A finite-element toolbox for the simulation of solid-liquid phase-change systems with natural convection

International audienceWe present and distribute a new numerical system using classical finite elements with mesh adaptivity for computing two-dimensional liquid-solid phase-change systems involving natural convection. The programs are written as a toolbox for FreeFem++ (www.freefem.org), a free finite-element software available for all existing operating systems. The code implements a single domain approach. The same set of equations is solved in both liquid and solid phases: the incompressible Navier-Stokes equations with Boussinesq approximation for thermal effects. This model describes naturally the evolution of the liquid flow which is dominated by convection effects. To make it valid also in the solid phase, a Carman-Kozeny-type penalty term is added to the momentum equations. The penalty term brings progressively (through an artificial mushy region) the velocity to zero into the solid. The energy equation is also modified to be valid in both phases using an enthalpy (temperature-transform) model introducing a regularized latent-heat term. Model equations are discretized using Galerkin triangular finite elements. Piecewise quadratic (P2) finite-elements are used for the velocity and piecewise linear (P1) for the pressure. For the temperature both P2 or P1 discretizations are possible. The coupled system of equations is integrated in time using a second-order Gear scheme. Non-linearities are treated implicitly and the resulting discrete equations are solved using a Newton algorithm. An efficient mesh adaptivity algorithm using metrics control is used to adapt the mesh every time step. This allows us to accurately capture multiple solid-liquid interfaces present in the domain, the boundary-layer structure at the walls and the unsteady convection cells in the liquid. We present several validations of the toolbox, by simulating benchmark cases of increasing difficulty: natural convection of air, natural convection of water, melting of a phase-change material, a melting-solidification cycle, and, finally, a water freezing case. Other similar cases could be easily simulated with this toolbox, since the code structure is extremely versatile and the syntax very close to the mathematical formulation of the model

FullSWOF_Paral: Comparison of two parallelization strategies (MPI and SKELGIS) on a software designed for hydrology applications

In this paper, we perform a comparison of two approaches for the parallelization of an existing, free software, FullSWOF 2D (http://www. univ-orleans.fr/mapmo/soft/FullSWOF/ that solves shallow water equations for applications in hydrology) based on a domain decomposition strategy. The first approach is based on the classical MPI library while the second approach uses Parallel Algorithmic Skeletons and more precisely a library named SkelGIS (Skeletons for Geographical Information Systems). The first results presented in this article show that the two approaches are similar in terms of performance and scalability. The two implementation strategies are however very different and we discuss the advantages of each one.Comment: 27 page

Identification of vortices in quantum fluids: finite element algorithms and programs

We present finite-element numerical algorithms for the identification of vortices in quantum fluids described by a macroscopic complex wave function. Their implementation using the free software FreeFem++ is distributed with this paper as a post-processing toolbox that can be used to analyse numerical or experimental data. Applications for Bose-Einstein condensates (BEC) and superfluid helium flows are presented. Programs are tested and validated using either numerical data obtained by solving the Gross-Pitaevskii equation or experimental images of rotating BEC. Vortex positions are computed as topological defects (zeros) of the wave function when numerical data are used. For experimental images, we compute vortex positions as local minima of the atomic density, extracted after a simple image processing. Once vortex centers are identified, we use a fit with a Gaussian to precisely estimate vortex radius. For vortex lattices, the lattice parameter (inter-vortex distance) is also computed. The post-processing toolbox offers a complete description of vortex configurations in superfluids. Tests for two-dimensional (giant vortex in rotating BEC, Abrikosov vortex lattice in experimental BEC) and three-dimensional (vortex rings, Kelvin waves and quantum turbulence fields in superfluid helium) configurations show the robustness of the software. The communication with programs providing the numerical or experimental wave function field is simple and intuitive. The post-processing toolbox can be also applied for the identification of vortices in superconductors

Numerical study of a family of dissipative KdV equations

The weak damped and forced KdV equation on the 1d Torus on $[0,L]$ have been analyzed by Ghidaglia\cite{Gh1,Gh2} Goubet\cite{G,GR}, Rosa and Cabral \cite{cabral-rosa} and asymptotic regularization effects have been proven and observed numerically. In this work, we consider a family of dampings that can be even weaker, particularly it can dissipate very few the high frequencies. We give here numerical evidences that point out dissipation of energy, regularization effect, and presence of special solutions that characterize a non trivial dynamics (steady states, time periodic solutions).Le comportement pour les grands temps de l'équation de KdV forcée et faiblement amortie sur le tore $[0,L]$ a été étudiée par Ghidaglia et Goubet qui ont prouvé l'effet régularisant asymptotique du flot ; Rosa et Cabral ont mis en évidence numériquement une dynamique non triviale. Dans ce travail, nous introduisons une famille d'amortissement permettant de dissiper plus faiblement encore. Nous établissons la dissipation de l'énergie et illustrons numériquement un effet de régularisation asymptotique ainsi que la présence de solutions spéciales (solutions stationnaires, solutions périodiques en temps)

On Damping Rates of dissipative KdV equations

We consider here different models of dissipative Korteweg-de Vries (KdV) equations on the torus. Using a proper wave function , we compare numerically the long time behavior effects of the damping models and we propose a hierarchy between these models. We also introduce a method based on the solution of an inverse problem to rebuild a posteriori the damping operator using only samples of the solution

Sediment transport modelling : Relaxation schemes for Saint-Venant -- Exner and three layer models

In this note we are interested in the modelling of sediment transport phenomena. We mostly focus on bedload transport and we do not consider suspension sediment processes. We first propose a coupled numerical scheme for the classical Saint-Venant -- Exner model. It is based on a relaxation approach and it works with all sediment flux function. We exhibit that this coupled approach is more stable than the splitting approach that is mostly used in industrial softwares. Then we derive an original three layers model in order to overcome the difficulties that are encountered when using the classical Exner approach and we write a related relaxation model

SOLUTION OF 2D BOUSSINESQ SYSTEMS WITH FREEFEM++: THE FLAT BOTTOM CASE

We consider here different family of Boussinesq systems in two space dimensions. These systems approximate the three-dimensional Euler equations and consist of three coupled nonlinear dispersive wave equations that describe propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom. We present here a FreeFem++ code aimed at solving numerically these systems where a discretization using P1 finite element for these systems was taken in space and a second order Runge-Kutta scheme in time. We give the detail of our code where we use a mesh adaptation technique. An optimization of the used algorithm is done and a comparison of the solution for different Boussinesq family is done too. The results we obtained agree with those of the literature