<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-1.grid-1.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-1.grid-1.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-1.grid-2r2.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-1.grid-2r2.png" alt="">
</td>
</tr>
</table>
<table style="width:80%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-10.ball_mapping_q1_ref0.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-10.ball_mapping_q1_ref1.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-10.ball_mapping_q1_ref0.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-10.ball_mapping_q1_ref1.png" alt=""></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-10.ball_mapping_q2_ref0.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-10.ball_mapping_q2_ref1.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-10.ball_mapping_q2_ref0.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-10.ball_mapping_q2_ref1.png" alt=""></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-10.ball_mapping_q3_ref0.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-10.ball_mapping_q3_ref1.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-10.ball_mapping_q3_ref0.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-10.ball_mapping_q3_ref1.png" alt=""></td>
</tr>
</table>
<table style="width:80%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-10.quarter-q1.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-10.quarter-q2.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-10.quarter-q3.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-10.quarter-q1.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-10.quarter-q2.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-10.quarter-q3.png" alt=""></td>
</tr>
</table>
We show the solutions on the initial mesh, the mesh after two
and after five adaptive refinement steps.
-<img src="http://www.dealii.org/images/steps/developer/step-12.sol-0.png" alt="">
-<img src="http://www.dealii.org/images/steps/developer/step-12.sol-2.png" alt="">
-<img src="http://www.dealii.org/images/steps/developer/step-12.sol-5.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-12.sol-0.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-12.sol-2.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-12.sol-5.png" alt="">
Then we show the final grid (after 5 refinement steps) and the solution again,
sharpness of the jump on the refined mesh and the over- and undershoots of the
solution along the interface:
-<img src="http://www.dealii.org/images/steps/developer/step-12.grid-5.png" alt="">
-<img src="http://www.dealii.org/images/steps/developer/step-12.3d-solution.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-12.grid-5.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-12.3d-solution.png" alt="">
And finally we show a plot of a 3d computation.
-<img src="http://www.dealii.org/images/steps/developer/step-12.sol-5-3d.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-12.sol-5-3d.png" alt="">
<a name="dg-vs-cg"></a>
0
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-12.cg.sol-0.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-12.cg.sol-0.png" alt="">
</td>
<td valign="top">
1
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-12.cg.sol-1.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-12.cg.sol-1.png" alt="">
</td>
</tr>
2
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-12.cg.sol-2.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-12.cg.sol-2.png" alt="">
</td>
<td valign="top">
3
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-12.cg.sol-3.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-12.cg.sol-3.png" alt="">
</td>
</tr>
4
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-12.cg.sol-4.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-12.cg.sol-4.png" alt="">
</td>
<td valign="top">
5
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-12.cg.sol-5.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-12.cg.sol-5.png" alt="">
</td>
</tr>
</table>
<table width="80%" align="center">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-13.solution-kelly-0.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-13.solution-kelly-0.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-13.solution-kelly-1.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-13.solution-kelly-1.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-13.solution-kelly-2.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-13.solution-kelly-2.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-13.solution-kelly-3.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-13.solution-kelly-3.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-13.solution-kelly-4.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-13.solution-kelly-4.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-13.solution-kelly-5.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-13.solution-kelly-5.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-13.solution-kelly-6.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-13.solution-kelly-6.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-13.solution-kelly-7.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-13.solution-kelly-7.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-13.solution-kelly-8.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-13.solution-kelly-8.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-13.solution-kelly-9.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-13.solution-kelly-9.png" alt="">
</td>
</tr>
</table>
viewed from top:
-<img src="http://www.dealii.org/images/steps/developer/step-13.grid-kelly-8.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-13.grid-kelly-8.png" alt="">
However, we are not yet finished with evaluation the point value
refinement criteria yields the following picture:
-<img src="http://www.dealii.org/images/steps/developer/step-13.error.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-13.error.png" alt="">
<table align="center">
<tr>
<td width="50%">
- <img src="http://www.dealii.org/images/steps/developer/step-14.point-value.solution-5.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-14.point-value.solution-5.png" alt="">
</td>
<td width="50%">
- <img src="http://www.dealii.org/images/steps/developer/step-14.point-value.solution-5-dual.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-14.point-value.solution-5-dual.png" alt="">
</td>
</tr>
</table>
<table width="80%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-value.grid-0.png" alt="" width="100%"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-value.grid-2.png" alt="" width="100%"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-value.grid-4.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-0.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-2.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-4.png" alt="" width="100%"></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-value.grid-5.png" alt="" width="100%"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-value.grid-7.png" alt="" width="100%"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-value.grid-8.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-5.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-7.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.grid-8.png" alt="" width="100%"></td>
</tr>
</table>
<table width="80%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-value.error.png" alt="" width="100%"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-value.error-estimation.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.error.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-value.error-estimation.png" alt="" width="100%"></td>
</tr>
</table>
is the distance to the evaluation point; it can be shown that this is
the optimal weight if we neglect the effects of boundaries):
-<img src="http://www.dealii.org/images/steps/developer/step-14.point-value.error-comparison.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-14.point-value.error-comparison.png" alt="">
<table align="center">
<tr>
<td width="50%">
- <img src="http://www.dealii.org/images/steps/developer/step-14.point-derivative.solution-5-dual.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.solution-5-dual.png" alt="">
</td>
</table>
This time, the grids in refinement cycles 0, 5, 6, 7, 8, and 9 look
<table align="center" width="80%">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-0.png" alt="" width="100%"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-5.png" alt="" width="100%"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-6.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-0.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-5.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-6.png" alt="" width="100%"></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-7.png" alt="" width="100%"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-8.png" alt="" width="100%"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-9.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-7.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-8.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.grid-9.png" alt="" width="100%"></td>
</tr>
</table>
<table width="80%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-derivative.error.png" alt="" width="100%"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.point-derivative.error-estimation.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.error.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.point-derivative.error-estimation.png" alt="" width="100%"></td>
</tr>
</table>
<table width="80%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.step-13.grid-9.png" alt="" width="100%"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.step-13.grid-10.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.step-13.grid-9.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.step-13.grid-10.png" alt="" width="100%"></td>
</tr>
</table>
<table width="80%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-14.step-13.solution-7.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-14.step-13.solution-7.png" alt="" width="100%"></td>
</tr>
</table>
The next point is to compare the new (duality based) mesh refinement
criterion with the old ones. These are the results:
-<img src="http://www.dealii.org/images/steps/developer/step-14.step-13.error-comparison.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-14.step-13.error-comparison.png" alt="">
<TABLE WIDTH="100%">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-15.solution-0.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-15.solution-0.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-15.solution-1.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-15.solution-1.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-15.solution-2.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-15.solution-2.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-15.solution-3.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-15.solution-3.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-15.solution-4.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-15.solution-4.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-15.solution-5.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-15.solution-5.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-15.solution-6.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-15.solution-6.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-15.solution-7.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-15.solution-7.png" alt="">
</td>
</tr>
</table>
because there isn't much change in the solution. The final solution
and mesh are shown here:
-<img src="http://www.dealii.org/images/steps/developer/step-15.grid.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-15.grid.png" alt="">
On the finest mesh, the solution looks like this:
<p align="center">
- <img src="http://www.dealii.org/images/steps/developer/step-16.solution.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-16.solution.png" alt="">
</p>
More importantly, we would like to see if the multigrid method really
