[cig-commits] commit: Changed S&P profile figure to show cycle 2 rather than cycle 3, and made some text changes to the S&P section.

Tue May 1 14:53:19 PDT 2012

changeset:   117:a185e65d80d5
tag:         tip
user:        Charles Williams <C.Williams at gns.cri.nz>
date:        Wed May 02 09:53:00 2012 +1200
files:       faultRup.tex figs/savageprescott_profiles.pdf
description:
Changed S&P profile figure to show cycle 2 rather than cycle 3, and made some text changes to the S&P section.


diff -r c51a616f878f -r a185e65d80d5 faultRup.tex

--- a/faultRup.tex	Tue May 01 16:48:31 2012 +1200
+++ b/faultRup.tex	Wed May 02 09:53:00 2012 +1200
@@ -1308,21 +1308,21 @@ in PyLith, including the ability to spec
 in PyLith, including the ability to specify complex, time-varying
 boundary conditions on the fault. It also provides a good test of the
 Maxwell viscoelastic bulk rheology. The analytical solution for this
-problem provides surface displacements as a function of distance from
-the fault and time since the last earthquake. An infinite strike-slip
-fault is assumed, so there is no geometry along the strike of the
-fault. The solution is controlled by the ratio of the fault locking
-depth ($D$) to the thickness of the elastic layer ($H$), and by the
-ratio of the earthquake recurrence time ($T$) to the viscoelastic
-relaxation time:  $tau_0 = \mu T/2\eta$, where $\mu$ is the shear
-modulus and $\eta$ is the viscosity.
+problem provides surface displacements in the y-direction (along-strike)
+as a function of distance from the fault and time since the last
+earthquake. An infinite strike-slip fault is assumed, so there is no
+geometry along the strike of the fault. The solution is controlled by
+the ratio of the fault locking depth ($D$) to the thickness of the
+elastic layer ($H$), and by the ratio of the earthquake recurrence time
+($T$) to the viscoelastic relaxation time: $\tau_0 = \mu T/2\eta$, where
+$\mu$ is the shear modulus and $\eta$ is the viscosity.
 
 For our model comparison, we used a locking depth of 20 km, an elastic
 layer thickness of 40 km, an earthquake recurrence time of 200 years, a
 shear modulus of 30 GPa, and a viscosity of $2.37\times 10^{19}$
 Pa-s. We used a relative plate velocity of 2 cm/year, implying a
 coseismic offset of 4 m every 200 years. The viscosity and shear modulus
-values yield a viscoelastic relaxation time of 50 years, and a $tau_0$
+values yield a viscoelastic relaxation time of 50 years, and a $\tau_0$
 value of 4.
 
 The problem is simulated in PyLith using a 3D mesh with (x, y, z)
@@ -1330,28 +1330,29 @@ in the y-direction were applied to the -
 in the y-direction were applied to the -x and +x faces, and
 x-displacements were also constrained to zero on these faces. The
 z-displacements were constrained to zero on the -z face. Finally, we
-constrained the x-displacements to zero on the -y and +y faces to more
-accurately represent an infinite fault. For comparison with analytical
-results, we extracted the numerical results along an x-profile at y = 0,
-corresponding to the center of the mesh in the y-direction. We solved
-the problem using both trilinear hexahedral cells as well as linear
-tetrahedral cells, using two different resolutions for each cell
-type. In our coarsest hexahedral mesh we used a uniform resolution of
-20 km. In our higher resolution hexahedral mesh we refined an inner
-region (x-dimension = 480 km, y-dimension = 240 km, z-dimension = 100
-km) by a factor of 3, yielding a resolution near the center of the fault
-of 6.7 km. For the tetrahedral meshes, we maintained the same resolution
-near the center of the fault (20 km and 6.7 km); however, for these
-meshes we linearly increased the cell size to the outer edges of the
-mesh. This resulted in cells with maximum dimensions of approximately 60
-km for the coarser mesh and 40 km for the higher resolution mesh. Note
-that for both the hexahedral and tetrahedral coarse meshes, the cell
-size on the fault is the maximum allowable size that still allows us to
-represent the fault locking depth as a sharp boundary.
+constrained the x-displacements to zero on the -y and +y faces to
+represent more accurately an infinite fault. For comparison with
+analytical results, we extracted the numerical results along an
+x-profile at y = 0, corresponding to the center of the mesh in the
+y-direction. We solved the problem using both trilinear hexahedral cells
+as well as linear tetrahedral cells, using two different resolutions for
+each cell type. In our coarsest hexahedral mesh we used a uniform
+resolution of 20 km. In our higher resolution hexahedral mesh we refined
+an inner region (x-dimension = 480 km, y-dimension = 240 km, z-dimension
+= 100 km) by a factor of 3, yielding a resolution near the center of the
+fault of 6.7 km. For the tetrahedral meshes, we maintained the same
+resolution near the center of the fault (20 km and 6.7 km); however, for
+these meshes we linearly increased the cell size to the outer edges of
+the mesh. This resulted in cells with maximum dimensions of
+approximately 60 km for the coarser mesh and 40 km for the higher
+resolution mesh. Note that for both the hexahedral and tetrahedral
+coarse meshes, the cell size on the fault is the maximum allowable size
+that still allows us to represent the fault locking depth as a sharp
+boundary.
 
