(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 51266, 2603]*) (*NotebookOutlinePosition[ 51932, 2626]*) (* CellTagsIndexPosition[ 51888, 2622]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Formulation of finite element matrices for a 2d elastic \ isoparametric 4-node quadrilateral element - Eigenvalue analysis of stiffness \ matrix (energy modes)\ \>", "Title", FontSize->36], Cell[BoxData[ \(Clear["\"]; Off[General::spell, General::spell1];\)], "Input",\ FontSize->36], Cell[CellGroupData[{ Cell["Shape functions and derivatives", "Subtitle", FontSize->36], Cell[BoxData[{ \(\(\[Phi]\_1\ = \ \(1\/4\) \((1 + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\);\)\), \ "\[IndentingNewLine]", \(\(\[Phi]\_2\ = \ \(1\/4\) \((1 - \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\);\)\), \ "\[IndentingNewLine]", \(\(\[Phi]\_3\ = \ \(1\/4\) \((1 - \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\);\)\), \ "\[IndentingNewLine]", \(\(\[Phi]\_4\ = \ \(1\/4\) \((1 + \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\);\)\), \ "\[IndentingNewLine]", \(\(d\[Phi]1\[Xi]1\ = \ D[\[Phi]\_1, \[Xi]\_1];\)\), "\[IndentingNewLine]", \(\(d\[Phi]1\[Xi]2\ = \ D[\[Phi]\_1, \[Xi]\_2];\)\), "\[IndentingNewLine]", \(\(d\[Phi]2\[Xi]1\ = \ D[\[Phi]\_2, \[Xi]\_1];\)\), "\[IndentingNewLine]", \(\(d\[Phi]2\[Xi]2\ = \ D[\[Phi]\_2, \[Xi]\_2];\)\), "\[IndentingNewLine]", \(\(d\[Phi]3\[Xi]1\ = \ D[\[Phi]\_3, \[Xi]\_1];\)\), "\[IndentingNewLine]", \(\(d\[Phi]3\[Xi]2\ = \ D[\[Phi]\_3, \[Xi]\_2];\)\), "\[IndentingNewLine]", \(\(d\[Phi]4\[Xi]1\ = \ D[\[Phi]\_4, \[Xi]\_1];\)\), "\[IndentingNewLine]", \(\(d\[Phi]4\[Xi]2\ = \ D[\[Phi]\_4, \[Xi]\_2];\)\)}], "Input", FontSize->36] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(h\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[BoxData[ \(h\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Element displacement interpolation matrix", "Subtitle", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"H", " ", "=", " ", RowBox[{"(", GridBox[{ {\(\[Phi]\_1\), "0", \(\[Phi]\_2\), "0", \(\[Phi]\_3\), "0", \(\[Phi]\_4\), "0"}, {"0", \(\[Phi]\_1\), "0", \(\[Phi]\_2\), "0", \(\[Phi]\_3\), "0", \(\[Phi]\_4\)} }], ")"}]}], ";"}], "\[IndentingNewLine]", \(MatrixForm[ Transpose[H]]\)}], "Input", FontSize->36], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/4\ \((1 + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\), "0"}, {"0", \(1\/4\ \((1 + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\)}, {\(1\/4\ \((1 - \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\), "0"}, {"0", \(1\/4\ \((1 - \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\)}, {\(1\/4\ \((1 - \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\)\), "0"}, {"0", \(1\/4\ \((1 - \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\)\)}, {\(1\/4\ \((1 + \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\)\), "0"}, {"0", \(1\/4\ \((1 + \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Element nodal displacement vector and dispacement \ interpolation\ \>", "Subtitle", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[{ \(Un\ = \ Table[U\_\(a, i\), {a, 4}, {i, 2}] // Flatten\), "\n", \(ue\ = \ H . Un // Simplify\)}], "Input", FontSize->36], Cell[BoxData[ \({U\_\(1, 1\), U\_\(1, 2\), U\_\(2, 1\), U\_\(2, 2\), U\_\(3, 1\), U\_\(3, 2\), U\_\(4, 1\), U\_\(4, 2\)}\)], "Output"], Cell[BoxData[ \({1\/4\ \((\((1 + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\ U\_\(1, 1\) - \((\(-1\ \) + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\ U\_\(2, 1\) + \((\(-1\) + \[Xi]\_2)\)\ \ \((\((\(-1\) + \[Xi]\_1)\)\ U\_\(3, 1\) - \((1 + \[Xi]\_1)\)\ U\_\(4, 1\))\))\ \), 1\/4\ \((\((1 + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\ U\_\(1, 2\) - \((\(-1\) \ + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\ U\_\(2, 2\) + \((\(-1\) + \[Xi]\_2)\)\ \((\ \((\(-1\) + \[Xi]\_1)\)\ U\_\(3, 2\) - \((1 + \[Xi]\_1)\)\ U\_\(4, 2\))\))\)}\ \)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Nodal coordinates and element coordinates interpolation (mapping)\ \ \>", "Subtitle", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[{ \(Xn\ = \ Table[X\_\(a, i\), {a, 4}, {i, 2}] // Flatten\), "\[IndentingNewLine]", \(xe\ = \ H . Xn\)}], "Input", FontSize->36], Cell[BoxData[ \({X\_\(1, 1\), X\_\(1, 2\), X\_\(2, 1\), X\_\(2, 2\), X\_\(3, 1\), X\_\(3, 2\), X\_\(4, 1\), X\_\(4, 2\)}\)], "Output"], Cell[BoxData[ \({1\/4\ \((1 + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\ X\_\(1, 1\) + 1\/4\ \((1 - \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\ X\_\(2, 1\) + 1\/4\ \((1 - \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\)\ X\_\(3, 1\) + 1\/4\ \((1 + \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\)\ X\_\(4, 1\), 1\/4\ \((1 + \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\ X\_\(1, 2\) + 1\/4\ \((1 - \[Xi]\_1)\)\ \((1 + \[Xi]\_2)\)\ X\_\(2, 2\) + 1\/4\ \((1 - \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\)\ X\_\(3, 2\) + 1\/4\ \((1 + \[Xi]\_1)\)\ \((1 - \[Xi]\_2)\)\ X\_\(4, 2\)}\)], \ "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Jacobian", "Subtitle", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"J", " ", "=", " ", RowBox[{"(", GridBox[{ {\(\[PartialD]\_\(\[Xi]\_1\)xe[\([1]\)]\), \(\[PartialD]\_\(\ \[Xi]\_1\)xe[\([2]\)]\)}, {\(\[PartialD]\_\(\[Xi]\_2\)xe[\([1]\)]\), \(\[PartialD]\_\(\ \[Xi]\_2\)xe[\([2]\)]\)} }], ")"}]}], ";"}], "\[IndentingNewLine]", \(MatrixForm[ Simplify[Expand[J]]]\)}], "Input", FontSize->36], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/4\ \((\((1 + \[Xi]\_2)\)\ X\_\(1, 1\) - \((1 + \[Xi]\_2)\)\ \ X\_\(2, 1\) + \((\(-1\) + \[Xi]\_2)\)\ \((X\_\(3, 1\) - X\_\(4, 1\))\))\)\), \(1\/4\ \((\((1 + \[Xi]\_2)\)\ \ X\_\(1, 2\) - \((1 + \[Xi]\_2)\)\ X\_\(2, 2\) + \((\(-1\) + \[Xi]\_2)\)\ \((X\ \_\(3, 2\) - X\_\(4, 2\))\))\)\)}, {\(1\/4\ \((\((1 + \[Xi]\_1)\)\ X\_\(1, 1\) + X\_\(2, 1\) - \[Xi]\_1\ X\_\(2, 1\) - X\_\(3, 