docu

mtex-toolbox · Jul 3, 2023 · 970e94b · 970e94b
1 parent 6791809
commit 970e94b
Show file tree

Hide file tree

Showing 9 changed files with 240 additions and 30 deletions.
diff --git a/_data/sidebars/function_reference_sidebar.yml b/_data/sidebars/function_reference_sidebar.yml
@@ -5953,6 +5953,10 @@ entries:
           url: /SO3VectorField.display.html
           output: web
 
+        - title: "div"
+          url: /SO3VectorField.div.html
+          output: web
+
         - title: "dot"
           url: /SO3VectorField.dot.html
           output: web
@@ -6070,6 +6074,10 @@ entries:
           url: /SO3VectorFieldHarmonic.display.html
           output: web
 
+        - title: "div"
+          url: /SO3VectorFieldHarmonic.div.html
+          output: web
+
         - title: "dot"
           url: /SO3VectorFieldHarmonic.dot.html
           output: web

diff --git a/images/SingleSlipModel_01.png b/images/SingleSlipModel_01.png
diff --git a/images/SingleSlipModel_02.png b/images/SingleSlipModel_02.png
diff --git a/images/SingleSlipModel_07.png b/images/SingleSlipModel_07.png
diff --git a/images/SingleSlipModel_08.png b/images/SingleSlipModel_08.png
diff --git a/pages/documentation_matlab/SingleSlipModel.html b/pages/documentation_matlab/SingleSlipModel.html
@@ -12,35 +12,16 @@
 This HTML was auto-generated from MATLAB code.
 To make changes, update the MATLAB code and republish this document.
       --><title>Sigle Slip Model</title><link rel="schema.DC" href="http://purl.org/dc/elements/1.1/"><meta name="DC.source" content="script_SingleSlipModel.m"></head><body><font size="2"><a href="https://github.com/mtex-toolbox/mtex/blob/develop/doc/Plasticity/SingleSlipModel.m">
-    edit page</a></font><div><!--introduction--><p>Details to this model can be found in</p><div><ul><li><a href="https://doi.org/10.1093/gji/ggy442">An analytical finite-strain parametrization for texture evolution in deforming olivine polycrystals</a>, Geoph. J. Intern. 216, 2019.</li></ul></div><!--/introduction--><h2 id="1">The Continuity Equation</h2><p>The evolution of the orientation distribution function (ODF) \(f(g)\) with respect to a crystallopgraphic spin \(\Omega(g)\) is governed by the continuity equation</p><p>\[\frac{\partial}{\partial t} f + \nabla f \cdot \Omega + f \div \Omega = 0\]</p><p>The solution of this equation depends on the initial texture \(f_0(g)\) at time zero and the crystallographic spin \(\Omega(g)\). In this model we assume the initial texture to be isotrope, i.e., \(f_0 = 1\) and the crystallopgraphic spin be associated with a single slip system. The full ODF will be later modelled as a superposition of the single slip models.</p><p>In this example we consider Olivine with has orthorhombic symmetry</p>
+    edit page</a></font><div><!--introduction--><p>Details to this model can be found in</p><div><ul><li><a href="https://doi.org/10.1093/gji/ggy442">An analytical finite-strain parametrization for texture evolution in deforming olivine polycrystals</a>, Geoph. J. Intern. 216, 2019.</li></ul></div><!--/introduction--><h2 id="1">The Continuity Equation</h2><p>The evolution of the orientation distribution function (ODF) \(f(g)\) with respect to a crystallopgraphic spin \(\Omega(g)\) is governed by the continuity equation</p><p>\[\frac{\partial}{\partial t} f + \nabla f \cdot \Omega + f \text{div} \Omega = 0\]</p><p>The solution of this equation depends on the initial texture \(f_0(g)\) at time zero and the crystallographic spin \(\Omega(g)\). In this model we assume the initial texture to be isotrope, i.e., \(f_0 = 1\) and the crystallopgraphic spin be associated with a single slip system. The full ODF will be later modelled as a superposition of the single slip models.</p><p>In this example we consider Olivine with has orthorhombic symmetry</p>
 {% highlight matlab %}
-cs = crystalSymmetry('222',[4.779 10.277 5.995],'mineral','olivine')
-{% endhighlight %}
-
-{% highlight plaintext %}
-cs = crystalSymmetry
-
-  mineral : olivine   
-  symmetry: 222       
-  elements: 4         
-  a, b, c : 4.8, 10, 6
+cs = crystalSymmetry('222',[4.779 10.277 5.995],'mineral','olivine');
 {% endhighlight %}
 <p>and the three basic slip systems</p>
 {% highlight matlab %}
 sS = slipSystem(Miller({1,0,0},{1,0,0},{0,0,1},cs,'uvw'),...
