From: Wolfgang Bangerth Date: Wed, 29 Mar 2006 00:20:09 +0000 (+0000) Subject: Doxygenize most of step-14 and 18. X-Git-Tag: v8.0.0~11969 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=abdac5568b0bd00913dca9c86949f3a6ecde7548;p=dealii.git Doxygenize most of step-14 and 18. git-svn-id: https://svn.dealii.org/trunk@12746 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-14/doc/point_derivative/error-estimation.eps b/deal.II/examples/step-14/doc/point_derivative/error-estimation.eps deleted file mode 100644 index d85bfa2a3e..0000000000 --- a/deal.II/examples/step-14/doc/point_derivative/error-estimation.eps +++ /dev/null @@ -1,621 +0,0 @@ -%!PS-Adobe-2.0 EPSF-2.0 -%%Title: error-estimation.eps -%%Creator: gnuplot 3.7 patchlevel 1 -%%CreationDate: Wed May 22 09:12:26 2002 -%%DocumentFonts: (atend) -%%BoundingBox: 50 50 266 201 -%%Orientation: Portrait -%%EndComments -/gnudict 256 dict def -gnudict begin -/Color true def -/Solid false def -/gnulinewidth 5.000 def -/userlinewidth gnulinewidth def -/vshift -46 def -/dl {10 mul} def -/hpt_ 31.5 def -/vpt_ 31.5 def -/hpt hpt_ def -/vpt vpt_ def -/M {moveto} bind def -/L {lineto} bind def -/R {rmoveto} bind def -/V {rlineto} bind def -/vpt2 vpt 2 mul def -/hpt2 hpt 2 mul def -/Lshow { currentpoint stroke M - 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0 vshift R show } def -/Rshow { currentpoint stroke M - dup stringwidth pop neg vshift R show } def -/Cshow { currentpoint stroke M - dup stringwidth pop -2 div vshift R show } def -/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def - /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def -/DL { Color {setrgbcolor Solid {pop []} if 0 setdash } - {pop pop pop Solid {pop []} if 0 setdash} ifelse } def -/BL { stroke userlinewidth 2 mul setlinewidth } def -/AL { stroke userlinewidth 2 div setlinewidth } def -/UL { dup gnulinewidth mul /userlinewidth exch def - 10 mul /udl exch def } def -/PL { stroke userlinewidth setlinewidth } def -/LTb { BL [] 0 0 0 DL } def -/LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def -/LT0 { PL [] 1 0 0 DL } def -/LT1 { PL [4 dl 2 dl] 0 1 0 DL } def -/LT2 { PL [2 dl 3 dl] 0 0 1 DL } def -/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def -/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def -/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def -/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def -/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def -/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def -/Pnt { stroke [] 0 setdash - 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2 copy vpt 0 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C4 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 180 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C5 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc - 2 copy moveto - 2 copy vpt 180 270 arc closepath fill - vpt 0 360 arc } bind def -/C6 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C7 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 270 arc closepath fill - vpt 0 360 arc closepath } bind def -/C8 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C9 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 450 arc closepath fill - vpt 0 360 arc closepath } bind def -/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill - 2 copy moveto - 2 copy vpt 90 180 arc closepath fill - vpt 0 360 arc closepath } bind def -/C11 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 180 arc closepath fill - 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C12 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C13 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc closepath fill - 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/C14 { BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 360 arc closepath fill - vpt 0 360 arc } bind def -/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill - vpt 0 360 arc closepath } bind def -/Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto - neg 0 rlineto closepath } bind def -/Square { dup Rec } bind def -/Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def -/S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def -/S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def -/S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def -/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def -/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def -/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill - 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0 vshift R show } def -/Rshow { currentpoint stroke M - dup stringwidth pop neg vshift R show } def -/Cshow { currentpoint stroke M - dup stringwidth pop -2 div vshift R show } def -/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def - /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def -/DL { Color {setrgbcolor Solid {pop []} if 0 setdash } - {pop pop pop Solid {pop []} if 0 setdash} ifelse } def -/BL { stroke userlinewidth 2 mul setlinewidth } def -/AL { stroke userlinewidth 2 div setlinewidth } def -/UL { dup gnulinewidth mul /userlinewidth exch def - 10 mul /udl exch def } def -/PL { stroke userlinewidth setlinewidth } def -/LTb { BL [] 0 0 0 DL } def -/LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def -/LT0 { PL [] 1 0 0 DL } def -/LT1 { PL [4 dl 2 dl] 0 1 0 DL } def -/LT2 { PL [2 dl 3 dl] 0 0 1 DL } def -/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def -/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def -/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def -/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def -/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def -/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def -/Pnt { stroke [] 0 setdash - 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Point values