<table align="center" style="width:80%">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-17.12-ux.png" alt="" width="100%"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-17.12-uy.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-17.12-ux.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-17.12-uy.png" alt="" width="100%"></td>
</tr>
</table>
<table align="center" width="80%">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-17.12-grid.png" alt="" width="100%"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-17.12-partition.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-17.12-grid.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-17.12-partition.png" alt="" width="100%"></td>
</tr>
</table>
<table width="80%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-17.4-3d-partition.png" width="100%" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-17.4-3d-ux.png" alt="" width="100%"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-17.4-3d-partition.png" width="100%" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-17.4-3d-ux.png" alt="" width="100%"></td>
</tr>
</table>
<table width="100%">
<tr width="100%">
<td width="33%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.sequential-0002.0000.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.sequential-0002.0000.png" alt="">
Time = 2
</td>
<td width="33%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.sequential-0005.0000.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.sequential-0005.0000.png" alt="">
Time = 5
</td>
<td width="33%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.sequential-0007.0000.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.sequential-0007.0000.png" alt="">
Time = 7
</td>
</tr>
<tr width="100%">
<td width="33%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.sequential-0008.0000.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.sequential-0008.0000.png" alt="">
Time = 8
</td>
<td width="33%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.sequential-0009.0000.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.sequential-0009.0000.png" alt="">
Time = 9
</td>
<td width="33%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.sequential-0010.0000.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.sequential-0010.0000.png" alt="">
Time = 10
</td>
</tr>
<table width="100%">
<tr width="100%">
<td width="49%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.parallel-000mesh.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.parallel-000mesh.png" alt="">
</td>
<td width="49%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.parallel-0002.p.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.parallel-0002.p.png" alt="">
</td>
</tr>
</table>
<table width="100%">
<tr width="100%">
<td width="33%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.parallel-0002.s.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.parallel-0002.s.png" alt="">
Time = 2
</td>
<td width="33%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.parallel-0005.s.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.parallel-0005.s.png" alt="">
Time = 5
</td>
<td width="33%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.parallel-0007.s.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.parallel-0007.s.png" alt="">
Time = 7
</td>
</tr>
<tr width="100%">
<td width="33%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.parallel-0008.s.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.parallel-0008.s.png" alt="">
Time = 8
</td>
<td width="33%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.parallel-0009.s.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.parallel-0009.s.png" alt="">
Time = 9
</td>
<td width="33%">
- <img src="http://www.dealii.org/images/steps/developer/step-18.parallel-0010.s.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-18.parallel-0010.s.png" alt="">
Time = 10
</td>
</tr>
We can then visualize this one file with gnuplot, obtaining something like
this:
-<img src="http://www.dealii.org/images/steps/developer/step-19.solution-0005.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-19.solution-0005.png" alt="">
That's not particularly exciting, but the file we're looking at has only one
32nd of the entire domain anyway, so we can't expect much.
equation):
<table style="width:60%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-2.sparsity-1.svg" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-2.sparsity-2.svg" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-2.sparsity-1.svg" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-2.sparsity-2.svg" alt=""></td>
</tr>
</table>
solution, let us take a look at it. The following three images show (from left
to right) the x-velocity, the y-velocity, and the pressure:
-<img src="http://www.dealii.org/images/steps/developer/step-20.u.png" alt="">
-<img src="http://www.dealii.org/images/steps/developer/step-20.v.png" alt="">
-<img src="http://www.dealii.org/images/steps/developer/step-20.p.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-20.u.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-20.v.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-20.p.png" alt="">
<table style="width:60%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-20.u-wiggle.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-20.v-wiggle.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-20.u-wiggle.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-20.v-wiggle.png" alt=""></td>
</tr>
</table>
inverse of this tensor (i.e., of <code>normalized_permeability</code>)
looks as follows:
-<img src="http://www.dealii.org/images/steps/developer/step-20.k-random.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-20.k-random.png" alt="">
With a permeability field like this, we would get x-velocities and pressures as
<table style="width:60%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-20.u-random.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-20.p-random.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-20.u-random.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-20.p-random.png" alt=""></td>
</tr>
</table>
over time. First, this is for the single crack model, as implemented in the
<code>SingleCurvingCrack::KInverse</code> class:
-<img src="http://www.dealii.org/images/steps/developer/step-21.centerline.gif" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-21.centerline.gif" alt="">
As can be seen, the water rich fluid snakes its way mostly along the
high-permeability zone in the middle of the domain, whereas the rest of the
centers of high permeability and fluid hops from one of these zones to
the next:
-<img src="http://www.dealii.org/images/steps/developer/step-21.random2d.gif" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-21.random2d.gif" alt="">
Finally, here is the same situation in three space dimensions, on a mesh with
<code>n_refinement_steps=5</code>, which produces a mesh of some 32,000 cells
and 167,000 degrees of freedom:
-<img src="http://www.dealii.org/images/steps/developer/step-21.random3d.gif" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-21.random3d.gif" alt="">
To repeat these computations, all you have to do is to change the line
@code
different places in the program during refinement cycles
zero through three in 3d is shown here:
-<img src="http://www.dealii.org/images/steps/developer/step-22.profile-3.png" alt="">
+<img src="https://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:
-<img src="http://www.dealii.org/images/steps/developer/step-22.profile-3.original.png" alt="">
+<img src="https://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
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.2d.mesh-0.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-0.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.2d.mesh-1.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-1.png" alt="">
</td>
</tr>
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.2d.mesh-2.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-2.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.2d.mesh-3.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-3.png" alt="">
</td>
</tr>
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.2d.mesh-4.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-4.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.2d.mesh-5.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.2d.mesh-5.png" alt="">
</td>
</tr>
</table>
transported along with the moving upper boundary and being replaced by
material coming from below:
-<img src="http://www.dealii.org/images/steps/developer/step-22.2d.solution.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-22.2d.solution.png" alt="">
This plot uses the capability of VTK-based visualization programs (in
this case of VisIt) to show vector data; this is the result of us
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.3d.mesh-0.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-0.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.3d.mesh-1.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-1.png" alt="">
</td>
</tr>
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.3d.mesh-2.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-2.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.3d.mesh-3.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-3.png" alt="">
</td>
</tr>
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.3d.mesh-4.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-4.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.3d.mesh-5.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.3d.mesh-5.png" alt="">
</td>
</tr>
</table>
by boundary conditions. The vector field computed makes for an
interesting graph:
-<img src="http://www.dealii.org/images/steps/developer/step-22.3d.solution.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-22.3d.solution.png" alt="">
The isocontours shown here as well are those of the pressure
variable, showing the singularity at the point of discontinuous
vector), then we get the following image after the first adaptive
refinement in two dimensions:
-<img src="http://www.dealii.org/images/steps/developer/step-22.2d.sparsity-nor.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-22.2d.sparsity-nor.png" alt="">
In order to generate such a graph, you have to insert a piece of
code like the following to the end of the setup step.
components. The renumbering with DoFRenumbering::Cuthill_McKee and grouping
the components into velocity and pressure yields the following output:
-<img src="http://www.dealii.org/images/steps/developer/step-22.2d.sparsity-ren.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-22.2d.sparsity-ren.png" alt="">
It is apparent that the situation has improved a lot. Most of the elements are
now concentrated around the diagonal in the (0,0) block in the matrix. Similar
is not a good choice in 3D - a full decomposition needs many new entries that
eventually won't fit into the physical memory (RAM):
-<img src="http://www.dealii.org/images/steps/developer/step-22.3d.sparsity_uu-ren.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-22.3d.sparsity_uu-ren.png" alt="">
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.3d-extension.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.3d-extension.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-22.3d-grid-extension.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-22.3d-grid-extension.png" alt="">
</td>
</tr>
</table>
step to an output file. If we process them adequately and paste them into a
movie, we get the following:
-<img src="http://www.dealii.org/images/steps/developer/step-23.movie.gif" alt="Animation of the solution of step 23.">
+<img src="https://www.dealii.org/images/steps/developer/step-23.movie.gif" alt="Animation of the solution of step 23.">
The movie shows the generated wave nice traveling through the domain and back,
being reflected at the clamped boundary. Some numerical noise is trailing the
propagating in the medium (in our simulation, we assume the medium is mineral
oil, which has a acoustic speed of 1.437 $\frac{mm}{\mu s}$):
-<img src="http://www.dealii.org/images/steps/developer/step-24.one_movie.gif" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-24.one_movie.gif" alt="">
For a single absorber, we of course have to change the
<code>InitialValuesP</code> class accordingly.
