deploy: dd51e5d

.buildinfo

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+# Sphinx build info version 1
+# This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done.
+config: 6533338d5e651ee36cbea8b7186bcef5
+tags: 645f666f9bcd5a90fca523b33c5a78b7

_sources/contents/examples/example_airfoil.rst.txt

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+.. _section-airfoil:
+
+Airfoil (MH-104)
+================
+
+Problem description
+-------------------
+
+..  figure:: /figures/examplemh1040.png
+    :name: fig_mh104_draw
+    :width: 5.5in
+    :align: center
+
+    Sketch of a cross section for a typical wind turbine blade [CHEN2010]_.
+
+This example demonstrates the capability of building a cross section having an airfoil shape, which is commonly seen on wind turbine blades or helicopter rotor blades.
+This example is also studied in [CHEN2010]_.
+A sketch of a cross section for a typical wind turbine blade is shown in :numref:`Fig. %s <fig_mh104_draw>`.
+The airfoil is MH 104 (http://m-selig.ae.illinois.edu/ads/coord_database.html#M).
+In this example, the chord length :math:`CL=1.9` m.
+The origin O is set to the point at 1/4 of the chord.
+Twist angle :math:`\theta` is :math:`0^\circ`.
+There are two webs, whose right boundaries are at the 20% and 50% location of the chord, respectively.
+Both low pressure and high pressure surfaces have four segments.
+The dividing points between segments are listed in :numref:`Table %s <table_div_pts>`.
+Materials are given in :numref:`Table %s <table_mh104_materials>` and layups are given in :numref:`Table %s <table_mh104_layups>`.
+A complete :math:`6\times 6` stiffness matrix is given in :numref:`Table %s <table_airfoil_result>`.
+Complete input files can be found in ``examples\ex_airfoil\``, including ``mh104.xml``, ``basepoints.dat``, ``baselines.xml``, ``materials.xml``, and ``layups.xml``.
+
+..  csv-table:: Dividing points
+    :name: table_div_pts
+    :header-rows: 2
+    :align: center
+
+    "Between segments", "Low pressure surface", "High pressure surface"
+    , ":math:`(x, y)`", ":math:`(x, y)`"
+    "1 and 2", "(0.004053940, 0.011734800)", "(0.006824530, -0.009881650)"
+    "2 and 3", "(0.114739930, 0.074571970)", "(0.126956710, -0.047620490)"
+    "3 and 4", "(0.536615950, 0.070226120)", "(0.542952100, -0.044437080)"
+
+..  figure:: /figures/examplemh1041.png
+    :name: fig_mh1041
+    :width: 6.5in
+    :align: center
+
+    *Base point*\ s of the tube cross section.
+
+..  figure:: /figures/examplemh1042.png
+    :name: fig_mh1042
+    :width: 6.5in
+    :align: center
+
+    *Base line*\ s of the tube cross section.
+
+..  figure:: /figures/examplemh1043.png
+    :name: fig_mh1043
+    :width: 6.5in
+    :align: center
+
+    *Segment*\ s of the tube cross section.
+
+..  figure:: /figures/examplemh1044.png
+    :name: fig_mh104
+    :width: 6.5in
+    :align: center
+
+    Meshed cross section viewed in Gmsh.
+
+..  csv-table:: Material properties
+    :name: table_mh104_materials
+    :header-rows: 2
+    :align: center
+
+    "Name", "Type", "Density", |e1|, |e2|, |e3|, |g12|, |g13|, |g23|, |nu12|, |nu13|, |nu23|
+    , , |den_si_k|, |mod_si_g|, |mod_si_g|, |mod_si_g|, |mod_si_g|, |mod_si_g|, |mod_si_g|, , ,
+    "Uni-directional FRP", "orthotropic", 1.86, 37.00, 9.00, 9.00, 4.00, 4.00, 4.00, 0.28, 0.28, 0.28
+    "Double-bias FRP", "orthotropic", 1.83, 10.30, 10.30, 10.30, 8.00, 8.00, 8.00, 0.30, 0.30, 0.30
+    "Gelcoat", "orthotropic", 1.83, 1e-8, 1e-8, 1e-8, 1e-9, 1e-9, 1e-9, 0.30, 0.30, 0.30
+    "Nexus", "orthotropic", 1.664, 10.30, 10.30, 10.30, 8.00, 8.00, 8.00, 0.30, 0.30, 0.30
+    "Balsa", "orthotropic", 0.128, 0.01, 0.01, 0.01, 2e-4, 2e-4, 2e-4, 0.30, 0.30, 0.30
+
+..  csv-table:: Layups
+    :name: table_mh104_layups
+    :header-rows: 2
+    :align: center
+
+    "Name", "Layer", "Material", "Ply thickness", "Orientation", "Number of plies"
+    , , , |len_si|, :math:`\circ`,
+    "layup_1", 1, "Gelcoat",         0.000381, 0, 1
+    ,          2, "Nexus",           0.00051, 0, 1
+    ,          3, "Double-bias FRP", 0.00053, 20, 18
+    "layup_2", 1, "Gelcoat",         0.000381, 0, 1
+    ,          2, "Nexus",           0.00051, 0, 1
+    ,          3, "Double-bias FRP", 0.00053, 20, 33
+    "layup_3", 1, "Gelcoat",             0.000381, 0, 1
+    ,          2, "Nexus",               0.00051, 0, 1
+    ,          3, "Double-bias FRP",     0.00053, 20, 17
+    ,          4, "Uni-directional FRP", 0.00053, 30, 38
+    ,          5, "Balsa",               0.003125, 0, 1
+    ,          6, "Uni-directional FRP", 0.00053, 30, 37
+    ,          7, "Double-bias FRP",     0.00053, 20, 16
