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xgseP6mEx;Eg9QIYSiB+_2=#xYD+qSaKo-o>cN#V3eL*?Red~@73pWQQ`~fn#0;vE1 literal 0 HcmV?d00001 diff --git a/funciones_forma/OEBPS/Text/Section0000.xhtml b/funciones_forma/OEBPS/Text/Section0000.xhtml new file mode 100644 index 0000000..ba23685 --- /dev/null +++ b/funciones_forma/OEBPS/Text/Section0000.xhtml @@ -0,0 +1,16 @@ + + + + + + + + + +

Funciones de forma Lagrangianas
+ Generación de funciones usando matrices

+ +

C.V. Zapata

+ + diff --git a/funciones_forma/OEBPS/Text/Section0001.xhtml b/funciones_forma/OEBPS/Text/Section0001.xhtml new file mode 100644 index 0000000..af732af --- /dev/null +++ b/funciones_forma/OEBPS/Text/Section0001.xhtml @@ -0,0 +1,29 @@ + + + + + + + + + +

Índice

+ +
    +
  1. Funciones de forma Lagrangianas
  2. + +
  3. Notación
  4. + +
  5. Funciones de forma unidimensionales
  6. + +
  7. Funciones de forma bidimensionales
  8. + +
  9. Funciones de forma tridimensionales
  10. + +
  11. Conclusiones
  12. + +
  13. Referencias
  14. +
+ + diff --git a/funciones_forma/OEBPS/Text/Section0002.xhtml b/funciones_forma/OEBPS/Text/Section0002.xhtml new file mode 100644 index 0000000..5bc423f --- /dev/null +++ b/funciones_forma/OEBPS/Text/Section0002.xhtml @@ -0,0 +1,29 @@ + + + + + + + + + + +

Notación

+ +
    +
  • +

    \(l^{(1)}\) Función de forma unidimensional Lagrangiana

    +
  • + +
  • +

    \(l^{(2)}\) Función de forma bidimensional Lagrangiana

    +
  • + +
  • +

    \(l^{(3)}\) Función de forma trididimensional Lagrangiana

    +
  • +
+ + diff --git a/funciones_forma/OEBPS/Text/Section0003.xhtml b/funciones_forma/OEBPS/Text/Section0003.xhtml new file mode 100644 index 0000000..747aecd --- /dev/null +++ b/funciones_forma/OEBPS/Text/Section0003.xhtml @@ -0,0 +1,35 @@ + + + + + + + + + + +

Funciones de forma unidimensionales

+ +

Definición

+ +

Las funciones de forma Lagrangianas [1] son polinomios con dominio \(-1 \leqslant x \leqslant 1\), la fórmula para generarlos es la siguiente:

+ +

\[l^{(n)}_{j} = \prod_{i=0, i \neq j}^k \frac{x - x_{i}}{x_{j} - x_{i}}\]

+ +

Ejemplo \(l^{(1)}\)

+ +

Función de forma Lagrangiana unidimensional de 2 nodos:

+ +

\[\begin{align*} l^{(1)}_{1} &= - \frac{1}{2} \xi + \frac{1}{2} \\ l^{(1)}_{2} &= \frac{1}{2} \xi + \frac{1}{2}\end{align*}\]

+ +
image
+ +

Función de forma Lagrangiana unidimensional de 3 nodos:

+ +

\[\begin{align*} l^{(1)}_{1} &= \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \\ l^{(1)}_{2} &= - \xi^{2} + 1 \\ l^{(1)}_{3} &= \frac{1}{2} \xi^{2} + \frac{1}{2} \xi\end{align*}\]

+ +
image
+ + diff --git a/funciones_forma/OEBPS/Text/Section0004.xhtml b/funciones_forma/OEBPS/Text/Section0004.xhtml new file mode 100644 index 0000000..d1c567c --- /dev/null +++ b/funciones_forma/OEBPS/Text/Section0004.xhtml @@ -0,0 +1,83 @@ + + + + + + + + + + +

Funciones de forma bidimensionales

+ +

Método

+ +

Las \(l^{(2)}\) se generar a partir de \(l^{(1)}\), multiplicando matrices:

+ +

\[\left [ l^{(1)} \right ] \left [ l^{(1)} \right ]^{T} = \left [ l^{(2)} \right ]\]

+ +

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.

+ +

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.