-

+ This program offers a lot of possibilities to play around. We can thus only show a small part of all possible results that can be obtained with the help of this program. However, you are encouraged to just try it out, by changing the settings in the main program. Here, we start by simply letting it run, unmodified: - -

+@code
 Refinement cycle: 0
    Number of degrees of freedom=72
    Point value=0.03243
@@ -46,27 +45,19 @@ Refinement cycle: 7
 Refinement cycle: 8
    Number of degrees of freedom=22303
    Point value=0.0334445
-
- -

+@endcode + -

First let's look what the program actually computed. On the fifth grid, primal and dual numerical solutions look like this:
- Primal solution - + @image html step-14.point-value.solution-5.png - Dual solution - + @image html step-14.point-value.solution-5-dual.png
@@ -77,57 +68,39 @@ to the symmetry in right hand side and domain, the solution should actually look like at the top right in all four corners, but the mesh refinement criterion involving the dual solution chose to refine them differently. -

-

+ + Looking at the grids that are produced in the course of subsequent refinement, these are some of them:
- - Grid 0 - + @image html step-14.point-value.grid-0.png - - Grid 2 - + @image html step-14.point-value.grid-2.png
- - Grid 4 - + @image html step-14.point-value.grid-4.png - - Grid 5 - + @image html step-14.point-value.grid-5.png
- - Grid 7 - + @image html step-14.point-value.grid-7.png - - Grid 8 - + @image html step-14.point-value.grid-8.png
@@ -137,9 +110,9 @@ difficult to generate such a mesh by hand, as this would involve to judge quantitatively how much which of the four corner singularities shall be resolved, and to set the weight compared to the vicinity of the evaluation point. -

-

+ + The program prints the point value and the estimated error in this quantity. From extrapolating it, we can guess that the exact value is somewhat like 0.0334473, plus or minus 0.0000001 (note that we get @@ -150,39 +123,33 @@ estimator is mostly exact, and extrapolated the computed value plus the estimated error, to get an approximation of the true value. Computing with more degrees of freedom shows that this assumption is indeed valid. -

-

+ + From the computed results, we can generate two graphs: one that shows -the convergence of the error J(u)-J(uh) (taking the +the convergence of the error $J(u)-J(u_h)$ (taking the extrapolated value as correct) in the point value, and the value that -we get by adding up computed value J(uh) and estimated -error eta (if the error estimator eta were exact, then the value -J(uh)+eta would equal the exact point value, and the error +we get by adding up computed value $J(u_h)$ and estimated +error eta (if the error estimator $eta$ were exact, then the value +$J(u_h)+\eta$ would equal the exact point value, and the error in this quantity would always be zero; however, since the error estimator is only a - good - approximation to the true error, we can by this only reduce the size of the error). In this graph, we also -indicate the complexity O(1/N) to show that mesh refinement +indicate the complexity ${\cal O}(1/N)$ to show that mesh refinement acts optimal in this case. The second chart compares true and estimated error, and shows that the two are actually very close to each other, even for such a complicated quantity as the point value: -

+
- - Error in point value - + @image html step-14.point-value.error.png - - Error in point value - + @image html step-14.point-value.error-estimation.png
@@ -190,7 +157,7 @@ value:

Comparing refinement criteria

-

+ Since we have accepted quite some effort when using the mesh refinement driven by the dual weighted error estimator (for solving the dual problem, and for evaluating the error representation), it is @@ -199,21 +166,17 @@ first compare the achieved error levels for different mesh refinement criteria. To generate this data, simply change the value of the mesh refinement criterion variable in the main program. The results are thus (for the weight in the Kelly indicator, we have chosen the -function 1/(r2+0.12), where r +function $1/(r^2+0.1^2)$, where $r$ is the distance to the evaluation point; it can be shown that this is the optimal weight if we neglect the effects of boundaries): -

