From 11938436b910bdd8cf6635acd09197b1f3ce5216 Mon Sep 17 00:00:00 2001 From: wolf Date: Tue, 28 Mar 2006 06:15:27 +0000 Subject: [PATCH] More doxygenization work git-svn-id: https://svn.dealii.org/trunk@12708 0785d39b-7218-0410-832d-ea1e28bc413d --- .../step-10.data/ball_mapping_q1_ref0.jpg | Bin 13387 -> 0 bytes .../step-10.data/ball_mapping_q1_ref1.jpg | Bin 20246 -> 0 bytes .../step-10.data/ball_mapping_q2_ref0.jpg | Bin 17337 -> 0 bytes .../step-10.data/ball_mapping_q2_ref1.jpg | Bin 20057 -> 0 bytes .../step-10.data/ball_mapping_q3_ref0.jpg | Bin 16295 -> 0 bytes .../step-10.data/ball_mapping_q3_ref1.jpg | Bin 20084 -> 0 bytes .../step-10.data/{intro.html => intro.dox} | 36 +++++----- .../step-10.data/quarter-q1.jpg | Bin 7951 -> 0 bytes .../step-10.data/quarter-q2.jpg | Bin 7688 -> 0 bytes .../step-10.data/quarter-q3.jpg | Bin 8013 -> 0 bytes .../{results.html => results.dox} | 68 +++++++++--------- .../step-10.ball_mapping_q1_ref0.png | Bin 0 -> 13197 bytes 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Introduction

-

+ This is a rather short example which only shows some aspects of using -higher order mappings. By mapping we mean the transformation +higher order mappings. By mapping we mean the transformation between the unit cell (i.e. the unit line, square, or cube) to the cells in real space. In all the previous examples, we have implicitly used linear or d-linear mappings; you will not have noticed this at @@ -14,16 +14,16 @@ cases where the piecewise linear approximation of the boundary your computational domain is an approximation to the real domain using curved boundaries as well. If the boundary approximation uses piecewise quadratic parabolas to approximate the true boundary, then -we say that this is a quadratic or Q2 approximation. If we -use piecewise graphs of cubic polynomials, then this is a Q3 +we say that this is a quadratic or $Q_2$ approximation. If we +use piecewise graphs of cubic polynomials, then this is a $Q_3$ approximation, and so on. -

-

+ + For some differential equations, it is known that piecewise linear -approximations of the boundary, i.e. Q1 mappings, are not +approximations of the boundary, i.e. $Q_1$ mappings, are not sufficient if the boundary of the domain is curved. Examples are the -biharmonic equation using C1 elements, or the Euler +biharmonic equation using $C^1$ elements, or the Euler equation on domains with curved reflective boundaries. In these cases, it is necessary to compute the integrals using a higher order mapping. The reason, of course, is that if we do not use a higher @@ -31,31 +31,31 @@ order mapping, the order of approximation of the boundary dominates the order of convergence of the entire numerical scheme, irrespective of the order of convergence of the discretization in the interior of the domain. -

-

+ + Rather than demonstrating the use of higher order mappings with one of these more complicated examples, we do only a brief computation: -calculating the value of pi=3.141592653589793238462643... by two +calculating the value of $\pi=3.141592653589793238462643\ldots$ by two different methods. -

-

+ + The first method uses a triangulated approximation of the circle with unit radius and integrates the unit function over it. Of course, if the domain were the exact unit circle, then the area would be pi, but since we only use an approximation by piecewise polynomial segments, the value of the area is not exactly pi. However, it is known that as -we refine the triangulation, a Qp mapping approximates the boundary -with an order hp+1, where h is the mesh +we refine the triangulation, a $Q_p$ mapping approximates the boundary +with an order $h^{p+1}$, where $h$ is the mesh width. We will check the values of the computed area of the circle and their convergence towards pi under mesh refinement for different mappings. We will also find a convergence behavior that is surprising at first, but has a good explanation. -

-

+ + The second method works similarly, but this time does not use the area of the triangulated unit circle, but rather its perimeter. Pi is then approximated by half of the perimeter, as the radius is equal to one. -

