different places in the program during refinement cycles
zero through three in 3d is shown here:
-@image html step-22.profile-3.png
+<img src="http://www.dealii.org/images/steps/developer/step-22.profile-3.png" alt="">
As can be seen, at this refinement level approximately three quarters of the
instruction count is spent on the actual solver (the SparseILU::vmult calls
As a final point, and as a point of reference, the following picture also
shows how the profile looked at an early stage of optimizing this program:
-@image html step-22.profile-3.original.png
+<img src="http://www.dealii.org/images/steps/developer/step-22.profile-3.original.png" alt="">
As mentioned above, the runtime of this version was about four times as long
as for the first profile, with the SparseILU decomposition taking up about
velocities and therefore time step sizes) in 2d will produce the
following graph:
-@image html step-32.beta.2d.png
+<img src="http://www.dealii.org/images/steps/developer/step-32.beta.2d.png" alt="">
As can be seen, values $\beta \le 0.05$ are too small whereas
$\beta=0.052$ appears to work, at least to the time horizon shown
as possible in order to make $\nu$ as small as possible. This gives us
a picture like this:
-@image html doc/step-32.beta_cr.2d.png
+<img src="http://www.dealii.org/images/steps/developer/doc/step-32.beta_cr.2d.png" alt="">
Consequently, $c_R=0.1$ would appear to be the right value here. While this
graph has been obtained for an exponent $\alpha=1$, in the program we use
href="http://www.wolframalpha.com/input/?i=plot+%28sqrt%28x^2%2By^2%29%2B0.2*%28sqrt%28x^2%2By^2%29*%281-sqrt%28x^2%2By^2%29%29*sin%286*atan2%28x%2Cy%29%29%29%2C+x%3D-1+to+1%2C+y%3D-1+to+1">this
page</a>):
- @image html step-32.2d-initial.png
+ <img src="http://www.dealii.org/images/steps/developer/step-32.2d-initial.png" alt="">
The point of this profile is that if we had used $s$ instead of $\tau$ in
the definition of $T(\mathbf x)$ then it would simply be a linear
The testcase that we use for this program consists of the flow around a square obstacle. The geometry is
as follows:
-@image html step-35.geometry.png
+<img src="http://www.dealii.org/images/steps/developer/step-35.geometry.png" alt="">
with $H=4.1$, making the geometry slightly non-symmetric.
<table>
<tr>
- <td> @image html step-35.Re_100.vorticity.0.png </td>
- <td> @image html step-35.Re_100.velocity.0.png </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_100.vorticity.0.png" alt=""> </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_100.velocity.0.png" alt=""> </td>
</tr>
<tr>
- <td> @image html step-35.Re_100.vorticity.1.png </td>
- <td> @image html step-35.Re_100.velocity.1.png </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_100.vorticity.1.png" alt=""> </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_100.velocity.1.png" alt=""> </td>
</tr>
<tr>
- <td> @image html step-35.Re_100.vorticity.2.png </td>
- <td> @image html step-35.Re_100.velocity.2.png </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_100.vorticity.2.png" alt=""> </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_100.velocity.2.png" alt=""> </td>
</tr>
<tr>
- <td> @image html step-35.Re_100.vorticity.3.png </td>
- <td> @image html step-35.Re_100.velocity.3.png </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_100.vorticity.3.png" alt=""> </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_100.velocity.3.png" alt=""> </td>
</tr>
<tr>
- <td> @image html step-35.Re_100.vorticity.4.png </td>
- <td> @image html step-35.Re_100.velocity.4.png </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_100.vorticity.4.png" alt=""> </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_100.velocity.4.png" alt=""> </td>
</tr>
</table>
<table>
<tr>
- <td> @image html step-35.Re_500.vorticity.0.png </td>
- <td> @image html step-35.Re_500.velocity.0.png </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_500.vorticity.0.png" alt=""> </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_500.velocity.0.png" alt=""> </td>
</tr>
<tr>
- <td> @image html step-35.Re_500.vorticity.1.png </td>
- <td> @image html step-35.Re_500.velocity.1.png </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_500.vorticity.1.png" alt=""> </td>
+ <td> <img src="http://www.dealii.org/images/steps/developer/step-35.Re_500.velocity.1.png" alt=""> </td>
</tr>
</table>
however, by looking at a zoom at the region behind the obstacle, and
the mesh size we have there:
-@image html step-35.Re_500.zoom.png
+<img src="http://www.dealii.org/images/steps/developer/step-35.Re_500.zoom.png" alt="">
It is clear here that the flow is still reasonably well resolved in
the immediate wake of the obstacle, where the mesh is relatively fine,
A rerun with one more mesh refinement set in the parameter file
confirms that results look much more sensible now:
-@image html step-35.Re_500.zoom_2.png
+<img src="http://www.dealii.org/images/steps/developer/step-35.Re_500.zoom_2.png" alt="">
Here, the zone where the mesh becomes coarser is immediately
downstream of the big red blob at the bottom of the wake behind the
with its functionality. Here's what I have come up with for the 2d solution:
<p align="center">
