Elementos finitos (Métodos de solución Ecuaciones diferenciales) (página 2)

• Componente 1-2

Así tenemos:

De lo anterior se llega a la siguiente solución, además de reemplazar los resultados de las cuatro integrales e igualar el promedio a cero.

Matriz global

Comprobación de resultados

A continuación se presenta la comprobación de los resultados obtenidos, por medio del software de cálculo Matlab.

5.1 Método de colocación

% SOLUCIONES APROXIMADAS DE LA ECUACION DIFERENCIAL %

% Ay"(x)+By(x)-Cy(x)-x=0 %

% EN EL INTERVALO 0 In polyfit at 72

| x=0.30000| y=0.683821 | y=0.138980 | y=0.126681 | y=0.090000 | y=0.358167 | y=0.563439 |

| Error(e)| e=0.000000 | e=-0.544842 | e=-0.557140 | e=-0.593821 | e=-0.325654 | e=-0.120382 |

===============================

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

| x=0.32000| y=0.714882 | y=0.153152 | y=0.140409 | y=0.102400 | y=0.379027 | y=0.600997 |

| Error(e)| e=0.000000 | e=-0.561730 | e=-0.574473 | e=-0.612482 | e=-0.335855 | e=-0.113885 |

===============================

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

| x=0.34000| y=0.744276 | y=0.167938 | y=0.154797 | y=0.115600 | y=0.399510 | y=0.638552 |

| Error(e)| e=0.000000 | e=-0.576338 | e=-0.589480 | e=-0.628676 | e=-0.344766 | e=-0.105724 |

===============================

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

| x=0.36000| y=0.772034 | y=0.183338 | y=0.169845 | y=0.129600 | y=0.419617 | y=0.676101 |

| Error(e)| e=0.000000 | e=-0.588697 | e=-0.602190 | e=-0.642434 | e=-0.352417 | e=-0.095933 |

===============================

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

| x=0.38000| y=0.798188 | y=0.199350 | y=0.185553 | y=0.144400 | y=0.439348 | y=0.713641 |

| Error(e)| e=0.000000 | e=-0.598838 | e=-0.612635 | e=-0.653788 | e=-0.358840 | e=-0.084547 |

===============================

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

| x=0.40000| y=0.822769 | y=0.215977 | y=0.201921 | y=0.160000 | y=0.458704 | y=0.751167 |

| Error(e)| e=0.000000 | e=-0.606793 | e=-0.620848 | e=-0.662769 | e=-0.364065 | e=-0.071602 |

===============================

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

| x=0.42000| y=0.845811 | y=0.233216 | y=0.218950 | y=0.176400 | y=0.477686 | y=0.788672 |

| Error(e)| e=0.000000 | e=-0.612595 | e=-0.626861 | e=-0.669411 | e=-0.368126 | e=-0.057139 |

===============================

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

| x=0.44000| y=0.867347 | y=0.251069 | y=0.236639 | y=0.193600 | y=0.496295 | y=0.826146 |

| Error(e)| e=0.000000 | e=-0.616278 | e=-0.630708 | e=-0.673747 | e=-0.371052 | e=-0.041201 |

===============================

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

| x=0.46000| y=0.887410 | y=0.269536 | y=0.254989 | y=0.211600 | y=0.514534 | y=0.863576 |

| Error(e)| e=0.000000 | e=-0.617874 | e=-0.632422 | e=-0.675810 | e=-0.372876 | e=-0.023835 |

===============================

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

| x=0.48000| y=0.906035 | y=0.288616 | y=0.273998 | y=0.230400 | y=0.532405 | y=0.900941 |

| Error(e)| e=0.000000 | e=-0.617419 | e=-0.632037 | e=-0.675635 | e=-0.373630 | e=-0.005094 |

======================================

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

Warning: Polynomial is not unique; degree >= number of data points.

> In polyfit at 72

| x=0.50000| y=0.923256 | y=0.308309 | y=0.293668 | y=0.250000 | y=0.549913 | y=0.938216 |

| Error(e)| e=0.000000 | e=-0.614947 | e=-0.629588 | e=-0.673256 | e=-0.373342 | e=0.014961 |

=====================================

Gráficas

A continuación se presentan los gráficos correspondientes a las distintas soluciones obtenidas por los métodos descritos anteriormente.

Conclusiones

• Aunque no se determino la solución exacta de la ecuación diferencial dada, podemos notar que al aumentar el número de elementos para el análisis por elementos finitos, según la teoría el contorno de la curva se aproximara a la solución exacta, para el caso del análisis por elementos finitos se realizo una prueba para 50 y 300.

• El análisis de resolución de la ecuación diferencial dada, realizado con los métodos de mínimos cuadrados, galerkin y método de colocación; los gráficos referentes a cada método obtenidos por medio del software de cálculo Matlab, difieren el uno del otro, la solución que más se aproxima a el método de elementos finitos, es el análisis realizado por el método de colocación como se puede observar en los anteriores gráficos presentados.

Autor:

Edwin Andrés Correa Quintana

Wilson Ferney Galeano

UNIVERSIDAD DE ANTIOQUIA

FACULTAD DE INGENIERÍA

DEPARTAMENTO DE INGENIERÍA MECÁNICA

MEDELLÍN

2010