff --git a/docs/content/inner_products_differential.rst b/docs/content/inner_products_differential.rst index 759dcc1f3..41214e795 100644 --- a/docs/content/inner_products_differential.rst +++ b/docs/content/inner_products_differential.rst @@ -29,72 +29,66 @@ we are interested in approximating the following inner products: **Gradient:** For the inner product between a vector (:math:`\vec{u}`) and the gradient of a scalar (:math:`\phi`), -there are two options. Either: +there are two options for where the variables should live. For :math:`\boldsymbol{\phi}` defined on the nodes +and :math:`\boldsymbol{u}` defined on cell edges: - (1) :math:`\boldsymbol{\phi}` is defined on the nodes and :math:`\boldsymbol{u}` is defined on cell edges, or - (2) :math:`\boldsymbol{\phi}` is defined at cell centers and :math:`\boldsymbol{u}` is defined on cell faces +.. math:: + \int_\Omega \vec{u} \cdot \nabla \phi \, dv \approx \boldsymbol{u^T M_e G_n \, \phi + B.C} -For each case, the approximation to the inner product is given by: +And for :math:`\boldsymbol{\phi}` defined at cell centers and :math:`\boldsymbol{u}` defined on cell faces: .. math:: - \begin{align} - \textrm{Case 1:} \; (\vec{u}, \nabla \phi ) &\approx \boldsymbol{u^T M_e G_n \, \phi + B.C.} \\ - \textrm{Case 2:} \; (\vec{u}, \nabla \phi ) &\approx - \boldsymbol{u^T D^T M_c \, \phi + B.C.} - \end{align} + \int_\Omega \vec{u} \cdot \nabla \phi \, dv \approx - \boldsymbol{u^T D^T M_c \, \phi + B.C.} where - - :math:`\mathbf{G_n}` is a discrete gradient operator that maps from nodes to edges - - :math:`\mathbf{D}` is a discrete divergence operator whose transpose acts as a gradient operator from faces to cell centers - - :math:`\mathbf{M_c}` is the :ref:`basic inner product matrix for vectors at cell centers ` - - :math:`\mathbf{M_e}` is the :ref:`basic inner product matrix for vectors at edges ` - - :math:`\mathbf{BC}` represents an additional contribution accounting for the boundary conditions + - :math:`\boldsymbol{G_n}` is a discrete gradient operator that maps from nodes to edges + - :math:`\boldsymbol{D}` is a discrete divergence operator whose transpose acts as a gradient operator from faces to cell centers + - :math:`\boldsymbol{M_c}` is the :ref:`basic inner product matrix for vectors at cell centers ` + - :math:`\boldsymbol{M_e}` is the :ref:`basic inner product matrix for vectors at edges ` + - :math:`\boldsymbol{BC}` represents an additional contribution accounting for the boundary conditions **Divergence:** For the inner product between a scalar (:math:`\psi`) and the divergence of a vector (:math:`\vec{w}`), -there are two options. Either: +there are two options for where the variables should live. For :math:`\boldsymbol{\psi}` defined at cell centers +and :math:`\boldsymbol{w}` defined on cell faces: - (1) :math:`\boldsymbol{\psi}` is defined at cell centers :math:`\boldsymbol{w}` is defined on cell faces - (2) :math:`\boldsymbol{\psi}` is defined on the nodes and :math:`\boldsymbol{w}` is defined on cell edges, or +.. math:: + \int_\Omega \psi \; (\nabla \cdot \vec{w}) \, dv \approx \boldsymbol{\psi^T M_c D \, w + B.C.} -For each case, the approximation to the inner product is given by: +And for :math:`\boldsymbol{\psi}` defined on the nodes and :math:`\boldsymbol{w}` defined on cell edges: .. math:: - \begin{align} - \textrm{Case 1:} \; (\psi, \nabla \cdot \vec{w} ) &\approx \boldsymbol{\psi^T M_c D \, w + B.C.} \\ - \textrm{Case 2:} \; (\psi, \nabla \cdot \vec{w} ) &\approx - \boldsymbol{\psi^T G_n^T M_e \, w + B.C.} - \end{align} + \int_\Omega \psi \; (\nabla \cdot \vec{w}) \, dv \approx - \boldsymbol{\psi^T G_n^T M_e \, w + B.C.} where - - :math:`\mathbf{D}` is a discrete divergence operator from faces to cell centers - - :math:`\mathbf{G_n}` is a discrete gradient operator whose transpose acts as a divergence operator from edges to nodes - - :math:`\mathbf{M_c}` is the :ref:`basic inner product matrix for vectors at cell centers ` - - :math:`\mathbf{M_e}` is the :ref:`basic inner product matrix for vectors at edges ` - - :math:`\mathbf{BC}` represents an additional contribution accounting for the boundary conditions + - :math:`\boldsymbol{D}` is a discrete divergence operator from faces to cell centers + - :math:`\boldsymbol{G_n}` is a discrete gradient operator whose transpose acts as a divergence operator from edges to nodes + - :math:`\boldsymbol{M_c}` is the :ref:`basic inner product matrix for vectors at cell centers ` + - :math:`\boldsymbol{M_e}` is the :ref:`basic inner product matrix for vectors at edges ` + - :math:`\boldsymbol{BC}` represents an additional contribution accounting for the boundary conditions **Curl:** For the inner product between a vector (:math:`\vec{u}`) and the curl of another vector (:math:`\vec{w}`), -there are two options. Either: +there are two options for where the variables should live. For :math:`\boldsymbol{u}` defined on the faces +and :math:`\boldsymbol{w}` defined on cell edges: - (1) :math:`\boldsymbol{u}` is defined on the faces and :math:`\boldsymbol{w}` is defined on cell edges, or - (2) :math:`\boldsymbol{u}` is defined on the edges :math:`\boldsymbol{w}` is defined on cell faces +.. math:: + \int_\Omega \vec{u} \cdot (\nabla \times \vec{w} ) \, dv \approx \boldsymbol{u^T M_f C \, w + B.C.} -For each case, the approximation to the inner product is given by: +And for :math:`\boldsymbol{u}` defined on the edges and :math:`\boldsymbol{w}` defined on cell faces: .. math:: - \begin{align} - \textrm{Case 1:} \; (\vec{u}, \nabla \times \vec{w} ) &\approx \boldsymbol{u^T M_f C \, w + B.C.} \\ - \textrm{Case 2:} \; (\vec{u}, \nabla \times \vec{w} ) &\approx \boldsymbol{u^T C^T M_f \, w + B.C.} - \end{align} + \int_\Omega \vec{u} \cdot (\nabla \times \vec{w} ) \, dv \approx \boldsymbol{u^T C^T \! M_f \, w + B.C.} where - - :math:`\mathbf{C}` is a discrete curl operator from edges to faces, whose transpose acts as a gradient operator from faces to edges - - :math:`\mathbf{M_f}` is the :ref:`basic inner product matrix for vectors on cell faces ` - - :math:`\mathbf{BC}` represents an additional contribution accounting for the boundary conditions + - :math:`\boldsymbol{C}` is a discrete curl operator from edges to faces, whose transpose acts as a curl operator from faces to edges + - :math:`\boldsymbol{M_f}` is the :ref:`basic inner product matrix for vectors on cell faces ` + - :math:`\boldsymbol{BC}` represents an additional contribution accounting for the boundary conditions Vector and the Gradient of a Scalar ----------------------------------- diff --git a/docs/content/inner_products_isotropic.rst b/docs/content/inner_products_isotropic.rst index b21b1b069..0e03e6027 100644 --- a/docs/content/inner_products_isotropic.rst +++ b/docs/content/inner_products_isotropic.rst @@ -26,7 +26,7 @@ equation :eq:`inner_product_isotropic` is approximated by: (\vec{u}, \sigma \vec{w} ) = \int_\Omega \vec{u} \cdot \sigma \vec{w} \, dv \approx \boldsymbol{u^T M w} where :math:`\boldsymbol{M}` represents an *inner-product matrix*, and vectors -:math:`\mathbf{u}` and :math:`\mathbf{w}` are discrete variables that live +:math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` are discrete variables that live on the mesh. It is important to note a few things about the inner-product matrix in this case: @@ -34,21 +34,21 @@ inner-product matrix in this case: 2. It depends on where the discrete variables live; e.g. edges, faces 3. It depends on the spacial variation of the material property :math:`\sigma` -For this class of inner products, the corresponding form of the inner product matricies for -discrete quantities living on various parts of the mesh are given by +For this class of inner products, the corresponding inner product matricies for +discrete quantities living on various parts of the mesh are given by: .. math:: - \textrm{Vectors on faces:} \; \mathbf{M_{\sigma f}} &= \frac{1}{4} \mathbf{P_f^T } \textrm{diag} \boldsymbol{\big ( e_k \otimes (v \odot \sigma ) \big )} \mathbf{P_f} \\ - \textrm{Vectors on edges:} \; \mathbf{M_{\sigma e}} &= \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} \boldsymbol{\big ( e_k \otimes (v \odot \sigma) \big )} \mathbf{P_e} + \textrm{Vectors on faces:} \; \boldsymbol{M_{\sigma f}} &= \frac{1}{4} \boldsymbol{P_f^T } \textrm{diag} \boldsymbol{\big ( e_k \otimes (v \odot \sigma ) \big )} \boldsymbol{P_f} \\ + \textrm{Vectors on edges:} \; \boldsymbol{M_{\sigma e}} &= \frac{1}{4^{k-1}} \boldsymbol{P_e^T } \textrm{diag} \boldsymbol{\big ( e_k \otimes (v \odot \sigma) \big )} \boldsymbol{P_e} where - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) - - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\boldsymbol{e_k}` is a vector of 1s of length :math:`k` - :math:`\odot` is the Hadamard product - :math:`\otimes` is the kronecker product - - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (faces, edges) to cell centers - - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{P_f}` and :math:`\boldsymbol{P_e}` are projection matricies that map quantities from faces and edges to cell centers, respectively + - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells - :math:`\boldsymbol{\sigma}` is a vector containing the physical property values for the cells .. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` @@ -58,11 +58,11 @@ Vectors on Cell Faces --------------------- We want to approximate the inner product between a vector quantity :math:`\vec{u}` and the product of -:math:`\sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined -to live on cess faces. Our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: +:math:`\sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` are defined +to live on cess faces. Our goal is to construct the inner product matrix :math:`\boldsymbol{M}` in the expression below: .. math:: - (\vec{u}, \sigma \vec{w}) = \int_\Omega \vec{u} \cdot \sigma \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + (\vec{u}, \sigma \vec{w}) = \int_\Omega \vec{u} \cdot \sigma \vec{w} \, dv \approx \boldsymbol{u^T M \, w} :label: inner_product_isotropic_faces We must respect the dot product. For vectors defined on cell faces, we discretize such that the @@ -94,35 +94,35 @@ where superscripts :math:`(1)` and :math:`(2)` denote face 1 and face 2, respect Using the contribution for each cell described in expression :eq:`inner_product_isotropic_faces_1`, we want to approximate the inner product in the form described by equation :eq:`inner_product_isotropic_faces`. To accomlish this, we construct a sparse matrix -:math:`\mathbf{P_f}` which projects quantities on the x, y and z faces separately to the +:math:`\boldsymbol{P_f}` which projects quantities on the x, y and z faces separately to the the cell centers. -For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +For discretize vectors :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` whose x, y (and z) components are organized on cell faces as follows: .. math:: - \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_y} \\ \end{bmatrix} \;\;\;\; \textrm{and} \;\;\;\; - \mathbf{w} = \begin{bmatrix} \mathbf{e_x} \\ \mathbf{e_y} \\ \mathbf{e_y} \\ \end{bmatrix} + \boldsymbol{w} = \begin{bmatrix} \boldsymbol{e_x} \\ \boldsymbol{e_y} \\ \boldsymbol{e_y} \\ \end{bmatrix} the approximation to the inner product is given by: .. math:: - (\vec{u}, \sigma \vec{w}) = \int_\Omega \vec{u} \cdot \sigma \vec{w} \, dv \approx \boldsymbol{\mathbf{u} \, M_{\sigma f}} \, \mathbf{w} + (\vec{u}, \sigma \vec{w}) = \int_\Omega \vec{u} \cdot \sigma \vec{w} \, dv \approx \boldsymbol{\boldsymbol{u} M_{\sigma f}} \, \boldsymbol{w} where the mass matrix has the form: .. math:: - \mathbf{M_{\sigma f}} = \frac{1}{4} \mathbf{P_f^T } \textrm{diag} \boldsymbol{\big ( e_k \otimes (v \odot \sigma ) \big )} \mathbf{P_f} + \boldsymbol{M_{\sigma f}} = \frac{1}{4} \boldsymbol{P_f^T } \textrm{diag} \boldsymbol{\big ( e_k \otimes (v \odot \sigma ) \big )} \boldsymbol{P_f} and - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) - - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\boldsymbol{e_k}` is a vector of 1s of length :math:`k` - :math:`\odot` is the Hadamard product - :math:`\otimes` is the kronecker product - - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers - - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{P_f}` is a projection matrix that maps from faces to cell centers + - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells - :math:`\boldsymbol{\sigma}` is a vector containing the physical property values for the cells @@ -133,11 +133,11 @@ Vectors on Cell Edges --------------------- We want to approximate the inner product between a vector quantity :math:`\vec{u}` and the product of -:math:`\sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\mathbf{u}` and :math:`\mathbf{w}` are defined -to live at cell edges. Our goal is to construct the inner product matrix :math:`\mathbf{M}` in the expression below: +:math:`\sigma` and :math:`\vec{w}`. Here, we discretize such that :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` are defined +to live at cell edges. Our goal is to construct the inner product matrix :math:`\boldsymbol{M}` in the expression below: .. math:: - (\vec{u}, \sigma \vec{w}) = \int_\Omega \vec{u} \cdot \sigma \vec{w} \, dv \approx \mathbf{u^T \, M \, w} + (\vec{u}, \sigma \vec{w}) = \int_\Omega \vec{u} \cdot \sigma \vec{w} \, dv \approx \boldsymbol{u^T M \, w} :label: inner_product_isotropic_edges We must respect the dot product. For vectors defined on cell edges, we discretize such that the @@ -169,35 +169,35 @@ where the superscript :math:`(n)` denotes a particular edge. Using the contribution for each cell described in expression :eq:`inner_product_isotropic_edges_1`, we want to approximate the inner product in the form described by equation :eq:`inner_product_isotropic_edges`. To accomlish this, we construct a sparse matrix -:math:`\mathbf{P_e}` which projects quantities on the x, y and z edges separately to the +:math:`\boldsymbol{P_e}` which projects quantities on the x, y and z edges separately to the the cell centers. -For discretize vectors :math:`\mathbf{u}` and :math:`\mathbf{w}` whose x, y (and z) components +For discretize vectors :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` whose x, y (and z) components are organized on cell edges as follows: .. math:: - \mathbf{u} = \begin{bmatrix} \mathbf{u_x} \\ \mathbf{u_y} \\ \mathbf{u_y} \\ \end{bmatrix} + \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_y} \\ \end{bmatrix} \;\;\;\; \textrm{and} \;\;\;\; - \mathbf{w} = \begin{bmatrix} \mathbf{e_x} \\ \mathbf{e_y} \\ \mathbf{e_y} \\ \end{bmatrix} + \boldsymbol{w} = \begin{bmatrix} \boldsymbol{e_x} \\ \boldsymbol{e_y} \\ \boldsymbol{e_y} \\ \end{bmatrix} the approximation to the inner product is given by: .. math:: - (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \mathbf{\mathbf{u} \, M_e \, \mathbf{w}} + (\vec{u}, \vec{w}) = \int_\Omega \vec{u} \cdot \vec{w} \, dv \approx \boldsymbol{\boldsymbol{u} M_e \, \boldsymbol{w}} where the mass matrix for face quantities has the form: .. math:: - \mathbf{M_{\sigma e}} = \frac{1}{4^{k-1}} \mathbf{P_e^T } \textrm{diag} \boldsymbol{\big ( e_k \otimes (v \odot \sigma) \big )} \mathbf{P_e} + \boldsymbol{M_{\sigma e}} = \frac{1}{4^{k-1}} \boldsymbol{P_e^T } \textrm{diag} \boldsymbol{\big ( e_k \otimes (v \odot \sigma) \big )} \boldsymbol{P_e} and - :math:`k = 1,2,3` represent the dimension (1D, 2D or 3D) - - :math:`\mathbf{e_k}` is a vector of 1s of length :math:`k` + - :math:`\boldsymbol{e_k}` is a vector of 1s of length :math:`k` - :math:`\odot` is the Hadamard product - :math:`\otimes` is the kronecker product - - :math:`\mathbf{P}` are projection matricies that map quantities from one part of the cell (nodes, faces, edges) to cell centers - - :math:`\mathbf{v}` is a vector that stores all of the volumes of the cells + - :math:`\boldsymbol{P_e}` is a projection matrix that maps from edges to cell centers + - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells - :math:`\boldsymbol{\sigma}` is a vector containing the physical property values for the cells .. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` From ddaec41f7ea792c695468a3ec64d9d81f19fc68a Mon Sep 17 00:00:00 2001 From: dccowan Date: Fri, 20 Nov 2020 17:13:44 -0800 Subject: [PATCH 07/35] Draft of differential operators --- docs/content/differential_operators.rst | 193 ++++++++++++++++++++++ docs/images/approximate_derivative.PNG | Bin 0 -> 19218 bytes docs/images/center_discretization.PNG | Bin 26747 -> 26488 bytes docs/images/curl_discretization.PNG | Bin 0 -> 83385 bytes docs/images/divergence_discretization.PNG | Bin 0 -> 47979 bytes docs/images/edge_discretization.PNG | Bin 43975 -> 43855 bytes docs/images/face_discretization.PNG | Bin 52889 -> 52300 bytes docs/images/gradient_discretization.PNG | Bin 0 -> 47386 bytes docs/images/node_discretization.PNG | Bin 35649 -> 35317 bytes docs/index.rst | 1 + 10 files changed, 194 insertions(+) create mode 100644 docs/content/differential_operators.rst create mode 100644 docs/images/approximate_derivative.PNG create mode 100644 docs/images/curl_discretization.PNG create mode 100644 docs/images/divergence_discretization.PNG create mode 100644 docs/images/gradient_discretization.PNG diff --git a/docs/content/differential_operators.rst b/docs/content/differential_operators.rst new file mode 100644 index 000000000..39d3a7193 --- /dev/null +++ b/docs/content/differential_operators.rst @@ -0,0 +1,193 @@ +.. _differential_operators: + +Discrete Differential Operators +******************************* + +Here, we provide the background theory for how discrete differential operators are formed. +For geophysical problems, the relationship between two physical quantities may include one of several differential operators: + + - **Divergence:** :math:`\nabla \cdot \vec{u} = \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_y}{\partial y}` + - **Gradient:** :math:`\nabla \phi = \frac{\partial \phi}{\partial x}\hat{x} + \frac{\partial \phi}{\partial y}\hat{y} + \frac{\partial \phi}{\partial z}\hat{z}` + - **Curl:** :math:`\nabla \times \vec{u} = \Big ( \frac{\partial u_y}{\partial z} - \frac{\partial u_z}{\partial y} \Big )\hat{x} - \Big ( \frac{\partial u_x}{\partial z} - \frac{\partial u_z}{\partial x} \Big )\hat{y} + \Big ( \frac{\partial u_x}{\partial y} - \frac{\partial u_y}{\partial x} \Big )\hat{z}` + +When implementing the finite volume method, continuous variables are discretized to live at the cell centers, nodes, edges or faces of a mesh. +Thus for each differential operator, we need a discrete approximation that acts on the discrete variables living on the mesh. + +Our approximations are derived using finite differentiation (figure below). So long as a function :math:`f(x)` is sufficiently smooth +within the interval :math:`[x-h/2, \; x+h/2]`, then the derivative of the function at :math:`x` is approximated by: + +.. math:: + \frac{df(x)}{dx} \approx \frac{f(x+h/2) \; - \; f(x-h/2)}{h} + +where the approximation becomes increasingly accurate as :math:`h \rightarrow 0`. In subsequent sections, we will show how +the gradient, divergence and curl can be computed for discrete variables. + +.. image:: ../images/approximate_derivative.png + :align: center + :width: 300 + + + + + +Divergence +---------- + +Let us define a continuous scalar function :math:`\phi` and a continuous vector function :math:`\vec{u}` such that: + +.. math:: + \phi = \nabla \cdot \vec{u} + +And let :math:`\boldsymbol{\phi}` and :math:`\boldsymbol{u}` be the discrete representations of :math:`\phi` and :math:`\vec{u}` +that live on the mesh (centers, nodes, edges or