The finite element method is an important, widely used numerical technique for solving partial differential equations. This technique utilizes basis functions for approximating the geometry and the variation of the solution field over finite regions, or elements, of the domain. These basis functions are generally formed by combinations of polynomials. In the past, the polynomial order of the basis has been low-typically of linear and quadratic order. However, in recent years so-called p and hp methods have been developed, which may elevate the order of the basis to arbitrary levels with the aim of accelerating the convergence of the numerical solution. The increasing complexity of numerical basis functions poses a significant challenge to visualization systems. In the past, such systems have been loosely coupled to simulation packages, exchanging data via file transfer, and internally reimplementing the basis functions in order to perform interpolation and implement visualization algorithms. However, as the basis functions become more complex and, in some cases, proprietary in nature, it becomes increasingly difficult if not impossible to reimplement them within the visualization system. Further, most visualization systems typically process linear primitives, in part to take advantage of graphics hardware and, in part, due to the inherent simplicity of the resulting algorithms. Thus, visualization of higher-order finite elements requires tessellating the basis to produce data compatible with existing visualization systems. In this paper, we describe adaptive methods that automatically tessellate complex finite element basis functions using a flexible and extensible software framework. These methods employ a recursive, edge-based subdivision algorithm driven by a set of error metrics including geometric error, solution error, and error in image space. Further, we describe advanced pretessellation techniques that guarantees capture of the critical points of the polynomial basis. The framework has been designed using the adaptor design pattern, meaning that the visualization system need not reimplement basis functions, rather it communicates with the simulation package via simple programmatic queries. We demonstrate our method on several examples, and have implemented the framework in the open-source visualization system VTK.
Este trabajo tiene como objetivo el desarrollo de una herramienta de software bajo la plataforma Matlab® que permite al estudiante la apropiación del conocimiento acerca de lógica difusa aplicada en el área de control industrial. Diversas investigaciones han demostrado que la implementación de algoritmos inteligentes mejora los resultados en ciertos procesos como optimización de parámetros, procesamiento de señales, y reconocimiento de patrones, entre otros. Sin embargo, su uso todavía no es muy difundido pues se trata de metodologías relativamente recientes y no todas las plataformas computacionales incluyen herramientas con algoritmos inteligentes o en su defecto son muy costosas. La herramienta presenta una metodología guiada paso a paso para facilitar la aplicación de los conceptos. Con esta herramienta el estudiante puede relacionar los conceptos teóricos sobre lógica difusa con la realidad industrial, poner a prueba sus conocimientos adquiridos y trabajar varios casos prácticos con el diseño del sistema difuso. Los estudiantes a través de una encuesta, asignaron una valoración positiva a la implementación de la herramienta.
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