+ +

Ejemplo \(l^{(2)}\)

+ +

Elemento rectangular de 4 nodos:

+ +

\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \end{matrix} \right ] = \left [ \begin{matrix} - \frac{1}{2} \xi + \frac{1}{2} \\ \frac{1}{2} \xi + \frac{1}{2} \end{matrix} \right ]\]

+ +

\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} & l^{(1)}_{2} \end{matrix} \right ] = \left [ \begin{matrix} l^{(2)}_{1} & l^{(2)}_{4} \\ l^{(2)}_{2} & l^{(2)}_{3} \end{matrix} \right ]\]

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\[\begin{align*} l^{(2)}_{1} &= \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta - \frac{1}{4} \xi \eta^{2} + \frac{1}{4} \xi \eta \\ l^{(2)}_{2} &= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} \eta + \frac{1}{2} \eta^{2} - \frac{1}{2} \eta \\ l^{(2)}_{3} &= \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta + \frac{1}{4} \xi \eta^{2} - \frac{1}{4} \xi \eta \\ l^{(2)}_{4} &= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \eta^{2} + \frac{1}{2} \xi\end{align*}\]

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Elemento rectangular de 9 nodos:

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\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \\ l^{(1)}_{3} \end{matrix} \right ] = \left [ \begin{matrix} \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \\ - \xi^{2} + 1 \\ \frac{1}{2} \xi^{2} + \frac{1}{2} \xi \end{matrix} \right ]\]

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\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \\ l^{(1)}_{3} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} & l^{(1)}_{2} & l^{(1)}_{3} \end{matrix} \right ] = \left [ \begin{matrix} l^{(2)}_{1} & l^{(2)}_{8} & l^{(2)}_{7} \\ l^{(2)}_{2} & l^{(2)}_{9} & l^{(2)}_{6} \\ l^{(2)}_{3} & l^{(2)}_{4} & l^{(2)}_{5} \end{matrix} \right ]\]

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\[\begin{align*} l^{(2)}_{1} &= \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta - \frac{1}{4} \xi \eta^{2} + \frac{1}{4} \xi \eta \\ l^{(2)}_{2} &= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} \eta + \frac{1}{2} \eta^{2} - \frac{1}{2} \eta \\ l^{(2)}_{3} &= \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta + \frac{1}{4} \xi \eta^{2} - \frac{1}{4} \xi \eta \\ l^{(2)}_{4} &= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \eta^{2} + \frac{1}{2} \xi \\ l^{(2)}_{5} &= \frac{1}{4} \xi^{2} \eta^{2} + \frac{1}{4} \xi^{2} \eta + \frac{1}{4} \xi \eta^{2} + \frac{1}{4} \xi \eta \\ l^{(2)}_{6} &= - \frac{1}{2} \xi^{2} \eta^{2} - \frac{1}{2} \xi^{2} \eta + \frac{1}{2} \eta^{2} + \frac{1}{2} \eta \\ l^{(2)}_{7} &= \frac{1}{4} \xi^{2} \eta^{2} + \frac{1}{4} \xi^{2} \eta - \frac{1}{4} \xi \eta^{2} - \frac{1}{4} \xi \eta \\ l^{(2)}_{8} &= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} + \frac{1}{2} \xi \eta^{2} - \frac{1}{2} \xi \\ l^{(2)}_{9} &= \xi^{2} \eta^{2} - \xi^{2} - \eta^{2} + 1\end{align*}\]

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Funciones de forma tridimensionales

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Método

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Las \(l^{(3)}\) también se generar a partir de \(l^{(1)}\), pero realizando una multiplicación con matrices 3D [2].

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\[\left [ l^{(1)} \right ] \left [ l^{(1)} \right ]^{T} \left [ l^{(1)} \right ]^{T} = \left [ l^{(3)} \right ]\]

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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 \(\left [ l^{(3)} \right ]\) es una matriz 3D cúbica.

