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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[ 749171, 13277]*) (*NotebookOutlinePosition[ 751031, 13332]*) (* CellTagsIndexPosition[ 750951, 13326]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Solution of Laplace's equation using interpolation functions rather than FEM \ / FEA\ \>", "Title"], Cell["\<\ Ken Hill kohill@rogers.com (613) 592-2809\ \>", "Subtitle"], Cell["\<\ Ref. Silvester and Ferrari \"Finite Elements for Electrical Engineers\" 3rd \ Edition\ \>", "Subsubtitle"], Cell[BoxData[{ \(Needs["\"]\), "\n", \(Needs["\"]\), "\n", \(Needs["\"]\), "\n", \(<< Calculus`VectorAnalysis`\), "\[IndentingNewLine]", \(<< Graphics`MultipleListPlot`\), "\[IndentingNewLine]", \(<< Graphics`Legend`\)}], "Input"], Cell["\<\ The potential of two long parallel wires a distance 2a apart with charge \ densities \[PlusMinus]q, respectively, is given by U(x, y) =(q / 2\[Pi]) ln( ((x + a)^2 + y^2) / ((x - a)^2 + y^2 ) )\ \>", "Text"], Cell["\<\ Turn off warning messages triggered by similar variable names.\ \>", "Text"], Cell["\<\ Off[General::spell]; Off[General::spell1];\ \>", "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(SetCoordinates[Cartesian[x, y, z]]\)], "Input"], Cell[BoxData[ \(Cartesian[x, y, z]\)], "Output"] }, Open ]], Cell[BoxData[{ \(\(dim1 = 50;\)\), "\[IndentingNewLine]", \(\(dim2 = 50;\)\)}], "Input"], Cell[CellGroupData[{ Cell["\<\ The potential of two long parallel wires a distance 2a apart with charge \ densities \[PlusMinus]q\ \>", "Subsection"], Cell[BoxData[ \(ThePot[x_, y_, a_, q_] := \((q/\((2 \[Pi])\))\) Log[\((\((x + a)\)^2 + y^2)\)/\((\((x - a)\)^2 + y^2)\)]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The symbolic \"Potential\" (note that some of the a's are changed to b's; in \ this example two a's are changed, each to represent a wire)\ \>", "Subsection"], Cell[BoxData[ \(\(fn = Table[{i, j, a[i, j]}, {i, 1, dim1}, {j, 1, dim2}] /. {a[23, 25] \[Rule] b[23, 25], a[27, 25] \[Rule] b[27, 25]};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Dimensions[fn]\)], "Input"], Cell[BoxData[ \({50, 50, 3}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["The Flattened \"Potential\"", "Subsection"], Cell[BoxData[ \(\(ne = Flatten[fn, 1];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Selecting all the a's data (at some point some of the a[i,j]s will be inputed \ as b[i,j]s, i.e. constants -- see above)\ \>", "Subsection"], Cell[BoxData[ \(\(avalues = Cases[ne, \(({_, _, a[x_, y_]} \[Rule] a[x, y])\)];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["The interpolation function on the potential", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(re = Interpolation[ne, InterpolationOrder \[Rule] 1]\)], "Input"], Cell[BoxData[ TagBox[\(InterpolatingFunction[{{1, 50}, {1, 50}}, "<>"]\), False, Editable->False]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Gradient of re using Mathematica", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Grad[re[x, y]]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ TagBox[\(InterpolatingFunction[{{1, 50}, {1, 50}}, "<>"]\), False, Editable->False], "[", \(x, y\), "]"}], ",", RowBox[{ TagBox[\(InterpolatingFunction[{{1, 50}, {1, 50}}, "<>"]\), False, Editable->False], "[", \(x, y\), "]"}], ",", "0"}], "}"}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Dot product of the gradient of re with itself", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(ss[x_, y_] = DotProduct[Grad[re[x, y]], Grad[re[x, y]]]\)], "Input"], Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{ TagBox[\(InterpolatingFunction[{{1, 50}, {1, 50}}, "<>"]\), False, Editable->False], "[", \(x, y\), "]"}], "2"], "+", SuperscriptBox[ RowBox[{ TagBox[\(InterpolatingFunction[{{1, 50}, {1, 50}}, "<>"]\), False, Editable->False], "[", \(x, y\), "]"}], "2"]}]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We work out the gradient and evaluate the integral of the dot product of the \ gradient with itself (see Silvester)\ \>", "Section"], Cell["\<\ Using the well-known principle of minimum potential energy we minimize the \ stored energy over the whole region. We need to integrate the dot product of \ the gradient with itself to compute the stored energy. We have been able to work out the stored energy over the whole region by \ defining a new interpolation function over the dot product of the gradient of \ the potential with itself and integrating the function. \ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Generate a new interpolation function for the dot product of the gradient \ with itself (proportional to the energy density) -- we should then be able to \ integrate it symbolically\ \>", "Subsection"], Cell[BoxData[ \(\(energyFN = Table[{i, j, ss[i, j]}, {i, 1, dim1}, {j, 1, dim2}];\)\)], "Input"], Cell[BoxData[ \(\(flatenergyFN = Flatten[energyFN, 1];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(energyfunction = Interpolation[flatenergyFN, InterpolationOrder \[Rule] 1]\)], "Input"], Cell[BoxData[ TagBox[\(InterpolatingFunction[{{1, 50}, {1, 50}}, "<>"]\), False, Editable->False]], "Output"] }, Open ]], Cell[BoxData[ \(\(StoredEnergy = Integrate[ energyfunction[x, y], {x, 1, dim1}, {y, 1, dim2}];\)\)], "Input"], Cell[CellGroupData[{ Cell["The equations for determining the potential values", "Subsubsection"], Cell[BoxData[ \(\(TheEquationsForExtremum = Table[D[StoredEnergy, avalues[\([i]\)]] \[Equal] 0, {i, 1, \(Dimensions[avalues]\)[\([1]\)]}];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Lets give values to the b's -- in this case we are looking at a dipole\ \>", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[{ \(b[23, 25] = 1\), "\[IndentingNewLine]", \(b[27, 25] = \(-1\)\)}], "Input"], Cell[BoxData[ \(1\)], "Output"], Cell[BoxData[ \(\(-1\)\)], "Output"] }, Open ]], Cell[BoxData[ \(\(TheSol = Flatten[Solve[TheEquationsForExtremum, avalues], 1];\)\)], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Let's Plot the interpolation function on the potential", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(DensityPlot[re[x, y] /. 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