(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 5.1'

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Cell[TextData[{
  "Author: Evgenii Rudnyi, ",
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Cell[CellGroupData[{

Cell["Imtek`Post4MOR`: 2nd Order System Example", "Section",
  CellDingbat->"\[FilledSquare]"],

Cell[TextData[{
  "The goal of the package is to make transient or harmonic simulations for a \
system of ordinary differential equations of the first or second order. I \
have written it to support simulation for a reduced model obtained by ",
  ButtonBox["mor4ansys",
    ButtonData:>{
      URL[ "http://www.imtek.uni-freiburg.de/simulation/mor4ansys"], None},
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    ButtonStyle->"Hyperlink"],
  ".\nThis notebook contains an example for a second order system. The manual \
is in Post4MORmanual.nb.\nWe will consider ",
  ButtonBox["Microgyroscope Benchmark",
    ButtonData:>{
      URL[ 
      "http://www.imtek.de/simulation/benchmark/index.php?li=30&nr=35889"], 
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    ButtonNote->
      "http://www.imtek.de/simulation/benchmark/index.php?li=30&nr=35889"],
  "."
}], "Text"],

Cell[CellGroupData[{

Cell["Setting Up Directory and Loading Functions", "SubsectionIcon"],

Cell["You may need to change the directory name.", "Text"],

Cell[BoxData[
    \(SetDirectory["\<.\>"]\)], "Input",
  CellLabel->"In[1]:="],

Cell["\<\
Loading functions (if file is in another directory, use the fule \
file name).\
\>", "Text"],

Cell[BoxData[
    \(<< Post4MOR.m\)], "Input",
  CellLabel->"In[2]:="],

Cell[TextData[{
  "An alternative way to load functions directly from my site (provided you \
are connected to Internet and starting ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " 5.1)."
}], "Text"],

Cell[BoxData[
    \(ToExpression[
      Import["\<http://Evgenii.Rudnyi.Ru/soft/Post4MOR/Post4MOR.m\>", "\<Text\
\>"]]\)], "Input"],

Cell["Path to examples (you may need to change it.", "Text"],

Cell[BoxData[
    \(\(filePath\  = \ ToFileName[{\ "\<ex\>"\ }];\)\)], "Input",
  CellLabel->"In[3]:="],

Cell["Read a reduced model", "Text"],

Cell[BoxData[
    \(sys\  = \ ReadSystem[\ filePath\  <> \ "\<gyro\>"\ \ ]\)], "Input",
  CellLabel->"In[4]:="]
}, Open  ]],

Cell[CellGroupData[{

Cell["Transient Simulation", "SubsectionIcon"],

Cell["Read in ANSYS results", "Text"],

Cell[BoxData[
    \(transient\  = \ ReadResult[filePath <> "\<gyro.transient\>"]\)], "Input",\

  CellLabel->"In[5]:="],

Cell["Add damping", "Text"],

Cell[BoxData[
    \(sysDamped\  = \ AddDamping[sys, 0. , 1*^-6]\)], "Input",
  CellLabel->"In[6]:="],

Cell[CellGroupData[{

Cell["Using NDSolve with TransientSolution Function", "SubsubsectionIcon"],

Cell["\<\
Make transient simulation for the reduced model of dimension \
10\
\>", "Text"],

Cell[BoxData[
    \(res10\  = \ 
      TransientSolution[XSeries[transient], TakeSystem[sysDamped, 10], 
        Verbose \[Rule] True]\)], "Input",
  CellLabel->"In[7]:="],

Cell["\<\
Make transient simulation for the reduced model of dimension \
5\
\>", "Text"],

Cell[BoxData[
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  CellLabel->"In[9]:="]
}, Open  ]],

Cell[CellGroupData[{

Cell["Using Ansys Integrator with AnsysTransientSolution Function", \
"SubsubsectionIcon"],

Cell["\<\
Make transient simulation for the reduced model of dimension \
10\
\>", "Text"],

Cell[BoxData[
    \(res10\  = \ 
      AnsysTransientSolution[XSeries[transient], 
        TakeSystem[sysDamped, 10]]\)], "Input",
  CellLabel->"In[10]:="],

Cell["\<\
Make transient simulation for the reduced model of dimension \
5\
\>", "Text"],

Cell[BoxData[
    \(res5\  = \ 
      AnsysTransientSolution[XSeries[transient], 
        TakeSystem[sysDamped, 5]]\)], "Input",
  CellLabel->"In[11]:="],

Cell["Make plot (ansys is red, res5 is green, res10 is blue)", "Text"],

Cell[BoxData[
    \(\(PlotResult[{transient, res5, res10}, 
        PlotStyle \[Rule] {RGBColor[1, 0, 0], RGBColor[0, 1, 0], 
            RGBColor[0, 0, 1]}];\)\)], "Input",
  CellLabel->"In[12]:="]
}, Open  ]],