- @image html step-4.solution-2d.png
+ <img src="http://www.dealii.org/images/steps/developer/step-4.solution-2d.png" alt="">
</p>
The picture shows the solution of the problem under consideration as
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- @image html step-4.solution-3d.png
+ <img src="http://www.dealii.org/images/steps/developer/step-4.solution-3d.png" alt="">
</td>
<td ALIGN="center">
- @image html step-4.contours-3d.png
+ <img src="http://www.dealii.org/images/steps/developer/step-4.contours-3d.png" alt="">
</td>
</tr>
</table>
<table align="center" class="tutorial" cellspacing="3" cellpadding="3">
<tr>
<td align="center">
- @image html "step-41.displacement.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.png" alt="">
</td>
<td align="center">
- @image html "step-41.active-set.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.png" alt="">
</td>
</tr>
</table>
0
</td>
<td valign="top">
- @image html step-41.displacement.00.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.00.png" alt="">
</td>
<td valign="top">
- @image html step-41.active-set.00.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.00.png" alt="">
</td>
<td valign="top">
- @image html step-41.displacement.3d.00.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.00.png" alt="">
</td>
</tr>
3
</td>
<td valign="top">
- @image html step-41.displacement.03.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.03.png" alt="">
</td>
<td valign="top">
- @image html step-41.active-set.03.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.03.png" alt="">
</td>
<td valign="top">
- @image html step-41.displacement.3d.03.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.03.png" alt="">
</td>
</tr>
6
</td>
<td valign="top">
- @image html step-41.displacement.06.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.06.png" alt="">
</td>
<td valign="top">
- @image html step-41.active-set.06.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.06.png" alt="">
</td>
<td valign="top">
- @image html step-41.displacement.3d.06.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.06.png" alt="">
</td>
</tr>
9
</td>
<td valign="top">
- @image html step-41.displacement.09.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.09.png" alt="">
</td>
<td valign="top">
- @image html step-41.active-set.09.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.09.png" alt="">
</td>
<td valign="top">
- @image html step-41.displacement.3d.09.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.09.png" alt="">
</td>
</tr>
12
</td>
<td valign="top">
- @image html step-41.displacement.12.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.12.png" alt="">
</td>
<td valign="top">
- @image html step-41.active-set.12.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.12.png" alt="">
</td>
<td valign="top">
- @image html step-41.displacement.3d.12.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.12.png" alt="">
</td>
</tr>
15
</td>
<td valign="top">
- @image html step-41.displacement.15.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.15.png" alt="">
</td>
<td valign="top">
- @image html step-41.active-set.15.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.15.png" alt="">
</td>
<td valign="top">
- @image html step-41.displacement.3d.15.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.15.png" alt="">
</td>
</tr>
18
</td>
<td valign="top">
- @image html step-41.displacement.18.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.18.png" alt="">
</td>
<td valign="top">
- @image html step-41.active-set.18.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.18.png" alt="">
</td>
<td valign="top">
- @image html step-41.displacement.3d.18.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.18.png" alt="">
</td>
</tr>
</table>
<table align="center">
<tr>
<td valign="top">
- @image html step-41.forces.01.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.forces.01.png" alt="">
</td>
<td valign="top">
- @image html step-41.forces.09.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.forces.09.png" alt="">
</td>
<td valign="top">
- @image html step-41.forces.18.png
+ <img src="http://www.dealii.org/images/steps/developer/step-41.forces.18.png" alt="">
</td>
</tr>
<table align="center" class="tutorial" cellspacing="3" cellpadding="3">
<tr>
<td align="center">
- @image html "step-43.3d.velocity.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-43.3d.velocity.png" alt="">
<p align="center">
Velocity vectors of flow through the porous medium with random
permeability model. Streaming paths of high permeability and resulting
</p>
</td>
<td align="center">
- @image html "step-43.3d.streamlines.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-43.3d.streamlines.png" alt="">
<p align="center">
Streamlines colored by the saturation along the streamline path. Blue
streamlines indicate low saturations, i.e., the flow along these
<tr>
<td align="center">
- @image html "step-43.3d.saturation.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-43.3d.saturation.png" alt="">
<p align="center">
Streamlines with a volume rendering of the saturation, showing how far
the fluid front has advanced at this time.