faces), respectively. Provided we know the discrete values :math:`\boldsymbol{u}`, +our goal is to use discrete differentiation to approximate the values of :math:`\boldsymbol{\phi}`. +We begin by considering a single cell (2D or 3D). We let the indices :math:`i`, :math:`j` and :math:`k` +denote positions along the x, y and z axes, respectively. + +.. image:: ../images/divergence_discretization.png + :align: center + :width: 600 + ++-------------+-------------------------------------------------+-----------------------------------------------------+ +| | **2D** | **3D** | ++-------------+-------------------------------------------------+-----------------------------------------------------+ +| **center** | :math:`(i,j)` | :math:`(i,j,k)` | ++-------------+-------------------------------------------------+-----------------------------------------------------+ +| **x faces** | :math:`(i-\frac{1}{2},j)\;\; (i+\frac{1}{2},j)` | :math:`(i-\frac{1}{2},j,k)\;\; (i+\frac{1}{2},j,k)` | ++-------------+-------------------------------------------------+-----------------------------------------------------+ +| **y faces** | :math:`(i,j-\frac{1}{2})\;\; (i,j+\frac{1}{2})` | :math:`(i,j-\frac{1}{2},k)\;\; (i,j+\frac{1}{2},k)` | ++-------------+-------------------------------------------------+-----------------------------------------------------+ +| **z faces** | N/A | :math:`(i,j,k-\frac{1}{2})\;\; (i,j,k+\frac{1}{2})` | ++-------------+-------------------------------------------------+-----------------------------------------------------+ + +As we will see, it makes the most sense for :math:`\boldsymbol{\phi}` to live at the cell centers and +for the components of :math:`\boldsymbol{u}` to live on the faces. If :math:`u_x` lives on x faces, then its discrete +derivative with respect to :math:`x` lives at the cell center. And if :math:`u_y` lives on y faces its discrete +derivative with respect to :math:`y` lives at the cell center. Likewise for :math:`u_z`. Thus to approximate the +divergence of :math:`\vec{u}` at the cell center, we simply need to sum the discrete derivatives of :math:`u_x`, :math:`u_y` +and :math:`u_z` living on the faces. Where :math:`h_x`, :math:`h_y` and :math:`h_z` represent the dimension of the cell along the x, y and +z directions, respectively: + +.. math:: + \begin{align} + \mathbf{In \; 2D:} \;\; \phi(i,j) \approx \; & \frac{u_x(i,j+\frac{1}{2}) - u_x(i,j-\frac{1}{2})}{h_x} \\ + & + \frac{u_y(i+\frac{1}{2},j) - u_y(i-\frac{1}{2},j)}{h_y} + \end{align} + +| + +.. math:: + \begin{align} + \mathbf{In \; 3D:} \;\; \phi(i,j,k) \approx \; & \frac{u_x(i+\frac{1}{2},j,k) - u_x(i-\frac{1}{2},j,k)}{h_x} \\ + & + \frac{u_y(i,j+\frac{1}{2},k) - u_y(i,j-\frac{1}{2},k)}{h_y} \\ + & + \frac{u_z(i,j,k+\frac{1}{2}) - u_z(i,j,k-\frac{1}{2})}{h_z} + \end{align} + + + + +Gradient +-------- + +Let us define a continuous scalar function :math:`\phi` and a continuous vector function :math:`\vec{u}` such that: + +.. math:: + \vec{u} = \nabla \phi + +And let :math:`\boldsymbol{\phi}` and :math:`\boldsymbol{u}` be the discrete representations of :math:`\phi` and :math:`\vec{u}` +that live on the mesh (centers, nodes, edges or faces), respectively. Provided we know the discrete values :math:`\boldsymbol{\phi}`, +our goal is to use discrete differentiation to approximate the vector components of :math:`\boldsymbol{u}`. +We begin by considering a single cell (2D or 3D). We let the indices :math:`i`, :math:`j` and :math:`k` +denote positions along the x, y and z axes, respectively. + +.. image:: ../images/gradient_discretization.png + :align: center + :width: 600 + +As we will see, it makes the most sense for :math:`\boldsymbol{\phi}` to live at the cell nodes and +for the components of :math:`\boldsymbol{u}` to live on corresponding edges. If :math:`\phi` lives on the nodes, then: + + - the partial derivative :math:`\frac{\partial \phi}{\partial x}\hat{x}` lives on x edges, + - the partial derivative :math:`\frac{\partial \phi}{\partial y}\hat{y}` lives on y edges, and + - the partial derivative :math:`\frac{\partial \phi}{\partial z}\hat{z}` lives on z edges + +Thus to approximate the gradient of :math:`\phi`, +we simply need to take discrete derivatives of :math:`\phi` with respect to :math:`x`, :math:`y` and :math:`z`, +and organize the resulting vector components on the corresponding edges. +Let :math:`h_x`, :math:`h_y` and :math:`h_z` represent the dimension of the cell along the x, y and +z directions, respectively. + +**In 2D**, the value of :math:`\phi` at 4 node locations is used to approximate the vector components of the +gradient at 4 edges locations (2 x-edges and 2 y-edges) as follows: + +.. math:: + \begin{align} + u_x \Big ( i+\frac{1}{2},j \Big ) \approx \; & \frac{\phi (i+1,j) - \phi (i,j)}{h_x} \\ + u_x \Big ( i+\frac{1}{2},j+1 \Big ) \approx \; & \frac{\phi (i+1,j+1) - \phi (i,j+1)}{h_x} \\ + u_y \Big ( i,j+\frac{1}{2} \Big ) \approx \; & \frac{\phi (i,j+1) - \phi (i,j)}{h_y} \\ + u_y \Big ( i+1,j+\frac{1}{2} \Big ) \approx \; & \frac{\phi (i+1,j+1) - \phi (i+1,j)}{h_y} + \end{align} + +**In 3D**, the value of :math:`\phi` at 8 node locations is used to approximate the vector components of the +gradient at 12 edges locations (4 x-edges, 4 y-edges and 4 z-edges). An example of the approximation +for each vector component is given below: + +.. math:: + \begin{align} + u_x \Big ( i+\frac{1}{2},j,k \Big ) \approx \; & \frac{\phi (i+1,j,k) - \phi (i,j,k)}{h_x} \\ + u_y \Big ( i,j+\frac{1}{2},k \Big ) \approx \; & \frac{\phi (i,j+1,k) - \phi (i,j,k)}{h_y} \\ + u_z \Big ( i,j,k+\frac{1}{2} \Big ) \approx \; & \frac{\phi (i,j,k+1) - \phi (i,j,k)}{h_z} + \end{align} + + + + + + +Curl +---- + +Let us define two continuous vector functions :math:`\vec{u}` and :math:`\vec{w}` such that: + +.. math:: + \vec{w} = \nabla \times \vec{u} + +And let :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` be the discrete representations of :math:`\vec{u}` and :math:`\vec{w}` +that live on the mesh (centers, nodes, edges or faces), respectively. Provided we know the discrete values :math:`\boldsymbol{u}`, +our goal is to use discrete differentiation to approximate the vector components of :math:`\boldsymbol{w}`. +We begin by considering a single 3D cell. We let the indices :math:`i`, :math:`j` and :math:`k` +denote positions along the x, y and z axes, respectively. + + + +.. image:: ../images/curl_discretization.png + :align: center + :width: 800 + + +As we will see, it makes the most sense for the vector components of :math:`\boldsymbol{u}` to live on the edges +for the vector components of :math:`\boldsymbol{w}` to live the faces. In this case, we need to approximate: + + + - the partial derivatives :math:`\frac{\partial u_y}{\partial z}` and :math:`\frac{\partial u_z}{\partial y}` to compute :math:`w_x`, + - the partial derivatives :math:`\frac{\partial u_x}{\partial z}` and :math:`\frac{\partial u_z}{\partial x}` to compute :math:`w_y`, and + - the partial derivatives :math:`\frac{\partial u_x}{\partial y}` and :math:`\frac{\partial u_y}{\partial x}` to compute :math:`w_z` + +**In 3D**, the value of :math:`\phi` at 8 node locations is used to approximate the vector components of the +gradient at 12 edges locations (4 x-edges, 4 y-edges and 4 z-edges). An example of the approximation +for each vector component is given below: + +.. math:: + \begin{align} + w_x \Big ( i,j+\frac{1}{2},k+\frac{1}{2} \Big ) \approx \; & + \Bigg( \frac{u_y (i,j+\frac{1}{2},k+1) - u_y (i,j+\frac{1}{2},k)}{h_z} \Bigg ) + - \Bigg( \frac{u_z (i,j+1,k+\frac{1}{2}) - u_z (i,j,k+\frac{1}{2})}{h_y} \Bigg ) \\ + & \\ + w_y \Big ( i+\frac{1}{2},j,k+\frac{1}{2} \Big ) \approx \; & + - \Bigg( \frac{u_x (i+\frac{1}{2},j,k+1) - u_x (i+\frac{1}{2},j,k)}{h_z} \Bigg ) + + \Bigg( \frac{u_z (i+1,j,k+\frac{1}{2}) - u_z (i,j,k+\frac{1}{2})}{h_x} \Bigg ) \\ + & \\ + w_z \Big ( i+\frac{1}{2},j+\frac{1}{2},k \Big ) \approx \; & + \Bigg( \frac{u_x (i+\frac{1}{2},j+1,k) - u_x (i+\frac{1}{2},j,k)}{h_y} \Bigg ) + - \Bigg( \frac{u_y (i+1,j+\frac{1}{2},k) - u_y (i,j+\frac{1}{2},k)}{h_z} \Bigg ) + \end{align} diff --git a/docs/images/approximate_derivative.PNG b/docs/images/approximate_derivative.PNG new file mode 100644 index 0000000000000000000000000000000000000000..9f4e5ee614f2ce111494e1ce5321d75f1e7f282e GIT binary patch literal 19218 zcmdVCgsmWXUsr>efSv#i4UJe6rfPtOhCT`YXyW03 z|1p$7}%{rr}mu;FE6l%#gRsK>_T=w5oc}sUEoJdoZJQ^>9mlNXgD)M$T zPPBnRQshf^6$j#FHl3oalS_rcyX#Z_2h%H49s}}SGAjWyl|Ovk$1EIYJqD+J0}p)0 zM9(zHp?GYNLVXu}HbmkL6(lEu3<@XekwB;7HG-YDT^?SJqy92@efLQp=UVNEuHohJ zVBpflCN}t;xdavP;p5}=$~L1EUSn0KVT|e7oAvlXa4kHxV&w#8c|T~T>~q)XavjtK zAjcVsfvtK@!-cxpw~=-D(E@A;h@|4>SvwwK`iy(me~ zDTY?4f`nO)=N^@1%_nTjDSn2^1yxn{=QS^z31S0v~B0P zyQF!nLYW*rH+8X?ptzKm9HQm&tB2O+FTTLfjftB5NJVhlcxo>1U8i{^S)tX0QnScT z&Ta__m+{xTOMQ!!KL@g9*OOxPT?E*vkYNwvO3jnA+mGY-a>z2>_KfIeelpn{UXq|w-xq6n1jh}T|x@&nV5ouHpcI=H?y%iYiF-)g0maOu1%%2$h7%UJ!p z{zGrMKTF#1umI^gEt3M`dGxh1=KfaO<=TvxVCxYa#GJr+^f*CJDkPecTlbTE=TzwJ z6yCQaOfm%;DpyZ6`Ls%skee%O>T9NC+POt7c%*3VIf3_6`HqF5e^VNG24vRxHFt2y z?5&Y-wVTUlBYKLOxi4h5nhkk985IMp4#^nu#*uUmph^q{$zbh{TZSf%il{)^)6a2FHZMi>V?`F>>f0N zRz<#%abN+Pm6?5vwmZ(z1(N+OxUWL&WRbJp#(zKMZ`=(L=%JTKFVpi+&7V{FVoW_) zx%6`Q_hF4%;+}5ud~u3FzwiKrO}Pd;YFW+6CylzbTrQ7SSVaCzc`e2@adQY*)apw= zS<(kDf=b99^0EE2Ip~KdQ^7_ATx+4Hw6b-L(n8FvEg$_lV7#IPf`-`r-_RPq)7|$S zEYN(hv|nY{wN%FtegVOyvZNrDaGfgJ{q<#q@~26;g<5&R&Fc?hQB2Z_I=~dO-e&n- zo*x>ll?0rh1n^Bbe0^06Ufx}hPP7jT&~fn?BQyn27l#vfV$CnwA?l%F1Z(eS14iDM z{|k7GD1j{6D%#068l=-NQaj1RvWD9n2brfcaWICoUh=SY{MO~hEqj(xAtL>HJlQN{ zv{W>Sq&_#1i2D^t86j{689lij#KyFGav!!*l~BVR1`Rum!fWP**{w%E#lRM6Ia-ME zf%yFU<-+cnE>guZ-%iYl;>7iroCnF6IfoDv)+{pSUUjD=V?m44T3sI0hT!aG06(E& zlyXz5Pm3J_&u?5`0KZi5*XedAd__J=LM5DN2VcPA z_1Wrh$kon3k z7-09~2@s^TGf0zPEoWwnu(3L98Qf*kP>rKr?$Gk5Bo+Uo`3N7##q}@5H!49N@bowm zY9uflCn%g0rI#dL5Qi0Mr3se5l6X{-6L{e*&qo~n=r`C`4b5k}Ujh!JxO4ps5+FEN zcQ%qC{xH>M{byaNmGIFI`2((4n;@AbwPgSgbnI|1*dcxSPWw_I1<~sK)7o>?*!j=TI)e^O>|b{E z*eVeuDTiJe8DCGQ{6M-{d-bq_oYJkrsu6_?>^rz(%pHs6 zymyOUi9alnob@wI_(=#JuM<4=(pj%(aQM2{ms9~CG30far`y6U3$^Ktt8A4=L7@}} zJ3;)}h8Z29smspHh_0;8XtAM&{Qh@CioYboT_Pt3e@Y#(ake_9aGAZ71W)?6DJDlt zO=D~qxPq5{Q*vpO?esR+Kbl-Ke~^C2aXvL_Y9rvW!z|~fq&=Ol=rqKd5jgGnLvzP3 zy^%r8iPirFm(I7E4Fp^ZI^TiI@^WvY7MqafcT%?sDy$#bhC~UROb=+VG9u4^rP

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docs/content/finite_volume_index.rst | 28 +- .../theory/derivation_examples_index.rst | 5 + .../theory/inner_products_anisotropic.rst | 10 +- docs/content/theory/inner_products_basic.rst | 18 +- .../theory/inner_products_differential.rst | 2 +- docs/content/theory/inner_products_index.rst | 309 +----------------- .../theory/inner_products_isotropic.rst | 9 +- .../theory/{meshes.rst => meshes_index.rst} | 18 +- docs/content/theory/operators_averaging.rst | 54 +++ .../content/theory/operators_differential.rst | 9 +- docs/content/theory/operators_index.rst | 16 +- docs/index.rst | 1 + 12 files changed, 117 insertions(+), 362 deletions(-) create mode 100644 docs/content/theory/derivation_examples_index.rst rename docs/content/theory/{meshes.rst => meshes_index.rst} (72%) diff --git a/docs/content/finite_volume_index.rst b/docs/content/finite_volume_index.rst index fda7ee525..9cffe36f7 100644 --- a/docs/content/finite_volume_index.rst +++ b/docs/content/finite_volume_index.rst @@ -3,8 +3,8 @@ Intoduction to Finite Volume **************************** -What Finite Volume? -------------------- +What is Finite Volume? +---------------------- The finite volume method is a method for numerically approximating the solution to partial differential equations. Implementation of the finite volume method requires the discretization of continuous functions and variables. @@ -23,18 +23,28 @@ This method is different from finite difference methods, as the final linear system is constructed by approximating the inner products between test functions and partial differential equations. +**Contents:** + + - :ref:`Meshes ` + - :ref:`Averaging and Differential Operators ` + - :ref:`Inner Products ` + - :ref:`Discretizing PDEs Derivation Examples ` + +**Tutorials and Examples Gallery:** + + - :ref:`Mesh Generation ` + - :ref:`Interpolation, Averaging and Differential Operators ` + - :ref:`Inner Products ` + - :ref:`Discretizing PDEs Derivation Examples ` + - :ref:`Examples Gallery ` + Examples -------- Below are several examples of the final linear system obtained using the finite volume approach. -A comprehensive derivation of the final result is not provided here. Instead, the necessary -background theory is parsed into several section, including: - - - :ref:`meshes ` - - :ref:`interpolation, averaging and differential operators ` - - :ref:`inner products ` - - finite volume full examples +A comprehensive derivation of the final result is not provided here. The full derivations are +provide in the :ref:`discretizing PDEs derivation examples ` theory section. Direct Current Resistivity ^^^^^^^^^^^^^^^^^^^^^^^^^^ diff --git a/docs/content/theory/derivation_examples_index.rst b/docs/content/theory/derivation_examples_index.rst new file mode 100644 index 000000000..45513aaaa --- /dev/null +++ b/docs/content/theory/derivation_examples_index.rst @@ -0,0 +1,5 @@ +.. _derivation_examples_index: + +Solving PDEs Examples +********************* + diff --git a/docs/content/theory/inner_products_anisotropic.rst b/docs/content/theory/inner_products_anisotropic.rst index 8843119fb..6d0e92458 100644 --- a/docs/content/theory/inner_products_anisotropic.rst +++ b/docs/content/theory/inner_products_anisotropic.rst @@ -3,8 +3,6 @@ Anisotropic Constitutive Relationships ************************************** -.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` - Summary ------- @@ -52,6 +50,8 @@ inner-product matrix in this case: For this class of inner products, the corresponding form of the inner product matricies for discrete quantities living on various parts of the mesh are shown below. +.. important:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` + **Diagonal Anisotropic:** For the diagonal anisotropic case, the tensor characterzing the material properties @@ -213,10 +213,6 @@ and - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells - :math:`\boldsymbol{P_f}` is a projection matrix that maps quantities from faces to cell centers - -.. note:: To see a validation of the discrete approximation, see our tutorials section (link) - - Vectors on Cell Edges ^^^^^^^^^^^^^^^^^^^^^ @@ -299,8 +295,6 @@ and - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells - :math:`\boldsymbol{P_e}` is a projection matrix that maps quantities from edges to cell centers -.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` - Fully Anisotropic Case ---------------------- diff --git a/docs/content/theory/inner_products_basic.rst b/docs/content/theory/inner_products_basic.rst index 47f44b164..ff1260ef4 100644 --- a/docs/content/theory/inner_products_basic.rst +++ b/docs/content/theory/inner_products_basic.rst @@ -47,7 +47,7 @@ where - :math:`\boldsymbol{P_n}`, :math:`\boldsymbol{P_f}` and :math:`\boldsymbol{P_e}` are projection matricies that map quantities from nodes, faces and edges to cell centers, respectively - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells -.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` +.. important:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` Scalars at Cell Centers @@ -92,9 +92,6 @@ the cell volumes (:math:`\boldsymbol{v}`), i.e.: \boldsymbol{M_c} = diag(\boldsymbol{v}) -.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` - - Scalars at Nodes ---------------- @@ -155,8 +152,6 @@ where - :math:`\boldsymbol{P_n}` is a projection matrix that maps quantities from nodes to cell centers - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells -.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` - Vectors on Cell Faces --------------------- @@ -230,8 +225,6 @@ and - :math:`\boldsymbol{P_f}` is the projection matrix that maps quantities from faces to cell centers - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells -.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` - Vectors on Cell Edges --------------------- @@ -305,12 +298,3 @@ and - :math:`\boldsymbol{P_e}` is the projection matrix that maps quantities from edges to cell centers - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells -.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on basic inner products ` - - - - - - - - diff --git a/docs/content/theory/inner_products_differential.rst b/docs/content/theory/inner_products_differential.rst index 41214e795..941f25bc9 100644 --- a/docs/content/theory/inner_products_differential.rst +++ b/docs/content/theory/inner_products_differential.rst @@ -24,7 +24,7 @@ we are interested in approximating the following inner products: (\vec{u}, \nabla \times \vec{w} ) &= \int_\Omega \vec{u} \cdot (\nabla \times \vec{w} ) \, dv \end{align} -.. note:: To construct differential operators and/or approximate inner products of this type, see the :ref:`tutorial on inner products with differential operators ` +.. important:: To construct differential operators and/or approximate inner products of this type, see the :ref:`tutorial on inner products with differential operators ` **Gradient:** diff --git a/docs/content/theory/inner_products_index.rst b/docs/content/theory/inner_products_index.rst index 56798f835..b09aa2d9b 100644 --- a/docs/content/theory/inner_products_index.rst +++ b/docs/content/theory/inner_products_index.rst @@ -1,4 +1,4 @@ -.. _inner_products: +.. _inner_products_index: Inner Products ************** @@ -32,306 +32,9 @@ discrete approximation for several classes of inner products. inner_products_anisotropic inner_products_differential +**Tutorials:** - - -xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx - - - - -By using the weak formulation of many of the PDEs in geophysical applications, -we can rapidly develop discretizations. Much of this work, however, needs a -good understanding of how to approximate inner products on our discretized -meshes. We will define the inner product as: - -.. math:: - - \left(a,b\right) = \int_\Omega{a \cdot b}{\partial v} - -where a and b are either scalars or vectors. - -.. note:: - - The InnerProducts class is a base class providing inner product matrices - for meshes and cannot run on its own. - - - - - - - - -**Example problem for DC resistivity** - -We will start with the formulation of the Direct Current (DC) resistivity -problem in geophysics. - - -.. math:: - - \frac{1}{\sigma}\vec{j} = \nabla \phi \\ - - \nabla\cdot \vec{j} = q - -In the following discretization, :math:`\sigma` and :math:`\phi` -will be discretized on the cell-centers and the flux, :math:`\vec{j}`, -will be on the faces. We will use the weak formulation to discretize -the DC resistivity equation. - -We can define in weak form by integrating with a general face function -:math:`\vec{f}`: - -.. math:: - - \int_{\Omega}{\sigma^{-1}\vec{j} \cdot \vec{f}} = \int_{\Omega}{\nabla \phi \cdot \vec{f}} - -Here we can integrate the right side by parts, - -.. math:: - - \nabla\cdot(\phi\vec{f})=\nabla\phi\cdot\vec{f} + \phi\nabla\cdot\vec{f} - -and rearrange it, and apply the Divergence theorem. - -.. math:: - - \int_{\Omega}{\sigma^{-1}\vec{j} \cdot \vec{f}} = - - \int_{\Omega}{(\phi \nabla \cdot \vec{f})} - + \int_{\partial \Omega}{ \phi \vec{f} \cdot \mathbf{n}} - -We can then discretize for every cell: - -.. math:: - - v_{\text{cell}} \sigma^{-1} (\mathbf{J}_x \mathbf{F}_x +\mathbf{J}_y \mathbf{F}_y + \mathbf{J}_z \mathbf{F}_z ) = -\phi^{\top} v_{\text{cell}} \mathbf{D}_{\text{cell}} \mathbf{F} + \text{BC} - -.. note:: - - We have discretized the dot product above, but remember that we do not - really have a single vector :math:`\mathbf{J}`, but