-  Miller({0,1,0},{0,0,1},{0,1,0},cs,'hkl'))
-{% endhighlight %}
-
-{% highlight plaintext %}
-sS = slipSystem (olivine)
- size: 1 x 3
-
-  u    v    w  | h    k    l CRSS
-  1    0    0    0    1    0    1
-  1    0    0    0    0    1    1
-  0    0    1    0    1    0    1
+  Miller({0,1,0},{0,0,1},{0,1,0},cs,'hkl'));
 {% endhighlight %}
-<p>To each of the slip systems we can associate an orientation dependent Schmid or deformation tensor</p>
+<p>To each of the slip systems we can associate an orientation dependent Schmid or deformation tensor \(S(g)\)</p>
 {% highlight matlab %}
 S = sS.deformationTensor
 {% endhighlight %}
@@ -50,7 +31,7 @@
   size: 1 x 3    
   rank: 2 (3 x 3)
 {% endhighlight %}
-<p>and make the for the orientation dependent strain rate tensor \(e\) the ansatz \(e_{ij}(g) = \gamma(g) S_{ij}\). Fitting this ansatz to a given a macroscopic strain tensor</p>
+<p>and make for the orientation dependent strain rate tensor \(e(g)\) the ansatz \(e_{ij}(g) = \gamma(g) S_{ij}(g)\). Fitting this ansatz to a given a macroscopic strain tensor</p>
 {% highlight matlab %}
 E = strainRateTensor([1 0 0; 0 0 0; 0 0 -1])
 {% endhighlight %}
@@ -63,14 +44,12 @@
   0  0  0
   0  0 -1
 {% endhighlight %}
-<p>via minimizing the square difference</p><p>\[\int \sum_{i,j} (e_{i,j}(g) - E_{i,j}) dg \to \text{min}\]</p><p>the orientation dependent strain rate tensor is identified as</p><p>\[e_{i,j}(g) = 10/3 \left&lt; S(g), E right&gt; S(g)\]</p><p>and the corresponding crystallographic spin tensor as</p><p>\[\Omega_i(g) = \epsilon_{ijk} e_{jk}(g)\]</p><p>This can be modeled in MTEX via</p>
+<p>via minimizing the square difference</p><p>\[\int_{SO(3)} \sum_{i,j} (e_{i,j}(g) - E_{i,j})^2 dg \to \text{min}\]</p><p>the orientation dependent strain rate tensor is identified as</p><p>\[e_{i,j}(g) = 10/3 \left&lt; S(g), E \right&gt; S(g)\]</p><p>and the corresponding crystallographic spin tensor as</p><p>\[\Omega_i(g) = \epsilon_{ijk} e_{jk}(g)\]</p><p>This can be modeled in MTEX via</p>
 {% highlight matlab %}
 % explicite version
 % Omega = @(ori) 0.5 * EinsteinSum(tensor.leviCivita,[1 -1 -2],(ori * S(1) : E) * (ori * S(1)),[-1 -2])
 
 Omega = @(ori) vector3d(spinTensor(((ori * S(1)) : E) .* (ori * S(1))));
-
-% does not yet work
 %Omega = @(ori) vector3d(spinTensor((S(1) : (inv(ori) * E)) .* S(1)));
 
 Omega = SO3VectorFieldHarmonic.quadrature(Omega,cs)
@@ -88,13 +67,13 @@
 {% endhighlight %}
 <center>
 {% include inline_image.html file="SingleSlipModel_01.png" %}
-</center><p>or the total amount of spin by</p>
