[cig-commits] r11612 - in long/3D/Gale/trunk: . documentation documentation/images
walter at geodynamics.org
walter at geodynamics.org
Thu Mar 27 16:23:56 PDT 2008
Author: walter
Date: 2008-03-27 16:23:56 -0700 (Thu, 27 Mar 2008)
New Revision: 11612
Removed:
long/3D/Gale/trunk/documentation/images/inclusion_strain_v.png
Modified:
long/3D/Gale/trunk/
long/3D/Gale/trunk/documentation/gale.lyx
Log:
r2067 at earth: boo | 2008-03-27 16:23:25 -0700
More doc updates
Property changes on: long/3D/Gale/trunk
___________________________________________________________________
Name: svk:merge
- 3a629746-de10-0410-b17b-fd6ecaaa963e:/cig:2062
+ 3a629746-de10-0410-b17b-fd6ecaaa963e:/cig:2067
Modified: long/3D/Gale/trunk/documentation/gale.lyx
===================================================================
--- long/3D/Gale/trunk/documentation/gale.lyx 2008-03-27 22:39:32 UTC (rev 11611)
+++ long/3D/Gale/trunk/documentation/gale.lyx 2008-03-27 23:23:56 UTC (rev 11612)
@@ -1,4 +1,4 @@
-#LyX 1.5.1 created this file. For more info see http://www.lyx.org/
+#LyX 1.5.3 created this file. For more info see http://www.lyx.org/
\lyxformat 276
\begin_document
\begin_header
@@ -6084,11 +6084,11 @@
\begin_layout Standard
The top cover image shows the strain rate invariant after the model has
extended 30 km.
- The resolution is 2048
+ The resolution is 2048
\begin_inset Formula $\times$
\end_inset
- 512, and we used a direct solver (Mumps).
+512, and we used a direct solver (Mumps).
The most prominent faults occur near where the crust thickens, although
smaller faults occur throughout the crust.
The depth of the faults is limited by the relatively low viscosity deeper
@@ -6097,11 +6097,11 @@
\end_layout
\begin_layout Standard
-The 3D input file models a region 1000km
+The 3D input file models a region 1000km
\begin_inset Formula $\times$
\end_inset
- 1000km.
+1000km.
Topography is imported from a data file.
Underneath, the crust extends further down 32 km, and the mantle is 68
km thick beyond that.
@@ -6149,15 +6149,15 @@
\begin_layout Standard
The bottom cover image shows the strain rate invariant after the model has
extended 24 km.
- The resolution is 128
+ The resolution is 128
\begin_inset Formula $\times$
\end_inset
- 128
+128
\begin_inset Formula $\times$
\end_inset
- 16, and we used an iterative solver (GMRES).
+16, and we used an iterative solver (GMRES).
The fault locations are determined by the variations in topography.
\end_layout
@@ -14915,189 +14915,246 @@
\begin_layout Standard
Gale has been tested against a number of different benchmarks.
-\end_layout
-
-\begin_layout Section
-\begin_inset LatexCommand label
-name "sec:Circular-Inclusion"
-
-\end_inset
-
-Circular Inclusion
-\end_layout
-
-\begin_layout Standard
-Schmid and Podladchikov
-\begin_inset LatexCommand cite
-key "Clast"
-
-\end_inset
-
- derived a simple analytic solution for the pressure and velocity fields
- for a circular inclusion under simple shear.
- The file
-\family typewriter
-input/benchmarks/circular_inclusion/README
-\family default
- has instructions on how to run this benchmark.
- Figure
+ Each benchmark tests different parts of the code, although there is some
+ overlap.
+ Specifically, Table
\begin_inset LatexCommand ref
-reference "fig:inclusion-strain-v"
+reference "tab:test-coverage"
\end_inset
- plots the strain rate and velocities of a medium resolution run.
+ summarizes which parts of the code are tested by which benchmark.
