Add example for mesh refinement

Author: robbievanleeuwen

.github/ISSUE_TEMPLATE/bug_report.md

@@ -8,13 +8,13 @@ about: Create a report to help us improve
 A clear and concise description of what the bug is.
 
 **To Reproduce**
-Steps to reproduce the behavior:
+Steps to reproduce the behaviour:
 1. Go to '...'
 2. Click on '....'
 3. Scroll down to '....'
 4. See error
 
-**Expected behavior**
+**Expected behaviour**
 A clear and concise description of what you expected to happen.
 
 **Screenshots**

docs/build/doctrees/environment.pickle

@@ -11,11 +11,10 @@
 Documentation
 =============
 
-*sectionproperties* is a python package for the analysis of arbitrary cross-sections
-using the finite element method written by Robbie van Leeuwen. *sectionproperties*
-can be used to determine section properties to be used in structural design and
-visualise cross-sectional stresses resulting from combinations of applied forces and
-bending moments.
+*sectionproperties* is a python package for the analysis of arbitrary cross-sections using the
+finite element method written by Robbie van Leeuwen. *sectionproperties* can be used to determine
+section properties to be used in structural design and visualise cross-sectional stresses resulting
+from combinations of applied forces and bending moments.
 
 A list of the `current features of the package and implementation goals for future releases
 <https://github.com/robbievanleeuwen/section-properties/tree/master/README.md>`_

docs/build/doctrees/index.doctree

@@ -829,14 +829,17 @@ accuracy of the result compared with the time taken to obtain the solution::
 
     Plot of the torsion constant as a function of the solution time.
 
-Advanced Example
-----------------
+Advanced Examples
+-----------------
 
-The following example demonstrates an example in which *sectionproperties* can
-be used for more academic purposes. In this example, the aspect ratio of a
-rectangular section is varied whilst keeping a constant cross-sectional area
-and the torsion constant calculated. The variation of the torsion constant with
-the aspect ratio is then plotted::
+The following examples demonstrates how *sectionproperties* can be used for more academic purposes.
+
+Torsion Constant of a Rectangle
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+In this example, the aspect ratio of a rectangular section is varied whilst keeping a constant
+cross-sectional area and the torsion constant calculated. The variation of the torsion constant
+with the aspect ratio is then plotted::
 
   import numpy as np
   import matplotlib.pyplot as plt
@@ -880,8 +883,87 @@ the aspect ratio is then plotted::
   ax.set_title("Rectangular Section Torsion Constant")
   plt.show()
 
-..  figure:: ../images/examples/advanced.png
+..  figure:: ../images/examples/advanced1.png
     :align: center
     :scale: 75 %
 
     Plot of the torsion constant as a function of the aspect ratio.
+
+Mesh Refinement
+^^^^^^^^^^^^^^^
+
+In this example the convergence of the torsion constant is investigated through an analysis of an
+I-section. The mesh is refined both by modifying the mesh size and by specifying the number of
+points making up the root radius. The figure below the example code shows that mesh refinement
+adjacent to the root radius is a far more efficient method in obtaining fast convergence when
+compared to reducing the mesh area size for the entire section::
+
+  import numpy as np
+  import matplotlib.pyplot as plt
+  import sectionproperties.pre.sections as sections
+  from sectionproperties.analysis.cross_section import CrossSection
+
+  # define mesh sizes
+  mesh_size_list = [50, 20, 10, 5, 3, 2, 1]
+  nr_list = [4, 8, 12, 16, 20, 24, 32, 64]
+
+  # initialise result lists
+  mesh_results = []
+  mesh_elements = []
+  nr_results = []
+  nr_elements = []
+
+  # calculate reference solution
+  geometry = sections.ISection(d=203, b=133, t_f=7.8, t_w=5.8, r=8.9, n_r=64)
+  mesh = geometry.create_mesh(mesh_sizes=[0.5])  # create mesh
+  section = CrossSection(geometry, mesh)  # create a CrossSection object
+  section.calculate_geometric_properties()
+  section.calculate_warping_properties()
+  j_reference = section.get_j()  # get the torsion constant
+
+  # run through mesh_sizes with n_r = 16
+  for mesh_size in mesh_size_list:
+      geometry = sections.ISection(d=203, b=133, t_f=7.8, t_w=5.8, r=8.9, n_r=16)
+      mesh = geometry.create_mesh(mesh_sizes=[mesh_size])  # create mesh
+      section = CrossSection(geometry, mesh)  # create a CrossSection object
+      section.calculate_geometric_properties()
+      section.calculate_warping_properties()
+
+      mesh_elements.append(len(section.elements))
+      mesh_results.append(section.get_j())
+
+  # run through n_r with mesh_size = 3
+  for n_r in nr_list:
+      geometry = sections.ISection(d=203, b=133, t_f=7.8, t_w=5.8, r=8.9, n_r=n_r)
+      mesh = geometry.create_mesh(mesh_sizes=[3])  # create mesh
+      section = CrossSection(geometry, mesh)  # create a CrossSection object
+      section.calculate_geometric_properties()
+      section.calculate_warping_properties()
+
+      nr_elements.append(len(section.elements))
+      nr_results.append(section.get_j())
+
+  # convert results to a numpy array
+  mesh_results = np.array(mesh_results)
+  nr_results = np.array(nr_results)
+
+  # compute the error
+  mesh_error_vals = (mesh_results - j_reference) / mesh_results * 100
+  nr_error_vals = (nr_results - j_reference) / nr_results * 100
+
+  # plot the results
+  (fig, ax) = plt.subplots()
+  ax.loglog(mesh_elements, mesh_error_vals, 'kx-', label='Mesh Size Refinement')
+  ax.loglog(nr_elements, nr_error_vals, 'rx-', label='Root Radius Refinement')
+  plt.xlabel("Number of Elements")
+  plt.ylabel("Torsion Constant Error [%]")
+  plt.legend(loc='center left', bbox_to_anchor=(1, 0.5))
+  plt.tight_layout()
+  plt.show()
+
+..  figure:: ../images/examples/advanced2.png
+    :align: center
+    :scale: 75 %
+
+    Plot of the torsion constant error as a function of number of elements used in the analysis for
+    both general mesh refinement and root radius refinement.

docs/build/doctrees/rst/examples.doctree

@@ -161,11 +161,10 @@
   <a class="reference internal image-reference" href="_images/logo.png"><img alt="sectionproperties" class="align-left" src="_images/logo.png" style="width: 100%;" /></a>
 <div class="section" id="documentation">
 <h1>Documentation<a class="headerlink" href="#documentation" title="Permalink to this headline">¶</a></h1>
-<p><em>sectionproperties</em> is a python package for the analysis of arbitrary cross-sections
-using the finite element method written by Robbie van Leeuwen. <em>sectionproperties</em>
-can be used to determine section properties to be used in structural design and
-visualise cross-sectional stresses resulting from combinations of applied forces and
-bending moments.</p>
+<p><em>sectionproperties</em> is a python package for the analysis of arbitrary cross-sections using the
+finite element method written by Robbie van Leeuwen. <em>sectionproperties</em> can be used to determine
+section properties to be used in structural design and visualise cross-sectional stresses resulting
+from combinations of applied forces and bending moments.</p>
 <p>A list of the <a class="reference external" href="https://github.com/robbievanleeuwen/section-properties/tree/master/README.md">current features of the package and implementation goals for future releases</a>
 can be found in the README file on github.</p>
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