approximations of - :math:`\vec{j}` on each face of our cell. In 2D that means 2 - approximations of :math:`\mathbf{J}_x` and 2 approximations of - :math:`\mathbf{J}_y`. In 3D we also have 2 approximations of - :math:`\mathbf{J}_z`. - -Regardless of how we choose to approximate this dot product, we can represent -this in vector form (again this is for every cell), and will generalize for -the case of anisotropic (tensor) sigma. - -.. math:: - - \mathbf{F}_c^{\top} (\sqrt{v_{\text{cell}}} \Sigma^{-1} \sqrt{v_{\text{cell}}}) \mathbf{J}_c = - -\phi^{\top} v_{\text{cell}} \mathbf{D}_{\text{cell}} \mathbf{F}) - + \text{BC} - -We multiply by square-root of volume on each side of the tensor conductivity -to keep symmetry in the system. Here :math:`\mathbf{J}_c` is the Cartesian -:math:`\mathbf{J}` (on the faces that we choose to use in our approximation) -and must be calculated differently depending on the mesh: - -.. math:: - \mathbf{J}_c = \mathbf{Q}_{(i)}\mathbf{J}_\text{TENSOR} \\ - \mathbf{J}_c = \mathbf{N}_{(i)}^{-1}\mathbf{Q}_{(i)}\mathbf{J}_\text{Curv} - -Here the :math:`i` index refers to where we choose to approximate this integral, as discussed in the note above. -We will approximate this integral by taking the fluxes clustered around every node of the cell, there are 8 combinations in 3D, and 4 in 2D. We will use a projection matrix :math:`\mathbf{Q}_{(i)}` to pick the appropriate fluxes. So, now that we have 8 approximations of this integral, we will just take the average. For the TensorMesh, this looks like: - -.. math:: - - \mathbf{F}^{\top} - {1\over 8} - \left(\sum_{i=1}^8 - \mathbf{Q}_{(i)}^{\top} \sqrt{v_{\text{cell}}} \Sigma^{-1} \sqrt{v_{\text{cell}}} \mathbf{Q}_{(i)} - \right) - \mathbf{J} - = - -\mathbf{F}^{\top} \mathbf{D}_{\text{cell}}^{\top} v_{\text{cell}} \phi + \text{BC} - -Or, when generalizing to the entire mesh and dropping our general face function: - -.. math:: - - \mathbf{M}^f_{\Sigma^{-1}} \mathbf{J} - = - - \mathbf{D}^{\top} \text{diag}(\mathbf{v}) \phi + \text{BC} - -By defining the faceInnerProduct (8 combinations of fluxes in 3D, 4 in 2D, 2 in 1D) to be: - -.. math:: - - \mathbf{M}^f_{\Sigma^{-1}} = - \sum_{i=1}^{2^d} - \mathbf{P}_{(i)}^{\top} \Sigma^{-1} \mathbf{P}_{(i)} - -Where :math:`d` is the dimension of the mesh. -The :math:`\mathbf{M}^f` is returned when given the input of :math:`\Sigma^{-1}`. - -Here each :math:`\mathbf{P} ~ \in ~ \mathbb{R}^{(d*nC, nF)}` is a combination -of the projection, volume, and any normalization to Cartesian coordinates -(where the dot product is well defined): - -.. math:: - - \mathbf{P}_{(i)} = \sqrt{ \frac{1}{2^d} \mathbf{I}^d \otimes \text{diag}(\mathbf{v})} \overbrace{\mathbf{N}_{(i)}^{-1}}^{\text{Curv only}} \mathbf{Q}_{(i)} - -.. note:: - - This is actually completed for each cell in the mesh at the same time, and the full matrices are returned. - -If ``returnP=True`` is requested in any of these methods the projection matrices are returned as a list ordered by nodes around which the fluxes were approximated:: - - # In 3D - P = [P000, P100, P010, P110, P001, P101, P011, P111] - # In 2D - P = [P00, P10, P01, P11] - # In 1D - P = [P0, P1] - -The derivation for ``edgeInnerProducts`` is exactly the same, however, when we -approximate the integral using the fields around each node, the projection -matrices look a bit different because we have 12 edges in 3D instead of just 6 -faces. The interface to the code is exactly the same. - - -.. Defining Tensor Properties -.. -------------------------- - -.. **For 3D:** - -.. Depending on the number of columns (either 1, 3, or 6) of mu, the material -.. property is interpreted as follows: - -.. .. math:: - -.. \vec{\mu} = \left[\begin{matrix} \mu_{1} & 0 & 0 \\ 0 & \mu_{1} & 0 \\ 0 & 0 & \mu_{1} \end{matrix}\right] - -.. \vec{\mu} = \left[\begin{matrix} \mu_{1} & 0 & 0 \\ 0 & \mu_{2} & 0 \\ 0 & 0 & \mu_{3} \end{matrix}\right] - -.. \vec{\mu} = \left[\begin{matrix} \mu_{1} & \mu_{4} & \mu_{5} \\ \mu_{4} & \mu_{2} & \mu_{6} \\ \mu_{5} & \mu_{6} & \mu_{3} \end{matrix}\right] - -.. **For 2D:** - -.. Depending on the number of columns (either 1, 2, or 3) of mu, the material property is interpreted as follows: - -.. .. math:: -.. \vec{\mu} = \left[\begin{matrix} \mu_{1} & 0 \\ 0 & \mu_{1} \end{matrix}\right] - -.. \vec{\mu} = \left[\begin{matrix} \mu_{1} & 0 \\ 0 & \mu_{2} \end{matrix}\right] - -.. \vec{\mu} = \left[\begin{matrix} \mu_{1} & \mu_{3} \\ \mu_{3} & \mu_{2} \end{matrix}\right] - - -.. Structure of Matrices -.. --------------------- - -.. Both the isotropic, and anisotropic material properties result in a diagonal mass matrix. -.. Which is nice and easy to invert if necessary, however, in the fully anisotropic case which is not aligned with the grid, the matrix is not diagonal. This can be seen for a 3D mesh in the figure below. - -.. .. plot:: - -.. import discretize -.. import numpy as np -.. import matplotlib.pyplot as plt -.. mesh = discretize.TensorMesh([10,50,3]) -.. m1 = np.random.rand(mesh.nC) -.. m2 = np.random.rand(mesh.nC,3) -.. m3 = np.random.rand(mesh.nC,6) -.. M = list(range(3)) -.. M[0] = mesh.getFaceInnerProduct(m1) -.. M[1] = mesh.getFaceInnerProduct(m2) -.. M[2] = mesh.getFaceInnerProduct(m3) -.. plt.figure(figsize=(13,5)) -.. for i, lab in enumerate(['Isotropic','Anisotropic','Tensor']): -.. plt.subplot(131 + i) -.. plt.spy(M[i],ms=0.5,color='k') -.. plt.tick_params(axis='both',which='both',labeltop='off',labelleft='off') -.. plt.title(lab + ' Material Property') -.. plt.show() - - -.. Taking Derivatives -.. ------------------ - -.. We will take the derivative of the fully anisotropic tensor for a 3D mesh, the -.. other cases are easier and will not be discussed here. Let us start with one -.. part of the sum which makes up :math:`\mathbf{M}^f_\Sigma` and take the -.. derivative when this is multiplied by some vector :math:`\mathbf{v}`: - -.. .. math:: - -.. \mathbf{P}^\top \boldsymbol{\Sigma} \mathbf{Pv} - -.. Here we will let :math:`\mathbf{Pv} = \mathbf{y}` and :math:`\mathbf{y}` will have the form: - -.. .. math:: - -.. \mathbf{y} = \mathbf{Pv} = -.. \left[ -.. \begin{matrix} -.. \mathbf{y}_1\\ -.. \mathbf{y}_2\\ -.. \mathbf{y}_3\\ -.. \end{matrix} -.. \right] - -.. .. math:: - -.. \mathbf{P}^\top\Sigma\mathbf{y} = -.. \mathbf{P}^\top -.. \left[\begin{matrix} -.. \boldsymbol{\sigma}_{1} & \boldsymbol{\sigma}_{4} & \boldsymbol{\sigma}_{5} \\ -.. \boldsymbol{\sigma}_{4} & \boldsymbol{\sigma}_{2} & \boldsymbol{\sigma}_{6} \\ -.. \boldsymbol{\sigma}_{5} & \boldsymbol{\sigma}_{6} & \boldsymbol{\sigma}_{3} -.. \end{matrix}\right] -.. \left[ -.. \begin{matrix} -.. \mathbf{y}_1\\ -.. \mathbf{y}_2\\ -.. \mathbf{y}_3\\ -.. \end{matrix} -.. \right] -.. = -.. \mathbf{P}^\top -.. \left[ -.. \begin{matrix} -.. \boldsymbol{\sigma}_{1}\circ \mathbf{y}_1 + \boldsymbol{\sigma}_{4}\circ \mathbf{y}_2 + \boldsymbol{\sigma}_{5}\circ \mathbf{y}_3\\ -.. \boldsymbol{\sigma}_{4}\circ \mathbf{y}_1 + \boldsymbol{\sigma}_{2}\circ \mathbf{y}_2 + \boldsymbol{\sigma}_{6}\circ \mathbf{y}_3\\ -.. \boldsymbol{\sigma}_{5}\circ \mathbf{y}_1 + \boldsymbol{\sigma}_{6}\circ \mathbf{y}_2 + \boldsymbol{\sigma}_{3}\circ \mathbf{y}_3\\ -.. \end{matrix} -.. \right] - -.. Now it is easy to take the derivative with respect to any one of the -.. parameters, for example, -.. :math:`\frac{\partial}{\partial\boldsymbol{\sigma}_1}` - -.. .. math:: -.. \frac{\partial}{\partial \boldsymbol{\sigma}_1}\left(\mathbf{P}^\top\Sigma\mathbf{y}\right) -.. = -.. \mathbf{P}^\top -.. \left[ -.. \begin{matrix} -.. \text{diag}(\mathbf{y}_1)\\ -.. 0\\ -.. 0\\ -.. \end{matrix} -.. \right] - -.. Whereas :math:`\frac{\partial}{\partial\boldsymbol{\sigma}_4}`, for -.. example, is: - -.. .. math:: -.. \frac{\partial}{\partial \boldsymbol{\sigma}_4}\left(\mathbf{P}^\top\Sigma\mathbf{y}\right) -.. = -.. \mathbf{P}^\top -.. \left[ -.. \begin{matrix} -.. \text{diag}(\mathbf{y}_2)\\ -.. \text{diag}(\mathbf{y}_1)\\ -.. 0\\ -.. \end{matrix} -.. \right] - -.. These are computed for each of the 8 projections, horizontally concatenated, -.. and returned. + - :ref:`Basic Inner Products ` + - :ref:`Inner Products with Constitutive Relationships ` + - :ref:`Inner Products with Differential Operators ` + - :ref:`Advanced Inner Product Examples ` \ No newline at end of file diff --git a/docs/content/theory/inner_products_isotropic.rst b/docs/content/theory/inner_products_isotropic.rst index 5426c2e84..bf73e15b7 100644 --- a/docs/content/theory/inner_products_isotropic.rst +++ b/docs/content/theory/inner_products_isotropic.rst @@ -51,7 +51,8 @@ where - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells - :math:`\boldsymbol{\sigma}` is a vector containing the physical property values for the cells -.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` + +.. important:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` Vectors on Cell Faces @@ -126,9 +127,6 @@ and - :math:`\boldsymbol{\sigma}` is a vector containing the physical property values for the cells -.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` - - Vectors on Cell Edges --------------------- @@ -199,6 +197,3 @@ and - :math:`\boldsymbol{P_e}` is a projection matrix that maps from edges to cell centers - :math:`\boldsymbol{v}` is a vector that stores all of the volumes of the cells - :math:`\boldsymbol{\sigma}` is a vector containing the physical property values for the cells - -.. note:: To construct the inner product matrix and/or approximate inner products of this type, see the :ref:`tutorial on inner products with constitutive relationships ` - diff --git a/docs/content/theory/meshes.rst b/docs/content/theory/meshes_index.rst similarity index 72% rename from docs/content/theory/meshes.rst rename to docs/content/theory/meshes_index.rst index 37a5661c1..66ef010eb 100644 --- a/docs/content/theory/meshes.rst +++ b/docs/content/theory/meshes_index.rst @@ -6,12 +6,22 @@ Meshes A mesh is a numerical grid on which discrete approximations of continuous functions and variables live. Meshes can be created to discretize and solve 1D, 2D or 3D problems. For a given system of partial differential equations, the mesh is a discrete representation of the domain and its boundaries. -Here, we demonstrate: +Here, we demonstrate mesh types supported by the *discretize* package, and where discrete quantities live (mesh locations). - - mesh types supported by the *discretize* package - - where discrete quantities live (mesh locations) +**Contents:** + - :ref:`Mesh Types` + - :ref:`Where Quantities Live ` +**Tutorials:** + + - :ref:`Mesh Types Overview ` + - :ref:`Tensor Meshes ` + - :ref:`Cylindrical Meshes ` + - :ref:`Tree Meshes ` + + +.. _meshes_index_types: Mesh Types ========== @@ -29,6 +39,8 @@ Examples are illustrate below. :align: center :width: 700 +.. _meshes_index_quantities: + Where Quantities Live ===================== diff --git a/docs/content/theory/operators_averaging.rst b/docs/content/theory/operators_averaging.rst index 71de6cecb..064f05de5 100644 --- a/docs/content/theory/operators_averaging.rst +++ b/docs/content/theory/operators_averaging.rst @@ -3,3 +3,57 @@ Interpolation and Averaging *************************** +Here, we provide the background theory for how discrete interpolation and averaging matricies are formed. +Interpolation is required when a discrete quantity is known on the mesh (centers, nodes, edges or faces), but we would like to estimate its +value at a point within the continuous domain. Averaging is required when quantities that live on different +parts of the mesh need to be multiplied. + +**Tutorials:** + + - :ref:`tutorial for constructing and applying averaging operators ` + + + +Interpolation +============= + +Presently there is an extensive set of interpolation methods (e.g. polynomial, spline, piecewise constant). +One of the most effective and widely used interpolation methods is linear interpolation. +The *discretize* package primarily uses linear interpolation because 1) it is very fast, and 2) higher order +interpolation methods require the construction of matricies which are less sparse. +Here, we discuss how a sparse matrix can be formed which interpolates the discrete values to +a set of points in continuous space. + +The formulation for linear interpolation is adequately presented on Wikipedia, see: + + - `Linear Interpolation (1D) `__ + - `Bilinear Interpolation (2D) `__ + - `Trilinear Interpolation (3D) `__ + +Interpolation Matrix in 1D +^^^^^^^^^^^^^^^^^^^^^^^^^^ + +**Basics for constructing the 1D interpolation matrix** + + + +Interpolation Matrix in 3D +^^^^^^^^^^^^^^^^^^^^^^^^^^ + +**Basics for constructing the 1D interpolation matrix** + + + +Scalars vs. Vectors +^^^^^^^^^^^^^^^^^^^ + +**Describe how components are treated separately** + + +Averaging +========= + + + + + diff --git a/docs/content/theory/operators_differential.rst b/docs/content/theory/operators_differential.rst index baf2ed030..a1442efd8 100644 --- a/docs/content/theory/operators_differential.rst +++ b/docs/content/theory/operators_differential.rst @@ -27,7 +27,7 @@ the gradient, divergence and curl can be computed for discrete variables. :width: 300 -.. note:: To construct discrete differential operators and apply them in a set of examples, see our :ref:`tutorial for differential operators ` +.. important:: To construct and apply differential operators to discrete quantities, see our* :ref:`tutorial for differential operators ` @@ -101,9 +101,6 @@ and :math:`\boldsymbol{u}` stores the components of :math:`\vec{u}` on cell face \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_z} \end{bmatrix} -.. note:: To construct discrete differential operators and apply them in a set of examples, see our :ref:`tutorial for differential operators ` - - Gradient -------- @@ -173,9 +170,6 @@ and :math:`\boldsymbol{u}` stores the components of :math:`\vec{u}` on cell edge \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_z} \end{bmatrix} -.. note:: To construct discrete differential operators and apply them in a set of examples, see our :ref:`tutorial for differential operators ` - - Curl ---- @@ -239,4 +233,3 @@ and :math:`\boldsymbol{w}` is a vector that stores the components of :math:`\vec \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_z} \end{bmatrix} \;\;\;\; \textrm{and} \;\;\;\; \begin{bmatrix} \boldsymbol{w_x} \\ \boldsymbol{w_y} \\ \boldsymbol{w_z} \end{bmatrix} -.. note:: To construct discrete differential operators and apply them in a set of examples, see our :ref:`tutorial for differential operators ` diff --git a/docs/content/theory/operators_index.rst b/docs/content/theory/operators_index.rst index 8c2e4891a..d892fead2 100644 --- a/docs/content/theory/operators_index.rst +++ b/docs/content/theory/operators_index.rst @@ -3,12 +3,11 @@ Operators ********* -A mathematical operator is function or operation that acts on elements within -a space to produce elements in the same or another space. - -For our purpose - +To approximate numerical solutions to partial differential equations using the *discretize* package, +we must be able to construct discrete approximations to the following types of mathematical operators: + - Averaging + - Differential (gradient, divergence, curl) **Contents:** @@ -16,4 +15,9 @@ For our purpose :maxdepth: 1 operators_averaging - operators_differential \ No newline at end of file + operators_differential + +**Tutorials:** + +- :ref:`Averaging Operators ` +- :ref:`Differential Operators ` \ No newline at end of file diff --git a/docs/index.rst b/docs/index.rst index c08ee3169..b157a5d63 100644 --- a/docs/index.rst +++ b/docs/index.rst @@ -36,6 +36,7 @@ content/theory/meshes content/theory/operators_index content/theory/inner_products_index + content/theory/derivation_examples_index .. toctree:: :maxdepth: 2 From 0e891b282f7c3f2fde76337479affe1d7b227bd3 Mon Sep 17 00:00:00 2001 From: dccowan Date: Mon, 23 Nov 2020 23:10:04 -0800 Subject: [PATCH 13/35] Adding interpolation matrix theory --- docs/content/theory/meshes_index.rst | 4 +- docs/content/theory/operators_averaging.rst | 131 +++++++++++++++++++- docs/content/theory/operators_index.rst | 18 ++- docs/images/interpolation_1d.PNG | Bin 0 -> 4282 bytes docs/images/interpolation_1d_sparse.PNG | Bin 0 -> 6630 bytes docs/images/interpolation_2d.PNG | Bin 0 -> 12213 bytes docs/index.rst | 2 +- 7 files changed, 144 insertions(+), 11 deletions(-) create mode 100644 docs/images/interpolation_1d.PNG create mode 100644 docs/images/interpolation_1d_sparse.PNG create mode 100644 docs/images/interpolation_2d.PNG diff --git a/docs/content/theory/meshes_index.rst b/docs/content/theory/meshes_index.rst index 66ef010eb..bc28dda54 100644 --- a/docs/content/theory/meshes_index.rst +++ b/docs/content/theory/meshes_index.rst @@ -49,9 +49,9 @@ Shashkov, 1999 `_). This approach requires the definitions of variables at either cell-centers, nodes, faces, or edges. Below, we illustrate the valid locations for discrete quantities for a single cell where: - - **Nodes** represent location of intersection between grid lines defining the mesh. + - **Nodes** represent locations of intersection between grid lines defining the mesh. - **X, Y and Z edges** are edges whose tangent lines are parallel to the X, Y and Z axis, respectively. - - **X, Y and Z faces** are faces which are normal to the X, Y and Z axis, respectively. + - **X, Y and Z faces** are faces which are normal to orientation of the X, Y and Z axis, respectively. .. image:: ../../images/cell_locations.png diff --git a/docs/content/theory/operators_averaging.rst b/docs/content/theory/operators_averaging.rst index 064f05de5..c60420f3a 100644 --- a/docs/content/theory/operators_averaging.rst +++ b/docs/content/theory/operators_averaging.rst @@ -6,7 +6,7 @@ Interpolation and Averaging Here, we provide the background theory for how discrete interpolation and averaging matricies are formed. Interpolation is required when a discrete quantity is known on the mesh (centers, nodes, edges or faces), but we would like to estimate its value at a point within the continuous domain. Averaging is required when quantities that live on different -parts of the mesh need to be multiplied. +parts of the mesh need to be added, subtracted, multiplied or divided. **Tutorials:** @@ -33,21 +33,140 @@ The formulation for linear interpolation is adequately presented on Wikipedia, s Interpolation Matrix in 1D ^^^^^^^^^^^^^^^^^^^^^^^^^^ -**Basics for constructing the 1D interpolation matrix** +Let us define a 1D mesh that contains 8 cells of arbitrary width; +the mesh is shown in the figure below. The width of each cell is +given by :math:`h_i`. The location of each node is given by :math:`x_i`. +.. image:: ../../images/interpolation_1d.png + :align: center + :width: 600 +Now let :math:`u(x)` be a function whose values are known at the nodes; +i.e. :math:`u_i = u(x_i)`. +The approximate value of the function at location :math:`x^*` +using linear interpolation is given by: -Interpolation Matrix in 3D -^^^^^^^^^^^^^^^^^^^^^^^^^^ +.. math:: + u(x^*) \approx u_3 + \Bigg ( \frac{u_4 - u_3}{h_3} \Bigg ) (x^* - x_3) + :label: operators_averaging_interpolation_1d + + +Suppose now that we organize the known values of :math:`u(x)` at the nodes +into a vector of the form: + +.. math:: + \boldsymbol{u} = \begin{bmatrix} u_0 & u_1 & u_2 & u_3 & u_4 & u_5 & u_6 & u_7 & u_8 \end{bmatrix}^T + +If we define a matrix consisting of a single row: + +.. math:: + A_0 = \begin{bmatrix} 0 & 0 & 0 & a_3 & a_4 & 0 & 0 & 0 & 0 \end{bmatrix} + +where + +.. math:: + a_3 = 1 - \frac{x^* - x_3}{h_3} \;\;\;\;\; \textrm{and} \;\;\;\;\; a_4 = \frac{x^* - x_3}{h_3} + +then + +.. math:: + u(x^*) \approx A_0 \, \boldsymbol{u} + +For a single location, we have just seen how a linear operator can be constructed to +compute the interpolation using a matrix vector-product. + +Now consider the case where you would like to interpolate the function from the nodes to +*M* arbitrary locations within the boundaries of the mesh. +For each location, we simply construct the corresponding row in the interpolation matrix. +Where :math:`\boldsymbol{u^*}` is a vector containing the approximations of :math:`u(x)` at :math:`M` +locations: + +.. math:: + (\boldsymbol{u^*}) \approx \boldsymbol{A\, u} \;\;\;\; \textrm{where} \;\;\;\; \begin{bmatrix} A_0 \\ A_1 \\ \vdots \\ A_{M-1} \end{bmatrix} + :label: operators_averaging_interpolation_matrix + +where :math:`\boldsymbol{A}` is a sparse matrix whose rows contain a maximum of 2 non-zero elements. +The size of :math:`\boldsymbol{A}` is the number of locations by the number of nodes. +For seven locations (:math:`x^* = 3,1,9,2,5,2,8`) and our mesh (9 nodes), +the non-zero elements of the interpolation matrix are illustrated below. + +.. image:: ../../images/interpolation_1d_sparse.png + :align: center + :width: 250 + + +**What if the function is defined at cell centers?