<h1>Results</h1>
-Running the program (with a ball as obstacle) on 12 cores produces output like this:
+The directory that contains this program contains a number of input
+parameter files that can be used to create various different
+simulations. For example, running the program with the
+<code>p1_adaptive.prm</code> parameter file (using a ball as obstacle and the
+box as domain) on 16 cores produces output like this:
@code
+ Using output directory 'p1adaptive/'
+ FE degree 1
+ transfer solution false
+
Cycle 0:
- Number of active cells: 218
- Total number of cells: 249
+ Number of active cells: 512
Number of degrees of freedom: 2187
Newton iteration 1
Size of active set: 1
Assembling system...
Solving system...
- Error: 173.076 -> 0.157131 in 8 FGMRES iterations.
- Number of elastic quadrature points: 4096 and plastic quadrature points: 0
- Residual of the non-contact part of the system: 0.157131
- with a damping parameter alpha = 1
+ Error: 173.076 -> 3.06875e-07 in 8 Bicgstab iterations.
+ Accepting Newton solution with residual: 3.06875e-07
+ Active set did not change!
Newton iteration 2
Updating active set...
Size of active set: 1
Assembling system...
Solving system...
- Error: 57.3852 -> 0.0426777 in 9 FGMRES iterations.
- Number of elastic quadrature points: 3944 and plastic quadrature points: 152
- Residual of the non-contact part of the system: 24.9209
- with a damping parameter alpha = 1
+ Error: 57.3622 -> 2.46266e-07 in 8 Bicgstab iterations.
+ Accepting Newton solution with residual: 24.9028
+ Active set did not change!
Newton iteration 3
Updating active set...
Size of active set: 1
Assembling system...
Solving system...
- Error: 24.9209 -> 0.0157852 in 8 FGMRES iterations.
- Number of elastic quadrature points: 3960 and plastic quadrature points: 136
- Residual of the non-contact part of the system: 1.63601
- with a damping parameter alpha = 1
-
- Newton iteration 4
- Updating active set...
- Size of active set: 1
- Assembling system...
- Solving system...
- Error: 1.63601 -> 0.00131281 in 8 FGMRES iterations.
- Number of elastic quadrature points: 3960 and plastic quadrature points: 136
- Residual of the non-contact part of the system: 0.00479668
- with a damping parameter alpha = 1
-
- Newton iteration 5
- Updating active set...
- Size of active set: 1
- Assembling system...
- Solving system...
- Error: 0.00479668 -> 2.27976e-06 in 9 FGMRES iterations.
- Number of elastic quadrature points: 3960 and plastic quadrature points: 136
- Residual of the non-contact part of the system: 2.29051e-06
+ Error: 24.9028 -> 1.83021e-07 in 7 Bicgstab iterations.
+ Residual of the non-contact part of the system: 1.63333
with a damping parameter alpha = 1
-
- Newton iteration 6
- Updating active set...
- Size of active set: 1
- Assembling system...
- Solving system...
- Error: 2.29051e-06 -> 1.31066e-09 in 9 FGMRES iterations.
- Number of elastic quadrature points: 3960 and plastic quadrature points: 136
- Residual of the non-contact part of the system: 1.31067e-09
- with a damping parameter alpha = 1
-
- Number of assembled systems = 6
- Number of Solver-Iterations = 51
- Writing graphical output...
-
-
-+---------------------------------------------+------------+------------+
-| Total wallclock time elapsed since start | 6.23s | |
-| | | |
-| Section | no. calls | wall time | % of total |
-+---------------------------------+-----------+------------+------------+
-| Assembling | 6 | 0.859s | 14% |
-| Graphical output | 1 | 0.22s | 3.5% |
-| Residual and lambda | 6 | 0.189s | 3% |
-| Setup | 1 | 0.265s | 4.3% |
-| Setup: distribute DoFs | 1 | 0.0549s | 0.88% |
-| Setup: matrix | 1 | 0.0142s | 0.23% |
-| Solve | 6 | 4.51s | 72% |
-| Solve: iterate | 6 | 1.79s | 29% |
-| Solve: setup preconditioner | 6 | 2.7s | 43% |
-| Update solution and constraints | 6 | 0.0246s | 0.4% |
-+---------------------------------+-----------+------------+------------+
-
+ Active set did not change!
-Cycle 1:
- Number of active cells: 463
- Total number of cells: 529
- Number of degrees of freedom: 6294
+...