\end_layout
\begin_layout Standard
-\noindent
-\align center
-\begin_inset Float figure
+\begin_inset Float table
placement H
wide false
sideways false
-status collapsed
+status open
\begin_layout Standard
\align center
-\begin_inset Graphics
- filename images/inclusion_strain_v.png
- scale 50
+\begin_inset Tabular
+<lyxtabular version="3" rows="14" columns="2">
+<features>
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+\begin_layout Standard
+Code Functionality
+\end_layout
+
\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+\begin_layout Standard
+Benchmark
+\end_layout
-\begin_inset Caption
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
\begin_layout Standard
-\begin_inset LatexCommand label
-name "fig:inclusion-strain-v"
+Stokes solver in 2D
+\end_layout
\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
-Strain rate invariant and velocities for the circular inclusion benchmark
-\end_layout
+\begin_layout Standard
+\begin_inset LatexCommand ref
+reference "sec:Circular-Inclusion"
\end_inset
+,
+\begin_inset LatexCommand ref
+reference "sec:Relaxation-of-Topography"
-\end_layout
+\end_inset
+,
+\begin_inset LatexCommand ref
+reference "sec:Geomod-2004"
+
\end_inset
\end_layout
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
\begin_layout Standard
-Because of the symmetry of the problem, we only have to solve over a quarter
- of the domain.
- The analytic solution is for an infinite plane, so we moved the boundaries
- to 80 times the radius of the inclusion.
- The differences between having the boundaries at 40 versus 80 times the
- radius are small, so we feel confident that boundary effects are not significan
-t for the resolutions we ran.
+Stokes solver in 3D
\end_layout
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
\begin_layout Standard
-A characteristic of the analytic solution is that the pressure is zero inside
- the inclusion, while outside it follows the relation
+\begin_inset LatexCommand ref
+reference "sec:Falling-Sphere"
+
+\end_inset
+
+
\end_layout
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
\begin_layout Standard
-\begin_inset Formula \[
-p_{m}=4\dot{\epsilon}\frac{\mu_{m}\left(\mu_{i}-\mu_{m}\right)}{\mu_{i}+\mu_{m}}\frac{r_{i}^{2}}{r^{2}}\cos\left(2\theta\right),\]
+Interpolation of viscosities from particles to the mesh in 2D
+\end_layout
\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
-where
-\begin_inset Formula $\dot{\epsilon}=1$
+\begin_layout Standard
+\begin_inset LatexCommand ref
+reference "sec:Circular-Inclusion"
+
\end_inset
- is the magnitude of the shear,
-\begin_inset Formula $r_{i}=0.1$
+,
+\begin_inset LatexCommand ref
+reference "sec:Relaxation-of-Topography"
+
\end_inset
- is the radius of the inclusion,
-\begin_inset Formula $\mu_{i}=2$
+,
+\begin_inset LatexCommand ref
+reference "sec:Geomod-2004"
+
\end_inset
- is the viscosity of the inclusion, and
-\begin_inset Formula $\mu_{m}=1$
+
+\end_layout
+
\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
- is the viscosity of the background media.
- This pressure discontinuity at the surface of the inclusion creates special
- difficulties for numerical codes, so the error tends to concentrate around
- that surface.
-
+\begin_layout Standard
+Interpolation of viscosities from particles to the mesh in 3D
\end_layout
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
\begin_layout Standard
-Figure
\begin_inset LatexCommand ref
-reference "fig:Pressure-inclusion"
+reference "sec:Falling-Sphere"
\end_inset
- plots the error in the pressure along the line
-\begin_inset Formula $y=x/2$
-\end_inset
- for different resolutions.
- Inside the inclusion near the surface, the pressure is consistently underestima
-ted.
- The pressure does not converge with higher resolution, giving us a clue
- that the numerical scheme is not completely accurate.
\end_layout
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
\begin_layout Standard
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status collapsed
+Time stepping
+\end_layout
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
\begin_layout Standard
-\align center
-\begin_inset Graphics
- filename images/inclusion_r8_p.png
- width 100col%
+\begin_inset LatexCommand ref
+reference "sec:Falling-Sphere"
\end_inset
+,
+\begin_inset LatexCommand ref
+reference "sec:Circular-Inclusion"
-\begin_inset Caption
+\end_inset
-\begin_layout Standard
-\begin_inset LatexCommand label
-name "fig:Pressure-inclusion"
+,
+\begin_inset LatexCommand ref
+reference "sec:Relaxation-of-Topography"
\end_inset
-Pressure along the line
-\begin_inset Formula $y=x/2$
+,
+\begin_inset LatexCommand ref
+reference "sec:Geomod-2004"
+
\end_inset
- for resolutions of 128
-\begin_inset Formula $\times$
+
+\end_layout
+
\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
-128 (blue), 256
-\begin_inset Formula $\times$
+\begin_layout Standard
+Gravity
+\end_layout
+
\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
-256 (green), and 512
-\begin_inset Formula $\times$
+\begin_layout Standard
+\begin_inset LatexCommand ref
+reference "sec:Falling-Sphere"
+
\end_inset
-512 (red).