** + +Here we let :math:`\bar{x}_i` define the center locations +for cells 0 through 7, and we let :math:`\bar{u}_i = u(\bar{x}_i)`. +In this case, the approximation defined in expression :eq:`operators_averaging_interpolation_1d` is replaced by: + +.. math:: + u(x^*) \approx \bar{u}_3 + 2 \Bigg ( \frac{\bar{u}_4 - \bar{u}_3}{h_3 + h_4} \Bigg ) (x^* - \bar{x}_3) + +For an arbitrary number of locations, we can construct an interpolation matrix similar to that shown +in expression :eq:`operators_averaging_interpolation_1d`. In this case however, the size of +:math:`\boldsymbol{A}` is the number of locations by the number of cells. Note that we **cannot** +interpolate at locations between the first or last cell center and the boundaries of the mesh +for quantities defined at cell centers. + + +Interpolation Matrix in 2D and 3D +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +In 1D, the location of the interpolated quantity lies between 2 nodes or cell centers. +In 2D however, the location of the interpolated quantity lies within 4 nodes or cell centers. + +.. image:: ../../images/interpolation_2d.png + :align: center + :width: 300 + +Let :math:`(x^*, y^*)` be within a cell whose nodes are located at +:math:`(x_1, y_1)`, :math:`(x_2, y_1)`, :math:`(x_1, y_2)` and :math:`(x_2, y_2)`. +If we define :math:`u_0 = u(x_1, y_1)`, :math:`u_1 = u(x_2, y_1)`, :math:`u_2 = u(x_1, y_2)` and +:math:`u_3 = u(x_2, y_2)`, then + +.. math:: + u(x^*, y^*) \approx a_0 u_0 + a_1 u_1 + a_2 u_2 + a_3 u_3 + +where :math:`a_0`, :math:`a_1`, :math:`a_2` and :math:`a_3` are coefficients determined from equations +governing `bilinear interpolation `__ . +These coefficients represent the 4 non-zero values within the corresponding row of the interpolation matrix :math:`\boldsymbol{A}`. + +Where the values of :math:`u(x,y)` at all nodes are organized into a single vector :math:`\boldsymbol{u}`, +and :math:`\boldsymbol{u^*}` is a vector containing the approximations of :math:`u(x,y)` at an arbitrary number of locations: + +.. math:: + \boldsymbol{u^*} \approx \boldsymbol{A\, u} + +In each row, the position of the non-zero elements :math:`a_0`, :math:`a_1`, :math:`a_2` and :math:`a_3` +corresponds to the indecies of the 4 nodes comprising a specific cell. +Once again the shape of :math:`\boldsymbol{A}` is the number of locations by the number of nodes. + +**What if the function is defined at cell centers?** + +A similar result can be obtained by interpolating a function define at cell centers. +In this case, we let :math:`(x^*, y^*)` lie within 4 cell centers located at +:math:`(\bar{x}_1, \bar{y}_1)`, :math:`(\bar{x}_2, \bar{y}_1)`, :math:`(\bar{x}_1, \bar{y}_2)` and :math:`(\bar{x}_2, \bar{y}_2)`. + +.. math:: + u(x^*, y^*) \approx a_0 \bar{u}_0 + a_1 \bar{u}_1 + a_2 \bar{u}_2 + a_3 \bar{u}_3 + +The size of the resulting interpolation matrix is the number of locations by number of cells. + +**What about the 3D case?** -**Basics for constructing the 1D interpolation matrix** +The derivation for the 3D case is effectively the same, except each cell is defined by 8 nodes and +thus: +.. math:: + u(x^*, y^*, z^*) \approx \sum_{k=0}^7 a_k u_k +This creates an interpolation matrix with 8 non-zero entries per row. Scalars vs. Vectors ^^^^^^^^^^^^^^^^^^^ -**Describe how components are treated separately** Averaging diff --git a/docs/content/theory/operators_index.rst b/docs/content/theory/operators_index.rst index d892fead2..d9987d1f0 100644 --- a/docs/content/theory/operators_index.rst +++ b/docs/content/theory/operators_index.rst @@ -3,12 +3,26 @@ Operators ********* -To approximate numerical solutions to partial differential equations using the *discretize* package, -we must be able to construct discrete approximations to the following types of mathematical operators: +To approximate numerical solutions to partial differential equations using the finite volume method, +we must be able to construct discrete approximations to the following mathematical operators: + - Interpolation - Averaging - Differential (gradient, divergence, curl) +Let any one of the aforementioned operations be defined as a mapping such that :math:`w = \mathbb{M}[u]`. +Where :math:`\boldsymbol{u}` is a vector containing the discrete representation of :math:`u` on the mesh, +and :math:`\boldsymbol{w}` is a vector containing the discrete represention of :math:`w` on the mesh, +we can approximate the mapping as: + +.. math:: + \boldsymbol{w} \approx \boldsymbol{M \, u} + +where :math:`\boldsymbol{M}` is a sparse matrix. Thus for each operator, the mapping is approximated by +constructing a sparse matrix and performing a matrix-vector product. Subsequent sections are devoted to +the general formation of these matrices. + + **Contents:** .. toctree:: diff --git a/docs/images/interpolation_1d.PNG b/docs/images/interpolation_1d.PNG new file mode 100644 index 0000000000000000000000000000000000000000..4f4d8ada1021dfb1c45c1cd01860dc850c61d7da GIT binary patch literal 4282 zcmd^?i8ou_`o}}7166&iHMFKy)y!K{4aKXaD2?eg##pHsifRx-P<>k+lo~?KN>QRQ ziU7wk+ ze0*877|LI7&P*^mHp4PDk@`*bRk!5@t0D6fazmen%s-lw;w>9rrFe!C=2sbHMkeet zJOYOB=%cK3BNmy_vzcj5W7;{30G_9Bk|N$LxGSS7_b>3~xEZX$Kgu|HmeIaE*Bl7IrB2U7H&{^;hFZaZLIJ+)_dS zt6k1v9uf{c7dOrxyuCh4oe5jpQ9XR2Y`wk2qg@^e&D2$T!MDRFn)bxKHp+LV5npOo z@v-c}vCzkcHmW!L`o-t=66~wnaog=GLP;DeIgu>s8t>u6VT^4TyiZ$_M`t^?+IP~X z)_3wp&cm<-+YqR=@;slmQm<&5!He)0@uscH_AL>NRUG!=(GvnmRVu?qKTWcotq@~3CKN}gZENK^PK+EV|Gk@t|r zoQJtOBZSDvRQpk^)7{=o4ZrhL-sFmU6r+|#8jR_yfd&~J&JA1vhR(Rf6$dhnevs-eR-M0 z?-Jf$X8l?NUMQ>JH`7Ssw3S!GI^w_Us272=sC~DJ97CnGRVxDKZZ^|cNwTu5h79@| 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Averaging is required when quantities that live on different +Here, we provide the background theory for how discrete averaging matricies are formed. +Averaging matrices are required when quantities that live on different parts of the mesh need to be added, subtracted, multiplied or divided. **Tutorials:** @@ -13,165 +12,184 @@ parts of the mesh need to be added, subtracted, multiplied or divided. - :ref:`tutorial for constructing and applying averaging operators ` +Averaging Matrices in 1D +======================== -Interpolation -============= - -Presently there is an extensive set of interpolation methods (e.g. polynomial, spline, piecewise constant). -One of the most effective and widely used interpolation methods is linear interpolation. -The *discretize* package primarily uses linear interpolation because 1) it is very fast, and 2) higher order -interpolation methods require the construction of matricies which are less sparse. -Here, we discuss how a sparse matrix can be formed which interpolates the discrete values to -a set of points in continuous space. - -The formulation for linear interpolation is adequately presented on Wikipedia, see: - - - `Linear Interpolation (1D) `__ - - `Bilinear Interpolation (2D) `__ - - `Trilinear Interpolation (3D) `__ +Nodes to Cell Centers +^^^^^^^^^^^^^^^^^^^^^ -Interpolation Matrix in 1D -^^^^^^^^^^^^^^^^^^^^^^^^^^ +Let us define a 1D mesh where cell center locations are defined at :math:`x_i` +and node locations are defined at :math:`x_{i+\frac{1}{2}}`. +The widths of the cells are given by :math:`\Delta x_i`. -Let us define a 1D mesh that contains 8 cells of arbitrary width; -the mesh is shown in the figure below. The width of each cell is -given by :math:`h_i`. The location of each node is given by :math:`x_i`. - -.. image:: ../../images/interpolation_1d.png +.. image:: ../../images/averaging_1d.png :align: center :width: 600 -Now let :math:`u(x)` be a function whose values are known at the nodes; -i.e. :math:`u_i = u(x_i)`. -The approximate value of the function at location :math:`x^*` -using linear interpolation is given by: +Now let :math:`u(x)` be a scalar function whose values are known at the nodes; i.e. :math:`u_{i+\frac{1}{2}} = u \big (x_{i+\frac{1}{2}} \big )`. +The average at the center of cell :math:`i` is given by: .. math:: - u(x^*) \approx u_3 + \Bigg ( \frac{u_4 - u_3}{h_3} \Bigg ) (x^* - x_3) - :label: operators_averaging_interpolation_1d - + \bar{u}_i = \frac{u_{i-\frac{1}{2}} + u_{i+\frac{1}{2}}}{2} -Suppose now that we organize the known values of :math:`u(x)` at the nodes -into a vector of the form: +Our goal is to construct a matrix :math:`\boldsymbol{A}` that averages the values +at the nodes and places them at the cell centers: .. math:: - \boldsymbol{u} = \begin{bmatrix} u_0 & u_1 & u_2 & u_3 & u_4 & u_5 & u_6 & u_7 & u_8 \end{bmatrix}^T + \bar{\boldsymbol{u}} = \boldsymbol{A \, u} -If we define a matrix consisting of a single row: +where :math:`\boldsymbol{u}` is a vector that stores the known values of :math:`u(x)` at the nodes, +and :math:`\bar{\boldsymbol{u}}` is a vector that stores the averages at cell centers. +For the entire mesh, the averaging matrix is given by .. math:: - A_0 = \begin{bmatrix} 0 & 0 & 0 & a_3 & a_4 & 0 & 0 & 0 & 0 \end{bmatrix} - -where + \boldsymbol{A} = \begin{bmatrix} + \frac{1}{2} & \frac{1}{2} & 0 & 0 & \cdots & 0 & 0 \\ + 0 & \frac{1}{2} & \frac{1}{2} & 0 & \cdots & 0 & 0 \\ + 0 & 0 & \frac{1}{2} & \frac{1}{2} & \cdots & 0 & 0 \\ + \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ + 0 & 0 & 0 & 0 & \cdots & \frac{1}{2} & \frac{1}{2} + \end{bmatrix} + :label: operators_averaging_n2c_1d + +where :math:`\boldsymbol{A}` is a sparse matrix. Where :math:`N` is the number of cells in the mesh, +:math:`\boldsymbol{A}` is :math:`N` by :math:`N+1`. + +Cell Centers to Nodes +^^^^^^^^^^^^^^^^^^^^^ + +Let us re-examine the figure illustrating the 1D mesh and assume we know the value of the function :math:`u(x)` at cell centers; i.e. :math:`u_i = u(x_i)`. +Note that the nodes are not equal distance from the cell centers on either side. +Therefore we cannot simply sum the values and divide by half. Instead we need to implement a weighted averaging. +At node :math:`i+\frac{1}{2}`, the average value is given by: .. math:: - a_3 = 1 - \frac{x^* - x_3}{h_3} \;\;\;\;\; \textrm{and} \;\;\;\;\; a_4 = \frac{x^* - x_3}{h_3} + \bar{u}_{i+\frac{1}{2}} = \Bigg ( \frac{\Delta x_{i+1}}{\Delta x_i + \Delta x_{i+1}} \Bigg ) u_{i} + + \Bigg ( \frac{\Delta x_i}{\Delta x_i + \Delta x_{i+1}} \Bigg ) u_{i+1} -then +Our goal is to construct a matrix :math:`\bar{\boldsymbol{A}}` that averages the values +at the cell centers and places them at the nodes: .. math:: - u(x^*) \approx A_0 \, \boldsymbol{u} + \bar{\boldsymbol{u}} = \bar{\boldsymbol{A}} \, \boldsymbol{u} -For a single location, we have just seen how a linear operator can be constructed to -compute the interpolation using a matrix vector-product. - -Now consider the case where you would like to interpolate the function from the nodes to -*M* arbitrary locations within the boundaries of the mesh. -For each location, we simply construct the corresponding row in the interpolation matrix. -Where :math:`\boldsymbol{u^*}` is a vector containing the approximations of :math:`u(x)` at :math:`M` -locations: +where :math:`\boldsymbol{u}` is a vector that stores the known values of :math:`u(x)` at cell centers, +and :math:`\bar{\boldsymbol{u}}` is a vector that stores the averages at the nodes. +For the entire mesh, the averaging matrix is given by: .. math:: - (\boldsymbol{u^*}) \approx \boldsymbol{A\, u} \;\;\;\; \textrm{where} \;\;\;\; \begin{bmatrix} A_0 \\ A_1 \\ \vdots \\ A_{M-1} \end{bmatrix} - :label: operators_averaging_interpolation_matrix - -where :math:`\boldsymbol{A}` is a sparse matrix whose rows contain a maximum of 2 non-zero elements. -The size of :math:`\boldsymbol{A}` is the number of locations by the number of nodes. -For seven locations (:math:`x^* = 3,1,9,2,5,2,8`) and our mesh (9 nodes), -the non-zero elements of the interpolation matrix are illustrated below. - -.. image:: ../../images/interpolation_1d_sparse.png + \bar{\boldsymbol{A}} = \begin{bmatrix} + 1 & 0 & 0 & 0 & \cdots & 0 & 0 \\ + a_1 & b_1 & 0 & 0 & \cdots & 0 & 0 \\ + 0 & a_2 & b_2 & 0 & \cdots & 0 & 0 \\ + \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ + 0 & 0 & 0 & 0 & \cdots & a_{N-1} & b_{N-1} \\ + 0 & 0 & 0 & 0 & \cdots & 0 & 1 + \end{bmatrix} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \begin{split} + a_i &= \frac{\Delta x_{i+1}}{\Delta x_i + \Delta x_{i+1}} \\ + & \\ + b_i &= \frac{\Delta x_i}{\Delta x_i + \Delta x_{i+1}} + \end{split} + :label: operators_averaging_c2n_1d + +where :math:`\bar{\boldsymbol{A}}` is a sparse matrix. Where :math:`N` is the number of cells in the mesh, +:math:`\bar{\boldsymbol{A}}` is :math:`N+1` by :math:`N`. Note that :math:`\bar{A}_{0,0}` and :math:`\bar{A}_{N,N-1}` +are equal to 1. This is because cell center locations need to compute the average lie outside the mesh. +Therefore we take the nearest neighbour instead. + + +Averaging Matrices in 2D and 3D +=============================== + +Scalars from Nodes to Cell Centers +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +In 1D, the value of a function at 2 locations is used to compute the average. +In 2D however, the value of the function at 4 locations is needed. +Let us define a 2D mesh where cell center locations are defined by :math:`(x_i,y_j)` +and the node locations are defined by :math:`\Big ( x_{i+\frac{1}{2}}, y_{i+\frac{1}{2}} \Big )`. +The widths of the cells in :math:`x` and :math:`y` are given by :math:`\Delta x_i` and :math:`\Delta y_j`, respectively. + +.. image:: ../../images/averaging_2d.png :align: center - :width: 250 + :width: 300 +Let :math:`u(x,y)` be a scalar function whose values are known at the nodes. +Where :math:`\bar{u}_{i,j}` is the average at the center of cell :math:`i,j`: -**What if the function is defined at cell centers?** +.. math:: + \bar{u}_{i,j} = \frac{1}{4} \Big ( u_{i-\frac{1}{2},j-\frac{1}{2}} + u_{i+\frac{1}{2},j-\frac{1}{2}} + u_{i-\frac{1}{2},j+\frac{1}{2}} + u_{i+\frac{1}{2}, j+\frac{1}{2}} \Big ) -Here we let :math:`\bar{x}_i` define the center locations -for cells 0 through 7, and we let :math:`\bar{u}_i = u(\bar{x}_i)`. -In this case, the approximation defined in expression :eq:`operators_averaging_interpolation_1d` is replaced by: +Our goal is to construct a matrix :math:`\boldsymbol{A}` that averages the values +at the nodes and places them at the cell centers: .. math:: - u(x^*) \approx \bar{u}_3 + 2 \Bigg ( \frac{\bar{u}_4 - \bar{u}_3}{h_3 + h_4} \Bigg ) (x^* - \bar{x}_3) + \bar{\boldsymbol{u}} = \boldsymbol{A \, u} -For an arbitrary number of locations, we can construct an interpolation matrix similar to that shown -in expression :eq:`operators_averaging_interpolation_1d`. In this case however, the size of -:math:`\boldsymbol{A}` is the number of locations by the number of cells. Note that we **cannot** -interpolate at locations between the first or last cell center and the boundaries of the mesh -for quantities defined at cell centers. +where :math:`\boldsymbol{u}` is a vector that stores the known values of :math:`u(x,y)` at the nodes, +and :math:`\bar{\boldsymbol{u}}` is a vector that stores the averages at cell centers. +For tensor meshes, the averaging matrix is rather easy to construct. Where the number of nodes in the :math:`x` and :math:`y` +directions are known, we use expression :eq:`operators_averaging_n2c_1d` to construct 1D averaging matrices +:math:`\boldsymbol{A_x}` and :math:`\boldsymbol{A_y}`. The averaging matrix in 2D is -Interpolation Matrix in 2D and 3D -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +.. math:: + \boldsymbol{A} = \boldsymbol{A_y} \otimes \boldsymbol{A_x} -In 1D, the location of the interpolated quantity lies between 2 nodes or cell centers. -In 2D however, the location of the interpolated quantity lies within 4 nodes or cell centers. +And in 3D: -.. image:: ../../images/interpolation_2d.png - :align: center - :width: 300 +.. math:: + \boldsymbol{A} = \boldsymbol{A_z} \otimes (\boldsymbol{A_y} \otimes \boldsymbol{A_x}) -Let :math:`(x^*, y^*)` be within a cell whose nodes are located at -:math:`(x_1, y_1)`, :math:`(x_2, y_1)`, :math:`(x_1, y_2)` and :math:`(x_2, y_2)`. -If we define :math:`u_0 = u(x_1, y_1)`, :math:`u_1 = u(x_2, y_1)`, :math:`u_2 = u(x_1, y_2)` and -:math:`u_3 = u(x_2, y_2)`, then +where :math:`\otimes` is the kronecker product. -.. math:: - u(x^*, y^*) \approx a_0 u_0 + a_1 u_1 + a_2 u_2 + a_3 u_3 -where :math:`a_0`, :math:`a_1`, :math:`a_2` and :math:`a_3` are coefficients determined from equations -governing `bilinear interpolation `__ . -These coefficients represent the 4 non-zero values within the corresponding row of the interpolation matrix :math:`\boldsymbol{A}`. +.. important:: For mesh with more complicated structures (e.g. tree mesh), the general theory is the same. However, the averaging matrix cannot be constructed directly using kronecker products. -Where the values of :math:`u(x,y)` at all nodes are organized into a single vector :math:`\boldsymbol{u}`, -and :math:`\boldsymbol{u^*}` is a vector containing the approximations of :math:`u(x,y)` at an arbitrary number of locations: -.. math:: - \boldsymbol{u^*} \approx \boldsymbol{A\, u} +Scalars from Cell Centers to Nodes +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Kronecker products can also be used to construct the averaging matrix from cell centers to nodes. +In this case, expression :eq:`operators_averaging_c2n_1d` is used to construct the 1D averaging +matricies in the :math:`x`, :math:`y` (and :math:`z`) directions using the dimensions of the cells +along each axis. Once again, nearest neighbour is used to assign a value to cell centers which lie outside the mesh. -In each row, the position of the non-zero elements :math:`a_0`, :math:`a_1`, :math:`a_2` and :math:`a_3` -corresponds to the indecies of the 4 nodes comprising a specific cell. -Once again the shape of :math:`\boldsymbol{A}` is the number of locations by the number of nodes. +For 2D averaging from cell centers to nodes: -**What if the function is defined at cell centers?** +.. math:: + \bar{\boldsymbol{A}} = \bar{\boldsymbol{A_y}} \otimes \bar{\boldsymbol{A_x}} -A similar result can be obtained by interpolating a function define at cell centers. -In this case, we let :math:`(x^*, y^*)` lie within 4 cell centers located at -:math:`(\bar{x}_1, \bar{y}_1)`, :math:`(\bar{x}_2, \bar{y}_1)`, :math:`(\bar{x}_1, \bar{y}_2)` and :math:`(\bar{x}_2, \bar{y}_2)`. +And for 3D averaging from cell centers to nodes: .. math:: - u(x^*, y^*) \approx a_0 \bar{u}_0 + a_1 \bar{u}_1 + a_2 \bar{u}_2 + a_3 \bar{u}_3 + \bar{\boldsymbol{A}} = \bar{\boldsymbol{A_z}} \otimes (\bar{\boldsymbol{A_y}} \otimes \bar{\boldsymbol{A_x}}) -The size of the resulting interpolation matrix is the number of locations by number of cells. -**What about the 3D case?