[cig-commits] r14526 - in long/3D/Gale/trunk: . documentation documentation/images

walter at geodynamics.org walter at geodynamics.org
Sun Mar 29 04:37:37 PDT 2009


Author: walter
Date: 2009-03-29 04:37:36 -0700 (Sun, 29 Mar 2009)
New Revision: 14526

Added:
   long/3D/Gale/trunk/documentation/images/Geomod2008_brittle_particles128x32.png
   long/3D/Gale/trunk/documentation/images/Geomod2008_brittle_particles256x64.png
   long/3D/Gale/trunk/documentation/images/Geomod2008_brittle_particles512x128.png
   long/3D/Gale/trunk/documentation/images/Geomod2008_brittle_setup.eps
   long/3D/Gale/trunk/documentation/images/Geomod2008_brittle_sri128x32.png
   long/3D/Gale/trunk/documentation/images/Geomod2008_brittle_sri256x64.png
   long/3D/Gale/trunk/documentation/images/Geomod2008_brittle_sri512x128.png
   long/3D/Gale/trunk/documentation/images/Geomod2008_unstable_particles128x32.png
   long/3D/Gale/trunk/documentation/images/Geomod2008_unstable_particles256x64.png
   long/3D/Gale/trunk/documentation/images/Geomod2008_unstable_particles512x128.png
   long/3D/Gale/trunk/documentation/images/Geomod2008_unstable_setup.eps
   long/3D/Gale/trunk/documentation/images/Geomod2008_unstable_sri128x32.png
   long/3D/Gale/trunk/documentation/images/Geomod2008_unstable_sri256x64.png
   long/3D/Gale/trunk/documentation/images/Geomod2008_unstable_sri512x128.png
   long/3D/Gale/trunk/documentation/images/Geomod2008_wedge_setup.eps
   long/3D/Gale/trunk/documentation/images/Stable_wedge_particles.png
   long/3D/Gale/trunk/documentation/images/Stable_wedge_sri.png
   long/3D/Gale/trunk/documentation/images/Stable_wedge_unstable.png
   long/3D/Gale/trunk/documentation/images/divergence_2D_error.eps
   long/3D/Gale/trunk/documentation/images/divergence_3D_error.eps
   long/3D/Gale/trunk/documentation/images/divergence_v.png
Modified:
   long/3D/Gale/trunk/
   long/3D/Gale/trunk/documentation/gale.lyx
Log:
 r2615 at dante:  boo | 2009-03-29 04:36:45 -0700
 Lots of doc updates.



Property changes on: long/3D/Gale/trunk
___________________________________________________________________
Name: svk:merge
   - 3a629746-de10-0410-b17b-fd6ecaaa963e:/cig:2613
   + 3a629746-de10-0410-b17b-fd6ecaaa963e:/cig:2615

Modified: long/3D/Gale/trunk/documentation/gale.lyx
===================================================================
--- long/3D/Gale/trunk/documentation/gale.lyx	2009-03-29 02:23:05 UTC (rev 14525)
+++ long/3D/Gale/trunk/documentation/gale.lyx	2009-03-29 11:37:36 UTC (rev 14526)
@@ -1,4 +1,4 @@
-#LyX 1.5.1 created this file. For more info see http://www.lyx.org/
+#LyX 1.5.5 created this file. For more info see http://www.lyx.org/
 \lyxformat 276
 \begin_document
 \begin_header
@@ -886,9 +886,19 @@
 In Gale, the velocity and pressure are represented by linear (Q1) finite
  elements (formally, this is called a Q1-Q1 scheme).
  Normally, this formulation is unstable.
- To stabilize it, we follow Dohrmann & Bochev () [TODO: add bib item and
- put ref.
- here] and add a compressibility term to the divergence equation
+ To stabilize it, we follow Dohrmann & Bochev 
+\begin_inset LatexCommand cite
+key "Stable Stokes"
+
+\end_inset
+
+ 
+\begin_inset LatexCommand cite
+key "Fast Solvers"
+
+\end_inset
+
+ and add a compressibility term to the divergence equation
 \end_layout
 
 \begin_layout Standard
@@ -1061,7 +1071,7 @@
 \end_inset
 
  are different.
- In particular, Equation 
+ Equation 
 \begin_inset LatexCommand ref
 reference "eq:stabilized"
 
@@ -1080,27 +1090,11 @@
  be unable to solve.
  One workaround is to scale the units of time and mass (e.g., seconds and
  kg) so that the viscosities are around 1.
- In particular, if the viscosities are around 
+ So if the viscosities are around 
 \begin_inset Formula $10^{25}$
 \end_inset
 
-, then scale time and mass as [TODO: do you mean 
-\begin_inset Quotes sld
-\end_inset
-
-is
-\begin_inset Quotes srd
-\end_inset
-
- instead of 
-\begin_inset Quotes sld
-\end_inset
-
-as
-\begin_inset Quotes srd
-\end_inset
-
-?]
+, then scale time and mass as
 \end_layout
 
 \begin_layout Standard
@@ -1279,7 +1273,7 @@
 \end_layout
 
 \begin_layout Standard
-Using standard finite-element techniques, we can collect all of the terms
+Using standard finite-element techniques, you can collect all of the terms
  together and represent them in matrix form
 \end_layout
 
@@ -2843,7 +2837,7 @@
 \end_layout
 
 \begin_layout Enumerate
-Using Octave to visualize the 10th step of 
+Using Octave to visualize the last step of 
 \family typewriter
 input/cookbook/yielding.xml
 \family default
@@ -2951,8 +2945,8 @@
 \begin_layout Standard
 In Gale, there are two kinds of resolution: grid and particles.
  The grid is where the Stokes equations are solved, and defines the resolution
- of everything in the ``fields'' and ``pressure'' output files (e.g., velocity,
- pressure, strain rate, etc.).
+ of everything in the ``fields'' output files (e.g., velocity, pressure, strain
+ rate, etc.).
  The resolution of the grid is determined by 
 \family typewriter
 elementResI
@@ -3386,20 +3380,6 @@
 input/cookbook/viscous.xml
 \family default
 .
- Figure 
-\begin_inset LatexCommand ref
-reference "fig:Strain-viscous"
-
-\end_inset
-
- [TODO: which figure? this label doesn't work now.] shows the velocity (see
- Section 
-\begin_inset LatexCommand ref
-reference "sub:Visualization-with-ParaView"
-
-\end_inset
-
-).
 \end_layout
 
 \begin_layout Section
@@ -5529,7 +5509,7 @@
 \family default
  component and a 
 \family typewriter
-MohrCoulomb
+DruckerPrager
 \family default
  component
 \end_layout
@@ -5580,7 +5560,7 @@
 \end_layout
 
 \begin_layout LyX-Code
-  <param name="Type">MohrCoulomb</param>
+  <param name="Type">DruckerPrager</param>
 \end_layout
 
