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Geomod 2008 Benchmarks
Using the lessons learned from the Geomod 2004 benchmarks, new benchmarks were created that would make it easier to compare numerical experiments with each other and with analog experiments [Geomod 2008].
Stable Wedge
This benchmark simulates a wall pushing a wedge as in Figure 1. There is an analytic solution [Dahlen Wedge] which details what the friction on the bottom and sides should be to provide enough resistance so that the wedge does not collapse under its own weight, but not so much as to cause any internal deformation as it slides. The derivation of the solution assumes that the friction along the sides has no cohesion. So the force at the tip will go to zero as the thickness of the material goes to zero. However, analog experiments suggest a finite cohesion, so this benchmark specifies a boundary cohesion.
We modeled the wedge using a relatively low viscosity (1 Pa-s) air layer on top. This low viscosity region does not, for the most part, affect the dynamics.
We modeled boundary friction by first fixing the sand to the boundary. We then modify the material properties in the element next to the boundary so that it provides the correct resistance. So in the bulk, the sand's internal angle of friction is 36° weakening to 31°, while in the element at the boundary the internal angle of friction is 16° weakening to 14°. Similarly, in the bulk, the cohesion is 10 [;Pa;], while at the boundary the cohesion is 10 [;Pa;] weakening to 0.01 [;Pa;]. If we do not weaken the cohesion, when we try to model an unstable wedge by reducing the internal angle of friction, the wedge never collapses on itself.
Figure 2 shows the strain rate invariant after the wall has moved 4 cm, and Figure 3 shows the particles. The bulk translates with almost no deformation, although, as expected, the tip deforms. The odd structures at the tip are below the grid resolution. Figure 4 shows a simulation when we reduce the boundary friction to 1°. The wedge quickly becomes unstable and collapses.
Figure 1: Setup for the stable wedge benchmark. Image courtesy of Susanne Buiter.
Figure 2: Strain rate invariant for the stable wedge benchmark within the wedge. Outside the wedge, the strain rates are large because of the air's low viscosity. The resolution is 512 × 128, and the wedge has translated 4 cm.
Figure 3: Material particles for the stable wedge benchmark. The deformation at the tip is caused by a finite boundary cohesion, although the actual structure is not resolved. The resolution is 512 × 128, and the wedge has translated 4 cm.
Figure 4: Strain rate invariant and velocity arrows for the stable wedge benchmark, but with the boundary friction angle reduced to 1°. Note that the strain rates are much higher than in Figure 2. The wedge collapses almost immediately. The resolution is 512 × 128, and the wedge has translated 0.17 cm.
Unstable Shortening
This benchmark simulates a wall pushing against a wall of sand as in Figure 5. There are three layers of sand, with the middle layer being a little heavier and sticking a little more to the boundary. Otherwise it is identical. Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, and Figure 11 show results at different times and different resolutions.
Figure 5: Setup for the unstable shortening benchmark. Image courtesy of Susanne Buiter.
Figure 6: Strain rate invariant for the unstable shortening benchmark with a resolution of 128 × 32. The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
Figure 7: Strain rate invariant for the unstable shortening benchmark with a resolution of 256 × 64. The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
Figure 8: Strain rate invariant for the unstable shortening benchmark with a resolution of 512 × 128. The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
Figure 9: Material particles for the unstable shortening benchmark with a resolution of 128 × 32. The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
Figure 10: Material particles for the unstable shortening benchmark with a resolution of 256 × 64. The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
Figure 11: Material particles for the unstable shortening benchmark with a resolution of 512 × 128. The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
Brittle Shortening
This benchmark is very similar to unstable shortening. The only difference is that part of the bottom is also moving along as shown in Figure Figure 12. This causes the deformation to start from inside the sand box, rather than along the walls. Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, and Figure 18 show results at different times and different resolutions.
Figure 12: Setup for the brittle shortening benchmark. Image courtesy of Susanne Buiter.
Figure 13: Strain rate invariant for the brittle shortening benchmark with a resolution of 128 × 32. The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
Figure 14: Strain rate invariant for the brittle shortening benchmark with a resolution of 256 × 64. The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
Figure 15: Strain rate invariant for the brittle shortening benchmark with a resolution of 512 × 128. The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
Figure 16: Material particles for the brittle shortening benchmark with a resolution of 128 × 32. The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
Figure 17: Material particles for the brittle shortening benchmark with a resolution of 256 × 64. The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
Figure 18: Material particles for the brittle shortening benchmark with a resolution of 512 × 128. The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
