\(l^{(1)}\) Función de forma unidimensional Lagrangiana
+\(l^{(2)}\) Función de forma bidimensional Lagrangiana
+\(l^{(3)}\) Función de forma trididimensional Lagrangiana
+Las funciones de forma Lagrangianas [1] son polinomios con dominio \(-1 \leqslant x \leqslant 1\), la fórmula para generarlos es la siguiente:
+ +\[l^{(n)}_{j} = \prod_{i=0, i \neq j}^k \frac{x - x_{i}}{x_{j} - x_{i}}\]
+ +Función de forma Lagrangiana unidimensional de 2 nodos:
+ +\[\begin{align*} l^{(1)}_{1} &= - \frac{1}{2} \xi + \frac{1}{2} \\ l^{(1)}_{2} &= \frac{1}{2} \xi + \frac{1}{2}\end{align*}\]
+ +Función de forma Lagrangiana unidimensional de 3 nodos:
+ +\[\begin{align*} l^{(1)}_{1} &= \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \\ l^{(1)}_{2} &= - \xi^{2} + 1 \\ l^{(1)}_{3} &= \frac{1}{2} \xi^{2} + \frac{1}{2} \xi\end{align*}\]
+ +
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.
+ +Elemento rectangular de 4 nodos:
+ +\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \end{matrix} \right ] = \left [ \begin{matrix} - \frac{1}{2} \xi + \frac{1}{2} \\ \frac{1}{2} \xi + \frac{1}{2} \end{matrix} \right ]\]
+ +\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} & l^{(1)}_{2} \end{matrix} \right ] = \left [ \begin{matrix} l^{(2)}_{1} & l^{(2)}_{4} \\ l^{(2)}_{2} & l^{(2)}_{3} \end{matrix} \right ]\]
+ +\[\begin{align*} l^{(2)}_{1} &= \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta - \frac{1}{4} \xi \eta^{2} + \frac{1}{4} \xi \eta \\ l^{(2)}_{2} &= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} \eta + \frac{1}{2} \eta^{2} - \frac{1}{2} \eta \\ l^{(2)}_{3} &= \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta + \frac{1}{4} \xi \eta^{2} - \frac{1}{4} \xi \eta \\ l^{(2)}_{4} &= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \eta^{2} + \frac{1}{2} \xi\end{align*}\]
+ +
Elemento rectangular de 9 nodos:
+ +\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \\ l^{(1)}_{3} \end{matrix} \right ] = \left [ \begin{matrix} \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \\ - \xi^{2} + 1 \\ \frac{1}{2} \xi^{2} + \frac{1}{2} \xi \end{matrix} \right ]\]
+ +\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \\ l^{(1)}_{3} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} & l^{(1)}_{2} & l^{(1)}_{3} \end{matrix} \right ] = \left [ \begin{matrix} l^{(2)}_{1} & l^{(2)}_{8} & l^{(2)}_{7} \\ l^{(2)}_{2} & l^{(2)}_{9} & l^{(2)}_{6} \\ l^{(2)}_{3} & l^{(2)}_{4} & l^{(2)}_{5} \end{matrix} \right ]\]
+ +\[\begin{align*} l^{(2)}_{1} &= \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta - \frac{1}{4} \xi \eta^{2} + \frac{1}{4} \xi \eta \\ l^{(2)}_{2} &= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} \eta + \frac{1}{2} \eta^{2} - \frac{1}{2} \eta \\ l^{(2)}_{3} &= \frac{1}{4} \xi^{2} \eta^{2} - \frac{1}{4} \xi^{2} \eta + \frac{1}{4} \xi \eta^{2} - \frac{1}{4} \xi \eta \\ l^{(2)}_{4} &= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \eta^{2} + \frac{1}{2} \xi \\ l^{(2)}_{5} &= \frac{1}{4} \xi^{2} \eta^{2} + \frac{1}{4} \xi^{2} \eta + \frac{1}{4} \xi \eta^{2} + \frac{1}{4} \xi \eta \\ l^{(2)}_{6} &= - \frac{1}{2} \xi^{2} \eta^{2} - \frac{1}{2} \xi^{2} \eta + \frac{1}{2} \eta^{2} + \frac{1}{2} \eta \\ l^{(2)}_{7} &= \frac{1}{4} \xi^{2} \eta^{2} + \frac{1}{4} \xi^{2} \eta - \frac{1}{4} \xi \eta^{2} - \frac{1}{4} \xi \eta \\ l^{(2)}_{8} &= - \frac{1}{2} \xi^{2} \eta^{2} + \frac{1}{2} \xi^{2} + \frac{1}{2} \xi \eta^{2} - \frac{1}{2} \xi \\ l^{(2)}_{9} &= \xi^{2} \eta^{2} - \xi^{2} - \eta^{2} + 1\end{align*}\]
+ +
Las \(l^{(3)}\) también se generar a partir de \(l^{(1)}\), pero realizando una multiplicación con matrices 3D [2].
+ +\[\left [ l^{(1)} \right ] \left [ l^{(1)} \right ]^{T} \left [ l^{(1)} \right ]^{T} = \left [ l^{(3)} \right ]\]
+ +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 \(\left [ l^{(3)} \right ]\) es una matriz 3D cúbica.