** -The derivation for the 3D case is effectively the same, except each cell is defined by 8 nodes and -thus: +Vectors from Faces to Cell Centers +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -.. math:: - u(x^*, y^*, z^*) \approx \sum_{k=0}^7 a_k u_k +Let us define a 2D mesh where cell center locations are defined by :math:`(x_i,y_j)`. +The widths of the cells in :math:`x` and :math:`y` are given by :math:`\Delta x_i` and :math:`\Delta y_j`, respectively. + +.. image:: ../../images/averaging_2d_faces.png + :align: center + :width: 400 -This creates an interpolation matrix with 8 non-zero entries per row. -Scalars vs. Vectors -^^^^^^^^^^^^^^^^^^^ +Vectors from Edges to Cell Centers +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -Averaging -========= +Let us define a 2D mesh where node locations are defined by :math:`(x_i,y_j)`. +The widths of the cells in :math:`x` and :math:`y` are given by :math:`\Delta x_i` and :math:`\Delta y_j`, respectively. +.. image:: ../../images/averaging_2d_edges.png + :align: center + :width: 400 diff --git a/docs/content/theory/operators_index.rst b/docs/content/theory/operators_index.rst index d9987d1f0..c78a869e0 100644 --- a/docs/content/theory/operators_index.rst +++ b/docs/content/theory/operators_index.rst @@ -28,6 +28,7 @@ the general formation of these matrices. .. toctree:: :maxdepth: 1 + operators_interpolation operators_averaging operators_differential diff --git a/docs/content/theory/operators_interpolation.rst b/docs/content/theory/operators_interpolation.rst new file mode 100644 index 000000000..9bb4001a2 --- /dev/null +++ b/docs/content/theory/operators_interpolation.rst @@ -0,0 +1,244 @@ +.. _operators_interpolation: + +Interpolation +************* + +Here, we provide the background theory for how discrete interpolation matricies are formed. +Interpolation is required when a discrete quantity is known on the mesh (centers, nodes, edges or faces), but we would like to estimate its +value at a point within the continuous domain. + +Presently there is an extensive set of interpolation methods (e.g. polynomial, spline, piecewise constant). +One of the most effective and widely used interpolation methods is linear interpolation. +The *discretize* package primarily uses linear interpolation because 1) it is very fast, and 2) higher order +interpolation methods require the construction of matricies which are less sparse. +Here, we discuss how a sparse matrix can be formed which interpolates the discrete values to +a set of points in continuous space. + +The formulation for linear interpolation is adequately presented on Wikipedia, see: + + - `Linear Interpolation (1D) `__ + - `Bilinear Interpolation (2D) `__ + - `Trilinear Interpolation (3D) `__ + + +Interpolation Matrix in 1D +========================== + +Let us define a 1D mesh that contains 8 cells of arbitrary width; +the mesh is shown in the figure below. The width of each cell is +given by :math:`h_i`. The location of each node is given by :math:`x_i`. + +.. image:: ../../images/interpolation_1d.png + :align: center + :width: 600 + +Now let :math:`u(x)` be a function whose values are known at the nodes; +i.e. :math:`u_i = u(x_i)`. +The approximate value of the function at location :math:`x^*` +using linear interpolation is given by: + +.. math:: + u(x^*) \approx u_3 + \Bigg ( \frac{u_4 - u_3}{h_3} \Bigg ) (x^* - x_3) + :label: operators_averaging_interpolation_1d + + +Suppose now that we organize the known values of :math:`u(x)` at the nodes +into a vector of the form: + +.. math:: + \boldsymbol{u} = \begin{bmatrix} u_0 & u_1 & u_2 & u_3 & u_4 & u_5 & u_6 & u_7 & u_8 \end{bmatrix}^T + +If we define a row: + +.. math:: + \boldsymbol{p_0} = \begin{bmatrix} 0 & 0 & 0 & a_3 & a_4 & 0 & 0 & 0 & 0 \end{bmatrix} + +where + +.. math:: + a_3 = 1 - \frac{x^* - x_3}{h_3} \;\;\;\;\; \textrm{and} \;\;\;\;\; a_4 = \frac{x^* - x_3}{h_3} + +then + +.. math:: + u(x^*) \approx \boldsymbol{p_0 \, u} + +For a single location, we have just seen how a linear operator can be constructed to +compute the interpolation using a matrix vector-product. + +Now consider the case where you would like to interpolate the function from the nodes to +an arbitrary number of locations within the boundaries of the mesh. +For each location, we simply construct the corresponding row in the interpolation matrix. +Where :math:`\boldsymbol{u^*}` is a vector containing the approximations of :math:`u(x)` at :math:`M` +locations: + +.. math:: + (\boldsymbol{u^*}) \approx \boldsymbol{P\, u} \;\;\;\;\;\; \textrm{where} \;\;\;\;\;\; + \boldsymbol{P} = \begin{bmatrix} \cdots \;\; \boldsymbol{p_0} \;\; \cdots \\ + \cdots \;\; \boldsymbol{p_1} \;\; \cdots \\ \vdots \\ + \cdots \, \boldsymbol{p_{M-1}} \, \cdots \end{bmatrix} + :label: operators_averaging_interpolation_matrix + +where :math:`\boldsymbol{P}` is a sparse matrix whose rows contain a maximum of 2 non-zero elements. +The size of :math:`\boldsymbol{P}` is the number of locations by the number of nodes. +For seven locations (:math:`x^* = 3,1,9,2,5,2,8`) and our mesh (9 nodes), +the non-zero elements of the interpolation matrix are illustrated below. + +.. image:: ../../images/interpolation_1d_sparse.png + :align: center + :width: 250 + + +**What if the function is defined at cell centers?** + +Here we let :math:`\bar{x}_i` define the center locations +for cells 0 through 7, and we let :math:`\bar{u}_i = u(\bar{x}_i)`. +In this case, the approximation defined in expression :eq:`operators_averaging_interpolation_1d` is replaced by: + +.. math:: + u(x^*) \approx \bar{u}_3 + 2 \Bigg ( \frac{\bar{u}_4 - \bar{u}_3}{h_3 + h_4} \Bigg ) (x^* - \bar{x}_3) + +For an arbitrary number of locations, we can construct an interpolation matrix similar to that shown +in expression :eq:`operators_averaging_interpolation_1d`. In this case however, the size of +:math:`\boldsymbol{P}` is the number of locations by the number of cells. Note that we **cannot** +interpolate at locations between the first or last cell center and the boundaries of the mesh +for quantities defined at cell centers. + + +Interpolation Matrix in 2D and 3D +================================= + +In 1D, the location of the interpolated quantity lies between 2 nodes or cell centers. +In 2D however, the location of the interpolated quantity lies within 4 nodes or cell centers. + +.. image:: ../../images/interpolation_2d.png + :align: center + :width: 300 + +Let :math:`(x^*, y^*)` be within a cell whose nodes are located at +:math:`(x_1, y_1)`, :math:`(x_2, y_1)`, :math:`(x_1, y_2)` and :math:`(x_2, y_2)`. +If we define :math:`u_0 = u(x_1, y_1)`, :math:`u_1 = u(x_2, y_1)`, :math:`u_2 = u(x_1, y_2)` and +:math:`u_3 = u(x_2, y_2)`, then + +.. math:: + u(x^*, y^*) \approx a_0 u_0 + a_1 u_1 + a_2 u_2 + a_3 u_3 + +where :math:`a_0`, :math:`a_1`, :math:`a_2` and :math:`a_3` are coefficients determined from equations +governing `bilinear interpolation `__ . +These coefficients represent the 4 non-zero values within the corresponding row of the interpolation matrix :math:`\boldsymbol{P}`. + +Where the values of :math:`u(x,y)` at all nodes are organized into a single vector :math:`\boldsymbol{u}`, +and :math:`\boldsymbol{u^*}` is a vector containing the approximations of :math:`u(x,y)` at an arbitrary number of locations: + +.. math:: + \boldsymbol{u^*} \approx \boldsymbol{P\, u} + :label: operators_interpolation_general + +In each row of :math:`\boldsymbol{P}`, the position of the non-zero elements :math:`a_0`, :math:`a_1`, :math:`a_2` and :math:`a_3` +corresponds to the indecies of the 4 nodes comprising a specific cell. +Once again the shape of :math:`\boldsymbol{P}` is the number of locations by the number of nodes. + +**What if the function is defined at cell centers?** + +A similar result can be obtained by interpolating a function define at cell centers. +In this case, we let :math:`(x^*, y^*)` lie within 4 cell centers located at +:math:`(\bar{x}_1, \bar{y}_1)`, :math:`(\bar{x}_2, \bar{y}_1)`, :math:`(\bar{x}_1, \bar{y}_2)` and :math:`(\bar{x}_2, \bar{y}_2)`. + +.. math:: + u(x^*, y^*) \approx a_0 \bar{u}_0 + a_1 \bar{u}_1 + a_2 \bar{u}_2 + a_3 \bar{u}_3 + +The resulting interpolation is defined similar to expression :eq:`operators_interpolation_general`. +However the size of the resulting interpolation matrix is the number of locations by number of cells. + +**What about for 3D case?** + +The derivation for the 3D case is effectively the same, except 8 node or center locations must +be used in the interpolation. Thus: + +.. math:: + u(x^*, y^*, z^*) \approx \sum_{k=0}^7 a_k u_k + +This creates an interpolation matrix :math:`\boldsymbol{P}` with 8 non-zero entries per row. + +Interpolation of Vectors +======================== + +Scalar quantities are discretized to live at nodes or cell centers, whereas the +components of vectors are discretized to live on their respective faces or edges; +see :ref:`where quantities live `. + +.. image:: ../../images/interpolation_2d_vectors.png + :align: center + :width: 600 + +Let :math:`\vec{u} (x,y)` be a 2D vector function that is known on the faces of the mesh; +that is, :math:`u_x` lives on the x-faces and :math:`u_y` lives on the y-faces. +Note that in the above figure, the x-faces and y-faces both form tensor grids. +If we want to approximate the components of the vector at a location :math:`(x^*,y^*)`, +we simply need to treat each component as a scalar function and interpolate it separately. + +Where :math:`u_{x,i}` represents the x-component of :math:`\vec{u} (x,y)` on a face :math:`i` being used for the interpolation, +the approximation of the x-component at :math:`(x^*, y^*)` has the form: + +.. math:: + u_x(x^*, y^*) \approx a_0 u_{x,0} + a_1 u_{x,1} + a_2 u_{x,2} + a_3 u_{x,3} + :label: operators_interpolation_xvec_coef + +For the the y-component, we have a similar representation: + +.. math:: + u_y(x^*, y^*) \approx b_0 u_{y,0} + b_1 u_{y,1} + b_2 u_{y,2} + b_3 u_{y,3} + +Where :math:`\boldsymbol{u}` is a vector that organizes the discrete components of :math:`\vec{u} (x,y)` on cell faces, +and :math:`\boldsymbol{u^*}` is a vector organizing the components of the approximations of :math:`\vec{u}(x,y)` at an arbitrary number of locations, +the interpolation matrix :math:`\boldsymbol{P}` is defined by: + +.. math:: + \boldsymbol{u^*} \approx \boldsymbol{P \, u} + :label: operators_interpolation_2d_sys + +where + +.. math:: + \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \end{bmatrix} + \;\;\textrm{,}\;\;\;\; + \boldsymbol{u^*} = \begin{bmatrix} \boldsymbol{u_x^*} \\ \boldsymbol{u_y^*} \end{bmatrix} + \;\;\;\;\textrm{and}\;\;\;\; + \boldsymbol{P} = \begin{bmatrix} \boldsymbol{P_x} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{P_y} \end{bmatrix} + +The interpolation matrix :math:`\boldsymbol{P}` is a sparse block-diagonal matrix. +The size of the interpolation matrix is the number of locations by the number of faces in the mesh. + +**What if we want to interpolate from edges?** + +In this case, the derivation is effectively the same. +However, the locations used for the interpolation are different and +:math:`\boldsymbol{u}` is now a vector that organizes the discrete components of :math:`\vec{u} (x,y)` on cell edges. + + +**What if we are interpolating a 3D vector?** + +In this case, there are 8 face locations or 8 edge locations that are used to approximate +:math:`\vec{u}(x,y,z)` at each location :math:`(x^*, y^*, z^*)`. +Similar to expression :eq:`operators_interpolation_xvec_coef` we have: + +.. math:: + \begin{align} + u_x(x^*, y^*, z^*) & \approx \sum_{i=1}^7 a_i u_{x,i} \\ + u_y(x^*, y^*, z^*) & \approx \sum_{i=1}^7 b_i u_{y,i} \\ + u_z(x^*, y^*, z^*) & \approx \sum_{i=1}^7 c_i u_{z,i} + \end{align} + +The interpolation can be expressed similar to that in equation :eq:`operators_interpolation_2d_sys`, +however: + +.. math:: + \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_z} \end{bmatrix} + \;\;\textrm{,}\;\;\;\; + \boldsymbol{u^*} = \begin{bmatrix} \boldsymbol{u_x^*} \\ \boldsymbol{u_y^*} \\ \boldsymbol{u_z^*} \end{bmatrix} + \;\;\;\;\textrm{and}\;\;\;\; + \boldsymbol{P} = \begin{bmatrix} \boldsymbol{P_x} & \boldsymbol{0} & \boldsymbol{0} \\ + \boldsymbol{0} & \boldsymbol{P_y} & \boldsymbol{0} \\ + \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{P_z} + \end{bmatrix} + diff --git a/docs/images/averaging_1d.PNG b/docs/images/averaging_1d.PNG new file mode 100644 index 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zV!6F<`k$h;VM{v0<~!lsZK82(h6na(q3xy83o-pZ{aVZ3RUTOU+k4h=reth`O%X0T zzXB^}G0l%x_J@Pla(Meg1RTTyS#+1-72>bME-g`tBZ%iYE%(hlbakyunk+L6YvC(f z5md7@S;Yk-p2pb2YoF|Tc#sRbfuCzQZjAdV0rLSFQPW4rt?#dwtaU!Dqrgk>E^f!S z(})YY*^2uDAKBBUy{9mi&37!YyMLkbMqG9~thywpb}cf9Kz_(MpH)ME($sh3cyXV+ z{>#%X-Ua=i74W{5wn*OolvRzBJ78Mk>l!Wwr96WMYFpw%%c?W8mxCoP9_X=xU?&e$ zkI{}@Ri3k-z{_`Nd1IsZiNUKIlV=Zg$OX|G$9;v}UgMZ?IK0d6&V8F6|(RWY0M zc?=x$=Jb5xt6U_tR^fW@2@frWeCUt6HW>&hL-pJ>?3r@K5*b zvj>>Y2p}(yeV72-F`lo;1|o|gyPQB8PpY?T;f;Fxkk!IPdHIq*ZZ%po&weIR=fWbj zSKZoH{8)wLkl``gg6tH^^o6F0nxEcjRF;i&QZp0}ye`!pC!!eat5pdqmhmP|I0>El zz6Sh>#aK{|rMOeq`!$VJz|P+GxW(;(o49aIS5y;y4)Yr+cMi8aDsc{nQ8%pG4hI=5S^sn| z<%gb4m#$Hp=62kZ;^yzFI&*DfM_Bu`KpA9FUji$}_8Tt+@s04$eNe(O|6w-*BA;?^ zB58r_=M1DqA?80~w439k(0f*cm;G^ej lytEP5=>OUO!s$4~dPq4m;X#&i?-+dD7nKtEd{@i;{{t4I)eryx literal 0 HcmV?d00001 From 8c4e7ab1afb0ad592a61d13d773864177e023247 Mon Sep 17 00:00:00 2001 From: dccowan Date: Wed, 25 Nov 2020 21:53:51 -0800 Subject: [PATCH 15/35] Finish averaging operators section small edits to notation --- docs/content/theory/operators_averaging.rst | 352 ++++++++++++++---- .../content/theory/operators_differential.rst | 18 +- docs/content/theory/operators_index.rst | 2 +- .../theory/operators_interpolation.rst | 26 +- docs/images/averaging_2d_edges.PNG | Bin 33940 -> 32226 bytes docs/images/averaging_2d_faces.PNG | Bin 33857 -> 30160 bytes docs/images/interpolation_1d_sparse.PNG | Bin 6630 -> 5835 bytes docs/images/interpolation_2d.PNG | Bin 12213 -> 14516 bytes docs/images/interpolation_2d_vectors.PNG | Bin 56524 -> 53553 bytes 9 files changed, 314 insertions(+), 84 deletions(-) diff --git a/docs/content/theory/operators_averaging.rst b/docs/content/theory/operators_averaging.rst index cfcd520d2..3da64a33d 100644 --- a/docs/content/theory/operators_averaging.rst +++ b/docs/content/theory/operators_averaging.rst @@ -7,6 +7,14 @@ Here, we provide the background theory for how discrete averaging matricies are Averaging matrices are required when quantities that live on different parts of the mesh need to be added, subtracted, multiplied or divided. +Where :math:`\boldsymbol{u}` is a discrete representation of a vector living somewhere on the mesh (nodes, edges, faces), +and :math:`\bar{\boldsymbol{u}}` is the vector containing the averages mapped to another part of the mesh, +we look to constructs an averaging matrix :math:`\boldsymbol{A}` such that: + +.. math:: + \bar{\boldsymbol{u}} = A \, \boldsymbol{u} + :label: operators_averaging_general + **Tutorials:** - :ref:`tutorial for constructing and applying averaging operators ` @@ -18,72 +26,77 @@ Averaging Matrices in 1D Nodes to Cell Centers ^^^^^^^^^^^^^^^^^^^^^ -Let us define a 1D mesh where cell center locations are defined at :math:`x_i` -and node locations are defined at :math:`x_{i+\frac{1}{2}}`. +Let us define a 1D mesh where cell center and node locations are defined according to the figure below. The widths of the cells are given by :math:`\Delta x_i`. .. image:: ../../images/averaging_1d.png :align: center :width: 600 -Now let :math:`u(x)` be a scalar function whose values are known at the nodes; i.e. :math:`u_{i+\frac{1}{2}} = u \big (x_{i+\frac{1}{2}} \big )`. -The average at the center of cell :math:`i` is given by: +If :math:`u(x)` is a scalar function whose values are known at the nodes +and :math:`\bar{u}_i` is the average value at the center of cell :math:`i`, +then: .. math:: - \bar{u}_i = \frac{u_{i-\frac{1}{2}} + u_{i+\frac{1}{2}}}{2} + \bar{u}_i = \frac{u_{i-\tfrac{1}{2}} + u_{i+\tfrac{1}{2}}}{2} -Our goal is to construct a matrix :math:`\boldsymbol{A}` that averages the values -at the nodes and places them at the cell centers: +Our goal is to construct a matrix :math:`A` that averages the values +at the nodes and places them at the cell centers, i.e.: .. math:: - \bar{\boldsymbol{u}} = \boldsymbol{A \, u} + \bar{\boldsymbol{u}} = A \, \boldsymbol{u} where :math:`\boldsymbol{u}` is a vector that stores the known values of :math:`u(x)` at the nodes, and :math:`\bar{\boldsymbol{u}}` is a vector that stores the averages at cell centers. -For the entire mesh, the averaging matrix is given by + +For the entire mesh, the averaging matrix is given by: .. math:: - \boldsymbol{A} = \begin{bmatrix} - \frac{1}{2} & \frac{1}{2} & 0 & 0 & \cdots & 0 & 0 \\ - 0 & \frac{1}{2} & \frac{1}{2} & 0 & \cdots & 0 & 0 \\ - 0 & 0 & \frac{1}{2} & \frac{1}{2} & \cdots & 0 & 0 \\ + A = \frac{1}{2} \begin{bmatrix} + 1 & 1 & 0 & 0 & \cdots & 0 & 0 \\ + 0 & 1 & 1 & 0 & \cdots & 0 & 0 \\ + 0 & 0 & 1 &1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ - 0 & 0 & 0 & 0 & \cdots & \frac{1}{2} & \frac{1}{2} + 0 & 0 & 0 & 0 & \cdots & 1 & 1 \end{bmatrix} :label: operators_averaging_n2c_1d -where :math:`\boldsymbol{A}` is a sparse matrix. Where :math:`N` is the number of cells in the mesh, -:math:`\boldsymbol{A}` is :math:`N` by :math:`N+1`. +where :math:`A` is a sparse matrix. Defining :math:`nc` as the number of cells in the mesh, +:math:`A` is has shape :math:`nc` by :math:`nc \! + \! 1`. Cell Centers to Nodes ^^^^^^^^^^^^^^^^^^^^^ -Let us re-examine the figure illustrating the 1D mesh and assume we know the value of the function :math:`u(x)` at cell centers; i.e. :math:`u_i = u(x_i)`. +Let us re-examine the figure illustrating the 1D mesh and assume we know the value of the function :math:`u(x)` at cell centers. Note that the nodes are not equal distance from the cell centers on either side. Therefore we cannot simply sum the values and divide by half. Instead we need to implement a weighted averaging. -At node :math:`i+\frac{1}{2}`, the average value is given by: + +If :math:`u(x)` is a scalar function whose values are known at the cell centers +and :math:`\bar{u}_i` is the average value at the nodes, +then: .. math:: \bar{u}_{i+\frac{1}{2}} = \Bigg ( \frac{\Delta x_{i+1}}{\Delta x_i + \Delta x_{i+1}} \Bigg ) u_{i} + \Bigg ( \frac{\Delta x_i}{\Delta x_i + \Delta x_{i+1}} \Bigg ) u_{i+1} -Our goal is to construct a matrix :math:`\bar{\boldsymbol{A}}` that averages the values -at the cell centers and places them at the nodes: +Our goal is to construct a matrix :math:`\bar{A}` that averages the values +at the cell centers and places them at the nodes, i.e.: .. math:: \bar{\boldsymbol{u}} = \bar{\boldsymbol{A}} \, \boldsymbol{u} where :math:`\boldsymbol{u}` is a vector that stores the known values of :math:`u(x)` at cell centers, and :math:`\bar{\boldsymbol{u}}` is a vector that stores the averages at the nodes. + For the entire mesh, the averaging matrix is given by: .. math:: - \bar{\boldsymbol{A}} = \begin{bmatrix} + \bar{A} = \frac{1}{2} \begin{bmatrix} 1 & 0 & 0 & 0 & \cdots & 0 & 0 \\ a_1 & b_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & a_2 & b_2 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ - 0 & 0 & 0 & 0 & \cdots & a_{N-1} & b_{N-1} \\ + 0 & 0 & 0 & 0 & \cdots & a_{nc-1} & b_{nc-1} \\ 0 & 0 & 0 & 0 & \cdots & 0 & 1 \end{bmatrix} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \begin{split} @@ -93,36 +106,39 @@ For the entire mesh, the averaging matrix is given by: \end{split} :label: operators_averaging_c2n_1d -where :math:`\bar{\boldsymbol{A}}` is a sparse matrix. Where :math:`N` is the number of cells in the mesh, -:math:`\bar{\boldsymbol{A}}` is :math:`N+1` by :math:`N`. Note that :math:`\bar{A}_{0,0}` and :math:`\bar{A}_{N,N-1}` -are equal to 1. This is because cell center locations need to compute the average lie outside the mesh. -Therefore we take the nearest neighbour instead. +where :math:`\bar{A}` is a sparse matrix. Defining :math:`nc` as the number of cells in the mesh, +:math:`\bar{A}` has shape :math:`nc \! + \! 