propagating in the medium (in our simulation, we assume the medium is mineral
oil, which has a acoustic speed of 1.437 $\frac{mm}{\mu s}$):
-@image html step-24.one_movie.gif
+<img src="http://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:
-@image html step-24.traces.png
+<img src="http://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>
-@image html step-24.one.png
+<img src="http://www.dealii.org/images/steps/developer/step-24.one.png" alt="">
</td>
<td>
-@image html step-24.one_s.png
+<img src="http://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:
-@image html step-24.one_sf.png
+<img src="http://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:
-@image html step-24.one_s2.png
+<img src="http://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:
-@image html step-24.multi_movie.gif
+<img src="http://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>
-@image html step-24.multi.png
+<img src="http://www.dealii.org/images/steps/developer/step-24.multi.png" alt="">
</td>
<td>
-@image html step-24.multi_s.png
+<img src="http://www.dealii.org/images/steps/developer/step-24.multi_s.png" alt="">
</td>
</tr>
</table>
<TABLE WIDTH="100%">
<tr>
<td>
-@image html step-24.multi_sf.png
+<img src="http://www.dealii.org/images/steps/developer/step-24.multi_sf.png" alt="">
</td>
<td>
-@image html step-24.multi_s2.png
+<img src="http://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.
-@image html step-25.1d-breather.gif "Animation of the 1D stationary breather." width=5cm
+<img src="http://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.
-@image html step-25.1d-breather_stp.png "A space-time plot of the 1D stationary breather." width=5cm
+<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.">
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.
-@image html step-25.2d-kink.png "Stationary 2D kink." width=5cm
+<img src="http://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.
-@image html step-25.2d-angled_kink.gif "Animation of a moving 2D kink, at 45 degrees to the axes of the grid, showing boundary effects." width=5cm
+<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.">
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.
-@image html step-25.2d-pseudobreather.gif "Animation of a 2D pseudobreather." width=5cm
+<img src="http://www.dealii.org/images/steps/developer/step-25.2d-pseudobreather.gif" alt="Animation of a 2D pseudobreather.">
<a name="extensions"></a>
can pass to our visualization program of choice. When we collate them into a
movie, the results looks like this:
-@image html step-33.slide.gif
+<img src="http://www.dealii.org/images/steps/developer/step-33.slide.gif" alt="">
As we see, the heavy mass of fluid is drawn down the slope by gravity, where
it collides with the ski lodge and is flung into the air! Hopefully everyone
We can also visualize the evolution of the adaptively refined grid:
-@image html step-33.slide_adapt.gif
+<img src="http://www.dealii.org/images/steps/developer/step-33.slide_adapt.gif" alt="">
The adaptivity follows and precedes the flow pattern, based on the heuristic
refinement scheme discussed above.
projects / dealii-svn.git / commitdiff
