[cig-commits] r15780 - doc/geodynamics.org/benchmarks/trunk/long
luis at geodynamics.org
luis at geodynamics.org
Wed Oct 7 12:15:21 PDT 2009
Author: luis
Date: 2009-10-07 12:15:20 -0700 (Wed, 07 Oct 2009)
New Revision: 15780
Modified:
doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.html
doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.rst
Log:
Updated formatting of figures in long/circular-inclusion.rst
Figures now centered, with 80% width, and hyperlinked.
Modified: doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.html
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.html 2009-10-07 19:15:14 UTC (rev 15779)
+++ doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.html 2009-10-07 19:15:20 UTC (rev 15780)
@@ -5,7 +5,7 @@
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<meta name="generator" content="Docutils 0.5: http://docutils.sourceforge.net/" />
<title>Circular Inclusion</title>
-<link rel="stylesheet" href="../css/default.css" type="text/css" />
+<link rel="stylesheet" href="../css/voidspace.css" type="text/css" />
</head>
<body>
<div class="document" id="circular-inclusion">
@@ -13,10 +13,11 @@
<p>Schmid and Podladchikov [Clast] derived a simple analytic solution for
the pressure and velocity fields for a circular inclusion under simple
-shear as in Figure [fig:inclusion-setup].</p>
-<div class="figure">
-<img alt="images/inclusion_setup.png" src="images/inclusion_setup.png" />
-<p class="caption">Figure [fig:inclusion-setup]</p>
+shear as in <a class="reference internal" href="#figure-inclusion-setup">Figure [Inclusion-Setup]</a>.</p>
+<!-- fig:Inclusion-Setup -->
+<div align="center" class="figure">
+<img alt="Circular Inclusion Geometry" src="images/inclusion_setup.png" />
+<p class="caption"><span class="target" id="figure-inclusion-setup">Figure [Inclusion-Setup]</span></p>
<div class="legend">
Schematic for the circular inclusion benchmark</div>
</div>
@@ -28,62 +29,67 @@
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>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>
<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]),
+<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
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.</p>
-<p>Figure [fig:Pressure-inclusion] plots the error in the pressure along the
+<p><a class="reference internal" href="#figure-pressure-inclusion">Figure [Pressure-Inclusion]</a> plots the error in the pressure along 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.png" src="images/inclusion_r8_p.png" />
-<p class="caption">Figure [fig:Pressure-inclusion]</p>
+<!-- fig:Pressure-Inclusion -->
+<div align="center" class="figure">
+<img alt="Pressure Field" src="images/inclusion_r8_p.png" style="width: 80%;" />
+<p class="caption"><span class="target" id="figure-pressure-inclusion">Figure [Pressure-Inclusion]</span></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
+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
-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
-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 [FD Stokes].</p>
-<div class="figure">
-<img alt="images/inclusion_r8_p_error.png" src="images/inclusion_r8_p_error.png" />
-<p class="caption">Figure [fig:Pressure-error]</p>
+<p>Outside the inclusion, the error is better behaved.
+<a class="reference internal" href="#figure-pressure-error">Figure [Pressure-Error]</a> plots the error in the pressure along
+the line [;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. <a class="reference internal" href="#figure-scaled-pressure-error">Figure [Scaled-Pressure-Error]</a>
+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 [FD Stokes].</p>
+<!-- fig:Pressure-Error -->
+<div align="center" class="figure">
+<img alt="Pressure Error" src="images/inclusion_r8_p_error.png" style="width: 80%;" />
+<p class="caption"><span class="target" id="figure-pressure-error">Figure [Pressure-Error]</span></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 $sr_i=0.1$.</div>
+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.png" src="images/inclusion_r8_p_scaled_error.png" />
-<p class="caption">Figure [fig:Scaled-pressure-error]</p>
+<!-- fig:Scaled-pressure-error -->
+<div align="center" class="figure">
+<img alt="Scaled Pressure Error" src="images/inclusion_r8_p_scaled_error.png" style="width: 80%;" />
+<p class="caption"><span class="target" id="figure-scaled-pressure-error">Figure [Scaled-Pressure-Error]</span></p>
<div class="legend">
-As in Figure [fig:Pressure-error], but with the error scaled with $h$.
+As in <a class="reference internal" href="#figure-pressure-error">Figure [Pressure-Error]</a>, 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>
Modified: doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.rst
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.rst 2009-10-07 19:15:14 UTC (rev 15779)
+++ doc/geodynamics.org/benchmarks/trunk/long/circular-inclusion.rst 2009-10-07 19:15:20 UTC (rev 15780)
@@ -4,11 +4,14 @@
Schmid and Podladchikov [Clast] derived a simple analytic solution for
the pressure and velocity fields for a circular inclusion under simple
-shear as in Figure [fig:inclusion-setup].
+shear as in `Figure [Inclusion-Setup]`_.
+.. fig:Inclusion-Setup
.. figure:: images/inclusion_setup.png
+ :alt: Circular Inclusion Geometry
+ :align: center
- Figure [fig:inclusion-setup]
+ _`Figure [Inclusion-Setup]`
Schematic for the circular inclusion benchmark
@@ -22,68 +25,82 @@
[;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.
+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
-viscosity of the background media. Many numerical codes that solve Stokes
-flow (Eq. [eq:simple momentum conservation] and [eq:continuity]),
+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
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.
-Figure [fig:Pressure-inclusion] plots the error in the pressure along the
+`Figure [Pressure-Inclusion]`_ plots the error in the pressure along 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.
+.. fig:Pressure-Inclusion
.. figure:: images/inclusion_r8_p.png
+ :alt: Pressure Field
+ :align: center
+ :width: 80%
- Figure [fig:Pressure-inclusion]
+ _`Figure [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
+ 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.
-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
-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
-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 [FD Stokes].
+Outside the inclusion, the error is better behaved.
+`Figure [Pressure-Error]`_ plots the error in the pressure along
+the line [;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 [Scaled-Pressure-Error]`_
+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 [FD Stokes].
+.. fig:Pressure-Error
.. figure:: images/inclusion_r8_p_error.png
+ :alt: Pressure Error
+ :align: center
+ :width: 80%
- Figure [fig:Pressure-error]
+ _`Figure [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 $sr_i=0.1$.
+ for resolutions of [;128 \\times 128;] (blue), [;256 \\times 256;] (red),
+ and [;512 \\times 512;] (black). The inclusion has radius [;r_i=0.1;].
+.. fig:Scaled-pressure-error
.. figure:: images/inclusion_r8_p_scaled_error.png
+ :alt: Scaled Pressure Error
+ :align: center
+ :width: 80%
- Figure [fig:Scaled-pressure-error]
+ _`Figure [Scaled-Pressure-Error]`
- As in Figure [fig:Pressure-error], but with the error scaled with $h$.
+ As in `Figure [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.
More information about the CIG-COMMITS
mailing list