In this tutorial we will go over creating a mesh, and the basics of numbering, counting, and naming conventions. With these skills you can move faster, and move between different meshes (hopefully) with ease!
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SimPEG Tutorial - Creating a Mesh
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| <h3 id="objective-">Objective:</h3> | |
| <p>In this tutorial we will go over creating a mesh, and the basics of numbering, counting, and naming conventions. With these skills you can move faster, and move between different meshes (hopefully) with ease!</p> | |
| <p>We will go over the basics of using a <em>TensorMesh</em>, but these skills are transferable to the other meshes available in SimPEG. All of the mesh generation code is located in the <em>mesh</em> package in SimPEG (i.e. <em>SimPEG.mesh</em>). The following meshes are available for use: </p> | |
| <pre><code>'TensorMesh' | |
| 'Cyl1DMesh' | |
| 'LogicallyOrthogonalMesh' | |
| 'TreeMesh' | |
| </code></pre><p>Each mesh code follows the guiding principles that are present in this tutorial, but the details, advantages and disadvantages differ between the implementations. Please read up on the documentation at <a href="http://simpeg.rtfd.org">http://simpeg.rtfd.org</a>.</p> | |
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| <div class="highlight"><pre><span class="kn">from</span> <span class="nn">SimPEG</span> <span class="kn">import</span> <span class="o">*</span> | |
| <span class="k">print</span> <span class="s">'</span><span class="se">\n</span><span class="s">'</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="n">m</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">dir</span><span class="p">(</span><span class="n">Mesh</span><span class="p">)</span> <span class="k">if</span> <span class="n">m</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">:]</span> <span class="o">==</span> <span class="s">'Mesh'</span> <span class="ow">and</span> <span class="s">'Base'</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">m</span><span class="p">])</span> | |
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| <pre> | |
| Warning: mumps solver not available. | |
| Cyl1DMesh | |
| LogicallyOrthogonalMesh | |
| TensorMesh | |
| TreeMesh | |
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| <h3 id="variable-locations-and-terminology">Variable Locations and Terminology</h3> | |
| <p>To create a <em>TensorMesh</em> 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 <em>TensorMesh</em>.</p> | |
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| <div class="highlight"><pre><span class="n">hx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">]</span> | |
| <span class="n">hy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">]</span> | |
| <span class="n">M</span> <span class="o">=</span> <span class="n">Mesh</span><span class="o">.</span><span class="n">TensorMesh</span><span class="p">([</span><span class="n">hx</span><span class="p">,</span> <span class="n">hy</span><span class="p">])</span> | |
| <span class="n">M</span><span class="o">.</span><span class="n">plotGrid</span><span class="p">(</span><span class="n">centers</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> | |
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| n4PYo4ZeAAAAAElFTkSuQmCC | |
| " | |
| > | |
| </div> | |
| </div> | |
| </div> | |
| </div> | |
| </div> | |
| <div class="cell border-box-sizing text_cell rendered"> | |
| <div class="prompt input_prompt"> | |
| </div> | |
| <div class="inner_cell"> | |
| <div class="text_cell_render border-box-sizing rendered_html"> | |
| <p>In this simple mesh, the $h_x$ 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:</p> | |
| <ul> | |
| <li>cell-centers</li> | |
| <li>nodes</li> | |
| <li>faces</li> | |
| <li>edges</li> | |
| </ul> | |
| </div> | |
| </div> | |
| </div> | |
| <div class="cell border-box-sizing code_cell rendered"> | |
| <div class="input"> | |
| <div class="prompt input_prompt"> | |
| In [3]: | |
| </div> | |
| <div class="inner_cell"> | |
| <div class="input_area"> | |
| <div class="highlight"><pre><span class="n">M</span><span class="o">.</span><span class="n">plotGrid</span><span class="p">(</span><span class="n">faces</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">nodes</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> | |