1\) + \[Xi]\_1\ X\_\(3, 1\) - X\_\(4, 1\) - \[Xi]\_1\ X\_\(4, 1\))\)\), \(1\/4\ \((\((1 \ + \[Xi]\_1)\)\ X\_\(1, 2\) + X\_\(2, 2\) - \[Xi]\_1\ X\_\(2, 2\) - X\_\(3, 2\) + \[Xi]\_1\ X\_\(3, 2\) - X\_\(4, 2\) - \[Xi]\_1\ X\_\(4, 2\))\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{\(Jinv\ \ = \ Inverse[J] // Simplify;\), "\[IndentingNewLine]", \(detJ\ = \ Det[J] // Simplify\), "\[IndentingNewLine]", RowBox[{ RowBox[{"A", " ", "=", " ", RowBox[{\(1\/detJ\), RowBox[{"(", GridBox[{ {\(\(J[\([2]\)]\)[\([2]\)]\), \(-\(J[\([1]\)]\)[\([2]\)]\), "0", "0"}, {"0", "0", \(-\(J[\([2]\)]\)[\([1]\)]\), \ \(\(J[\([1]\)]\)[\([1]\)]\)}, {\(-\(J[\([2]\)]\)[\([1]\)]\), \(\(J[\([1]\)]\)[\([1]\)]\), \ \(\(J[\([2]\)]\)[\([2]\)]\), \(-\(J[\([1]\)]\)[\([2]\)]\)} }], ")"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"G", " ", "=", " ", RowBox[{"(", GridBox[{ {"d\[Phi]1\[Xi]1", "0", "d\[Phi]2\[Xi]1", "0", "d\[Phi]3\[Xi]1", "0", "d\[Phi]4\[Xi]1", "0"}, {"d\[Phi]1\[Xi]2", "0", "d\[Phi]2\[Xi]2", "0", "d\[Phi]3\[Xi]2", "0", "d\[Phi]4\[Xi]2", "0"}, {"0", "d\[Phi]1\[Xi]1", "0", "d\[Phi]2\[Xi]1", "0", "d\[Phi]3\[Xi]1", "0", "d\[Phi]4\[Xi]1"}, {"0", "d\[Phi]1\[Xi]2", "0", "d\[Phi]2\[Xi]2", "0", "d\[Phi]3\[Xi]2", "0", "d\[Phi]4\[Xi]2"} }], ")"}]}]}], "Input", FontSize->36], Cell[BoxData[ \(1\/8\ \((\(-X\_\(2, 2\)\)\ X\_\(3, 1\) + \[Xi]\_1\ X\_\(2, 2\)\ X\_\(3, \ 1\) + X\_\(2, 1\)\ X\_\(3, 2\) - \[Xi]\_1\ X\_\(2, 1\)\ X\_\(3, 2\) - \ \[Xi]\_1\ X\_\(2, 2\)\ X\_\(4, 1\) - \[Xi]\_2\ X\_\(2, 2\)\ X\_\(4, 1\) - X\_\(3, 2\)\ X\_\(4, 1\) + \[Xi]\_2\ X\_\(3, 2\)\ X\_\(4, 1\) + X\_\(1, 2\)\ \((\(-\((1 + \[Xi]\_2)\)\)\ X\_\(2, 1\) + \[Xi]\_2\ \ X\_\(3, 1\) + X\_\(4, 1\) + \[Xi]\_1\ \((\(-X\_\(3, 1\)\) + X\_\(4, 1\))\))\) + \[Xi]\_1\ X\_\(2, 1\)\ X\_\(4, 2\) \ + \[Xi]\_2\ X\_\(2, 1\)\ X\_\(4, 2\) + X\_\(3, 1\)\ X\_\(4, 2\) - \[Xi]\_2\ X\_\(3, 1\)\ X\_\(4, 2\) + X\_\(1, 1\)\ \((\((1 + \[Xi]\_2)\)\ X\_\(2, 2\) + \[Xi]\_1\ X\_\(3, \ 2\) - \[Xi]\_2\ X\_\(3, 2\) - X\_\(4, 2\) - \[Xi]\_1\ X\_\(4, 2\))\))\)\)], "Output"], Cell[BoxData[ \({{1\/4\ \((1 + \[Xi]\_2)\), 0, 1\/4\ \((\(-1\) - \[Xi]\_2)\), 0, 1\/4\ \((\(-1\) + \[Xi]\_2)\), 0, 1\/4\ \((1 - \[Xi]\_2)\), 0}, {1\/4\ \((1 + \[Xi]\_1)\), 0, 1\/4\ \((1 - \[Xi]\_1)\), 0, 1\/4\ \((\(-1\) + \[Xi]\_1)\), 0, 1\/4\ \((\(-1\) - \[Xi]\_1)\), 0}, {0, 1\/4\ \((1 + \[Xi]\_2)\), 0, 1\/4\ \((\(-1\) - \[Xi]\_2)\), 0, 1\/4\ \((\(-1\) + \[Xi]\_2)\), 0, 1\/4\ \((1 - \[Xi]\_2)\)}, {0, 1\/4\ \((1 + \[Xi]\_1)\), 0, 1\/4\ \((1 - \[Xi]\_1)\), 0, 1\/4\ \((\(-1\) + \[Xi]\_1)\), 0, 1\/4\ \((\(-1\) - \[Xi]\_1)\)}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Element strain-operator matrix ", "Subtitle", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[{ \(\(B\ = \ A . G // Simplify;\)\), "\[IndentingNewLine]", \(Dimensions[B]\)}], "Input", FontSize->36], Cell[BoxData[ \({3, 8}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(values\ = \ {X\_\(1, 1\) \[Rule] 1, \ X\_\(1, 2\) \[Rule] 1, \ X\_\(2, 1\) \[Rule] \(-2\), \ X\_\(2, 2\) \[Rule] 1, \ X\_\(3, 1\) \[Rule] \(-1\), \ X\_\(3, 2\) \[Rule] \(-1\), \ X\_\(4, 1\) \[Rule] 1, \ X\_\(4, 2\) \[Rule] \(-1\)}\), "\[IndentingNewLine]", \(\(J\ /. \ values\ // \ Simplify\) // MatrixForm\), "\[IndentingNewLine]", \(\(B\ /. \ values // Simplify\) // MatrixForm\)}], "Input", FontSize->36], Cell[BoxData[ \({X\_\(1, 1\) \[Rule] 1, X\_\(1, 2\) \[Rule] 1, X\_\(2, 1\) \[Rule] \(-2\), X\_\(2, 2\) \[Rule] 1, X\_\(3, 1\) \[Rule] \(-1\), X\_\(3, 2\) \[Rule] \(-1\), X\_\(4, 1\) \[Rule] 1, X\_\(4, 2\) \[Rule] \(-1\)}\)], "Output"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/4\ \((5 + \[Xi]\_2)\)\), "0"}, {\(1\/4\ \((\(-1\) + \[Xi]\_1)\)\), "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(1 + \[Xi]\_2\)\/\(5 + \[Xi]\_2\)\), "0", \(-\(\(1 + \[Xi]\_2\)\/\(5 + \[Xi]\_2\)\)\), "0", \(\(\(-1\) + \[Xi]\_2\)\/\(5 + \[Xi]\_2\)\), "0", \(\(1 - \[Xi]\_2\)\/\(5 + \[Xi]\_2\)\), "0"}, {"0", \(\(3 + 2\ \[Xi]\_1 + \[Xi]\_2\)\/\(10 + 2\ \[Xi]\_2\)\), "0", \(\(1 - \[Xi]\_1\)\/\(5 + \[Xi]\_2\)\), "0", \(\(3\ \((\(-1\) + \[Xi]\_1)\)\)\/\(2\ \((5 + \ \[Xi]\_2)\)\)\), "0", \(-\(\(2 + 3\ \[Xi]\_1 + \[Xi]\_2\)\/\(2\ \((5 + \ \[Xi]\_2)\)\)\)\)}, {\(\(3 + 2\ \[Xi]\_1 + \[Xi]\_2\)\/\(10 + 2\ \[Xi]\_2\)\), \(\(1 + \[Xi]\_2\)\/\(5 + \[Xi]\_2\)\), \ \(\(1 - \[Xi]\_1\)\/\(5 + \[Xi]\_2\)\), \(-\(\(1 + \[Xi]\_2\)\/\(5 + \[Xi]\_2\ \)\)\), \(\(3\ \((\(-1\) + \[Xi]\_1)\)\)\/\(2\ \((5 + \[Xi]\_2)\)\)\), \ \(\(\(-1\) + \[Xi]\_2\)\/\(5 + \[Xi]\_2\)\), \(-\(\(2 + 3\ \[Xi]\_1 + \[Xi]\_2\)\/\(2\ \((5 + \ \[Xi]\_2)\)\)\)\), \(\(1 - \[Xi]\_2\)\/\(5 + \[Xi]\_2\)\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Stiffness Matrix", "Subtitle", FontSize->36], Cell[CellGroupData[{ Cell["Elastic moduli (Plane strain)", "Subtitle", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"CC", " ", "=", " ", RowBox[{\(EE\/\((1 - \[Nu]\^2)\)\), RowBox[{"(", GridBox[{ {"1", "\[Nu]", "0"}, {"\[Nu]", "1", "0"}, {"0", "0", \(\((1 - \[Nu])\)\/2\)} }], ")"}]}]}]], "Input", FontSize->36], Cell[BoxData[ \({{EE\/\(1 - \[Nu]\^2\), \(EE\ \[Nu]\)\/\(1 - \[Nu]\^2\), 0}, {\(EE\ \[Nu]\)\/\(1 - \[Nu]\^2\), EE\/\(1 - \[Nu]\^2\), 0}, {0, 0, \(EE\ \((1 - \[Nu])\)\)\/\(2\ \((1 - \[Nu]\^2)\)\)}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(values\ = \ {X\_\(1, 1\) \[Rule] 1, \ X\_\(1, 2\) \[Rule] 1, \ X\_\(2, 1\) \[Rule] \(-1\), \ X\_\(2, 2\) \[Rule] 1, \ X\_\(3, 1\) \[Rule] \(-1\), \ X\_\(3, 2\) \[Rule] \(-1\), \ X\_\(4, 1\) \[Rule] 1, \ X\_\(4, 2\) \[Rule] \(-1\)}\), "\[IndentingNewLine]", \(\(integrand\ = \ detJ\ Transpose[B] . CC . B\ \ /. \ values // Simplify;\)\), "\[IndentingNewLine]", \(K\ = \ Integrate[integrand\ , {\[Xi]\_1, \(-1\), 1}, {\[Xi]\_2, \(-1\), 1}] // Simplify\)}], "Input", FontSize->36], Cell[BoxData[ \({X\_\(1, 1\) \[Rule] 1, X\_\(1, 2\) \[Rule] 1, X\_\(2, 1\) \[Rule] \(-1\), X\_\(2, 2\) \[Rule] 1, X\_\(3, 1\) \[Rule] \(-1\), X\_\(3, 2\) \[Rule] \(-1\), X\_\(4, 1\) \[Rule] 1, X\_\(4, 2\) \[Rule] \(-1\)}\)], "Output"], Cell[BoxData[ \({{\(EE\ \((\(-3\) + \[Nu])\)\)\/\(6\ \((\(-1\) + \[Nu]\^2)\)\), EE\/\(8 - 8\ \[Nu]\), \(EE\ \((3 + \[Nu])\)\)\/\(12\ \((\(-1\) + \ \[Nu]\^2)\)\), \(EE - 3\ EE\ \[Nu]\)\/\(\(-8\) + 8\ \[Nu]\^2\), \(-\(\(EE\ \ \((\(-3\) + \[Nu])\)\)\/\(12\ \((\(-1\) + \[Nu]\^2)\)\)\)\), EE\/\(\(-8\) + 8\ \[Nu]\), \(EE\ \[Nu]\)\/\(6 - 6\ \[Nu]\^2\), \(EE - \ 3\ EE\ \[Nu]\)\/\(8 - 8\ \[Nu]\^2\)}, {EE\/\(8 - 8\ \[Nu]\), \(EE\ \((\(-3\) \ + \[Nu])\)\)\/\(6\ \((\(-1\) + \[Nu]\^2)\)\), \(EE - 3\ EE\ \[Nu]\)\/\(8 - 8\ \ \[Nu]\^2\), \(EE\ \[Nu]\)\/\(6 - 6\ \[Nu]\^2\), EE\/\(\(-8\) + 8\ \[Nu]\), \(-\(\(EE\ \((\(-3\) + \[Nu])\)\)\/\(12\ \ \((\(-1\) + \[Nu]\^2)\)\)\)\), \(EE - 3\ EE\ \[Nu]\)\/\(\(-8\) + 8\ \ \[Nu]\^2\), \(EE\ \((3 + \[Nu])\)\)\/\(12\ \((\(-1\) + \[Nu]\^2)\)\)}, {\(EE\ \ \((3 + \[Nu])\)\)\/\(12\ \((\(-1\) + \[Nu]\^2)\)\), \(EE - 3\ EE\ \ \[Nu]\)\/\(8 - 8\ \[Nu]\^2\), \(EE\ \((\(-3\) + \[Nu])\)\)\/\(6\ \((\(-1\) + \ \[Nu]\^2)\)\), EE\/\(\(-8\) + 8\ \[Nu]\), \(EE\ \[Nu]\)\/\(6 - 6\ \[Nu]\^2\), \(EE - \ 3\ EE\ \[Nu]\)\/\(\(-8\) + 8\ \[Nu]\^2\), \(-\(\(EE\ \((\(-3\) + \[Nu])\)\)\/\ \(12\ \((\(-1\) + \[Nu]\^2)\)\)\)\), EE\/\(8 - 8\ \[Nu]\)}, {\(EE - 3\ EE\ \[Nu]\)\/\(\(-8\) + 8\ \[Nu]\^2\ \), \(EE\ \[Nu]\)\/\(6 - 6\ \[Nu]\^2\), EE\/\(\(-8\) + 8\ \[Nu]\), \(EE\ \((\(-3\) + \[Nu])\)\)\/\(6\ \((\(-1\ \) + \[Nu]\^2)\)\), \(EE - 3\ EE\ \[Nu]\)\/\(8 - 8\ \[Nu]\^2\), \(EE\ \((3 + \ \[Nu])\)\)\/\(12\ \((\(-1\) + \[Nu]\^2)\)\), EE\/\(8 - 8\ \[Nu]\), \(-\(\(EE\ \((\(-3\) + \[Nu])\)\)\/\(12\ \ \((\(-1\) + \[Nu]\^2)\)\)\)\)}, {\(-\(\(EE\ \((\(-3\) + \[Nu])\)\)\/\(12\ \((\ \(-1\) + \[Nu]\^2)\)\)\)\), EE\/\(\(-8\) + 8\ \[Nu]\), \(EE\ \[Nu]\)\/\(6 - 6\ \[Nu]\^2\), \(EE - \ 3\ EE\ \[Nu]\)\/\(8 - 8\ \[Nu]\^2\), \(EE\ \((\(-3\) + \[Nu])\)\)\/\(6\ \ \((\(-1\) + \[Nu]\^2)\)\), EE\/\(8 - 8\ \[Nu]\), \(EE\ \((3 + \[Nu])\)\)\/\(12\ \((\(-1\) + \ \[Nu]\^2)\)\), \(EE - 3\ EE\ \[Nu]\)\/\(\(-8\) + 8\ \[Nu]\^2\)}, \ {EE\/\(\(-8\) + 8\ \[Nu]\), \(-\(\(EE\ \((\(-3\) + \[Nu])\)\)\/\(12\ \ \((\(-1\) + \[Nu]\^2)\)\)\)\), \(EE - 3\ EE\ \[Nu]\)\/\(\(-8\) + 8\ \ \[Nu]\^2\), \(EE\ \((3 + \[Nu])\)\)\/\(12\ \((\(-1\) + \[Nu]\^2)\)\), EE\/\(8 - 8\ \[Nu]\), \(EE\ \((\(-3\) + \[Nu])\)\)\/\(6\ \((\(-1\) + \ \[Nu]\^2)\)\), \(EE - 3\ EE\ \[Nu]\)\/\(8 - 8\ \[Nu]\^2\), \(EE\ \[Nu]\)\/\(6 \ - 6\ \[Nu]\^2\)}, {\(EE\ \[Nu]\)\/\(6 - 6\ \[Nu]\^2\), \(EE - 3\ EE\ \ \[Nu]\)\/\(\(-8\) + 8\ \[Nu]\^2\), \(-\(\(EE\ \((\(-3\) + \[Nu])\)\)\/\(12\ \ \((\(-1\) + \[Nu]\^2)\)\)\)\), EE\/\(8 - 8\ \[Nu]\), \(EE\ \((3 + \[Nu])\)\)\/\(12\ \((\(-1\) + \ \[Nu]\^2)\)\), \(EE - 3\ EE\ \[Nu]\)\/\(8 - 8\ \[Nu]\^2\), \(EE\ \((\(-3\) + \ \[Nu])\)\)\/\(6\ \((\(-1\) + \[Nu]\^2)\)\), EE\/\(\(-8\) + 8\ \[Nu]\)}, {\(EE - 3\ EE\ \[Nu]\)\/\(8 - 8\ \[Nu]\^2\ \), \(EE\ \((3 + \[Nu])\)\)\/\(12\ \((\(-1\) + \[Nu]\^2)\)\), EE\/\(8 - 8\ \[Nu]\), \(-\(\(EE\ \((\(-3\) + \[Nu])\)\)\/\(12\ \ \((\(-1\) + \[Nu]\^2)\)\)\)\), \(EE - 3\ EE\ \[Nu]\)\/\(\(-8\) + 8\ \ \[Nu]\^2\), \(EE\ \[Nu]\)\/\(6 - 6\ \[Nu]\^2\), EE\/\(\(-8\) + 8\ \[Nu]\), \(EE\ \((\(-3\) + \[Nu])\)\)\/\(6\ \((\(-1\ \) + \[Nu]\^2)\)\)}}\)], "Output"] }, Open ]], Cell[BoxData[ \( (*Det[K] // Simplify*) \)], "Input", FontSize->36] }, Closed]], Cell[CellGroupData[{ Cell["Eigenvalue analysis of stiffness matrix", "Subtitle", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[{ \(k\ = K\/\(EE\/\((1 - \[Nu]\^2)\)\)\ /. \ \[Nu] \[Rule] 0.3 // Simplify\), "\[IndentingNewLine]", \(evalues\ = \ Eigenvalues[k]\), "\[IndentingNewLine]", \(evectors\ = \ Eigenvectors[k]\)}], "Input", FontSize->36], Cell[BoxData[ \({{0.44999999999999996`, 0.1625`, \(-0.27499999999999997`\), \(-0.01250000000000001`\), \ \(-0.22499999999999998`\), \(-0.1625`\), 0.05`, 0.01250000000000001`}, {0.1625`, 0.44999999999999996`, 0.01250000000000001`, 0.05`, \(-0.1625`\), \(-0.22499999999999998`\), \ \(-0.01250000000000001`\), \(-0.27499999999999997`\)}, \ {\(-0.27499999999999997`\), 0.01250000000000001`, 0.44999999999999996`, \(-0.1625`\), 0.05`, \(-0.01250000000000001`\), \(-0.22499999999999998`\), 0.1625`}, {\(-0.01250000000000001`\), 0.05`, \(-0.1625`\), 0.44999999999999996`, 0.01250000000000001`, \(-0.27499999999999997`\), 0.1625`, \(-0.22499999999999998`\)}, {\(-0.22499999999999998`\), \ \(-0.1625`\), 0.05`, 0.01250000000000001`, 0.44999999999999996`, 0.1625`, \(-0.27499999999999997`\), \(-0.01250000000000001`\)}, \ {\(-0.1625`\), \(-0.22499999999999998`\), \(-0.01250000000000001`\), \ \(-0.27499999999999997`\), 0.1625`, 0.44999999999999996`, 0.01250000000000001`, 0.05`}, {0.05`, \(-0.01250000000000001`\), \(-0.22499999999999998`\), 0.1625`, \(-0.27499999999999997`\), 0.01250000000000001`, 0.44999999999999996`, \(-0.1625`\)}, {0.01250000000000001`, \ \(-0.27499999999999997`\), 0.1625`, \(-0.22499999999999998`\), \(-0.01250000000000001`\), 0.05`, \(-0.1625`\), 0.44999999999999996`}}\)], "Output"], Cell[BoxData[ \({1.2999999999999998`, 0.7000000000000002`, 0.6999999999999998`, 0.45000000000000007`, 0.44999999999999996`, \(-1.3388795687364547`*^-16\), 1.0847371641737624`*^-16, \(-7.182839392716468`*^-18\)}\)], "Output"], Cell[BoxData[ \({{\(-0.3535533905932737`\), \(-0.35355339059327384`\), 0.35355339059327384`, \(-0.3535533905932739`\), 0.35355339059327373`, 0.3535533905932737`, \(-0.35355339059327395`\), 0.3535533905932738`}, {\(-0.15593671501555595`\), \ \(-0.47506182851304424`\), \(-0.47506182851304496`\), 0.1559367150155565`, 0.1559367150155571`, 0.47506182851304446`, 0.4750618285130437`, \(-0.15593671501555642`\)}, \ {0.4750618285130444`, \(-0.1559367150155565`\), \(-0.15593671501555662`\), \ \(-0.4750618285130444`\), \(-0.47506182851304407`\), 0.1559367150155564`, 0.15593671501555628`, 0.4750618285130444`}, {\(-0.48581183365859765`\), \ \(-0.11826606562015686`\), 0.4858118336585971`, 0.11826606562015624`, \(-0.4858118336585974`\), \ \(-0.11826606562015572`\), 0.48581183365859837`, 0.1182660656201562`}, {\(-0.11826606562015612`\), 0.48581183365859754`, 0.11826606562015625`, \(-0.48581183365859765`\), \ \(-0.11826606562015624`\), 0.4858118336585976`, 0.1182660656201565`, \(-0.4858118336585977`\)}, \ {\(-0.38273162117388043`\), 0.43920367116504033`, \(-0.3827316211738804`\), \ \(-0.2537410087976552`\), 0.31021305878881505`, \(-0.2537410087976553`\), 0.3102130587888151`, 0.43920367116504017`}, {\(-0.3554321687258649`\), \ \(-0.34594895889511623`\), \(-0.3554321687258649`\), \ \(-0.41401334084519775`\), \(-0.2873677867757832`\), \ \(-0.41401334084519775`\), \(-0.28736778677578323`\), \(-0.34594895889511607`\ \)}, {0.3196630719811747`, \(-0.24983885421314386`\), 0.3196630719811747`, \(-0.3731064915765291`\), 0.44293070934456025`, \(-0.37310649157652914`\), 0.44293070934456025`, \(-0.2498388542131438`\)}}\)], "Output"] }, Open ]], Cell[BoxData[{ \(U\_1[v_]\ := \ Table[\(evectors[\([v]\)]\)[\([j]\)], \ {j, 1, 8, 2}]\), "\[IndentingNewLine]", \(\ u1[v_]\ := \ Flatten[{U\_1[v], \(U\_1[v]\)[\([1]\)]}]\), "\[IndentingNewLine]", \(U\_2[v_]\ := \ Table[\(evectors[\([v]\)]\)[\([j]\)], \ {j, 2, 8, 2}]\), "\[IndentingNewLine]", \(u2[v_]\ := \ Flatten[{U\_2[v], \(U\_2[v]\)[\([1]\)]}]\), "\[IndentingNewLine]", \(Ut[v_]\ := \ {u1[v], \ u2[v]}\), "\[IndentingNewLine]", \(U[v_]\ := \ Transpose[Ut[v]]\)}], "Input", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[{ \(X\_1 = Table[Xn[\([j]\)] /. values, {j, 1, 8, 2}]; \ x1\ = \ Flatten[{X\_1, X\_1[\([1]\)]}]\), "\[IndentingNewLine]", \(X\_2 = Table[Xn[\([j]\)] /. values, {j, 2, 8, 2}]; \ x2\ = \ Flatten[{X\_2, X\_2[\([1]\)]}]\), "\[IndentingNewLine]", \(X\ = \ Transpose[{x1, x2}]\)}], "Input", FontSize->36], Cell[BoxData[ \({1, \(-1\), \(-1\), 1, 1}\)], "Output"], Cell[BoxData[ \({1, 1, \(-1\), \(-1\), 1}\)], "Output"], Cell[BoxData[ \({{1, 1}, {\(-1\), 1}, {\(-1\), \(-1\)}, {1, \(-1\)}, {1, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(b\ = \ B /. \ values\ // Simplify;\)\), "\[IndentingNewLine]", \(\(b . evectors[\([8]\)] // Simplify\) // Chop\)}], "Input", FontSize->36], Cell[BoxData[ \({0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(plots\ = \ Table[Show[ Graphics[{Thickness[0.01], RGBColor[1, 0, 0], Line[X], RGBColor[0, 0, 1], Line[X + U[j]]}, \ AspectRatio \[Rule] Automatic, Axes \[Rule] True]], {j, 8}];\)\)], "Input", FontSize->36], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.47619 0.5 0.47619 [ [.02381 .4875 -6 -9 ] [.02381 .4875 6 0 ] [.2619 .4875 -12 -9 ] [.2619 .4875 12 0 ] [.7381 .4875 -9 -9 ] [.7381 .4875 9 0 ] [.97619 .4875 -3 -9 ] [.97619 .4875 3 0 ] [.4875 .02381 -12 -4.5 ] [.4875 .02381 0 4.5 ] [.4875 .2619 -24 -4.5 ] [.4875 .2619 0 4.5 ] [.4875 .7381 -18 -4.5 ] [.4875 .7381 0 4.5 ] [.4875 .97619 -6 -4.5 ] [.4875 .97619 0 4.5 ] [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .5 m .02381 .50625 L s [(-1)] .02381 .4875 0 1 Mshowa .2619 .5 m .2619 .50625 L s [(-0.5)] .2619 .4875 0 1 Mshowa .7381 .5 m .7381 .50625 L s [(0.5)] .7381 .4875 0 1 Mshowa .97619 .5 m .97619 .50625 L s [(1)] .97619 .4875 0 1 Mshowa .125 Mabswid .07143 .5 m .07143 .50375 L s .11905 .5 m .11905 .50375 L s .16667 .5 m .16667 .50375 L s .21429 .5 m .21429 .50375 L s .30952 .5 m .30952 .50375 L s .35714 .5 m .35714 .50375 L s .40476 .5 m .40476 .50375 L s .45238 .5 m .45238 .50375 L s .54762 .5 m .54762 .50375 L s .59524 .5 m .59524 .50375 L s .64286 .5 m .64286 .50375 L s .69048 .5 m .69048 .50375 L s .78571 .5 m .78571 .50375 L s .83333 .5 m .83333 .50375 L s .88095 .5 m .88095 .50375 L s .92857 .5 m .92857 .50375 L s .25 Mabswid 0 .5 m 1 .5 L s .5 .02381 m .50625 .02381 L s [(-1)] .4875 .02381 1 0 Mshowa .5 .2619 m .50625 .2619 L s [(-0.5)] .4875 .2619 1 0 Mshowa .5 .7381 m .50625 .7381 L s [(0.5)] .4875 .7381 1 0 Mshowa .5 .97619 m .50625 .97619 L s [(1)] .4875 .97619 1 0 Mshowa .125 Mabswid .5 .07143 m .50375 .07143 L s .5 .11905 m .50375 .11905 L s .5 .16667 m .50375 .16667 L s .5 .21429 m .50375 .21429 L s .5 .30952 m .50375 .30952 L s .5 .35714 m .50375 .35714 L s .5 .40476 m .50375 .40476 L s .5 .45238 m .50375 .45238 L s .5 .54762 m .50375 .54762 L s .5 .59524 m .50375 .59524 L s .5 .64286 m .50375 .64286 L s .5 .69048 m .50375 .69048 L s .5 .78571 m .50375 .78571 L s .5 .83333 m .50375 .83333 L s .5 .88095 m .50375 .88095 L s .5 .92857 m .50375 .92857 L s .25 Mabswid .5 0 m .5 1 L s 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 1 0 0 r .01 w .97619 .97619 m .02381 .97619 L .02381 .02381 L .97619 .02381 L .97619 .97619 L s 0 0 1 r .80783 .80783 m .19217 .80783 L .19217 .19217 L .80783 .19217 L .80783 .80783 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 288}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .78365 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.322827 0.391827 0.322827 [ [.01576 .37933 -12 -9 ] [.01576 .37933 12 0 ] [.17717 .37933 -6 -9 ] [.17717 .37933 6 0 ] [.33859 .37933 -12 -9 ] [.33859 .37933 12 0 ] [.66141 .37933 -9 -9 ] [.66141 .37933 9 0 ] [.82283 .37933 -3 -9 ] [.82283 .37933 3 0 ] [.98424 .37933 -9 -9 ] [.98424 .37933 9 0 ] [.4875 .069 -12 -4.5 ] [.4875 .069 0 4.5 ] [.4875 .23041 -24 -4.5 ] [.4875 .23041 0 4.5 ] [.4875 .55324 -18 -4.5 ] [.4875 .55324 0 4.5 ] [.4875 .71465 -6 -4.5 ] [.4875 .71465 0 4.5 ] [ 0 0 0 0 ] [ 1 .78365 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .01576 .39183 m .01576 .39808 L s [(-1.5)] .01576 .37933 0 1 Mshowa .17717 .39183 m .17717 .39808 L s [(-1)] .17717 .37933 0 1 Mshowa .33859 .39183 m .33859 .39808 L s [(-0.5)] .33859 .37933 0 1 Mshowa .66141 .39183 m .66141 .39808 L s [(0.5)] .66141 .37933 0 1 Mshowa .82283 .39183 m .82283 .39808 L s [(1)] .82283 .37933 0 1 Mshowa .98424 .39183 m .98424 .39808 L s [(1.5)] .98424 .37933 0 1 Mshowa .125 Mabswid .04804 .39183 m .04804 .39558 L s .08032 .39183 m .08032 .39558 L s .11261 .39183 m .11261 .39558 L s .14489 .39183 m .14489 .39558 L s .20946 .39183 m .20946 .39558 L s .24174 .39183 m .24174 .39558 L s .27402 .39183 m .27402 .39558 L s .3063 .39183 m .3063 .39558 L s .37087 .39183 m .37087 .39558 L s .40315 .39183 