 \begin_layout LyX-Code
@@ -6664,21 +6644,20 @@
 \end_inset
 
  at the bottom.
- The crust's yielding behavior is modeled with a Mohr-Coulomb rheology,
- with a cohesion of 
+ The crust's yielding behavior is modeled with a Drucker-Prager rheology,
+ with an internal angle of friction of 
+\begin_inset Formula $30$
+\end_inset
+
+ degrees and a cohesion of 
 \begin_inset Formula $4.4\cdot10^{6}kg\, m^{-1}s^{-2}$
 \end_inset
 
-, weakening to 
+ weakening to 
 \begin_inset Formula $4\cdot10^{5}kg\, m^{-1}s^{-2}$
 \end_inset
 
 .
- The internal angle of friction is 
-\begin_inset Formula $30$
-\end_inset
-
- degrees.
 \end_layout
 
 \begin_layout Standard
@@ -7283,21 +7262,174 @@
 \end_inset
 
 ), then the easiest way is to copy and paste an existing function.
- Specifically
+ For example, suppose you wanted to create a function 
+\family typewriter
+xSquared
+\family default
+ that returns 
+\begin_inset Formula $Ax^{2}$
+\end_inset
+
+, where A is a constant provided in the input file.
+ You start by opening the file
 \end_layout
 
 \begin_layout LyX-Code
 src/StgFEM/plugins/StandardConditionFunctions/StandardConditionFunctions.c
 \end_layout
 
+\begin_layout Standard
+and finding the function
+\end_layout
+
 \begin_layout LyX-Code
+StgFEM_StandardConditionFunctions_Extension
+\end_layout
+
+\begin_layout Standard
+Make a copy of that function and rename it to
+\end_layout
+
+\begin_layout LyX-Code
+StgFEM_StandardConditionFunctions_xSquared
+\end_layout
+
+\begin_layout Standard
+Then you only need to modify the last 6 lines of the function to get the
+ desired behavior.
+ Specifically, it will become
+\end_layout
+
+\begin_layout LyX-Code
+void StgFEM_StandardConditionFunctions_xSquared( Node_LocalIndex node_lI,
+ Variable_Index var_I, void* _context, void* _result ) {
+\end_layout
+
+\begin_layout LyX-Code
+        DiscretisationContext*  context            = (DiscretisationContext*)_co
+ntext;
+\end_layout
+
+\begin_layout LyX-Code
+        Dictionary*             dictionary         = context->dictionary;
+\end_layout
+
+\begin_layout LyX-Code
+        FeVariable*             feVariable         = NULL;
+\end_layout
+
+\begin_layout LyX-Code
+        FeMesh*                 mesh               = NULL;
+\end_layout
+
+\begin_layout LyX-Code
+        double*                 result             = (double*) _result;
+\end_layout
+
+\begin_layout LyX-Code
+        double*                 coord;
+\end_layout
+
+\begin_layout LyX-Code
+        double                  factor;
+\end_layout
+
+\begin_layout LyX-Code
+        
+\end_layout
+
+\begin_layout LyX-Code
+        feVariable = (FeVariable*)FieldVariable_Register_GetByName( context->fie
+ldVariable_Register, "VelocityField" );
+\end_layout
+
+\begin_layout LyX-Code
+        mesh       = feVariable->feMesh;
+\end_layout
+
+\begin_layout LyX-Code
+        /* Find Centre of Solid Body Rotation */
+\end_layout
+
+\begin_layout LyX-Code
+        factor = Dictionary_GetDouble_WithDefault( dictionary, "xSquaredFactor",
+ 1.0 );
+\end_layout
+
+\begin_layout LyX-Code
+        /* Find coordinate of node */
+\end_layout
+
+\begin_layout LyX-Code
+        coord = Mesh_GetVertex( mesh, node_lI );
+\end_layout
+
+\begin_layout LyX-Code
+        *result = factor * (coord[ I_AXIS ]*coord[ I_AXIS ];
+\end_layout
+
+\begin_layout LyX-Code
+}
+\end_layout
+
+\begin_layout Standard
+Then you need to register your function at the top of the file.
+ After the lines
+\end_layout
+
+\begin_layout LyX-Code
+condFunc = ConditionFunction_New( StgFEM_StandardConditionFunctions_Extension,
+ "Velocity_Extension" );
+\end_layout
+
+\begin_layout LyX-Code
+ConditionFunction_Register_Add( context->condFunc_Register, condFunc );
+\end_layout
+
+\begin_layout Standard
+add the lines
+\end_layout
+
+\begin_layout LyX-Code
+condFunc = ConditionFunction_New( StgFEM_StandardConditionFunctions_xSquared,
+ "xSquared" );
+\end_layout
+
+\begin_layout LyX-Code
+ConditionFunction_Register_Add( context->condFunc_Register, condFunc );
+\end_layout
+
+\begin_layout Standard
+The last thing to do is to modify
+\end_layout
+
+\begin_layout LyX-Code
 src/StgFEM/plugins/StandardConditionFunctions/StandardConditionFunctions.h
 \end_layout
 
 \begin_layout Standard
-The 
+After the line
 \end_layout
 
+\begin_layout LyX-Code
+void StgFEM_StandardConditionFunctions_Extension( Node_LocalIndex node_lI,
+ Variable_Index var_I, void* _context, void* _result ) ;
+\end_layout
+
+\begin_layout Standard
+add the line
+\end_layout
+
+\begin_layout LyX-Code
+void StgFEM_StandardConditionFunctions_xSquared( Node_LocalIndex node_lI,
+ Variable_Index var_I, void* _context, void* _result ) ;
+\end_layout
+
+\begin_layout Standard
+Now recompile the code, and you should be able to use your new function
+ from input files.
+\end_layout
+
 \begin_layout Chapter
 \start_of_appendix
 Input File Format
@@ -7440,106 +7572,86 @@
 \end_layout
 
 \begin_layout Standard
-Gale also has the ability to load modules during runtime.
- To use these plugins, simply list them in a section outside of the 
-\family typewriter
-components
-\family default
- structure.
- For example, the following lines will enable the 
-\family typewriter
-DumpSwarm
-\family default
-, 
-\family typewriter
-EulerDeform
-\family default
-, and 
-\family typewriter
-VTKOutput
-\family default
- plugins
+Gale nominally has the ability to load modules during runtime.
+ Dynamically loading modules is, in general, difficult to get working on
+ multiple platforms.
+ To simplify things, Gale compiles a number of plugins into the code:
 \end_layout
 
 \begin_layout LyX-Code
-<list name="plugins">
+StgFEM_FrequentOutput
 \end_layout
 
 \begin_layout LyX-Code
-  <param>Underworld_EulerDeform</param>
+StgFEM_CPUTime
 \end_layout
 
 \begin_layout LyX-Code
-  <param>Underworld_DumpSwarm</param>
+Underworld_MovingMesh
 \end_layout
 
 \begin_layout LyX-Code
-  <param>Underworld_VTKOutput</param>
+Underworld_Vrms
 \end_layout
 