program, to make things a bit more interesting compared to the present case of
only a single source:
-<img src="http://www.dealii.org/images/steps/developer/step-24.traces.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-24.traces.png" alt="">
One thing that can be seen, for example, is that the arrival of the second and
fourth signals shifts to earlier times for greater detector numbers (i.e. the
<TABLE WIDTH="100%">
<tr>
<td>
-<img src="http://www.dealii.org/images/steps/developer/step-24.one.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-24.one.png" alt="">
</td>
<td>
-<img src="http://www.dealii.org/images/steps/developer/step-24.one_s.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-24.one_s.png" alt="">
</td>
</tr>
</table>
the simulated results can be made to look closer to the experimental
data:
-<img src="http://www.dealii.org/images/steps/developer/step-24.one_sf.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-24.one_sf.png" alt="">
In our simulations, we see spurious signals behind the main wave that
result from numerical artifacts. This problem can be alleviated by using finer
mesh, resulting in the following plot:
-<img src="http://www.dealii.org/images/steps/developer/step-24.one_s2.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-24.one_s2.png" alt="">
in the program. The following movie shows the propagation of the generated
thermoacoustic waves in the medium by multiple absorbers:
-<img src="http://www.dealii.org/images/steps/developer/step-24.multi_movie.gif" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-24.multi_movie.gif" alt="">
Experimental data and our simulated data are compared in the following two
figures:
<TABLE WIDTH="100%">
<tr>
<td>
-<img src="http://www.dealii.org/images/steps/developer/step-24.multi.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-24.multi.png" alt="">
</td>
<td>
-<img src="http://www.dealii.org/images/steps/developer/step-24.multi_s.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-24.multi_s.png" alt="">
</td>
</tr>
</table>
<TABLE WIDTH="100%">
<tr>
<td>
-<img src="http://www.dealii.org/images/steps/developer/step-24.multi_sf.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-24.multi_sf.png" alt="">
</td>
<td>
-<img src="http://www.dealii.org/images/steps/developer/step-24.multi_s2.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-24.multi_s2.png" alt="">
</td>
</tr>
</table>
\f]
where $c_1$, $c_2$ and $m<1$ are constants. In the simulation below, we have chosen $c_1=0$, $c_2=0$, $m=0.5$. Moreover, it is know that the period of oscillation of the breather is $2\pi\sqrt{1-m^2}$, hence we have chosen $t_0=-5.4414$ and $t_f=2.7207$ so that we can observe three oscillations of the solution. Then, taking $u_0(x) = u_{\mathrm{breather}}(x,t_0)$, $\theta=0$ and $k=h/10$, the program computed the following solution.
-<img src="http://www.dealii.org/images/steps/developer/step-25.1d-breather.gif" alt="Animation of the 1D stationary breather.">
+<img src="https://www.dealii.org/images/steps/developer/step-25.1d-breather.gif" alt="Animation of the 1D stationary breather.">
Though not shown how to do this in the program, another way to visualize the
(1+1)-d solution is to use output generated by the DataOutStack class; it
solutions. This produces the space-time plot below instead of the animation
above.
-<img src="http://www.dealii.org/images/steps/developer/step-25.1d-breather_stp.png" alt="A space-time plot of the 1D stationary breather.">
+<img src="https://www.dealii.org/images/steps/developer/step-25.1d-breather_stp.png" alt="A space-time plot of the 1D stationary breather.">
Furthermore, since the breather is an analytical solution of the sine-Gordon
equation, we can use it to validate our code, although we have to assume that
The simulation shown below was performed with $u_0(x) = u_{\mathrm{kink}}(x,t_0)$, $\theta=\frac{1}{2}$, $k=20h$, $t_0=1$ and $t_f=500$. The $L^2$ norm of the error of the finite element solution at each time step remained on the order of $10^{-2}$, showing that the program is working correctly in 2D, as well as 1D. Unfortunately, the solution is not very interesting, nonetheless we have included a snapshot of it below for completeness.
-<img src="http://www.dealii.org/images/steps/developer/step-25.2d-kink.png" alt="Stationary 2D kink.">
+<img src="https://www.dealii.org/images/steps/developer/step-25.2d-kink.png" alt="Stationary 2D kink.">
Now that we have validated the code in 1D and 2D, we move to a problem where the analytical solution is unknown.
the "exponentially damped" spectrum of the time stepping schemes
assures these oscillations are not created.
-<img src="http://www.dealii.org/images/steps/developer/step-25.2d-angled_kink.gif" alt="Animation of a moving 2D kink, at 45 degrees to the axes of the grid, showing boundary effects.">
+<img src="https://www.dealii.org/images/steps/developer/step-25.2d-angled_kink.gif" alt="Animation of a moving 2D kink, at 45 degrees to the axes of the grid, showing boundary effects.">
Another interesting solution to the sine-Gordon equation (which cannot be
obtained analytically) can be produced by using two 1D breathers to construct
--- it acts like a breather (as far as the pictures are concerned); however,
it appears to break up and reassemble, rather than just oscillate.
-<img src="http://www.dealii.org/images/steps/developer/step-25.2d-pseudobreather.gif" alt="Animation of a 2D pseudobreather.">
+<img src="https://www.dealii.org/images/steps/developer/step-25.2d-pseudobreather.gif" alt="Animation of a 2D pseudobreather.">
<a name="extensions"></a>
Maybe of more interest is a visualization of the solution and the mesh on which
it was computed:
-<img src="http://www.dealii.org/images/steps/developer/step-26.movie.gif" alt="Animation of the solution of step 26.">
+<img src="https://www.dealii.org/images/steps/developer/step-26.movie.gif" alt="Animation of the solution of step 26.">
The movie shows how the two sources switch on and off and how the mesh reacts
to this. It is quite obvious that the mesh as is is probably not the best we
Of maybe more interest is to look at the graphical output. First, here is the
solution of the problem:
-<img src="http://www.dealii.org/images/steps/developer/step-27.solution.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-27.solution.png" alt="">
Secondly, let us look at the sequence of meshes generated:
<table width="60%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.mesh-0.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.mesh-1.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.mesh-2.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.mesh-0.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.mesh-1.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.mesh-2.png" alt=""></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.mesh-3.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.mesh-4.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.mesh-5.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.mesh-3.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.mesh-4.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.mesh-5.png" alt=""></td>
</tr>
</table>
<table width="60%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.fe_degree-0.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.fe_degree-1.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.fe_degree-2.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.fe_degree-0.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.fe_degree-1.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.fe_degree-2.png" alt=""></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.fe_degree-3.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.fe_degree-4.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.fe_degree-5.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.fe_degree-3.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.fe_degree-4.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.fe_degree-5.png" alt=""></td>
</tr>
</table>
<table width="60%" align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.smoothness-0.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.smoothness-1.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.smoothness-2.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.smoothness-0.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.smoothness-1.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.smoothness-2.png" alt=""></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.smoothness-3.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.smoothness-4.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-27.smoothness-5.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.smoothness-3.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.smoothness-4.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-27.smoothness-5.png" alt=""></td>
</tr>
</table>
The grids of fast and thermal energy groups at mesh iteration #9 are shown
in following figure.
-<img src="http://www.dealii.org/images/steps/developer/step-28.grid-0.9.order2.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-28.grid-0.9.order2.png" alt="">
-<img src="http://www.dealii.org/images/steps/developer/step-28.grid-1.9.order2.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-28.grid-1.9.order2.png" alt="">
We see that the grid of thermal group is much finer than the one of fast group.
