From: Wolfgang Bangerth Date: Mon, 13 Feb 2012 10:08:33 +0000 (+0000) Subject: Get rid of unneeded pictures. Write the results section. X-Git-Tag: v8.0.0~2921 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=be34d43faa9cc938081aacf44fe7392cb71c5037;p=dealii.git Get rid of unneeded pictures. Write the results section. git-svn-id: https://svn.dealii.org/trunk@25058 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-41/doc/results.dox b/deal.II/examples/step-41/doc/results.dox index f4c6feefb5..15abbbf7de 100644 --- a/deal.II/examples/step-41/doc/results.dox +++ b/deal.II/examples/step-41/doc/results.dox @@ -1 +1,193 @@

Results

+ +Running the program produces output like this: +@code +Number of active cells: 16384 +Total number of cells: 21845 +Number of degrees of freedom: 16641 + +Newton iteration 0 + Assembling system... + Solving system... + Error: 0.310059 -> 5.16619e-05 in 5 CG iterations. + Updating active set... + Size of active set: 13164 + Residual of the non-contact part of the system: 1.61863e-05 + Writing graphical output... + +Newton iteration 1 + Assembling system... + Solving system... + Error: 1.11987 -> 0.00109377 in 6 CG iterations. + Updating active set... + Size of active set: 12363 + Residual of the non-contact part of the system: 3.9373 + Writing graphical output... + +... + +Newton iteration 17 + Assembling system... + Solving system... + Error: 0.00713308 -> 2.29249e-06 in 4 CG iterations. + Updating active set... + Size of active set: 5399 + Residual of the non-contact part of the system: 0.000957525 + Writing graphical output... + +Newton iteration 18 + Assembling system... + Solving system... + Error: 0.000957525 -> 2.8033e-07 in 4 CG iterations. + Updating active set... + Size of active set: 5399 + Residual of the non-contact part of the system: 2.8033e-07 + Writing graphical output... +@endcode + +The iterations end once the active set doesn't change any more (it has +5,399 constrained degrees of freedom at that point). More revealing is +to look at a sequence of graphical output files (every third step is +shown, with the number of the iteration in the leftmost column): + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ 0   + + @image html step-41.displacement.00.png + + @image html step-41.active-set.00.png + + @image html step-41.displacement.3d.00.png +
+ 3   + + @image html step-41.displacement.03.png + + @image html step-41.active-set.03.png + + @image html step-41.displacement.3d.03.png +
+ 6   + + @image html step-41.displacement.06.png + + @image html step-41.active-set.06.png + + @image html step-41.displacement.3d.06.png +
+ 9   + + @image html step-41.displacement.09.png + + @image html step-41.active-set.09.png + + @image html step-41.displacement.3d.09.png +
+ 12   + + @image html step-41.displacement.12.png + + @image html step-41.active-set.12.png + + @image html step-41.displacement.3d.12.png +
+ 15   + + @image html step-41.displacement.15.png + + @image html step-41.active-set.15.png + + @image html step-41.displacement.3d.15.png +
+ 18   + + @image html step-41.displacement.18.png + + @image html step-41.active-set.18.png + + @image html step-41.displacement.3d.18.png +
+ +The pictures show that in the first step, the solution (which has been +computed without any of the constraints active) bends through so much +that pretty much every interior point has to be bounced back to the +stairstep function, producing a discontinuous solution. Over the +course of the active set iterations, this unphysical membrane shape is +smoothed out, the contact with the lower-most stair step disappears, +and the solution stabilizes. + + + +

Possibilities for extensions

+ +As with any of the programs of this tutorial, there are a number of +obvious possibilities for extensions and experiments. The first one is +clear: introduce adaptivity. Contact problems are prime candidates for +adaptive meshes because the solution has lines along which it is less +regular (the places where contact is established between membrane and +obstacle) and other areas where the solution is very smooth (or, in +the present context, constant wherever it is in contact with the +obstacle). Adding this to the current program should not pose too many +difficulties. + +A more challenging task would be an extension to 3d. The problem here +is not so much to simply make everything run in 3d. Rather, it is that +when a 3d body is deformed and gets into contact with an obstacle, +then the obstacle does not act as a constraining body force within the +domain as is the case here. Rather, the contact force only acts on the +boundary of the object. The inequality then is not in the differential +equation but in fact in the (Neumann-type) boundary conditions, though +this leads to a similar kind of variational +inequality. Mathematically, this means that the Lagrange multiplier +only lives on the surface, though it can of course be extended by zero +into the domain if that is convenient. As in the current program, one +does not need to form and store this Lagrange multiplier explicitly.