]> https://gitweb.dealii.org/ - dealii-svn.git/commitdiff
Update with images.
authorbangerth <bangerth@0785d39b-7218-0410-832d-ea1e28bc413d>
Tue, 8 Sep 2009 20:51:55 +0000 (20:51 +0000)
committerbangerth <bangerth@0785d39b-7218-0410-832d-ea1e28bc413d>
Tue, 8 Sep 2009 20:51:55 +0000 (20:51 +0000)
git-svn-id: https://svn.dealii.org/trunk@19419 0785d39b-7218-0410-832d-ea1e28bc413d

27 files changed:
deal.II/examples/step-32/doc/intro.dox
deal.II/examples/step-32/doc/results.dox
deal.II/examples/step-32/doc/step-32.2d.grid.2100.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.partition.2100.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.t_vs_vmax.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.0000.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.0100.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.0200.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.0300.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.0400.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.0500.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.0600.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.0700.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.0800.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.0900.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.1000.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.1100.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.1200.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.1300.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.1400.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.1500.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.1600.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.1700.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.1800.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.1900.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.2000.png [new file with mode: 0644]
deal.II/examples/step-32/doc/step-32.2d.temperature.2100.png [new file with mode: 0644]

index b878c070d3d2a5e649edccc7480d328a9e9b6bfb..e85a7676143961c8f0ad981da6d41fe42322e7c5 100644 (file)
@@ -116,7 +116,7 @@ non-sensical, nothing bad happened. On the other hand, as we will explain
 below, things here will not be that simply scaled: $\eta$ will be around
 $10^{21}$, velocities on the order of $10^{-8}$, pressure around $10^8$, and
 the diameter of the domain is $10^7$. In other words, the order of magnitude
-for the first equation is going to be 
+for the first equation is going to be
 $\eta\text{div}\varepsilon(\mathbf u) \approx 10^{21} \frac{10^{-8}}{(10^7)^2}
 \approx 10^{-1}$, whereas the second equation will be around
 $\text{div}{\mathbf u}\approx \frac{10^{-8}}{10^7} \approx 10^{-15}$. Well, so
@@ -128,7 +128,7 @@ extremely small values, the resulting flow field is definitely not divergence
 free. As an auxiliary problem, it turns out that it is difficult to find a
 tolerance that always works; in practice, one often ends up with a tolerance
 that requires 30 or 40 iterations for most time steps, and 10,000 for some
-others. 
+others.
 
 So what's a numerical analyst to do in a case like this? The answer is to
 start at the root and first make sure that everything is mathematically
@@ -151,7 +151,7 @@ $\frac{\eta}{L} \nabla \cdot$ at the bottom left, but not its transpose
 operator at the top right). This, however, can be cured by introducing a
 scaled pressure $\hat p = \frac{L}{\eta}p$, and we get the scaled equations
 @f{eqnarray*}
-  -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + 
+  -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) +
   \nabla \left(\frac{\eta}{L} \hat p\right) &=&
   -\rho \; \beta \; T \mathbf{g},
   \\
@@ -228,7 +228,7 @@ As a final note, let us remark that in @ref step_31 "step-31" we computed the
 Schur complement $S=B A^{-1} B^T$ by approximating
 $-\text{div}(-\eta\Delta)^{-1}\nabla \approx \frac 1{\eta} \mathbf{1}$. Now,
 however, we have re-scaled the $B$ and $B^T$ operators. So $S$ should now
-approximate 
+approximate
 $-\frac{\eta}{L}\text{div}(-\eta\Delta)^{-1}\nabla \frac{\eta}{L} \approx
 \left(\frac{\eta}{L}\right)^2 \frac 1{\eta} \mathbf{1}$.
 This is exactly the operator we use to approximate $S$.
@@ -563,7 +563,7 @@ SI system) even if we list them here in other units.
 
 As a reminder, let us again state the equations we want to solve are these:
 @f{eqnarray*}
-  -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + 
+  -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) +
   \nabla \left( \frac{\eta}{L} \hat p\right) &=&
   -\rho \; \beta \; T \mathbf{g},
   \\
@@ -814,7 +814,7 @@ the following quantities:
 
 All of these pieces of equation data are defined in the program in the
 <code>EquationData</code> namespace. When run, the program produces
-maximal velocities around 10 centimeters per year, approximately the
-physically correct order of magnitude. We will set the end time to 30
-million years.
+long-term maximal velocities around 10-40 centimeters per year (see
+the results section below), approximately the physically correct order
+of magnitude. We will set the end time to 1 billion years.
 
index 3457c694e11e6c03a901cff74ffc44bbe1179f1b..d0d8d0726dfe1583d218daeded3810d5d94ead67 100644 (file)
@@ -1,7 +1,7 @@
 <h1>Results</h1>
 
 When run, the program simulates convection in 3d in much the same way
-as @ref step_31 "step-31" did, though with an entirely different testcase. 
+as @ref step_31 "step-31" did, though with an entirely different testcase.
 