The solutions on these grids are, (Note: flux are normalized with total fission
source equal to 1)
-<img src="http://www.dealii.org/images/steps/developer/step-28.solution-0.9.order2.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-28.solution-0.9.order2.png" alt="">
-<img src="http://www.dealii.org/images/steps/developer/step-28.solution-1.9.order2.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-28.solution-1.9.order2.png" alt="">
Then we plot the convergence data with polynomial order being equal to 1,2 and 3.
-<img src="http://www.dealii.org/images/steps/developer/step-28.convergence.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-28.convergence.png" alt="">
The estimated ``exact'' k-effective = 0.906834721253 which is simply from last
mesh iteration of polynomial order 3 minus 2e-10. We see that h-adaptive calculations
<table align="center" border="1" cellspacing="3" cellpadding="3">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-29.v.png" alt="v = Re(u)">
+ <img src="https://www.dealii.org/images/steps/developer/step-29.v.png" alt="v = Re(u)">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-29.w.png" alt="w = Im(u)">
+ <img src="https://www.dealii.org/images/steps/developer/step-29.w.png" alt="w = Im(u)">
</td>
</tr>
<tr>
<td colspan="2">
- <img src="http://www.dealii.org/images/steps/developer/step-29.intensity.png" alt="|u|">
+ <img src="https://www.dealii.org/images/steps/developer/step-29.intensity.png" alt="|u|">
</td>
</tr>
</table>
<table align="center" border="1" cellspacing="3" cellpadding="3">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-29.surface.png" alt="|u|">
+ <img src="https://www.dealii.org/images/steps/developer/step-29.surface.png" alt="|u|">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-29.contours.png" alt="|u|">
+ <img src="https://www.dealii.org/images/steps/developer/step-29.contours.png" alt="|u|">
</td>
</tr>
</table>
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-3.solution-1.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-3.solution-1.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-3.solution-2.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-3.solution-2.png" alt="">
</td>
</tr>
</table>
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-30.sol-1.iso.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-30.sol-1.iso.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-30.sol-1.aniso.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-30.sol-1.aniso.png" alt="">
</td>
</tr>
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-30.sol-5.iso.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-30.sol-5.iso.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-30.sol-5.aniso.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-30.sol-5.aniso.png" alt="">
</td>
</tr>
</table>
refinement. This observation is strengthened by the plot of the an adapted
anisotropic grid, e.g. the grid after three refinement steps.
-<img src="http://www.dealii.org/images/steps/developer/step-30.grid-3.aniso.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-30.grid-3.aniso.png" alt="">
In the whole left part of the domain refinement is only performed along the
y-axis of cells. In the right part of the domain the refinement is dominated by
<table align="center" border="1" cellspacing="3" cellpadding="3">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.solution.00.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.solution.00.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.solution.01.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.solution.01.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.solution.02.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.solution.02.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.solution.03.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.solution.03.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.solution.04.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.solution.04.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.solution.05.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.solution.05.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.solution.06.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.solution.06.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.solution.07.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.solution.07.png" alt="">
</td>
</tr>
</table>
<table align="center" border="1" cellspacing="3" cellpadding="3">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.grid.00.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.grid.00.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.grid.01.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.grid.01.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.grid.02.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.grid.02.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.grid.03.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.grid.03.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.grid.04.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.grid.04.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.grid.05.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.grid.05.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.grid.06.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.grid.06.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.2d.grid.07.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.2d.grid.07.png" alt="">
</td>
</tr>
</table>
<table align="center" border="1" cellspacing="3" cellpadding="3">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.3d.solution.00.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.3d.solution.00.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.3d.solution.01.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.3d.solution.01.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.3d.solution.02.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.3d.solution.02.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.3d.solution.03.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.3d.solution.03.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.3d.solution.04.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.3d.solution.04.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.3d.solution.05.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.3d.solution.05.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.3d.solution.06.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.3d.solution.06.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.3d.solution.07.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.3d.solution.07.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.3d.solution.08.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.3d.solution.08.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.3d.solution.09.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.3d.solution.09.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.3d.solution.10.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.3d.solution.10.png" alt="">
</td>
<td>
</td>
<table align="center" border="1" cellspacing="3" cellpadding="3">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.timestep.q1.beta=0.01.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q1.beta=0.01.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.timestep.q1.beta=0.03.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q1.beta=0.03.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.timestep.q1.beta=0.1.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q1.beta=0.1.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.timestep.q1.beta=0.5.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q1.beta=0.5.png" alt="">
</td>
</tr>
</table>
<table align="center" border="1" cellspacing="3" cellpadding="3">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.timestep.q2.beta=0.01.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q2.beta=0.01.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.timestep.q2.beta=0.03.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q2.beta=0.03.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-31.timestep.q2.beta=0.1.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-31.timestep.q2.beta=0.1.png" alt="">
</td>
</tr>
</table>
velocities and therefore time step sizes) in 2d will produce the
following graph:
-<img src="http://www.dealii.org/images/steps/developer/step-32.beta.2d.png" alt="">
+<img src="https://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:
-<img src="http://www.dealii.org/images/steps/developer/step-32.beta_cr.2d.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/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>):
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d-initial.png" alt="">
+ <img src="https://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
<table align="center" border="1" cellspacing="3" cellpadding="3">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.3d.cube.0.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.3d.cube.0.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.3d.cube.1.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.3d.cube.1.png" alt="">
</td>
</tr>
</table>
<table>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.0000.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.0000.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.0100.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.0100.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.0200.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.0200.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.0300.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.0300.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.0400.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.0400.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.0500.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.0500.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.0600.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.0600.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.0700.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.0700.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.0800.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.0800.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.0900.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.0900.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.1000.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.1000.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.1100.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.1100.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.1200.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.1200.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.1300.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.1300.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.1400.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.1400.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.1500.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.1500.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.1600.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.1600.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.1700.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.1700.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.1800.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.1800.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.1900.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.1900.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.2000.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.2000.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.temperature.2100.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.temperature.2100.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.grid.2100.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.grid.2100.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-32.2d.partition.2100.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-32.2d.partition.2100.png" alt="">
</td>
</tr>
</table>
These different phases can also be identified if we look at the
maximal velocity as a function of time in the simulation:
-<img src="http://www.dealii.org/images/steps/developer/step-32.2d.t_vs_vmax.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-32.2d.t_vs_vmax.png" alt="">
Here, the velocity (shown in centimeters per year) becomes very large,
to the order of several meters per year) at the beginning when the
with the mesh) onto 512 processors:
<p align="center">
-<img src="http://www.dealii.org/images/steps/developer/step-32.3d-sphere.solution.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-32.3d-sphere.solution.png" alt="">
-<img src="http://www.dealii.org/images/steps/developer/step-32.3d-sphere.partition.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-32.3d-sphere.partition.png" alt="">
</p>
domain. The combination of the two for the two dimensional case looks
like
-<img src="http://www.dealii.org/images/steps/developer/step-34_2d.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-34_2d.png" alt="">
while in three dimensions we show first the potential on the surface,
together with a contour plot,
-<img src="http://www.dealii.org/images/steps/developer/step-34_3d.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-34_3d.png" alt="">
and then the external contour plot of the potential, with opacity set to 25%:
-<img src="http://www.dealii.org/images/steps/developer/step-34_3d-2.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-34_3d-2.png" alt="">
<a name="extensions"></a>
The testcase that we use for this program consists of the flow around a square obstacle. The geometry is
as follows:
-<img src="http://www.dealii.org/images/steps/developer/step-35.geometry.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-35.geometry.png" alt="">
with $H=4.1$, making the geometry slightly non-symmetric.
