futurepress · ClaudioVZ · Dec 27, 2013 · Dec 27, 2013
diff --git a/funciones_forma.epub b/funciones_forma.epub
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+<?xml version="1.0" encoding="UTF-8"?>
+<container version="1.0" xmlns="urn:oasis:names:tc:opendocument:xmlns:container">
+    <rootfiles>
+        <rootfile full-path="OEBPS/content.opf" media-type="application/oebps-package+xml"/>
+   </rootfiles>
+</container>
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diff --git a/funciones_forma/OEBPS/Text/Section0000.xhtml b/funciones_forma/OEBPS/Text/Section0000.xhtml
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+<?xml version="1.0" encoding="utf-8" standalone="no"?>
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1//EN"
+  "http://www.w3.org/TR/xhtml11/DTD/xhtml11.dtd">
+
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+  <title></title>
+</head>
+
+<body>
+  <h1>Funciones de forma Lagrangianas<br />
+  Generación de funciones usando matrices</h1>
+
+  <h2 class="sigil_not_in_toc">C.V. Zapata</h2>
+</body>
+</html>
diff --git a/funciones_forma/OEBPS/Text/Section0001.xhtml b/funciones_forma/OEBPS/Text/Section0001.xhtml
@@ -0,0 +1,29 @@
+<?xml version="1.0" encoding="utf-8" standalone="no"?>
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1//EN"
+  "http://www.w3.org/TR/xhtml11/DTD/xhtml11.dtd">
+
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+  <title></title>
+</head>
+
+<body>
+  <h1>Índice</h1>
+
+  <ol>
+    <li><a href="../Text/Section0000.xhtml">Funciones de forma Lagrangianas</a></li>
+
+    <li><a href="../Text/Section0002.xhtml">Notación</a></li>
+
+    <li><a href="../Text/Section0003.xhtml">Funciones de forma unidimensionales</a></li>
+
+    <li><a href="../Text/Section0004.xhtml">Funciones de forma bidimensionales</a></li>
+
+    <li><a href="../Text/Section0005.xhtml">Funciones de forma tridimensionales</a></li>
+
+    <li><a href="../Text/Section0006.xhtml">Conclusiones</a></li>
+
+    <li><a href="../Text/Section0007.xhtml">Referencias</a></li>
+  </ol>
+</body>
+</html>
diff --git a/funciones_forma/OEBPS/Text/Section0002.xhtml b/funciones_forma/OEBPS/Text/Section0002.xhtml
@@ -0,0 +1,29 @@
+<?xml version="1.0" encoding="utf-8" standalone="no"?>
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1//EN"
+  "http://www.w3.org/TR/xhtml11/DTD/xhtml11.dtd">
+
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+  <title></title>
+  <script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript">
+</script>
+</head>
+
+<body>
+  <h1>Notación</h1>
+
+  <ul>
+    <li>
+      <p>\(l^{(1)}\) Función de forma unidimensional Lagrangiana</p>
+    </li>
+
+    <li>
+      <p>\(l^{(2)}\) Función de forma bidimensional Lagrangiana</p>
+    </li>
+
+    <li>
+      <p>\(l^{(3)}\) Función de forma trididimensional Lagrangiana</p>
+    </li>
+  </ul>
+</body>
+</html>
diff --git a/funciones_forma/OEBPS/Text/Section0003.xhtml b/funciones_forma/OEBPS/Text/Section0003.xhtml
@@ -0,0 +1,35 @@
+<?xml version="1.0" encoding="utf-8" standalone="no"?>
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1//EN"
+  "http://www.w3.org/TR/xhtml11/DTD/xhtml11.dtd">
+
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+  <title></title>
+  <script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript">
+</script>
+</head>
+
+<body>
+  <h1>Funciones de forma unidimensionales</h1>
+
+  <h2 id="sigil_toc_id_1">Definición</h2>
+
+  <p>Las funciones de forma Lagrangianas <a href="../Text/Section0007.xhtml#bib01" style="text-decoration:none">[1]</a> son polinomios con dominio \(-1 \leqslant x \leqslant 1\), la fórmula para generarlos es la siguiente:</p>
