Merge pull request #19 from pyroteus/combine-metrics-demo

Update website to add metric combination demo

_modules/animate/metric.html

@@ -85,8 +85,8 @@ <h1>Source code for animate.metric</h1><div class="highlight"><pre>
 <span class="sd">        &quot;&quot;&quot;</span>
         <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">function_space</span><span class="p">,</span> <span class="n">fmesh</span><span class="o">.</span><span class="n">MeshGeometry</span><span class="p">):</span>
             <span class="n">function_space</span> <span class="o">=</span> <span class="n">ffs</span><span class="o">.</span><span class="n">TensorFunctionSpace</span><span class="p">(</span><span class="n">function_space</span><span class="p">,</span> <span class="s2">&quot;CG&quot;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">function_space</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
         <span class="bp">self</span><span class="o">.</span><span class="n">metric_parameters</span> <span class="o">=</span> <span class="p">{}</span>
+        <span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">function_space</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
 
         <span class="c1"># Check that we have an appropriate tensor P1 function</span>
         <span class="n">fs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">function_space</span><span class="p">()</span>
@@ -997,7 +997,7 @@ <h3>Navigation</h3>
       </ul>
     </div>
     <div class="footer" role="contentinfo">
-    &#169; Copyright 2021-2023, Joseph G. Wallwork et al..
+    &#169; Copyright 2021-2024, Joseph G. Wallwork et al..
       Created using <a href="https://www.sphinx-doc.org/">Sphinx</a> 7.2.6.
     </div>
   </body>

_modules/index.html

@@ -88,7 +88,7 @@ <h3>Navigation</h3>
       </ul>
     </div>
     <div class="footer" role="contentinfo">
-    &#169; Copyright 2021-2023, Joseph G. Wallwork et al..
+    &#169; Copyright 2021-2024, Joseph G. Wallwork et al..
       Created using <a href="https://www.sphinx-doc.org/">Sphinx</a> 7.2.6.
     </div>
   </body>