<table>
<tr>
- <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>
+ <td> <img src="https://www.dealii.org/images/steps/developer/step-35.Re_100.vorticity.0.png" alt=""> </td>
+ <td> <img src="https://www.dealii.org/images/steps/developer/step-35.Re_100.velocity.0.png" alt=""> </td>
</tr>
<tr>
- <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>
+ <td> <img src="https://www.dealii.org/images/steps/developer/step-35.Re_100.vorticity.1.png" alt=""> </td>
+ <td> <img src="https://www.dealii.org/images/steps/developer/step-35.Re_100.velocity.1.png" alt=""> </td>
</tr>
<tr>
- <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>
+ <td> <img src="https://www.dealii.org/images/steps/developer/step-35.Re_100.vorticity.2.png" alt=""> </td>
+ <td> <img src="https://www.dealii.org/images/steps/developer/step-35.Re_100.velocity.2.png" alt=""> </td>
</tr>
<tr>
- <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>
+ <td> <img src="https://www.dealii.org/images/steps/developer/step-35.Re_100.vorticity.3.png" alt=""> </td>
+ <td> <img src="https://www.dealii.org/images/steps/developer/step-35.Re_100.velocity.3.png" alt=""> </td>
</tr>
<tr>
- <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>
+ <td> <img src="https://www.dealii.org/images/steps/developer/step-35.Re_100.vorticity.4.png" alt=""> </td>
+ <td> <img src="https://www.dealii.org/images/steps/developer/step-35.Re_100.velocity.4.png" alt=""> </td>
</tr>
</table>
<table>
<tr>
- <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>
+ <td> <img src="https://www.dealii.org/images/steps/developer/step-35.Re_500.vorticity.0.png" alt=""> </td>
+ <td> <img src="https://www.dealii.org/images/steps/developer/step-35.Re_500.velocity.0.png" alt=""> </td>
</tr>
<tr>
- <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>
+ <td> <img src="https://www.dealii.org/images/steps/developer/step-35.Re_500.vorticity.1.png" alt=""> </td>
+ <td> <img src="https://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:
-<img src="http://www.dealii.org/images/steps/developer/step-35.Re_500.zoom.png" alt="">
+<img src="https://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:
-<img src="http://www.dealii.org/images/steps/developer/step-35.Re_500.zoom_2.png" alt="">
+<img src="https://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
<table width="80%">
<tr>
-<td><img src="http://www.dealii.org/images/steps/developer/step-36.default.eigenfunction.0.png" alt=""></td>
-<td><img src="http://www.dealii.org/images/steps/developer/step-36.default.eigenfunction.1.png" alt=""></td>
-<td><img src="http://www.dealii.org/images/steps/developer/step-36.default.eigenfunction.2.png" alt=""></td>
+<td><img src="https://www.dealii.org/images/steps/developer/step-36.default.eigenfunction.0.png" alt=""></td>
+<td><img src="https://www.dealii.org/images/steps/developer/step-36.default.eigenfunction.1.png" alt=""></td>
+<td><img src="https://www.dealii.org/images/steps/developer/step-36.default.eigenfunction.2.png" alt=""></td>
</tr>
<tr>
-<td><img src="http://www.dealii.org/images/steps/developer/step-36.default.eigenfunction.3.png" alt=""></td>
-<td><img src="http://www.dealii.org/images/steps/developer/step-36.default.eigenfunction.4.png" alt=""></td>
+<td><img src="https://www.dealii.org/images/steps/developer/step-36.default.eigenfunction.3.png" alt=""></td>
+<td><img src="https://www.dealii.org/images/steps/developer/step-36.default.eigenfunction.4.png" alt=""></td>
<td></td>
</tr>
</table>
potential is the lowest, i.e. in the top right and bottom left sector of inner
circle of the potential):
-<img src="http://www.dealii.org/images/steps/developer/step-36.mod.potential.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-36.mod.potential.png" alt="">
The first five eigenfunctions are now like this:
<table width="80%">
<tr>
-<td><img src="http://www.dealii.org/images/steps/developer/step-36.mod.eigenfunction.0.png" alt=""></td>
-<td><img src="http://www.dealii.org/images/steps/developer/step-36.mod.eigenfunction.1.png" alt=""></td>
-<td><img src="http://www.dealii.org/images/steps/developer/step-36.mod.eigenfunction.2.png" alt=""></td>
+<td><img src="https://www.dealii.org/images/steps/developer/step-36.mod.eigenfunction.0.png" alt=""></td>
+<td><img src="https://www.dealii.org/images/steps/developer/step-36.mod.eigenfunction.1.png" alt=""></td>
+<td><img src="https://www.dealii.org/images/steps/developer/step-36.mod.eigenfunction.2.png" alt=""></td>
</tr>
<tr>
-<td><img src="http://www.dealii.org/images/steps/developer/step-36.mod.eigenfunction.3.png" alt=""></td>
-<td><img src="http://www.dealii.org/images/steps/developer/step-36.mod.eigenfunction.4.png" alt=""></td>
+<td><img src="https://www.dealii.org/images/steps/developer/step-36.mod.eigenfunction.3.png" alt=""></td>
+<td><img src="https://www.dealii.org/images/steps/developer/step-36.mod.eigenfunction.4.png" alt=""></td>
<td></td>
</tr>
</table>
solution. We show a picture anyway, illustrating the size of the
solution through both isocontours and volume rendering:
-<img src="http://www.dealii.org/images/steps/developer/step-37.solution.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-37.solution.png" alt="">
Of more interest is to evaluate some aspects of the multigrid solver.
