A local projection stabilization/continuous Galerkin--Petrov method for incompressible flow problems

The local projection stabilization (LPS) method in space is consid-ered to approximate the evolutionary Oseen equations. Optimal error bounds independent of the viscosity parameter are obtained in the continuous-in-time case for the approximations of both velocity and pressure. In addition, the fully discrete case in combination with higher order continuous Galerkin--Petrov (cGP) methods is studied. Error estimates of order k + 1 are proved, where k denotes the polynomial degree in time, assuming that the convective term is time-independent. Numerical results show that the predicted order is also achieved in the general case of time-dependent convective terms

On really locking-free mixed finite element methods for the transient incompressible Stokes equations

Inf-sup stable mixed methods for the steady incompressible Stokes equations that relax the divergence constraint are often claimed to deliver locking-free discretizations. However, this relaxation leads to a pressure-dependent contribution in the velocity error, which is proportional to the inverse of the viscosity, thus giving rise to a (different) locking phenomenon. However, a recently proposed modification of the right hand side alone leads to a discretization that is really locking-free, i.e., its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity. In this contribution, we extend this approach to the transient incompressible Stokes equations, where besides the right hand side also the velocity time derivative requires an improved space discretization. Semi-discrete and fully-discrete a-priori velocity and pressure error estimates are derived, which show beautiful robustness properties. Two numerical examples illustrate the superior accuracy of pressure-robust space discretizations in the case of small viscosities

On really locking-free mixed finite element methods for the transient incompressible Stokes equations

Inf-sup stable mixed methods for the steady incompressible Stokes equations that relax the divergence constraint are often claimed to deliver locking-free discretizations. However, this relaxation leads to a pressure-dependent contribution in the velocity error, which is proportional to the inverse of the viscosity, thus giving rise to a (different) locking phenomenon. However, a recently proposed modification of the right hand side alone leads to a discretization that is really locking-free, i.e., its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity. In this contribution, we extend this approach to the transient incompressible Stokes equations, where besides the right hand side also the velocity time derivative requires an improved space discretization. Semi-discrete and fully-discrete a-priori velocity and pressure error estimates are derived, which show beautiful robustness properties. Two numerical examples illustrate the superior accuracy of pressure-robust space discretizations in the case of small viscosities

Development and applications of the Finite Point Method to compressible aerodynamics problems

This work deals with the development and application of the Finite Point Method (FPM) to compressible aerodynamics problems. The research focuses mainly on investigating the capabilities of the meshless technique to address practical problems, one of the most outstanding issues in meshless methods. The FPM spatial approximation is studied firstly, with emphasis on aspects of the methodology that can be improved to increase its robustness and accuracy. Suitable ranges for setting the relevant approximation parameters and the performance likely to be attained in practice are determined. An automatic procedure to adjust the approximation parameters is also proposed to simplify the application of the method, reducing problem- and user-dependence without affecting the flexibility of the meshless technique. The discretization of the flow equations is carried out following wellestablished approaches, but drawing on the meshless character of the methodology. In order to meet the requirements of practical applications, the procedures are designed and implemented placing emphasis on robustness and efficiency (a simplification of the basic FPM technique is proposed to this end). The flow solver is based on an upwind spatial discretization of the convective fluxes (using the approximate Riemann solver of Roe) and an explicit time integration scheme. Two additional artificial diffusion schemes are also proposed to suit those cases of study in which computational cost is a major concern. The performance of the flow solver is evaluated in order to determine the potential of the meshless approach. The accuracy, computational cost and parallel scalability of the method are studied in comparison with a conventional FEM-based technique. Finally, practical applications and extensions of the flow solution scheme are presented. The examples provided are intended not only to show the capabilities of the FPM, but also to exploit meshless advantages. Automatic hadaptive procedures, moving domain and fluid-structure interaction problems, as well as a preliminary approach to solve high-Reynolds viscous flows, are a sample of the topics explored. All in all, the results obtained are satisfactorily accurate and competitive in terms of computational cost (if compared with a similar mesh-based implementation). This indicates that meshless advantages can be exploited with efficiency and constitutes a good starting point towards more challenging applications.En este trabajo se aborda el desarrollo del Método de Puntos Finitos (MPF) y su aplicación a problemas de aerodinámica de flujos compresibles. El objetivo principal es investigar el potencial de la técnica sin malla para la solución de problemas prácticos, lo cual constituye una de las limitaciones más importantes de los métodos sin malla. En primer lugar se estudia la aproximación espacial en el MPF, haciendo hincapié en aquéllos aspectos que pueden ser mejorados para incrementar la robustez y exactitud de la metodología. Se determinan rangos adecuados para el ajuste de los parámetros de la aproximación y su comportamiento en situaciones prácticas. Se propone además un procedimiento de ajuste automático de estos parámetros a fin de simplificar la aplicación del método y reducir la dependencia de factores como el tipo de problema y la intervención del usuario, sin afectar la flexibilidad de la técnica sin malla. A continuación se aborda el esquema de solución de las ecuaciones del flujo. La discretización de las mismas se lleva a cabo siguiendo métodos estándar, pero aprovechando las características de la técnica sin malla. Con el objetivo de abordar problemas prácticos, se pone énfasis en la robustez y eficiencia de la implementación numérica (se propone además una simplificación del procedimiento de solución). El comportamiento del esquema se estudia en detalle para evaluar su potencial y se analiza su exactitud, coste computacional y escalabilidad, todo ello en comparación con un método convencional basado en Elementos Finitos. Finalmente se presentan distintas aplicaciones y extensiones de la metodología desarrollada. Los ejemplos numéricos pretenden demostrar las capacidades del método y también aprovechar las ventajas de la metodología sin malla en áreas en que la misma puede ser de especial interés. Los problemas tratados incluyen, entre otras características, el refinamiento automático de la discretización, la presencia de fronteras móviles e interacción fluido-estructura, como así también una aplicación preliminar a flujos compresibles de alto número de Reynolds. Los resultados obtenidos muestran una exactitud satisfactoria. Además, en comparación con una técnica similar basada en Elementos Finitos, demuestran ser competitivos en términos del coste computacional. Esto indica que las ventajas de la metodología sin malla pueden ser explotadas con eficiencia, lo cual constituye un buen punto de partida para el desarrollo de ulteriores aplicaciones.Postprint (published version