</p>
</td>
<td align="center">
- @image html "step-43.3d.mesh.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-43.3d.mesh.png" alt="">
<p align="center">
Surface of the mesh showing the adaptive refinement along the front.
</p>
<strong> 75 </strong>,
291-304.
- @image html "step-44.setup.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-44.setup.png" alt="">
The material is quasi-incompressible neo-Hookean with <a href="http://en.wikipedia.org/wiki/Shear_modulus">shear modulus</a> $\mu = 80.194e6$ and $\nu = 0.4999$.
For such a choice of material properties a conventional single-field $Q_1$ approach would lock.
<table align="center" class="tutorial" cellspacing="3" cellpadding="3">
<tr>
<td align="center">
- @image html "step-44.Q1-P0_convergence.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-44.Q1-P0_convergence.png" alt="">
<p align="center">
Convergence of the $Q_1-DGPM_0-DGPM_0$ formulation.
</p>
</td>
<td align="center">
- @image html "step-44.Q2-P1_convergence.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-44.Q2-P1_convergence.png" alt="">
<p align="center">
Convergence of the $Q_2-DGPM_1-DGPM_1$ formulation.
</p>
<table align="center" class="tutorial" cellspacing="3" cellpadding="3">
<tr>
<td align="center">
- @image html "step-44.Q1-P0_gr_1_p_ratio_80-displacement.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-44.Q1-P0_gr_1_p_ratio_80-displacement.png" alt="">
<p align="center">
Z-displacement solution.
</p>
</td>
<td align="center">
- @image html "step-44.Q1-P0_gr_1_p_ratio_80-pressure.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-44.Q1-P0_gr_1_p_ratio_80-pressure.png" alt="">
<p align="center">
Discontinuous piece-wise constant pressure field.
</p>
</td>
<td align="center">
- @image html "step-44.Q1-P0_gr_1_p_ratio_80-dilatation.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-44.Q1-P0_gr_1_p_ratio_80-dilatation.png" alt="">
<p align="center">
Discontinuous piece-wise constant dilatation field.
</p>
<tr>
<td align="center">
- @image html "step-44.Q2-P1_gr_3_p_ratio_80-displacement.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-44.Q2-P1_gr_3_p_ratio_80-displacement.png" alt="">
<p align="center">
Z-displacement solution.
</p>
</td>
<td align="center">
- @image html "step-44.Q2-P1_gr_3_p_ratio_80-pressure.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-44.Q2-P1_gr_3_p_ratio_80-pressure.png" alt="">
<p align="center">
Discontinuous linear pressure field.
</p>
</td>
<td align="center">
- @image html "step-44.Q2-P1_gr_3_p_ratio_80-dilatation.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-44.Q2-P1_gr_3_p_ratio_80-dilatation.png" alt="">
<p align="center">
Discontinuous linear dilatation field.
</p>
<table align="center" class="tutorial" cellspacing="3" cellpadding="3">
<tr>
<td align="center">
- @image html "step-44.Normalised_runtime.png"
+ <img src="http://www.dealii.org/images/steps/developer/step-44.Normalised_runtime.png" alt="">
<p align="center">
Runtime on a 4-core machine, normalised against the lowest grid resolution $Q_1-DGPM_0-DGPM_0$ solution that utilised a SSOR preconditioner.
</p>
<TABLE WIDTH="60%" ALIGN="center">
<tr valign="top">
<td valign="top" align="center">
- @image html step-46.velocity-magnitude.png
+ <img src="http://www.dealii.org/images/steps/developer/step-46.velocity-magnitude.png" alt="">
<p align="center">
Magnitude of the fluid velocity.
</td>
<td valign="top" align="center">
- @image html step-46.pressure.png
+ <img src="http://www.dealii.org/images/steps/developer/step-46.pressure.png" alt="">
<p align="center">
Fluid pressure. The dynamic range has been truncated to cut off the
</tr>
<tr valign="top">
<td valign="top" align="center">
- @image html step-46.velocity.png
+ <img src="http://www.dealii.org/images/steps/developer/step-46.velocity.png" alt="">
<p align="center">
Fluid velocity.
