Initial commit

7aa80c51 · Praetorius, Simon · 7aa80c51 · 7aa80c51 · 419b0a6e · 7aa80c51
Commit 7aa80c51 authored Oct 04, 2018 by Praetorius, Simon
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--- a/.gitattributes
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+*.pdf filter=lfs diff=lfs merge=lfs -text
+*.png filter=lfs diff=lfs merge=lfs -text
+*.jpg filter=lfs diff=lfs merge=lfs -text
--- a/.gitmodules
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+[submodule "MathJax"]
+	path = MathJax
+	url = https://github.com/mathjax/MathJax.git
--- a/MathJax @ 419b0a6e
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+Subproject commit 419b0a6eee7eefc0f85e47f7d4f8227ec28b8e57
--- a/adaptivity.html
+++ b/adaptivity.html
--- a/amdis-workshop_2016.0.1.pdf
+++ b/amdis-workshop_2016.0.1.pdf
--- a/boundary_conditions.html
+++ b/boundary_conditions.html
--- a/complex_geometries.html
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--- a/coupled_problems.html
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--- a/cpp11.html
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--- a/data_processing.html
+++ b/data_processing.html
+<!DOCTYPE html>
+<html>
+  <head>
+    <title>AMDiS - Adaptive Multi-Dimensional Simulations</title>
+    <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/>
+    <style type="text/css">
+      @import url(https://fonts.googleapis.com/css?family=Yanone+Kaffeesatz);
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+      @import url(https://fonts.googleapis.com/css?family=Ubuntu);
+      @import url(https://fonts.googleapis.com/css?family=Droid+Serif:400,700,400italic);
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+    </style>
+    <!--<link rel="stylesheet" type="text/css" href="style_display.css" />-->
+    <link rel="stylesheet" type="text/css" href="style_print.css" />
+  </head>
+  <body>
+    <textarea id="source">
+
+class: center, middle
+
+# Session 2
+## Handling data on unstructured grids
+
+---
+
+# Agenda
+
+### Monday
+- Scalar linear second order PDEs
+- **Handling data on unstructured grids**
+- Adaptivity and systems of equations
+- Introduction to Student Projects
+
+---
+
+# Motivation
+### Extract maxima of solution
+<img src="images/ball500.png" width="45%" /><img src="images/pfc_sphere.png" width="45%" />
+
+### Calculate integrals
+<img src="images/energy.png" width="45%" />
+
+---
+
+# Working with the discrete solution
+Finite-Element function `\(u_h\)` expressed as linear combination of basis-functions `\(\phi_i\)`:
+\\[
+u\_h(x) = \sum\_i u\_i \phi\_i(x),
+\\]
+with `\(u_i:=\)` Degrees of Freedom, stored in `DOFVector<value_type> U`:
+- Evaluate at DOF index: `U[idx]`
+- Iterate over all DOFs:
+```
+DOFIterator<value_type> it(&U [, USED_DOFS]);
+for (it.reset(); !it.end(); ++it)
+      *it = value; // idx = it.getDOFIndex()
+```
+
+- Evaluate DOFVector at coordinate `\(x\)`:
+```
+WorldVector<double> x;
+x[0] = 0.2; x[1] = 0.3;
+value_type value = U(x);
+```
+
+---
+
+# Fill a DOFVector}
+```
+U << EXPRESSION;
+```
+where `EXPRESSION` can contain e.g.