+    "layup_4", 1, "Gelcoat",         0.000381, 0, 1
+    ,          2, "Nexus",           0.00051, 0, 1
+    ,          3, "Double-bias FRP", 0.00053, 20, 17
+    ,          4, "Balsa",           0.003125, 0, 1
+    ,          5, "Double-bias FRP", 0.00053, 0, 16
+    "layup_web", 1, "Uni-directional FRP", 0.00053, 0, 38
+    ,            2, "Balsa",               0.003125, 0, 1
+    ,            3, "Uni-directional FRP", 0.00053, 0, 38
+
+
+
+
+
+Result
+------
+
+..  table:: Effective Timoshenko stiffness matrix
+    :name: table_airfoil_result
+
+    =================================== =================================== =================================== =================================== =================================== ===================================
+    :math:`\phantom{-}2.395\times 10^9` :math:`\phantom{-}1.588\times 10^6` :math:`\phantom{-}7.215\times 10^6` :math:`-3.358\times 10^7`           :math:`\phantom{-}6.993\times 10^7` :math:`-5.556\times 10^8`
+    :math:`\phantom{-}1.588\times 10^6` :math:`\phantom{-}4.307\times 10^8` :math:`-3.609\times 10^6`           :math:`-1.777\times 10^7`           :math:`\phantom{-}1.507\times 10^7` :math:`\phantom{-}2.652\times 10^5`
+    :math:`\phantom{-}7.215\times 10^6` :math:`-3.609\times 10^6`           :math:`\phantom{-}2.828\times 10^7` :math:`\phantom{-}8.440\times 10^5` :math:`\phantom{-}2.983\times 10^5` :math:`-5.260\times 10^6`
+    :math:`-3.358\times 10^7`           :math:`-1.777\times 10^7`           :math:`\phantom{-}8.440\times 10^5` :math:`\phantom{-}2.236\times 10^7` :math:`-2.024\times 10^6`           :math:`\phantom{-}2.202\times 10^6`
+    :math:`\phantom{-}6.993\times 10^7` :math:`\phantom{-}1.507\times 10^7` :math:`\phantom{-}2.983\times 10^5` :math:`-2.024\times 10^6`           :math:`\phantom{-}2.144\times 10^7` :math:`-9.137\times 10^6`
+    :math:`-5.556\times 10^8`           :math:`\phantom{-}2.652\times 10^5` :math:`-5.260\times 10^6`           :math:`\phantom{-}2.202\times 10^6` :math:`-9.137\times 10^6`           :math:`\phantom{-}4.823\times 10^8`
+    =================================== =================================== =================================== =================================== =================================== ===================================
+
+..  table:: Results from reference [CHEN2010]_
+    :name: table_airfoil_result_ref
+
+    =================================== =================================== =================================== =================================== =================================== ===================================
+    :math:`\phantom{-}2.389\times 10^9` :math:`\phantom{-}1.524\times 10^6` :math:`\phantom{-}6.734\times 10^6` :math:`-3.382\times 10^7`           :math:`-2.627\times 10^7`           :math:`-4.736\times 10^8`
+    :math:`\phantom{-}1.524\times 10^6` :math:`\phantom{-}4.334\times 10^8` :math:`-3.741\times 10^6`           :math:`-2.935\times 10^5`           :math:`\phantom{-}1.527\times 10^7` :math:`\phantom{-}3.835\times 10^5`
+    :math:`\phantom{-}6.734\times 10^6` :math:`-3.741\times 10^6`           :math:`\phantom{-}2.743\times 10^7` :math:`-4.592\times 10^4`           :math:`-6.869\times 10^2`           :math:`-4.742\times 10^6`
+    :math:`-3.382\times 10^7`           :math:`-2.935\times 10^5`           :math:`-4.592\times 10^4`           :math:`\phantom{-}2.167\times 10^7` :math:`-6.279\times 10^4`           :math:`\phantom{-}1.430\times 10^6`
+    :math:`-2.627\times 10^7`           :math:`\phantom{-}1.527\times 10^7` :math:`-6.869\times 10^2`           :math:`-6.279\times 10^4`           :math:`\phantom{-}1.970\times 10^7` :math:`\phantom{-}1.209\times 10^7`
+    :math:`-4.736\times 10^8`           :math:`\phantom{-}3.835\times 10^5` :math:`-4.742\times 10^6`           :math:`\phantom{-}1.430\times 10^6` :math:`\phantom{-}1.209\times 10^7` :math:`\phantom{-}4.406\times 10^8`
+    =================================== =================================== =================================== =================================== =================================== ===================================
+
+..  note:: The errors between the result and the reference are caused by the difference of modeling of the trailing edge. If reduce the trailing edge skin to a single thin layer, then the difference between the trailing edge shapes is minimized, and the two resulting stiffness matrices are basically the same, as shown in :numref:`Fig. %s <fig_mh104_comparison>`.
+
+..  figure:: /figures/examplemh104_comparison.png
+    :name: fig_mh104_comparison
+    :width: 6in
+    :align: center
+
+    Comparison of stiffness matrices after modifying the trailing edge.
+