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Results

-

+ The program performs two tasks, the first being to generate a visualization of the mapped domain, the second to compute pi by the two methods described. Let us first take a look at the generated graphics. They are generated in Gnuplot format, and can be viewed with the commands -

+@code
 set data style lines
 set size 0.721, 1
 set nokey
 plot [-1:1][-1:1] "ball0_mapping_q1.dat"
-
+@endcode or using one of the other filenames. The second line makes sure that the aspect ratio of the generated output is actually 1:1, i.e. a circle is drawn as a circle on your screen, rather than as an ellipse. The third line switches off the key in the graphic, as that will only print information (the filename) which is not that important right now. -

-

+ + The following table shows the triangulated computational domain for Q1, Q2, and Q3 mappings, for the original coarse grid (left), and a once uniformly refined grid (right). If your browser does not display @@ -29,28 +29,28 @@ these pictures in acceptable quality, view them one by one.
- Q1 mapping, coarse grid + @image html step-10.ball_mapping_q1_ref0.png - Q1 mapping, once refined grid + @image html step-10.ball_mapping_q1_ref1.png
- Q2 mapping, coarse grid + @image html step-10.ball_mapping_q2_ref0.png - Q2 mapping, once refined grid + @image html step-10.ball_mapping_q2_ref1.png
- Q3 mapping, coarse grid + @image html step-10.ball_mapping_q3_ref0.png - Q3 mapping, once refined grid + @image html step-10.ball_mapping_q3_ref1.png
@@ -62,13 +62,13 @@ with dashed lines the exact circle:
- Q1 mapping, coarse grid + @image html step-10.quarter-q1.png - Q2 mapping, coarse grid + @image html step-10.quarter-q2.png - Q3 mapping, coarse grid + @image html step-10.quarter-q3.png
@@ -82,12 +82,12 @@ only on cells at the boundary. Higher order mappings are therefore usually not noticably slower than lower order ones, because the additional computations are only performed on a small subset of all cells. -

-

+ + The second purpose of the program was to compute the value of pi to good accuracy. This is the output of this part of the program: -

+@code
 Computation of Pi by the area:
 ==============================
 Degree = 1
@@ -164,31 +164,31 @@ cells         eval.pi         error
 320     3.1415926540544636      4.6467e-10      6.00
 1280    3.1415926535970535      7.2602e-12      6.00
 5120    3.1415926535899010      1.0805e-13      6.07
-
-

+@endcode + + -

One of the immediate observations from the output is that in all cases the values converge quickly to the true value of -pi=3.141592653589793238462643. Note that for the Q4 mapping, the last +$\pi=3.141592653589793238462643$. Note that for the $Q_4$ mapping, the last number is correct to 13 digits in both computations, which is already -quite a lot. However, also note that for the Q1 mapping, even on the +quite a lot. However, also note that for the $Q_1$ mapping, even on the finest grid the accuracy is significantly worse than on the coarse -grid for a Q4 mapping! -

+grid for a $Q_4$ mapping! + + -

The last column of the output shows the convergence order, in powers -of the mesh width h. In the introduction, we had stated that -the convergence order for a Qp mapping should be -hp+1. However, in the example shown, the Q2 and Q4 -mappings show a convergence order of hp+2! This at +of the mesh width $h$. In the introduction, we had stated that +the convergence order for a $Q_p$ mapping should be +$h^{p+1}$. However, in the example shown, the $Q_2$ and $Q_4$ +mappings show a convergence order of $h^{p+2}$! This at first surprising fact is readily explained by the particular boundary we have chosen in this example. In fact, the circle is described by the function -sqrt(1-x2), which has the series expansion -1-x2/2-x4/8-x6/16+... -around x=0. Thus, for the quadratic mapping where the +$\sqrt{1-x^2}$, which has the series expansion +$1-x^2/2-x^4/8-x^6/16+\ldots$ +around $x=0$. Thus, for the quadratic mapping where the truncation error of the quadratic approximation should be cubic, there is no such term but only a quartic one, which raises the convergence -order to 4, instead of 3. The same happens for the Q4 mapping. -

+order to 4, instead of 3. 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Results

-

+ The output of this program consist of the console output, the eps files including the grids, and the solutions given in gnuplot format. - -

+@code
 Cycle 0:
    Number of active cells:       64
    Number of degrees of freedom: 256
 Time of assemble_system1: 0.05
 Time of assemble_system2: 0.04
 solution1 and solution2 coincide.
-Writing grid to <grid-0.eps>...
-Writing solution to <sol-0.gnuplot>...
+Writing grid to ...
+Writing solution to ...
 
