(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 545378, 7893] NotebookOptionsPosition[ 541957, 7781] NotebookOutlinePosition[ 542323, 7797] CellTagsIndexPosition[ 542280, 7794] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Illustrating the Finite Element Method By Solving Laplace' s Equation In Two \ Dimensions For A Non-Rectangular Region\ \>", "Subtitle", CellChangeTimes->{{3.392112710109375*^9, 3.39211272421875*^9}, { 3.392112987125*^9, 3.39211299590625*^9}, {3.3921422011875*^9, 3.39214222871875*^9}, {3.392312105682641*^9, 3.392312119260766*^9}, { 3.392316644979516*^9, 3.392316657276391*^9}, {3.392540553234375*^9, 3.392540587265625*^9}}, TextAlignment->Center], Cell[" Frank Kampas (fkampas@msn.com)", "Subsubtitle", CellChangeTimes->{{3.392542377484375*^9, 3.392542388578125*^9}}, TextAlignment->Center], Cell[CellGroupData[{ Cell[" Introduction", "Section", CellChangeTimes->{{3.392112736375*^9, 3.39211274834375*^9}}], Cell[TextData[{ " The finite element method is the standard approach for solving partial \ differential equations in geometries in which the boundaries are not simple \ variable ranges. This article illustrates the basic approach by solving \ Laplace's equation in two dimensions ( ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[PartialD]", RowBox[{"x", ",", "x"}]], "u"}], " ", "+", RowBox[{ SubscriptBox["\[PartialD]", RowBox[{"y", ",", "y"}]], "u"}]}], " ", "=", " ", "0"}], TraditionalForm]]], " ) for a region consisting of two unequal rectangles joined together, for \ a Dirichlet boundary condition (the function is specified on the boundary)." }], "Text", CellChangeTimes->{{3.39211289878125*^9, 3.392112967875*^9}, { 3.39211302796875*^9, 3.392113028140625*^9}, {3.39211328709375*^9, 3.3921134085625*^9}, {3.392113445328125*^9, 3.3921134455*^9}, { 3.39211348328125*^9, 3.392113510203125*^9}, {3.392113571734375*^9, 3.392113579390625*^9}, {3.3921136478125*^9, 3.3921137180625*^9}, { 3.392113922203125*^9, 3.39211396490625*^9}, {3.3921176581875*^9, 3.392117680875*^9}, {3.392313784323266*^9, 3.392313822698266*^9}, { 3.392313860104516*^9, 3.392313895932641*^9}, {3.392540745671875*^9, 3.392540901265625*^9}, {3.39254094596875*^9, 3.3925410058125*^9}}, FontSize->14], Cell[TextData[{ " The method used minimizes the square of the gradient of the function u, \ \[Del]u \[CenterDot] \[Del]u = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox["\[PartialD]", "x"], " ", "u"}], ")"}], "2"], "+", " ", SuperscriptBox[ RowBox[{"(", RowBox[{ SubscriptBox["\[PartialD]", "y"], " ", "u"}], ")"}], "2"], " "}], TraditionalForm]]], "in the region. The equivalence of this minimization to the 2-d Laplace \ equation can be seen using the package VariationalMethods. 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The minimization variables are the values of the function at the \ nodes of the mesh. \ \>", "Text", CellChangeTimes->{{3.392117531328125*^9, 3.392117593046875*^9}, { 3.39211823478125*^9, 3.3921182895625*^9}, {3.392316692588891*^9, 3.392316703838891*^9}, {3.392541041703125*^9, 3.39254106690625*^9}, { 3.3925411193125*^9, 3.39254115228125*^9}}, FontSize->14], Cell["\<\ In actual practice, a more complicated function would be used to for the \ solution and the process would be carried out iteratively, with the elements \ subdivided repeatedly in order to lower the error to below some threshold. \ Despite the simplifications used, the result obtained is in good agreement \ with a more sophisticated calculation except near the boundary between two \ closely spaced right-angle bends.\ \>", "Text", CellChangeTimes->{{3.39254116246875*^9, 3.392541406734375*^9}}, FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell[" Mathematical Formalism", "Section", CellChangeTimes->{{3.39211834971875*^9, 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square of the gradient in the triangle, as a function of the coordinates \ of the vertices and the function values at them, is \ \>", "Text", CellChangeTimes->{{3.392119384828125*^9, 3.392119447171875*^9}, { 3.392541545875*^9, 3.392541564484375*^9}}, FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"grad2", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x1_", ",", "y1_", ",", "u1_"}], "}"}], ",", RowBox[{"{", RowBox[{"x2_", ",", "y2_", ",", "u2_"}], "}"}], ",", RowBox[{"{", RowBox[{"x3_", ",", "y3_", ",", "u3_"}], "}"}]}], "}"}], "]"}], "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]}], ")"}], "/.", " ", RowBox[{"fsoln", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], "//", "Simplify"}]}]], "Input", CellChangeTimes->{{3.39211945984375*^9, 