- The inclusion has a radius
-\begin_inset Formula $r_{i}=0.1.$
+,
+\begin_inset LatexCommand ref
+reference "sec:Relaxation-of-Topography"
+
\end_inset
- Note that the pressure should be zero inside the inclusion, but the numerical
- solutions consistently underestimate the pressure.
-\end_layout
+,
+\begin_inset LatexCommand ref
+reference "sec:Geomod-2004"
\end_inset
@@ -15105,170 +15162,275 @@
\end_layout
\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
-
+\begin_layout Standard
+Internal constraints
\end_layout
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
\begin_layout Standard
-Outside the inclusion, the error is better behaved.
- Figure
\begin_inset LatexCommand ref
-reference "fig:Pressure-Error"
+reference "sec:Falling-Sphere"
\end_inset
- plots the error in the pressure along the line
-\begin_inset Formula $y=x/2$
+
+\end_layout
+
\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
- outside the inclusion for different resolutions.
- While there are still problems near the surface, away from the surface
- the solutions are quite good.
- Figure
+\begin_layout Standard
+No-slip boundary conditions
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
\begin_inset LatexCommand ref
-reference "fig:Scaled-pressure-error"
+reference "sec:Falling-Sphere"
\end_inset
- plots the error scaled with resolution, and we can see that the error scales
- linearly with resolution.
- This gives us confidence that, at least away from the inclusion, the code
- is giving the right answer.
- This kind of result, where the solution is bad close to the surface, but
- good otherwise, is typical for numerical solutions of this problem
-\begin_inset LatexCommand cite
-key "FD Stokes"
+,
+\begin_inset LatexCommand ref
+reference "sec:Geomod-2004"
\end_inset
-.
-
+
\end_layout
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
\begin_layout Standard
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status collapsed
+Non-zero velocity boundary conditions
+\end_layout
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
\begin_layout Standard
-\align center
-\begin_inset Graphics
- filename images/inclusion_r8_p_error.png
- width 100col%
+\begin_inset LatexCommand ref
+reference "sec:Circular-Inclusion"
\end_inset
+,
+\begin_inset LatexCommand ref
+reference "sec:Geomod-2004"
+\end_inset
+
+
\end_layout
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
\begin_layout Standard
-\begin_inset Caption
+Free slip boundary conditions
+\end_layout
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
\begin_layout Standard
-\begin_inset LatexCommand label
-name "fig:Pressure-Error"
+\begin_inset LatexCommand ref
+reference "sec:Circular-Inclusion"
\end_inset
-Error in the pressure outside the inclusion along the line
-\begin_inset Formula $y=x/2$
+,
+\begin_inset LatexCommand ref
+reference "sec:Relaxation-of-Topography"
+
\end_inset
- for resolutions of 128
-\begin_inset Formula $\times$
+
+\end_layout
+
\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
-128 (blue), 256
-\begin_inset Formula $\times$
+\begin_layout Standard
+Free surface
+\end_layout
+
\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
-256 (green), and 512
-\begin_inset Formula $\times$
+\begin_layout Standard
+\begin_inset LatexCommand ref
+reference "sec:Relaxation-of-Topography"
+
\end_inset
-512 (red).
- The inclusion has a radius
-\begin_inset Formula $r_{i}=0.1.$
+,
+\begin_inset LatexCommand ref
+reference "sec:Geomod-2004"
+
\end_inset
\end_layout
\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
-
+\begin_layout Standard
+Mohr Coulomb rheology
\end_layout
\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+\begin_layout Standard
+\begin_inset LatexCommand ref
+reference "sec:Geomod-2004"
+\end_inset
+
+
\end_layout
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
\begin_layout Standard
-\begin_inset Float figure
-placement H
-wide false
-sideways false
-status collapsed
+Friction boundary conditions
+\end_layout
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
\begin_layout Standard
-\align center
-\begin_inset Graphics
- filename images/inclusion_r8_p_scaled_error.png
- width 100col%
+\begin_inset LatexCommand ref
+reference "sec:Geomod-2004"
\end_inset
\end_layout
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
\begin_inset Caption
\begin_layout Standard
\begin_inset LatexCommand label
-name "fig:Scaled-pressure-error"
+name "tab:test-coverage"
\end_inset
-Scaled error in the pressure outside the inclusion along the line
-\begin_inset Formula $y=x/2$
-\end_inset
+Summary of which parts of the code are tested by benchmarks
+\end_layout
- for resolutions of 128
-\begin_inset Formula $\times$
\end_inset
-128 (blue), 256
-\begin_inset Formula $\times$
-\end_inset
-256 (green), and 512
-\begin_inset Formula $\times$
-\end_inset
+\end_layout
-512 (red).
