simpeg/docs/api_Mesh.rst

.. _api_Mesh:

SimPEG Meshes
*************

The Mesh objects in SimPEG provide a numerical grid on which to solve
differential equations. Each mesh type has a similar API to make switching
between different meshes relatively simple.

Overview of Meshes Available
============================

The following meshes are available for use:

.. toctree::
   :maxdepth: 2

   api_MeshCode

Each mesh code follows the guiding principles that are present in this
tutorial, but the details, advantages and disadvantages differ between
the implementations.




Variable Locations and Terminology
==================================

We will go over the basics of using a TensorMesh, but these skills are transferable
to the other meshes available in SimPEG. All of the mesh generation code is located
in the Mesh package in SimPEG (i.e. SimPEG.Mesh).


To create a TensorMesh we need to create mesh tensors, the widths of
each cell of the mesh in each dimension. We will call these tensors h,
and these will be define the constant widths of cells in each dimension
of the TensorMesh.

.. plot::
    :include-source:

    from SimPEG import Mesh, np
    hx = np.r_[3,2,1,1,1,1,2,3]
    hy = np.r_[3,1,1,3]
    M = Mesh.TensorMesh([hx, hy])
    M.plotGrid(centers=True)


In this simple mesh, the hx vector defines the widths of the cell
in the x dimension, and starts counting from the origin (0,0). The
resulting mesh is divided into cells, and the cell-centers are
plotted above as red circles. Other terminology for this mesh are:

- cell-centers
- nodes
- faces
- edges

.. plot::
    :include-source:

    from SimPEG import Mesh, np
    import matplotlib.pyplot as plt
    hx = np.r_[3,2,1,1,1,1,2,3]
    hy = np.r_[3,1,1,3]
    M = Mesh.TensorMesh([hx, hy])
    M.plotGrid(faces=True, nodes=True)
    plt.title('Cell faces in the x- and y-directions.')
    plt.legend(('Nodes', 'X-Faces', 'Y-Faces'))

Generally, the faces are used to discretize fluxes, quantities that
leave or enter the cells. As such, these fluxes have a direction to
them, which is normal to the cell (i.e. directly out of the cell face).
The plot above shows that x-faces point in the x-direction, and
y-faces point in the y-direction. The nodes are shown in blue,
and lie at the intersection of the grid lines. In a two-dimensional
mesh, the edges actually live in the same location as the faces,
however, they align (or are tangent to) the face. This is easier to
see in 3D, when the edges do not live in the same location as the faces.
In the 3D plot below, the edge variables are seen as black triangles,
and live on the edges(!) of the cell.

.. plot::
    :include-source:

    from SimPEG import Mesh
    Mesh.TensorMesh([1,1,1]).plotGrid(faces=True, edges=True, centers=True)

How many of each?
-----------------

When making variables that live in each of these locations, it is
important to know how many of each variable type you are dealing with.
SimPEG makes this pretty easy:

::

    In [1]: print M
            ---- 2-D TensorMesh ----
             x0: 0.00
             y0: 0.00
            nCx: 8
            nCy: 4
             hx: 3.00, 2.00, 4*1.00, 2.00, 3.00
             hy: 3.00, 2*1.00, 3.00

    In [2]: count = {'numCells': M.nC,
      ....:          'numCells_xDir': M.nCx,
      ....:          'numCells_yDir': M.nCy,
      ....:          'numCells_vector': M.vnC}

    In [3]: print 'This mesh has %(numCells)d cells, which is %(numCells_xDir)d*%(numCells_yDir)d!!' % count

            This mesh has 32 cells, which is 8*4!!

    In [4]: print count

            {
             'numCells_vector': array([8, 4]),
             'numCells_yDir': 4,
             'numCells_xDir': 8,
             'numCells': 32
            }

SimPEG also counts the nodes, faces, and edges.

::

    Nodes: M.nN, M.nNx, M.nNy, M.nNz, M.vnN
    Faces: M.nF, M.nFx, M.nFy, M.nFz, M.vnF, M.vnFx, M.vnFy, M.vnFz
    Edges: M.nE, M.nEx, M.nEy, M.nEz, M.vnE, M.vnEx, M.vnEy, M.vnEz

Face and edge variables have different counts depending on
the dimension of the direction that you are interested in.
In a 4x5 mesh, for example, there is a 5x5 grid of x-faces,
and a 4x6 grid of y-faces. You can count them below!
As such, the vnF(x,y,z) and vnE(x,y,z) properties give the
vector grid size.

.. plot::
    :include-source:

    from SimPEG import Mesh
    Mesh.TensorMesh([4,5]).plotGrid(faces=True)


Making Tensors
--------------

For tensor meshes, there are some additional functions that can come
in handy. For example, creating mesh tensors can be a bit time
consuming, these can be created speedily by just giving numbers
and sizes of padding. See the example below, that follows this
notation::

    h1 = (
           (numPad, sizeStart [, increaseFactor]),
           (numCore, sizeCode),
           (numPad, sizeStart [, increaseFactor])
         )

.. plot::
    :include-source:

    from SimPEG import Mesh, Utils
    h1 = (5, 10, 1.5), (20, 5), (3, 10)
    M = Mesh.TensorMesh(Utils.meshTensors(h1, h1))
    M.plotGrid()

Hopefully, you now know how to create TensorMesh objects in SimPEG,
and by extension you are also familiar with how to create and use
other types of meshes in this SimPEG framework.


The API
=======

.. toctree::
   :maxdepth: 2

   api_MeshCode