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Ejemplo \(l^{(3)}\)

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Elemento hexaédrico de 8 nodos:

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\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \end{matrix} \right ] = \left [ \begin{matrix} - \frac{1}{2} \xi + \frac{1}{2} \\ \frac{1}{2} \xi + \frac{1}{2} \end{matrix} \right ]\]

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\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} & l^{(1)}_{2} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} & l^{(1)}_{2} \end{matrix} \right ] = \left [ \left [ \begin{matrix} l^{(3)}_{1} & l^{(3)}_{5} \\ l^{(3)}_{2} & l^{(3)}_{6} \end{matrix} \right ] \left [ \begin{matrix} l^{(3)}_{4} & l^{(3)}_{8} \\ l^{(3)}_{3} & l^{(3)}_{7} \end{matrix} \right ] \right ]\]

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\[\begin{align*} l^{(3)}_{1} &= - \frac{1}{8} \xi \eta \zeta + \frac{1}{8} \xi \eta + \frac{1}{8} \xi \zeta - \frac{1}{8} \xi + \frac{1}{8} \eta \zeta - \frac{1}{8} \eta - \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{2} &= \frac{1}{8} \xi \eta \zeta - \frac{1}{8} \xi \eta - \frac{1}{8} \xi \zeta + \frac{1}{8} \xi + \frac{1}{8} \eta \zeta - \frac{1}{8} \eta - \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{6} &= - \frac{1}{8} \xi \eta \zeta - \frac{1}{8} \xi \eta + \frac{1}{8} \xi \zeta + \frac{1}{8} \xi - \frac{1}{8} \eta \zeta - \frac{1}{8} \eta + \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{5} &= \frac{1}{8} \xi \eta \zeta + \frac{1}{8} \xi \eta - \frac{1}{8} \xi \zeta - \frac{1}{8} \xi - \frac{1}{8} \eta \zeta - \frac{1}{8} \eta + \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{4} &= \frac{1}{8} \xi \eta \zeta - \frac{1}{8} \xi \eta + \frac{1}{8} \xi \zeta - \frac{1}{8} \xi - \frac{1}{8} \eta \zeta + \frac{1}{8} \eta - \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{3} &= - \frac{1}{8} \xi \eta \zeta + \frac{1}{8} \xi \eta - \frac{1}{8} \xi \zeta + \frac{1}{8} \xi - \frac{1}{8} \eta \zeta + \frac{1}{8} \eta - \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{7} &= \frac{1}{8} \xi \eta \zeta + \frac{1}{8} \xi \eta + \frac{1}{8} \xi \zeta + \frac{1}{8} \xi + \frac{1}{8} \eta \zeta + \frac{1}{8} \eta + \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{8} &= - \frac{1}{8} \xi \eta \zeta - \frac{1}{8} \xi \eta - \frac{1}{8} \xi \zeta - \frac{1}{8} \xi + \frac{1}{8} \eta \zeta + \frac{1}{8} \eta + \frac{1}{8} \zeta + \frac{1}{8}\end{align*}\]

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Elemento hexaédrico de 27 nodos:

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\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \\ l^{(1)}_{3} \end{matrix} \right ] = \left [ \begin{matrix} \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \\ - \xi^{2} + 1 \\ \frac{1}{2} \xi^{2} + \frac{1}{2} \xi \end{matrix} \right ]\]

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\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \\ l^{(1)}_{3} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} & l^{(1)}_{2} & l^{(1)}_{3} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} & l^{(1)}_{2} & l^{(1)}_{3} \end{matrix} \right ] = \left [ \left [ \begin{matrix} l^{(3)}_{1} & l^{(3)}_{10} & l^{(3)}_{19} \\ l^{(3)}_{2} & l^{(3)}_{11} & l^{(3)}_{20} \\ l^{(3)}_{3} & l^{(3)}_{12} & l^{(3)}_{21} \end{matrix} \right ] \left [ \begin{matrix} l^{(3)}_{8} & l^{(3)}_{17} & l^{(3)}_{26} \\ l^{(3)}_{9} & l^{(3)}_{18} & l^{(3)}_{27} \\ l^{(3)}_{4} & l^{(3)}_{13} & l^{(3)}_{22} \end{matrix} \right ] \left [ \begin{matrix} l^{(3)}_{7} & l^{(3)}_{16} & l^{(3)}_{25} \\ l^{(3)}_{6} & l^{(3)}_{15} & l^{(3)}_{24} \\ l^{(3)}_{5} & l^{(3)}_{14} & l^{(3)}_{23} \end{matrix} \right ] \right ]\]