 \begin_layout LyX-Code
-</list> 
+Underworld_EulerDeform
 \end_layout
 
-\begin_layout Standard
-Dynamically loading modules is, in general, difficult to get working on
- multiple platforms.
- To make things easier, Gale is bundled with a number of plugins compiled
- into the code and ready to use:
-\end_layout
-
 \begin_layout LyX-Code
-StgFEM_FrequentOutput
+StgFEM_PrintFeVariableDiscreteValues
 \end_layout
 
 \begin_layout LyX-Code
-StgFEM_CPUTime
+Underworld_VTKOutput
 \end_layout
 
-\begin_layout LyX-Code
-Underworld_MovingMesh
+\begin_layout Standard
+To use these plugins, list them in a 
+\family typewriter
+plugins
+\family default
+ section outside of the 
+\family typewriter
+components
+\family default
+ structure.
+ For example, the following lines will enable the 
+\family typewriter
+EulerDeform
+\family default
+ and 
+\family typewriter
+VTKOutput
+\family default
+ plugins
 \end_layout
 
 \begin_layout LyX-Code
-Underworld_Vrms
+<list name="plugins">
 \end_layout
 
 \begin_layout LyX-Code
-Underworld_EulerDeform
+  <param>Underworld_EulerDeform</param>
 \end_layout
 
 \begin_layout LyX-Code
-StgFEM_PrintFeVariableDiscreteValues
+  <param>Underworld_VTKOutput</param>
 \end_layout
 
 \begin_layout LyX-Code
-Underworld_VTKOutput
+</list> 
 \end_layout
 
 \begin_layout Standard
-Note that you must still include those plugins in the list of plugins in
- order to use them.
- You can add additional plugins by modifying the list of static plugins
- in 
+You can add additional plugins by modifying the list of static plugins in
+ 
 \family typewriter
 src/Gale/src/main.c
 \family default
  and making sure that the plugin is compiled into the Gale executable.
 \end_layout
 
-\begin_layout Standard
-If you want to use dynamically loaded plugins, make sure that you compiled
- with shared libraries (use the 
-\family typewriter
---with-shared
-\family default
- option when compiling) and set your 
-\family typewriter
-LD_LIBRARY_PATH
-\family default
- environment variable appropriately.
-\end_layout
-
 \begin_layout Subsection
 EulerDeform
 \begin_inset LatexCommand label
@@ -7555,7 +7667,7 @@
 \family typewriter
 Underworld_EulerDeform
 \family default
- component allows the upper surface to move freely or stay rigidly in place.
+ plugin allows the upper surface to move freely or stay rigidly in place.
  An example 
 \family typewriter
 EulerDeform
@@ -7648,7 +7760,7 @@
  the new mesh.
  This interpolation can sometimes go awry.
  Often, this is because something else is going wrong.
- For example, if your velocity solution is bad and gets large velocities,
+ For example, if your velocity solution is bad and you get large velocities,
  then the mesh can turn itself inside out.
  This, in turn, will cause 
 \family typewriter
@@ -7737,6 +7849,23 @@
 \end_layout
 
 \begin_layout Standard
+If you set 
+\family typewriter
+staticRight
+\family default
+ or 
+\family typewriter
+staticLeft
+\family default
+ but do not fix the upper corners, then Gale will move the top right or
+ left corner to the boundary and interpolate the height.
+ This is useful if material is flowing out and you want the boundary of
+ the mesh to vary as lumps go through.
+ If material is actually flowing in, Gale will be unable to interpolate
+ and will complain.
+\end_layout
+
+\begin_layout Standard
 The 
 \family typewriter
 floatRightTop
@@ -7745,20 +7874,14 @@
 \family typewriter
 floatLeftTop
 \family default
- variables are useful when material flows out of the right or left boundary,
- and you want the height of the boundary to follow the material.
- These two variables tell Gale to move the top right or left corner (2D)
- or edge (3D) back to 
-\family typewriter
-xRightCoord
-\family default
- or 
-\family typewriter
-xLeftCoord
-\family default
-, but to interpolate the height.
- If material is actually flowing in through the top, Gale will be unable
- to interpolate and will complain.
+ variables are useful when you are using a boundary layer (see Sections
+ 
+\begin_inset LatexCommand ref
+reference "sub:DruckerPrager"
+
+\end_inset
+
+), and you want the height of the boundary layer to match the interior.
 \end_layout
 
 \begin_layout Standard
@@ -9101,46 +9224,6 @@
 \end_layout
 
 \begin_layout LyX-Code
-<struct name="StrainRateXXField">
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="Type">OperatorFeVariable</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="Operator">TakeFirstComponent</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="FeVariable">StrainRateField</param>
-\end_layout
-
-\begin_layout LyX-Code
-</struct>
-\end_layout
-
-\begin_layout LyX-Code
-<struct name="StrainRateYYField">
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="Type">OperatorFeVariable</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="Operator">TakeSecondComponent</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="FeVariable">StrainRateField</param>
-\end_layout
-
-\begin_layout LyX-Code
-</struct>
-\end_layout
-
-\begin_layout LyX-Code
 <struct name="pressure">
 \end_layout
 
@@ -9957,6 +10040,62 @@
 \end_layout
 
 \begin_layout LyX-Code
+<struct name="c_matrix">
+\end_layout
+
+\begin_layout LyX-Code
+  <param name="Type">StiffnessMatrix</param>
+\end_layout
+
+\begin_layout LyX-Code
+  <param name="RowVariable">PressureField</param>
+\end_layout
+
+\begin_layout LyX-Code
+  <param name="ColumnVariable">PressureField</param>
+\end_layout
+
+\begin_layout LyX-Code
+  <param name="RHS">cont_force</param>
+\end_layout
+
+\begin_layout LyX-Code
+  <param name="allowZeroElementContributions">True</param>
+\end_layout
+
+\begin_layout LyX-Code
+</struct>
+\end_layout
+
+\begin_layout LyX-Code
+<struct name="mixedStabiliser">
+\end_layout
+
+\begin_layout LyX-Code
+  <param name="Type">MixedStabiliserTerm</param>
+\end_layout
+
+\begin_layout LyX-Code
+  <param name="Swarm">gaussSwarm</param>
+\end_layout
+
+\begin_layout LyX-Code
+  <param name="picSwarm">picIntegrationPoints</param>
+\end_layout
+
+\begin_layout LyX-Code
+  <param name="storeVisc">storeViscosity</param>
+\end_layout
+
+\begin_layout LyX-Code
+  <param name="StiffnessMatrix">c_matrix</param>
+\end_layout
+
+\begin_layout LyX-Code
+</struct>
+\end_layout
+
+\begin_layout LyX-Code
 <struct name="background">
 \end_layout
 