-Since this is a viscoelastic problem, it is necessary to 'spin-up' the
+Since this is a viscoelastic problem, it is necessary to spin-up the
 solution for several earthquake cycles until a near steady-state is
-obtained. There is an additional issue for the numerical solution. In
+achieved. There is an additional issue for the numerical solution. In
 the analytical solution, steady plate motion is simply
 superimposed. This is not possible with the numerical solution; however,
 after several earthquake cycles we approach this state. We run both the
@@ -1362,18 +1363,18 @@ rigorous test of the accuracy of the vis
 rigorous test of the accuracy of the viscoelastic solution for
 moderately large time steps. The fault slip is specified using two
 different kinematic slip functions: the default StepSlipFn, which allows
-us to specify slip for each rupture event, and the ConstRateSlipFn,
-which allows us to specify steady slip on the creeping portion of the
-fault.
+us to specify coseismic slip for each rupture event, and the
+ConstRateSlipFn, which allows us to specify steady slip on the creeping
+portion of the fault.
 
 A comparison of the analytical and numerical results is shown in Figure
 ~\ref{fig:savage:prescott:profiles}. To examine the differences very
 close to the fault, we have used a logarithmic scale for the x
 axis. Further from the fault, all models show virtually identical
 results. The top portion of the figure shows the results during the
-third earthquake cycle, while the bottom portion shows the results
+second earthquake cycle, while the bottom portion shows the results
 during the tenth earthquake cycle. Close to the fault, all the
-simulations show similar behavior between the third and tenth
+simulations show similar behavior between the second and tenth
 cycle. Further from the fault, although all of the numerical solutions
 predict identical displacements, the viscoelastic solution has not yet
 achieved steady state, and thus under-predicts the displacement.
@@ -1384,7 +1385,10 @@ numerical models predict nearly identica
 numerical models predict nearly identical results, and provide a very
 close fit to the analytical solution. Close to the fault, the effects of
 inadequate discretization become apparent for the coarse meshes, as the
-details of the complex fault slip cannot be accurately represented.
+details of the complex fault slip cannot be accurately
+represented. Thus, with propert discretization size, PyLith is able to
+represent complex spatial and temporal kinematic slip distributions for
+quasi-static problems extremely accurately.
 
 Benchmarks such as this can be quite helpful when designing meshes for
 real-world problems. As seen in this benchmark, a coarse discretization
@@ -1643,14 +1647,20 @@ MGK acknowledges partial support from NS
   \noindent\includegraphics{figs/savageprescott_profiles}
   \caption{Comparison of displacement profiles perpendicular to the
     fault in the Savage and Prescott benchmark during earthquake
-    cycles 3 and 10. The displacements values shown are
-    relative to the values at the beginning of the earthquake cycle to
-  facilitate comparison between the analytical solution and the
-  numerical models, which require spin-up to reach the steady-state
-  solution. Both the hexahedral (Hex8) and tetrahedral (Tet4)
-  discretizations resolve the viscoelastic deformation and display
-  excellent agreement with the steady-state solution by the tenth
-  earthquake cycle.}
+    cycles 2 and 10. The displacement values shown are relative to the
+    values at the beginning of the earthquake cycle to facilitate
+    comparison between the analytical solution and the numerical
+    models. Both the analytical and numerical simulations require
+    spin-up to reach the steady-state solution, and the numerical
+    models also require spin-up to achieve steady plate motion, which
+    is superimposed on the analytical solution. Both the hexahedral
+    (Hex8) and tetrahedral (Tet4) discretizations resolve the
+    viscoelastic deformation and display excellent agreement with the
+    steady-state solution by the tenth earthquake cycle. The coarser
+    (20 km) resolutions are unable to match the details of the
+    displacement field very close to the fault, but all of the
+    numerical models provide a good fit to the analytical solution far
+    from the fault.}
   \label{fig:savage:prescott:profiles}
 \end{figure}
 
diff -r c51a616f878f -r a185e65d80d5 figs/savageprescott_profiles.pdf
Binary file figs/savageprescott_profiles.pdf has changed