- Newton iteration 1
- Updating active set...
- Size of active set: 8
- Assembling system...
- Solving system...
- Error: 69.7055 -> 0.0512854 in 11 FGMRES iterations.
- Number of elastic quadrature points: 11400 and plastic quadrature points: 984
- Residual of the non-contact part of the system: 16.4859
- with a damping parameter alpha = 1
-
- Newton iteration 2
- Updating active set...
- Size of active set: 9
- Assembling system...
- Solving system...
- Error: 142.778 -> 0.0754216 in 8 FGMRES iterations.
- Number of elastic quadrature points: 11252 and plastic quadrature points: 1132
- Residual of the non-contact part of the system: 17.2965
- with a damping parameter alpha = 1
-
- Newton iteration 3
- Updating active set...
- Size of active set: 9
- Assembling system...
- Solving system...
- Error: 17.2965 -> 0.0126474 in 10 FGMRES iterations.
- Number of elastic quadrature points: 11032 and plastic quadrature points: 1352
- Residual of the non-contact part of the system: 14.2312
- with a damping parameter alpha = 1
-
- Newton iteration 4
- Updating active set...
- Size of active set: 9
- Assembling system...
- Solving system...
- Error: 14.2312 -> 0.0130557 in 10 FGMRES iterations.
- Number of elastic quadrature points: 11189 and plastic quadrature points: 1195
- Residual of the non-contact part of the system: 12.6684
- with a damping parameter alpha = 1
-
- Newton iteration 5
+ Newton iteration 6
Updating active set...
- Size of active set: 9
- Assembling system...
- Solving system...
- Error: 12.6684 -> 0.00951713 in 10 FGMRES iterations.
- Number of elastic quadrature points: 10987 and plastic quadrature points: 1397
- Residual of the non-contact part of the system: 13.9042
- with a damping parameter alpha = 1
- Number of elastic quadrature points: 11161 and plastic quadrature points: 1223
- Residual of the non-contact part of the system: 8.7982
- with a damping parameter alpha = 0.5
-
- Newton iteration 6
- Updating active set...
- Size of active set: 9
- Assembling system...
- Solving system...
- Error: 8.7982 -> 0.0057014 in 10 FGMRES iterations.
- Number of elastic quadrature points: 11068 and plastic quadrature points: 1316
- Residual of the non-contact part of the system: 9.05367
- with a damping parameter alpha = 1
- Number of elastic quadrature points: 11160 and plastic quadrature points: 1224
- Residual of the non-contact part of the system: 4.7814
- with a damping parameter alpha = 0.5
-
- Newton iteration 7
- Updating active set...
- Size of active set: 9
- Assembling system...
- Solving system...
- Error: 4.7814 -> 0.00316863 in 10 FGMRES iterations.
- Number of elastic quadrature points: 11152 and plastic quadrature points: 1232
- Residual of the non-contact part of the system: 1.71796
- with a damping parameter alpha = 1
-
- Newton iteration 8
- Updating active set...
- Size of active set: 9
- Assembling system...
- Solving system...
- Error: 1.71796 -> 0.00104906 in 7 FGMRES iterations.
- Number of elastic quadrature points: 11144 and plastic quadrature points: 1240
- Residual of the non-contact part of the system: 0.013944
- with a damping parameter alpha = 1
-
- Newton iteration 9
- Updating active set...
- Size of active set: 9
- Assembling system...
- Solving system...
- Error: 0.013944 -> 1.26708e-05 in 11 FGMRES iterations.
- Number of elastic quadrature points: 11144 and plastic quadrature points: 1240
- Residual of the non-contact part of the system: 1.38354e-05
- with a damping parameter alpha = 1
-
- Newton iteration 10
- Updating active set...
- Size of active set: 9
+ Size of active set: 1
Assembling system...
Solving system...
- Error: 1.38354e-05 -> 7.16927e-09 in 14 FGMRES iterations.
- Number of elastic quadrature points: 11144 and plastic quadrature points: 1240
- Residual of the non-contact part of the system: 7.16925e-09
+ Error: 1.43188e-07 -> 4.61326e-16 in 8 Bicgstab iterations.
+ Residual of the non-contact part of the system: 5.67016e-14
with a damping parameter alpha = 1
-
- Number of assembled systems = 10
- Number of Solver-Iterations = 101
- Writing graphical output...
+ Active set did not change!
