163 research outputs found
A mesh adaptivity scheme on the Landau-de Gennes functional minimization case in 3D, and its driving efficiency
This paper presents a 3D mesh adaptivity strategy on unstructured tetrahedral
meshes by a posteriori error estimates based on metrics, studied on the case of
a nonlinear finite element minimization scheme for the Landau-de Gennes free
energy functional of nematic liquid crystals. Newton's iteration for tensor
fields is employed with steepest descent method possibly stepping in.
Aspects relating the driving of mesh adaptivity within the nonlinear scheme
are considered. The algorithmic performance is found to depend on at least two
factors: when to trigger each single mesh adaptation, and the precision of the
correlated remeshing. Each factor is represented by a parameter, with its
values possibly varying for every new mesh adaptation. We empirically show that
the time of the overall algorithm convergence can vary considerably when
different sequences of parameters are used, thus posing a question about
optimality.
The extensive testings and debugging done within this work on the simulation
of systems of nematic colloids substantially contributed to the upgrade of an
open source finite element-oriented programming language to its 3D meshing
possibilities, as also to an outer 3D remeshing module
A finite-element toolbox for the stationary Gross-Pitaevskii equation with rotation
We present a new numerical system using classical finite elements with mesh
adaptivity for computing stationary solutions of the Gross-Pitaevskii equation.
The programs are written as a toolbox for FreeFem++ (www.freefem.org), a free
finite-element software available for all existing operating systems. This
offers the advantage to hide all technical issues related to the implementation
of the finite element method, allowing to easily implement various numerical
algorithms.Two robust and optimised numerical methods were implemented to
minimize the Gross-Pitaevskii energy: a steepest descent method based on
Sobolev gradients and a minimization algorithm based on the state-of-the-art
optimization library Ipopt. For both methods, mesh adaptivity strategies are
implemented to reduce the computational time and increase the local spatial
accuracy when vortices are present. Different run cases are made available for
2D and 3D configurations of Bose-Einstein condensates in rotation. An optional
graphical user interface is also provided, allowing to easily run predefined
cases or with user-defined parameter files. We also provide several
post-processing tools (like the identification of quantized vortices) that
could help in extracting physical features from the simulations. The toolbox is
extremely versatile and can be easily adapted to deal with different physical
models
Overlapping Domain Decomposition Methods with FreeFem++
International audienceIn this note, the performances of a framework for two-level overlapping domain decomposition methods are assessed. Numerical experiments are run on Curie, a Tier-0 system for PRACE, for two second order elliptic PDE with highly heterogeneous coefficients: a scalar equation of diffusivity and the system of linear elasticity. Those experiments yield systems with up to ten billion unknowns in 2D and one billion unknowns in 3D, solved on few thousands cores
Scalable Domain Decomposition Preconditioners for Heterogeneous Elliptic Problems
International audienceDomain decomposition methods are, alongside multigrid methods, one of the dominant paradigms in contemporary large-scale partial differential equation simulation. In this paper, a lightweight implementation of a theoretically and numerically scalable preconditioner is presented in the context of overlapping methods. The performance of this work is assessed by numerical simulations executed on thousands of cores, for solving various highly heterogeneous elliptic problems in both 2D and 3D with billions of degrees of freedom. Such problems arise in computational science and engineering, in solid and fluid mechanics. While focusing on overlapping domain decomposition methods might seem too restrictive, it will be shown how this work can be applied to a variety of other methods, such as non-overlapping methods and abstract deflation based preconditioners. It is also presented how multilevel preconditioners can be used to avoid communication during an iterative process such as a Krylov method
Frictionless contact problem for hyperelastic materials with interior point optimizer
This paper presents a method to solve the mechanical problems undergoing finite deformations and the unilateral contact problems without friction for hyperelastic materials. We apply it to an industrial application: contact between a mechanical gasket and an obstacle. The main idea is to formulate the contact problem into an optimization's one, in order to use the Interior Point OPTimizer (IPOPT) to solve it. Finally, the FreeFEM software is used to compute and solve the contact problem. Our method is validated against several benchmarks and used on an industrial application example
R-adaptation par l'estimateur d'erreur hiérarchique
International audienceThe aim of this work is to devise a method to determine the optimal position of the nodes in a finite element discretization for a boundary value problem. The node displacement procedure (also called R-adaptation) is a crucial step in a global mesh adaptation procedure. In the present approch, we determine the nodal position by minimizing the approximation error. This error is evaluated using a hierarchical estimator. A numerical test is presented.L'objectif de ce travail est de déterminer la meilleure position des noeuds d'un maillage, utilisé lors de la discrétisation d'un problème aux limites par la méthode des éléments finis. La procédure de déplacement des noeuds (appelé aussi R-adaptation) est une étape importante dans la stratégie globale d'adaptation de maillage. La position optimale des noeuds est déterminée en minimisant l'erreur d'approximation. Pour évaluer cette erreur nous utilisons l'estimateur d'erreur hiérarchique. Un test numérique est présenté
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
Mortar finite element discretization of a model coupling Darcy and Stokes equations
As a first draft of a model for a river flowing on a homogeneous porous ground, we consider a system where the Darcy and Stokes equations are coupled via appropriate matching conditions on the interface. We propose a discretization of this problem which combines the mortar method with
standard finite elements, in order to handle separately the flow inside and outside the porous medium. We prove a priori and a posteriori error estimates for the resulting discrete problem. Some numerical experiments confirm the interest of the discretization.EU Marie CurieMinisterio de Educación y Cienci
- …