When we run this program in 2D for quadratic ($Q_2$) elements, we get the
Finally, the program produces graphical output that we can visualize. Here is
a plot of the results:
-<img src="http://www.dealii.org/images/steps/developer/step-38.solution-3d.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-38.solution-3d.png" alt="">
The program also works for 1d curves in 2d, not just 2d surfaces in 3d. You
can test this by changing the template argument in <code>main()</code> like
solution extruded into the third dimension, clearly showing the change in sign
as the curve moves from one quadrant of the domain into the adjacent one):
-<img src="http://www.dealii.org/images/steps/developer/step-38.solution-2d.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-38.solution-2d.png" alt="">
<a name="extensions"></a>
the x- and y-coordinates a bit. Let's show the computational domain and the
solution first before we go into details of the implementation below:
-<img src="http://www.dealii.org/images/steps/developer/step-38.warp-1.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-38.warp-1.png" alt="">
-<img src="http://www.dealii.org/images/steps/developer/step-38.warp-2.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-38.warp-2.png" alt="">
The way to produce such a mesh is by using the GridTools::transform
function. It needs a way to transform each individual mesh point to a
<h2>Postprocessing of the logfile</h2>
-<img src="http://www.dealii.org/images/steps/developer/step-39-convergence.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-39-convergence.png" alt="">
Using the perl script <tt>postprocess.pl</tt>, we extract relevant
data into <tt>output.dat</tt>, which can be used to plot graphs with
<tt>gnuplot</tt>. The graph above for instance was produced with
with its functionality. Here's what I have come up with for the 2d solution:
<p align="center">
- <img src="http://www.dealii.org/images/steps/developer/step-4.solution-2d.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-4.solution-2d.png" alt="">
</p>
(@dealiiVideoLectureSeeAlso{11,32})
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-4.solution-3d.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-4.solution-3d.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-4.contours-3d.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-4.contours-3d.png" alt="">
</td>
</tr>
</table>
<TABLE WIDTH="100%">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-40.mesh.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-40.mesh.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-40.solution.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-40.solution.png" alt="">
</td>
</tr>
</table>
<TABLE WIDTH="100%">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-40.strong2.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-40.strong2.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-40.strong.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-40.strong.png" alt="">
</td>
</tr>
</table>
<TABLE WIDTH="100%">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-40.256.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-40.256.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-40.4096.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-40.4096.png" alt="">
</td>
</tr>
</table>
<table align="center" class="tutorial" cellspacing="3" cellpadding="3">
<tr>
<td align="center">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.png" alt="">
</td>
<td align="center">
- <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.active-set.png" alt="">
</td>
</tr>
</table>
0
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.00.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.00.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.00.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.active-set.00.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.00.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.3d.00.png" alt="">
</td>
</tr>
3
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.03.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.03.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.03.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.active-set.03.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.03.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.3d.03.png" alt="">
</td>
</tr>
6
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.06.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.06.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.06.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.active-set.06.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.06.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.3d.06.png" alt="">
</td>
</tr>
9
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.09.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.09.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.09.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.active-set.09.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.09.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.3d.09.png" alt="">
</td>
</tr>
12
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.12.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.12.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.12.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.active-set.12.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.12.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.3d.12.png" alt="">
</td>
</tr>
15
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.15.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.15.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.15.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.active-set.15.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.15.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.3d.15.png" alt="">
</td>
</tr>
18
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.18.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.18.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.active-set.18.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.active-set.18.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.displacement.3d.18.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.displacement.3d.18.png" alt="">
</td>
</tr>
</table>
<table align="center">
<tr>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.forces.01.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.forces.01.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.forces.09.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.forces.09.png" alt="">
</td>
<td valign="top">
- <img src="http://www.dealii.org/images/steps/developer/step-41.forces.18.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-41.forces.18.png" alt="">
</td>
</tr>
which is actually shown -, the obstacle corresponds
to a Chinese character that is discussed below):
-<img src="http://www.dealii.org/images/steps/developer/step-42.CellConstitutionLi2.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-42.CellConstitutionLi2.png" alt="">
This problem description implies that we have to take care of an additional
looks as follows (see also the picture at
the top of this section on how the end result looks like):
- <img src="http://www.dealii.org/images/steps/developer/step-42.character.png" alt="" width="25%">
+ <img src="https://www.dealii.org/images/steps/developer/step-42.character.png" alt="" width="25%">
</ul>
Other than that, let us comment only on the following aspects:
<table align="center">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-42.CellConstitutionColorbar.png">
+ <img src="https://www.dealii.org/images/steps/developer/step-42.CellConstitutionColorbar.png">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-42.CellConstitutionBall2.png" alt="" width="70%">
+ <img src="https://www.dealii.org/images/steps/developer/step-42.CellConstitutionBall2.png" alt="" width="70%">
</td>
<td valign="top">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-42.CellConstitutionLi2.png" alt="" alt="" width="70%">
+ <img src="https://www.dealii.org/images/steps/developer/step-42.CellConstitutionLi2.png" alt="" alt="" width="70%">
</td>
</tr>
</table>
<table align="center" class="tutorial" cellspacing="3" cellpadding="3">
<tr>
<td align="center">
- <img src="http://www.dealii.org/images/steps/developer/step-43.3d.velocity.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-43.3d.streamlines.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-43.3d.saturation.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-43.3d.mesh.png" alt="">
+ <img src="https://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.
- <img src="http://www.dealii.org/images/steps/developer/step-44.setup.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-44.Q1-P0_convergence.png" alt="">
+ <img src="https://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 in 3-d.
</p>
</td>
<td align="center">
- <img src="http://www.dealii.org/images/steps/developer/step-44.Q2-P1_convergence.png" alt="">
+ <img src="https://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 in 3-d.
</p>
<table align="center" class="tutorial" cellspacing="3" cellpadding="3">
<tr>
<td align="center">
- <img src="http://www.dealii.org/images/steps/developer/step-44.Q1-P0_gr_1_p_ratio_80-displacement.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-44.Q1-P0_gr_1_p_ratio_80-displacement.png" alt="">
<p align="center">
Z-displacement solution for the 3-d problem.
</p>
</td>
<td align="center">
- <img src="http://www.dealii.org/images/steps/developer/step-44.Q1-P0_gr_1_p_ratio_80-pressure.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-44.Q1-P0_gr_1_p_ratio_80-dilatation.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-44.Q2-P1_gr_3_p_ratio_80-displacement.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-44.Q2-P1_gr_3_p_ratio_80-displacement.png" alt="">
<p align="center">
Z-displacement solution for the 3-d problem.
</p>
</td>
<td align="center">
- <img src="http://www.dealii.org/images/steps/developer/step-44.Q2-P1_gr_3_p_ratio_80-pressure.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-44.Q2-P1_gr_3_p_ratio_80-dilatation.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-44.Normalised_runtime.png" alt="">
+ <img src="https://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>
<tr>
<td align="center">
- <img src="http://www.dealii.org/images/steps/developer/step-44.2d-gr_2.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-44.2d-gr_2.png" alt="">
<p align="center">
Y-displacement solution in 2-d for 2 global grid refinement levels.
</p>
</td>
<td align="center">
- <img src="http://www.dealii.org/images/steps/developer/step-44.2d-gr_5.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-44.2d-gr_5.png" alt="">
<p align="center">
Y-displacement solution in 2-d for 5 global grid refinement levels.
</p>
The created output is not very surprising. We simply see that the solution is
periodic with respect to the left and lower boundary:
-<img src="http://www.dealii.org/images/steps/developer/step-45.periodic.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-45.periodic.png" alt="">
Without the periodicity constraints we would have ended up with the following solution:
-<img src="http://www.dealii.org/images/steps/developer/step-45.non_periodic.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-45.non_periodic.png" alt="">
We will consider the following situation as a testcase:
-<img src="http://www.dealii.org/images/steps/developer/step-46.layout.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-46.layout.png" alt="">
As discussed at the top of this document, we need to assume in a few places
that a cell is either entirely in the fluid or solid part of the domain and,
<TABLE WIDTH="60%" ALIGN="center">
<tr valign="top">
<td valign="top" align="center">
- <img src="http://www.dealii.org/images/steps/developer/step-46.velocity-magnitude.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-46.pressure.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-46.velocity.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-46.velocity.png" alt="">
<p align="center">
Fluid velocity.
</td>
<td valign="top" align="center">
- <img src="http://www.dealii.org/images/steps/developer/step-46.displacement.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-46.3d.velocity.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-46.3d.streamlines.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-46.3d.displacement.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-46.3d.displacement.png" alt="">
<p align="center">
Solid displacement.
