Showing papers by "Ramon Codina published in 2009"

Unified Stabilized Finite Element Formulations for the Stokes and the Darcy Problems

Abstract: In this paper we propose stabilized finite element methods for both Stokes' and Darcy's problems that accommodate any interpolation of velocities and pressures. Apart from the interest of this fact, the important issue is that we are able to deal with both problems at the same time, in a completely unified manner, in spite of the fact that the functional setting is different. Concerning the stabilization formulation, we discuss the effect of the choice of the length scale appearing in the expression of the stabilization parameters, both in what refers to stability and to accuracy. This choice is shown to be crucial in the case of Darcy's problem. As an additional feature of this work, we treat two types of stabilized formulations, showing that they have a very similar behavior.

The fixed‐mesh ALE approach applied to solid mechanics and fluid–structure interaction problems

Abstract: In this paper we propose a method to solve Solid Mechanics and fluid–structure interaction problems using always a fixed background mesh for the spatial discretization The main feature of the method is that it properly accounts for the advection of information as the domain boundary evolves To achieve this, we use an Arbitrary Lagrangian–Eulerian (ALE) framework, the distinctive characteristic being that at each time step results are projected onto a fixed, background mesh For solid mechanics problems subject to large strains, the fixed-mesh (FM)-ALE method avoids the element stretching found in fully Lagrangian approaches For FSI problems, FM-ALE allows for the use of a single background mesh to solve both the fluid and the structure Copyright © 2009 John Wiley & Sons, Ltd

Approximate imposition of boundary conditions in immersed boundary methods

Abstract: We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. As basic approximation technique we consider the finite element method with a mesh that does not match the boundary of the computational domain, and therefore Dirichlet boundary conditions need to be prescribed in an approximate way. As starting variational approach we consider Nitsche's methods, and we then move to two options that yield non-symmetric problems but that turned out to be robust and efficient. The essential idea is to use the degrees of freedom of certain nodes of the finite element mesh to minimize the difference between the exact and the approximated boundary condition.

Mathematical Models for Thermally Coupled Low Speed Flows

Abstract: In this paper we review and clarify some aspects of the asymptotic analysis of the compressible Navier Stokes equations in the low Mach number limit. In the absence of heat exchange (the isentropic regime) this limit is well understood and rigorous results are available. When heat exchange is considered different simplified models can be obtained, the most famous being the Boussinesq approximation. Here a unified formal justification of these models is presented, paying special attention to the relation between the low Mach number and the Boussinesq approximations. Precise conditions for their validity are given for classical problems in bounded domains.

A heuristic argument for the sole use of numerical stabilization with no physical LES modeling in the simulation of incompressible turbulent flows

Abstract: We aim at giving support to the idea that no physical Large Eddy Simulation (LES) model should be used in the simulation of turbulent flows. It is heuristically shown that the rate of transfer of subgrid kinetic energy provided by the stabilization terms of the Orthogonal Subgrid Scale (OSS) finite element method is already proportional to the molecular physical dissipation rate (for an appropriate choice of the stabilization parameter). This precludes the necessity of including an extra LES physical model to achieve this behavior and somehow justifies the purely numerical approach to solve turbulent flows. The argumentation is valid for a fine enough mesh with characteristic element size, $h$, so that $h$ lies in the inertial subrange of a turbulent flow.

Computational aeroacoustics of viscous low speed flows using subgrid scale finite element methods

Abstract: A methodology to perform computational aeroacoustics (CAA) of viscous low speed flows in the framework of stabilized finite element methods is presented. A hybrid CAA procedure is followed that makes use of Lighthill's acoustic analogy in the frequency domain. The procedure has been conceptually divided into three steps. In the first one, the incompressible Navier–Stokes equations are solved to obtain the flow velocity field. In the second step, Lighthill's acoustic source term is computed from this velocity field and then Fourier transformed to the frequency domain. Finally, the acoustic pressure field is obtained by solving the corresponding inhomogeneous Helmholtz equation. All equations in the formulation are solved using subgrid scale stabilized finite element methods. The main ideas of the subgrid scale numerical strategy are outlined and its benefits when compared to the Galerkin approach are described. As numerical examples, the aerodynamic noise generated by flow past a two-dimensional cylinder and by flow past two cylinders in parallel arrangement are addressed.

