(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 5.2'

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  ButtonData:>{
    URL[ "http://www.imtek.uni-freiburg.de/simulation/mor4ansys"], None},
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Cell[TextData[{
  "Author: Evgenii Rudnyi, ",
  ButtonBox["http://Evgenii.Rudnyi.Ru/",
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      URL[ "http://Evgenii.Rudnyi.Ru/"], None},
    ButtonStyle->"Hyperlink"]
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Cell[CellGroupData[{

Cell["Imtek`Post4MOR`: Manual", "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},
    Active->True,
    ButtonStyle->"Hyperlink"],
  ".\nExamples can be found in two separate notebooks, FirstOrderSystem.nb \
and SecondOrderSystem.nb. This notebook contains formal description of \
objects DynamicSystem and SimulationResult as well as simulation an plotting \
functions."
}], "Text"],

Cell[CellGroupData[{

Cell["DynamicSystem", "SubsectionIcon"],

Cell[TextData[{
  "The object represent a system of ordinary differential equations in the \
form as follows\n\n",
  Cell[BoxData[GridBox[{
          {\(dx\/dt = Ax + Bu\)},
          {\(y = Cx + Du\)}
          }]]],
  "\t\t\t\t(1)\n",
  Cell[BoxData[GridBox[{
          {\(E  dx\/dt = Ax + Bu\)},
          {\(y = Cx + Du\)}
          }]]],
  "\t\t\t(2)\n",
  Cell[BoxData[GridBox[{
          {\(E  dx\/dt + Kx = Bu\)},
          {\(y = Cx + Du\)}
          }]]],
  "\t\t\t(3)\n",
  Cell[BoxData[GridBox[{
          {\(M \(\( d\^2\) x\)\/dt\^2 + E  dx\/dt + Kx = Bu\)},
          {\(y = Cx + Du\)}
          }]]],
  "\t\t(4)\n\nThe object contains the system matrices. Matrices M, E and D \
are optional, matrices A(K), B and C must be present. In addition the object \
contains output names for the vector y."
}], "Text"],

Cell[CellGroupData[{

Cell["Constructors", "SubsubsectionIcon"],

Cell[BoxData[
    StyleBox[GridBox[{
          {
            RowBox[{
              
              StyleBox[\(MakeDynamicSystem[matM, matE, matK, matB, matC, 
                  namesC, matD]\),
                "DefinitionBox",
                FontVariations->{"CompatibilityType"->0}], 
              StyleBox[" ",
                "DefinitionBox",
                FontVariations->{"CompatibilityType"->0}], 
              StyleBox["and",
                "DefinitionBox",
                FontVariations->{"CompatibilityType"->0}], 
              StyleBox[" ",
                "DefinitionBox",
                FontVariations->{
                "CompatibilityType"->0}], \(MakeDynamicSystem[matM, matE, 
                matK, matB, matD]\)}], 
            
            StyleBox[\(give\ a\ DynamicSystem\ object . \ matM, matE, 
              and\ matD\ should\ be\ matrices\ or\ Null . \ matK, matB, \ 
              and\ matC\ must\ be\ matrices . \ 
                  namesC\ is\ a\ list\ of\ strings . \ 
                  matD\ is\ optional\ and\ by\ default\ is\ Null . \ 
                  The\ second\ form\ makes\ matC\ equal\ to\ the\ identity\ \
\(\(matrix\)\(.\)\)\),
              "DefinitionBox"]},
          {\(MakeFirstOrderSystem[matA, matB, matC, namesC, matD], \ 
            MakeFirstOrderSystem[matE, matA, matB, matC, namesC, matD], \ 
            and\ MakeFirstOrderSystem[matE, matA, matB, 
                matD]\), \(give\ a\ DynamicSystem\ object\ for\ a\ first\ \
order\ system . \ matE, and\ matD\ should\ be\ matrices\ or\ Null . \ matA, 
            matB, and\ matC\ must\ be\ matrices . 
                namesC\ is\ a\ list\ of\ strings . \ 
                matD\ by\ default\ is\ Null . \ 
                The\ last\ form\ makes\ matC\ equal\ to\ the\ identity\ \
\(\(matrix\)\(.\)\)\)},
          {\(MakeSecondOrderSystem[matM, matK, matB, matC, namesC, matD], \ 
            MakeSecondOrderSystem[matM, matE, matK, matB, matC, namesC, matD] 
              and\ MakeSecondOrderSystem[matM, matE, matK, matB, 
                matD]\), \(give\ a\ DynamicSystem\ object\ for\ a\ second\ \
order\ system . \ matM, matE, \ 
            and\ matD\ should\ be\ matrices\ or\ Null . \ matK, \ matB, 
            and\ matC\ must\ be\ matrices . \ 
                namesC\ is\ a\ list\ of\ strings . \ 
                matD\ is\ optional\ and\ by\ default\ is\ Null . \ 
                The\ last\ form\ makes\ matC\ equal\ to\ the\ identity\ \
\(\(matrix\)\(.\)\)\)}
          },
        GridFrame->True],
      "DefinitionBox"]], "Text"],