-

- Error comparison - -

- -

+ +@image html step-14.point-value.error-comparison.png + + + Checking these numbers, we see that for global refinement, the error -is proportional to O(1/(sqrt(N) log(N))), and for the dual -estimator O(1/N). Generally speaking, we see that the dual +is proportional to $O(1/(sqrt(N) log(N)))$, and for the dual +estimator $O(1/N)$. Generally speaking, we see that the dual weighted error estimator is better than the other refinement indicators, at least when compared with those that have a similarly regular behavior. The Kelly indicator produces smaller errors, but @@ -225,20 +188,19 @@ results can be improved by adding the estimated error to the computed values. In terms of reliability, the weighted estimator is thus better than the Kelly indicator, although the latter sometimes produces smaller errors. -

+

Evaluation of point stresses

-

+ Besides evaluating the values of the solution at a certain point, the program also offers the possibility to evaluate the x-derivatives at a certain point, and also to tailor mesh refinement for this. To let the program compute these quantities, simply replace the two occurences of PointValueEvaluation in the main function by PointXDerivativeEvaluation, and let the program run: - -

+@code
 Refinement cycle: 0
    Number of degrees of freedom=72
    Point x-derivative=-0.0719397
@@ -278,11 +240,10 @@ Refinement cycle: 8
 Refinement cycle: 9
    Number of degrees of freedom=31438
    Point x-derivative=-0.0528145
-
- -

+@endcode + + -

The solution looks roughly the same as before (the exact solution of course is the same, only the grid changed a little), but the dual solution is now different. A close-up around the point of @@ -290,10 +251,7 @@ evaluation shows this:
- Dual solution - + @image html step-14.point-derivative.solution-5-dual.png
This time, the grids in refinement cycles 0, 5, 6, 7, 8, and 9 look @@ -301,123 +259,93 @@ like this:
- - Grid 0 - + @image html step-14.point-derivative.grid-0.png - - Grid 5 - + @image html step-14.point-derivative.grid-5.png
- - Grid 6 - + @image html step-14.point-derivative.grid-6.png - - Grid 7 - + @image html step-14.point-derivative.grid-7.png
- - Grid 8 - + @image html step-14.point-derivative.grid-8.png - - Grid 9 - + @image html step-14.point-derivative.grid-9.png
Note the assymetry of the grids compared with those we obtained for the point evaluation, which is due to the directionality of the x-derivative for which we tailored the refinement criterion. -

-

+ + Then, it is interesting to compare actually computed values of the quantity of interest (i.e. the x-derivative of the solution at one point) with a reference value of -0.0528223... plus or minus 0.0000005. We get this reference value by computing on finer grid after some more mesh refinements, with approximately 130,000 cells. -Recall that if the error is O(1/N) in the optimal case, then +Recall that if the error is $O(1/N)$ in the optimal case, then taking a mesh with ten times more cells gives us one additional digit in the result. -

-

+ + In the left part of the following chart, you again see the convergence of the error towards this extrapolated value, while on the right you see a comparison of true and estimated error:
- - Error in point derivative - + @image html step-14.point-derivative.error.png - - Error in point derivative - + @image html step-14.point-derivative.error-estimation.png
After an initial phase where the true error changes its sign, the estimated error matches it quite well, again. Also note the dramatic improvement in the error when using the estimated error to correct the -computed value of J(uh). -

+computed value of $J(u_h)$. +

Step-13 revisited

-

+ If instead of the Exercise_2_3 data set, we choose CurvedRidges in the main function, we can redo the computations of the previous example program, to compare whether the results obtained with the help of the dual weighted error estimator are better than those we had previously. -

-

+ + First, the meshes after 9 and 10 adaptive refinement cycles, respectively, look like this:
- Grid 9 - + @image html step-14.step-13.grid-9.png - Grid 10 - + @image html step-14.step-13.grid-10.png
@@ -429,31 +357,24 @@ like so:
- Solution 7 - + @image html step-14.step-13.solution-7.png
Obviously, the solution is worse at some places, but the mesh refinement process should have taken care that these places are not important for computing the point value. -

-

+ + The next point is to compare the new (duality based) mesh refinement criterion with the old ones. These are the results: -

-

- Error comparison - -

- -

+ +@image html step-14.step-13.error-comparison.png + + + The results are, well, somewhat mixed. First, the Kelly indicator disqualifies itself by its unsteady behavior, changing the sign of the error several times, and with increasing errors under mesh @@ -468,13 +389,13 @@ approximate curved boundaries, or if the coefficients were not constant, then this advantage of globally refinement meshes would vanish, while the good performance of the duality based estimator would remain. -