_sources/contents/examples/example_airfoil_recover.rst.txt

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+.. _example-airfoil-recover:
+
+Airfoil (Recover)
+=================
+
+Problem description
+-------------------
+
+This example continues from the  previous one to demonstrate the
+dehomogenization analysis. It is assumed that a 1D beam analysis has
+been finished, and data of global deformations and loads have been
+added to the main input file correctly
+(:ref:`See the recover section <section-recover>`). Suppose that the
+results in :numref:`Table %s <table_mh104_1dresults>` are used and
+default values are kept for others. All files generated from the VABS
+homogenization analysis are kept in the same place as input files.
+Final visualization is required and the contour plot of one of the
+stress components is shown in :numref:`Fig. %s <fig_mh104r>`.
+Complete files can be found in ``examples\ex_airfoil_r\``, including
+``mh104.xml``, ``basepoints.dat``, ``baselines.xml``, ``materials.xml``,
+``layups.xml``, ``mh104.sg``, ``mh104.sg.ech``,
+``mh104.sg.K``, ``mh104.sg.opt``, ``mh104.sg.v0``,
+``mh104.sg.v1S``, and ``mh104.sg.v22``.
+
+..  csv-table:: Sectional forces and moments
+    :name: table_mh104_1dresults
+    :header-rows: 1
+    :align: center
+
+    Quantity, Value
+    ":math:`F_1`, N", 1
+    ":math:`F_2`, N", 2
+    ":math:`F_3`, N", 3
+    ":math:`M_1`, Nm", 4
+    ":math:`M_2`, Nm", 5
+    ":math:`M_3`, Nm", 6
+
+
+
+
+
+Result
+------
+
+..  figure:: /figures/examplemh104r.png
+    :name: fig_mh104r
+    :width: 6in
+    :align: center
+
+    Contour plot of recovered nodal stress SN11.
+

