From: Wolfgang Bangerth Date: Tue, 8 Sep 2009 20:51:55 +0000 (+0000) Subject: Update with images. X-Git-Tag: v8.0.0~7154 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=273f8efd522000a61b9c5b13e14044d8b85fba39;p=dealii.git Update with images. git-svn-id: https://svn.dealii.org/trunk@19419 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-32/doc/intro.dox b/deal.II/examples/step-32/doc/intro.dox index b878c070d3..e85a767614 100644 --- a/deal.II/examples/step-32/doc/intro.dox +++ b/deal.II/examples/step-32/doc/intro.dox @@ -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 EquationData 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. diff --git a/deal.II/examples/step-32/doc/results.dox b/deal.II/examples/step-32/doc/results.dox index 3457c694e1..d0d8d0726d 100644 --- a/deal.II/examples/step-32/doc/results.dox +++ b/deal.II/examples/step-32/doc/results.dox @@ -1,7 +1,7 @@

Results

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.

Comparison of results with step-31

@@ -41,7 +41,7 @@ cluster at Texas A&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:
@@ -125,8 +125,8 @@ Timestep 53906:  t=9.99985e+08 years
 
 
 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:
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+ @image html step-32.2d.temperature.0000.png + + + + @image html step-32.2d.temperature.0100.png + + + + @image html step-32.2d.temperature.0200.png + +
+ @image html step-32.2d.temperature.0300.png + + + + @image html step-32.2d.temperature.0400.png + + + + @image html step-32.2d.temperature.0500.png + +
+ @image html step-32.2d.temperature.0600.png + + + + @image html step-32.2d.temperature.0700.png + + + + @image html step-32.2d.temperature.0800.png + +
+ @image html step-32.2d.temperature.0900.png + + + + @image html step-32.2d.temperature.1000.png + + + + @image html step-32.2d.temperature.1100.png + +
+ @image html step-32.2d.temperature.1200.png + + + + @image html step-32.2d.temperature.1300.png + + + + @image html step-32.2d.temperature.1400.png + +
+ @image html step-32.2d.temperature.1500.png + + + + @image html step-32.2d.temperature.1600.png + + + + @image html step-32.2d.temperature.1700.png + +
+ @image html step-32.2d.temperature.1800.png + + + + @image html step-32.2d.temperature.1900.png + + + + @image html step-32.2d.temperature.2000.png + +
+ @image html step-32.2d.temperature.2100.png + + + + @image html step-32.2d.grid.2100.png + + + + @image html step-32.2d.partition.2100.png + +
+ +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. At +this site 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.