+

Conclusions and outlook

-

+ The results here are not too clearly indicating the superiority of the dual weighted error estimation approach for mesh refinement over other mesh refinement criteria, such as the Kelly indicator. This is due to @@ -485,9 +406,9 @@ plenty of examples are provided where the dual weighted approach can reduce the necessary numerical work by orders of magnitude, making this the only way to compute certain quantities to reasonable accuracies at all. -

-

+ + Besides the objections you may raise against its use as a mesh refinement criterion, consider that accurate knowledge of the error in the quantity one might want to compute is of great use, since we can @@ -499,15 +420,15 @@ satisfies any requirements on its accuracy. Also, as was shown for the evaluation of point values and derivatives, the error estimate can be used to extrapolate the results, yielding much higher accuracy in the quantity we want to know. -

-

+ + Leaving these mathematical considerations, we tried to write the program in a modular way, such that implementing another test case, or another evaluation and dual functional is simple. You are encouraged to take the program as a basis for your own experiments, and to play a little. -

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exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def -/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def -/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill - 2 copy vpt Square fill - Bsquare } bind def -/S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def -/S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def -/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill - Bsquare } bind def -/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill - Bsquare } bind def -/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def -/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy vpt Square fill Bsquare } bind def -/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def -/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def -/D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def -/D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def -/D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def -/D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def -/D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def -/D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def -/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def -/D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def -/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def -/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def -/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def -/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def -/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def -/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def -/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def -/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def -/DiaE { stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke } def -/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke } def -/TriUE { stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath stroke } def -/TriDE { stroke [] 0 setdash vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V closepath stroke } def -/PentE { stroke [] 0 setdash gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - closepath stroke grestore } def -/CircE { stroke [] 0 setdash - hpt 0 360 arc stroke } def -/Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def -/DiaW { stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V Opaque stroke } def -/BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V Opaque stroke } def -/TriUW { stroke [] 0 setdash vpt 1.12 mul add M - 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Results

-

+ Running the program takes a good while if one doesn't change the flags in the Makefile: in debug mode (the default) and on only a single machine, it takes about 3h45min on my Athlon XP 2GHz. Fortunately, but setting debug-mode = off in the Makefile, this can be reduced significantly, to about 23 minutes, a much more reasonable time. -

-

+ + If run, the program prints the following output, explaining what it is doing during all that time: - -

+@code
 examples/step-18> time make run
 ============================ Running step-18
 Timestep 1 at time 1
@@ -85,14 +84,12 @@ Timestep 10 at time 10
     Solver converged in 149 iterations.
     Updating quadrature point data...
     Moving mesh...
-
- +@endcode In other words, it is computing on 12,000 cells and with some 52,000 unknowns. Not a whole lot, but enough for a coupled three-dimensional problem to keep a computer busy for a while. At the end of the day, this is what we have for output: - -
+@code
 examples/step-18> ls -l *.d2
 -rw-r--r--  1 bangerth wheeler 8797414 May 25 09:10 solution-0001.0000.d2
 -rw-r--r--  1 bangerth wheeler 8788500 May 25 09:32 solution-0002.0000.d2
@@ -104,16 +101,13 @@ examples/step-18> ls -l *.d2
 -rw-r--r--  1 bangerth wheeler 8585146 May 25 11:45 solution-0008.0000.d2
 -rw-r--r--  1 bangerth wheeler 8489764 May 25 12:07 solution-0009.0000.d2
 -rw-r--r--  1 bangerth wheeler 8405388 May 25 12:29 solution-0010.0000.d2
-
- -

+@endcode + -

Let us convert these files in deal.II intermediate format to gmv -format (this assumes that you have already compiled the step-19 example program): - -

+format (this assumes that you have already compiled the 
+@ref step_19 "step-19" example program):
+@code
 examples/step-18> ../step-19/step-19
 
 Converter from deal.II intermediate format to other graphics formats.
@@ -123,69 +117,56 @@ Usage: ./step-19 [-p parameter_file] list_of_input_files [-x output_format] outp
 examples/step-18> ../step-19/step-19 solution-0001.0000.d2 -x gmv solution-0001.0000.gmv
 examples/step-18> ../step-19/step-19 solution-0002.0000.d2 -x gmv solution-0002.0000.gmv
 [...]
-
- +@endcode Of course, since we have run the program only in sequential mode, we do have only one intermediate file for each time step that we have to take as input. -

