Reconstrucción de tensiones para el método de elementos finitos con mallas poligonales
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Universidad Industrial de Santander
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El método de los elementos finitos es una de las herramientas numéricas más utilizadas para el diseño en ingeniería. En los últimos años se han desarrollado nuevas aproximaciones numéricas para extender el uso del método de elementos finitos a mallas poligonales. Dichas aproximaciones mejoran la precisión de la solución y aumentan la flexibilidad en el mallado. Sin embargo, como toda aproximación, es necesario cuantificar el valor del error inducido para poder validar los resultados obtenidos. En este trabajo se presenta el uso de una técnica de estimación del error de discretización para mallas de elementos finitos poligonales. La técnica está basada en la reconstrucción de la solución en tensiones mediante un procedimiento de mínimos cuadrados ponderados que considera la influencia de las ecuaciones de equilibrio. Se ha utilizado un problema con solución exacta para evaluar la efectividad del estimador, obteniendo buenos resultados a nivel local y global.
The finite element method is one of the most used numerical tools for engineering design. In recent years, novel numerical approximations have been proposed to extend the finite element method to meshes using arbitrary polygons. Such approximations are aimed to improve accuracy and increase the flexibility during the meshing process. However, as any approximation, they exhibit an error that requires to be quantified in order to validate the numerical results. In this paper, we present a technique to estimate the discretization error in energy norm for arbitrary polygonal finite element meshes. This recovery-based technique uses a moving least squares approach that considers constraints to represent the equilibrium equations. We use two benchmark problems to evaluate the effectivity of the error estimator, with good results both locally and globally.
The finite element method is one of the most used numerical tools for engineering design. In recent years, novel numerical approximations have been proposed to extend the finite element method to meshes using arbitrary polygons. Such approximations are aimed to improve accuracy and increase the flexibility during the meshing process. However, as any approximation, they exhibit an error that requires to be quantified in order to validate the numerical results. In this paper, we present a technique to estimate the discretization error in energy norm for arbitrary polygonal finite element meshes. This recovery-based technique uses a moving least squares approach that considers constraints to represent the equilibrium equations. We use two benchmark problems to evaluate the effectivity of the error estimator, with good results both locally and globally.
Keywords
Finite element method, polygonal elements, error estimation, estress recovery, adaptivity, moving least squares, Método de elementos finitos, elementos poligonales, estimación del error, adaptatividad, reconstrucción de tensiones, ajuste por mínimos cuadrados