[cig-commits] commit: Fixed some problems with S&P section.

[Mercurial]

changeset:   124:dc1f86ee9c6b
tag:         tip
user:        Charles Williams <C.Williams at gns.cri.nz>
date:        Mon May 14 14:03:40 2012 +1200
files:       faultRup.tex
description:
Fixed some problems with S&P section.


diff -r 9a339f379222 -r dc1f86ee9c6b faultRup.tex
--- a/faultRup.tex	Thu May 10 17:17:47 2012 -0400
+++ b/faultRup.tex	Mon May 14 14:03:40 2012 +1200
@@ -1288,7 +1288,7 @@ benchmarks.
 
 As a test of our quasi-static solution, we compare our numerical
 results against the analytical solution of
-\citep{Savage:Prescott:1978}. This problem consists of a infinitely
+\citep{Savage:Prescott:1978}. This problem consists of an infinitely
 long strike-slip fault in an elastic layer overlying a Maxwell
 viscoelastic half-space. Figure~\ref{fig:savage:prescott::solution}
 illustrates the geometry of the problem with an exaggerated view of
@@ -1300,8 +1300,8 @@ creeping portion so the cumulative slip 
 creeping portion so the cumulative slip over an earthquake cycle is
 uniform.
 
-This problem tests the ability the kinematic fault implementation to
-include steady aseismic creep and multiple earthquake ruptures complex
+This problem tests the ability of the kinematic fault implementation to
+include steady aseismic creep and multiple earthquake ruptures 
 along with viscoelastic relaxation. The analytical solution for this
 problem provides the along-strike component of surface displacement as
 a function of distance perpendicular to the fault. The solution is
@@ -1313,7 +1313,7 @@ viscosity.
 
 For this benchmark we use 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,
+shear modulus of 30 GPa, a viscosity of $2.37\times 10^{19}$ Pa-s,
 and 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 $\tau_0 =4$.
@@ -1341,43 +1341,50 @@ that still allows us to represent the fa
 that still allows us to represent the fault locking depth as a sharp
 boundary.
 
-In this viscoelastic problem the numerical model does not achieve
-steady-state behavior until after several earthquake cycles. The
-duration of this model spin-up depends on how close the initial
-conditions of the match the steady-state behavior and the relaxation
-time of the viscoelastic material. Shorter relaxation will converge to
-the steady-state behavior in fewer earthquake cycles. We simulate ten
-earthquake cycles for each of the numerical models for a total
-duration of 2000 years with a time step of five years. This time step
-corresponds to one tenth of the viscoelastic relaxation time; hence it
-tests the accuracy of the viscoelastic solution for moderately large
-time steps relative to the relaxation time. 
+In this viscoelastic problem neither the analytical or numerical
+models approach steady-state behavior until after several earthquake
+cycles. There is also a difference in how steady plate motion is
+applied for the two models. For the analytical solution, steady plate
+motion is simply superimposed, while for the numerical solution steady
+plate motion is approached after several earthquake cycles, once the
+applied fault slip and velocity boundary conditions have produced
+nearly steady flow in the viscoelastic half-space. It is therefore
+necessary to spin-up both solutions for several earthquake cycles to
+allow a comparison between the two. In this way, both models will have
+achieved steady-state behavior, and both models will have
+approximately the same component of steady plate motion. We simulate
+ten earthquake cycles for both the analytical and numerical models for
+a total duration of 2000 years. For the numerical solution we use a
+constant time step size of five years. This time step corresponds to
+one tenth of the viscoelastic relaxation time; hence it tests the
+accuracy of the viscoelastic solution for moderately large time steps
+relative to the relaxation time.
 
 Figure~\ref{fig:savage:prescott:profiles} compares the numerical
 results extracted on the ground surface along the center of the model
 perpendicular to the fault with the analytic solution. Using a
-logarithmic scale with distance from the fault facilitates examing the
-solution both close to the fault and in the far-field. In the
-far-field the simulations show virtually identical results; however,
-the viscoelastic solution has not yet achieved steady state and the
-simulations underpredict the displacement. By the tenth earthquake
-cycle, the simulations reach a steady state and match the analytical
-solution.
+logarithmic scale with distance from the fault facilitates examining
+the solution both close to the fault and in the far-field. For the
+second earthquake cycle, the far-field numerical solution does not yet
+accurately represent steady plate motion and the numerical simulations
+underpredict the displacement. By the tenth earthquake cycle, steady
+plate motion is accurately simulated and the numerical results thus
+match the analytical solution.
 
 Within about one elastic thickness of the fault the effect of the
 resolution of the numerical models becomes apparent. We find large
 errors for the coarse models, which have discretization sizes matching
-the elastic thickness. The finer resolution models (6.7 km
+the fault locking depth. The finer resolution models (6.7 km
 discretization size) provide a close fit to the analytical
 solution. The 6.7 km hexahedral solution is indistinguishable from the
 analytical solution in Figure~\ref{fig:savage:prescott:profiles}(b);
 the 6.7 km tetrahedral solution slightly underpredicts the analytical
-solution.  The greater accuracy of the hexahedral cells relative to
-the tetrahedral cells with the same nominal discretization size for
-quasi-static solutions is consistent with our findings for other
-benchmarks. The greater number of polynomial terms in the basis
-functions of the hexahedral allows the model to capture a more complex
-deformation field.
+solution for times later in the earthquake cycle.  The greater
+accuracy of the hexahedral cells relative to the tetrahedral cells
+with the same nominal discretization size for quasi-static solutions
+is consistent with our findings for other benchmarks. The greater
+number of polynomial terms in the basis functions of the hexahedra
+allows the model to capture a more complex deformation field.
 
 \subsection{Dynamic}
 \label{sec:verification:dynamic}
@@ -1618,10 +1625,10 @@ simulations of earthquake rupture propag
     way through earthquake cycle 10 of the Savage and Prescott
     benchmark, which involves viscoelastic relaxation over multiple
     earthquake cycles on a vertical, strike-slip fault. The
-    coordinates are in units of locking depth and the displacement
-    field is in units of coseismic slip. The locking depth is one-half
-    of the thickness of the elastic layer. We refine the mesh by a
-    factor of three near the center of the
+    coordinates are in units of elastic layer thickness and the
+    displacement field is in units of coseismic slip. The locking
+    depth is one-half of the thickness of the elastic layer. We refine
+    the hexahedral mesh by a factor of three near the center of the
     domain. Figure~\ref{fig:savage:prescott:profiles} compares
     profiles along y=0 with the analytic solution.}
   \label{fig:savage:prescott::solution}
@@ -1645,7 +1652,7 @@ simulations of earthquake rupture propag
     coarser (20 km) resolutions are unable to match the details of the
     displacement field at distances less than about one elastic
     thickness, but all of the numerical models provide a good fit to
-    the analytical solution at distances greater than a 2-3 times the
+    the analytical solution at distances greater than 2-3 times the
     elastic thickness.}
   \label{fig:savage:prescott:profiles}
 \end{figure}