Thesis UN MÉTODO DE ELEMENTOS FINITOS ESTABILIZADOS DE BAJO ORDEN A DIVERGENCIA NULA PARA EL PROBLEMA DE BOUSSINESQ ESTACIONARIO
Loading...
Date
2018
Authors
NARANJO PEÑALOZA, CESAR IGNACIO
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
En este trabajo se propone una familia de Métodos de Elementos Finitos Mixtos estabilizadospara la simulación numérica de un problema generalizado de Boussinesq en dos ytres dimensiones, el cual describe el movimiento de un fluido incompresible sujeto a unafuente de calor, lo que resulta en un problema de Navier-Stokes acoplado con una ecuaciónde Advección-Difusión. El método utiliza elementos continuos de orden 1 para la velocidad,elementos discontinuos de grado 0 para la presión y elementos continuos de grado1 para la temperatura. Mediante un post-proceso, es posible obtener un campo de velocidada divergencia exactamente nula, a través del uso de funciones de Raviart–Thomas,para ser utilizado como campo convectivo en las ecuaciones del problema. El esquema propuestoestá basado en métodos estabilizados de bajo orden que permiten el uso de espacioselementos finitos de bajo costo computacional y que son construidos para asegurar una mayorestabilidad en presencia de capas límites. Estas estabilizaciones pueden corresponder acualquier método estabilizado sujeto a ciertas condiciones y restricciones que se proveen. Seprueba la existencia y estabilidad de soluciones para las soluciones discretas y se obtienenestimaciones de error dependientes del tamaño de la malla para soluciones suficientementepequeñas y suaves. Finalmente, se realizan pruebas numéricas considerando métodos estabilizadosadecuados existentes en la literatura, para validar los análisis y el desempeñode las estimaciones de error, los cuales muestran una buena estabilidad incluso en casoslímites.
In this work, a family of Stabilized Finite Element Methods is proposed for the numericalsimulation of a generalized Boussinesq problem in two and three dimensions, whichdescribes the motion of an incompressible flow under a heat source, which results in a Navier-Stokes problem coupled with an Advection-Difussion Equation. The method useorder 1 continous elements for the velocity, order 0 discontinous elements for the pressureand order 1 continuos elements for the temperature. Through a post-process, is possibleto obtain an exaclty divergence-free velocity field through the use of Raviart–Thomas functions,which is used as the advective field in the equations of the problem. The proposedscheme is based on low order stabilized methods which allow the utilization of finite elementspaces whit a low computational cost which are built to ensure a higher stabilitywhen boundary layers appear. This stabilization terms can be taken form any stabilizedmethod satisfying some conditions and restrictions provided in this work. Existence andstability for the discrete solutions are proven and error estimations depending on the meshsize are obtained for small and soft enough solutions. Finally, numerical tests are doneconsidering suitable stabilized methods from the literature to validate the analysis and theperformance of the error estimations, which show a good stabilty even in limit cases.
In this work, a family of Stabilized Finite Element Methods is proposed for the numericalsimulation of a generalized Boussinesq problem in two and three dimensions, whichdescribes the motion of an incompressible flow under a heat source, which results in a Navier-Stokes problem coupled with an Advection-Difussion Equation. The method useorder 1 continous elements for the velocity, order 0 discontinous elements for the pressureand order 1 continuos elements for the temperature. Through a post-process, is possibleto obtain an exaclty divergence-free velocity field through the use of Raviart–Thomas functions,which is used as the advective field in the equations of the problem. The proposedscheme is based on low order stabilized methods which allow the utilization of finite elementspaces whit a low computational cost which are built to ensure a higher stabilitywhen boundary layers appear. This stabilization terms can be taken form any stabilizedmethod satisfying some conditions and restrictions provided in this work. Existence andstability for the discrete solutions are proven and error estimations depending on the meshsize are obtained for small and soft enough solutions. Finally, numerical tests are doneconsidering suitable stabilized methods from the literature to validate the analysis and theperformance of the error estimations, which show a good stabilty even in limit cases.
Description
Catalogado desde la version PDF de la tesis.
Keywords
BOUSSINESQ , DIVERGENCIA NULA , FEM , METODOS ESTABILIZADOS