 Cycle 1:
    Number of active cells:       112
@@ -21,8 +20,8 @@ Cycle 1:
 Time of assemble_system1: 0.09
 Time of assemble_system2: 0.07
 solution1 and solution2 coincide.
-Writing grid to <grid-1.eps>...
-Writing solution to <sol-1.gnuplot>...
+Writing grid to ...
+Writing solution to ...
 
 Cycle 2:
    Number of active cells:       214
@@ -30,8 +29,8 @@ Cycle 2:
 Time of assemble_system1: 0.17
 Time of assemble_system2: 0.14
 solution1 and solution2 coincide.
-Writing grid to <grid-2.eps>...
-Writing solution to <sol-2.gnuplot>...
+Writing grid to ...
+Writing solution to ...
 
 Cycle 3:
    Number of active cells:       415
@@ -39,8 +38,8 @@ Cycle 3:
 Time of assemble_system1: 0.32
 Time of assemble_system2: 0.28
 solution1 and solution2 coincide.
-Writing grid to <grid-3.eps>...
-Writing solution to <sol-3.gnuplot>...
+Writing grid to ...
+Writing solution to ...
 
 Cycle 4:
    Number of active cells:       796
@@ -48,8 +47,8 @@ Cycle 4:
 Time of assemble_system1: 0.62
 Time of assemble_system2: 0.52
 solution1 and solution2 coincide.
-Writing grid to <grid-4.eps>...
-Writing solution to <sol-4.gnuplot>...
+Writing grid to ...
+Writing solution to ...
 
 Cycle 5:
    Number of active cells:       1561
@@ -57,35 +56,34 @@ Cycle 5:
 Time of assemble_system1: 1.23
 Time of assemble_system2: 1.03
 solution1 and solution2 coincide.
-Writing grid to <grid-5.eps>...
-Writing solution to <sol-5.gnuplot>...
+Writing grid to ...
+Writing solution to ...
+@endcode
 
-
- We see that, as expected, on each refinement step the two solutions coincide. The difference measured in time of treating each face only once (second version of the DG method) in comparison with treating each face twice within a nested loop over all cells and all faces of each cell (first version), is much less than one might have expected. The gain is less than 20% on the last few refinement steps. -

-

First we show the solutions on the initial mesh, the mesh after two + + First we show the solutions on the initial mesh, the mesh after two and after five adaptive refinement steps. -

- sol-0.gnuplot - sol-2.gnuplot - sol-5.gnuplot -

-

Then we show the final grid (after 5 refinement steps). The +@image step-12.sol-0.png +@image step-12.sol-2.png +@image step-12.sol-5.png + + +Then we show the final grid (after 5 refinement steps). The grid is largely concentrated in the vicinity of the jump of the solution. -