</td>
<td valign="top" align="center">
- @image html step-46.displacement.png
+ <img src="http://www.dealii.org/images/steps/developer/step-46.displacement.png" alt="">
<p align="center">
Solid displacement.
<TABLE WIDTH="60%" ALIGN="center">
<tr valign="top">
<td valign="top" align="center">
- @image html step-46.3d.velocity.png
+ <img src="http://www.dealii.org/images/steps/developer/step-46.3d.velocity.png" alt="">
<p align="center">
Vectors of the fluid velocity and magnitude of the
</td>
<td valign="top" align="center">
- @image html step-46.3d.streamlines.png
+ <img src="http://www.dealii.org/images/steps/developer/step-46.3d.streamlines.png" alt="">
<p align="center">
Streamlines of the velocity, with the mesh superimposed.
</tr>
<tr valign="top">
<td valign="top" align="center" colspan="2">
- @image html step-46.3d.displacement.png
+ <img src="http://www.dealii.org/images/steps/developer/step-46.3d.displacement.png" alt="">
<p align="center">
Solid displacement.
<TABLE WIDTH="100%">
<tr>
<td>
- @image html step-5.solution-0.png
+ <img src="http://www.dealii.org/images/steps/developer/step-5.solution-0.png" alt="">
</td>
<td>
- @image html step-5.solution-1.png
+ <img src="http://www.dealii.org/images/steps/developer/step-5.solution-1.png" alt="">
</td>
</tr>
<tr>
<td>
- @image html step-5.solution-2.png
+ <img src="http://www.dealii.org/images/steps/developer/step-5.solution-2.png" alt="">
</td>
<td>
- @image html step-5.solution-3.png
+ <img src="http://www.dealii.org/images/steps/developer/step-5.solution-3.png" alt="">
</td>
</tr>
<tr>
<td>
- @image html step-5.solution-4.png
+ <img src="http://www.dealii.org/images/steps/developer/step-5.solution-4.png" alt="">
</td>
<td>
- @image html step-5.solution-5.png
+ <img src="http://www.dealii.org/images/steps/developer/step-5.solution-5.png" alt="">
</td>
</tr>
</table>
-@image html step-6.solution.png
+<img src="http://www.dealii.org/images/steps/developer/step-6.solution.png" alt="">
<TABLE WIDTH="100%">
<tr>
<td>
- @image html step-6.grid-0.png
+ <img src="http://www.dealii.org/images/steps/developer/step-6.grid-0.png" alt="">
</td>
<td>
- @image html step-6.grid-1.png
+ <img src="http://www.dealii.org/images/steps/developer/step-6.grid-1.png" alt="">
</td>
</tr>
<tr>
<td>
- @image html step-6.grid-2.png
+ <img src="http://www.dealii.org/images/steps/developer/step-6.grid-2.png" alt="">
</td>
<td>
- @image html step-6.grid-3.png
+ <img src="http://www.dealii.org/images/steps/developer/step-6.grid-3.png" alt="">
</td>
</tr>
<tr>
<td>
- @image html step-6.grid-4.png
+ <img src="http://www.dealii.org/images/steps/developer/step-6.grid-4.png" alt="">
</td>
<td>
- @image html step-6.grid-5.png
+ <img src="http://www.dealii.org/images/steps/developer/step-6.grid-5.png" alt="">
</td>
</tr>
<tr>
<td>
- @image html step-6.grid-6.png
+ <img src="http://www.dealii.org/images/steps/developer/step-6.grid-6.png" alt="">
</td>
<td>
- @image html step-6.grid-7.png
+ <img src="http://www.dealii.org/images/steps/developer/step-6.grid-7.png" alt="">
</td>
</tr>
</table>
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- @image html step-6.q2.dofs_vs_iterations.png
+ <img src="http://www.dealii.org/images/steps/developer/step-6.q2.dofs_vs_iterations.png" alt="">
</td>
<td ALIGN="center">
- @image html step-6.q2.dofs_vs_time.png
+ <img src="http://www.dealii.org/images/steps/developer/step-6.q2.dofs_vs_time.png" alt="">
</td>
</tr>
</table>
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- @image html step-6.q1.dofs_vs_iterations.png
+ <img src="http://www.dealii.org/images/steps/developer/step-6.q1.dofs_vs_iterations.png" alt="">
</td>
<td ALIGN="center">
- @image html step-6.q1.dofs_vs_time.png
+ <img src="http://www.dealii.org/images/steps/developer/step-6.q1.dofs_vs_time.png" alt="">
</td>
</tr>
</table>