- So the medium resolution error is multiplied by 2 and the high resolution
- error is multiplied by 4.
- The inclusion has a radius
-\begin_inset Formula $r_{i}=0.1.$
\end_inset
\end_layout
+\begin_layout Standard
+With the exception of the GeoMod 2004 benchmark (Section
+\begin_inset LatexCommand ref
+reference "sec:Geomod-2004"
+
\end_inset
-
+), the benchmarks can be carried out to high precision (~1%).
+ In particular, the error should follow the relation
\end_layout
+\begin_layout Standard
+\begin_inset Formula \[
+error\propto h+O(h^{2}),\]
+
\end_inset
+ where
+\begin_inset Formula $h$
+\end_inset
+ is the size of the element.
+ This means that if we plot the error from three different resolutions (high,
+ medium and low) and scale it by
+\begin_inset Formula $h$
+\end_inset
+
+, we should see that the medium resolution error is closer to the high resolutio
+n error than the low resolution error.
+ In practice, this may be difficult to achieve because there are almost
+ always other sources of error besides resolution.
+
\end_layout
+\begin_layout Standard
+Altogether, these benchmarks give us a high degree of confidence in the
+ code.
+\end_layout
+
\begin_layout Section
\begin_inset LatexCommand label
name "sec:Falling-Sphere"
@@ -15442,7 +15604,7 @@
\end_inset
- where
+where
\begin_inset Formula $\eta'$
\end_inset
@@ -15508,7 +15670,7 @@
\end_inset
- The walls reduce the speed by a factor of two.
+ The walls reduce the speed by about a factor of two.
\end_layout
\begin_layout Standard
@@ -15613,6 +15775,7 @@
\noindent
\align center
\begin_inset Float figure
+placement H
wide false
sideways false
status open
@@ -15651,8 +15814,7 @@
\end_layout
\begin_layout Standard
-If we scale the error by multiplying the medium resolution error by 2 and
- the high resolution error by 4, we get Figure
+Scaling the error with resolution gives Figure
\begin_inset LatexCommand ref
reference "fig:Scaled-error-velocity"
@@ -15667,6 +15829,7 @@
\noindent
\align center
\begin_inset Float figure
+placement H
wide false
sideways false
status open
@@ -15690,8 +15853,19 @@
\end_inset
-Scaled error in computed velocity vs.
- resolution
+As in Figure
+\begin_inset LatexCommand ref
+reference "fig:Error-in-velocity"
+
+\end_inset
+
+, but with the error scaled with
+\begin_inset Formula $h$
+\end_inset
+
+.
+ So the medium resolution error is multiplied by 2 and the high resolution
+ error is multiplied by 4.
\end_layout
\end_inset
@@ -15699,16 +15873,401 @@
\end_layout
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+\begin_inset LatexCommand label
+name "sec:Circular-Inclusion"
+
+\end_inset
+
+Circular Inclusion
+\end_layout
+
\begin_layout Standard
+Schmid and Podladchikov
+\begin_inset LatexCommand cite
+key "Clast"
+\end_inset
+
+ derived a simple analytic solution for the pressure and velocity fields
+ for a circular inclusion under simple shear as in Figure
+\begin_inset LatexCommand ref
+reference "fig:inclusion-setup"
+
+\end_inset
+
+.
\end_layout
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Standard
+\align center
+\begin_inset Graphics
+ filename images/inclusion_setup.fig
+
\end_inset
\end_layout
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:inclusion-setup"
+
+\end_inset
+
+Schematic for the circular inclusion benchmark
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The file
+\family typewriter
+input/benchmarks/circular_inclusion/README
+\family default
+ has instructions on how to run this benchmark.