1` by :math:`nc`. Note that :math:`\bar{A}_{0,0}` and :math:`\bar{A}_{nc,nc-1}` +are equal to 1. This is because cell center locations needed to compute the average lie outside the mesh +and we must extrapolate using the nearest neighbour. -Averaging Matrices in 2D and 3D -=============================== +Averaging Scalars in 2D and 3D +============================== -Scalars from Nodes to Cell Centers -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +Nodes to Cell Centers +^^^^^^^^^^^^^^^^^^^^^ -In 1D, the value of a function at 2 locations is used to compute the average. -In 2D however, the value of the function at 4 locations is needed. -Let us define a 2D mesh where cell center locations are defined by :math:`(x_i,y_j)` -and the node locations are defined by :math:`\Big ( x_{i+\frac{1}{2}}, y_{i+\frac{1}{2}} \Big )`. +For a 2D mesh, the value of the function at 4 locations is needed to average from nodes to cell centers. +Let us define a 2D mesh where cell center locations :math:`(x_i, y_j)` are represented using indices :math:`(i,j)`. The widths of the cells in :math:`x` and :math:`y` are given by :math:`\Delta x_i` and :math:`\Delta y_j`, respectively. .. image:: ../../images/averaging_2d.png :align: center :width: 300 -Let :math:`u(x,y)` be a scalar function whose values are known at the nodes. -Where :math:`\bar{u}_{i,j}` is the average at the center of cell :math:`i,j`: +If :math:`u(x,y)` is a scalar function whose values are known at the nodes +and :math:`\bar{u} (i,j)` is the average at the center of cell :math:`i,j`, +then: .. math:: - \bar{u}_{i,j} = \frac{1}{4} \Big ( u_{i-\frac{1}{2},j-\frac{1}{2}} + u_{i+\frac{1}{2},j-\frac{1}{2}} + u_{i-\frac{1}{2},j+\frac{1}{2}} + u_{i+\frac{1}{2}, j+\frac{1}{2}} \Big ) + \bar{u}(i,j) = \frac{1}{4} \Big [ + u \Big ( i-\tfrac{1}{2}, j-\tfrac{1}{2} \Big ) + + u \Big ( i+\tfrac{1}{2}, j-\tfrac{1}{2} \Big ) + + u \Big ( i-\tfrac{1}{2}, j+\tfrac{1}{2} \Big ) + + u \Big ( i+\tfrac{1}{2}, j+\tfrac{1}{2} \Big ) \Big ] Our goal is to construct a matrix :math:`\boldsymbol{A}` that averages the values -at the nodes and places them at the cell centers: +at the nodes and places them at the cell centers, i.e.: .. math:: \bar{\boldsymbol{u}} = \boldsymbol{A \, u} @@ -130,66 +146,276 @@ at the nodes and places them at the cell centers: where :math:`\boldsymbol{u}` is a vector that stores the known values of :math:`u(x,y)` at the nodes, and :math:`\bar{\boldsymbol{u}}` is a vector that stores the averages at cell centers. -For tensor meshes, the averaging matrix is rather easy to construct. Where the number of nodes in the :math:`x` and :math:`y` -directions are known, we use expression :eq:`operators_averaging_n2c_1d` to construct 1D averaging matrices -:math:`\boldsymbol{A_x}` and :math:`\boldsymbol{A_y}`. The averaging matrix in 2D is +For tensor meshes, the averaging matrix is rather easy to construct. +Using equation :eq:`operators_averaging_n2c_1d`, the number of cells in the x-direction +can be used to construct a matrix :math:`A_x`. And the number of cells in the y-direction +can be used to construct a matrix :math:`A_y`. The averaging matrix in 2D is given by: .. math:: - \boldsymbol{A} = \boldsymbol{A_y} \otimes \boldsymbol{A_x} + \boldsymbol{A} = A_y \otimes A_x -And in 3D: +where :math:`\otimes` is the kronecker product. For a 3D tensor mesh, the averaging matrix +from nodes to cell centers would be given by: .. math:: - \boldsymbol{A} = \boldsymbol{A_z} \otimes (\boldsymbol{A_y} \otimes \boldsymbol{A_x}) - -where :math:`\otimes` is the kronecker product. - - -.. important:: For mesh with more complicated structures (e.g. tree mesh), the general theory is the same. However, the averaging matrix cannot be constructed directly using kronecker products. + \boldsymbol{A} = A_z \otimes (A_y \otimes A_x) -Scalars from Cell Centers to Nodes -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +Cell Centers to Nodes +^^^^^^^^^^^^^^^^^^^^^ -Kronecker products can also be used to construct the averaging matrix from cell centers to nodes. +A nearly identical approach can be implemented to average from cell centers to nodes. In this case, expression :eq:`operators_averaging_c2n_1d` is used to construct the 1D averaging matricies in the :math:`x`, :math:`y` (and :math:`z`) directions using the dimensions of the cells along each axis. Once again, nearest neighbour is used to assign a value to cell centers which lie outside the mesh. -For 2D averaging from cell centers to nodes: +Our goal is to construct a matrix :math:`\boldsymbol{A}` that averages the values +at the cell centers and places them at the nodes, i.e.: + +.. math:: + \bar{\boldsymbol{u}} = \boldsymbol{A \, u} + +where :math:`\boldsymbol{u}` is a vector that stores the known values of :math:`u(x,y)` at the cell centers, +and :math:`\bar{\boldsymbol{u}}` is a vector that stores the averages at nodes. + +For 2D averaging from cell centers to nodes, we use equation :eq:`operators_averaging_c2n_1d` and the cell widths +in the x and y directions to construct 1D averaging matrices :math:`\bar{A}_x` and :math:`\bar{A}_y`, respectively. +The averaging operator for a 2D tensor mesh is given by: .. math:: - \bar{\boldsymbol{A}} = \bar{\boldsymbol{A_y}} \otimes \bar{\boldsymbol{A_x}} + \boldsymbol{A} = \bar{A}_y \otimes \bar{A}_x And for 3D averaging from cell centers to nodes: .. math:: - \bar{\boldsymbol{A}} = \bar{\boldsymbol{A_z}} \otimes (\bar{\boldsymbol{A_y}} \otimes \bar{\boldsymbol{A_x}}) + \boldsymbol{A} = \bar{A}_z \otimes (\bar{A}_y \otimes \bar{A}_x) + + +Faces to Cell Centers +^^^^^^^^^^^^^^^^^^^^^ + +Let us define a 2D mesh where cell center locations :math:`(x_i, y_j)` are represented using indices :math:`(i,j)`. +The widths of the cells in :math:`x` and :math:`y` are given by :math:`\Delta x_i` and :math:`\Delta y_j`, respectively. + +.. image:: ../../images/averaging_2d_faces.png + :align: center + :width: 350 + +If :math:`u(x,y)` is a scalar quantity whose values are known on the faces. +and :math:`\bar{u}(i,j)` is the average at the center of cell :math:`i,j`, +then: + +.. math:: + \bar{u}(i,j) = \frac{1}{4} \Big [ + u \Big ( i-\tfrac{1}{2}, j \Big ) + + u \Big ( i+\tfrac{1}{2}, j \Big ) + + u \Big ( i, j-\tfrac{1}{2} \Big ) + + u \Big ( i, j+\tfrac{1}{2} \Big ) \Big ] + +Our goal is to construct a matrix :math:`\boldsymbol{A}` that averages the values +on the faces and places them at the cell centers, i.e.: + +.. math:: + \bar{\boldsymbol{u}} = \boldsymbol{A \, u} + +where :math:`\boldsymbol{u}` is a vector that stores the known values of :math:`u(x,y)` on the x and y-faces, +and :math:`\bar{\boldsymbol{u}}` is a vector that stores the averages at cell centers. + +Let :math:`I_n` be an :math:`n` by :math:`n` identity matrix. +And use equation :eq:`operators_averaging_n2c_1d` to construct 1D averaging matrices :math:`A_x` and :math:`A_y`. +Then for a 2D tensor mesh, the averaging matrix has the form: + +.. math:: + \boldsymbol{A} = \frac{1}{2} \begin{bmatrix} \boldsymbol{A_x} & \boldsymbol{A_y} \end{bmatrix} + +where + +.. math:: + \begin{align} + \boldsymbol{A_x} &= I_{ny} \otimes A_x \\ + \boldsymbol{A_y} &= A_y \otimes I_{nx} + \end{align} + :label: operators_averaging_Ai_f2c_2d + +For a 3D tensor mesh, the averaging matrix takes the form: + +.. math:: + \boldsymbol{A} = \frac{1}{3} \begin{bmatrix} \boldsymbol{A_x} & \boldsymbol{A_y} & \boldsymbol{A_z} \end{bmatrix} + +where + +.. math:: + \begin{align} + \boldsymbol{A_x} &= I_{nz} \otimes ( I_{ny} \otimes A_x ) \\ + \boldsymbol{A_y} &= I_{nz} \otimes ( A_y \otimes I_{nx} ) \\ + \boldsymbol{A_z} &= A_z \otimes ( I_{ny} \otimes I_{nx} ) + \end{align} + :label: operators_averaging_Ai_f2c_3d + + +Edges to Cell Centers +^^^^^^^^^^^^^^^^^^^^^ + +Let us define a 2D mesh where cell center locations :math:`(x_i, y_j)` are represented using indices :math:`(i,j)`. +The widths of the cells in :math:`x` and :math:`y` are given by :math:`\Delta x_i` and :math:`\Delta y_j`, respectively. + +.. image:: ../../images/averaging_2d_edges.png + :align: center + :width: 350 + +If :math:`u(x,y)` is a scalar quantity whose values are known on the edges. +and :math:`\bar{u}(i,j)` is the average at the center of cell :math:`i,j`, +then: + +.. math:: + \bar{u}(i,j) = \frac{1}{4} \Big [ + u \Big ( i, j-\tfrac{1}{2} \Big ) + + u \Big ( i, j+\tfrac{1}{2} \Big ) + + u \Big ( i-\tfrac{1}{2}, j \Big ) + + u \Big ( i+\tfrac{1}{2}, j \Big ) \Big ] + +Our goal is to construct a matrix :math:`\boldsymbol{A}` that averages the values +on the edges and places them at the cell centers, i.e.: + +.. math:: + \bar{\boldsymbol{u}} = \boldsymbol{A \, u} + +where :math:`\boldsymbol{u}` is a vector that stores the known values of :math:`u(x,y)` on the x and y-edges, +and :math:`\bar{\boldsymbol{u}}` is a vector that stores the averages at cell centers. + +Let :math:`I_n` be an :math:`n` by :math:`n` identity matrix. +And use equation :eq:`operators_averaging_n2c_1d` to construct 1D averaging matrices :math:`A_x` and :math:`A_y`. +Then for a 2D tensor mesh, the averaging matrix has the form: + +.. math:: + \boldsymbol{A} = \frac{1}{2} \begin{bmatrix} \boldsymbol{A_x} & \boldsymbol{A_y} \end{bmatrix} + +where + +.. math:: + \begin{align} + \boldsymbol{A_x} &= A_x \otimes I_{ny} \\ + \boldsymbol{A_y} &= I_{nx} \otimes A_y + \end{align} + :label: operators_averaging_Ai_e2c_2d + +For a 3D tensor mesh, the averaging matrix takes the form: + +.. math:: + \boldsymbol{A} = \frac{1}{3} \begin{bmatrix} \boldsymbol{A_x} & \boldsymbol{A_y} & \boldsymbol{A_z} \end{bmatrix} +where + +.. math:: + \begin{align} + \boldsymbol{A_x} &= I_{nz} \otimes ( A_y \otimes A_x ) \\ + \boldsymbol{A_y} &= A_z \otimes ( I_{ny} \otimes A_x ) \\ + \boldsymbol{A_z} &= A_z \otimes ( A_y \otimes I_{nx} ) + \end{align} + :label: operators_averaging_Ai_e2c_3d -Vectors from Faces to Cell Centers -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +Averaging Vectors in 2D and 3D +============================== + +Faces to Cell Centers +^^^^^^^^^^^^^^^^^^^^^ + +Let :math:`\vec{u}(x,y)` be a vector function that is known on the faces. +That is, :math:`u_x (x,y)` lives on x-faces and :math:`u_y(x,y)` lives on y-faces. +In this case, the x-faces are used to average the x-component +to cell centers and the y-faces are used to average the y-component to cell centers separately. -Let us define a 2D mesh where cell center locations are defined by :math:`(x_i,y_j)`. +Let us define a 2D mesh where cell center locations :math:`(x_i, y_j)` are represented using indices :math:`(i,j)`. The widths of the cells in :math:`x` and :math:`y` are given by :math:`\Delta x_i` and :math:`\Delta y_j`, respectively. .. image:: ../../images/averaging_2d_faces.png :align: center - :width: 400 + :width: 350 +Where :math:`\bar{u}_x (i,j)` is the average x-component at the center of cell :math:`i,j`: +.. math:: + \bar{u}_x (i,j) = \frac{1}{2} \Big [ u_x \Big ( i-\tfrac{1}{2},j \Big ) + u_x \Big ( i+\tfrac{1}{2},j \Big ) \Big ] + +And where :math:`\bar{u}_y (i,j)` is the average at the center of cell :math:`i,j`: + +.. math:: + \bar{u}_y (i,j) = \frac{1}{2} \Big [ u_y \Big ( i,j-\tfrac{1}{2} \Big ) + u_y \Big ( i,j+\tfrac{1}{2} \Big ) \Big ] + +Our goal is to construct a matrix :math:`\boldsymbol{A}` that averages the vector components living +on the faces separately and places them at the cell centers. +Here, the operation :math:`\bar{\boldsymbol{u}} = \boldsymbol{A \, u}` takes the form: + +.. math:: + \begin{bmatrix} \bar{\boldsymbol{u}}_{\boldsymbol{x}} \\ \bar{\boldsymbol{u}}_{\boldsymbol{y}} \end{bmatrix} = + \begin{bmatrix} \boldsymbol{A_x} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{A_y} \end{bmatrix} + \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \end{bmatrix} +where :math:`\boldsymbol{u}` is a vector that stores the known values of :math:`u_x(x,y)` and :math:`u_y(x,y)` on their respective faces, +and :math:`\bar{\boldsymbol{u}}` is a vector that stores the x component and y component averages at cell centers. +Matrices :math:`\boldsymbol{A_x}` and :math:`\boldsymbol{A_y}` are defined in expression :eq:`operators_averaging_Ai_f2c_2d`. + +In 3D, the corresponding averaging matrix is defined by: + +.. math:: + \begin{bmatrix} \bar{\boldsymbol{u}}_{\boldsymbol{x}} \\ \bar{\boldsymbol{u}}_{\boldsymbol{y}} \\ \bar{\boldsymbol{u}}_{\boldsymbol{z}} \end{bmatrix} = + \begin{bmatrix} \boldsymbol{A_x} & \boldsymbol{0} & \boldsymbol{0} \\ + \boldsymbol{0} & \boldsymbol{A_y} & \boldsymbol{0} \\ + \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{A_z} \end{bmatrix} + \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_z} \end{bmatrix} + +where matrices :math:`\boldsymbol{A_x}`, :math:`\boldsymbol{A_y}` and :math:`\boldsymbol{A_z}` are defined in expression :eq:`operators_averaging_Ai_f2c_3d`. + + +Edges to Cell Centers +^^^^^^^^^^^^^^^^^^^^^ -Vectors from Edges to Cell Centers -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +Let :math:`\vec{u}(x,y)` be a vector function that is known on the edges. +That is, :math:`u_x (x,y)` lives on x-edges and :math:`u_y(x,y)` lives on y-edges. +In this case, the x-edges are used to average the x-component +to cell centers and the y-edges are used to average the y-component to cell centers separately. -Let us define a 2D mesh where node locations are defined by :math:`(x_i,y_j)`. +Let us define a 2D mesh where cell center locations :math:`(x_i, y_j)` are represented using indices :math:`(i,j)`. The widths of the cells in :math:`x` and :math:`y` are given by :math:`\Delta x_i` and :math:`\Delta y_j`, respectively. .. image:: ../../images/averaging_2d_edges.png :align: center - :width: 400 + :width: 350 + +Where :math:`\bar{u}_x (i,j)` is the average x-component at the center of cell :math:`i,j`: + +.. math:: + \bar{u}_x (i,j) = \frac{1}{2} \Big [ u_x \Big ( i,j-\tfrac{1}{2} \Big ) + u_x \Big ( i,j+\tfrac{1}{2} \Big ) \Big ] + +And where :math:`\bar{u}_y (i,j)` is the average at the center of cell :math:`i,j`: + +.. math:: + \bar{u}_y (i,j) = \frac{1}{2} \Big [ u_y \Big ( i-\tfrac{1}{2},j \Big ) + u_y \Big ( i+\tfrac{1}{2},j \Big ) \Big ] + +Our goal is to construct a matrix :math:`\boldsymbol{A}` that averages the vector components living +on the edges separately and places them at the cell centers. +Here, the operation :math:`\bar{\boldsymbol{u}} = \boldsymbol{A \, u}` takes the form: + +.. math:: + \begin{bmatrix} \bar{\boldsymbol{u}}_{\boldsymbol{x}} \\ \bar{\boldsymbol{u}}_{\boldsymbol{y}} \end{bmatrix} = + \begin{bmatrix} \boldsymbol{A_x} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{A_y} \end{bmatrix} + \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \end{bmatrix} + +where :math:`\boldsymbol{u}` is a vector that stores the known values of :math:`u_x(x,y)` and :math:`u_y(x,y)` on their respective edges, +and :math:`\bar{\boldsymbol{u}}` is a vector that stores the x component and y component averages at cell centers. +Matrices :math:`\boldsymbol{A_x}` and :math:`\boldsymbol{A_y}` are defined in expression :eq:`operators_averaging_Ai_e2c_2d`. + +In 3D, the corresponding averaging matrix is defined by: + +.. math:: + \begin{bmatrix} \bar{\boldsymbol{u}}_{\boldsymbol{x}} \\ \bar{\boldsymbol{u}}_{\boldsymbol{y}} \\ \bar{\boldsymbol{u}}_{\boldsymbol{z}} \end{bmatrix} = + \begin{bmatrix} \boldsymbol{A_x} & \boldsymbol{0} & \boldsymbol{0} \\ + \boldsymbol{0} & \boldsymbol{A_y} & \boldsymbol{0} \\ + \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{A_z} \end{bmatrix} + \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_z} \end{bmatrix} + +where matrices :math:`\boldsymbol{A_x}`, :math:`\boldsymbol{A_y}` and :math:`\boldsymbol{A_z}` are defined in expression :eq:`operators_averaging_Ai_e2c_3d`. diff --git a/docs/content/theory/operators_differential.rst b/docs/content/theory/operators_differential.rst index a1442efd8..76e76408e 100644 --- a/docs/content/theory/operators_differential.rst +++ b/docs/content/theory/operators_differential.rst @@ -27,7 +27,7 @@ the gradient, divergence and curl can be computed for discrete variables. :width: 300 -.. important:: To construct and apply differential operators to discrete quantities, see our* :ref:`tutorial for differential operators ` +.. important:: To construct and apply differential operators to discrete quantities, see our :ref:`tutorial for differential operators ` @@ -54,16 +54,16 @@ denote positions along the x, y and z axes, respectively. +-------------+-------------------------------------------------+-----------------------------------------------------+ | **center** | :math:`(i,j)` | :math:`(i,j,k)` | +-------------+-------------------------------------------------+-----------------------------------------------------+ -| **x faces** | :math:`(i-\frac{1}{2},j)\;\; (i+\frac{1}{2},j)` | :math:`(i-\frac{1}{2},j,k)\;\; (i+\frac{1}{2},j,k)` | +| **x-faces** | :math:`(i-\frac{1}{2},j)\;\; (i+\frac{1}{2},j)` | :math:`(i-\frac{1}{2},j,k)\;\; (i+\frac{1}{2},j,k)` | +-------------+-------------------------------------------------+-----------------------------------------------------+ -| **y faces** | :math:`(i,j-\frac{1}{2})\;\; (i,j+\frac{1}{2})` | :math:`(i,j-\frac{1}{2},k)\;\; (i,j+\frac{1}{2},k)` | +| **y-faces** | :math:`(i,j-\frac{1}{2})\;\; (i,j+\frac{1}{2})` | :math:`(i,j-\frac{1}{2},k)\;\; (i,j+\frac{1}{2},k)` | +-------------+-------------------------------------------------+-----------------------------------------------------+ -| **z faces** | N/A | :math:`(i,j,k-\frac{1}{2})\;\; (i,j,k+\frac{1}{2})` | +| **z-faces** | N/A | :math:`(i,j,k-\frac{1}{2})\;\; (i,j,k+\frac{1}{2})` | +-------------+-------------------------------------------------+-----------------------------------------------------+ As we will see, it makes the most sense for :math:`\boldsymbol{\phi}` to live at the cell centers and -for the components of :math:`\boldsymbol{u}` to live on the faces. If :math:`u_x` lives on x faces, then its discrete -derivative with respect to :math:`x` lives at the cell center. And if :math:`u_y` lives on y faces its discrete +for the components of :math:`\boldsymbol{u}` to live on the faces. If :math:`u_x` lives on x-faces, then its discrete +derivative with respect to :math:`x` lives at the cell center. And if :math:`u_y` lives on y-faces its discrete derivative with respect to :math:`y` lives at the cell center. Likewise for :math:`u_z`. Thus to approximate the divergence of :math:`\vec{u}` at the cell center, we simply need to sum the discrete derivatives of :math:`u_x`, :math:`u_y` and :math:`u_z` that are defined at the cell center. Where :math:`h_x`, :math:`h_y` and :math:`h_z` represent the dimension of the cell along the x, y and @@ -122,9 +122,9 @@ denote positions along the x, y and z axes, respectively. As we will see, it makes the most sense for :math:`\boldsymbol{\phi}` to live at the cell nodes and for the components of :math:`\boldsymbol{u}` to live on corresponding edges. If :math:`\phi` lives on the nodes, then: - - the partial derivative :math:`\dfrac{\partial \phi}{\partial x}\hat{x}` lives on x edges, - - the partial derivative :math:`\dfrac{\partial \phi}{\partial y}\hat{y}` lives on y edges, and - - the partial derivative :math:`\dfrac{\partial \phi}{\partial