Cell[CellGroupData[{

Cell["Discussion", "SubsubsectionIcon"],

Cell["\<\
You can see that AnsysTransientSolution is much faster and gives \
better agreement with ANSYS results. However, remember that results produced \
by TransientSolution is much accurate! AnsysTransientSolution makes the same \
integration errors as ANSYS and what you see is cancelation of error. As a \
result, it makes sense to use TransientSolution to check if the number of \
integration points chosen in ANSYS was good enough.\
\>", "Text"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Harmonic Simulation", "SubsectionIcon"],

Cell["Read ANSYS results", "Text"],

Cell[BoxData[
    \(harmonic\  = \ ReadResult[filePath <> "\<gyro.harmonic\>"]\)], "Input",
  CellLabel->"In[13]:="],

Cell["Add damping", "Text"],

Cell[BoxData[
    \(sysDamped\  = \ AddDamping[sys, 0. , 1*^-7]\)], "Input",
  CellLabel->"In[14]:="],

Cell["\<\
Make transient simulation for the reduced model of dimension \
30\
\>", "Text"],

Cell[BoxData[
    \(res30\  = \ 
      HarmonicSolution[XSeries[harmonic], 
        TakeSystem[sysDamped, 30]]\)], "Input",
  CellLabel->"In[15]:="],

Cell["\<\
Make transient simulation for the reduced model of dimension \
20\
\>", "Text"],

Cell[BoxData[
    \(res20\  = \ 
      HarmonicSolution[XSeries[harmonic], 
        TakeSystem[sysDamped, 20]]\)], "Input",
  CellLabel->"In[16]:="],

Cell["\<\
Make transient simulation for the reduced model of dimension \
10\
\>", "Text"],

Cell[BoxData[
    \(res10\  = \ 
      HarmonicSolution[XSeries[harmonic], 
        TakeSystem[sysDamped, 10]]\)], "Input",
  CellLabel->"In[17]:="],

Cell["Generate plots", "Text"],

Cell[BoxData[
    \(\(PlotResult[{harmonic, res10, res20, res30}, \ 
        PlotStyle \[Rule] {RGBColor[1, 0, 0], RGBColor[0, 1, 0], 
            RGBColor[0, 0, 1], RGBColor[0, 1, 1]}, \ 
        FunctionX \[Rule] \((Log10[#] &)\), 
        FunctionY \[Rule] \((Log10[Abs[#]] &)\)];\)\)], "Input",
  CellLabel->"In[18]:="]
}, Open  ]],

Cell[CellGroupData[{

Cell["Local Error Indicator", "SubsectionIcon"],

Cell[TextData[{
  "We can estimate local error and choose an optimal dimension of the reduced \
system as described in paper ",
  ButtonBox["Error indicators for fully automatic extraction of heat-transfer \
macromodels for MEMS",
    ButtonData:>{
      URL[ "http://stacks.iop.org/0960-1317/15/430"], None},
    ButtonStyle->"Hyperlink"],
  "."
}], "Text"],

Cell[TextData[{
  "Compute an error indicator for ",
  Cell[BoxData[
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  ", ",
  Cell[BoxData[
      \(TraditionalForm\`10\^4\)]],
  ", and ",
  Cell[BoxData[
      \(TraditionalForm\`10\^5\)]],
  " Hz."
}], "Text"],

Cell[BoxData[
    \(\(er1 = 
        LocalErrorIndicator[
          FrequencyConvergence[1*^3, sysDamped, 1, 30]];\)\)], "Input",
  CellLabel->"In[19]:="],

Cell[BoxData[
    \(\(er2 = 
        LocalErrorIndicator[
          FrequencyConvergence[1*^4, sysDamped, 1, 30]];\)\)], "Input",
  CellLabel->"In[20]:="],

Cell[BoxData[
    \(\(er3 = 
        LocalErrorIndicator[
          FrequencyConvergence[1*^5, sysDamped, 1, 30]];\)\)], "Input",
  CellLabel->"In[21]:="],

Cell["Make a plot Log10[Relative Error] vs. system dimension", "Text"],

Cell[BoxData[
    \(\(MultipleListPlot[{er1, er2, er3}, 
        SymbolShape \[Rule] {PlotSymbol[Triangle, Filled \[Rule] False], 
            PlotSymbol[Box, Filled \[Rule] False], 
            PlotSymbol[Diamond, Filled \[Rule] False]}, 
        SymbolStyle \[Rule] {RGBColor[1, 0, 0], RGBColor[0, 1, 0], 
            RGBColor[0, 0, 1]}, 
        PlotLegend \[Rule] {"\<1e3 Hz\>", "\<1e4 Hz\>", "\<1e5 Hz\>"}, 
        LegendBorder \[Rule] {}, LegendPosition \[Rule] {0.45, \(-0.3\)}, 
        LegendSize \[Rule] {0.8, 0.28}, 
        Evaluate[Imtek`Post4MOR`Private`defaultPlotOptions]];\)\)], "Input",
  CellLabel->"In[22]:="],

Cell["\<\
Now you can choose the dimension of the reduced system based on \
your requirements. 
As we have ANSYS results, we can check if error estimates are accurate \
enough. We do it for the next frequency\
\>", "Text"],

Cell[BoxData[
    \(freq\  = \(XSeries[harmonic]\)[\([50]\)]\)], "Input",
  CellLabel->"In[23]:="],

Cell[BoxData[
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