@@ -12294,95 +12433,6 @@
 \end_layout
 
 \begin_layout Standard
-To output the effective isotropic viscosity and stress, you have to add
- 
-\family typewriter
-StoreVisc
-\family default
- and 
-\family typewriter
-StoreStress
-\family default
- components to the material
-\end_layout
-
-\begin_layout LyX-Code
-<struct name="storeViscosity">
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="Type">StoreVisc</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="MaterialPointsSwarm">materialSwarm</param>
-\end_layout
-
-\begin_layout LyX-Code
-</struct>
-\end_layout
-
-\begin_layout LyX-Code
-<struct name="storeStress">
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="Type">StoreStress</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="MaterialPointsSwarm">materialSwarm</param>
-\end_layout
-
-\begin_layout LyX-Code
-</struct>
-\end_layout
-
-\begin_layout Standard
-and then add them to the material
-\end_layout
-
-\begin_layout LyX-Code
-<struct name="yieldingMaterial">
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="Type">RheologyMaterial</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="Shape">yieldingShape</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <list name="Rheology">
-\end_layout
-
-\begin_layout LyX-Code
-    <param>storeViscosity</param>
-\end_layout
-
-\begin_layout LyX-Code
-    <param>storeStress</param>
-\end_layout
-
-\begin_layout LyX-Code
-    <param>viscousRheology</param>
-\end_layout
-
-\begin_layout LyX-Code
-    <param>yieldingRheology</param>
-\end_layout
-
-\begin_layout LyX-Code
-  </list>
-\end_layout
-
-\begin_layout LyX-Code
-</struct>
-\end_layout
-
-\begin_layout Standard
 For each material, you can specify a density, a coefficient of thermal expansivi
 ty (
 \begin_inset Formula $\alpha$
@@ -13041,11 +13091,11 @@
 
  is the softening strain.
  This allows us to define quantities like the effective cohesion 
-\begin_inset Formula $C_{effective}\equiv C_{pristine}\left(1-\alpha\right)+C_{yielded}\alpha$
+\begin_inset Formula $C'\equiv C_{pristine}\left(1-\alpha\right)+C_{yielded}\alpha$
 \end_inset
 
  and effective friction coefficient 
-\begin_inset Formula $\phi_{effective}=\phi_{pristine}\left(1-\alpha\right)+\phi_{yielded}\alpha$
+\begin_inset Formula $\tan\phi'=\tan\phi_{pristine}\left(1-\alpha\right)+\tan\phi_{yielded}\alpha$
 \end_inset
 
 .
@@ -13302,10 +13352,24 @@
 
 \begin_layout Standard
 This is the simplest yielding rheology in Gale.
- The yielding stress is simply the effective cohesion, so this rheology
- only has a few input parameters:
+ The yielding stress is simply the effective cohesion.
+ Specifically, the yielding condition specifies
 \end_layout
 
+\begin_layout Standard
+\begin_inset Formula \[
+\sqrt{J_{2}}=C'\]
+
+\end_inset
+
+where 
+\begin_inset Formula $J_{2}$
+\end_inset
+
+ is the second invariant of the deviatoric stress tensor.
+ This rheology only has a few input parameters:
+\end_layout
+
 \begin_layout Itemize
 
 \family typewriter
@@ -13344,22 +13408,19 @@
 \family default
 , then the viscosity is set to 
 \begin_inset Formula \[
-\eta_{effective}=2\sigma_{yield}^{2}\eta/\left(\sigma_{yield}^{2}+I_{2}^{2}\right),\]
+\eta_{new}=2C'^{2}\eta/\left(C'^{2}+J_{2}\right).\]
 
 \end_inset
 
- where 
-\begin_inset Formula $I_{2}$
-\end_inset
-
- is the second invariant of the stress tensor.
+ This is a way of creeping up on the correct viscosity to avoid setting
+ the viscosity too low.
  Otherwise the viscosity is set to
 \end_layout
 
 \begin_deeper
 \begin_layout Standard
 \begin_inset Formula \[
-\eta_{effective}=\eta\sigma_{yield}/I_{2},\]
+\eta_{new}=\eta C'/\sqrt{J_{2}},\]
 
 \end_inset
 
@@ -13490,35 +13551,142 @@
 \end_layout
 
 \begin_layout Subsubsection
+\begin_inset LatexCommand label
+name "sub:DruckerPrager"
+
+\end_inset
+
 DruckerPrager
 \end_layout
 
 \begin_layout Standard
 This rheology uses the same parameters as Von Mises, but also adds a friction
  coefficient that can soften.
- Specifically, the yield stress becomes
+ Specifically, the yield condition is
 \begin_inset Formula \[
-\sigma_{yield}=\alpha p+C_{p},\]
+\sqrt{J_{2}}=Ap+B,\]
 
 \end_inset
 
 where 
+\begin_inset Formula $p$
+\end_inset
+
+ is the pressure.
+ The value of the constants 
+\begin_inset Formula $A$
+\end_inset
+
+ and 
+\begin_inset Formula $B$
+\end_inset
+
+ are different from 2D and 3D.
+ In 2D, Drucker-Prager and Mohr-Coulomb are identical.
+ Specifically,
+\end_layout
+
+\begin_layout Standard
 \begin_inset Formula \[
 \begin{array}{ccc}
-C_{p} & = & \frac{6C_{effective}\cos\phi_{effective}}{\sqrt{3}\left(3-\sin\phi_{effective}\right)}\\
-\alpha & = & \frac{6\sin\phi_{effective}}{\sqrt{3}\left(3-\sin\phi_{effective}\right)}\end{array},\]
+A & = & \sin\phi'\\
+B & = & C'\cos\phi'\end{array}.\]
 
 \end_inset
 
-and 
-\begin_inset Formula $p$
+ In 3D, the mapping between friction angles and cohesion to 
+\begin_inset Formula $A$
 \end_inset
 
- is the pressure.
+ and 
+\begin_inset Formula $B$
+\end_inset
+
+ is more complicated 
+\begin_inset Formula \[
+\begin{array}{ccc}
+A & = & \frac{2\sin\phi'}{\sqrt{3}\left(3-\sin\phi'\right)}\\
+B & = & \frac{6C'\cos\phi'}{\sqrt{3}\left(3-\sin\phi'\right)}\end{array}.\]
+
+\end_inset
+
+ You can also write a Mohr-Coulomb rheology in this form, but then the constants
  
+\begin_inset Formula $A$
+\end_inset
+
+ and 
+\begin_inset Formula $B$
+\end_inset
+
+ depend on 
+\begin_inset Formula $J_{2}$
+\end_inset
+
+.
+ So reducing the viscosity does not result in a linear decrease in 
+\begin_inset Formula $J_{2}$
+\end_inset
+
+.
+ This makes it difficult for the code to find a solution.
+ In practice, the yield surface for Drucker-Prager and Mohr-Coulomb are
+ not too dissimilar.
+ Mohr-Coulomb's yield surface is a six-sided cone, while Drucker-Prager's
+ yield surface is the smooth cone inscribing the Mohr-Coulomb segmented
+ cone.
 \end_layout
 