Large scale finite element solvers for the large eddy simulation of incompressible turbulent flows

Premi extraordinari doctorat UPC curs 2015-2016, àmbit d’Enginyeria CivilIn this thesis we have developed a path towards large scale Finite Element simulations of turbulent incompressible flows. We have assessed the performance of residual-based variational multiscale (VMS) methods for the large eddy simulation (LES) of turbulent incompressible flows. We consider VMS models obtained by different subgrid scale approximations which include either static or dynamic subscales, linear or nonlinear multiscale splitting, and different choices of the subscale space. We show that VMS thought as an implicit LES model can be an alternative to the widely used physical-based models. This method is traditionally combined with equal-order velocity-pressure pairs, since it provides pressure stabilization. In this work, we also consider a different approach, based on inf-sup stable elements and convection-only stabilization. In order to do so, we define a symmetric projection stabilization of the convective term using an orthogonal subscale decomposition. The accuracy and efficiency of this method compared with residual-based algebraic subgrid scales and orthogonal subscales methods for equal-order interpolation is also assessed in this thesis. Furthermore, we propose Runge-Kutta time integration schemes for the incompressible Navier-Stokes equations with two salient properties. First, velocity and pressure computations are segregated at the time integration level, without the need to perform additional fractional step techniques that spoil high orders of accuracy. Second, the proposed methods keep the same order of accuracy for both velocities and pressures. Precisely, the symmetric projection stabilization approach is suitable for segregated Runge-Kutta time integration schemes. This combination, together with the use of block-preconditioning techniques, lead to elasticity-type and Laplacian-type problems that can be optimally preconditioned using the balancing domain decomposition by constraints preconditioners. The weak scalability of this formulation have been demonstrated in this document. Additionally, we also contemplate the weak imposition of the Dirichlet boundary conditions for wall-bounded turbulent flows. Four well known problems have been mainly considered for the numerical experiments: the decay of homogeneous isotropic turbulence, the Taylor-Green vortex problem, the turbulent flow in a channel and the turbulent flow around an airfoil.En aquesta tesi s'han desenvolupat diferents algoritmes per la simulació a gran escala de fluxos turbulents incompressibles mitjançant el mètode dels Elements Finits. En primer lloc s'ha avaluat el comportament dels mètodes de multiescala variacional (VMS) basats en el residu, per la simulació de grans vòrtexs (LES) de fluxos turbulents. S'han considerat diferents models VMS tenint en compte diferents aproximacions de les subescales, que inclouen tant subescales estàtiques o dinàmiques, una definicó lineal o nolineal, i diferents seleccions de l'espai de les subescales. S'ha demostrat que els mètodes VMS pensats com a models LES poden ser una alternativa als models basats en la física del problema. Aquest tipus de mètode normalment es combina amb l'ús de parelles de velocitat i pressió amb igual ordre d'interpolació. En aquest treball, també s'ha considerat un enfocament diferent, basat en l'ús d'elements inf-sup estables conjuntament amb estabilització del terme convectiu. Amb aquest objectiu, s'ha definit un mètode d'estabilització amb projecció simètrica del terme convectiu mitjançant una descomposició ortogonal de les subescales. En aquesta tesi també s'ha valorat la precisió i eficiència d'aquest mètode comparat amb mètodes basats en el residu fent servir interpolacions amb igual ordre per velocitats i pressions. A més, s'ha proposat un esquema d'integració en temps basat en els mètodes de Runge-Kutta que té dues propietats destacables. En primer lloc, el càlcul de la velocitat i la pressió es segrega al nivell de la integració temporal, sense la necessitat d'introduir tècniques de fraccionament del pas de temps. En segon lloc, els esquemes segregats de Runge-Kutta proposats, mantenen el mateix ordre de precisió tant per les velocitats com per les pressions. Precisament, els mètodes d'estabilització amb projecció simètrica són adequats per ser integrats en temps mitjançant esquemes segregats de Runge-Kutta. Aquesta combinació, juntament amb l'ús de tècniques de precondicionament en blocs, dóna lloc a problemes tipus elasticitat i Laplacià que poden ser òptimament precondicionats fent servir els anomenats \textit{balancing domain decomposition by constraints preconditioners}. La escalabilitat dèbil d'aquesta formulació s'ha demostrat en aquest document. Adicionalment, també s'ha contemplat la imposició de forma dèbil de les condicions de contorn de Dirichlet en problemes de fluxos turbulents delimitats per parets. En aquesta tesi principalment s'han considerat quatre problemes ben coneguts per fer els experiments numèrics: el decaïment de turbulència isotròpica i homogènia, el problema del vòrtex de Taylor-Green, el flux turbulent en un canal i el flux turbulent al voltant d'una ala.Award-winningPostprint (published version