<code>untitled.geo</code> looks like in gmsh (displaying the boundary
indicators as well as the mesh discussed further down below):
-<img src="http://www.dealii.org/images/steps/developer/step-49.gmsh_picture.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-49.gmsh_picture.png" alt="">
You might want to open the <code>untitled.geo</code> file in a text editor (it
is located in the same directory as the <code>step-49.cc</code> source file) to
and saved again by deal.II as an image (see the <code>grid_1</code> function
of the current program):
-<img src="http://www.dealii.org/images/steps/developer/step-49.grid-1.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-49.grid-1.png" alt="">
<h3>Modifying a Mesh</h3>
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-5a.png" alt=""> regular input mesh
+ <img src="https://www.dealii.org/images/steps/developer/step-49.grid-5a.png" alt=""> regular input mesh
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-5.png" alt=""> output mesh
+ <img src="https://www.dealii.org/images/steps/developer/step-49.grid-5.png" alt=""> output mesh
</td>
</tr>
</TABLE>
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-6a.png" alt=""> regular input mesh
+ <img src="https://www.dealii.org/images/steps/developer/step-49.grid-6a.png" alt=""> regular input mesh
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-6.png" alt=""> wall-adapted output mesh
+ <img src="https://www.dealii.org/images/steps/developer/step-49.grid-6.png" alt=""> wall-adapted output mesh
</td>
</tr>
</TABLE>
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-7a.png" alt=""> regular input mesh
+ <img src="https://www.dealii.org/images/steps/developer/step-49.grid-7a.png" alt=""> regular input mesh
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-7.png" alt=""> perturbed output mesh
+ <img src="https://www.dealii.org/images/steps/developer/step-49.grid-7.png" alt=""> perturbed output mesh
</td>
</tr>
</TABLE>
<table width="80%" align="center">
<tr>
- <td ALIGN="center"><img src="http://www.dealii.org/images/steps/developer/step-49.grid-2a.png" alt="" height="200px">input mesh 1</td>
- <td ALIGN="center"><img src="http://www.dealii.org/images/steps/developer/step-49.grid-2b.png" alt="" height="200px">input mesh 2</td>
- <td ALIGN="center"><img src="http://www.dealii.org/images/steps/developer/step-49.grid-2.png" alt="" height="200px">merged mesh</td>
+ <td ALIGN="center"><img src="https://www.dealii.org/images/steps/developer/step-49.grid-2a.png" alt="" height="200px">input mesh 1</td>
+ <td ALIGN="center"><img src="https://www.dealii.org/images/steps/developer/step-49.grid-2b.png" alt="" height="200px">input mesh 2</td>
+ <td ALIGN="center"><img src="https://www.dealii.org/images/steps/developer/step-49.grid-2.png" alt="" height="200px">merged mesh</td>
</tr>
</table>
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-3a.png" alt="" height="200px"> input mesh
+ <img src="https://www.dealii.org/images/steps/developer/step-49.grid-3a.png" alt="" height="200px"> input mesh
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-3.png" alt="" height="200px"> top vertices moved upwards
+ <img src="https://www.dealii.org/images/steps/developer/step-49.grid-3.png" alt="" height="200px"> top vertices moved upwards
</td>
</tr>
</table>
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-4base.png" alt=""> input mesh
+ <img src="https://www.dealii.org/images/steps/developer/step-49.grid-4base.png" alt=""> input mesh
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-49.grid-4.png" alt=""> extruded output mesh
+ <img src="https://www.dealii.org/images/steps/developer/step-49.grid-4.png" alt=""> extruded output mesh
</td>
</tr>
</table>
into this category. For example, for this mesh the central hole is supposed to
be round:
-<img src="http://www.dealii.org/images/steps/developer/step-49.grid-2a.png" alt="" height="200px">
+<img src="https://www.dealii.org/images/steps/developer/step-49.grid-2a.png" alt="" height="200px">
On the other hand, if you simply refine it, the Triangulation class can not
know whether you wanted the hole to be round or to be an octagon. The default
is to place new points along existing edges. After two mesh refinement steps,
this would yield the following mesh, which is not what we wanted:
-<img src="http://www.dealii.org/images/steps/developer/step-49.grid-2d-refined.png" alt="" height="200px">
+<img src="https://www.dealii.org/images/steps/developer/step-49.grid-2d-refined.png" alt="" height="200px">
What needs to happen is that you tell the triangulation that you in fact want
to use a curved boundary. The way to do this requires three steps:
microstructured electric device. In a CAD program, the geometry looks like
this:
-<img src="http://www.dealii.org/images/steps/developer/step-49.yuhan.1.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-49.yuhan.1.png" alt="">
In the following, we will walk you through the entire process of creating a
mesh for this geometry, including a number of common pitfalls by showing the
With this code, you get a mesh that looks like this:
-<img src="http://www.dealii.org/images/steps/developer/step-49.yuhan.2.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-49.yuhan.2.png" alt="">
The next step is to teach each of the top surfaces that they should be
curved. We can do this by creating CylinderBoundary objects that
With this code, we get a mesh that looks like this:
-<img src="http://www.dealii.org/images/steps/developer/step-49.yuhan.3.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-49.yuhan.3.png" alt="">
This is clearly not correct: The new vertices that have been entered at
mid-edge and mid-face points are not where they should have been. Upon some
This yields an improvement, though it is still not quite correct:
-<img src="http://www.dealii.org/images/steps/developer/step-49.yuhan.4.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-49.yuhan.4.png" alt="">
Looking closely at this mesh, we realize that the new points on mid-face
vertices are where they should be, though the new vertices inserted at
instead of TriaAccessor::set_boundary_id() used above. With this change,
the grid now looks like this:
-<img src="http://www.dealii.org/images/steps/developer/step-49.yuhan.5.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-49.yuhan.5.png" alt="">
This is already better. However, something is still going wrong on the
front left face. On second look, we can also see that the faces where
With this, we finally get a mesh that looks good:
-<img src="http://www.dealii.org/images/steps/developer/step-49.yuhan.6.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-49.yuhan.6.png" alt="">
We can then refine the mesh two more times to see in more detail what
happens to the curved part of the boundary:
- <img src="http://www.dealii.org/images/steps/developer/step-49.yuhan.7.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-49.yuhan.7.png" alt="">
So, yes!, this is finally what we were looking for!
<TABLE WIDTH="100%">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-5.solution-0.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-5.solution-0.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-5.solution-1.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-5.solution-1.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-5.solution-2.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-5.solution-2.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-5.solution-3.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-5.solution-3.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-5.solution-4.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-5.solution-4.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-5.solution-5.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-5.solution-5.png" alt="">
</td>
</tr>
</table>
<table align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.sol_2.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.sol_3.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.sol_2.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.sol_3.png" alt=""></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.sol_4.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.sol_8.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.sol_4.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.sol_8.png" alt=""></td>
</tr>
</table>
<table align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.post_2.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.post_3.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.post_2.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.post_3.png" alt=""></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.post_4.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.post_8.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.post_4.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.post_8.png" alt=""></td>
</tr>
</table>
<table align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.sol_q3_2.png" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.post_q3_2.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.sol_q3_2.png" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.post_q3_2.png" alt=""></td>
</tr>
</table>
<table align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.2d_plain.png" width="400" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.2dt_plain.png" width="400" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.2d_plain.png" width="400" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.2dt_plain.png" width="400" alt=""></td>
</tr>
</table>
<table align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.2d_post.png" width="400" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.2dt_post.png" width="400" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.2d_post.png" width="400" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.2dt_post.png" width="400" alt=""></td>
</tr>
</table>
<table align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.2d_postb.png" width="400" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.2dt_postb.png" width="400" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.2d_postb.png" width="400" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.2dt_postb.png" width="400" alt=""></td>
</tr>
</table>
<table align="center">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.3d_plain.png" width="400" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.3dt_plain.png" width="400" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.3d_plain.png" width="400" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.3dt_plain.png" width="400" alt=""></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.3d_post.png" width="400" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.3dt_post.png" width="400" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.3d_post.png" width="400" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.3dt_post.png" width="400" alt=""></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.3d_postb.png" width="400" alt=""></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-51.3dt_postb.png" width="400" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.3d_postb.png" width="400" alt=""></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-51.3dt_postb.png" width="400" alt=""></td>
</tr>
</table>
zone where two continental plates drift apart. Not to beat around the bush,
the geometry we want to describe looks like this:
-<img src="http://www.dealii.org/images/steps/developer/step-53.topo.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-53.topo.png" alt="">
In particular, though you cannot see this here, the top surface is not
just colored by the elevation but is, in fact, deformed to follow the
their shading algorithms to (because the top surfaces of the cells are now no
longer just tangential to a sphere but tilted):
-<img src="http://www.dealii.org/images/steps/developer/step-53.mesh.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-53.mesh.png" alt="">
Yet, at least as far as visualizations are concerned, this is still not too
impressive. Rather, let us visualize things in a way so that we show the
actual elevation along the top surface. In other words, we want a picture like
this, with an incredible amount of detail:
-<img src="http://www.dealii.org/images/steps/developer/step-53.topo.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-53.topo.png" alt="">
A zoom-in of this picture shows the vertical displacement quite clearly (here,
looking from the West-Northwest over the rift valley, the triple peaks
and toward the great flatness of
<a href="http://en.wikipedia.org/wiki/Lake_Victoria">Lake Victoria</a>):
-<img src="http://www.dealii.org/images/steps/developer/step-53.topozoom.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-53.topozoom.png" alt="">
These image were produced with three small modifications:
description in which we enlarge the vertical topography to enhance the effect
(courtesy of Alexander Grayver):
-<img src="http://www.dealii.org/images/steps/developer/step-53.smooth-geometry.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-53.smooth-geometry.png" alt="">
So what is happening here? Partly, this is only a result of visualization, but
there is an underlying real cause as well:
parts the curvilinear length of the CAD curve portion that lies between the
vertices of the original edge.
