# [cig-commits] r15722 - doc/geodynamics.org/benchmarks/trunk/long

luis at geodynamics.org luis at geodynamics.org
Wed Sep 30 15:07:39 PDT 2009

Author: luis
Date: 2009-09-30 15:07:38 -0700 (Wed, 30 Sep 2009)
New Revision: 15722

Modified:
doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.html
doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.rst
Log:
Fixes to long

Modified: doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.html
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.html	2009-09-30 22:07:30 UTC (rev 15721)
+++ doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.html	2009-09-30 22:07:38 UTC (rev 15722)
@@ -298,18 +298,21 @@
<p>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 complicated. So we used the simple
-relation $$v_x = -dot{epsilon}y, v_y=dot{epsilon}x,$$ for the
-boundaries, where $dot{epsilon}$ is the magnitude of the shear and $x$
-and $y$ are the coordinates. This induces an error of order $r_i^2/r^2$,
-where $r_i=0.1$ is the radius of the inclusion, and $r$ 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
+relation</p>
+<blockquote>
+[;v_x = -\dot{\epsilon}y, v_y=\dot{\epsilon}x,;]</blockquote>
+<p>for the boundaries, where [;dot{epsilon};] is the magnitude of the shear
+and [;x;] and [;y;] are the coordinates. This induces an error of order
+[;r_i^2/r^2;], where [;r_i=0.1;] is the radius of the inclusion, and [;r;]
+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
were looking at. Just to make sure, we did runs with boundaries at 40
times the radius of the inclusion and got very similar results.</p>
<p>A characteristic of the analytic solution is that the pressure is zero
-inside the inclusion, while outside it follows the relation
-$$p_m=4dot{epsilon}frac{mu_m(mu_i-mu_m)}{mu_i+mu_m}frac{r_i^2}{r^2}cos(2theta),$$
-where $mu_i=2$ is the viscosity of the inclusion and $mu_m=1$ is the
+inside the inclusion, while outside it follows the relation</p>
+<blockquote>
+[;p_m=4\dot{\epsilon}\frac{\mu_m(\mu_i-\mu_m)}{\mu_i+\mu_m}\frac{r_i^2}{r^2}\cos(2\theta),;]</blockquote>
+<p>where [;\mu_i=2;] is the viscosity of the inclusion and [;\mu_m=1;] is the
viscosity of the background media. Many numerical codes that solve Stokes
flow (Eq. [eq:simple momentum conservation] and [eq:continuity]),
including Gale, assume that pressure, velocity, and viscosity are
@@ -317,16 +320,23 @@
violates that assumption, so the error tends to concentrate near the
surface of the inclusion.</p>
<p>Figure [fig:Pressure-inclusion] plots the error in the pressure along the
-line $y=x/2$ for different resolutions. Inside the inclusion near the
+line [;y=x/2;] 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 default
numerical scheme is not accurate.</p>
<div class="figure">
-<img alt="images/inclusion_r8_p.pngFigure[fig:Pressure-inclusion]Pressurealongtheline$y=x/2$forresolutionsof$128\times128$(blue),$256\times256$(red),and$512\times512$(black).Theinclusionhasradius$r_i=0.1$.Notethatthepressureshouldbezeroinsidetheinclusion,butthenumericalsolutionsconsistentlyunderestimatethepressure." src="images/inclusion_r8_p.pngFigure[fig:Pressure-inclusion]Pressurealongtheline$y=x/2$forresolutionsof$128\times128$(blue),$256\times256$(red),and$512\times512$(black).Theinclusionhasradius$r_i=0.1$.Notethatthepressureshouldbezeroinsidetheinclusion,butthenumericalsolutionsconsistentlyunderestimatethepressure." />
+<img alt="images/inclusion_r8_p.png" src="images/inclusion_r8_p.png" />
+<p class="caption">Figure [fig:Pressure-inclusion]</p>
+<div class="legend">
+Pressure along the line $y=x/2$ for resolutions of $128 times 128$
+(blue), $256 times 256$ (red), and $512 times 512$ (black). The
+inclusion has radius $r_i=0.1$. Note that the pressure should be zero
+inside the inclusion, but the numerical solutions consistently
+underestimate the pressure.</div>
</div>
<p>Outside the inclusion, the error is better behaved. Figure
[fig:Pressure-error] plots the error in the pressure along the line
-$y=x/2$ outside the inclusion for different resolutions. While there are
+[;y=x/2;] outside the inclusion for different resolutions. While there are
still problems near the surface, away from the surface the solutions are
quite good. Figure [fig:Scaled-pressure-error] plots the error scaled
with resolution, and we can see that the error scales linearly with
@@ -335,10 +345,20 @@
where the solution is bad close to the surface, but good otherwise, is
typical for numerical solutions of this problem [FD Stokes].</p>
<div class="figure">
-<img alt="images/inclusion_r8_p_error.pngFigure[fig:Pressure-error]Errorinthepressureoutsidetheinclusionalongtheline$y=x/2$forresolutionsof$128\times128$(blue),$256\times256$(red),and$512\times512$(black).Theinclusionhasradius$r_i=0.1$." src="images/inclusion_r8_p_error.pngFigure[fig:Pressure-error]Errorinthepressureoutsidetheinclusionalongtheline$y=x/2$forresolutionsof$128\times128$(blue),$256\times256$(red),and$512\times512$(black).Theinclusionhasradius$r_i=0.1$." />
+<img alt="images/inclusion_r8_p_error.png" src="images/inclusion_r8_p_error.png" />
+<p class="caption">Figure [fig:Pressure-error]</p>
+<div class="legend">
+Error in the pressure outside the inclusion along the line $y=x/2$ for
+resolutions of $128 times 128$ (blue), $256 times 256$ (red), and
+$512 times 512$ (black). The inclusion has radius $r_i=0.1$.</div>
</div>
<div class="figure">
-<img alt="images/inclusion_r8_p_scaled_error.pngFigure[fig:Scaled-pressure-error]AsinFigure[fig:Pressure-error],butwiththeerrorscaledwith$h$.Sothemedium-resolutionerrorismultipliedby2andthehigh-resolutionerrorismultipliedby4." src="images/inclusion_r8_p_scaled_error.pngFigure[fig:Scaled-pressure-error]AsinFigure[fig:Pressure-error],butwiththeerrorscaledwith$h$.Sothemedium-resolutionerrorismultipliedby2andthehigh-resolutionerrorismultipliedby4." />
+<img alt="images/inclusion_r8_p_scaled_error.png" src="images/inclusion_r8_p_scaled_error.png" />
+<p class="caption">Figure [fig:Scaled-pressure-error]</p>
+<div class="legend">
+As in Figure [fig:Pressure-error], but with the error scaled with $h$.
+So the medium-resolution error is multiplied by 2 and the
+high-resolution error is multiplied by 4.</div>
</div>
</div>
</body>