 \begin_layout Standard
+Note that 
+\family typewriter
+minimumYieldStress
+\family default
+ is interpreted differently.
+ If it is zero (the default), then the actual minimum yield stress will
+ be the effective cohesion.
+ This is because there tends to be numerical problems when using a very
+ small minimum yield stress under tension.
+\end_layout
+
+\begin_layout Standard
+Also, the Drucker-Prager implementation allows you to specify that material
+ near the boundary will have different yielding properties.
+ This is useful for simulating frictional boundaries.
+ For example, if 
+\family typewriter
+boundaryLeft
+\family default
+ is 
+\family typewriter
+True
+\family default
+, then in the element on the left boundary, Gale will use 
+\family typewriter
+boundaryCohesion
+\family default
+ instead of 
+\family typewriter
+cohesion
+\family default
+, 
+\family typewriter
+boundaryFrictionCoefficient
+\family default
+ instead of 
+\family typewriter
+frictionCoefficient
+\family default
+, etc.
+\end_layout
+
+\begin_layout Standard
+Finally, DruckerPrager requires a pressure.
+ If you are using HydrostaticTerm (see Section ), you have to remember to
+ give it that as well.
+\end_layout
+
+\begin_layout Standard
 \begin_inset VSpace defskip
 \end_inset
 
@@ -13529,7 +13697,7 @@
 \noindent
 \align center
 \begin_inset Tabular
-<lyxtabular version="3" rows="3" columns="2">
+<lyxtabular version="3" rows="16" columns="2">
 <features>
 <column alignment="center" valignment="top" leftline="true" width="0">
 <column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
@@ -13553,6 +13721,46 @@
 \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
+PressureField
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+none
+\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
+HydrostaticTerm
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+none
+\end_layout
+
+\end_inset
+</cell>
+</row>
 <row topline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
@@ -13593,183 +13801,221 @@
 \end_inset
 </cell>
 </row>
-</lyxtabular>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
 
+\begin_layout Standard
+minimumYieldStress
+\end_layout
+
 \end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
 
+\begin_layout Standard
+0 (see above)
+\end_layout
 
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+boundaryCohesion
 \end_layout
 
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
 \begin_layout Standard
-See also Section 
-\begin_inset LatexCommand ref
-reference "sub:VonMises"
+0
+\end_layout
 
 \end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
 
-.
+\begin_layout Standard
+boundaryCohesionAfterSoftening
 \end_layout
 
-\begin_layout Subsubsection
-MohrCoulomb
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+0
 \end_layout
 
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
 \begin_layout Standard
-This rheology is like Drucker-Prager, but it uses the actual normal stress
- instead of inferring a normal stress from the pressure.
- More explicitly, the yield shear stress is 
-\begin_inset Formula \[
-\sigma_{yield}=\sigma_{n}\tan\phi_{effective}+C_{effective}.\]
+boundaryFrictionCoefficient
+\end_layout
 
 \end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
 
- When the material is yielding, the viscosity is modified so that the stress
- on the particle is the yield stress.
- Note that 
-\family typewriter
-minimumYieldStress
-\family default
- is interpreted differently.
- If it is zero (the default), then actual minimum yield stress will be the
- effective cohesion.
- This is because there tends to be numerical problems when using a very
- small minimum yield stress under tension.
+\begin_layout Standard
+0
 \end_layout
 
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
 \begin_layout Standard
-One of the parameters that this rheology takes is a pressure field.
+boundaryFrictionCoefficientAfterSoftening
 \end_layout
 
-\begin_layout Standard
-\begin_inset VSpace defskip
 \end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
 
-
+\begin_layout Standard
+0
 \end_layout
 
-\begin_layout Standard
-\noindent
-\align center
-\begin_inset Tabular
-<lyxtabular version="3" rows="6" 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">
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
 \begin_layout Standard
-Defaults
+boundaryLeft
 \end_layout
 
 \end_inset
 </cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
 \begin_layout Standard
-
+False
 \end_layout
 
 \end_inset
 </cell>
 </row>
-<row topline="true">
+<row bottomline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
 \begin_layout Standard
-cohesion
+boundaryRight
 \end_layout
 
 \end_inset
 </cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
 \begin_layout Standard
-0
+False
 \end_layout
 
 \end_inset
 </cell>
 </row>
-<row topline="true">
+<row bottomline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
 \begin_layout Standard
-cohesionAfterSoftening
+boundaryTop
 \end_layout
 
 \end_inset
 </cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
 \begin_layout Standard
-0
+False
 \end_layout
 
 \end_inset
 </cell>
 </row>
-<row topline="true">
+<row bottomline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
 \begin_layout Standard
-frictionCoefficient
+boundaryBottom
 \end_layout
 
 \end_inset
 </cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
 \begin_layout Standard
-0
+False
 \end_layout
 
 \end_inset
 </cell>
 </row>
-<row topline="true">
+<row bottomline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
 \begin_layout Standard
-frictionCoefficientAfterSoftening
+boundaryFront
 \end_layout
 
 \end_inset
 </cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
 \begin_layout Standard
-0
+False
 \end_layout
 
 \end_inset
 </cell>
 </row>
-<row topline="true" bottomline="true">
+<row bottomline="true">
 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
 \begin_layout Standard
-minimumYieldStress
+boundaryBack
 \end_layout
 
 \end_inset
 </cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 \begin_inset Text
 
 \begin_layout Standard
-0 (see above)
+False
 \end_layout
 
 \end_inset
@@ -13782,6 +14028,16 @@
 
 \end_layout
 
+\begin_layout Standard
+See also Section 
+\begin_inset LatexCommand ref
+reference "sub:VonMises"
+
+\end_inset
+
+.
+\end_layout
+
 \begin_layout Subsubsection
 FaultingMoresiMulhaus2006
 \end_layout
@@ -13852,7 +14108,7 @@
 \begin_layout Standard
 Otherwise, it adds one variable not present in 
 \family typewriter
-MohrCoulomb
+DruckerPrager
 \family default
 : 
 \family typewriter
@@ -14791,161 +15047,6 @@
 \end_layout
 