- grid-5.eps -

- -

And finally we show a plot of a 3d computation. -

- sol-5-3d.eps -

+ +@image step-12.grid-5.png + + +And finally we show a plot of a 3d computation. + +@image html step-12.sol-5-3d.png + diff --git a/deal.II/doc/tutorial/chapter-2.step-by-step/step-12.data/sol-0.gif b/deal.II/doc/tutorial/chapter-2.step-by-step/step-12.data/sol-0.gif deleted file mode 100644 index d4d0b524692d37da03263422b50f302b9830dd07..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 103727 zcmd^o4Rl?_edp-u>*>R`Yzrg2!oF68^WZqn!!fN=TzqX55W!9#0e3}Fk|z*CJp}Rq z7kdwj7|F7a>JqG^uCZuSKYx_CE{V-YST8zTKO0(`6R|dJjt`L4`gz&Sk|tHfl5L%a z)!mu`_H{I|M}mUxwqeOTgMlA2P!@V|9S-|=m&r@ zIjJmMsI;{yQ&UQ3r?PFE(%-L)k1G=s%9bt4=%_M1t$hCT%CTb;%EYuXy<6FROgXkq z**2<-jw|C+%G7hpbCb&C)DQPg?|J6ZTQ9z>ysX@Gld^B0a`rX0AD#wp2 zPd#;BIloug`~AtM&W(QM()9U#%Dxwr7yfa4=ZjN2l{06QyYE)IyC=`>n|l7qi9hRp z_08=EUfUuo^3;nv$1gsn#A4k__cO{f$``*lpbTt7a-Ki0y!`S3<-m_dw;WfFD_{9a zN=Z$8uTR;tXZ-tJN+NN5`oQ%0QW>C~IJH+faA10R8X%p>vhw7U%HF*b6BFA829%y2C6yY#|9<7Z`;>TG 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Introduction

-

+ This example shows the basic usage of the multilevel functions in -deal.II. It solves the Helmholtz equation with Neumann boundary conditions +deal.II. It solves the Helmholtz equation with Neumann boundary conditions to avoid additional complications due to Dirichlet boundary conditions (there, some library functions are missing). Therefore, the solution is the constant function with value unity. In all other respects, it is similar to step 5. -

-

In order to allow sufficient flexibility in conjunction with systems of + + In order to allow sufficient flexibility in conjunction with systems of differential equations and block preconditioners, quite a few different objects have to be created before starting the multilevel method. These are

    @@ -19,17 +19,17 @@ have to be created before starting the multilevel method. These are
  • MGMatrix, the matrix object having a special level multiplication, i.e. we basically store one matrix per grid level and allow multiplication with it.
-

-

+ + These objects are combined in an object of type Multigrid, containing the implementation of the V-cycle, which is in turn used by the preconditioner PreconditionMG, ready for plug-in into a linear solver of the LAC library. -

-

-The multilevel method in deal.II follows in many respects the outlines + + +The multilevel method in deal.II follows in many respects the outlines of the various publications by James Bramble, Joseph Pasciak and Jinchao Xu. In order to understand many of the options, a rough familiarity with their work is quite helpful. -

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Results

-

+ There is not much to be said about the results of this program, apart from that they look nice. All images were made using GMV from the output files that the program wrote to disk. The first picture shows the displacement as a vector field, where one vector is shown at each vertex of the grid: -

-

-displacement-vectors -

-

+@image html step-8.vectors.png + + You can clearly see the sources of x-displacement around x=0.5 and x=-0.5, and of y-displacement at the origin. The next image shows the final grid after eight steps of refinement: -

-

-final-grid -

-

+@image html step-8.grid.png + + + Finally, the x-displacement and y-displacement are displayed separately: -

-

+ +
-displacement-x +@image html step-8.x.png -displacement-y +@image html step-8.y.png
-

-

+ + It should be noted that intuitively one would have expected the solution to be symmetric about the x- and y-axes since the x- and y-forces are symmetric with respect to these axes. However, the force considered as a vector is not symmetric and consequently neither is the solution. -

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Results

-

+ The results of this program are not particularly spectacular. They consist of the console output, some grid files, and the solution on the finest grid. First for the console output: - -

+@code
 Cycle 0:
    Number of active cells:       256
    Number of degrees of freedom: 289
@@ -25,31 +24,27 @@ Cycle 4:
 Cycle 5:
    Number of active cells:       26002
    Number of degrees of freedom: 30035
-
- +@endcode As can be seen, quite a number of cells is used on the finest level to resolve the features of the solution. The final grid showing this is displayed in the following picture: -

-

- final grid -

-

+@image html step-9.grid.png + + + The structure of the grid will be understandable by looking at the solution itself: -

-

- final solution -

-

+@image html step-9.solution.png + + + Note that the solution is created by that part that is transported along the wiggled advection field from the left and lower boundaries to the top right, and the part that is created by the source in the lower left corner, and the results of which are also transported along. The grid shown above is well-adapted to resolve these features. -

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