+
+\end_layout
+
+\begin_layout Standard
+Because of the symmetry of the problem, we only have to solve over the top
+ right quarter of the domain.
+ For the velocity boundary conditions, the analytic solution is a bit complicate
+d.
+ So we used the simple relation
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+v_{x} & = & -\dot{\epsilon}y,\\
+v_{y} & = & \dot{\epsilon}x,\end{eqnarray*}
+
+\end_inset
+
+ for the boundaries, where
+\begin_inset Formula $\dot{\epsilon}=1$
+\end_inset
+
+ is the magnitude of the shear and
+\begin_inset Formula $x$
+\end_inset
+
+ and
+\begin_inset Formula $y$
+\end_inset
+
+ are the coordinates.
+ This induces an error of order
+\begin_inset Formula $r_{i}^{2}/r^{2}$
+\end_inset
+
+, where
+\begin_inset Formula $r_{i}=0.1$
+\end_inset
+
+ is the radius of the inclusion, and
+\begin_inset Formula $r$
+\end_inset
+
+ is the radius.
+ We have the boundaries at 80 times the radius of the inclusion, giving
+ an error of about 0.01%, which is much smaller than the other errors we
+ are looking at.
+ Just to make sure, we did runs with the boundaries at 40 times the radius
+ of the inclusion and got very similar results.
+\end_layout
+
+\begin_layout Standard
+A characteristic of the analytic solution is that the pressure is zero inside
+ the inclusion, while outside it follows the relation
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+p_{m}=4\dot{\epsilon}\frac{\mu_{m}\left(\mu_{i}-\mu_{m}\right)}{\mu_{i}+\mu_{m}}\frac{r_{i}^{2}}{r^{2}}\cos\left(2\theta\right),\]
+
+\end_inset
+
+where
+\begin_inset Formula $\mu_{i}=2$
+\end_inset
+
+ is the viscosity of the inclusion and
+\begin_inset Formula $\mu_{m}=1$
+\end_inset
+
+ is the viscosity of the background media.
+ Many numerical codes that solve Stokes flow (eqns.
+
+\begin_inset LatexCommand ref
+reference "eq:simple momentum conservation"
+
+\end_inset
+
+ and
+\begin_inset LatexCommand ref
+reference "eq:continuity"
+
+\end_inset
+
+), including Gale, assume that pressure, velocity, and viscosity are continuous.
+ The pressure discontinuity at the surface of the inclusion violates that
+ assumption, so the error tends to concentrate near the surface of the inclusion.
+\end_layout
+
+\begin_layout Standard
+Figure
+\begin_inset LatexCommand ref
+reference "fig:Pressure-inclusion"
+
+\end_inset
+
+ plots the error in the pressure along the line
+\begin_inset Formula $y=x/2$
+\end_inset
+
+ for different resolutions.
+ Inside the inclusion near the surface, the pressure is consistently wrong.
+ The pressure does not converge with higher resolution, giving us a clue
+ that the numerical scheme is not completely accurate.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status collapsed
+
+\begin_layout Standard
+\align center
+\begin_inset Graphics
+ filename images/inclusion_r8_p.png
+ width 100col%
+
+\end_inset
+
+
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Pressure-inclusion"
+
+\end_inset
+
+Pressure along the line
+\begin_inset Formula $y=x/2$
+\end_inset
+
+ for resolutions of 128
+\begin_inset Formula $\times$
+\end_inset
+
+128 (blue), 256
+\begin_inset Formula $\times$
+\end_inset
+
+256 (green), and 512
+\begin_inset Formula $\times$
+\end_inset
+
+512 (red).
+ The inclusion has a radius
+\begin_inset Formula $r_{i}=0.1.$
+\end_inset
+
+ Note that the pressure should be zero inside the inclusion, but the numerical
+ solutions consistently underestimate the pressure.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Outside the inclusion, the error is better behaved.
+ Figure
+\begin_inset LatexCommand ref
+reference "fig:Pressure-Error"
+
+\end_inset
+
+ plots the error in the pressure along the line
+\begin_inset Formula $y=x/2$
+\end_inset
+
+ outside the inclusion for different resolutions.
+ While there are still problems near the surface, away from the surface
+ the solutions are quite good.
+ Figure
+\begin_inset LatexCommand ref
+reference "fig:Scaled-pressure-error"
+
+\end_inset
+
+ plots the error scaled with resolution, and we can see that the error scales
+ linearly with resolution.