 \begin_layout Subsection
-Static Friction Boundary Conditions
-\end_layout
-
-\begin_layout Standard
-Static friction looks like a selectively applied velocity condition.
- Gale computes the normal and shear stresses on the boundary from the previous
- time step and uses that to determine whether a particular node sticks or
- slips.
- To be exact, if
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\sigma_{ns}<StaticFriction\cdot\sigma_{nn},\]
-
-\end_inset
-
- where 
-\begin_inset Formula $\sigma_{ns}$
-\end_inset
-
- is the shear stress and 
-\begin_inset Formula $\sigma_{nn}$
-\end_inset
-
- is the normal stress, then that node is locked for that time step.
- Otherwise, the node is allowed to move freely, although a kinetic friction
- boundary condition may induce stresses.
-\end_layout
-
-\begin_layout Standard
-The 
-\family typewriter
-StaticFriction
-\family default
- structure is used just like a normal velocity condition, except that there
- are some extra parameters.
- For example, to apply static friction to a non-moving bottom, you could
- add the structure
-\end_layout
-
-\begin_layout LyX-Code
-<struct>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="type">StaticFrictionVC</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="wall">bottom</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="StaticFriction">0.34432761329</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="includeUpperX">False</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="StickOnFirstStep">True</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="StressField">StressField</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <param name="PressureField">PressureField</param>
-\end_layout
-
-\begin_layout LyX-Code
-  <list name="variables">
-\end_layout
-
-\begin_layout LyX-Code
-    <struct>
-\end_layout
-
-\begin_layout LyX-Code
-      <param name="name">vx</param>
-\end_layout
-
-\begin_layout LyX-Code
-      <param name="type">double</param>
-\end_layout
-
-\begin_layout LyX-Code
-      <param name="value">0.0</param>
-\end_layout
-
-\begin_layout LyX-Code
-    </struct>
-\end_layout
-
-\begin_layout LyX-Code
-  </list>
-\end_layout
-
-\begin_layout LyX-Code
-</struct>
-\end_layout
-
-\begin_layout Standard
-The most important parameter is 
-\family typewriter
-StaticFriction
-\family default
-, which tells Gale the strength of the friction.
- The parameter 
-\family typewriter
-includeUpperX
-\family default
- tells Gale whether it should apply static friction on the right corner.
- If you have incompatible boundary conditions for the wall (e.g., if the wall
- is moving inwards), then you need to tell Gale not to apply static friction
- there.
- You may also set the parameters 
-\family typewriter
-includeLowerX
-\family default
-, 
-\family typewriter
-includeUpperY
-\family default
-, 
-\family typewriter
-includeLowerY
-\family default
-, 
-\family typewriter
-includeUpperZ
-\family default
-, and 
-\family typewriter
-includeLowerZ
-\family default
-.
-\end_layout
-
-\begin_layout Standard
-Finally, for the first step, Gale does not have any information from a previous
- step as to whether to stick or slip.
- You can tell it to have the entire boundary either stick or slip on the
- first step by setting the parameter 
-\family typewriter
-StickOnFirstStep
-\family default
-.
- The default is to have the boundary stick.
-\end_layout
-
-\begin_layout Subsection
 Temperature Boundary Conditions
 \begin_inset LatexCommand label
 name "sub:Temperature-Boundary-Conditions"
@@ -15264,6 +15365,14 @@
 \end_layout
 
 \begin_layout Section
+HydrostaticTerm
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Section
 Buoyancy Forces
 \begin_inset LatexCommand label
 name "sec:Buoyancy-Forces"
@@ -15948,7 +16057,13 @@
 
 , and all names with CamelCase capitalization are variables from the input
  file.
- If you need to write your own function, see Section [TODO - which section?].
+ If you need to write your own function, see Section 
+\begin_inset LatexCommand ref
+reference "sec:Standard-Condition-Functions-modify"
+
+\end_inset
+
+.
 \end_layout
 
 \begin_layout Description
@@ -16810,23 +16925,7 @@
 \end_inset
 
 .
- Finally, there will be 
-\family typewriter
-.txt
-\family default
- files from 
-\family typewriter
-DumpSwarm
-\family default
-, explained below in Section 
-\begin_inset LatexCommand ref
-reference "sec:.txt-Files-(Particle"
-
-\end_inset
-
-.
- [TODO -- this reference does not point to the right place.
- what section did you want?]
+ 
 \end_layout
 
 \begin_layout Section
@@ -17292,7 +17391,7 @@
 \begin_inset Text
 
 \begin_layout Standard
-Mohr Coulomb rheology
+Drucker Prager rheology in 2D
 \end_layout
 
 \end_inset
@@ -18494,6 +18593,218 @@
 
 \begin_layout Section
 \begin_inset LatexCommand label
+name "sec:Divergence"
+
+\end_inset
+
+Divergence
+\end_layout
+
+\begin_layout Standard
+This benchmarks tests the implementation of the divergence term in equation
+ 
+\begin_inset LatexCommand ref
+reference "eq:divergence"
+
+\end_inset
+
+.
+ In 2D, a constant divergence is applied to a square domain, and the velocity
+ on the corners is set to enforce a spreading from the center of the square.
+ Figure 
+\begin_inset LatexCommand ref
+reference "fig:Divergence_v_sri"
+
+\end_inset
+
+ shows the velocity and strain rate invariant for a numerical solution.
+ For a constant divergence 
+\begin_inset Formula $d$
+\end_inset
+
+, the analytic solution for this setup is
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\begin{array}{ccc}
+v_{x} & = & x\cdot d/2\\
+v_{y} & = & y\cdot d/2\end{array}.\]
+
+\end_inset
+
+ In 3D, the analytic solution is
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\begin{array}{ccc}
+v_{x} & = & x\cdot d/3\\
+v_{y} & = & y\cdot d/3\\
+v_{z} & = & z\cdot d/3\end{array}.\]
+
+\end_inset
+
+ In both cases, the strain rate invariant equals 
+\begin_inset Formula $\sqrt{d/2}$
+\end_inset
+
+.
+ As shown in Figure 
+\begin_inset LatexCommand ref
+reference "fig:Divergence_2D_error"
+
+\end_inset
+
+, the main source of error in 2D comes from inaccuracies in the solver.
+ Figure 
+\begin_inset LatexCommand ref
+reference "fig:Divergence_3D_error"
+
+\end_inset
+
+ paints a different picture in 3D, where the main source of error comes
+ from having a finite number of particles.
+ 
+\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/divergence_v.png
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Divergence_v_sri"
+
+\end_inset
+
+Velocity and Strain Rate Invariant solution for the 2D Divergence benchmark.
+ The variation in the strain rate invariant is uniformly small.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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/divergence_2D_error.eps
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Divergence_2D_error"
+
+\end_inset
+
+Maximum error in the strain rate invariant for the 2D Divergence benchmark
+ vs tolerance in the linear solver.
+ The resolution is kept at 32
+\begin_inset Formula $\times$
+\end_inset
+
+32, and the number of particles per cell is kept at 30.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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/divergence_3D_error.eps
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Divergence_3D_error"
+
+\end_inset
+
+Maximum error in the strain rate invariant for the 3D Divergence benchmark
+ vs the number of particles in each cell.
+ The resolution is kept at 16
+\begin_inset Formula $\times$
+\end_inset
+
+16
+\begin_inset Formula $\times$
+\end_inset
+
+16, and the tolerance in the linear solver is kept at 
+\begin_inset Formula $10^{-7}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+\begin_inset LatexCommand label
 name "sec:Shear"
 
 \end_inset
@@ -18586,16 +18897,6 @@
 \end_inset
 
  shows the error scaled with resolution.
- In this case, the pressure on the boundary completely determines the forces
- normal to the boundary.
- Because Gale uses constant pressure elements, the pressure on the boundary
- has an error of order 
-\begin_inset Formula $h$
-\end_inset
-
-.
- This error in the pressure dominates the error in the friction boundary
- condition.
  The observed error decreases linearly with resolution, giving us confidence
  that the implementation of static friction is correct.
 \end_layout
@@ -18829,7 +19130,7 @@
 \end_layout
 
 \begin_layout Section
-Mohr-Coulomb
+Drucker-Prager
 \end_layout
 
 \begin_layout Subsection
@@ -18837,7 +19138,8 @@
 \end_layout
 
 \begin_layout Standard
-For the Mohr-Coulomb rheology, we can write the yielding relation as 
+For the Drucker-Prager rheology in 2D, we can write the yielding relation
+ as 
 \end_layout
 