 
 <h3>Comparison of results with step-31</h3>
@@ -41,7 +41,7 @@ cluster at Texas A&amp;M University.
 
 If we run the program as shown above, the output will look roughly like this,
 producing the final part of the output after some 2 days when run on 10
-processors: 
+processors:
 
 <code>
 <pre>
@@ -125,8 +125,8 @@ Timestep 53906:  t=9.99985e+08 years
 </code>
 
 As can be seen here, we spend most of the compute time in assembling linear
-systems, refining the mesh, and in particular in solving the Stokes and 
-temperature linear systems. 
+systems, refining the mesh, and in particular in solving the Stokes and
+temperature linear systems.
 
 The 50% spent on solving the linear systems are affected in large part
 because the Brazos cluster has a relatively slow ethernet interconnect. A
@@ -136,3 +136,155 @@ with assembling the linear systems, though one could do significantly better
 with estimating the error by making sure that each processor only estimates
 the error on those cells it owns.
 
+
+The program writes output every 25th time step, but we won't show all
+2100 or so images this produces. Rather, let us only show the output
+from every 2500th time step here, even though this does, of course,
+not do full justice to the dynamics that are going on:
+<table>
+  <tr>
+    <td>
+      @image html step-32.2d.temperature.0000.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.0100.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.0200.png
+    <td>
+  </tr>
+
+  <tr>
+    <td>
+      @image html step-32.2d.temperature.0300.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.0400.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.0500.png
+    <td>
+  </tr>
+
+  <tr>
+    <td>
+      @image html step-32.2d.temperature.0600.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.0700.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.0800.png
+    <td>
+  </tr>
+
+  <tr>
+    <td>
+      @image html step-32.2d.temperature.0900.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.1000.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.1100.png
+    <td>
+  </tr>
+
+  <tr>
+    <td>
+      @image html step-32.2d.temperature.1200.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.1300.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.1400.png
+    <td>
+  </tr>
+
+  <tr>
+    <td>
+      @image html step-32.2d.temperature.1500.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.1600.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.1700.png
+    <td>
+  </tr>
+
+  <tr>
+    <td>
+      @image html step-32.2d.temperature.1800.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.1900.png
+    <td>
+
+    <td>
+      @image html step-32.2d.temperature.2000.png
+    <td>
+  </tr>
+
+
+  <tr>
+    <td>
+      @image html step-32.2d.temperature.2100.png
+    <td>
+
+    <td>
+      @image html step-32.2d.grid.2100.png
+    <td>
+
+    <td>
+      @image html step-32.2d.partition.2100.png
+    <td>
+  </tr>
+</table>
+
+The last two images show the grid as well as the partitioning of the
+mesh for the last timestep shown into the 10 subdomains used for this
+computation. The full dynamics are really only visible by looking at
+an animation. <a
+href="http:/www.math.tamu.edu/~bangerth/images/pictures/convection-outward/step-32.2d.convection.gif">At
+this site</a> is such an animation. Beware that this animation is
+about 20MB large, though it is well worth watching due to its almost
+artistic quality.
+
+If you watch the movie, you'll see that the convection pattern goes
+through several stages: First, it gets rid of the instable temperature
+layering with the hot material overlaid by the dense cold
+material. After this great driver is removed and we have a sort of
+stable situation, a few blobs start to separate from the hot boundary
+layer at the inner ring and rise up, with a few cold fingers also
+dropping down from the outer ring. During this phase, the solution
+remains mostly symmetric, reflecting the 12-fold symmetry of the
+original mesh. In a final phase, the fluid enters vigorous chaotic
+stirring in which all symmetries are lost. This is a pattern that then
+continues to dominate flow.
+
+These different phases can also be identified if we look at the
+maximal velocity as a function of time in the simulation:
+
+@image html step-32.2d.t_vs_vmax.png
+
+Here, the velocity (shown in centimeters per year) becomes very large,
+to the order of several meters per year) at the beginning when the
+temperature layering is instable. It then calms down to relatively
+small values before picking up again in the chaotic stirring
+regime. There, it remains in the range of 10-40 centimeters per year,
+quite within the physically expected region.
diff --git a/deal.II/examples/step-32/doc/step-32.2d.grid.2100.png b/deal.II/examples/step-32/doc/step-32.2d.grid.2100.png
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