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https://hal.archives-ouvertes.fr/hal-01513346/document

Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

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Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Finite element methods for Maxwell's equations.

A stabilized finite element (FEM) formulation for the wave equation in mixed form with convection is presented, which permits using the same interpolation fields for the acoustic pressure and the acoustic particle velocity. The formulation is based on a variational multiscale approach, in which the problem unknowns are split into a large scale component that can be captured by the computational mesh, and a small, subgrid scale component, whose influence into the large scales
has to be modelled. A suitable option is that of taking the subgrid scales, or subscales, as being related to the finite element residual by means of a matrix of stabilization parameters. The design of the later turns to be the key for the good performance of the method. In addition, the mixed convected wave equation has been set in an arbitrary Lagrangian-Eulerian (ALE) frame of reference to account for domains with moving boundaries. The movement of the boundaries in the present work consists of two components, an external prescribed motion and a motion related to the boundary elastic back reaction to the acoustic pressure, in the normal direction. A mass-damper-stiffness auxiliary equation is solved for each boundary node to include this effect. As a first benchmark example, we have considered the case of 2D simple duct acoustics with mean flow. More complex 3D examples are also presented consisting of vowel and diphthong generation, following a numerical approach to voice production. The numerical simulation of voice not only allows one to see how waves propagate inside the vocal tract, but also to collect the
acoustic pressure at a node close to the mouth exit, convert it to an audio file and listen to it.

Stabilized finite element formulation for the mixed convected wave equation in domains with driven flexible boundaries

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.

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

Mould filling problems with slow inlet velocities lead to complex two phase flow simulations. As the Froude number decreases the coupling between the position of the inteface and the resulting flow increases. For such flows we have developed a model that enriches the finite element pressure shape functions in elements cut by the interface so that a discontinuous pressure gradient can be represented. In this work we show that the model can be applied to complex mould filling problems obtained directly from the foundry.

Ramon Codina

Papers

Stabilized finite element formulation for the mixed convected wave equation in domains with driven flexible boundaries

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

An enriched pressure interpolation finite element model for mould filling problems

Numerical simulation of Fluid–Structure Interaction problems with viscoelastic fluids using a log-conformation reformulation

A variational multiscale stabilized finite element formulation for Reissner–Mindlin plates and Timoshenko beams