| <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s">'Cell faces in the x- and y-directions.'</span><span class="p">)</span> | |
| <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">((</span><span class="s">'Nodes'</span><span class="p">,</span> <span class="s">'X-Faces'</span><span class="p">,</span> <span class="s">'Y-Faces'</span><span class="p">))</span> | |
| </pre></div> | |
| </div> | |
| </div> | |
| </div> | |
| <div class="output_wrapper"> | |
| <div class="output"> | |
| <div class="output_area"><div class="prompt output_prompt"> | |
| Out[3]:</div> | |
| <div class="output_text output_subarea output_pyout"> | |
| <pre> | |
| <matplotlib.legend.Legend at 0x10f501690> | |
| </pre> | |
| </div> | |
| </div> | |
| <div class="output_area"><div class="prompt"></div> | |
| <div class="output_png output_subarea "> | |
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| <p>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.</p> | |
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| In [4]: | |
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| <div class="highlight"><pre><span class="n">Mesh</span><span class="o">.</span><span class="n">TensorMesh</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">plotGrid</span><span class="p">(</span><span class="n">faces</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">edges</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">centers</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> | |
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| QmCC | |
| " | |
| > | |
| </div> | |
| </div> | |
| </div> | |
| </div> | |
| </div> | |
| <div class="cell border-box-sizing text_cell rendered"> | |
| <div class="prompt input_prompt"> | |
| </div> | |
| <div class="inner_cell"> | |
| <div class="text_cell_render border-box-sizing rendered_html"> | |
| <h3 id="how-many-of-each-">How many of each?</h3> | |
| <p>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:</p> | |
| </div> | |
| </div> | |
| </div> | |
| <div class="cell border-box-sizing code_cell rendered"> | |
| <div class="input"> | |
| <div class="prompt input_prompt"> | |
| In [5]: | |
| </div> | |
| <div class="inner_cell"> | |
| <div class="input_area"> | |
| <div class="highlight"><pre><span class="k">print</span> <span class="n">M</span> | |
| </pre></div> | |
| </div> | |
| </div> | |
| </div> | |
| <div class="output_wrapper"> | |
| <div class="output"> | |
| <div class="output_area"><div class="prompt"></div> | |
| <div class="output_subarea output_stream output_stdout output_text"> | |
| <pre> | |
| ---- 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 | |
| </pre> | |
| </div> | |
| </div> | |
| </div> | |
| </div> | |
| </div> | |
| <div class="cell border-box-sizing code_cell rendered"> | |
| <div class="input"> | |
| <div class="prompt input_prompt"> | |
| In [6]: | |
| </div> | |
| <div class="inner_cell"> | |
| <div class="input_area"> | |
| <div class="highlight"><pre><span class="n">count</span> <span class="o">=</span> <span class="p">{</span><span class="s">'numCells'</span><span class="p">:</span> <span class="n">M</span><span class="o">.</span><span class="n">nC</span><span class="p">,</span> | |
| <span class="s">'numCells_xDir'</span><span class="p">:</span> <span class="n">M</span><span class="o">.</span><span class="n">nCx</span><span class="p">,</span> | |
| <span class="s">'numCells_yDir'</span><span class="p">:</span> <span class="n">M</span><span class="o">.</span><span class="n">nCy</span><span class="p">,</span> | |
| <span class="s">'numCells_vector'</span><span class="p">:</span> <span class="n">M</span><span class="o">.</span><span class="n">vnC</span><span class="p">}</span> | |
| <span class="k">print</span> <span class="s">'This mesh has </span><span class="si">%(numCells)d</span><span class="s"> cells, which is </span><span class="si">%(numCells_xDir)d</span><span class="s">*</span><span class="si">%(numCells_yDir)d</span><span class="s">!!'</span> <span class="o">%</span> <span class="n">count</span> | |
| <span class="k">print</span> <span class="n">count</span> | |
| </pre></div> | |
| </div> | |
| </div> | |
| </div> | |
| <div class="output_wrapper"> | |
| <div class="output"> | |
| <div class="output_area"><div class="prompt"></div> | |
| <div class="output_subarea output_stream output_stdout output_text"> | |
| <pre> | |