3.3921194700625*^9}, { 3.392120031609375*^9, 3.392120034046875*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"u2", " ", 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The signed area of a triangle (see ", StyleBox["MathWorld", FontSlant->"Italic"], " article TriangleArea) is given by: " }], "Text", CellChangeTimes->{{3.39211954703125*^9, 3.392119609546875*^9}, { 3.392119772234375*^9, 3.3921198008125*^9}}, FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{"area", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x1_", ",", "y1_"}], "}"}], ",", RowBox[{"{", RowBox[{"x2_", ",", "y2_"}], "}"}], ",", RowBox[{"{", RowBox[{"x3_", ",", "y3_"}], "}"}]}], "}"}], "]"}], "=", RowBox[{ FractionBox["1", "2"], RowBox[{"Det", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x1", ",", "y1", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"x2", ",", "y2", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"x3", ",", "y3", ",", "1"}], "}"}]}], "}"}], "]"}]}]}]}]], "Input", CellChangeTimes->{ 3.392119769015625*^9, {3.3921198189375*^9, 3.392119836328125*^9}, { 3.392120045828125*^9, 3.3921200473125*^9}}], Cell[BoxData[ RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "x2"}], " ", "y1"}], "+", RowBox[{"x3", " ", "y1"}], "+", RowBox[{"x1", " ", "y2"}], "-", RowBox[{"x3", " ", "y2"}], "-", RowBox[{"x1", " ", "y3"}], "+", RowBox[{"x2", " ", "y3"}]}], ")"}]}]], "Output", CellChangeTimes->{{3.3921198220625*^9, 3.392119837296875*^9}, 3.39212004828125*^9, 3.392130871640625*^9, 3.392314323885766*^9, 3.392316056932641*^9, 3.39254204384375*^9}] }, Open ]], Cell["\<\ Therefore the product of the square of the gradient and the area is \ \>", "Text", CellChangeTimes->{{3.39211986640625*^9, 3.392119887234375*^9}}, FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"grad2Area", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x1_", ",", "y1_", ",", "u1_"}], "}"}], ",", RowBox[{"{", RowBox[{"x2_", ",", "y2_", ",", "u2_"}], "}"}], ",", RowBox[{"{", RowBox[{"x3_", ",", "y3_", ",", "u3_"}], "}"}]}], "}"}], "]"}], "=", RowBox[{ RowBox[{ RowBox[{"grad2", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x1", ",", "y1", ",", "u1"}], "}"}], ",", RowBox[{"{", RowBox[{"x2", ",", "y2", ",", "u2"}], "}"}], ",", RowBox[{"{", RowBox[{"x3", ",", "y3", ",", "u3"}], "}"}]}], "}"}], "]"}], "*", RowBox[{"Abs", "[", " ", RowBox[{"area", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x1", ",", "y1"}], "}"}], ",", RowBox[{"{", RowBox[{"x2", ",", "y2"}], "}"}], ",", RowBox[{"{", RowBox[{"x3", ",", "y3"}], "}"}]}], "}"}], "]"}], " ", "]"}]}], "//", "Simplify"}]}]], "Input", CellChangeTimes->{{3.392119898796875*^9, 3.392119942734375*^9}, { 3.3921201279375*^9, 3.392120143*^9}, {3.392315940323266*^9, 3.392315940948266*^9}, {3.392316026010766*^9, 3.392316026510766*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"u2", " ", "x1"}], "-", RowBox[{"u3", " ", "x1"}], "-", RowBox[{"u1", " ", "x2"}], "+", RowBox[{"u3", " ", "x2"}], "+", RowBox[{"u1", " ", "x3"}], "-", RowBox[{"u2", " ", "x3"}]}], ")"}], "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"u2", " ", "y1"}], "-", RowBox[{"u3", " ", "y1"}], "-", RowBox[{"u1", " ", "y2"}], "+", RowBox[{"u3", " ", "y2"}], "+", RowBox[{"u1", " ", "y3"}], "-", RowBox[{"u2", " ", "y3"}]}], ")"}], "2"]}], ")"}], " ", RowBox[{"Abs", "[", RowBox[{ RowBox[{"x2", " ", "y1"}], "-", RowBox[{"x3", " ", "y1"}], "-", RowBox[{"x1", " ", "y2"}], "+", RowBox[{"x3", " ", "y2"}], "+", RowBox[{"x1", " ", "y3"}], "-", RowBox[{"x2", " ", "y3"}]}], "]"}]}], ")"}], "/", RowBox[{"(", RowBox[{"2", " ", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"x2", " ", "y1"}], "-", RowBox[{"x3", " ", "y1"}], "-", RowBox[{"x1", " ", "y2"}], "+", RowBox[{"x3", " ", "y2"}], "+", RowBox[{"x1", " ", "y3"}], "-", RowBox[{"x2", " ", "y3"}]}], ")"}], "2"]}], ")"}]}]], "Output", CellChangeTimes->{{3.39211990815625*^9, 3.392119944296875*^9}, { 3.392120136390625*^9, 3.392120144171875*^9}, 3.392130874859375*^9, 3.392314323948266*^9, 3.392315949854516*^9, 3.392316057010766*^9, 3.392542043953125*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[" Creating the Mesh of Triangles", "Section", CellChangeTimes->{{3.392111389859375*^9, 3.392111394515625*^9}, { 3.3921400895625*^9, 3.392140092234375*^9}}], Cell[TextData[{ "In this example, the region in question is defined by a list of vertices \ and the boundary condition is that the value on the boundary is equal to the \ ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "y"}], ")"}], "2"], TraditionalForm]]], ". 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