 \begin_layout Standard
@@ -19002,8 +19304,8 @@
 
  shows a plot of the numerical vs.
  analytic results for several different angles for medium resolution.
- This gives us confidence that, at least in compression, our Mohr-Coulomb
- implementation gives the correct results.
+ This gives us confidence that, at least in compression in 2D, our Drucker-Prage
+r implementation gives the correct results.
 \end_layout
 
 \begin_layout Standard
@@ -19063,10 +19365,6 @@
 
 \end_layout
 
-\begin_layout Standard
-
-\end_layout
-
 \end_inset
 
 
@@ -19076,12 +19374,12 @@
 \begin_inset Float figure
 wide false
 sideways false
-status collapsed
+status open
 
 \begin_layout Standard
 \align center
 \begin_inset Graphics
-	filename images/mohr_coulomb_angles.pdf
+	filename images/mohr_coulomb_angles.eps
 
 \end_inset
 
@@ -19099,7 +19397,6 @@
 
 Numerical vs analytic results for fault angles as a function of internal
  angle of friction.
- [TODO - this image file is still missing in svn]
 \end_layout
 
 \end_inset
@@ -19576,7 +19873,19 @@
 \end_inset
 
 Strain rate invariant for the shortening model after 14 cm of shortening
- for three different resolutions: (a) 128x32, (b) 256x64, and (c) 512x128.
+ for three different resolutions: (a) 128
+\begin_inset Formula $\times$
+\end_inset
+
+32, (b) 256
+\begin_inset Formula $\times$
+\end_inset
+
+64, and (c) 512
+\begin_inset Formula $\times$
+\end_inset
+
+128.
 \end_layout
 