+ Writing graphical output... p1adaptive/solution-00.pvtu
+---------------------------------------------+------------+------------+
-| Total wallclock time elapsed since start | 12.1s | |
+| Total wallclock time elapsed since start | 1.13s | |
| | | |
| Section | no. calls | wall time | % of total |
+---------------------------------+-----------+------------+------------+
-| Assembling | 10 | 4.07s | 34% |
-| Graphical output | 1 | 0.282s | 2.3% |
-| Residual and lambda | 12 | 0.819s | 6.8% |
-| Setup | 1 | 0.895s | 7.4% |
-| Setup: distribute DoFs | 1 | 0.139s | 1.1% |
-| Setup: matrix | 1 | 0.0272s | 0.22% |
-| Setup: refine mesh | 1 | 0.579s | 4.8% |
-| Solve | 10 | 5.68s | 47% |
-| Solve: iterate | 10 | 3.44s | 28% |
-| Solve: setup preconditioner | 10 | 2.18s | 18% |
-| Update solution and constraints | 10 | 0.0952s | 0.79% |
+| Assembling | 6 | 0.463s | 41% |
+| Graphical output | 1 | 0.0257s | 2.3% |
+| Residual and lambda | 4 | 0.0754s | 6.7% |
+| Setup | 1 | 0.227s | 20% |
+| Setup: constraints | 1 | 0.0347s | 3.1% |
+| Setup: distribute DoFs | 1 | 0.0441s | 3.9% |
+| Setup: matrix | 1 | 0.0119s | 1.1% |
+| Setup: vectors | 1 | 0.00155s | 0.14% |
+| Solve | 6 | 0.246s | 22% |
+| Solve: iterate | 6 | 0.0631s | 5.6% |
+| Solve: setup preconditioner | 6 | 0.167s | 15% |
+| update active set | 6 | 0.0401s | 3.6% |
+---------------------------------+-----------+------------+------------+
+Peak virtual memory used, resident in kB: 541884 77464
+Contact force = 37.3058
-Cycle 2:
- Number of active cells: 1170
- Total number of cells: 1337
- Number of degrees of freedom: 18207
-
- .
- .
- .
-
+...
Cycle 3:
- Number of active cells: 2955
- Total number of cells: 3377
+ Number of active cells: 14652
Number of degrees of freedom: 52497
-
- .
- .
- .
-
-
-Cycle 4:
- Number of active cells: 7400
- Total number of cells: 8457
- Number of degrees of freedom: 154647
-
- .
- .
- .
-
-
-Cycle 5:
- Number of active cells: 19279
- Total number of cells: 22033
- Number of degrees of freedom: 461106
-
- Newton iteration 1
- Updating active set...
- Size of active set: 1672
- Assembling system...
- Solving system...
- Error: 3.36904 -> 0.00314153 in 28 FGMRES iterations.
- Number of elastic quadrature points: 57144 and plastic quadrature points: 1065608
- Residual of the non-contact part of the system: 0.174753
- with a damping parameter alpha = 1
- Newton iteration 2
- Updating active set...
- Size of active set: 2335
- Assembling system...
- Solving system...
- Error: 2.84616 -> 0.00266184 in 25 FGMRES iterations.
- Number of elastic quadrature points: 56415 and plastic quadrature points: 1066337
- Residual of the non-contact part of the system: 0.0845109
- with a damping parameter alpha = 1
-
- Newton iteration 3
- Updating active set...
- Size of active set: 2313
- Assembling system...
- Solving system...
- Error: 0.0870898 -> 8.60248e-05 in 97 FGMRES iterations.
- Number of elastic quadrature points: 56802 and plastic quadrature points: 1065950
- Residual of the non-contact part of the system: 0.0148214
- with a damping parameter alpha = 1
-
- Newton iteration 4
+ Newton iteration 1
Updating active set...
- Size of active set: 2313
+ Size of active set: 145
Assembling system...
Solving system...
- Error: 0.0148214 -> 1.46054e-05 in 72 FGMRES iterations.
- Number of elastic quadrature points: 56828 and plastic quadrature points: 1065924
- Residual of the non-contact part of the system: 0.00125281
- with a damping parameter alpha = 1
+ Error: 296.309 -> 2.45532e-06 in 9 Bicgstab iterations.
+ Accepting Newton solution with residual: 2.45532e-06
+ Active set did not change!
- Newton iteration 5
- Updating active set...
- Size of active set: 2313
- Assembling system...
- Solving system...
- Error: 0.00125281 -> 1.22184e-06 in 68 FGMRES iterations.