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\[\begin{align*} l^{(3)}_{1} &= \frac{1}{8} \xi^{2} \eta^{2} \zeta^{2} - \frac{1}{8} \xi^{2} \eta^{2} \zeta - \frac{1}{8} \xi^{2} \eta \zeta^{2} + \frac{1}{8} \xi^{2} \eta \zeta - \frac{1}{8} \xi \eta^{2} \zeta^{2} + \frac{1}{8} \xi \eta^{2} \zeta + \frac{1}{8} \xi \eta \zeta^{2} - \frac{1}{8} \xi \eta \zeta \\ l^{(3)}_{2} &= - \frac{1}{4} \xi^{2} \eta^{2} \zeta^{2} + \frac{1}{4} \xi^{2} \eta^{2} \zeta + \frac{1}{4} \xi^{2} \eta \zeta^{2} - \frac{1}{4} \xi^{2} \eta \zeta + \frac{1}{4} \eta^{2} \zeta^{2} - \frac{1}{4} \eta^{2} \zeta - \frac{1}{4} \eta \zeta^{2} + \frac{1}{4} \eta \zeta \\ l^{(3)}_{3} &= \frac{1}{8} \xi^{2} \eta^{2} \zeta^{2} - \frac{1}{8} \xi^{2} \eta^{2} \zeta - \frac{1}{8} \xi^{2} \eta \zeta^{2} + \frac{1}{8} \xi^{2} \eta \zeta + \frac{1}{8} \xi \eta^{2} \zeta^{2} - \frac{1}{8} \xi \eta^{2} \zeta - \frac{1}{8} \xi \eta \zeta^{2} + \frac{1}{8} \xi \eta \zeta \\ l^{(3)}_{12} &= - \frac{1}{4} \xi^{2} \eta^{2} \zeta^{2} + \frac{1}{4} \xi^{2} \eta^{2} + \frac{1}{4} \xi^{2} \eta \zeta^{2} - \frac{1}{4} \xi^{2} \eta - \frac{1}{4} \xi \eta^{2} \zeta^{2} + \frac{1}{4} \xi \eta^{2} + \frac{1}{4} \xi \eta \zeta^{2} - \frac{1}{4} \xi \eta \\ l^{(3)}_{21} &= \frac{1}{8} \xi^{2} \eta^{2} \zeta^{2} + \frac{1}{8} \xi^{2} \eta^{2} \zeta - \frac{1}{8} \xi^{2} \eta \zeta^{2} - \frac{1}{8} \xi^{2} \eta \zeta + \frac{1}{8} \xi \eta^{2} \zeta^{2} + \frac{1}{8} \xi \eta^{2} \zeta - \frac{1}{8} \xi \eta \zeta^{2} - \frac{1}{8} \xi \eta \zeta \\ l^{(3)}_{20} &= - \frac{1}{4} \xi^{2} \eta^{2} \zeta^{2} - \frac{1}{4} \xi^{2} \eta^{2} \zeta + \frac{1}{4} \xi^{2} \eta \zeta^{2} + \frac{1}{4} \xi^{2} \eta \zeta + \frac{1}{4} \eta^{2} \zeta^{2} + \frac{1}{4} \eta^{2} \zeta - \frac{1}{4} \eta \zeta^{2} - \frac{1}{4} \eta \zeta \\ l^{(3)}_{19} &= \frac{1}{8} \xi^{2} \eta^{2} \zeta^{2} + \frac{1}{8} \xi^{2} \eta^{2} \zeta - \frac{1}{8} \xi^{2} \eta \zeta^{2} - \frac{1}{8} \xi^{2} \eta \zeta - \frac{1}{8} \xi \eta^{2} \zeta^{2} - \frac{1}{8} \xi \eta^{2} \zeta + \frac{1}{8} \xi \eta \zeta^{2} + \frac{1}{8} \xi \eta \zeta \\ l^{(3)}_{10} &= - \frac{1}{4} \xi^{2} \eta^{2} \zeta^{2} + \frac{1}{4} \xi^{2} \eta^{2} + \frac{1}{4} \xi^{2} \eta \zeta^{2} - \frac{1}{4} \xi^{2} \eta + \frac{1}{4} \xi \eta^{2} \zeta^{2} - \frac{1}{4} \xi \eta^{2} - \frac{1}{4} \xi \eta \zeta^{2} + \frac{1}{4} \xi \eta \\ l^{(3)}_{11} &= \frac{1}{2} \xi^{2} \eta^{2} \zeta^{2} - \frac{1}{2} \xi^{2} \eta^{2} - \frac{1}{2} \xi^{2} \eta \zeta^{2} + \frac{1}{2} \xi^{2} \eta - \frac{1}{2} \eta^{2} \zeta^{2} + \frac{1}{2} \eta^{2} + \frac{1}{2} \eta \zeta^{2} - \frac{1}{2} \eta \\ l^{(3)}_{8} &= - \frac{1}{4} \xi^{2} \eta^{2} \zeta^{2} + \frac{1}{4} \xi^{2} \eta^{2} \zeta + \frac{1}{4} \xi^{2} \zeta^{2} - \frac{1}{4} \xi^{2} \zeta + \frac{1}{4} \xi \eta^{2} \zeta^{2} - \frac{1}{4} \xi \eta^{2} \zeta - \frac{1}{4} \xi \zeta^{2} + \frac{1}{4} \xi \zeta \\ l^{(3)}_{9} &= \frac{1}{2} \xi^{2} \eta^{2} \zeta^{2} - \frac{1}{2} \xi^{2} \eta^{2} \zeta - \frac{1}{2} \xi^{2} \zeta^{2} + \frac{1}{2} \xi^{2} \zeta - \frac{1}{2} \eta^{2} \zeta^{2} + \frac{1}{2} \eta^{2} \zeta + \frac{1}{2} \zeta^{2} - \frac{1}{2} \zeta \\ l^{(3)}_{13} &= - \frac{1}{4} \xi^{2} \eta^{2} \zeta^{2} + \frac{1}{4} \xi^{2} \eta^{2} \zeta + \frac{1}{4} \xi^{2} \zeta^{2} - \frac{1}{4} \xi^{2} \zeta - \frac{1}{4} \xi \eta^{2} \zeta^{2} + \frac{1}{4} \xi \eta^{2} \zeta + \frac{1}{4} \xi \zeta^{2} - \frac{1}{4} \xi \zeta \\ l^{(3)}_{22} &= \frac{1}{2} \xi^{2} \eta^{2} \zeta^{2} - \frac{1}{2} \xi^{2} \eta^{2} - \frac{1}{2} \xi^{2} \zeta^{2} + \frac{1}{2} \xi^{2} + \frac{1}{2} \xi \eta^{2} \zeta^{2} - \frac{1}{2} \xi \eta^{2} - \frac{1}{2} \xi \zeta^{2} + \frac{1}{2} \xi \\ l^{(3)}_{27} &= - \frac{1}{4} \xi^{2} \eta^{2} \zeta^{2} - \frac{1}{4} \xi^{2} \eta^{2} \zeta + \frac{1}{4} \xi^{2} \zeta^{2} + \frac{1}{4} \xi^{2} \zeta - \frac{1}{4} \xi \eta^{2} \zeta^{2} - \frac{1}{4} \xi \eta^{2} \zeta + \frac{1}{4} \xi \zeta^{2} + \frac{1}{4} \xi \zeta \\ l^{(3)}_{26} &= \frac{1}{2} \xi^{2} \eta^{2} \zeta^{2} + \frac{1}{2} \xi^{2} \eta^{2} \zeta - \frac{1}{2} \xi^{2} \zeta^{2} - \frac{1}{2} \xi^{2} \zeta - \frac{1}{2} \eta^{2} \zeta^{2} - \frac{1}{2} \eta^{2} \zeta + \frac{1}{2} \zeta^{2} + \frac{1}{2} \zeta \\ l^{(3)}_{17} &= - \frac{1}{4} \xi^{2} \eta^{2} \zeta^{2} - \frac{1}{4} \xi^{2} \eta^{2} \zeta + \frac{1}{4} \xi^{2} \zeta^{2} + \frac{1}{4} \xi^{2} \zeta + \frac{1}{4} \xi \eta^{2} \zeta^{2} + \frac{1}{4} \xi \eta^{2} \zeta - \frac{1}{4} \xi \zeta^{2} - \frac{1}{4} \xi \zeta \\ l^{(3)}_{18} &= \frac{1}{2} \xi^{2} \eta^{2} \zeta^{2} - \frac{1}{2} \xi^{2} \eta^{2} - \frac{1}{2} \xi^{2} \zeta^{2} + \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \eta^{2} \zeta^{2} + \frac{1}{2} \xi \eta^{2} + \frac{1}{2} \xi \zeta^{2} - \frac{1}{2} \xi \\ l^{(3)}_{14} &= - \xi^{2} \eta^{2} \zeta^{2} + \xi^{2} \eta^{2} + \xi^{2} \zeta^{2} - \xi^{2} + \eta^{2} \zeta^{2} - \eta^{2} - \zeta^{2} + 1 \\ l^{(3)}_{7} &= \frac{1}{8} \xi^{2} \eta^{2} \zeta^{2} - \frac{1}{8} \xi^{2} \eta^{2} \zeta + \frac{1}{8} \xi^{2} \eta \zeta^{2} - \frac{1}{8} \xi^{2} \eta \zeta - \frac{1}{8} \xi \eta^{2} \zeta^{2} + \frac{1}{8} \xi \eta^{2} \zeta - \frac{1}{8} \xi \eta \zeta^{2} + \frac{1}{8} \xi \eta \zeta \\ l^{(3)}_{6} &= - \frac{1}{4} \xi^{2} \eta^{2} \zeta^{2} + \frac{1}{4} \xi^{2} \eta^{2} \zeta - \frac{1}{4} \xi^{2} \eta \zeta^{2} + \frac{1}{4} \xi^{2} \eta \zeta + \frac{1}{4} \eta^{2} \zeta^{2} - \frac{1}{4} \eta^{2} \zeta + \frac{1}{4} \eta \zeta^{2} - \frac{1}{4} \eta \zeta \\ l^{(3)}_{5} &= \frac{1}{8} \xi^{2} \eta^{2} \zeta^{2} - \frac{1}{8} \xi^{2} \eta^{2} \zeta + \frac{1}{8} \xi^{2} \eta \zeta^{2} - \frac{1}{8} \xi^{2} \eta \zeta + \frac{1}{8} \xi \eta^{2} \zeta^{2} - \frac{1}{8} \xi \eta^{2} \zeta + \frac{1}{8} \xi \eta \zeta^{2} - \frac{1}{8} \xi \eta \zeta \\ l^{(3)}_{14} &= - \frac{1}{4} \xi^{2} \eta^{2} \zeta^{2} + \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta \zeta^{2} + \frac{1}{4} \xi^{2} \eta - \frac{1}{4} \xi \eta^{2} \zeta^{2} + \frac{1}{4} \xi \eta^{2} - \frac{1}{4} \xi \eta \zeta^{2} + \frac{1}{4} \xi \eta \\ l^{(3)}_{23} &= \frac{1}{8} \xi^{2} \eta^{2} \zeta^{2} + \frac{1}{8} \xi^{2} \eta^{2} \zeta + \frac{1}{8} \xi^{2} \eta \zeta^{2} + \frac{1}{8} \xi^{2} \eta \zeta + \frac{1}{8} \xi \eta^{2} \zeta^{2} + \frac{1}{8} \xi \eta^{2} \zeta + \frac{1}{8} \xi \eta \zeta^{2} + \frac{1}{8} \xi \eta \zeta \\ l^{(3)}_{24} &= - \frac{1}{4} \xi^{2} \eta^{2} \zeta^{2} - \frac{1}{4} \xi^{2} \eta^{2} \zeta - \frac{1}{4} \xi^{2} \eta \zeta^{2} - \frac{1}{4} \xi^{2} \eta \zeta + \frac{1}{4} \eta^{2} \zeta^{2} + \frac{1}{4} \eta^{2} \zeta + \frac{1}{4} \eta \zeta^{2} + \frac{1}{4} \eta \zeta \\ l^{(3)}_{25} &= \frac{1}{8} \xi^{2} \eta^{2} \zeta^{2} + \frac{1}{8} \xi^{2} \eta^{2} \zeta + \frac{1}{8} \xi^{2} \eta \zeta^{2} + \frac{1}{8} \xi^{2} \eta \zeta - \frac{1}{8} \xi \eta^{2} \zeta^{2} - \frac{1}{8} \xi \eta^{2} \zeta - \frac{1}{8} \xi \eta \zeta^{2} - \frac{1}{8} \xi \eta \zeta \\ l^{(3)}_{16} &= - \frac{1}{4} \xi^{2} \eta^{2} \zeta^{2} + \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta \zeta^{2} + \frac{1}{4} \xi^{2} \eta + \frac{1}{4} \xi \eta^{2} \zeta^{2} - \frac{1}{4} \xi \eta^{2} + \frac{1}{4} \xi \eta \zeta^{2} - \frac{1}{4} \xi \eta \\ l^{(3)}_{15} &= \frac{1}{2} \xi^{2} \eta^{2} \zeta^{2} - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} \eta \zeta^{2} - \frac{1}{2} \xi^{2} \eta - \frac{1}{2} \eta^{2} \zeta^{2} + \frac{1}{2} \eta^{2} - \frac{1}{2} \eta \zeta^{2} + \frac{1}{2} \eta\end{align*}\]

