From c4632919b4fc9e839a42b3a5c80f839793abc76b Mon Sep 17 00:00:00 2001 From: bangerth Date: Mon, 13 Feb 2012 10:08:33 +0000 Subject: [PATCH] Get rid of unneeded pictures. 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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. diff --git a/deal.II/examples/step-41/doc/step-41.active-set.01.png b/deal.II/examples/step-41/doc/step-41.active-set.01.png deleted file mode 100644 index a17098df507a708986ab00fa49da5e13ae16fbe7..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 4882 zcmd5=XEdB$yPh!&B4e}=M2$BgiRgl45WPi<61_8e8*L;EM(;t05)-_L!RWn(XhA|4 zy+)#UCQ6iV@|_=Nedm1NkMr-WXWh@W_OdFd$t3MB>wJ3$iP`WEAc@t5NXixwLH_-kM>7iouhgA8e zDfK%`{~>)no;nZ*)FB!Sz+t9E>`8Y=RpF_g|Md2pqd$XwUeErqo&Y5es#sQmfu7Mw zQJhREf3LyAOqSWy<0Dx&MDZhYXp}YQQ-_v^6l<+9U)X#Qd!a}+9a35dFOqzU3rM1Z zN($dv*KBIu7$mo~?H@-N8s1pFfCYJcEEy#nwKs2!eUnq8GgtDe2B1G=YK)6#x=52u;Kvf zKtl8x+ey5nu&jVjVF9f>zn|-3I#CqMS9BC>J3dryxNM;~W^En~XH2z1%m`X$m2Nc+ zrdI=>-3oiJhSmu|Cr2@{UkT-J(Mw{zABf{s*SM3dF<%5EDP8atGI3k|JcVL{c`eRL zN=1v(Q+>aUIm-{S%hKk%_ZZQQF$QbiyFtg{k)b2i{PKg>z1K~SK)o)cKzPS%PxT!n--quB$;CCuVyD=K5Iu{}o6Io6ExyEnBGw(* 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