z}\hat{z}` lives on z edges + - the partial derivative :math:`\dfrac{\partial \phi}{\partial x}\hat{x}` lives on x-edges, + - the partial derivative :math:`\dfrac{\partial \phi}{\partial y}\hat{y}` lives on y-edges, and + - the partial derivative :math:`\dfrac{\partial \phi}{\partial z}\hat{z}` lives on z-edges Thus to approximate the gradient of :math:`\phi`, we simply need to take discrete derivatives of :math:`\phi` with respect to :math:`x`, :math:`y` and :math:`z`, diff --git a/docs/content/theory/operators_index.rst b/docs/content/theory/operators_index.rst index c78a869e0..c68aa3626 100644 --- a/docs/content/theory/operators_index.rst +++ b/docs/content/theory/operators_index.rst @@ -4,7 +4,7 @@ Operators ********* To approximate numerical solutions to partial differential equations using the finite volume method, -we must be able to construct discrete approximations to the following mathematical operators: +we must be able to construct discrete approximations to the following mathematical operations: - Interpolation - Averaging diff --git a/docs/content/theory/operators_interpolation.rst b/docs/content/theory/operators_interpolation.rst index 9bb4001a2..57b4eca1f 100644 --- a/docs/content/theory/operators_interpolation.rst +++ b/docs/content/theory/operators_interpolation.rst @@ -3,17 +3,21 @@ Interpolation ************* -Here, we provide the background theory for how discrete interpolation matricies are formed. -Interpolation is required when a discrete quantity is known on the mesh (centers, nodes, edges or faces), but we would like to estimate its -value at a point within the continuous domain. +Interpolation is required when a discrete quantity is known on the mesh (centers, nodes, edges or faces), +but we would like to estimate its value at locations within the continuous domain. +Here, we discuss how a sparse matrix :math:`\boldsymbol{P}` can be formed which interpolates the discrete values to +a set of locations in continuous space. Where :math:`\boldsymbol{u}` is vector that stores +the values of a discrete quantity on the mesh (centers, nodes, faces or edges) and +:math:`\boldsymbol{w}` is a vector containing the interpolated quantity at a set of locations, +we look to construct an interpolation matrix such that: + +.. math:: + \boldsymbol{w} = \boldsymbol{P \, u} Presently there is an extensive set of interpolation methods (e.g. polynomial, spline, piecewise constant). One of the most effective and widely used interpolation methods is linear interpolation. The *discretize* package primarily uses linear interpolation because 1) it is very fast, and 2) higher order interpolation methods require the construction of matricies which are less sparse. -Here, we discuss how a sparse matrix can be formed which interpolates the discrete values to -a set of points in continuous space. - The formulation for linear interpolation is adequately presented on Wikipedia, see: - `Linear Interpolation (1D) `__ @@ -24,9 +28,9 @@ The formulation for linear interpolation is adequately presented on Wikipedia, s Interpolation Matrix in 1D ========================== -Let us define a 1D mesh that contains 8 cells of arbitrary width; -the mesh is shown in the figure below. The width of each cell is -given by :math:`h_i`. The location of each node is given by :math:`x_i`. +Let us define a 1D mesh that contains 8 cells of arbitrary width. +The mesh is illustrated in the figure below. The width of each cell is +defined as :math:`h_i`. The location of each node is defined as :math:`x_i`. .. image:: ../../images/interpolation_1d.png :align: center @@ -73,13 +77,13 @@ Where :math:`\boldsymbol{u^*}` is a vector containing the approximations of :mat locations: .. math:: - (\boldsymbol{u^*}) \approx \boldsymbol{P\, u} \;\;\;\;\;\; \textrm{where} \;\;\;\;\;\; + \boldsymbol{u^*} \approx \boldsymbol{P\, u} \;\;\;\;\;\; \textrm{where} \;\;\;\;\;\; \boldsymbol{P} = \begin{bmatrix} \cdots \;\; \boldsymbol{p_0} \;\; \cdots \\ \cdots \;\; \boldsymbol{p_1} \;\; \cdots \\ \vdots \\ \cdots \, \boldsymbol{p_{M-1}} \, \cdots \end{bmatrix} :label: operators_averaging_interpolation_matrix -where :math:`\boldsymbol{P}` is a sparse matrix whose rows contain a maximum of 2 non-zero elements. +:math:`\boldsymbol{P}` is a sparse matrix whose rows contain a maximum of 2 non-zero elements. The size of :math:`\boldsymbol{P}` is the number of locations by the number of nodes. 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diff --git a/docs/index.rst b/docs/index.rst index f813feb31..021c78269 100644 --- a/docs/index.rst +++ b/docs/index.rst @@ -33,6 +33,7 @@ :caption: Theory content/finite_volume + content/differential_operators content/inner_products .. toctree:: From 7a58a64b443d6ff90e29011b0a5f6b4d1241a3c4 Mon Sep 17 00:00:00 2001 From: dccowan Date: Fri, 20 Nov 2020 19:54:10 -0800 Subject: [PATCH 08/35] Final draft of discrete operators --- docs/content/differential_operators.rst | 97 +++++++++++++++++++------ 1 file changed, 73 insertions(+), 24 deletions(-) diff --git a/docs/content/differential_operators.rst b/docs/content/differential_operators.rst index 39d3a7193..2790be699 100644 --- a/docs/content/differential_operators.rst +++ b/docs/content/differential_operators.rst @@ -6,9 +6,9 @@ Discrete Differential Operators Here, we provide the background theory for how discrete differential operators are formed. For geophysical problems, the relationship between two physical quantities may include one of several differential operators: - - **Divergence:** :math:`\nabla \cdot \vec{u} = \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_y}{\partial y}` - - **Gradient:** :math:`\nabla \phi = \frac{\partial \phi}{\partial x}\hat{x} + \frac{\partial \phi}{\partial y}\hat{y} + \frac{\partial \phi}{\partial z}\hat{z}` - - **Curl:** :math:`\nabla \times \vec{u} = \Big ( \frac{\partial u_y}{\partial z} - \frac{\partial u_z}{\partial y} \Big )\hat{x} - \Big ( \frac{\partial u_x}{\partial z} - \frac{\partial u_z}{\partial x} \Big )\hat{y} + \Big ( \frac{\partial u_x}{\partial y} - \frac{\partial u_y}{\partial x} \Big )\hat{z}` + - **Divergence:** :math:`\nabla \cdot \vec{u} = \dfrac{\partial u_x}{\partial x} + \dfrac{\partial u_y}{\partial y} + \dfrac{\partial u_y}{\partial y}` + - **Gradient:** :math:`\nabla \phi = \dfrac{\partial \phi}{\partial x}\hat{x} + \dfrac{\partial \phi}{\partial y}\hat{y} + \dfrac{\partial \phi}{\partial z}\hat{z}` + - **Curl:** :math:`\nabla \times \vec{u} = \Bigg ( \dfrac{\partial u_y}{\partial z} - \dfrac{\partial u_z}{\partial y} \Bigg )\hat{x} - \Bigg ( \dfrac{\partial u_x}{\partial z} - \dfrac{\partial u_z}{\partial x} \Bigg )\hat{y} + \Bigg ( \dfrac{\partial u_x}{\partial y} - \dfrac{\partial u_y}{\partial x} \Bigg )\hat{z}` When implementing the finite volume method, continuous variables are discretized to live at the cell centers, nodes, edges or faces of a mesh. Thus for each differential operator, we need a discrete approximation that acts on the discrete variables living on the mesh. @@ -27,6 +27,7 @@ the gradient, divergence and curl can be computed for discrete variables. :width: 300 +.. note:: To construct discrete differential operators and apply them in a set of examples, see our :ref:`tutorial for differential operators ` @@ -65,7 +66,7 @@ for the components of :math:`\boldsymbol{u}` to live on the faces. If :math:`u_x derivative with respect to :math:`x` lives at the cell center. And if :math:`u_y` lives on y faces its discrete derivative with respect to :math:`y` lives at the cell center. Likewise for :math:`u_z`. Thus to approximate the divergence of :math:`\vec{u}` at the cell center, we simply need to sum the discrete derivatives of :math:`u_x`, :math:`u_y` -and :math:`u_z` living on the faces. Where :math:`h_x`, :math:`h_y` and :math:`h_z` represent the dimension of the cell along the x, y and +and :math:`u_z` that are defined at the cell center. Where :math:`h_x`, :math:`h_y` and :math:`h_z` represent the dimension of the cell along the x, y and z directions, respectively: .. math:: @@ -84,6 +85,23 @@ z directions, respectively: \end{align} +Ultimately we are trying to approximate the divergence at the center of every cell in a mesh. +Adjacent cells share faces. If the components :math:`u_x`, :math:`u_y` and :math:`u_z` are +continuous across their respective faces, then :math:`\boldsymbol{\phi}` and :math:`\boldsymbol{u}` +can be related by a sparse matrix-vector product: + +.. math:: + \boldsymbol{\phi} = \boldsymbol{D \, u} + +where :math:`\boldsymbol{D}` is the divergence matrix from faces to cell centers, +:math:`\boldsymbol{\phi}` is a vector containing the discrete approximations of :math:`\phi` at all cell centers, +and :math:`\boldsymbol{u}` stores the components of :math:`\vec{u}` on cell faces as a vector of the form: + +.. math:: + \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_z} \end{bmatrix} + + +.. note:: To construct discrete differential operators and apply them in a set of examples, see our :ref:`tutorial for differential operators ` Gradient @@ -107,9 +125,9 @@ denote positions along the x, y and z axes, respectively. As we will see, it makes the most sense for :math:`\boldsymbol{\phi}` to live at the cell nodes and for the components of :math:`\boldsymbol{u}` to live on corresponding edges. If :math:`\phi` lives on the nodes, then: - - the partial derivative :math:`\frac{\partial \phi}{\partial x}\hat{x}` lives on x edges, - - the partial derivative :math:`\frac{\partial \phi}{\partial y}\hat{y}` lives on y edges, and - - the partial derivative :math:`\frac{\partial \phi}{\partial z}\hat{z}` lives on z edges + - the partial derivative :math:`\dfrac{\partial \phi}{\partial x}\hat{x}` lives on x edges, + - the partial derivative :math:`\dfrac{\partial \phi}{\partial y}\hat{y}` lives on y edges, and + - the partial derivative :math:`\dfrac{\partial \phi}{\partial z}\hat{z}` lives on z edges Thus to approximate the gradient of :math:`\phi`, we simply need to take discrete derivatives of :math:`\phi` with respect to :math:`x`, :math:`y` and :math:`z`, @@ -140,8 +158,22 @@ for each vector component is given below: \end{align} +Ultimately we are trying to approximate the vector components of the gradient at all edges of a mesh. +Adjacent cells share nodes. If :math:`\phi` is continuous at the nodes, then :math:`\boldsymbol{\phi}` and :math:`\boldsymbol{u}` +can be related by a sparse matrix-vector product: + +.. math:: + \boldsymbol{u} = \boldsymbol{G_n \, \phi} + +where :math:`\boldsymbol{G_n}` is the gradient matrix that maps from nodes to edges, +:math:`\boldsymbol{\phi}` is a vector containing :math:`\phi` at all nodes, +and :math:`\boldsymbol{u}` stores the components of :math:`\vec{u}` on cell edges as a vector of the form: + +.. math:: + \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_z} \end{bmatrix} +.. note:: To construct discrete differential operators and apply them in a set of examples, see our :ref:`tutorial for differential operators ` Curl @@ -158,8 +190,6 @@ our goal is to use discrete differentiation to approximate the vector components We begin by considering a single 3D cell. We let the indices :math:`i`, :math:`j` and :math:`k` denote positions along the x, y and z axes, respectively. - - .. image:: ../images/curl_discretization.png :align: center :width: 800 @@ -169,25 +199,44 @@ As we will see, it makes the most sense for the vector components of :math:`\bol for the vector components of :math:`\boldsymbol{w}` to live the faces. In this case, we need to approximate: - - the partial derivatives :math:`\frac{\partial u_y}{\partial z}` and :math:`\frac{\partial u_z}{\partial y}` to compute :math:`w_x`, - - the partial derivatives :math:`\frac{\partial u_x}{\partial z}` and :math:`\frac{\partial u_z}{\partial x}` to compute :math:`w_y`, and - - the partial derivatives :math:`\frac{\partial u_x}{\partial y}` and :math:`\frac{\partial u_y}{\partial x}` to compute :math:`w_z` + - the partial derivatives :math:`\dfrac{\partial u_y}{\partial z}` and :math:`\dfrac{\partial u_z}{\partial y}` to compute :math:`w_x`, + - the partial derivatives :math:`\dfrac{\partial u_x}{\partial z}` and :math:`\dfrac{\partial u_z}{\partial x}` to compute :math:`w_y`, and + - the partial derivatives :math:`\dfrac{\partial u_x}{\partial y}` and :math:`\dfrac{\partial u_y}{\partial x}` to compute :math:`w_z` -**In 3D**, the value of :math:`\phi` at 8 node locations is used to approximate the vector components of the -gradient at 12 edges locations (4 x-edges, 4 y-edges and 4 z-edges). An example of the approximation -for each vector component is given below: +**In 3D**, discrete values at 12 edge locations (4 x-edges, 4 y-edges and 4 z-edges) are used to +approximate the vector components of the curl at 6 face locations (2 x-faces, 2-faces and 2 z-faces). +An example of the approximation for each vector component is given below: .. math:: \begin{align} - w_x \Big ( i,j+\frac{1}{2},k+\frac{1}{2} \Big ) \approx \; & - \Bigg( \frac{u_y (i,j+\frac{1}{2},k+1) - u_y (i,j+\frac{1}{2},k)}{h_z} \Bigg ) - - \Bigg( \frac{u_z (i,j+1,k+\frac{1}{2}) - u_z (i,j,k+\frac{1}{2})}{h_y} \Bigg ) \\ + w_x \Big ( i,j \! +\!\!\frac{1}{2},k \! +\!\!\frac{1}{2} \Big ) \!\approx\! \; & + \!\Bigg ( \! \frac{u_y (i,j \! +\!\!\frac{1}{2},k \! +\!\!1) \! -\! u_y (i,j \! +\!\!\frac{1}{2},k)}{h_z} \Bigg) \! + \! -\! \!\Bigg ( \! \frac{u_z (i,j \! +\!\!1,k \! +\!\!\frac{1}{2}) \! -\! u_z (i,j,k \! +\!\!\frac{1}{2})}{h_y} \Bigg) \! \\ & \\ - w_y \Big ( i+\frac{1}{2},j,k+\frac{1}{2} \Big ) \approx \; & - - \Bigg( \frac{u_x (i+\frac{1}{2},j,k+1) - u_x (i+\frac{1}{2},j,k)}{h_z} \Bigg ) - + \Bigg( \frac{u_z (i+1,j,k+\frac{1}{2}) - u_z (i,j,k+\frac{1}{2})}{h_x} \Bigg ) \\ + w_y \Big ( i \! +\!\!\frac{1}{2},j,k \! +\!\!\frac{1}{2} \Big ) \!\approx\! \; & + \!\Bigg ( \! \frac{u_z (i \! +\!\!1,j,k \! +\!\!\frac{1}{2}) \! -\! u_z (i,j,k \! +\!\!\frac{1}{2})}{h_x} \Bigg) \! + \! -\! \!\Bigg ( \! \frac{u_x (i \! +\!\!\frac{1}{2},j,k \! +\!\!1) \! -\! u_x (i \! +\!\!\frac{1}{2},j,k)}{h_z} \Bigg) \\ & \\ - w_z \Big ( i+\frac{1}{2},j+\frac{1}{2},k \Big ) \approx \; & - \Bigg( \frac{u_x (i+\frac{1}{2},j+1,k) - u_x (i+\frac{1}{2},j,k)}{h_y} \Bigg ) - - \Bigg( \frac{u_y (i+1,j+\frac{1}{2},k) - u_y (i,j+\frac{1}{2},k)}{h_z} \Bigg ) + w_z \Big ( i \! +\!\!\frac{1}{2},j \! +\!\!\frac{1}{2},k \Big ) \!\approx\! \; & + \!\Bigg ( \! \frac{u_x (i \! +\!\!\frac{1}{2},j \! +\!\!1,k) \! -\! u_x (i \! +\!\!\frac{1}{2},j,k)}{h_y} \Bigg) \! + \! -\! \!\Bigg ( \! \frac{u_y (i \! +\!\!1,j \! +\!\!\frac{1}{2},k) \! -\! u_y (i,j \! +\!\!\frac{1}{2},k)}{h_x} \Bigg ) \end{align} + + +Ultimately we are trying to approximate the curl on all the faces within a mesh. +Adjacent cells share edges. If the components :math:`u_x`, :math:`u_y` and :math:`u_z` are +continuous across at the edges, then :math:`\boldsymbol{u}` and :math:`\boldsymbol{w}` +can be related by a sparse matrix-vector product: + +.. math:: + \boldsymbol{w} = \boldsymbol{C \, u} + +where :math:`\boldsymbol{C}` is the curl matrix from edges to faces, +:math:`\boldsymbol{u}` is a vector that stores the components of :math:`\vec{u}` on cell edges, +and :math:`\boldsymbol{w}` is a vector that stores the components of :math:`\vec{w}` on cell faces such that: + +.. math:: + \boldsymbol{u} = \begin{bmatrix} \boldsymbol{u_x} \\ \boldsymbol{u_y} \\ \boldsymbol{u_z} \end{bmatrix} + \;\;\;\; \textrm{and} \;\;\;\; \begin{bmatrix} \boldsymbol{w_x} \\ \boldsymbol{w_y} \\ \boldsymbol{w_z} \end{bmatrix} + +.. note:: To construct discrete differential operators and apply them in a set of examples, see our :ref:`tutorial for differential operators ` From 826232aa7367ac88603be3bba7f8bd19316f0db4 Mon Sep 17 00:00:00 2001 From: dccowan Date: Sat, 21 Nov 2020 13:05:44 -0800 Subject: [PATCH 09/35] Improve finite volume theory title page --- docs/content/finite_volume.rst | 37 +++++++++++++------------- docs/images/finitevolumeschematic.png | Bin 0 -> 64631 bytes 2 files changed, 18 insertions(+), 19 deletions(-) create mode 100644 docs/images/finitevolumeschematic.png diff --git a/docs/content/finite_volume.rst b/docs/content/finite_volume.rst index 493bc8c99..5c9149d5c 100644 --- a/docs/content/finite_volume.rst +++ b/docs/content/finite_volume.rst @@ -1,41 +1,40 @@ -.. _api_FiniteVolume: +.. _finite_volume_index: -Finite Volume Method Concept +Intoduction to Finite Volume **************************** -Introduction ------------- +What is the Finite Volume Method? +--------------------------------- The finite volume method is a method for numerically solving partial differential equations. Implementation of the finite volume method requires the discretization of continuous functions, resulting in discrete approximations. The discrete approximations are typically organized on a numerical grid (or mesh), -which defines boundaries, locations, and connectivity. The final product of the approach -is a linear system of equations :math:`\mathbf{Ax=b}` that can be solved to compute -the discrete approximation of a desired quantity. +which defines the domain, its boundaries, and the locations of discrete quantities. +The final product of the approach is a linear system of equations :math:`\boldsymbol{A \phi=q}` +that can be solved to compute the discrete approximation of a desired quantity. -.. image:: ../images/finitevolrealestate.png - :width: 400 px - :alt: FiniteVolume +.. image:: ../images/finitevolumeschematic.png + :width: 700 :align: center In SimPEG, we have implemented a staggered mimetic finite volume approach (`Hyman and Shashkov, 1999 `_). This approach requires the definitions of variables at either cell-centers, nodes, -faces, or edges as seen in the figure above. For geophysical simulations, we also -require that the following operations be discretized: - - - Averaging between nodes, faces, edges and cell-centers - - Interpolation from nodes, faces, edges or cell-centers to other locations - - Differential operators such as the gradient, divergence and curl +faces, or edges. This method is different from finite difference methods, +as the final linear system is constructed by approximating the inner products between +test functions and partial differential equations. Examples -------- -Below are several examples of the final linear system resulting from the finite volume approach. -The details of constructing the final solution are not discussed here. Details for constructing -the final linear system for a PDE are discussed in :ref:`inner products ` +Below are several examples of the final linear system obtained using the finite volume approach. +A comprehensive derivation of the final result is not provided here. It in instead parsed into +a set of theory sections which include: + + - :ref:`discrete differential operators ` + - :ref:`inner products ` Direct Current Resistivity ^^^^^^^^^^^^^^^^^^^^^^^^^^ diff --git a/docs/images/finitevolumeschematic.png b/docs/images/finitevolumeschematic.png new file mode 100644 index 0000000000000000000000000000000000000000..f91392ea60a9d67363bf38481ed97ecc2222c5c7 GIT binary patch literal 64631 zcmeGDg;$i{`v#1XB2o${U6KNVbPuI;cZW1XcOxjFbazP#49(CjjUY7x3@IH0(lNw4 z`tkc)>%4!#IcuF~EtthTvG=q09oKzbH=os1WbvMmKS4o3!IPJJtAT=oX@!D<8vXbY z@|!x6Zg=Dds=J1)6iU@N!&hNfeaYIGkH^Oyu9#E^_+FiwHgb{Gcvd)A*pE z+~Uf=mDKVvIatE>A=-ps{SGEUvud$@{_Yuuov|MATS**_1MrvrFu%L9myd}_UU064 z`n^-1!STUqRwI7eUlGa@31-(2kdUk?Q~Um+=cU2Bidm_G_=~Jl>nWwFO2G4!cZk`OK@oXXIw_%Uo{@+3TG~p)r<^LT|P##%*j{M(&7AdL!f2Y2pyjM&8-Q?f?I_|L6X+|K49#StRV1F;u}+N1^^;OvCKcelS`Rhr=zRM0<r&yF1SZHAPi`M$8j+xda@X0P0@an;m^g-A!xYAUBn{*jh6(i%BL z?wI+(d=~I=;nTJJ@#9H`LE0v4qHt|`GZj=?afx4CJpjV}#gc2eFV(vQxvN&#Q~mUI 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z$ziPb!XuGIf;2i+ZDrIFG*GxX6P})4ik2_sW`Q?