m .40315 .39558 L s .43543 .39183 m .43543 .39558 L s .46772 .39183 m .46772 .39558 L s .53228 .39183 m .53228 .39558 L s .56457 .39183 m .56457 .39558 L s .59685 .39183 m .59685 .39558 L s .62913 .39183 m .62913 .39558 L s .6937 .39183 m .6937 .39558 L s .72598 .39183 m .72598 .39558 L s .75826 .39183 m .75826 .39558 L s .79054 .39183 m .79054 .39558 L s .85511 .39183 m .85511 .39558 L s .88739 .39183 m .88739 .39558 L s .91968 .39183 m .91968 .39558 L s .95196 .39183 m .95196 .39558 L s .25 Mabswid 0 .39183 m 1 .39183 L s .5 .069 m .50625 .069 L s [(-1)] .4875 .069 1 0 Mshowa .5 .23041 m .50625 .23041 L s [(-0.5)] .4875 .23041 1 0 Mshowa .5 .55324 m .50625 .55324 L s [(0.5)] .4875 .55324 1 0 Mshowa .5 .71465 m .50625 .71465 L s [(1)] .4875 .71465 1 0 Mshowa .125 Mabswid .5 .10128 m .50375 .10128 L s .5 .13356 m .50375 .13356 L s .5 .16585 m .50375 .16585 L s .5 .19813 m .50375 .19813 L s .5 .2627 m .50375 .2627 L s .5 .29498 m .50375 .29498 L s .5 .32726 m .50375 .32726 L s .5 .35954 m .50375 .35954 L s .5 .42411 m .50375 .42411 L s .5 .45639 m .50375 .45639 L s .5 .48867 m .50375 .48867 L s .5 .52096 m .50375 .52096 L s .5 .58552 m .50375 .58552 L s .5 .61781 m .50375 .61781 L s .5 .65009 m .50375 .65009 L s .5 .68237 m .50375 .68237 L s .5 .03672 m .50375 .03672 L s .5 .00443 m .50375 .00443 L s .5 .74694 m .50375 .74694 L s .5 .77922 m .50375 .77922 L s .25 Mabswid .5 0 m .5 .78365 L s 0 0 m 1 0 L 1 .78365 L 0 .78365 L closepath clip newpath 1 0 0 r .01 w .82283 .71465 m .17717 .71465 L .17717 .069 L .82283 .069 L .82283 .71465 L s 0 0 1 r .77249 .56129 m .02381 .76499 L .22751 .22236 L .97619 .01866 L .77249 .56129 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 225.688}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .67794 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.322827 0.338969 0.322827 [ [.01576 .32647 -12 -9 ] [.01576 .32647 12 0 ] [.17717 .32647 -6 -9 ] [.17717 .32647 6 0 ] [.33859 .32647 -12 -9 ] [.33859 .32647 12 0 ] [.66141 .32647 -9 -9 ] [.66141 .32647 9 0 ] [.82283 .32647 -3 -9 ] [.82283 .32647 3 0 ] [.98424 .32647 -9 -9 ] [.98424 .32647 9 0 ] [.4875 .01614 -12 -4.5 ] [.4875 .01614 0 4.5 ] [.4875 .17756 -24 -4.5 ] [.4875 .17756 0 4.5 ] [.4875 .50038 -18 -4.5 ] [.4875 .50038 0 4.5 ] [.4875 .6618 -6 -4.5 ] [.4875 .6618 0 4.5 ] [ 0 0 0 0 ] [ 1 .67794 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .01576 .33897 m .01576 .34522 L s [(-1.5)] .01576 .32647 0 1 Mshowa .17717 .33897 m .17717 .34522 L s [(-1)] .17717 .32647 0 1 Mshowa .33859 .33897 m .33859 .34522 L s [(-0.5)] .33859 .32647 0 1 Mshowa .66141 .33897 m .66141 .34522 L s [(0.5)] .66141 .32647 0 1 Mshowa .82283 .33897 m .82283 .34522 L s [(1)] .82283 .32647 0 1 Mshowa .98424 .33897 m .98424 .34522 L s [(1.5)] .98424 .32647 0 1 Mshowa .125 Mabswid .04804 .33897 m .04804 .34272 L s .08032 .33897 m .08032 .34272 L s .11261 .33897 m .11261 .34272 L s .14489 .33897 m .14489 .34272 L s .20946 .33897 m .20946 .34272 L s .24174 .33897 m .24174 .34272 L s .27402 .33897 m .27402 .34272 L s .3063 .33897 m .3063 .34272 L s .37087 .33897 m .37087 .34272 L s .40315 .33897 m .40315 .34272 L s .43543 .33897 m .43543 .34272 L s .46772 .33897 m .46772 .34272 L s .53228 .33897 m .53228 .34272 L s .56457 .33897 m .56457 .34272 L s .59685 .33897 m .59685 .34272 L s .62913 .33897 m .62913 .34272 L s .6937 .33897 m .6937 .34272 L s .72598 .33897 m .72598 .34272 L s .75826 .33897 m .75826 .34272 L s .79054 .33897 m .79054 .34272 L s .85511 .33897 m .85511 .34272 L s .88739 .33897 m .88739 .34272 L s .91968 .33897 m .91968 .34272 L s .95196 .33897 m .95196 .34272 L s .25 Mabswid 0 .33897 m 1 .33897 L s .5 .01614 m .50625 .01614 L s [(-1)] .4875 .01614 1 0 Mshowa .5 .17756 m .50625 .17756 L s [(-0.5)] .4875 .17756 1 0 Mshowa .5 .50038 m .50625 .50038 L s [(0.5)] .4875 .50038 1 0 Mshowa .5 .6618 m .50625 .6618 L s [(1)] .4875 .6618 1 0 Mshowa .125 Mabswid .5 .04842 m .50375 .04842 L s .5 .08071 m .50375 .08071 L s .5 .11299 m .50375 .11299 L s .5 .14527 m .50375 .14527 L s .5 .20984 m .50375 .20984 L s .5 .24212 m .50375 .24212 L s .5 .2744 m .50375 .2744 L s .5 .30669 m .50375 .30669 L s .5 .37125 m .50375 .37125 L s .5 .40353 m .50375 .40353 L s .5 .43582 m .50375 .43582 L s .5 .4681 m .50375 .4681 L s .5 .53267 m .50375 .53267 L s .5 .56495 m .50375 .56495 L s .5 .59723 m .50375 .59723 L s .5 .62951 m .50375 .62951 L s .25 Mabswid .5 0 m .5 .67794 L s 0 0 m 1 0 L 1 .67794 L 0 .67794 L closepath clip newpath 1 0 0 r .01 w .82283 .6618 m .17717 .6618 L .17717 .01614 L .82283 .01614 L .82283 .6618 L s 0 0 1 r .97619 .61146 m .12683 .50843 L .02381 .06648 L .87317 .1695 L .97619 .61146 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 195.188}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .75263 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.320492 0.376315 0.320492 [ [.01926 .36381 -12 -9 ] [.01926 .36381 12 0 ] [.17951 .36381 -6 -9 ] [.17951 .36381 6 0 ] [.33975 .36381 -12 -9 ] [.33975 .36381 12 0 ] [.66025 .36381 -9 -9 ] [.66025 .36381 9 0 ] [.82049 .36381 -3 -9 ] [.82049 .36381 3 0 ] [.98074 .36381 -9 -9 ] [.98074 .36381 9 0 ] [.4875 .05582 -12 -4.5 ] [.4875 .05582 0 4.5 ] [.4875 .21607 -24 -4.5 ] [.4875 .21607 0 4.5 ] [.4875 .53656 -18 -4.5 ] [.4875 .53656 0 4.5 ] [.4875 .69681 -6 -4.5 ] [.4875 .69681 0 4.5 ] [ 0 0 0 0 ] [ 1 .75263 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .01926 .37631 m .01926 .38256 L s [(-1.5)] .01926 .36381 0 1 Mshowa .17951 .37631 m .17951 .38256 L s [(-1)] .17951 .36381 0 1 Mshowa .33975 .37631 m .33975 .38256 L s [(-0.5)] .33975 .36381 0 1 Mshowa .66025 .37631 m .66025 .38256 L s [(0.5)] .66025 .36381 0 1 Mshowa .82049 .37631 m .82049 .38256 L s [(1)] .82049 .36381 0 1 Mshowa .98074 .37631 m .98074 .38256 L s [(1.5)] .98074 .36381 0 1 Mshowa .125 Mabswid .05131 .37631 m .05131 .38006 L s .08336 .37631 m .08336 .38006 L s .11541 .37631 m .11541 .38006 L s .14746 .37631 m .14746 .38006 L s .21156 .37631 m .21156 .38006 L s .24361 .37631 m .24361 .38006 L s .27566 .37631 m .27566 .38006 L s .3077 .37631 m .3077 .38006 L s .3718 .37631 m .3718 .38006 L s .40385 .37631 m .40385 .38006 L s .4359 .37631 m .4359 .38006 L s .46795 .37631 m .46795 .38006 L s .53205 .37631 m .53205 .38006 L s .5641 .37631 m .5641 .38006 L s .59615 .37631 m .59615 .38006 L s .6282 .37631 m .6282 .38006 L s .6923 .37631 m .6923 .38006 L s .72434 .37631 m .72434 .38006 L s .75639 .37631 m .75639 .38006 L s .78844 .37631 m .78844 .38006 L s .85254 .37631 m .85254 .38006 L s .88459 .37631 m .88459 .38006 L s .91664 .37631 m .91664 .38006 L s .94869 .37631 m .94869 .38006 L s .25 Mabswid 0 .37631 m 1 .37631 L s .5 .05582 m .50625 .05582 L s [(-1)] .4875 .05582 1 0 Mshowa .5 .21607 m .50625 .21607 L s [(-0.5)] .4875 .21607 1 0 Mshowa .5 .53656 m .50625 .53656 L s [(0.5)] .4875 .53656 1 0 Mshowa .5 .69681 m .50625 .69681 L s [(1)] .4875 .69681 1 0 Mshowa .125 Mabswid .5 .08787 m .50375 .08787 L s .5 .11992 m .50375 .11992 L s .5 .15197 m .50375 .15197 L s .5 .18402 m .50375 .18402 L s .5 .24812 m .50375 .24812 L s .5 .28017 m .50375 .28017 L s .5 .31222 m .50375 .31222 L s .5 .34427 m .50375 .34427 L s .5 .40836 m .50375 .40836 L s .5 .44041 m .50375 .44041 L s .5 .47246 m .50375 .47246 L s .5 .50451 m .50375 .50451 L s .5 .56861 m .50375 .56861 L s .5 .60066 m .50375 .60066 L s .5 .63271 m .50375 .63271 L s .5 .66476 m .50375 .66476 L s .5 .02377 m .50375 .02377 L s .5 .72886 m .50375 .72886 L s .25 Mabswid .5 0 m .5 .75263 L s 0 0 m 1 0 L 1 .75263 L 0 .75263 L closepath clip newpath 1 0 0 r .01 w .82049 .69681 m .17951 .69681 L .17951 .05582 L .82049 .05582 L .82049 .69681 L s 0 0 1 r .66479 .6589 m .33521 .73471 L .02381 .01792 L .97619 .09373 L .66479 .6589 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 216.75}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.32867 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.425829 0.664337 0.425829 [ [.07417 .65184 -6 -9 ] [.07417 .65184 6 0 ] [.28709 .65184 -12 -9 ] [.28709 .65184 12 0 ] [.71291 .65184 -9 -9 ] [.71291 .65184 9 0 ] [.92583 .65184 -3 -9 ] [.92583 .65184 3 0 ] [.4875 .02559 -24 -4.5 ] [.4875 .02559 0 4.5 ] [.4875 .23851 -12 -4.5 ] [.4875 .23851 0 4.5 ] [.4875 .45142 -24 -4.5 ] [.4875 .45142 0 4.5 ] [.4875 .87725 -18 -4.5 ] [.4875 .87725 0 4.5 ] [.4875 1.09017 -6 -4.5 ] [.4875 1.09017 0 4.5 ] [.4875 1.30308 -18 -4.5 ] [.4875 1.30308 0 4.5 ] [ 0 0 0 0 ] [ 1 1.32867 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .07417 .66434 m .07417 .67059 L s [(-1)] .07417 .65184 0 1 Mshowa .28709 .66434 m .28709 .67059 L s [(-0.5)] .28709 .65184 0 1 Mshowa .71291 .66434 m .71291 .67059 L s [(0.5)] .71291 .65184 0 1 Mshowa .92583 .66434 m .92583 .67059 L s [(1)] .92583 .65184 0 1 Mshowa .125 Mabswid .11675 .66434 m .11675 .66809 L s .15934 .66434 m .15934 .66809 L s .20192 .66434 m .20192 .66809 L s .2445 .66434 m .2445 .66809 L s .32967 .66434 m .32967 .66809 L s .37225 .66434 m .37225 .66809 L s .41483 .66434 m .41483 .66809 L s .45742 .66434 m .45742 .66809 L s .54258 .66434 m .54258 .66809 L s .58517 .66434 m .58517 .66809 L s .62775 .66434 m .62775 .66809 L s .67033 .66434 m .67033 .66809 L s .7555 .66434 m .7555 .66809 L s .79808 .66434 m .79808 .66809 L s .84066 .66434 m .84066 .66809 L s .88325 .66434 m .88325 .66809 L s .03159 .66434 m .03159 .66809 L s .96841 .66434 m .96841 .66809 L s .25 Mabswid 0 .66434 m 1 .66434 L s .5 .02559 m .50625 .02559 L s [(-1.5)] .4875 .02559 1 0 Mshowa .5 .23851 m .50625 .23851 L s [(-1)] .4875 .23851 1 0 Mshowa .5 .45142 m .50625 .45142 L s [(-0.5)] .4875 .45142 1 0 Mshowa .5 .87725 m .50625 .87725 L s [(0.5)] .4875 .87725 1 0 Mshowa .5 1.09017 m .50625 1.09017 L s [(1)] .4875 1.09017 1 0 Mshowa .5 1.30308 m .50625 1.30308 L s [(1.5)] .4875 1.30308 1 0 Mshowa .125 Mabswid .5 .06818 m .50375 .06818 L s .5 .11076 m .50375 .11076 L s .5 .15334 m .50375 .15334 L s .5 .19593 m .50375 .19593 L s .5 .28109 m .50375 .28109 L s .5 .32367 m .50375 .32367 L s .5 .36626 m .50375 .36626 L s .5 .40884 m .50375 .40884 L s .5 .49401 m .50375 .49401 L s .5 .53659 m .50375 .53659 L s .5 .57917 m .50375 .57917 L s .5 .62175 m .50375 .62175 L s .5 .70692 m .50375 .70692 L s .5 .7495 m .50375 .7495 L s .5 .79209 m .50375 .79209 L s .5 .83467 m .50375 .83467 L s .5 .91983 m .50375 .91983 L s .5 .96242 m .50375 .96242 L s .5 1.005 m .50375 1.005 L s .5 1.04758 m .50375 1.04758 L s .5 1.13275 m .50375 1.13275 L s .5 1.17533 m .50375 1.17533 L s .5 1.21792 m .50375 1.21792 L s .5 1.2605 m .50375 1.2605 L s .25 Mabswid .5 0 m .5 1.32867 L s 0 0 m 1 0 L 1 1.32867 L 0 1.32867 L closepath clip newpath 1 0 0 r .01 w .92583 1.09017 m .07417 1.09017 L .07417 .23851 L .92583 .23851 L .92583 1.09017 L s 0 0 1 r .87547 1.29704 m .12453 .88329 L .02381 .44538 L .97619 .03164 L .87547 1.29704 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{216.75, 287.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.512823 0.353658 0.467205 0.353658 [ [.15917 .4547 -6 -9 ] [.15917 .4547 6 0 ] [.33599 .4547 -12 -9 ] [.33599 .4547 12 0 ] [.68965 .4547 -9 -9 ] [.68965 .4547 9 0 ] [.86648 .4547 -3 -9 ] [.86648 .4547 3 0 ] [.50032 .11355 -12 -4.5 ] [.50032 .11355 0 4.5 ] [.50032 .29038 -24 -4.5 ] [.50032 .29038 0 4.5 ] [.50032 .64403 -18 -4.5 ] [.50032 .64403 0 4.5 ] [.50032 .82086 -6 -4.5 ] [.50032 .82086 0 4.5 ] [.50032 .99769 -18 -4.5 ] [.50032 .99769 0 4.5 ] [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .15917 .4672 m .15917 .47345 L s [(-1)] .15917 .4547 0 1 Mshowa .33599 .4672 m .33599 .47345 L s [(-0.5)] .33599 .4547 0 1 Mshowa .68965 .4672 m .68965 .47345 L s [(0.5)] .68965 .4547 0 1 Mshowa .86648 .4672 m .86648 .47345 L s [(1)] .86648 .4547 0 1 Mshowa .125 Mabswid .19453 .4672 m .19453 .47095 L s .2299 .4672 m .2299 .47095 L s .26526 .4672 m .26526 .47095 L s .30063 .4672 m .30063 .47095 L s .37136 .4672 m .37136 .47095 L s .40673 .4672 m .40673 .47095 L s .44209 .4672 m .44209 .47095 L s .47746 .4672 m .47746 .47095 L s .54819 .4672 m .54819 .47095 L s .58355 .4672 m .58355 .47095 L s .61892 .4672 m .61892 .47095 L s .65429 .4672 m .65429 .47095 L s .72502 .4672 m .72502 .47095 L s .76038 .4672 m .76038 .47095 L s .79575 .4672 m .79575 .47095 L s .83112 .4672 m .83112 .47095 L s .1238 .4672 m .1238 .47095 L s .08843 .4672 m .08843 .47095 L s .05307 .4672 m .05307 .47095 L s .0177 .4672 m .0177 .47095 L s .90185 .4672 m .90185 .47095 L s .93721 .4672 m .93721 .47095 L s .97258 .4672 m .97258 .47095 L s .25 Mabswid 0 .4672 m 1 .4672 L s .51282 .11355 m .51907 .11355 L s [(-1)] .50032 .11355 1 0 Mshowa .51282 .29038 m .51907 .29038 L s [(-0.5)] .50032 .29038 1 0 Mshowa .51282 .64403 m .51907 .64403 L s [(0.5)] .50032 .64403 1 0 Mshowa .51282 .82086 m .51907 .82086 L s [(1)] .50032 .82086 1 0 Mshowa .51282 .99769 m .51907 .99769 L s [(1.5)] .50032 .99769 1 0 Mshowa .125 Mabswid .51282 .14891 m .51657 .14891 L s .51282 .18428 m .51657 .18428 L s .51282 .21964 m .51657 .21964 L s .51282 .25501 m .51657 .25501 L s .51282 .32574 m .51657 .32574 L s .51282 .36111 m .51657 .36111 L s .51282 .39647 m .51657 .39647 L s .51282 .43184 m .51657 .43184 L s .51282 .50257 m .51657 .50257 L s .51282 .53794 m .51657 .53794 L s .51282 .5733 m .51657 .5733 L s .51282 .60867 m .51657 .60867 L s .51282 .6794 m .51657 .6794 L s .51282 .71477 m .51657 .71477 L s .51282 .75013 m .51657 .75013 L s .51282 .7855 m .51657 .7855 L s .51282 .85623 m .51657 .85623 L s .51282 .89159 m .51657 .89159 L s .51282 .92696 m .51657 .92696 L s .51282 .96233 m .51657 .96233 L s .51282 .07818 m .51657 .07818 L s .51282 .04282 m .51657 .04282 L s .51282 .00745 m .51657 .00745 L s .25 Mabswid .51282 0 m .51282 1 L s 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 1 0 0 r .01 w .86648 .82086 m .15917 .82086 L .15917 .11355 L .86648 .11355 L .86648 .82086 L s 0 0 1 r .73113 .97619 m .02381 .73113 L .26887 .02381 L .97619 .26887 L .73113 .97619 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 288}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.02487 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.571857 0.404334 0.596135 0.404334 [ [.16752 .58364 -6 -9 ] [.16752 .58364 6 0 ] [.36969 .58364 -12 -9 ] [.36969 .58364 12 0 ] [.77402 .58364 -9 -9 ] [.77402 .58364 9 0 ] [.97619 .58364 -3 -9 ] [.97619 .58364 3 0 ] [.55936 .1918 -12 -4.5 ] [.55936 .1918 0 4.5 ] [.55936 .39397 -24 -4.5 ] [.55936 .39397 0 4.5 ] [.55936 .7983 -18 -4.5 ] [.55936 .7983 0 4.5 ] [.55936 1.00047 -6 -4.5 ] [.55936 1.00047 0 4.5 ] [ 0 0 0 0 ] [ 1 1.02487 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .16752 .59614 m .16752 .60239 L s [(-1)] .16752 .58364 0 1 Mshowa .36969 .59614 m .36969 .60239 L s [(-0.5)] .36969 .58364 0 1 Mshowa .77402 .59614 m .77402 .60239 L s [(0.5)] .77402 .58364 0 1 Mshowa .97619 .59614 m .97619 .60239 L s [(1)] .97619 .58364 0 1 Mshowa .125 Mabswid .20796 .59614 m .20796 .59989 L s .24839 .59614 m .24839 .59989 L s .28882 .59614 m .28882 .59989 L s .32926 .59614 m .32926 .59989 L s .41012 .59614 m .41012 .59989 L s .45056 .59614 m .45056 .59989 L s .49099 .59614 m .49099 .59989 L s .53142 .59614 m .53142 .59989 L s .61229 .59614 m .61229 .59989 L s .65272 .59614 m .65272 .59989 L s .69316 .59614 m .69316 .59989 L s .73359 .59614 m .73359 .59989 L s .81446 .59614 m .81446 .59989 L s .85489 .59614 m .85489 .59989 L s .89532 .59614 m .89532 .59989 L s .93576 .59614 m .93576 .59989 L s .12709 .59614 m .12709 .59989 L s .08666 .59614 m .08666 .59989 L s .04622 .59614 m .04622 .59989 L s .00579 .59614 m .00579 .59989 L s .25 Mabswid 0 .59614 m 1 .59614 L s .57186 .1918 m .57811 .1918 L s [(-1)] .55936 .1918 1 0 Mshowa .57186 .39397 m .57811 .39397 L s [(-0.5)] .55936 .39397 1 0 Mshowa .57186 .7983 m .57811 .7983 L s [(0.5)] .55936 .7983 1 0 Mshowa .57186 1.00047 m .57811 1.00047 L s [(1)] .55936 1.00047 1 0 Mshowa .125 Mabswid .57186 .23223 m .57561 .23223 L s .57186 .27267 m .57561 .27267 L s .57186 .3131 m .57561 .3131 L s .57186 .35353 m .57561 .35353 L s .57186 .4344 m .57561 .4344 L s .57186 .47483 m .57561 .47483 L s .57186 .51527 m .57561 .51527 L s .57186 .5557 m .57561 .5557 L s .57186 .63657 m .57561 .63657 L s .57186 .677 m .57561 .677 L s .57186 .71744 m .57561 .71744 L s .57186 .75787 m .57561 .75787 L s .57186 .83874 m .57561 .83874 L s .57186 .87917 m .57561 .87917 L s .57186 .9196 m .57561 .9196 L s .57186 .96004 m .57561 .96004 L s .57186 .15137 m .57561 .15137 L s .57186 .11093 m .57561 .11093 L s .57186 .0705 m .57561 .0705 L s .57186 .03007 m .57561 .03007 L s .25 Mabswid .57186 0 m .57186 1.02487 L s 0 0 m 1 0 L 1 1.02487 L 0 1.02487 L closepath clip newpath 1 0 0 r .01 w .97619 1.00047 m .16752 1.00047 L .16752 .1918 L .97619 .1918 L .97619 1.00047 L s 0 0 1 r .83248 .86059 m .02381 .83307 L .05133 .0244 L .86 .05192 L .83248 .86059 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{281, 287.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .97142 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.413661 0.389852 0.558437 0.389852 [ [.02381 .54594 -6 -9 ] [.02381 .54594 6 0 ] [.21874 .54594 -12 -9 ] [.21874 .54594 12 0 ] [.60859 .54594 -9 -9 ] [.60859 .54594 9 0 ] [.80351 .54594 -3 -9 ] [.80351 .54594 3 0 ] [.99844 .54594 -9 -9 ] [.99844 .54594 9 0 ] [.40116 .16859 -12 -4.5 ] [.40116 .16859 0 4.5 ] [.40116 .36351 -24 -4.5 ] [.40116 .36351 0 4.5 ] [.40116 .75336 -18 -4.5 ] [.40116 .75336 0 4.5 ] [.40116 .94829 -6 -4.5 ] [.40116 .94829 0 4.5 ] [ 0 0 0 0 ] [ 1 .97142 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .55844 m .02381 .56469 L s [(-1)] .02381 .54594 0 1 Mshowa .21874 .55844 m .21874 .56469 L s [(-0.5)] .21874 .54594 0 1 Mshowa .60859 .55844 m .60859 .56469 L s [(0.5)] .60859 .54594 0 1 Mshowa .80351 .55844 m .80351 .56469 L s [(1)] .80351 .54594 0 1 Mshowa .99844 .55844 m .99844 .56469 L s [(1.5)] .99844 .54594 0 1 Mshowa .125 Mabswid .06279 .55844 m .06279 .56219 L s .10178 .55844 m .10178 .56219 L s .14077 .55844 m .14077 .56219 L s .17975 .55844 m .17975 .56219 L s .25772 .55844 m .25772 .56219 L s .29671 .55844 m .29671 .56219 L s .33569 .55844 m .33569 .56219 L s .37468 .55844 m .37468 .56219 L s .45265 .55844 m .45265 .56219 L s .49163 .55844 m .49163 .56219 L s .53062 .55844 m .53062 .56219 L s .5696 .55844 m .5696 .56219 L s .64757 .55844 m .64757 .56219 L s .68656 .55844 m .68656 .56219 L s .72554 .55844 m .72554 .56219 L s .76453 .55844 m .76453 .56219 L s .8425 .55844 m .8425 .56219 L s .88148 .55844 m .88148 .56219 L s .92047 .55844 m .92047 .56219 L s .95945 .55844 m .95945 .56219 L s .25 Mabswid 0 .55844 m 1 .55844 L s .41366 .16859 m .41991 .16859 L s [(-1)] .40116 .16859 1 0 Mshowa .41366 .36351 m .41991 .36351 L s [(-0.5)] .40116 .36351 1 0 Mshowa .41366 .75336 m .41991 .75336 L s [(0.5)] .40116 .75336 1 0 Mshowa .41366 .94829 m .41991 .94829 L s [(1)] .40116 .94829 1 0 Mshowa .125 Mabswid .41366 .20757 m .41741 .20757 L s .41366 .24656 m .41741 .24656 L s .41366 .28554 m .41741 .28554 L s .41366 .32453 m .41741 .32453 L s .41366 .4025 m .41741 .4025 L s .41366 .44148 m .41741 .44148 L s .41366 .48047 m .41741 .48047 L s .41366 .51945 m .41741 .51945 L s .41366 .59742 m .41741 .59742 L s .41366 .63641 m .41741 .63641 L s .41366 .67539 m .41741 .67539 L s .41366 .71438 m .41741 .71438 L s .41366 .79235 m .41741 .79235 L s .41366 .83133 m .41741 .83133 L s .41366 .87032 m .41741 .87032 L s .41366 .9093 m .41741 .9093 L s .41366 .1296 m .41741 .1296 L s .41366 .09061 m .41741 .09061 L s .41366 .05163 m .41741 .05163 L s .41366 .01264 m .41741 .01264 L s .25 Mabswid .41366 0 m .41366 .97142 L s 0 0 m 1 0 L 1 .97142 L 0 .97142 L closepath clip newpath 1 0 0 r .01 w .80351 .94829 m .02381 .94829 L .02381 .16859 L .80351 .16859 L .80351 .94829 L s 0 0 1 r .92813 .85089 m .14843 .80283 L .19649 .02313 L .97619 .07119 L .92813 .85089 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 279.75}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]] }, Closed]] }, FrontEndVersion->"5.0 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{1173, 945}, WindowMargins->{{-6, Automatic}, {22, Automatic}}, StyleDefinitions -> "DemoText.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1754, 51, 200, 5, 205, "Title"], Cell[1957, 58, 111, 3, 121, "Input"], Cell[CellGroupData[{ Cell[2093, 65, 67, 1, 67, "Subtitle"], Cell[2163, 68, 1152, 24, 829, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[3352, 97, 70, 2, 65, "Input"], Cell[3425, 101, 35, 1, 52, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3497, 107, 77, 1, 67, "Subtitle"], Cell[CellGroupData[{ Cell[3599, 112, 439, 10, 159, "Input"], Cell[4041, 124, 731, 13, 273, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[4821, 143, 109, 4, 67, "Subtitle"], Cell[CellGroupData[{ Cell[4955, 151, 149, 3, 121, "Input"], Cell[5107, 156, 145, 2, 52, "Output"], Cell[5255, 160, 502, 7, 115, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[5806, 173, 111, 4, 67, "Subtitle"], Cell[CellGroupData[{ Cell[5942, 181, 170, 5, 121, "Input"], Cell[6115, 188, 145, 2, 52, "Output"], Cell[6263, 192, 575, 9, 115, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[6887, 207, 44, 1, 67, "Subtitle"], Cell[CellGroupData[{ Cell[6956, 212, 424, 10, 170, "Input"], Cell[7383, 224, 957, 17, 100, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[8377, 246, 1274, 26, 1033, "Input"], Cell[9654, 274, 830, 12, 117, "Output"], Cell[10487, 288, 618, 9, 205, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[11154, 303, 67, 1, 67, "Subtitle"], Cell[CellGroupData[{ Cell[11246, 308, 128, 3, 121, "Input"], Cell[11377, 313, 40, 1, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[11454, 319, 487, 9, 283, "Input"], Cell[11944, 330, 259, 4, 70, "Output"], Cell[12206, 336, 282, 7, 70, "Output"], Cell[12491, 345, 1228, 23, 70, "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[13768, 374, 52, 1, 71, "Subtitle"], Cell[CellGroupData[{ Cell[13845, 379, 65, 1, 67, "Subtitle"], Cell[CellGroupData[{ Cell[13935, 384, 295, 8, 169, "Input"], Cell[14233, 394, 232, 3, 70, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[14514, 403, 586, 12, 445, "Input"], Cell[15103, 417, 259, 4, 70, "Output"], Cell[15365, 423, 3202, 48, 70, "Output"] }, Open ]], Cell[18582, 474, 73, 2, 67, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[18692, 481, 75, 1, 71, "Subtitle"], Cell[CellGroupData[{ Cell[18792, 486, 265, 6, 238, "Input"], Cell[19060, 494, 1457, 25, 70, "Output"], Cell[20520, 521, 249, 4, 70, "Output"], Cell[20772, 527, 1795, 31, 70, "Output"] }, Open ]], Cell[22582, 561, 578, 13, 445, "Input"], Cell[CellGroupData[{ Cell[23185, 578, 340, 6, 283, "Input"], Cell[23528, 586, 59, 1, 70, "Output"], Cell[23590, 589, 59, 1, 70, "Output"], Cell[23652, 592, 103, 2, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[23792, 599, 170, 3, 121, "Input"], Cell[23965, 604, 43, 1, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[24045, 610, 296, 7, 391, "Input"], Cell[24344, 619, 2770, 212, 70, 2651, 208, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False], Cell[27117, 833, 3523, 260, 70, 3400, 256, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False], Cell[30643, 1095, 3406, 248, 70, 3283, 244, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False], Cell[34052, 1345, 3468, 254, 70, 3346, 250, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False], Cell[37523, 1601, 3476, 254, 70, 3350, 250, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False], Cell[41002, 1857, 3541, 260, 70, 3422, 256, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False], Cell[44546, 2119, 3286, 236, 70, 3163, 232, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False], Cell[47835, 2357, 3403, 242, 70, 3281, 238, "GraphicsData", "PostScript", \ "Graphics", ImageCacheValid->False] }, Open ]] }, Closed]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)