+
+  <p>\[l^{(n)}_{j} = \prod_{i=0, i \neq j}^k \frac{x - x_{i}}{x_{j} - x_{i}}\]</p>
+
+  <h2 id="sigil_toc_id_2">Ejemplo \(l^{(1)}\)</h2>
+
+  <p>Función de forma Lagrangiana unidimensional de 2 nodos:</p>
+
+  <p>\[\begin{align*} l^{(1)}_{1} &amp;= - \frac{1}{2} \xi + \frac{1}{2} \\ l^{(1)}_{2} &amp;= \frac{1}{2} \xi + \frac{1}{2}\end{align*}\]</p>
+
+  <div><img alt="image" height="162" src="../Images/2_nodos.png" style="margin:0px auto;display:block" width="320" /></div>
+
+  <p>Función de forma Lagrangiana unidimensional de 3 nodos:</p>
+
+  <p>\[\begin{align*} l^{(1)}_{1} &amp;= \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \\ l^{(1)}_{2} &amp;= - \xi^{2} + 1 \\ l^{(1)}_{3} &amp;= \frac{1}{2} \xi^{2} + \frac{1}{2} \xi\end{align*}\]</p>
+
+  <div><img alt="image" height="162" src="../Images/3_nodos.png" style="margin:0px auto;display:block" width="320" /></div>
+</body>
+</html>
diff --git a/funciones_forma/OEBPS/Text/Section0004.xhtml b/funciones_forma/OEBPS/Text/Section0004.xhtml
@@ -0,0 +1,83 @@
+<?xml version="1.0" encoding="utf-8" standalone="no"?>
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1//EN"
+  "http://www.w3.org/TR/xhtml11/DTD/xhtml11.dtd">
+
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+  <title></title>
+  <script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript">
+</script>
+</head>
+
+<body>
+  <h1>Funciones de forma bidimensionales</h1>
+
+  <h2 id="sigil_toc_id_3">Método</h2>
+
+  <p>Las \(l^{(2)}\) se generar a partir de \(l^{(1)}\), multiplicando matrices:</p>
+
+  <p>\[\left [ l^{(1)} \right ] \left [ l^{(1)} \right ]^{T} = \left [ l^{(2)} \right ]\]</p>
+
+  <p>En donde \(\left [ l^{(1)} \right ]\) es una matriz columna con \(j\)-elementos, \(\left [ l^{(1)} \right ]^{T}\) es una matriz fila con \(j\)-elementos y \(\left [ l^{(2)} \right ]\) es una matriz cuadrada.</p>
+
+  <p>Si recorremos los elementos que forman parte de la matriz \(\left [ l^{(2)} \right ]\), uno a uno desde \(l^{(2)}_1\) hasta \(l^{(2)}_n\) forman una espiral, si graficamos las funciones que forman la matriz en el orden anterior observamos que el punto \((\xi, \eta)\) en el que la función vale \(1\) también hace un recorrido en espiral.</p>
+
+  <h2 id="sigil_toc_id_4">Ejemplo \(l^{(2)}\)</h2>
+
+  <p>Elemento rectangular de 4 nodos:</p>
+
+  <p>\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \end{matrix} \right ] = \left [ \begin{matrix} - \frac{1}{2} \xi + \frac{1}{2} \\ \frac{1}{2} \xi + \frac{1}{2} \end{matrix} \right ]\]</p>
+
+  <p>\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} &amp; l^{(1)}_{2} \end{matrix} \right ] = \left [ \begin{matrix} l^{(2)}_{1} &amp; l^{(2)}_{4} \\ l^{(2)}_{2} &amp; l^{(2)}_{3} \end{matrix} \right ]\]</p>
+
+  <p>\[\begin{align*} l^{(2)}_{1} &amp;= \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta - \frac{1}{4} \xi \eta^{2} + \frac{1}{4} \xi \eta \\ l^{(2)}_{2} &amp;= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} \eta + \frac{1}{2} \eta^{2} - \frac{1}{2} \eta \\ l^{(2)}_{3} &amp;= \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta + \frac{1}{4} \xi \eta^{2} - \frac{1}{4} \xi \eta \\ l^{(2)}_{4} &amp;= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \eta^{2} + \frac{1}{2} \xi\end{align*}\]</p>
+
+  <div style="display:table">
+    <div style="display:table-row">
+      <div style="display:table-cell"><img alt="image" height="162" src="../Images/4_nodos_1.png" width="251" /></div>