Cell["\<\
MakeFirstOrderSystem follows the convention of Eq (1) and (2) while \
MakeDynamicSystem and MakeSecondOrderSystem follow the convention of Eq (3) \
and (4). The difference is in the sign of MatrixK (MatrixA = - MatrixK). 
If  dimensions of different matrices are not compatible, the constructors \
throw an exception.\
\>", "Text"]
}, Closed]],

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            StyleBox[\(gives\ system\ matrix\ D\ of\ \(\(sys\)\(.\)\)\),
              "DefinitionBox"]},
          {\(OutputNames[
              sys]\), \(gives\ a\ list\ of\ output\ names\ of\ \(\(sys\)\(.\)\
\)\)}
          },
        GridFrame->True],
      "DefinitionBox"]], "Text"],

Cell["\<\
If a matrix does not present in the system, a selector returns a \
default matrix: the zero matrix for MatrixD; for a first order system: the \
zero matrix for MatrixM and the unity matrix for MatrixE; for a second order \
system: the unity matrix for MatrixM and the zero matrix for MatrixE.\
\>", \
"Text"]
}, Closed]],

Cell[CellGroupData[{

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          {
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            StyleBox[\(gives\ True\ if\ system\ matrix\ D\ is\ present\ in\ \
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          {\(SecondOrderSystemQ[
              sys]\), \(gives\ True\ if\ sys\ represent\ a\ second\ order\ \
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              sys]\), \(gives\ True\ if\ sys\ represent\ an\ explicit\ first\ \
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          },
        GridFrame->True],
      "DefinitionBox"]], "Text"],

Cell["", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Transformation Functions", "SubsubsectionIcon"],

Cell[BoxData[
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          {\(DeleteOutputs[sys, 
              names]\), \(gives\ a\ new\ system\ when\ outputs\ defined\ in\ \
a\ list\ of\ names\ are\ \(\(deleted\)\(.\)\)\)},
          {\(AddOutputs[sys, matCadd, 
              names]\), \(gives\ a\ new\ system\ with\ additional\ outputs\ \
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          {
            StyleBox[\(DeleteDamping[sys]\),
              "DefinitionBox",
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              "DefinitionBox"]},
          {\(AddDamping[sys, 
              matE]\), \(gives\ a\ new\ system\ with\ matrix\ E\ inreased\ by\
\ matE . \ sys\ must\ be\ a\ second\ order\ \(\(system\)\(.\)\)\)},
          {\(AddDamping[sys, alpha, 
              beta]\), \(gives\ a\ new\ system\ with\ matrix\ E\ increased\ \
by\ Rayleigh\ damping\ alpha*matM + 
              beta*matK . \ 
                  sys\ must\ be\ a\ second\ order\ \(\(system\)\(.\)\)\)},
          {\(AddDampingRatio[sys, 
              imatK]\), \(gives\ a\ new\ system\ with\ matrix\ K\ inreased\ \
by\ I\ imatK . \ sys\ must\ be\ a\ second\ order\ \(\(system\)\(.\)\)\)},
          {\(AddDampingRatio[sys, 
              dmprat]\), \(gives\ a\ new\ system\ with\ matrix\ K\ increased\ \
by\ I\ 2\ dmprat\ MatrixK . \ 
                sys\ must\ be\ a\ second\ order\ \(\(system\)\(.\)\)\)},
          {\(ToFirstOrderSystem[
              sys]\), \(gives\ a\ new\ system\ in\ the\ form\ of\ a\ first\ \
order\ \(\(system\)\(.\)\)\)},
          {\(ToExplicitSystem[
              sys]\), \(gives\ a\ new\ system\ in\ the\ form\ of\ an\ \
explicit\ first\ order\ \(\(system\)\(.\)\)\)},
          {\(TakeSystem[sys, 
              dim]\), \(gives\ a\ new\ system\ of\ dimension\ dim . \ 
                This\ operation\ makes\ sense\ only\ when\ an\ original\ \
system\ has\ been\ obtained\ by\ iterative\ model\ reduction\ \(\(method\)\(.\
\)\)\)}
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name\ \(\(baseName\)\(.\)\)\),
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          {\(WriteSystem[sys, 
              baseName]\), \(writes\ a\ system\ to\ files\ with\ the\ base\ \
name\ \(\(baseName\)\(.\)\)\)},
          {\(ReadSystem[file, 
              type]\), \(reads\ a\ system\ from\ a\ file\ written\ in\ the\ \
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    ButtonData:>{
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  "."
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Cell["\<\
The object represent results of transient or harmonic simulation \
for DynamicSystem. We have an X-axis (time or frequency) and results for all \
outputs for an Y-axis. As such, there are four components: XSeries, XName, \
YSeries, YNames. XSeries is a vector and YSeries is a matrix. Names are text \
strings and they allow us to decide if two objects are compatible with each \
other. \
\>", "Text"],

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\(\(strings\)\(.\)\)\),
              "DefinitionBox"]}
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      "DefinitionBox"]], "Text"],

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If  dimensions of different components are not compatible, the \
constructor throws an exception.\
\>", "Text"]
}, Closed]],