| This mesh has 32 cells, which is 8*4!! | |
| {'numCells_vector': array([8, 4]), 'numCells_yDir': 4, 'numCells_xDir': 8, 'numCells': 32} | |
| </pre> | |
| </div> | |
| </div> | |
| </div> | |
| </div> | |
| </div> | |
| <div class="cell border-box-sizing text_cell rendered"> | |
| <div class="prompt input_prompt"> | |
| </div> | |
| <div class="inner_cell"> | |
| <div class="text_cell_render border-box-sizing rendered_html"> | |
| <p>SimPEG also counts the nodes, faces, and edges.</p> | |
| <pre><code>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 | |
| </code></pre><p>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.</p> | |
| </div> | |
| </div> | |
| </div> | |
| <div class="cell border-box-sizing code_cell rendered"> | |
| <div class="input"> | |
| <div class="prompt input_prompt"> | |
| In [7]: | |
| </div> | |
| <div class="inner_cell"> | |
| <div class="input_area"> | |
| <div class="highlight"><pre><span class="n">Mesh</span><span class="o">.</span><span class="n">TensorMesh</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span><span class="o">.</span><span class="n">plotGrid</span><span class="p">(</span><span class="n">faces</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> | |
| <span class="k">print</span> <span class="n">Mesh</span><span class="o">.</span><span class="n">TensorMesh</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span><span class="o">.</span><span class="n">vnFx</span> | |
| <span class="k">print</span> <span class="n">Mesh</span><span class="o">.</span><span class="n">TensorMesh</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span><span class="o">.</span><span class="n">vnFy</span> | |
| </pre></div> | |
| </div> | |
| </div> | |
| </div> | |
| <div class="output_wrapper"> | |
| <div class="output"> | |
| <div class="output_area"><div class="prompt"></div> | |
| <div class="output_subarea output_stream output_stdout output_text"> | |
| <pre> | |
| [5 5] | |
| [4 6] | |
| </pre> | |
| </div> | |
| </div> | |
| <div class="output_area"><div class="prompt"></div> | |
| <div class="output_png output_subarea "> | |
| <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYYAAAEPCAYAAABGP2P1AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz | |
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| " | |
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| <h3 id="some-extra-things-">Some extra things!</h3> | |
| <p>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:</p> | |
| <pre><code>h1 = ( (numPad, sizeStart [, increaseFactor]), (numCore, sizeCode), (numPad, sizeStart [, increaseFactor]) ) | |
| </code></pre> | |
| </div> | |
| </div> | |
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| In [9]: | |
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| <div class="highlight"><pre><span class="n">h1</span> <span class="o">=</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">),</span> <span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> | |
| <span class="n">M</span> <span class="o">=</span> <span class="n">Mesh</span><span class="o">.</span><span class="n">TensorMesh</span><span class="p">(</span><span class="n">Utils</span><span class="o">.</span><span class="n">meshTensors</span><span class="p">(</span><span class="n">h1</span><span class="p">,</span> <span class="n">h1</span><span class="p">))</span> | |
| <span class="k">print</span> <span class="n">M</span> | |
| <span class="n">M</span><span class="o">.</span><span class="n">plotGrid</span><span class="p">()</span> | |
| </pre></div> | |
| </div> | |
| </div> | |
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| <pre> | |
| ---- 2-D TensorMesh ---- | |
| x0: 0.00 | |
| y0: 0.00 | |
| nCx: 28 | |
| nCy: 28 | |
| hx: 50.62, 33.75, 22.50, 15.00, 10.00, 20*5.00, 10.00, 13.00, 16.90 | |
| hy: 50.62, 33.75, 22.50, 15.00, 10.00, 20*5.00, 10.00, 13.00, 16.90 | |
| </pre> | |
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| " | |
| > | |
| </div> | |
| </div> | |
| </div> | |
| </div> | |
| </div> | |
| <div class="cell border-box-sizing text_cell rendered"> | |
| <div class="prompt input_prompt"> | |
| </div> | |
| <div class="inner_cell"> | |
| <div class="text_cell_render border-box-sizing rendered_html"> | |
| <p>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.</p> | |
| </div> | |
| </div> | |
| </div> |
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