Modified: doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.rst
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.rst	2009-09-30 22:07:30 UTC (rev 15721)
+++ doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.rst	2009-09-30 22:07:38 UTC (rev 15722)
@@ -6,8 +6,10 @@
the pressure and velocity fields for a circular inclusion under simple
shear as in Figure [fig:inclusion-setup].

-.. figure:: images/inclusion_setup.eps
+.. figure:: images/inclusion_setup.png
+
Figure [fig:inclusion-setup]
+
Schematic for the circular inclusion benchmark

The file input/benchmarks/circular_inclusion/README has instructions
@@ -16,19 +18,24 @@
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 complicated. So we used the simple
-relation $$v_x = -\dot{\epsilon}y, v_y=\dot{\epsilon}x,$$ for the
-boundaries, where $\dot{\epsilon}$ is the magnitude of the shear and $x$
-and $y$ are the coordinates. This induces an error of order $r_i^2/r^2$,
-where $r_i=0.1$ is the radius of the inclusion, and $r$ 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
+relation
+
+    [;v_x = -\\dot{\\epsilon}y, v_y=\\dot{\\epsilon}x,;]
+
+for the boundaries, where [;\\dot{\\epsilon};] is the magnitude of the shear
+and [;x;] and [;y;] are the coordinates. This induces an error of order
+[;r_i^2/r^2;], where [;r_i=0.1;] is the radius of the inclusion, and [;r;]
+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
were looking at. Just to make sure, we did runs with boundaries at 40
times the radius of the inclusion and got very similar results.

A characteristic of the analytic solution is that the pressure is zero
inside the inclusion, while outside it follows the relation
-$$p_m=4\dot{\epsilon}\frac{\mu_m(\mu_i-\mu_m)}{\mu_i+\mu_m}\frac{r_i^2}{r^2}\cos(2\theta),$$
-where $\mu_i=2$ is the viscosity of the inclusion and $\mu_m=1$ is the
+
+    [;p_m=4\\dot{\\epsilon}\\frac{\\mu_m(\\mu_i-\\mu_m)}{\\mu_i+\\mu_m}\\frac{r_i^2}{r^2}\\cos(2\\theta),;]
+
+where [;\\mu_i=2;] is the viscosity of the inclusion and [;\\mu_m=1;] is the
viscosity of the background media. Many numerical codes that solve Stokes
flow (Eq. [eq:simple momentum conservation] and [eq:continuity]),
including Gale, assume that pressure, velocity, and viscosity are
@@ -37,13 +44,15 @@
surface of the inclusion.

Figure [fig:Pressure-inclusion] plots the error in the pressure along the
-line $y=x/2$ for different resolutions. Inside the inclusion near the
+line [;y=x/2;] 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 default
numerical scheme is not accurate.

.. figure:: images/inclusion_r8_p.png
+
Figure [fig:Pressure-inclusion]
+
Pressure along the line $y=x/2$ for resolutions of $128 \times 128$
(blue), $256 \times 256$ (red), and $512 \times 512$ (black). The
inclusion has radius $r_i=0.1$. Note that the pressure should be zero
@@ -52,7 +61,7 @@

Outside the inclusion, the error is better behaved. Figure
[fig:Pressure-error] plots the error in the pressure along the line
-$y=x/2$ outside the inclusion for different resolutions. While there are
+[;y=x/2;] outside the inclusion for different resolutions. While there are
still problems near the surface, away from the surface the solutions are
quite good. Figure [fig:Scaled-pressure-error] plots the error scaled
with resolution, and we can see that the error scales linearly with
@@ -62,14 +71,20 @@
typical for numerical solutions of this problem [FD Stokes].

.. figure:: images/inclusion_r8_p_error.png
+
Figure [fig:Pressure-error]
-   Error in the pressure outside the inclusion along the line $y=x/2$ for
-   resolutions of $128 \times 128$ (blue), $256 \times 256$ (red), and
-   $512 \times 512$ (black). The inclusion has radius $r_i=0.1$.

+   Error in the pressure outside the inclusion along the line [;y=x/2;]
+   for resolutions of [;128 \times 128;] (blue), [;256 \times 256;] (red),
+   and [;512 \times 512;] (black). The inclusion has radius $sr_i=0.1$.
+
+
.. figure:: images/inclusion_r8_p_scaled_error.png
+
Figure [fig:Scaled-pressure-error]
+
As in Figure [fig:Pressure-error], but with the error scaled with $h$.
So the medium-resolution error is multiplied by 2 and the
high-resolution error is multiplied by 4.

+