+</center><p>or the divergence of this vectorfield</p>
 {% highlight matlab %}
-plot(sqrt(normSquare(Omega)),'sigma')
+plot(div(Omega),'sigma')
 {% endhighlight %}
 <center>
 {% include inline_image.html file="SingleSlipModel_02.png" %}
-</center><h2 id="9">Solutions of the Continuity Equation</h2><p>The solutions of the continuity equation can be analytically computed and are available via the command <a href="SO3FunSBF.SO3FunSBF.html"><code class="language-plaintext highlighter-rouge">SO3FunSBF</code></a>. This command takes as input the specific slips system <code class="language-plaintext highlighter-rouge">sS</code> and the makroscopic strain tensor.</p>
+</center><h2 id="9">Solutions of the Continuity Equation</h2><p>The solutions of the continuity equation can be analytically computed and are available via the command <a href="SO3FunSBF.SO3FunSBF.html"><code class="language-plaintext highlighter-rouge">SO3FunSBF</code></a>. This command takes as input the specific slips system <code class="language-plaintext highlighter-rouge">sS</code> and the makroscopic strain tensor <code class="language-plaintext highlighter-rouge">E</code></p>
 {% highlight matlab %}
 odf1 = SO3FunSBF(sS(1),E)
 odf2 = SO3FunSBF(sS(2),E)
@@ -146,4 +125,18 @@
 {% endhighlight %}
 <center>
 {% include inline_image.html file="SingleSlipModel_06.png" %}
+</center><h2 id="15">Checking the Continuity Equation</h2><p>We may now check wether the continuity equation is satisfied. In a stable state the time difference will be zero and hence \(f \text{div} \Omega\)</p>
+{% highlight matlab %}
+figure(1)
+plot(odf1 .* div(Omega),'sigma')
+{% endhighlight %}
+<center>
+{% include inline_image.html file="SingleSlipModel_07.png" %}
+</center><p>should be the negative of the inner product \(\nabla f \cdot \Omega\)</p>
+{% highlight matlab %}
+figure(2)
+plot(dot(grad(odf1),Omega),'sigma')
+{% endhighlight %}
+<center>
+{% include inline_image.html file="SingleSlipModel_08.png" %}
 </center></div></body></html>
diff --git a/pages/function_reference_matlab/SO3Fun.calcComponentsOld.html b/pages/function_reference_matlab/SO3Fun.calcComponentsOld.html
@@ -0,0 +1,93 @@
+---
+title: calcComponentsOld
+
+sidebar: function_reference_sidebar
+permalink: SO3Fun.calcComponentsOld.html
+folder: function_reference
+toc: false
+---
+    <html><head>
+      <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
+   <!--
+This HTML was auto-generated from MATLAB code.
+To make changes, update the MATLAB code and republish this document.