_sources/demos/combining_metrics.py.rst.txt

@@ -0,0 +1,127 @@
+Intersecting and averaging metric fields
+===========================================
+As we saw in the `previous demo <./simple_metric.py.html>`__, the metric
+tells the mesh adaptivity library what the desired edge lengths is in
+different directions, in different parts of the domain.  As we will see these
+desired edge lengths are often based on an estimate of the discretisation
+error we want to minimize for a certain solution field.  Therefore, if in a
+model we are solving for multiple fields, we may end up with multiple metric
+fields telling us about the desired edge lengths for the discretisation of
+each of these fields individually. If we want the model to employ a single
+optimal mesh for the discretisation of all of these fields however, we need a
+way to combine the metrics taking into account the different requirements for
+the different solution fields. The same functionality is needed if in a
+time-dependent model we have multiple metrics based on the solution at
+different timesteps, and we want to adapt the mesh optimally for all of these
+timesteps combined.
+
+A natural way to combine two metrics for mesh adaptation, is the so called
+intersection method. A natural, geometric interpretation exists if we
+represent the individual metrics by ellipsoids, which indicate the desired
+edge length in different directions through their principle axes. The
+intersection of two metrics produces a metric whose associated ellipsoid is
+the largest ellipsoid that can fit within the ellipsoids of the two metrics:
+
+.. figure:: combining_ellipse_intersection.jpg
+   :figwidth: 70%
+   :align: center
+
+
+Below we first set up two metrics: Metric 1 asks for a medium resolution of
+:math:`hm=0.025` in the left, and a coarse resolution of :math:`hc=0.1` in
+the right half of the domain. Metric 2 prescribes the coarse resolution for
+:math:`y<0.3` and :math:`y>0.7`, but asks for an anisotropic resolution,
+:math:`hc` in the :math:`x`-direction and :math:`hf=0.01` in the
+:math:`y`-direction, for :math:`0.3<y<0.7`
+
+::
+
+  import matplotlib.pyplot as plt
+  from firedrake import *
+  from animate import *
+
+  mesh = UnitSquareMesh(100, 100)
+  P1_ten = TensorFunctionSpace(mesh, "CG", 1)
+  metric1 = RiemannianMetric(P1_ten)
+  metric2 = RiemannianMetric(P1_ten)
+  x, y = SpatialCoordinate(mesh)
+  r = 0.2
+  hf = 0.01
+  hm = 0.025
+  hc = 0.1
+  anisotropic = as_matrix([[1 / hc**2, 0], [0, 1 / hf**2]])
+  medium = as_matrix([[1 / hm**2, 0], [0, 1 / hm**2]])
+  coarse = as_matrix([[1 / hc**2, 0], [0, 1 / hc**2]])
+  metric1.interpolate(conditional(x < 0.5, medium, coarse))
+  metric2.interpolate(
+      conditional(And(abs(x - 0.5) < r, abs(y - 0.5) < r), anisotropic, coarse)
+  )
+  metric2.interpolate(conditional(abs(y - 0.5) < r, anisotropic, coarse))
+  mesh1 = adapt(mesh, metric1)
+  mesh2 = adapt(mesh, metric2)
+  fig, axes = plt.subplots(figsize=(16, 8), ncols=2)
+  triplot(mesh1, axes=axes[0])
+  axes[0].set_aspect("equal")
+  axes[0].set_title("Mesh based on metric1")
+  triplot(mesh2, axes=axes[1])
+  axes[1].set_aspect("equal")
+  axes[1].set_title("Mesh based on metric2")
+  fig.show()
+  fig.savefig("combining_two_metrics.jpg")
+
+
+.. figure:: combining_two_metrics.jpg
+   :figwidth: 90%
+   :align: center
+
+We can intersect multiple metric by using the
+:meth:`~.RiemannianMetric.intersect()` method of a
+:class:`~.RiemannianMetric`, which intersects the given metric with another,
+or multiple, provided metrics. We therefore first make a copy of metric1 and
+then intersect it with metric2:
+
+::
+
+  intersected_metric = metric1.copy(deepcopy=True)
+  intersected_metric.intersect(metric2)
+  mesh_intersected = adapt(mesh, intersected_metric)
+  fig, axes = plt.subplots(figsize=(16, 16))
+  triplot(mesh_intersected, axes=axes)
+  axes.set_aspect("equal")
+  axes.set_title("Mesh based on intersected metric")
+  fig.show()
+  fig.savefig("combining_intersection.jpg")
+
+
+.. figure:: combining_intersection.jpg
+   :figwidth: 90%
+   :align: center
+
+As we can observe, in every region the resolution respects the minimum
+resolution in all directions asked for by both metrics. For example, in the
+region :math:`x<0.5`, :math:`0.3<y<0.7` the resolution in the :math:`x`-direction is :math:`hf=0.01`, as
+required by metric2, but in the :math:`y`-direction the resolution is :math:`hm=0.02` as
+required by metric1.
+
+If instead we average the metrics using the
+:meth:`~.RiemannianMetric.average()` method,
+
+::
+
+  averaged_metric = metric1.copy(deepcopy=True)
+  averaged_metric.average(metric2)
+  mesh_averaged = adapt(mesh, averaged_metric)
+  fig, axes = plt.subplots(figsize=(16, 16))
+  triplot(mesh_averaged, axes=axes)
+  axes.set_aspect("equal")
+  axes.set_title("Mesh based on averaged metric")
+  fig.show()
+  fig.savefig("combining_averaging.jpg")
+
+
+.. figure:: combining_averaging.jpg
+   :figwidth: 90%
+   :align: center
+
+the resolution in, e.g., the region :math:`x<0.5, y<0.3` is now based on an
+average of :math:`hm=0.02` and :math:`hc=0.1`, i.e. an edge length of 0.06.