-

+ + If we visualize these files with GMV, we get to see the full picture of the disaster our forced compression wreaks on the cylinder (click on the images for a larger version; colors in the images encode the norm of the stress in the material): -

+
- - + @image html step-18.sequential-0002.0000.png Time = 2 - - + @image html step-18.sequential-0005.0000.png Time = 5 - - + @image html step-18.sequential-0007.0000.png Time = 7
- - - Time = 8 + @image html step-18.sequential-0008.0000.png + Time = 8 - - - Time = 9 + @image html step-18.sequential-0009.0000.png + Time = 9 - - - Time = 10 + @image html step-18.sequential-0010.0000.png + Time = 10
-

+ As is clearly visible, as we keep compressing the cylinder, it starts to buckle and ultimately collapses. Towards the end of the simulation, the deflection pattern becomes nonsymmetric (the cylinder top slides @@ -199,18 +180,17 @@ by an iterative solver could have the same effect. Another typical source for asymmetries in adaptive computations is that only a certain fraction of cells is refined in each step, which may lead to asymmetric meshes even if the original coarse mesh was symmetric. -

-

+ + Whether the computation is fully converged is a different matter. In order to see whether it is, we ran the program again with one more global refinement at the beginning and with the time step halved. This would have taken a very long time on a single machine, so we used our cluster again and ran it on 16 processors (8 dual-processor machines) in parallel. The beginning of the output now looks like this: - -

+@code
 Timestep 1 at time 0.5
   Cycle 0:
     Number of active cells:       29696 (by partition: 1862+1890+1866+1850+1864+1850+1858+1842+1911+1851+1911+1804+1854+1816+1839+1828)
@@ -251,12 +231,10 @@ Timestep 20 at time 10
     Solver converged in 493 iterations.
     Updating quadrature point data...
     Moving mesh...
-
- +@endcode That's quite a good number of unknowns, given that we are in 3d. The output of this program are 16 files for each time step: - -
+@code
 examples/step-18> ls -l solution-0001.000*
 -rw-r--r--    1 bangerth mfw       4325219 Aug 11 09:44 solution-0001.0000-000.d2
 -rw-r--r--    1 bangerth mfw       4454460 Aug 11 09:44 solution-0001.0000-001.d2
@@ -274,12 +252,10 @@ examples/step-18> ls -l solution-0001.000*
 -rw-r--r--    1 bangerth mfw       4466037 Aug 11 09:43 solution-0001.0000-013.d2
 -rw-r--r--    1 bangerth mfw       4505679 Aug 11 09:44 solution-0001.0000-014.d2
 -rw-r--r--    1 bangerth mfw       4340488 Aug 11 09:44 solution-0001.0000-015.d2
-
- +@endcode We merge and convert these 16 intermediate files into a single gmv file as follows: - -
+@code
 examples/step-18> time ../step-19/step-19 solution-0001.0000-* -x gmv -o solution-0001.0000.gmv
 
 real    0m45.929s
@@ -287,86 +263,68 @@ user    0m41.290s
 sys     0m0.990s
 examples/step-18> ls -l solution-0001.0000.gmv
 -rw-r--r--    1 bangerth mfw      68925360 Aug 11 17:04 solution-0001.0000.gmv
-
- +@endcode -

Doing so for all time steps, we obtain gmv files that we can visualize (albeit with some difficulty, due to their size gmv isn't exactly fast when plotting them). Here are first the mesh on which we compute as well as the partitioning for the 16 processors: -

+
- - + @image html step-18.parallel-000mesh.png - - + @image html step-18.parallel-0002.p.png
-

+ Finally, here is the same output as we have shown before for the much smaller sequential case: -

+
- - + @image html step-18.parallel-0002.s.png Time = 2 - - + @image html step-18.parallel-0005.s.png Time = 5 - - + @image html step-18.parallel-0007.s.png Time = 7
- - + @image html step-18.parallel-0008.s.png Time = 8 - - + @image html step-18.parallel-0009.s.png Time = 9 - - + @image html step-18.parallel-0010.s.png Time = 10
-

+ If one compares this with the previous run, the results are qualitatively similar, but quantitatively definitely different. The previous computation was therefore certainly not converged, though we can't say for sure anything about @@ -377,4 +335,4 @@ deformation model we chose completely inadequate, but for a realistic simulation we would also need to make sure that the body does not intersects itself during deformation. Without such a formulation we cannot expect anything that make sense, even if it produces nice pictures! -

+