+- The coordinates: `X()`, `X(i)`
+- Scalar values: `1.0`, `constant(1.0)`
+- Matrix and vector expressions: `two_norm(...)`, `vec * vec`
+- Other DOFVectors: `valueOf(V)`, `gradientOf(V)`
+
+([Expressions manual](https://goo.gl/JK8EUI))
+
+### Example:
+Let `\(C\)` and `\(U\)` be of type `DOFVector<double>`:
+\\[
+U(x) := \frac{1}{2}\left(C(x)^2\,(1-C(x))^2 + \frac{1}{\epsilon}\|\nabla C(x)\|^2\right)
+\\]
+
+Assign expression on rhs to DOFVector `\(U\)`:
+```
+U << 0.5*( pow<2>(valueOf(C) * (1.0 - valueOf(C))
+      + (1.0/eps) * unary_dot(gradientOf(C)) );
+```
+
+---
+
+# Working with the discrete solution
+- Reduce the `DOFVector`, i.e. calc integrals, norms:
+```
+integrate( EXPRESSION [, Mesh*] )
+```
+```
+integrate( valueOf(U) );
+integrate( two_norm(gradientOf(U)) );
+integrate(pow<2>(valueOf(U)) + unary_dot(gradientOf(U)));
+```
+- Other reduction operations (over DOFs):
+  - `max( EXPRESSION )` `\(\Rightarrow\max\{ expr(x_i)\,:\, x_i\in\Omega \}\)`
+  - `min( EXPRESSION )` `\(\Rightarrow\min\{ expr(x_i)\,:\, x_i\in\Omega \}\)`
+  - `abs_max( EXPRESSION )` `\(\Rightarrow\max\{ |expr(x_i)|\,:\, x_i\in\Omega \}\)`
+  - `abs_min( EXPRESSION )` `\(\Rightarrow\min\{ |expr(x_i)|\,:\, x_i\in\Omega \}\)`
+
+---
+
+# Working with the mesh
+
+.center[
+<img src="images/ellipt_macro.png" width="30%" /><img src="images/torus_macro.jpg" width="30%" />
+]
+
+### Change position of mesh vertices:
+<img src="images/Fsi.png" width="70%" />
+
+---
+
+# Working with the mesh
+A Mesh holds all information about a triangulation. Get the mesh from `ProblemStat`:
+```
+Mesh&amp; mesh = *prob.getMesh();
+```
+- Number of vertices: `mesh.getNumberOfVertices()`
+- Number of triangles (elements): `mesh.getNumberOfElements()`
+- World coordinates are stored in `WorldVector<double>`.
+- Get the coordinates of all DOFs:
+```
+DOFVector<WorldVector<double> > Coords(prob.getFeSpace(), "coords");
+mesh.getDofCoords(Coords);
+```
+- Change coordinates of the mesh:
+```
+ParametricSimple parametric(Coords);
+mesh.setParametric(&amp;parametric);
+```
+
+---
+
+# Working with the mesh:
+- (Advanced) Traverse the mesh element-wise:
+
+```
+Flag traverseFlag = Mesh::CALL_LEAF_EL | (FILL_FLAGS);
+TraverseStack stack;
+ElInfo *elInfo = stack.traverseFirst(&amp;mesh, -1, traverseFlag);
+while (elInfo) {
+  // do something with elInfo
+  elInfo = stack.traverseNext(elInfo);
+}
+```
+- (Advanced) Global Refinement of the mesh: Use `RefinementManager`
+
+```
+RefinementManager* refManager = prob.getRefinementManager();
+
+Flag f = refManager->globalRefine(&amp;mesh, 5);
+if (f == MESH_REFINED) {
+  MSG("Mesh globally refined by 5 levels!\n");
+}
+```
+
+---
+
+# Example: Print solution with coordinates
+Writing out the solution to the screen:
+```
+DOFVector<WorldVector<double>> C(U.getFeSpace(), "c");
+mesh.getDofCoords(C);
+
+DOFIterator<double> it(U);
+DOFIterator<WorldVector<double>> c_it(C);
+for (it.reset(),c_it.reset();  !it.end();  ++it,++c_it)
+  std::cout << "U(" << *c_it << ") = " << *it << "\n";
+```
+This will print out:
+<pre><code style="background: #ddd;">U(0 0) = 0
+U(1 0) = 0
+U(1 1) = 0
+U(0 1) = 0
+U(0.5 0.5) = 0.0833333
+</code></pre>
+
+---
+class: center, middle
+
+# Input/Output
+
+---
+
+# Input/Output
+Write DOFVectors to file, for visualization, serialization, ...