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Conclusiones

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Se implementó la forma de cálculo en Python [3], usando IPython notebook [4], Numpy [5], Sympy [6], Matplotlib [7] y Mayavi [8]; también se realizo una tabla comparativa de los tiempos de ejecución de las funciones:

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NodosElemento 1D
20.003 seg.
30.003 seg.
40.055 seg.
50.074 seg.
60.167 seg.
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NodosElemento 2D
40.021 seg.
90.006 seg.
160.345 seg.
250.469 seg.
361.530 seg.
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NodosElemento 3D
80.060 seg.
270.014 seg.
644.637 seg.
1257.197 seg.
21657.337 seg.
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Para generar elementos de mayor grado y reducir el tiempo de cálculo se optimizará el código y se usara Numba [9].

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Se estudiarán las propiedades de las matrices \(l^{(2)}\):

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    Rango

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    Traza

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    Determinante

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En el caso de las matrices \(l^{(3)}\):

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    Hiperdeterminante

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Los archivos del trabajo se encuentran en https://github.com/ClaudioVZ/Teoria-FEM-Python.

+ + diff --git a/funciones_forma/OEBPS/Text/Section0007.xhtml b/funciones_forma/OEBPS/Text/Section0007.xhtml new file mode 100644 index 0000000..2a8a0d6 --- /dev/null +++ b/funciones_forma/OEBPS/Text/Section0007.xhtml @@ -0,0 +1,35 @@ + + + + + + + + + + +

Referencias

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    +
  1. O.C.Zienkiewicz, R.L.Taylor; El Método de los Elementos Finitos Volumen 1. McGrall-Hill; CIMNE, 1994.
  2. + +
  3. C.V.Zapata. Matrices 3D: Multiplicación. 2013.
  4. + +
  5. Lenguaje de programación http://www.python.org/
  6. + +
  7. Intérprete interactivo http://ipython.org/
  8. + +
  9. Biblioteca de funciones matemáticas y operaciones con arreglos http://www.numpy.org/
  10. + +
  11. Biblioteca para matemática simbólica http://sympy.org/en/index.html
  12. + +
  13. Biblioteca para gráficos 2D y 3D http://matplotlib.org/
  14. + +
  15. Biblioteca para gráficos 3D http://code.enthought.com/projects/mayavi/
  16. + +
  17. Compilador jit especializado http://numba.pydata.org/
  18. +
+ + diff --git a/funciones_forma/OEBPS/content.opf b/funciones_forma/OEBPS/content.opf new file mode 100644 index 0000000..dd538ac --- /dev/null +++ b/funciones_forma/OEBPS/content.opf @@ -0,0 +1,83 @@ + + + + urn:uuid:8641db26-0737-4eff-b877-f762a71848ae + Funcines de forma lagrangianas + ClaudioVZ + es-BO + 2013-12-27 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/funciones_forma/OEBPS/toc.ncx b/funciones_forma/OEBPS/toc.ncx new file mode 100644 index 0000000..2d1502f --- /dev/null +++ b/funciones_forma/OEBPS/toc.ncx @@ -0,0 +1,100 @@ + + + + + + + + + + + Unknown + + + + + Funciones de forma Lagrangianas Generación de funciones usando matrices + + + + + + Índice + + + + + + Notación + + + + + + Funciones de forma unidimensionales + + + + + Definición + + + + + + Ejemplo \(l^{(1)}\) + + + + + + + Funciones de forma bidimensionales + + + + + Método + + + + + + Ejemplo \(l^{(2)}\) + + + + + + + Funciones de forma tridimensionales + + + + + Método + + + + + + Ejemplo \(l^{(3)}\) + + + + + + + Conclusiones + + + + + + Referencias + + + + + \ No newline at end of file diff --git a/funciones_forma/mimetype b/funciones_forma/mimetype new file mode 100644 index 0000000..57ef03f --- /dev/null +++ b/funciones_forma/mimetype @@ -0,0 +1 @@ +application/epub+zip \ No newline at end of file

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