-dDiU&c4x?>YHLh_e<>?lbV7E7 zU!_5t0xo~~at#pB;no>Cy_P~1F~2cGd~${d{v75KgQH-WpPpVcpD^Q%zgK&}Nh^o> zBXD^Uom5 Date: Sun, 22 Nov 2020 12:42:57 -0800 Subject: [PATCH 10/35] small edit --- docs/content/finite_volume.rst | 22 +++++++++++----------- docs/content/inner_products.rst | 6 ------ 2 files changed, 11 insertions(+), 17 deletions(-) diff --git a/docs/content/finite_volume.rst b/docs/content/finite_volume.rst index 5c9149d5c..2a9c28855 100644 --- a/docs/content/finite_volume.rst +++ b/docs/content/finite_volume.rst @@ -3,8 +3,8 @@ Intoduction to Finite Volume **************************** -What is the Finite Volume Method? ---------------------------------- +What Finite Volume? +------------------- The finite volume method is a method for numerically solving partial differential equations. Implementation of the finite volume method requires the discretization @@ -30,11 +30,12 @@ Examples -------- Below are several examples of the final linear system obtained using the finite volume approach. -A comprehensive derivation of the final result is not provided here. It in instead parsed into -a set of theory sections which include: +A comprehensive derivation of the final result is not provided here. Instead, the necessary +background theory is parsed into several section, including: - - :ref:`discrete differential operators ` + - :ref:`discretizing differential operators ` - :ref:`inner products ` + - finite volume full examples Direct Current Resistivity ^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -56,13 +57,12 @@ If we choose to define the discrete representation of the electric potential on the solution for the electric potentials after applying the finite volume approach is given by: .. math:: - \boldsymbol{[G^T M_{\sigma e} G ]} \boldsymbol{\phi} = \mathbf{q_s} + \boldsymbol{[G^T \! M_{\sigma e} G ]} \boldsymbol{\phi} = \mathbf{q_s} -where :math:`\boldsymbol{G^T M_{\sigma e} G }` is a sparse matrix and +where :math:`\boldsymbol{G^T \! M_{\sigma e} G }` is a sparse matrix and - :math:`\boldsymbol{\phi}` is the discrete approximation to the electric potentials on the nodes - :math:`\boldsymbol{G}` is a discrete gradient operator - - :math:`\boldsymbol{G^T}` is the transpose of :math:`\boldsymbol{G}` but acts as a divergence operator - :math:`\boldsymbol{M_{\sigma e}}` is the mass matrix - :math:`\boldsymbol{q_s}` is the discrete representation of the source term on the nodes @@ -95,16 +95,16 @@ The solution for the electric potentials after applying the finite volume approa .. math:: \begin{align} - \boldsymbol{C^T M_{\mu f} \, b } - \boldsymbol{M_{\sigma e} \, e} = \mathbf{j_s} \\ + \boldsymbol{C^T \! M_{\mu f} \, b } - \boldsymbol{M_{\sigma e} \, e} = \mathbf{j_s} \\ \mathbf{C \, e} = -i \omega \mathbf{b} \end{align} which can be combined to form a single linear system: .. math:: - \boldsymbol{[C^T M_{\mu f} C } + i\omega \boldsymbol{M_{\sigma e}]} \mathbf{e} = -i \omega \mathbf{j_s} + \boldsymbol{[C^T \! M_{\mu f} C } + i\omega \boldsymbol{M_{\sigma e}]} \mathbf{e} = -i \omega \mathbf{j_s} -where :math:`\boldsymbol{C^T M_{\mu f} C } + i\omega \boldsymbol{M_{\sigma e}}` is a sparse matrix and +where :math:`\boldsymbol{C^T \! M_{\mu f} C } + i\omega \boldsymbol{M_{\sigma e}}` is a sparse matrix and - :math:`\boldsymbol{e}` is the discrete approximation to the electric field on edges - :math:`\boldsymbol{b}` is the discrete approximation to the magnetic flux density on faces diff --git a/docs/content/inner_products.rst b/docs/content/inner_products.rst index e5593ca9a..56798f835 100644 --- a/docs/content/inner_products.rst +++ b/docs/content/inner_products.rst @@ -16,12 +16,6 @@ given by: .. math:: (\vec{u}, \vec{v}) = \int_\Omega \vec{u} \cdot \vec{v} \, dv -.. To implement the finite volume method, we: - -.. 1. take the inner product between each differential equation and a test function -.. 2. re-express the inner product using the weak formulation by applying the divergence theorem when necessary -.. 3. formulate a discrete approximation to each inner product which respects the boundary conditions on our variables - When implementing the finite volume method, we construct discrete approximations to inner products. The approximations to inner products are combined and simplified to form a linear system in terms of an unknown variable, then solved. The approximation to each inner product depends on the quantities From 29ce0c7c1a10d00fe2e804a2db2fe6457fca65ff Mon Sep 17 00:00:00 2001 From: dccowan Date: Mon, 23 Nov 2020 15:34:28 -0800 Subject: [PATCH 11/35] Create mesh theory section --- ...ite_volume.rst => finite_volume_index.rst} | 21 ++++---- .../inner_products_anisotropic.rst | 8 +-- .../{ => theory}/inner_products_basic.rst | 8 +-- .../inner_products_differential.rst | 0 .../inner_products_index.rst} | 0 .../{ => theory}/inner_products_isotropic.rst | 4 +- docs/content/theory/meshes.rst | 48 ++++++++++++++++++ docs/content/theory/operators_averaging.rst | 5 ++ .../operators_differential.rst} | 14 ++--- docs/content/theory/operators_index.rst | 19 +++++++ docs/images/cell_locations.png | Bin 0 -> 54201 bytes docs/images/finitevolumeschematic.png | Bin 64631 -> 64627 bytes docs/images/mesh_types.png | Bin 0 -> 523256 bytes docs/index.rst | 7 +-- 14 files changed, 103 insertions(+), 31 deletions(-) rename docs/content/{finite_volume.rst => finite_volume_index.rst} (83%) rename docs/content/{ => theory}/inner_products_anisotropic.rst (99%) rename docs/content/{ => theory}/inner_products_basic.rst (98%) rename docs/content/{ => theory}/inner_products_differential.rst (100%) rename docs/content/{inner_products.rst => theory/inner_products_index.rst} (100%) rename docs/content/{ => theory}/inner_products_isotropic.rst (99%) create mode 100644 docs/content/theory/meshes.rst create mode 100644 docs/content/theory/operators_averaging.rst rename docs/content/{differential_operators.rst => theory/operators_differential.rst} (97%) create mode 100644 docs/content/theory/operators_index.rst create mode 100644 docs/images/cell_locations.png create mode 100644 docs/images/mesh_types.png diff --git a/docs/content/finite_volume.rst b/docs/content/finite_volume_index.rst similarity index 83% rename from docs/content/finite_volume.rst rename to docs/content/finite_volume_index.rst index 2a9c28855..fda7ee525 100644 --- a/docs/content/finite_volume.rst +++ b/docs/content/finite_volume_index.rst @@ -6,11 +6,9 @@ Intoduction to Finite Volume What Finite Volume? ------------------- -The finite volume method is a method for numerically solving partial differential -equations. Implementation of the finite volume method requires the discretization -of continuous functions, resulting in discrete approximations. The discrete -approximations are typically organized on a numerical grid (or mesh), -which defines the domain, its boundaries, and the locations of discrete quantities. +The finite volume method is a method for numerically approximating the solution to partial differential equations. +Implementation of the finite volume method requires the discretization of continuous functions and variables. +Discrete representations of functions and variables are organized on a numerical grid (or mesh). The final product of the approach is a linear system of equations :math:`\boldsymbol{A \phi=q}` that can be solved to compute the discrete approximation of a desired quantity. @@ -18,10 +16,10 @@ that can be solved to compute the discrete approximation of a desired quantity. :width: 700 :align: center -In SimPEG, we have implemented a staggered mimetic finite volume approach (`Hyman and -Shashkov, 1999 `_). This -approach requires the definitions of variables at either cell-centers, nodes, -faces, or edges. This method is different from finite difference methods, +In *discretize*, we use a staggered mimetic finite volume approach (`Hyman and +Shashkov, 1999 `_). +This approach requires the definitions of variables at either cell-centers, nodes, faces, or edges. +This method is different from finite difference methods, as the final linear system is constructed by approximating the inner products between test functions and partial differential equations. @@ -33,8 +31,9 @@ Below are several examples of the final linear system obtained using the finite A comprehensive derivation of the final result is not provided here. Instead, the necessary background theory is parsed into several section, including: - - :ref:`discretizing differential operators ` - - :ref:`inner products ` + - :ref:`meshes ` + - :ref:`interpolation, averaging and differential operators ` + - :ref:`inner products ` - finite volume full examples Direct Current Resistivity diff --git a/docs/content/inner_products_anisotropic.rst b/docs/content/theory/inner_products_anisotropic.rst similarity index 99% rename from docs/content/inner_products_anisotropic.rst rename to docs/content/theory/inner_products_anisotropic.rst index 56aa6ccb7..8843119fb 100644 --- a/docs/content/inner_products_anisotropic.rst +++ b/docs/content/theory/inner_products_anisotropic.rst @@ -150,7 +150,7 @@ x-component of the vectors live on the x-faces, the y-component lives y-faces an lives on the z-faces. For a single cell, this is illustrated in 2D and 3D below. By decomposing the domain into a set of finite cells, we assume the tensor properties are spacial invariant within each cell. -.. image:: ../images/face_discretization.png +.. image:: ../../images/face_discretization.png :align: center :width: 600 @@ -234,7 +234,7 @@ x-component of the vectors live on the x-edges, the y-component lives y-edges an lives on the z-edges. This is illustrated in 2D and 3D below. By decomposing the domain into a set of finite cells, we assume the tensor properties are spacial invariant within each cell. -.. image:: ../images/edge_discretization.png +.. image:: ../../images/edge_discretization.png :align: center :width: 600 @@ -321,7 +321,7 @@ x-component of the vectors live on the x-faces, the y-component lives y-faces an lives on the z-faces. For a single cell, this is illustrated in 2D and 3D below. By decomposing the domain into a set of finite cells, we assume the tensor properties are spacial invariant within each cell. -.. image:: ../images/face_discretization.png +.. image:: ../../images/face_discretization.png :align: center :width: 600 @@ -406,7 +406,7 @@ x-component of the vectors live on the x-edges, the y-component lives y-edges an lives on the z-edges. This is illustrated in 2D and 3D below. By decomposing the domain into a set of finite cells, we assume the tensor properties are spacial invariant within each cell. -.. image:: ../images/edge_discretization.png +.. image:: ../../images/edge_discretization.png :align: center :width: 600 diff --git a/docs/content/inner_products_basic.rst b/docs/content/theory/inner_products_basic.rst similarity index 98% rename from docs/content/inner_products_basic.rst rename to docs/content/theory/inner_products_basic.rst index bbbf4aba3..47f44b164 100644 --- a/docs/content/inner_products_basic.rst +++ b/docs/content/theory/inner_products_basic.rst @@ -62,7 +62,7 @@ our goal is to construct the inner product matrix :math:`\boldsymbol{M}` such th (\psi , \phi ) = \int_\Omega \psi \, \phi \, dv \approx \boldsymbol{\psi^T M \, \phi} -.. image:: ../images/center_discretization.png +.. image:: ../../images/center_discretization.png :align: center :width: 600 @@ -112,7 +112,7 @@ to live on cell nodes, it makes more sense for cell volumes to be considered a p which lives at cell centers. This makes evaluating the inner product more complicated as discrete quantities do not live at the same place. -.. image:: ../images/node_discretization.png +.. image:: ../../images/node_discretization.png :align: center :width: 600 @@ -177,7 +177,7 @@ We must respect the dot product. For vectors defined on cell faces, we discretiz x-component of the vectors live on the x-faces, the y-component lives y-faces and the z-component lives on the z-faces. For a single cell, this is illustrated in 2D and 3D below. -.. image:: ../images/face_discretization.png +.. image:: ../../images/face_discretization.png :align: center :width: 600 @@ -251,7 +251,7 @@ We must respect the dot product. For vectors defined on cell edges, we discretiz x-component of the vectors live on the x-edges, the y-component lives y-edges and the z-component lives on the z-edges. This is illustrated in 2D and 3D below. -.. image:: ../images/edge_discretization.png +.. image:: ../../images/edge_discretization.png :align: center :width: 600 diff --git a/docs/content/inner_products_differential.rst b/docs/content/theory/inner_products_differential.rst similarity index 100% rename from docs/content/inner_products_differential.rst rename to docs/content/theory/inner_products_differential.rst diff --git a/docs/content/inner_products.rst b/docs/content/theory/inner_products_index.rst similarity index 100% rename from docs/content/inner_products.rst rename to docs/content/theory/inner_products_index.rst diff --git a/docs/content/inner_products_isotropic.rst b/docs/content/theory/inner_products_isotropic.rst similarity index 99% rename from docs/content/inner_products_isotropic.rst rename to docs/content/theory/inner_products_isotropic.rst index 0e03e6027..5426c2e84 100644 --- a/docs/content/inner_products_isotropic.rst +++ b/docs/content/theory/inner_products_isotropic.rst @@ -70,7 +70,7 @@ x-component of the vectors live on the x-faces, the y-component lives y-faces an lives on the z-faces. For a single cell, this is illustrated in 2D and 3D below. By decomposing the domain into a set of finite cells, we assume the physical properties are spacial invariant within each cell. -.. image:: ../images/face_discretization.png +.. image:: ../../images/face_discretization.png :align: center :width: 600 @@ -145,7 +145,7 @@ x-component of the vectors live on the x-edges, the y-component lives y-edges an lives on the z-edges. This is illustrated in 2D and 3D below. By decomposing the domain into a set of finite cells, we assume the material properties are spacial invariant within each cell. -.. image:: ../images/edge_discretization.png +.. image:: ../../images/edge_discretization.png :align: center :width: 600 diff --git a/docs/content/theory/meshes.rst b/docs/content/theory/meshes.rst new file mode 100644 index 000000000..37a5661c1 --- /dev/null +++ b/docs/content/theory/meshes.rst @@ -0,0 +1,48 @@ +.. _meshes_index: + +Meshes +****** + +A mesh is a numerical grid on which discrete approximations of continuous functions and variables live. +Meshes can be created to discretize and solve 1D, 2D or 3D problems. +For a given system of partial differential equations, the mesh is a discrete representation of the domain and its boundaries. +Here, we demonstrate: + + - mesh types supported by the *discretize* package + - where discrete quantities live (mesh locations) + + + +Mesh Types +========== + +Mesh types supported by the *discretize* package include: + + - **Tensor Meshes:** A mesh where the grid locations are organized according to tensor products + - **Tree Meshes:** A mesh where the dimensions of cells are :math:`2^n` larger than the dimension of the smallest cell size + - **Curvilinear Meshes:** A tensor mesh where the axes are curvilinear + - **Cylindrical Meshes:** A pseudo-2D mesh for solving 3D problems with perfect symmetry in the radial direction + +Examples are illustrate below. + +.. image:: ../../images/mesh_types.png + :align: center + :width: 700 + +Where Quantities Live +===================== + +In *discretize*, we use a staggered mimetic finite volume approach (`Hyman and +Shashkov, 1999 `_). +This approach requires the definitions of variables at either cell-centers, nodes, faces, or edges. +Below, we illustrate the valid locations for discrete quantities for a single cell where: + + - **Nodes** represent location of intersection between grid lines defining the mesh. + - **X, Y and Z edges** are edges whose tangent lines are parallel to the X, Y and Z axis, respectively. + - **X, Y and Z faces** are faces which are normal to the X, Y and Z axis, respectively. + + +.. image:: ../../images/cell_locations.png + :align: center + :width: 700 + diff --git a/docs/content/theory/operators_averaging.rst b/docs/content/theory/operators_averaging.rst new file mode 100644 index 000000000..71de6cecb --- /dev/null +++ b/docs/content/theory/operators_averaging.rst @@ -0,0 +1,5 @@ +.. _operators_averaging: + +Interpolation and Averaging +*************************** + diff --git a/docs/content/differential_operators.rst b/docs/content/theory/operators_differential.rst similarity index 97% rename from docs/content/differential_operators.rst rename to docs/content/theory/operators_differential.rst index 2790be699..baf2ed030 100644 --- a/docs/content/differential_operators.rst +++ b/docs/content/theory/operators_differential.rst @@ -1,7 +1,7 @@ -.. _differential_operators: +.. _operators_differential: -Discrete Differential Operators -******************************* +Differential Operators +********************** Here, we provide the background theory for how discrete differential operators are formed. For geophysical problems, the relationship between two physical quantities may include one of several differential operators: @@ -22,7 +22,7 @@ within the interval :math:`[x-h/2, \; x+h/2]`, then the derivative of the functi where the approximation becomes increasingly accurate as :math:`h \rightarrow 0`. In subsequent sections, we will show how the gradient, divergence and curl can be computed for discrete variables. -.. image:: ../images/approximate_derivative.png +.. image:: ../../images/approximate_derivative.png :align: center :width: 300 @@ -45,7 +45,7 @@ our goal is to use discrete differentiation to approximate the values of :math:` We begin by considering a single cell (2D or 3D). We let the indices :math:`i`, :math:`j` and :math:`k` denote positions along the x, y and z axes, respectively. -.. image:: ../images/divergence_discretization.png +.. image:: ../../images/divergence_discretization.png :align: center :width: 600 @@ -118,7 +118,7 @@ our goal is to use discrete differentiation to approximate the vector components We begin by considering a single cell (2D or 3D). We let the indices :math:`i`, :math:`j` and :math:`k` denote positions along the x, y and z axes, respectively. -.. image:: ../images/gradient_discretization.png +.. image:: ../../images/gradient_discretization.png :align: center :width: 600 @@ -190,7 +190,7 @@ our goal is to use discrete differentiation to approximate the vector components We begin by considering a single 3D cell. We let the indices :math:`i`, :math:`j` and :math:`k` denote positions along the x, y and z axes, respectively. -.. image:: ../images/curl_discretization.png +.. image:: ../../images/curl_discretization.png :align: center :width: 800 diff --git a/docs/content/theory/operators_index.rst b/docs/content/theory/operators_index.rst new file mode 100644 index 000000000..8c2e4891a --- /dev/null +++ b/docs/content/theory/operators_index.rst @@ -0,0 +1,19 @@ +.. _operators_index: + +Operators +********* + +A mathematical operator is function or operation that acts on elements within +a space to produce elements in the same or another space. + +For our purpose + + + +**Contents:** + +.. toctree:: + :maxdepth: 1 + + operators_averaging + operators_differential \ No newline at end of file diff --git a/docs/images/cell_locations.png b/docs/images/cell_locations.png new file mode 100644 index 0000000000000000000000000000000000000000..b92a535661230438fb8cd3081706d64d12b0d144 GIT binary patch literal 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