+- File writer with automatic file-type detection:
+  ```
+  io::writeFile(DOFVECTOR, FILENAME);
+  ```
+  Detection by filename extension: ".vtu", ".arh", ".dat", ".2d", ".gnu",...
+  - Use **ParaView** for visualization
+  - Tool **MeshConv** can convert beween mesh formats
+- File reader with automatic file-type detection:
+```
+io::readFile(FILENAME, DOFVECTOR);
+```
+- Use ARH format to exchange data (can be converted to VTU by MeshConv.)
+
+
+---
+class: center, middle
+
+# Exercise2
+## Poisson equation
+
+---
+
+# Exercise2: Poisson equation
+We want to solve the Poisson equation
+\\[
+-\Delta u = f(x)\quad\text{ in }\Omega,\quad u|\_{\partial\Omega} = g
+\\]
+in a rectangular domain `\(\Omega\)`, with
+\\[
+f(x) = 40(1 - 10\|x\|^2)e^{-10.0\|x\|^2}\\\\
+g(x) = e^{-10.0\|x\|^2} \\\\
+\\]
+and with exact solution `\(u^\ast=g\)`.
+1. Assemble and solve the equation.
+2. Interpolate error `\(ERR := |u_h - g|\)` to DOFVector and write it to a file.
+3. Calculate the error-norms `\(\|u_h - g\|_{L_2(\Omega)}\)` and `\(\|u_h - g\|_{H_1(\Omega)}\)`
+4. Evaluate the error at the grid-point `\(x=(0.5, 0.5)\)`.
+5. Refine the mesh globally and compare error-norms to old error-norms.
+
+---
+
+# Advanced Exercise2: Mesh adaption
+The datastructure `AdaptInfo` provides parameters for tolerance and nr. of iterations for an adaption process:
+```
+AdaptInfo adaptInfo("adapt");
+adaptInfo.getMaxSpaceIteration(); // => adapt->max iteration
+adaptInfo.getSpaceTolerance(0);   // => adapt[0]->tolerance
+```
+
+1. Use AdaptInfo to write an adaption loop that refines the mesh globally to reduce the error until a tolerance is reached.
+2. Write the error DOFVector in every adaption iteration to a file.
+3. Calculate the exponent in the error estimate `\(\|u_h - I_h u^\ast\|_\#\leq C h^k\)`
+4. (optional) Transform the mesh, by `\(x\mapsto 2x\)` and solve again.
+
+---
+
+# Some hints
+
+### Used functions/classes:
+
+```
+DOFVector << EXPRESSION;
+integrate( EXPRESSION );
+
+// EXPRESSION: {valueOf(U), gradientOf(U), X(), X(i), +,-,*,/,
+//              unary_dot(EXPR.), pow<2>(EXPR.), absolute(EXPR.)}
+
+RefinementManager* refManager = prob.getRefinementManager();
+refManager->globalRefine(prob.getMesh(), NR_OF_REFINEMENTS);
+
+adaptInfo.getSpaceTolerance(0);
+adaptInfo.getMaxSpaceIteration();
+
+io::writeFile( DOFVECTOR, FILENAME );
+```
+
+### Parameters to modify:
+```matlab
+adapt[0]->tolerance: DOUBLE
+adapt->max iteration: INTEGER
+```
+
+    </textarea>
+    <script src="lib/remark.js" type="text/javascript"></script>
+    <script type="text/javascript" src="../MathJax/MathJax.js?config=TeX-AMS_HTML"></script>
+
+    <script type="text/javascript">
+      var slideshow = remark.create({
+        ratio: "4:3",
+        highlightLanguage: "cpp"
+      });
+
+      // Setup MathJax
+      MathJax.Hub.Config({
+          tex2jax: {
+          skipTags: ['script', 'noscript', 'style', 'textarea', 'pre']
+          }
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+      MathJax.Hub.Queue(function() {
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+</html>
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--- a/images/Fsi.png
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--- a/images/R9_energy.png
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--- a/images/Sphere_decomposition2.png
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--- a/images/partition3.png
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