+      --><title>calcComponentsOld</title><link rel="schema.DC" href="http://purl.org/dc/elements/1.1/"><meta name="DC.source" content="script_SO3Fun__calcComponentsOld.m"></head><body><font size="2"><a href="https://github.com/mtex-toolbox/mtex/blob/develop/SO3Fun/@SO3Fun/calcComponentsOld.m">
+    edit page</a></font><div><!--introduction--><!--/introduction--><p>heuristic to find modal orientations</p><h2 id="3">Syntax</h2>
+{% highlight matlab %}
+[modes, volume] = calcComponents(SO3F)
+[modes, volume, centerId] = calcComponents(SO3F,'seed',ori)
+{% endhighlight %}
+<h2 id="4">Input</h2><p>
+   <table cellpadding="4" cellspacing="0" class="funcref" width="100%">
+      <tr>
+         <td width="100px">
+            <tt>SO3F</tt>
+         </td>
+         <td>
+            <tt><a href="SO3Fun.SO3Fun.html">SO3Fun</a></tt>
+         </td>
+      </tr>
+      <tr>
+         <td width="100px">
+            <tt>ori</tt>
+         </td>
+         <td>
+            <tt>initial list of <a href="orientation.orientation.html">orientation</a></tt>
+         </td>
+      </tr>
+   </table>
+</p><h2 id="5">Output</h2><p>
+   <table cellpadding="4" cellspacing="0" class="funcref" width="100%">
+      <tr>
+         <td width="100px">
+            <tt>modes</tt>
+         </td>
+         <td>
+            <tt>modal <a href="orientation.orientation.html">orientation</a></tt>
+         </td>
+      </tr>
+      <tr>
+         <td width="100px">
+            <tt>volume</tt>
+         </td>
+         <td>
+            <tt>volume of the component</tt>
+         </td>
+      </tr>
+      <tr>
+         <td width="100px">
+            <tt>centerId</tt>
+         </td>
+         <td>
+            <tt>list of ids to which each initial ori converged to</tt>
+         </td>
+      </tr>
+   </table>
+</p><h2 id="6">Options</h2><p>
+   <table cellpadding="4" cellspacing="0" class="funcref" width="100%">
+      <tr>
+         <td width="100px">
+            <tt>resolution</tt>
+         </td>
+         <td>
+            <tt>search-grid resolution</tt>
+         </td>
+      </tr>
+      <tr>
+         <td width="100px">
+            <tt>angle</tt>
+         </td>
+         <td>
+            <tt>maximum component width used for volume computation</tt>
+         </td>
+      </tr>
+      <tr>
+         <td width="100px">
+            <tt>exact</tt>
+         </td>
+         <td>
+            <tt>do not dismiss very small modes at the end</tt>
+         </td>
+      </tr>
+   </table>
+</p><h2 id="7">See also</h2><p><a href="SO3Fun.max.html">SO3Fun.max</a></p></div></body></html>
diff --git a/pages/function_reference_matlab/SO3VectorField.div.html b/pages/function_reference_matlab/SO3VectorField.div.html
@@ -0,0 +1,58 @@
+---
+title: div
+
+sidebar: function_reference_sidebar
+permalink: SO3VectorField.div.html
+folder: function_reference
+toc: false
+---
+    <html><head>
+      <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
+   <!--
+This HTML was auto-generated from MATLAB code.
+To make changes, update the MATLAB code and republish this document.
