<a name="Results"></a>
<h1>Results</h1>
-The parameter file that comes with the tutorial code sets
-d=0.3, which amounts to a focus of the transducer lens
-at x=0.5, y=0.3. The coarse mesh is refined 5 times,
+The current program reads its run-time parameters from an input file
+called <code>step-29.prm</code> that looks like this:
+@code
+subsection Mesh & geometry parameters
+ # Distance of the focal point of the lens to the x-axis
+ set Focal distance = 0.3
+
+ # Number of global mesh refinement steps applied to initial coarse grid
+ set Number of refinements = 5
+end
+
+
+subsection Physical constants
+ # Wave speed
+ set c = 1.5e5
+
+ # Frequency
+ set omega = 3.0e7
+end
+
+
+subsection Output parameters
+ # Name of the output file (without extension)
+ set Output file = solution
+
+ # A name for the output format to be used
+ set Output format = gmv
+end
+@endcode
+
+As can be seen, we set
+$d=0.3$, which amounts to a focus of the transducer lens
+at $x=0.5$, $y=0.3$. The coarse mesh is refined 5 times,
resulting in 160x160 cells, and the output is written in gmv
-format.
+format. The parameter reader understands many more parameters
+pertaining in particular to the generation of output, see the
+explanation in @ref step_19 "step-19", but we need none of these
+parameters here and therefore stick with their default values.
Here's the console output of the program in debug mode:
@code
examples/step-29> make run
============================ Running step-29
-DEAL::Generating grid... done (1.34000s)
+DEAL::Generating grid... done (1.11607s)
DEAL:: Number of active cells: 25600
-DEAL::Setting up system... done (1.17000s)
+DEAL::Setting up system... done (1.10807s)
DEAL:: Number of degrees of freedom: 51842
-DEAL::Assembling system matrix... done (7.52000s)
-DEAL::Solving linear system... done (4.43000s)
-DEAL::Generating output... done (5.94000s)
+DEAL::Assembling system matrix... done (6.50841s)
+DEAL::Solving linear system... done (2.89218s)
+DEAL::Generating output... done (4.52428s)
@endcode
(Of course, execution times will differ if you run the program
locally.) The fact that most of the time is spent on assembling
-the system matrix and generating output is due to some extra checks
-done in debug mode, in optimized mode these parts of the program
-run much faster:
+the system matrix and generating output is due to the many assertion
+that need to be checked in debug mode. In optimized mode these parts
+of the program run much faster whereas solving the linear system is
+hardly sped up at all:
@code
-examples/step-29> make run
============================ Running step-29
-DEAL::Generating grid... done (0.0500000s)
+DEAL::Generating grid... done (0.0280020s)
DEAL:: Number of active cells: 25600
-DEAL::Setting up system... done (0.170000s)
+DEAL::Setting up system... done (0.112007s)
DEAL:: Number of degrees of freedom: 51842
-DEAL::Assembling system matrix... done (0.370000s)
-DEAL::Solving linear system... done (3.65000s)
-DEAL::Generating output... done (1.23000s)
+DEAL::Assembling system matrix... done (0.160010s)
+DEAL::Solving linear system... done (2.21614s)
+DEAL::Generating output... done (0.880055s)
@endcode
The graphical output of the program looks as follows:
-@image html step-29.v.png "v = Re(u)" width=5cm
+<table align="center" border="1" cellspacing="3" cellpadding="3">
+ <tr>
+ <td>
+ @image html step-29.v.png "v = Re(u)" width=4cm
+ </td>
+ <td>
+ @image html step-29.w.png "w = Im(u)" width=4cm
+ </td>
+ </tr>
-@image html step-29.w.png "w = Im(u)" width=5cm
+ <tr>
+ <td colspan="2">
+ @image html step-29.intensity.png "|u|" width=4cm
+ </td>
+ </tr>
+</table>
-@image html step-29.intensity.png "|u|" width=5cm
-
-The first two picturse show the real and imaginary parts of
+The first two pictures show the real and imaginary parts of
$u$, whereas the last shows the intensity $|u|$. One can clearly
see that the intensity is focussed around the focal point of the
lens (0.5, 0.3), and that the focus
-is rather sharp in x-direction but more blurred in y-direction, which is a
-consequence of the geometry and the wave nature of the problem.
+is rather sharp in $x$-direction but more blurred in $y$-direction, which is a
+consequence of the geometry of the focusing lens, its finite aperture,
+and the wave nature of the problem.
+
+Because colorful graphics are always fun, and to stress the focusing
+effects some more, here is another set of images highlighting how well
+the intensity is actually focused in $x$-direction:
+
+<table align="center" border="1" cellspacing="3" cellpadding="3">
+ <tr>
+ <td>
+ @image html step-29.surface.png "|u|" width=4cm
+ </td>
+ <td>
+ @image html step-29.contours.png "|u|" width=4cm
+ </td>
+ </tr>
+</table>
+As a final note, the structure of the program makes it easy to
+determine which parts of the program scale nicely as the mesh is
+refined and which parts don't. Here are the run times for 5, 6, and 7
+global refinements:
+
+@code
+DEAL::Generating grid... done (0.0320020s)
+DEAL:: Number of active cells: 25600
+DEAL::Setting up system... done (0.104006s)
+DEAL:: Number of degrees of freedom: 51842
+DEAL::Assembling system matrix... done (0.164011s)
+DEAL::Solving linear system... done (2.23214s)
+DEAL::Generating output... done (0.900056s)
+
+DEAL::Generating grid... done (0.132009s)
+DEAL:: Number of active cells: 102400
+DEAL::Setting up system... done (0.408025s)
+DEAL:: Number of degrees of freedom: 206082
+DEAL::Assembling system matrix... done (0.656041s)
+DEAL::Solving linear system... done (14.8849s)
+DEAL::Generating output... done (3.57222s)
+
+DEAL::Generating grid... done (0.504031s)
+DEAL:: Number of active cells: 409600
+DEAL::Setting up system... done (1.72011s)
+DEAL:: Number of degrees of freedom: 821762
+DEAL::Assembling system matrix... done (2.63216s)
+DEAL::Solving linear system... done (117.811s)
+DEAL::Generating output... done (15.1489s)
+@endcode
+
+Each time we refine the mesh once, so the number of cells and degrees
+of freedom roughlt quadruples from each step to the next. As can be seen,
+generating the grid, setting up degrees of freedom, assembling the
+linear system, and generating output scale pretty closely to linear,
+whereas solving the linear system is an operation that requires 8
+times more time each time the number of degrees of freedom is
+increased by a factor of 4, i.e. it is ${\cal O}(N^{3/2})$. This can
+be explained by the fact that (using optimal ordering) the
+bandwidth of a finite element matrix is $B={\cal O}(N^{(dim-1)/dim})$,
+and the effort to solve a banded linear system using LU decomposition
+is ${\cal O}(BN)$. This also explains why the program does run in 3d
+as well (after changing the dimension on the
+<code>UltrasoundProblem</code> object), but scales very badly and
+takes extraordinate patience before it finishes solving the linear
+system.
\ No newline at end of file
projects / dealii.git / commitdiff