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https://github.com/wassname/simpeg.git
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Changed LogicallyOrthogonalMesh to LogicallyRectMesh and updated all dependencies.
LOM --> LRM removed LomView.py, and put plot grid code inside Mesh code. Added tutorial style introduction to the mesh.
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rowanc1
+168
-24
@@ -1,38 +1,182 @@
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.. _api_Mesh:
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SimPEG Meshes
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*************
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Tensor Mesh
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===========
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The Mesh objects in SimPEG provide a numerical grid on which to solve
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differential equations. Each mesh type has a similar API to make switching
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between different meshes relatively simple.
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.. automodule:: SimPEG.Mesh.TensorMesh
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:show-inheritance:
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:members:
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:undoc-members:
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Overview of Meshes Available
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============================
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The following meshes are available for use:
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.. toctree::
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:maxdepth: 2
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api_MeshCode
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Each mesh code follows the guiding principles that are present in this
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tutorial, but the details, advantages and disadvantages differ between
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the implementations.
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Cylindrical 1D Mesh
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===================
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.. automodule:: SimPEG.Mesh.Cyl1DMesh
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:show-inheritance:
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:members:
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:undoc-members:
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Logically Orthogonal Mesh
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=========================
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Variable Locations and Terminology
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==================================
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.. automodule:: SimPEG.Mesh.LogicallyOrthogonalMesh
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:show-inheritance:
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:members:
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:undoc-members:
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We will go over the basics of using a TensorMesh, but these skills are transferable
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to the other meshes available in SimPEG. All of the mesh generation code is located
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in the Mesh package in SimPEG (i.e. SimPEG.Mesh).
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Base Mesh
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=========
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To create a TensorMesh we need to create mesh tensors, the widths of
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each cell of the mesh in each dimension. We will call these tensors h,
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and these will be define the constant widths of cells in each dimension
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of the TensorMesh.
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.. automodule:: SimPEG.Mesh.BaseMesh
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:members:
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:undoc-members:
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.. plot::
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:include-source:
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from SimPEG import Mesh, np
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hx = np.r_[3,2,1,1,1,1,2,3]
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hy = np.r_[3,1,1,3]
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M = Mesh.TensorMesh([hx, hy])
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M.plotGrid(centers=True)
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In this simple mesh, the hx vector defines the widths of the cell
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in the x dimension, and starts counting from the origin (0,0). The
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resulting mesh is divided into cells, and the cell-centers are
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plotted above as red circles. Other terminology for this mesh are:
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- cell-centers
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- nodes
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- faces
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- edges
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.. plot::
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:include-source:
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from SimPEG import Mesh, np
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import matplotlib.pyplot as plt
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hx = np.r_[3,2,1,1,1,1,2,3]
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hy = np.r_[3,1,1,3]
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M = Mesh.TensorMesh([hx, hy])
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M.plotGrid(faces=True, nodes=True)
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plt.title('Cell faces in the x- and y-directions.')
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plt.legend(('Nodes', 'X-Faces', 'Y-Faces'))
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Generally, the faces are used to discretize fluxes, quantities that
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leave or enter the cells. As such, these fluxes have a direction to
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them, which is normal to the cell (i.e. directly out of the cell face).
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The plot above shows that x-faces point in the x-direction, and
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y-faces point in the y-direction. The nodes are shown in blue,
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and lie at the intersection of the grid lines. In a two-dimensional
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mesh, the edges actually live in the same location as the faces,
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however, they align (or are tangent to) the face. This is easier to
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see in 3D, when the edges do not live in the same location as the faces.
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In the 3D plot below, the edge variables are seen as black triangles,
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and live on the edges(!) of the cell.
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.. plot::
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:include-source:
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from SimPEG import Mesh
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Mesh.TensorMesh([1,1,1]).plotGrid(faces=True, edges=True, centers=True)
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How many of each?
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-----------------
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When making variables that live in each of these locations, it is
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important to know how many of each variable type you are dealing with.
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SimPEG makes this pretty easy:
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::
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In [1]: print M
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---- 2-D TensorMesh ----
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x0: 0.00
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y0: 0.00
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nCx: 8
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nCy: 4
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hx: 3.00, 2.00, 4*1.00, 2.00, 3.00
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hy: 3.00, 2*1.00, 3.00
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In [2]: count = {'numCells': M.nC,
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....: 'numCells_xDir': M.nCx,
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....: 'numCells_yDir': M.nCy,
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....: 'numCells_vector': M.vnC}
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In [3]: print 'This mesh has %(numCells)d cells, which is %(numCells_xDir)d*%(numCells_yDir)d!!' % count
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This mesh has 32 cells, which is 8*4!!
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In [4]: print count
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{
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'numCells_vector': array([8, 4]),
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'numCells_yDir': 4,
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'numCells_xDir': 8,
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'numCells': 32
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}
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SimPEG also counts the nodes, faces, and edges.
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::
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Nodes: M.nN, M.nNx, M.nNy, M.nNz, M.vnN
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Faces: M.nF, M.nFx, M.nFy, M.nFz, M.vnF, M.vnFx, M.vnFy, M.vnFz
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Edges: M.nE, M.nEx, M.nEy, M.nEz, M.vnE, M.vnEx, M.vnEy, M.vnEz
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Face and edge variables have different counts depending on
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the dimension of the direction that you are interested in.
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In a 4x5 mesh, for example, there is a 5x5 grid of x-faces,
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and a 4x6 grid of y-faces. You can count them below!
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As such, the vnF(x,y,z) and vnE(x,y,z) properties give the
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vector grid size.
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.. plot::
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:include-source:
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from SimPEG import Mesh
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Mesh.TensorMesh([4,5]).plotGrid(faces=True)
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Making Tensors
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--------------
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For tensor meshes, there are some additional functions that can come
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in handy. For example, creating mesh tensors can be a bit time
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consuming, these can be created speedily by just giving numbers
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and sizes of padding. See the example below, that follows this
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notation::
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h1 = (
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(numPad, sizeStart [, increaseFactor]),
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(numCore, sizeCode),
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(numPad, sizeStart [, increaseFactor])
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)
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.. plot::
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:include-source:
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from SimPEG import Mesh, Utils
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h1 = (5, 10, 1.5), (20, 5), (3, 10)
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M = Mesh.TensorMesh(Utils.meshTensors(h1, h1))
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M.plotGrid()
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Hopefully, you now know how to create TensorMesh objects in SimPEG,
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and by extension you are also familiar with how to create and use
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other types of meshes in this SimPEG framework.
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The API
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=======
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.. toctree::
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:maxdepth: 2
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api_MeshCode
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@@ -0,0 +1,35 @@
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.. _api_MeshCode:
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Tensor Mesh
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===========
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.. automodule:: SimPEG.Mesh.TensorMesh
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:show-inheritance:
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:members:
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:undoc-members:
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Cylindrical 1D Mesh
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===================
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.. automodule:: SimPEG.Mesh.Cyl1DMesh
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:show-inheritance:
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:members:
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:undoc-members:
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Logically Rectangular Mesh
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==========================
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.. automodule:: SimPEG.Mesh.LogicallyRectMesh
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:show-inheritance:
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:members:
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:undoc-members:
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Base Mesh
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=========
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.. automodule:: SimPEG.Mesh.BaseMesh
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:members:
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:undoc-members:
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+2
-2
@@ -15,10 +15,10 @@ Matrix Utilities
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:members:
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:undoc-members:
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LOM Utilities
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LRM Utilities
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=============
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.. automodule:: SimPEG.Utils.lomutils
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.. automodule:: SimPEG.Utils.lrmutils
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:members:
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:undoc-members:
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