+      --><title>div</title><link rel="schema.DC" href="http://purl.org/dc/elements/1.1/"><meta name="DC.source" content="script_SO3VectorField__div.m"></head><body><font size="2"><a href="https://github.com/mtex-toolbox/mtex/blob/develop/SO3Fun/@SO3VectorField/div.m">
+    edit page</a></font><div><!--introduction--><!--/introduction--><p>divergence of an SO3VectorField</p><h2 id="3">Syntax</h2>
+{% highlight matlab %}
+D = SO3VF.div % compute the divergence
+d = SO3VF.div(rot) % evaluate the divergence in rot
+{% endhighlight %}
+<h2 id="4">Input</h2><p>
+   <table cellpadding="4" cellspacing="0" class="funcref" width="100%">
+      <tr>
+         <td width="100px">
+            <tt>SO3VF</tt>
+         </td>
+         <td>
+            <tt><a href="SO3VectorField.SO3VectorField.html">SO3VectorField</a></tt>
+         </td>
+      </tr>
+      <tr>
+         <td width="100px">
+            <tt>rot</tt>
+         </td>
+         <td>
+            <tt><a href="rotation.rotation.html">rotation</a> / <a href="orientation.orientation.html">orientation</a></tt>
+         </td>
+      </tr>
+   </table>
+</p><h2 id="5">Output</h2><p>
+   <table cellpadding="4" cellspacing="0" class="funcref" width="100%">
+      <tr>
+         <td width="100px">
+            <tt>D</tt>
+         </td>
+         <td>
+            <tt><a href="SO3Fun.SO3Fun.html">SO3Fun</a></tt>
+         </td>
+      </tr>
+      <tr>
+         <td width="100px">
+            <tt>d</tt>
+         </td>
+         <td>
+            <tt>@double divergence of |SO3VF| at rotation |rot|</tt>
+         </td>
+      </tr>
+   </table>
+</p><h2 id="6">See also</h2><p><a href="SO3VectorFieldHarmonic.div.html">SO3VectorFieldHarmonic.div</a> <a href="SO3Fun.grad.html">SO3Fun.grad</a></p></div></body></html>
diff --git a/pages/function_reference_matlab/SO3VectorFieldHarmonic.div.html b/pages/function_reference_matlab/SO3VectorFieldHarmonic.div.html
@@ -0,0 +1,58 @@
+---
+title: div
+
+sidebar: function_reference_sidebar
+permalink: SO3VectorFieldHarmonic.div.html
+folder: function_reference
+toc: false
+---
+    <html><head>
+      <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
+   <!--
+This HTML was auto-generated from MATLAB code.
+To make changes, update the MATLAB code and republish this document.
+      --><title>div</title><link rel="schema.DC" href="http://purl.org/dc/elements/1.1/"><meta name="DC.source" content="script_SO3VectorFieldHarmonic__div.m"></head><body><font size="2"><a href="https://github.com/mtex-toolbox/mtex/blob/develop/SO3Fun/@SO3VectorFieldHarmonic/div.m">
+    edit page</a></font><div><!--introduction--><!--/introduction--><p>divergence of an SO3VectorFieldHarmonic</p><h2 id="3">Syntax</h2>
+{% highlight matlab %}
+D = SO3VF.div % compute the divergence
+d = SO3VF.div(rot) % evaluate the divergence in rot
+{% endhighlight %}
+<h2 id="4">Input</h2><p>
+   <table cellpadding="4" cellspacing="0" class="funcref" width="100%">
+      <tr>
+         <td width="100px">
+            <tt>SO3VF</tt>
+         </td>
+         <td>
+            <tt><a href="SO3VectorFieldHarmonic.SO3VectorFieldHarmonic.html">SO3VectorFieldHarmonic</a></tt>
+         </td>
+      </tr>
+      <tr>
+         <td width="100px">
+            <tt>rot</tt>
+         </td>
+         <td>
+            <tt><a href="rotation.rotation.html">rotation</a> / <a href="orientation.orientation.html">orientation</a></tt>
+         </td>
+      </tr>
+   </table>
+</p><h2 id="5">Output</h2><p>
+   <table cellpadding="4" cellspacing="0" class="funcref" width="100%">
+      <tr>
+         <td width="100px">
+            <tt>D</tt>
+         </td>
+         <td>
+            <tt><a href="SO3FunHarmonic.SO3FunHarmonic.html">SO3FunHarmonic</a></tt>
+         </td>
+      </tr>
+      <tr>
+         <td width="100px">
+            <tt>d</tt>
+         </td>
+         <td>
+            <tt>@double divergence of |SO3VF| at rotation |rot|</tt>
+         </td>
+      </tr>
+   </table>
+</p><h2 id="6">See also</h2><p><a href="SO3VectorField.div.html">SO3VectorField.div</a> <a href="SO3FunHarmonic.grad.html">SO3FunHarmonic.grad</a></p></div></body></html>