_sources/contents/examples/example_box.rst.txt

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+.. _example-box-beam:
+
+Box beam
+========
+
+Problem description
+-------------------
+
+.. figure:: /figures/examplebox0.png
+  :name: fig_box_draw
+  :width: 4in
+  :align: center
+
+  Cross section of the box beam [YU2012]_.
+
+This example is a thin-walled box beam whose cross section is depicted in :numref:`Fig. %s <fig_box_draw>` [YU2012]_.
+The width :math:`a_2=0.953` in, height :math:`a_3=0.530` in, and thickness :math:`t=0.030` in.
+Each wall has six plies of the same composite material and the same fiber orientation of :math:`15^\circ`.
+Material properties and layup scheme are listed in :numref:`Table %s <table_box_materials>` and :numref:`Table %s <table_box_layups>`.
+Cross-sectional properties are given in :numref:`Table %s <table_box_result>` and compared with those in Ref. [YU2012]_.
+The tiny differences are due to different meshes.
+Complete input files can be found in ``examples\ex_box\``, including ``box.xml`` and ``materials.xml``.
+
+.. figure:: /figures/examplebox1.png
+  :name: fig_box1
+  :width: 6in
+  :align: center
+
+  *Base point*\ s, *Base line*\ s and *Segment*\ s of the box beam cross section.
+
+.. figure:: /figures/examplebox.png
+  :name: fig_box
+  :width: 6in
+  :align: center
+
+  Meshed cross section viewed in Gmsh.
+
+.. tabularcolumns:: |l|l|r|r|r|r|r|r|r|r|r|r|
+
+.. csv-table:: Material properties
+  :name: table_box_materials
+  :header-rows: 2
+  :align: center
+
+  "Name", "Density", |e1|, |e2|, |e3|, |g12|, |g13|, |g23|, |nu12|, |nu13|, |nu23|
+   , |den_im|, |mod_im_m|, |mod_im_m|, |mod_im_m|, |mod_im_m|, |mod_im_m|, |mod_im_m|, , ,
+  "mat_1", 0.0001353, 20.59, 1.42, 1.42, 0.87, 0.87, 0.696, 0.30, 0.30, 0.34
+
+.. csv-table:: Layups
+  :name: table_box_layups
+  :header-rows: 2
+  :align: center
+
+  "Name", "Layer", "Material", "Ply thickness", "Orientation", "Number of plies"
+  , , , |len_im|, :math:`\circ`,
+  "layup1", 1, "mat_1", 0.05, -15, 6
+
+
+
+
+
+Result
+------
+
+.. table:: Results
+   :name: table_box_result
+   :align: center
+
+   ========================== ==================================== ====================================
+   Component                  Value                                Reference [YU2012]_
+   ========================== ==================================== ====================================
+   :math:`S_{11}` [|stf0_im|] :math:`\phantom{-}1.437 \times 10^6` :math:`\phantom{-}1.437 \times 10^6`
+   :math:`S_{22}` [|stf0_im|] :math:`\phantom{-}9.026 \times 10^4` :math:`\phantom{-}9.027 \times 10^4`
+   :math:`S_{33}` [|stf0_im|] :math:`\phantom{-}3.941 \times 10^4` :math:`\phantom{-}3.943 \times 10^4`
+   :math:`S_{14}` [|stf1_im|] :math:`\phantom{-}1.074 \times 10^5` :math:`\phantom{-}1.074 \times 10^5`
+   :math:`S_{25}` [|stf1_im|] :math:`-5.201 \times 10^4`           :math:`-5.201 \times 10^4`
+   :math:`S_{36}` [|stf1_im|] :math:`-5.635 \times 10^4`           :math:`-5.635 \times 10^4`
+   :math:`S_{44}` [|stf2_im|] :math:`\phantom{-}1.679 \times 10^4` :math:`\phantom{-}1.679 \times 10^4`
+   :math:`S_{55}` [|stf2_im|] :math:`\phantom{-}6.621 \times 10^4` :math:`\phantom{-}6.621 \times 10^4`
+   :math:`S_{66}` [|stf2_im|] :math:`\phantom{-}1.725 \times 10^5` :math:`\phantom{-}1.725 \times 10^5`
+   ========================== ==================================== ====================================
+