A numerical approximation of the thermal coupling of fluids and solids

Abstract: In this article we analyze the problem of the thermal coupling of fluids and solids through a common interface. We state the global thermal problem in the whole domain, including the fluid part and the solid part. This global thermal problem presents discontinuous physical properties that depend on the solution of auxiliary problems on each part of the domain (a fluid flow problem and a solid state problem). We present a domain decomposition strategy to iteratively solve problems posed in both subdomains and discuss some implementation aspects of the algorithm. This domain decomposition framework is also used to revisit the use of wall function approaches used in this context. Copyright © 2008 John Wiley & Sons, Ltd.

Fixed Mesh Methods in Computational Mechanics

Abstract: In many coupled problems of practical interest the domain of at least one of the problems evolves in time. The Arbitrary Eulerian Lagrangian (ALE) approach is a tool very often employed to cope with this domain motion. However, in this work we aim at describing numerical techniques that allow us to use a fixed mesh for the approximation of moving boundary problems, particularly using the finite element approach. This type of formulations is often termed embedded or immersed boundary methods. Emphasis will be put in describing a particular version of the ALE formulation using fixed meshes that we have developed, and that we call fixed-mesh ALE method (FM-ALE) [1]. In the classical ALE approach, the mesh in which the computational domain is discretized is deformed. This is done according to a prescribed motion of part of its boundary, which is transmitted to the interior nodes in a way as smooth as possible so as to avoid mesh distortion. The FM-ALE formulation has a different motivation. Instead of assuming that the computational domain is defined by the mesh boundary, we assume that there is a function that defines the boundary of the domain where the flow takes place. We will refer to it as the boundary function. When this boundary function moves, the flow domain changes, and that must be taken into account at the moment of writing the conservation equations that govern the flow, which need to be cast in the ALE format. However, our purpose here is to explain how to use always a background fixed mesh. Other possibilities to use a single grid in the whole simulation can be found in the literature. They were designed as an alternative to body fitted meshes and can be divided into two main groups, corresponding in fact to two ways of prescribing the boundary conditions on the moving boundary:  Force term. The interaction of the fluid and the solid is taken into account through a force term, which appears either in the strong or in the weak form of the flow equations. Among this type of methods, let us cite for example the Immersed Boundary method as a variant of the Penalty method, where punctual forces are added to the momentum equation, and the Fictitious Domain method, where the solid boundary conditions are imposed through a Lagrange multiplier.  Approximate boundary conditions. Instead of adding a force term, these methods impose the boundary conditions in an approximate way once the discretization has been carried out, either by modifying the differential operators near the interface (in finite differences) or by modifying the unknowns near the interface.

Finite element variational multiscale formulation for low Mach number flows coupled with radiative heat transfer

Abstract: Problems of combustion in fire scenarios are usually mathematically described by the Low Mach number approximation of Navier Stokes equations [4]. Thermal radiation has direct effects on many industrial applications, such as fires in vehicular tunnels, combustion in furnaces, gas turbine models, etc. Growing concern with high temperature processes has emphasized the need for an evaluation of the effect of radiation heat transfer. We describe a finite element formulation for the coupling of the low Mach number model and the radiative transfer equations based on the variational multiscales method [1]. We extend the subgrid scales to the radiation intensity appearing in the energy equation. We compare a standard ASGS formulation [3] and a complete residual based formulation additionally containing all nonlinear terms as crossand Reynoldsstress terms. The Radiative transfer equation is modelled by the PN and the DOM models [2]. Both models are stabilized using the variational multiscale method. Implementation issues, such as the linearization procedure and the coupling of the different equations in play are described. Numerical simulations are presented for three dimensional fires in vehicular tunnels scenarios, we discuss the effect of the different stabilization techniques and radiative models.