- Number of elastic quadrature points: 56828 and plastic quadrature points: 1065924
- Residual of the non-contact part of the system: 4.38909e-06
- with a damping parameter alpha = 1
-
- Newton iteration 6
+...
+
+ Newton iteration 10
Updating active set...
- Size of active set: 2313
+ Size of active set: 145
Assembling system...
Solving system...
- Error: 4.38909e-06 -> 4.30005e-09 in 189 FGMRES iterations.
- Number of elastic quadrature points: 56828 and plastic quadrature points: 1065924
- Residual of the non-contact part of the system: 4.30364e-09
+ Error: 2.71542e-07 -> 2.69366e-15 in 21 Bicgstab iterations.
+ Residual of the non-contact part of the system: 1.73713e-13
with a damping parameter alpha = 1
-
- Number of assembled systems = 6
- Number of Solver-Iterations = 479
- Writing graphical output...
+ Active set did not change!
+ Writing graphical output... p1adaptive/solution-03.pvtu
+---------------------------------------------+------------+------------+
-| Total wallclock time elapsed since start | 445s | |
+| Total wallclock time elapsed since start | 38.4s | |
| | | |
| Section | no. calls | wall time | % of total |
+---------------------------------+-----------+------------+------------+
-| Assembling | 6 | 218s | 49% |
-| Graphical output | 1 | 4.47s | 1% |
-| Residual and lambda | 6 | 31.9s | 7.2% |
-| Setup | 1 | 33s | 7.4% |
-| Setup: distribute DoFs | 1 | 4.69s | 1.1% |
-| Setup: matrix | 1 | 1.29s | 0.29% |
-| Setup: refine mesh | 1 | 22.4s | 5% |
-| Solve | 6 | 154s | 35% |
-| Solve: iterate | 6 | 145s | 33% |
-| Solve: setup preconditioner | 6 | 9.12s | 2.1% |
-| Update solution and constraints | 6 | 1.24s | 0.28% |
+| Assembling | 10 | 22.5s | 58% |
+| Graphical output | 1 | 0.327s | 0.85% |
+| Residual and lambda | 9 | 3.75s | 9.8% |
+| Setup | 1 | 4.83s | 13% |
+| Setup: constraints | 1 | 0.578s | 1.5% |
+| Setup: distribute DoFs | 1 | 0.71s | 1.8% |
+| Setup: matrix | 1 | 0.111s | 0.29% |
+| Setup: refine mesh | 1 | 4.83s | 13% |
+| Setup: vectors | 1 | 0.00548s | 0.014% |
+| Solve | 10 | 5.49s | 14% |
+| Solve: iterate | 10 | 3.5s | 9.1% |
+| Solve: setup preconditioner | 10 | 1.84s | 4.8% |
+| update active set | 10 | 0.662s | 1.7% |
+---------------------------------+-----------+------------+------------+
+Peak virtual memory used, resident in kB: 566052 105788
+Contact force = 56.794
+
+...
@endcode
-For each adaptive refinement cycle the iterations end if the active set doesn't
-change anymore and if the residual is accurate enough. In the tables
-at the end of each cycle you find information about computing time and the
-number of calls of different parts of the program like Assembling or calculating
-the residual. In the second cycle there are 12 calls for the residual function
-and only 10 calls for Solving or Assembling what means that two damping steps were
-necessary in this cycle.
+The tables at the end of each cycle show information about computing time
+and the number of calls of different parts of the program like assembly or
+calculating the residual, for the most recent mesh refinement cycle. Some of
+the numbers above can be improved by transfering the solution from one mesh to
+the next, an option we have not exercised here.
-For this simulation we used the SolutionTranfer class to interpolate the solution
-of the previous cycle to new mesh of the current cycle (for the start solution of cycle 0
-we used a pure elastic solution). That worked out well for the ball but for the
-Chinese symbol as obstacle it turns out that an elastic start solution for each
-cycle results in fewer Newton iterations.
+In a typical run, you can observe that for every refinement step, the active
+set - the contact points - are iterated out at first. After that the Newton
+method has only to resolve the plasticity. For the finer meshes,
+quadratic convergence can be observed for the last 4 or 5 Newton iterations.
-In every refinement step you can observe that the active set - the contact points -
-are iterated out at first. After that the Newton method has only to struggle with the
-plasticity. For the finer meshes there is quadratic convergence not until the
-last 4 or 5 Newton iterations.