 \end_inset
@@ -19589,6 +19898,1022 @@
 
 \end_layout
 
+\begin_layout Section
+Geomod 2008
+\end_layout
+
+\begin_layout Standard
+Using the lessons learned from the Geomod 2004 benchmarks, new benchmarks
+ were created that would make it easier to compare numerical experiments
+ with each other and with analogue experiments 
+\begin_inset LatexCommand cite
+key "Geomod 2008"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Stable Wedge
+\end_layout
+
+\begin_layout Standard
+This benchmark simulates a wall pushing a wedge as in Figure 
+\begin_inset LatexCommand ref
+reference "fig:Wedge_setup"
+
+\end_inset
+
+.
+ There is an analytic solution 
+\color none
+
+\begin_inset LatexCommand cite
+key "Dahlen Wedge"
+
+\end_inset
+
+ which details what 
+\color inherit
+the friction on the bottom and sides should be to provide enough resistance
+ so that the wedge does not collapse under its own weight, but not so much
+ as to cause any internal deformation as it slides.
+ The derivation of the solution assumes that the friction along the sides
+ has no cohesion.
+ So the force at the tip will go to zero as the thickness of the material
+ goes to zero.
+ However, analogue experiments suggest a finite cohesion, so this benchmark
+ specifies a boundary cohesion.
+\end_layout
+
+\begin_layout Standard
+We modeled the wedge using a relatively low viscosity (
+\begin_inset Formula $1Pa\cdot s$
+\end_inset
+
+) air layer on top.
+ This low viscosity region does not, for the most part, affect the dynamics.
+\end_layout
+
+\begin_layout Standard
+We modeled boundary friction by first fixing the sand to the boundary.
+ We then modify the material properties in the element next to the boundary
+ so that it provides the correct resistance.
+ So in the bulk, the sand's internal angle of friction is 
+\begin_inset Formula $36\textdegree$
+\end_inset
+
+ weakening to 
+\begin_inset Formula $31\textdegree$
+\end_inset
+
+, while in the element at the boundary the internal angle of friction is
+ 
+\begin_inset Formula $16\textdegree$
+\end_inset
+
+ weakening to 
+\begin_inset Formula $14\textdegree$
+\end_inset
+
+.
+ Similarly, in the bulk, the cohesion is 
+\begin_inset Formula $10\, Pa$
+\end_inset
+
+, while at the boundary the cohesion is 
+\begin_inset Formula $10\, Pa$
+\end_inset
+
+ weakening to 
+\begin_inset Formula $0.01\, Pa$
+\end_inset
+
+.
+ If we do not weaken the cohesion, then when we try to model an unstable
+ wedge by reducing the internal angle of friction, then the wedge will never
+ collapse on itself.
+\end_layout
+
+\begin_layout Standard
+Figure 
+\begin_inset LatexCommand ref
+reference "fig:Stable_sri"
+
+\end_inset
+
+ shows the strain rate invariant after the wall has moved 4 cm, and Figure
+ 
+\begin_inset LatexCommand ref
+reference "fig:Stable_particles"
+
+\end_inset
+
+ shows the particles.
+ The bulk translates with almost no deformation, although, as expected,
+ the tip deforms.
+ The odd structures at the tip are below the grid resolution.
+ Figure 
+\begin_inset LatexCommand ref
+reference "fig:Stable_unstable"
+
+\end_inset
+
+ shows a simulation when we reduce the boundary friction to 
+\begin_inset Formula $1\textdegree$
+\end_inset
+
+.
+ The wedge quickly becomes unstable and collapses.
+\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/Geomod2008_wedge_setup.eps
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Wedge_setup"
+
+\end_inset
+
+Set up for the stable wedge benchmark.
+ Image courtesy of Susanne Buiter.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Standard
+\align center
+\begin_inset Graphics
+	filename images/Stable_wedge_sri.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Stable_sri"
+
+\end_inset
+
+Strain rate invariant for the stable wedge benchmark within the wedge.
+ Outside the wedge, the strain rates are large because of the air's low
+ viscosity.
+ The resolution is 512x128, and the wedge has translated 4 cm.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\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/Stable_wedge_particles.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Stable_particles"
+
+\end_inset
+
+Material particles for the stable wedge benchmark.
+ The deformation at the tip is caused by a finite boundary cohesion, although
+ the actual structure is not resolved.
+ The resolution is 512x128, and the wedge has translated 4 cm.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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/Stable_wedge_unstable.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Stable_unstable"
+
+\end_inset
+
+Strain rate invariant and velocity arrows for the stable wedge benchmark,
+ but with the boundary friction angle reduced to 
+\begin_inset Formula $1\textdegree$
+\end_inset
+
+.
+ Note that the strain rates are much higher than in Figure 
+\begin_inset LatexCommand ref
+reference "fig:Stable_sri"
+
+\end_inset
+
+.
+ The wedge collapses almost immediately.
+ The resolution is 512x128, and the wedge has translated 0.17 cm.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Unstable Shortening
+\end_layout
+
+\begin_layout Standard
+This benchmark simulates a wall pushing against a wall of sand as in Figure
+ 
+\begin_inset LatexCommand ref
+reference "fig:Unstable-setup"
+
+\end_inset
+
+.
+ There are three layers of sand, with the middle layer being a little heavier
+ and sticking a little more to the boundary.
+ Otherwise it is identical.
+ Figures 
+\begin_inset LatexCommand ref
+reference "fig:unstable_sri_128"
+
+\end_inset
+
+, 
+\begin_inset LatexCommand ref
+reference "fig:unstable_sri_256"
+
+\end_inset
+
+, 
+\begin_inset LatexCommand ref
+reference "fig:unstable_sri_512"
+
+\end_inset
+
+, 
+\begin_inset LatexCommand ref
+reference "fig:unstable_particles_128"
+
+\end_inset
+
+, 
+\begin_inset LatexCommand ref
+reference "fig:unstable_particles_256"
+
+\end_inset
+
+, and 
+\begin_inset LatexCommand ref
+reference "fig:unstable_particles_512"
+
+\end_inset
+
+ show results at different times and different resolutions.
+\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/Geomod2008_unstable_setup.eps
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Unstable-setup"
+
+\end_inset
+
+Set up for the unstable shortening benchmark.
+ Image courtesy of Susanne Buiter.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Standard
+\align center
+\begin_inset Graphics
+	filename images/Geomod2008_unstable_sri128x32.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:unstable_sri_128"
+
+\end_inset
+
+Strain rate invariant for the unstable shortening benchmark with a resolution
+ of 128x32.
+ The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\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/Geomod2008_unstable_sri256x64.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:unstable_sri_256"
+
+\end_inset
+
+Strain rate invariant for the unstable shortening benchmark with a resolution
+ of 256x64.
+ The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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/Geomod2008_unstable_sri512x128.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:unstable_sri_512"
+
+\end_inset
+
+Strain rate invariant for the unstable shortening benchmark with a resolution
+ of 512x128.
+ The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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/Geomod2008_unstable_particles128x32.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:unstable_particles_128"
+
+\end_inset
+
+Material particles for the unstable shortening benchmark with a resolution
+ of 128x32.
+ The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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/Geomod2008_unstable_particles256x64.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:unstable_particles_256"
+
+\end_inset
+
+Material particles for the unstable shortening benchmark with a resolution
+ of 256x64.
+ The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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/Geomod2008_unstable_particles512x128.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:unstable_particles_512"
+
+\end_inset
+
+Material particles for the unstable shortening benchmark with a resolution
+ of 512x128.
+ The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Brittle Shortening
+\end_layout
+
+\begin_layout Standard
+This benchmark is very similar to unstable shortening, but the bottom right
+ part translates along as well as shown in Figure 
+\begin_inset LatexCommand ref
+reference "fig:Brittle_setup"
+
+\end_inset
+
+.
+ This causes the deformation to start from inside the sand box, rather than
+ along the walls.
+ Figures 
+\begin_inset LatexCommand ref
+reference "fig:brittle_sri_128"
+
+\end_inset
+
+, 
+\begin_inset LatexCommand ref
+reference "fig:brittle_sri_256"
+
+\end_inset
+
+, 
+\begin_inset LatexCommand ref
+reference "fig:brittle_sri_512"
+
+\end_inset
+
+, 
+\begin_inset LatexCommand ref
+reference "fig:brittle_particles_128"
+
+\end_inset
+
+, 
+\begin_inset LatexCommand ref
+reference "fig:brittle_particles_256"
+
+\end_inset
+
+, and 
+\begin_inset LatexCommand ref
+reference "fig:brittle_particles_512"
+
+\end_inset
+
+ show results at different times and different resolutions.
+\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/Geomod2008_brittle_setup.eps
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:Brittle_setup"
+
+\end_inset
+
+Set up for the brittle shortening benchmark.
+ Image courtesy of Susanne Buiter.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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/Geomod2008_brittle_sri128x32.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:brittle_sri_128"
+
+\end_inset
+
+Strain rate invariant for the brittle shortening benchmark with a resolution
+ of 128x32.
+ The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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/Geomod2008_brittle_sri256x64.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:brittle_sri_256"
+
+\end_inset
+
+Strain rate invariant for the brittle shortening benchmark with a resolution
+ of 256x64.
+ The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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/Geomod2008_brittle_sri512x128.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:brittle_sri_512"
+
+\end_inset
+
+Strain rate invariant for the brittle shortening benchmark with a resolution
+ of 512x128.
+ The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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/Geomod2008_brittle_particles128x32.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:brittle_particles_128"
+
+\end_inset
+
+Material particles for the brittle shortening benchmark with a resolution
+ of 128x32.
+ The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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/Geomod2008_brittle_particles256x64.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:brittle_particles_256"
+
+\end_inset
+
+Material particles for the brittle shortening benchmark with a resolution
+ of 256x64.
+ The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\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/Geomod2008_brittle_particles512x128.png
+	lyxscale 50
+	width 100col%
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Caption
+
+\begin_layout Standard
+\begin_inset LatexCommand label
+name "fig:brittle_particles_512"
+
+\end_inset
+
+Material particles for the brittle shortening benchmark with a resolution
+ of 512x128.
+ The snapshots are taken at 0, 2.5, 5, 7.5, and 10 cm of shortening.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Chapter
 License 
 \end_layout
@@ -20466,12 +21791,78 @@
 
 \begin_layout Bibliography
 \begin_inset LatexCommand bibitem
-key "key-1"
+label "14"
+key "Stable Stokes"
 
 \end_inset
 
-TODO Dohrmann & Bochev reference?
+Dohrmann, C., and 
+\color none
+Bochev, P.
+ (2004), A stabilized finite element method for the Stokes problem based
+ on polynomial pressure projections, 
+\shape italic
+\color inherit
+Int.
+ J.
+ Num.
+ Meth.
+ Fluids.
+\shape default
+\color none
+, 46, 183-201
 \end_layout
 
+\begin_layout Bibliography
+
+\color none
+\begin_inset LatexCommand bibitem
+label "15"
+key "Fast Solvers"
+
+\end_inset
+
+Elman, H.
+ C., Silvester, D.
+ J., and Wathen, A.
+ J.
+ (2005), 
+\shape italic
+\color inherit
+Finite Elements and Fast Iterative Solvers: with Applications in Incompressible
+ Fluid Dynamics
+\shape default
+\color none
+, Oxford University Press
+\end_layout
+
+\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "16"
+key "Geomod 2008"
+
+\end_inset
+
+Buiter, S., and Schreurs, G., http://www.geodynamics.no/benchmarks/benchmark-annum20
+08.html
+\end_layout
+
+\begin_layout Bibliography
+\begin_inset LatexCommand bibitem
+label "17"
+key "Dahlen Wedge"
+
+\end_inset
+
+Dahlen, F.A.
+ (1984), Noncohesive Critical Wedges: An Exact Solution,
+\shape italic
+ J.
+ Geophys.
+ Res., 89
+\shape default
+, B12, 10125-10133
+\end_layout
+
 \end_body
 \end_document

Added: long/3D/Gale/trunk/documentation/images/Geomod2008_brittle_particles128x32.png
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===================================================================
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