_sources/index.rst.txt

@@ -74,6 +74,7 @@ source code for both packages is hosted on GitHub:
     :maxdepth: 1
 
     Defining a simple metric and adapting a mesh with respect to it <demos/simple_metric.py>
+    Combining metrics <demos/combining_metrics.py>
 
 .. rubric:: Goalie demos
 

_static/nature.css

@@ -13,7 +13,6 @@
 
 /* -- page layout ----------------------------------------------------------- */
 
-
 body {
     font-family: Arial, sans-serif;
     font-size: 100%;
@@ -29,7 +28,7 @@ div.documentwrapper {
 }
 
 div.bodywrapper {
-    margin: 0 0 0 400px; /*acse-ej321 : custom for sidebar*/
+    margin: 0 0 0 230px;
 }
 
 hr {
@@ -75,13 +74,10 @@ div.related a {
 div.sphinxsidebar {
     font-size: 0.75em;
     line-height: 1.5em;
-    width: 400px; /*acse-ej321 : custom for sidebar*/
 }
 
 div.sphinxsidebarwrapper{
     padding: 20px 0;
-    overflow: auto; /*acse-ej321 : custom for sidebar*/
-    width: 400px; /*acse-ej321 : custom for sidebar*/
 }
 
 div.sphinxsidebar h3,

animate.html

@@ -802,7 +802,7 @@ <h3>Navigation</h3>
       </ul>
     </div>
     <div class="footer" role="contentinfo">
-    &#169; Copyright 2021-2023, Joseph G. Wallwork et al..
+    &#169; Copyright 2021-2024, Joseph G. Wallwork et al..
       Created using <a href="https://www.sphinx-doc.org/">Sphinx</a> 7.2.6.
     </div>
   </body>

animate/1-metric-based.html

@@ -438,7 +438,7 @@ <h3>Navigation</h3>
       </ul>
     </div>
     <div class="footer" role="contentinfo">
-    &#169; Copyright 2021-2023, Joseph G. Wallwork et al..
+    &#169; Copyright 2021-2024, Joseph G. Wallwork et al..
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   </body>

animate/index.html

@@ -116,7 +116,7 @@ <h3>Navigation</h3>
       </ul>
     </div>
     <div class="footer" role="contentinfo">
-    &#169; Copyright 2021-2023, Joseph G. Wallwork et al..
+    &#169; Copyright 2021-2024, Joseph G. Wallwork et al..
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demos/burgers-hessian.py.html

@@ -321,7 +321,7 @@ <h3>Navigation</h3>
       </ul>
     </div>
     <div class="footer" role="contentinfo">
-    &#169; Copyright 2021-2023, Joseph G. Wallwork et al..
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demos/burgers.py.html

@@ -289,7 +289,7 @@ <h3>Navigation</h3>
       </ul>
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     <div class="footer" role="contentinfo">
-    &#169; Copyright 2021-2023, Joseph G. Wallwork et al..
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demos/burgers1.py.html

@@ -255,7 +255,7 @@ <h3>Navigation</h3>
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     <div class="footer" role="contentinfo">
-    &#169; Copyright 2021-2023, Joseph G. Wallwork et al..
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demos/burgers2.py.html

@@ -250,7 +250,7 @@ <h3>Navigation</h3>
       </ul>
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     <div class="footer" role="contentinfo">
-    &#169; Copyright 2021-2023, Joseph G. Wallwork et al..
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demos/burgers_ee.py.html

@@ -262,7 +262,7 @@ <h3>Navigation</h3>
       </ul>
     </div>
     <div class="footer" role="contentinfo">
-    &#169; Copyright 2021-2023, Joseph G. Wallwork et al..
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demos/burgers_oo.py.html

@@ -239,7 +239,7 @@ <h3>Navigation</h3>
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     <div class="footer" role="contentinfo">
-    &#169; Copyright 2021-2023, Joseph G. Wallwork et al..
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demos/burgers_time_integrated.py.html

@@ -245,7 +245,7 @@ <h3>Navigation</h3>
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