+We will not discuss here in all detail what happens with each of the input
+files. Rather, let us just show pictures of the solution (the left half of the
+domain is omitted if cells have zero quadrature points at which the plastic
+inequality is active):
<table align="center">
<tr>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-42.CellConstitutionColorbar.png" alt="">
+ <img src="http://www.dealii.org/images/steps/developer/step-42.CellConstitutionColorbar.png">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-42.CellConstitutionBall2.png" alt="">
+ <img src="http://www.dealii.org/images/steps/developer/step-42.CellConstitutionBall2.png" alt="" width="40%">
</td>
<td valign="top">
</td>
<td>
- <img src="http://www.dealii.org/images/steps/developer/step-42.CellConstitutionLi2.png" alt="">
+ <img src="http://www.dealii.org/images/steps/developer/step-42.CellConstitutionLi2.png" alt="" alt="" width="40%">
</td>
</tr>
</table>
The picture shows the adaptive refinement and as well how much a cell is
plastified during the contact with the ball. Remember that we consider the
norm of the deviator part of the stress in each quadrature point to
-see if there is elastic or plastic behavior. In the middle of the top -
-where the mesh is finest - you can see the hollow caused by the ball. The blue
+see if there is elastic or plastic behavior.
+The blue
color means that this cell contains only elastic quadrature points in
contrast to the red cells in which all quadrature points are plastified.
-Since we used trilinear ansatz functions each cell contains eight
-of these points which may or may not be elastic or plastic.
-For further details see Frohne, Heister, Bangerth: Efficient
-Numerical Methods For The Large-Scale, Parallel Solution Of Elastoplastic Contact
-Problems, preprint.
+In the middle of the top surface -
+where the mesh is finest - a very close look shows the dimple caused by the
+obstacle. This is the result of the <code>move_mesh()</code>
+function. However, because the indentation of the obstacles we consider here
+is so small, it is hard to discern this effect; one could play with displacing
+vertices of the mesh by a multiple of the computed displacement.
+
+Further discussion of results that can be obtained using this program is
+provided in the publication mentioned at the very top of this page.
<a name="extensions"></a>
<h1>Possibilities for extensions</h1>
-Extend the program from a static to a quasi-static problem, perhaps by choosing a
-backward-Euler-scheme for the time discretization (for theoretical results see Frohne: FEM-Simulation
-der Umformtechnik metallischer Oberflächen im Mikrokosmos, Ph.D. thesis,
-University of Siegen, Germany, 2011).
-
-Like mentioned in step-41 it would be an interesting advance to consider a contact
-problem with friction. In almost every mechanical process friction has a big influence.
-For the modelling we have to take into account tangential stresses at the contact
-surface. Also we have to observe that friction adds another nonlinearity to
-our problem.
-
-If we simulate a frictional contact the next step to consider heat development
-over the contact zone is not too far away. The heat that is caused by friction
-between two bodies rises up the temperature in the deformable body and entails an
-change of some material parameters.
-
-It might be of interest to implement more accurate, problem-adapted error
+There are, as always, multiple possibilities for extending this program. From
+an algorithmic perspective, this program goes about as far as one can at the
+time of writing, using the best available algorithms for the contact
+inequality, the plastic nonlinearity, and the linear solvers. However, there
+are things one would like to do with this program as far as more realistic
+situations are concerned:
+<ul>
+<li> Extend the program from a static to a quasi-static situation, perhaps by
+choosing a backward-Euler-scheme for the time discretization. Some theoretical
+results can be found in the PhD thesis by Jörg Frohne, <i>FEM-Simulation
+der Umformtechnik metallischer Oberflächen im Mikrokosmos</i>, University
+of Siegen, Germany, 2011.
+
+<li> It would also be an interesting advance to consider a contact problem
+with friction. In almost every mechanical process friction has a big
+influence. To model this situation, we have to take into account tangential
+stresses at the contact surface. Friction also adds another inequality to
+our problem since body and obstacle will typically stick together as long as
+the tangential stress does not exceed a certain limit, beyond which the two
+bodies slide past each other.
+
+<li> If we already simulate a frictional contact, the next step to consider
+is heat generation over the contact zone. The heat that is
+caused by friction between two bodies raises the temperature in the
+deformable body and entails an change of some material parameters.
+
+<li> It might be of interest to implement more accurate, problem-adapted error
estimators for contact as well as for the plasticity.
+</ul>
projects / dealii.git / commitdiff