-<img src="http://www.dealii.org/images/steps/developer/step-54.CurveSplit.png" alt="" width="500">
+<img src="https://www.dealii.org/images/steps/developer/step-54.CurveSplit.png" alt="" width="500">
A different projection strategy has been implemented in the
composed of several sub-shapes, the projection is carried out onto every
single sub-shape, and the closest projection point point is selected.
-<img src="http://www.dealii.org/images/steps/developer/step-54.NormalProjectionEdge.png" alt="" width="500">
-<img src="http://www.dealii.org/images/steps/developer/step-54.NormalProjection.png" alt="" width="500">
+<img src="https://www.dealii.org/images/steps/developer/step-54.NormalProjectionEdge.png" alt="" width="500">
+<img src="https://www.dealii.org/images/steps/developer/step-54.NormalProjection.png" alt="" width="500">
As we are about to experience, for some shapes, setting the projection
direction as that normal to the CAD surface will not lead to surface mesh
higher control on the projection direction to be enforced to ensure good mesh
quality.
-<img src="http://www.dealii.org/images/steps/developer/step-54.DirectionalProjection.png" alt="" width="500">
+<img src="https://www.dealii.org/images/steps/developer/step-54.DirectionalProjection.png" alt="" width="500">
Of course the latter approach is effective only when the orientation of the
which a more specific approach is known to be appropriate.
-<img src="http://www.dealii.org/images/steps/developer/step-54.ProjectionComparisons.png" alt="" width="700">
+<img src="https://www.dealii.org/images/steps/developer/step-54.ProjectionComparisons.png" alt="" width="700">
<h3> The testcase </h3>
try to do this here. To already give you an idea of the geometry we consider,
here is a picture:
-<img src="http://www.dealii.org/images/steps/developer/step-54.bare.png" alt="" width="500">
+<img src="https://www.dealii.org/images/steps/developer/step-54.bare.png" alt="" width="500">
In the program, we read both the geometry and a coarse mesh from files, and
then employ several of the options discussed above to place new vertices for a
<table style="width:90%">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.common_0.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_1.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_2.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.common_0.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_1.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_2.png" alt="" width="400"></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_3.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_4.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_5.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_3.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_4.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_5.png" alt="" width="400"></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_front_3.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_front_4.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_front_5.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_front_3.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_front_4.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_front_5.png" alt="" width="400"></td>
</tr>
</table>
<table style="width:90%">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.common_0.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.directional_1.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.directional_2.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.common_0.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.directional_1.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.directional_2.png" alt="" width="400"></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.directional_3.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.directional_4.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.directional_5.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.directional_3.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.directional_4.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.directional_5.png" alt="" width="400"></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.directional_front_3.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.directional_front_4.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.directional_front_5.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.directional_front_3.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.directional_front_4.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.directional_front_5.png" alt="" width="400"></td>
</tr>
</table>
<table style="width:90%">
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.common_0.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_1.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_2.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.common_0.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_1.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_2.png" alt="" width="400"></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_3.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_4.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_5.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_3.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_4.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_5.png" alt="" width="400"></td>
</tr>
<tr>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_front_3.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_front_4.png" alt="" width="400"></td>
- <td><img src="http://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_front_5.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_front_3.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_front_4.png" alt="" width="400"></td>
+ <td><img src="https://www.dealii.org/images/steps/developer/step-54.normal_to_mesh_front_5.png" alt="" width="400"></td>
</tr>
</table>
-<img src="http://www.dealii.org/images/steps/developer/step-6.solution.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-6.solution.png" alt="">
<TABLE WIDTH="100%">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.grid-0.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.grid-0.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.grid-1.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.grid-1.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.grid-2.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.grid-2.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.grid-3.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.grid-3.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.grid-4.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.grid-4.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.grid-5.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.grid-5.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.grid-6.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.grid-6.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.grid-7.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.grid-7.png" alt="">
</td>
</tr>
</table>
<TABLE WIDTH="60%" ALIGN="center">
<tr>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-6.q2.dofs_vs_iterations.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.q2.dofs_vs_iterations.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-6.q2.dofs_vs_time.png" alt="">
+ <img src="https://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">
- <img src="http://www.dealii.org/images/steps/developer/step-6.q1.dofs_vs_iterations.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.q1.dofs_vs_iterations.png" alt="">
</td>
<td ALIGN="center">
- <img src="http://www.dealii.org/images/steps/developer/step-6.q1.dofs_vs_time.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.q1.dofs_vs_time.png" alt="">
</td>
</tr>
</table>
After a few global refinement steps, this would lead to a mesh of the following
kind:
-<img src="http://www.dealii.org/images/steps/developer/step-6.manifold-grid-4-bad.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-6.manifold-grid-4-bad.png" alt="">
This is not a good mesh: the central cell has been refined in such a way that
the children located in the four corners of the original central cell
<TABLE WIDTH="100%">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.manifold-grid-0.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.manifold-grid-0.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.manifold-grid-1.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.manifold-grid-1.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.manifold-grid-2.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.manifold-grid-2.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.manifold-grid-3.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.manifold-grid-3.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.manifold-grid-4.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.manifold-grid-4.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.manifold-grid-5.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.manifold-grid-5.png" alt="">
</td>
</tr>
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.manifold-grid-6.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.manifold-grid-6.png" alt="">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-6.manifold-grid-7.png" alt="">
+ <img src="https://www.dealii.org/images/steps/developer/step-6.manifold-grid-7.png" alt="">
</td>
</tr>
</table>
here:
-<img src="http://www.dealii.org/images/steps/developer/step-7.solution.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-7.solution.png" alt="">
<TABLE WIDTH="100%">
<tr>
<td>
-<img src="http://www.dealii.org/images/steps/developer/step-8.x.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-8.x.png" alt="">
</td>
<td>
-<img src="http://www.dealii.org/images/steps/developer/step-8.y.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-8.y.png" alt="">
</td>
</tr>
</table>
$x=-0.5$, and of $y$-displacement at the origin. The next image shows the
final grid after eight steps of refinement:
-<img src="http://www.dealii.org/images/steps/developer/step-8.grid.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-8.grid.png" alt="">
What one frequently would like to do is to show the displacement as a vector
hundred vertices from which to draw the vectors; drawing them from each
individual vertex would make the picture unreadable):
-<img src="http://www.dealii.org/images/steps/developer/step-8.vectors.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-8.vectors.png" alt="">
We note that one may have intuitively expected the
displayed in the following picture:
-<img src="http://www.dealii.org/images/steps/developer/step-9.grid.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-9.grid.png" alt="">
solution itself:
-<img src="http://www.dealii.org/images/steps/developer/step-9.solution.png" alt="">
+<img src="https://www.dealii.org/images/steps/developer/step-9.solution.png" alt="">