+ This gives us confidence that, at least away from the inclusion, the code
+ is giving the right answer.
+ This kind of result, where the solution is bad close to the surface, but
+ good otherwise, is typical for numerical solutions of this problem
+\begin_inset LatexCommand cite
+key "FD Stokes"
+
+\end_inset
+
+.
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Standard
+\align center
+\begin_inset Graphics
+ filename images/inclusion_r8_p_error.png
+ width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Pressure-Error"
+
+\end_inset
+
+Error in the pressure outside the inclusion along the line
+\begin_inset Formula $y=x/2$
+\end_inset
+
+ for resolutions of 128
+\begin_inset Formula $\times$
+\end_inset
+
+128 (blue), 256
+\begin_inset Formula $\times$
+\end_inset
+
+256 (green), and 512
+\begin_inset Formula $\times$
+\end_inset
+
+512 (red).
+ The inclusion has a radius
+\begin_inset Formula $r_{i}=0.1.$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Standard
+\align center
+\begin_inset Graphics
+ filename images/inclusion_r8_p_scaled_error.png
+ width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Scaled-pressure-error"
+
+\end_inset
+
+As in Figure
+\begin_inset LatexCommand ref
+reference "fig:Pressure-Error"
+
+\end_inset
+
+, but with the error scaled with
+\begin_inset Formula $h$
+\end_inset
+
+.
+ So the medium resolution error is multiplied by 2 and the high resolution
+ error is multiplied by 4.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
\begin_layout Section
+\begin_inset LatexCommand label
+name "sec:Relaxation-of-Topography"
+
+\end_inset
+
Relaxation of Topography
\end_layout
@@ -15797,30 +16356,10 @@
\end_layout
\begin_layout Standard
-Running the code with multiple resolutions and measuring the error in the
- height in the trough gives Figure
-\begin_inset LatexCommand ref
-reference "fig:topo-error"
-
-\end_inset
-
-.
- Scaling the high resolution error by 4 and the medium resolution by 2 gives
- Figure
-\begin_inset LatexCommand ref
-reference "fig:scaled-topo-error"
-
-\end_inset
-
-.
- The error decreases linearly with increasing resolution, giving us confidence
- in our ability to accurately track topography.
-\end_layout
-
-\begin_layout Standard
\noindent
\align center
\begin_inset Float figure
+placement H
wide false
sideways false
status open
@@ -15829,6 +16368,7 @@
\align center
\begin_inset Graphics
filename images/Paraview_topography.png
+ scale 50
\end_inset
@@ -15855,23 +16395,39 @@
\end_layout
\begin_layout Standard
+Running the code with multiple resolutions and measuring the error in the
+ height in the trough gives Figure
+\begin_inset LatexCommand ref
+reference "fig:topo-error"
+
+\end_inset
+
+.
+ Scaling the error with resolution gives Figure
+\begin_inset LatexCommand ref
+reference "fig:scaled-topo-error"
+
+\end_inset
+
+.
+ The error decreases linearly with increasing resolution, giving us confidence
+ in our ability to accurately track topography.
+\end_layout
+
+\begin_layout Standard
\noindent
\align center
\begin_inset Float figure
+placement H
wide false
sideways false
status open
\begin_layout Standard
\align center
-
-\end_layout
-
-\begin_layout Standard
-\align center
\begin_inset Graphics
filename images/topo_error.eps
- scale 75
+ scale 50
\end_inset
@@ -15905,6 +16461,7 @@
\noindent
\align center
\begin_inset Float figure
+placement H
wide false
sideways false
status open
@@ -15913,7 +16470,7 @@
\align center
\begin_inset Graphics
filename images/topo_scaled_error.eps
- scale 75
+ scale 50
\end_inset
@@ -15926,8 +16483,19 @@
\end_inset
-Error in the height at the trough with the high resolution error scaled
- by 4 and the medium resolution error scaled by 2.
+As in Figure
+\begin_inset LatexCommand ref
+reference "fig:topo-error"
+
+\end_inset
+
+, but with the error scaled with
+\begin_inset Formula $h$
+\end_inset
+
+.
+ So the medium resolution error is multiplied by 2 and the high resolution
+ error is multiplied by 4.
\end_layout
\end_inset
Deleted: long/3D/Gale/trunk/documentation/images/inclusion_strain_v.png
===================================================================
(Binary files differ)
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