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          {\(YNames[res]\), \(gives\ YNames\ of\ \(\(res\)\(.\)\)\)}
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The criterion for compatibility is that all names are the same and \
in the same order.\
\>", "Text"]
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Cell["\<\
You may want to reduce the number of YSeries before plotting or to \
re-sort them. To this end, use TransformResult. TransformFunction is an \
option to define a function fun[y, x]. If it is a single function, it is \
applied to all YSeries. Alternatively, one can specify a list of pairs {name, \
fun}.
Another operation is to find a difference between different results for a \
difference plot. You define what the difference means by using an option \
ErrorFunction. In the case, if XSeries are different, Difference uses \
Interpolation. You can specify its options as additional arguments to \
Difference.\
\>", "Text"]
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Cell["\<\
PlotResult is a front end to MultipleListPlot to make a series of \
plots. Each plot contains a single output for all SimulationResult in \
listres.
Use Difference to make a difference plot.
Use options FunctionY and FunctionX to modify values before plotting. By \
default, they are equal to Identity.
Use MultipleListPlot options to make your plot look nicer. \
\>", "Text"]
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                file, \*"\""Table\)\*"\""]\), \(writes\ res\ to\ file\ as\ \
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                This\ is\ convenient\ if\ you\ would\ like\ to\ use\ another\ \
plotting\ \(\(software\)\(.\)\)\)},
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              XName]\), \(reads\ SimulationResult\ from\ file\ in\ my\ old\ \
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Cell["\<\
An external represenation of SimulationResult is based on the \
MatrixMarket format. Additionally, file file.names contains XName and YNames.\
\
\>", "Text"]
}, Closed]]
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              ops]\), \(gives\ SimulationResult\ for\ a\ given\ list\ of\ \
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          {\(AnsysTransientSolution[time, sys, 
              ops]\), \(gives\ SimulationResult\ for\ a\ given\ list\ time\ \
by\ using\ integrators\ like\ in\ \(ANSYS : \ 
                backward\ Euler\ for\ a\ first\ order\ system\ and\ Newmark\ \
for\ a\ second\ order\ system . \ \(Options : \ InputFunction\)\), 
            InitialState, \ 
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                TINTP \[Rule] {gamma, alpha, delta, 
                  theta}\ allows\ us\ to\ change\ integrator\ behavior\ \
\(\((see\ ANSYS\ manual)\)\(.\)\)\)},
          {\(TransientSolution[time, sys, 
              ops]\), \(gives\ SimulationResult\ for\ a\ given\ list\ time . \
\ \(Options : \ Verbose\), InputFunction, \ InitialState, 
            NDSolve\ options . \ 
                It\ uses\ different\ approaches\ for\ different\ types\ of\ \
DynamicSystem . \ Use\ ToFirstOrderSystem\ or\ ToExplicitSystem, 
            if\ TransienSolution\ \(\(fails\)\(.\)\)\)},
          {\(FrequencyConvergence[freq, sys, start, finish] 
              or\ FrequencyConvergence[freq, sys, start, finish, 
                step]\), \(gives\ an\ object\ for\ LocalError\ and\ \
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finish\ for\ frequency\ \(\(freq\)\(.\)\)\)},
          {\(LocalError[conv, true, 
              ops]\), \(gives\ a\ list\ of\ local\ errors\ computed\ from\ \
the\ object\ conv\ obtained\ by\ FrequencyConvergence\ and\ a\ list\ of\ true\
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                Option\ is\ ErrorFunction\ \(\((by\ default\ \
Log10RelativeError)\)\(.\)\)\)},
          {\(LocalErrorIndicator[conv, 
              ops]\), \(gives\ a\ list\ of\ local\ error\ estimates\ computed\
\ from\ the\ object\ conv\ obtained\ by\ FrequencyConvergence . \ 
                
                Option\ is\ ErrorFunction\ \(\((by\ default\ \
Log10RelativeError)\)\(.\)\)\)}
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Cell[TextData[StyleBox["HarmonicSolution sets XName to \"Frequency\". \
TransientSolution sets XName to \"Time\".\nBy default, an input function is a \
unit step function for each input. You can modify this, by using an option \
InputFunction. For HarmonicSolution, InputFunction must be a list of \
numerical values. For TransientSolution, InputFunction can be a list of \
numerical values or a list of functions in two arguments: time and a state \
vector. InitialState by default is zero. \nHarmonicSolution uses LinearSolve \
and you can use its options as additional arguments. TransientSolution uses \
NDSolve and you can use its options. AnsysTransientSolution implements \
integrators from ANSYS, use TINTP to set integrator constants.",
  FontVariations->{"CompatibilityType"->0}]], "Text",
  CellDingbat->None],

Cell["\<\
NDSolve in TransientSolution uses only the last value in list time. \
It chooses integration points by itself adaptively. Use Verbose->True to see \
how many timesteps NDSolve has made. AnsysTransientSolution makes timesteps \
according to list time. As a result, AnsysTransientSolution is much faster \
but TransientSolution gives more accurate result.\
\>", "Text"],

Cell[TextData[{
  "Functions LocalError and LocalErrorIndicator can be used to choose an \
optimal dimension for the reduced model ",
  "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"],
  "."
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