+ +Elemento hexaédrico de 8 nodos:
+ +\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \end{matrix} \right ] = \left [ \begin{matrix} - \frac{1}{2} \xi + \frac{1}{2} \\ \frac{1}{2} \xi + \frac{1}{2} \end{matrix} \right ]\]
+ +\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} & l^{(1)}_{2} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} & l^{(1)}_{2} \end{matrix} \right ] = \left [ \left [ \begin{matrix} l^{(3)}_{1} & l^{(3)}_{5} \\ l^{(3)}_{2} & l^{(3)}_{6} \end{matrix} \right ] \left [ \begin{matrix} l^{(3)}_{4} & l^{(3)}_{8} \\ l^{(3)}_{3} & l^{(3)}_{7} \end{matrix} \right ] \right ]\]
+ +\[\begin{align*} l^{(3)}_{1} &= - \frac{1}{8} \xi \eta \zeta + \frac{1}{8} \xi \eta + \frac{1}{8} \xi \zeta - \frac{1}{8} \xi + \frac{1}{8} \eta \zeta - \frac{1}{8} \eta - \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{2} &= \frac{1}{8} \xi \eta \zeta - \frac{1}{8} \xi \eta - \frac{1}{8} \xi \zeta + \frac{1}{8} \xi + \frac{1}{8} \eta \zeta - \frac{1}{8} \eta - \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{6} &= - \frac{1}{8} \xi \eta \zeta - \frac{1}{8} \xi \eta + \frac{1}{8} \xi \zeta + \frac{1}{8} \xi - \frac{1}{8} \eta \zeta - \frac{1}{8} \eta + \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{5} &= \frac{1}{8} \xi \eta \zeta + \frac{1}{8} \xi \eta - \frac{1}{8} \xi \zeta - \frac{1}{8} \xi - \frac{1}{8} \eta \zeta - \frac{1}{8} \eta + \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{4} &= \frac{1}{8} \xi \eta \zeta - \frac{1}{8} \xi \eta + \frac{1}{8} \xi \zeta - \frac{1}{8} \xi - \frac{1}{8} \eta \zeta + \frac{1}{8} \eta - \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{3} &= - \frac{1}{8} \xi \eta \zeta + \frac{1}{8} \xi \eta - \frac{1}{8} \xi \zeta + \frac{1}{8} \xi - \frac{1}{8} \eta \zeta + \frac{1}{8} \eta - \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{7} &= \frac{1}{8} \xi \eta \zeta + \frac{1}{8} \xi \eta + \frac{1}{8} \xi \zeta + \frac{1}{8} \xi + \frac{1}{8} \eta \zeta + \frac{1}{8} \eta + \frac{1}{8} \zeta + \frac{1}{8} \\ l^{(3)}_{8} &= - \frac{1}{8} \xi \eta \zeta - \frac{1}{8} \xi \eta - \frac{1}{8} \xi \zeta - \frac{1}{8} \xi + \frac{1}{8} \eta \zeta + \frac{1}{8} \eta + \frac{1}{8} \zeta + \frac{1}{8}\end{align*}\]
+ +
Elemento hexaédrico de 27 nodos:
+ +\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \\ l^{(1)}_{3} \end{matrix} \right ] = \left [ \begin{matrix} \frac{1}{2} \xi^{2} - \frac{1}{2} \xi \\ - \xi^{2} + 1 \\ \frac{1}{2} \xi^{2} + \frac{1}{2} \xi \end{matrix} \right ]\]
+ +\[\left [ \begin{matrix} l^{(1)}_{1} \\ l^{(1)}_{2} \\ l^{(1)}_{3} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} & l^{(1)}_{2} & l^{(1)}_{3} \end{matrix} \right ] \left [ \begin{matrix} l^{(1)}_{1} & l^{(1)}_{2} & l^{(1)}_{3} \end{matrix} \right ] = \left [ \left [ \begin{matrix} l^{(3)}_{1} & l^{(3)}_{10} & l^{(3)}_{19} \\ l^{(3)}_{2} & l^{(3)}_{11} & l^{(3)}_{20} \\ l^{(3)}_{3} & l^{(3)}_{12} & l^{(3)}_{21} \end{matrix} \right ] \left [ \begin{matrix} l^{(3)}_{8} & l^{(3)}_{17} & l^{(3)}_{26} \\ l^{(3)}_{9} & l^{(3)}_{18} & l^{(3)}_{27} \\ l^{(3)}_{4} & l^{(3)}_{13} & l^{(3)}_{22} \end{matrix} \right ] \left [ \begin{matrix} l^{(3)}_{7} & l^{(3)}_{16} & l^{(3)}_{25} \\ l^{(3)}_{6} & l^{(3)}_{15} & l^{(3)}_{24} \\ l^{(3)}_{5} & l^{(3)}_{14} & l^{(3)}_{23} \end{matrix} \right ] \right ]\]
+ +\[\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*}\]
+ +
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:
+ +| Nodos | + +Elemento 1D | +
|---|---|
| 2 | + +0.003 seg. | +
| 3 | + +0.003 seg. | +
| 4 | + +0.055 seg. | +
| 5 | + +0.074 seg. | +
| 6 | + +0.167 seg. | +
| Nodos | + +Elemento 2D | +
|---|---|
| 4 | + +0.021 seg. | +
| 9 | + +0.006 seg. | +
| 16 | + +0.345 seg. | +
| 25 | + +0.469 seg. | +
| 36 | + +1.530 seg. | +
| Nodos | + +Elemento 3D | +
|---|---|
| 8 | + +0.060 seg. | +
| 27 | + +0.014 seg. | +
| 64 | + +4.637 seg. | +
| 125 | + +7.197 seg. | +
| 216 | + +57.337 seg. | +
Para generar elementos de mayor grado y reducir el tiempo de cálculo se optimizará el código y se usara Numba [9].
+ +Se estudiarán las propiedades de las matrices \(l^{(2)}\):
+ +Rango
+Traza
+Determinante
+En el caso de las matrices \(l^{(3)}\):
+ +Hiperdeterminante
+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 @@ + + + + + +