_sources/contents/examples/example_channel.rst.txt

@@ -0,0 +1,84 @@
+.. _example-channel:
+
+Channel
+=======
+
+Problem description
+-------------------
+
+.. figure:: /figures/ex_channel_0.png
+  :name: fig_channel0
+  :width: 3in
+  :align: center
+
+  Cross section of the pipe [CHEN2010]_.
+
+This example has a cross section of a highly heterogeneous channel.
+This cross section geometry can be defined as shown in :numref:`Fig. %s <fig_channel0>` [CHEN2010]_.
+The isotropic material properties are given in :numref:`Table %s <table_channel_materials>`.
+The layup is defined having a single layer with the thickness 0.001524 m.
+The result is shown in :numref:`Table %s <table_channel_result>` and compared with those in [CHEN2010]_.
+Complete input files can be found in ``examples\ex_channel\``, including ``channel.xml`` and ``materials.xml``.
+
+.. figure:: /figures/ex_channel_1.png
+  :name: fig_channel1
+  :width: 6in
+  :align: center
+
+  *Base point*\ s, *Base line*\ s and *Segment*\ s of the channel cross section.
+
+.. figure:: /figures/ex_channel_mesh.png
+  :name: fig_channel_mesh
+  :width: 4in
+  :align: center
+
+  Meshed cross section viewed in Gmsh.
+
+.. csv-table:: Material properties
+  :name: table_channel_materials
+  :header-rows: 2
+  :align: center
+
+  "Name", "Type", "Density", |e|, |nu|
+   , , |den_si|, |mod_si_g|,
+  "mtr1", "isotropic", 1068.69, 206.843, 0.49
+
+.. csv-table:: Layups
+  :name: table_channel_layups
+  :header-rows: 2
+  :align: center
+
+  "Name", "Layer", "Material", "Ply thickness", "Orientation", "Number of plies"
+  , , , |len_si|, :math:`\circ`,
+  "layup1", 1, "mtr1", 0.001524, 0, 1
+
+
+
+
+Result
+------
+
+.. table:: Results
+   :name: table_channel_result
+
+   =================================== ======================== =================================== =================================== =================================== ===================================
+   :math:`\phantom{-}1.906\times 10^7` :math:`0.0`              :math:`\phantom{-}0.0`              :math:`\phantom{-}0.0`              :math:`-4.779\times 10^4`           :math:`-1.325\times 10^5`
+   :math:`\phantom{-}0.0`              :math:`2.804\times 10^6` :math:`\phantom{-}2.417\times 10^5` :math:`\phantom{-}2.128\times 10^4` :math:`\phantom{-}0.0`              :math:`\phantom{-}0.0`
+   :math:`\phantom{-}0.0`              :math:`2.417\times 10^5` :math:`\phantom{-}2.146\times 10^6` :math:`-7.663\times 10^3`           :math:`\phantom{-}0.0`              :math:`\phantom{-}0.0`
+   :math:`\phantom{-}0.0`              :math:`2.128\times 10^4` :math:`-7.663\times 10^3`           :math:`\phantom{-}2.091\times 10^2` :math:`\phantom{-}0.0`              :math:`\phantom{-}0.0`
+   :math:`-4.779\times 10^4`           :math:`0.0`              :math:`\phantom{-}0.0`              :math:`\phantom{-}0.0`              :math:`\phantom{-}2.011\times 10^3` :math:`\phantom{-}9.104\times 10^2`
+   :math:`-1.325\times 10^5`           :math:`0.0`              :math:`\phantom{-}0.0`              :math:`\phantom{-}0.0`              :math:`\phantom{-}9.104\times 10^2` :math:`\phantom{-}1.946\times 10^3`
+   =================================== ======================== =================================== =================================== =================================== ===================================
+
+.. table:: Results from reference [CHEN2010]_
+   :name: table_channel_result_ref
+
+   =================================== ======================== =================================== =================================== =================================== ===================================
+   :math:`\phantom{-}1.903\times 10^7` :math:`0.0`              :math:`\phantom{-}0.0`              :math:`\phantom{-}0.0`              :math:`-4.778\times 10^4`           :math:`-1.325\times 10^5`
+   :math:`\phantom{-}0.0`              :math:`2.791\times 10^6` :math:`\phantom{-}2.364\times 10^5` :math:`\phantom{-}2.122\times 10^4` :math:`\phantom{-}0.0`              :math:`\phantom{-}0.0`
+   :math:`\phantom{-}0.0`              :math:`2.364\times 10^5` :math:`\phantom{-}2.137\times 10^6` :math:`-7.679\times 10^3`           :math:`\phantom{-}0.0`              :math:`\phantom{-}0.0`
+   :math:`\phantom{-}0.0`              :math:`2.122\times 10^4` :math:`-7.679\times 10^3`           :math:`\phantom{-}2.086\times 10^2` :math:`\phantom{-}0.0`              :math:`\phantom{-}0.0`
+   :math:`-4.778\times 10^4`           :math:`0.0`              :math:`\phantom{-}0.0`              :math:`\phantom{-}0.0`              :math:`\phantom{-}2.010\times 10^3` :math:`\phantom{-}9.102\times 10^2`
+   :math:`-1.325\times 10^5`           :math:`0.0`              :math:`\phantom{-}0.0`              :math:`\phantom{-}0.0`              :math:`\phantom{-}9.102\times 10^2` :math:`\phantom{-}1.944\times 10^3`
+   =================================== ======================== =================================== =================================== =================================== ===================================
+