+
+      <div style="display:table-cell"><img alt="image" height="162" src="../Images/4_nodos_2.png" width="251" /></div>
+    </div>
+
+    <div style="display:table-row">
+      <div style="display:table-cell"><img alt="image" height="162" src="../Images/4_nodos_3.png" width="251" /></div>
+
+      <div style="display:table-cell"><img alt="image" height="162" src="../Images/4_nodos_4.png" width="251" /></div>
+    </div>
+  </div>
+
+  <p>Elemento rectangular de 9 nodos:</p>
+
+  <p>\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \\ l^{(1)}_{3} \end{matrix} \right ] = \left [ \begin{matrix} \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \\ - \xi^{2} + 1 \\ \frac{1}{2} \xi^{2} + \frac{1}{2} \xi \end{matrix} \right ]\]</p>
+
+  <p>\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \\ l^{(1)}_{3} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} &amp; l^{(1)}_{2} &amp; l^{(1)}_{3} \end{matrix} \right ] = \left [ \begin{matrix} l^{(2)}_{1} &amp; l^{(2)}_{8} &amp; l^{(2)}_{7} \\ l^{(2)}_{2} &amp; l^{(2)}_{9} &amp; l^{(2)}_{6} \\ l^{(2)}_{3} &amp; l^{(2)}_{4} &amp; l^{(2)}_{5} \end{matrix} \right ]\]</p>
+
+  <p>\[\begin{align*} l^{(2)}_{1} &amp;= \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta - \frac{1}{4} \xi \eta^{2} + \frac{1}{4} \xi \eta \\ l^{(2)}_{2} &amp;= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} \eta + \frac{1}{2} \eta^{2} - \frac{1}{2} \eta \\ l^{(2)}_{3} &amp;= \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta + \frac{1}{4} \xi \eta^{2} - \frac{1}{4} \xi \eta \\ l^{(2)}_{4} &amp;= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \eta^{2} + \frac{1}{2} \xi \\ l^{(2)}_{5} &amp;= \frac{1}{4} \xi^{2} \eta^{2} + \frac{1}{4} \xi^{2} \eta + \frac{1}{4} \xi \eta^{2} + \frac{1}{4} \xi \eta \\ l^{(2)}_{6} &amp;= - \frac{1}{2} \xi^{2} \eta^{2} - \frac{1}{2} \xi^{2} \eta + \frac{1}{2} \eta^{2} + \frac{1}{2} \eta \\ l^{(2)}_{7} &amp;= \frac{1}{4} \xi^{2} \eta^{2} + \frac{1}{4} \xi^{2} \eta - \frac{1}{4} \xi \eta^{2} - \frac{1}{4} \xi \eta \\ l^{(2)}_{8} &amp;= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} + \frac{1}{2} \xi \eta^{2} - \frac{1}{2} \xi \\ l^{(2)}_{9} &amp;= \xi^{2} \eta^{2} - \xi^{2} - \eta^{2} + 1\end{align*}\]</p>
+
+  <div style="display:table">
+    <div style="display:table-row">
+      <div style="display:table-cell"><img alt="image" height="162" src="../Images/9_nodos_1.png" width="251" /></div>
+
+      <div style="display:table-cell"><img alt="image" height="162" src="../Images/9_nodos_2.png" width="251" /></div>
+
+      <div style="display:table-cell"><img alt="image" height="162" src="../Images/9_nodos_3.png" width="251" /></div>
+    </div>
+
+    <div style="display:table-row">
+      <div style="display:table-cell"><img alt="image" height="162" src="../Images/9_nodos_4.png" width="251" /></div>
+
+      <div style="display:table-cell"><img alt="image" height="162" src="../Images/9_nodos_5.png" width="251" /></div>
+
+      <div style="display:table-cell"><img alt="image" height="162" src="../Images/9_nodos_6.png" width="251" /></div>
+    </div>
+
+    <div style="display:table-row">
+      <div style="display:table-cell"><img alt="image" height="162" src="../Images/9_nodos_7.png" width="251" /></div>
+
+      <div style="display:table-cell"><img alt="image" height="162" src="../Images/9_nodos_8.png" width="251" /></div>
+
+      <div style="display:table-cell"><img alt="image" height="162" src="../Images/9_nodos_9.png" width="251" /></div>
+    </div>
+  </div>
+</body>
+</html>