53 research outputs found
A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation
In this paper we improve traditional steepest descent methods for the direct
minimization of the Gross-Pitaevskii (GP) energy with rotation at two levels.
We first define a new inner product to equip the Sobolev space and derive
the corresponding gradient. Secondly, for the treatment of the mass
conservation constraint, we use a projection method that avoids more
complicated approaches based on modified energy functionals or traditional
normalization methods. The descent method with these two new ingredients is
studied theoretically in a Hilbert space setting and we give a proof of the
global existence and convergence in the asymptotic limit to a minimizer of the
GP energy. The new method is implemented in both finite difference and finite
element two-dimensional settings and used to compute various complex
configurations with vortices of rotating Bose-Einstein condensates. The new
Sobolev gradient method shows better numerical performances compared to
classical or gradient methods, especially when high rotation rates
are considered.Comment: to appear in SIAM J Sci Computin
Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization
In this paper we combine concepts from Riemannian Optimization and the theory
of Sobolev gradients to derive a new conjugate gradient method for direct
minimization of the Gross-Pitaevskii energy functional with rotation. The
conservation of the number of particles constrains the minimizers to lie on a
manifold corresponding to the unit norm. The idea developed here is to
transform the original constrained optimization problem to an unconstrained
problem on this (spherical) Riemannian manifold, so that fast minimization
algorithms can be applied as alternatives to more standard constrained
formulations. First, we obtain Sobolev gradients using an equivalent definition
of an inner product which takes into account rotation. Then, the
Riemannian gradient (RG) steepest descent method is derived based on projected
gradients and retraction of an intermediate solution back to the constraint
manifold. Finally, we use the concept of the Riemannian vector transport to
propose a Riemannian conjugate gradient (RCG) method for this problem. It is
derived at the continuous level based on the "optimize-then-discretize"
paradigm instead of the usual "discretize-then-optimize" approach, as this
ensures robustness of the method when adaptive mesh refinement is performed in
computations. We evaluate various design choices inherent in the formulation of
the method and conclude with recommendations concerning selection of the best
options. Numerical tests demonstrate that the proposed RCG method outperforms
the simple gradient descent (RG) method in terms of rate of convergence. While
on simple problems a Newton-type method implemented in the {\tt Ipopt} library
exhibits a faster convergence than the (RCG) approach, the two methods perform
similarly on more complex problems requiring the use of mesh adaptation. At the
same time the (RCG) approach has far fewer tunable parameters.Comment: 28 pages, 13 figure
Sobolev gradients and image interpolation
We present here a new image inpainting algorithm based on the Sobolev
gradient method in conjunction with the Navier-Stokes model. The original model
of Bertalmio et al is reformulated as a variational principle based on the
minimization of a well chosen functional by a steepest descent method. This
provides an alternative of the direct solving of a high-order partial
differential equation and, consequently, allows to avoid complicated numerical
schemes (min-mod limiters or anisotropic diffusion). We theoretically analyze
our algorithm in an infinite dimensional setting using an evolution equation
and obtain global existence and uniqueness results as well as the existence of
an -limit. Using a finite difference implementation, we demonstrate
using various examples that the Sobolev gradient flow, due to its smoothing and
preconditioning properties, is an effective tool for use in the image
inpainting problem
Three-dimensional vortex structure of a fast rotating Bose-Einstein condensate with harmonic-plus-quartic confinement
We address the challenging proposition of using real experimental parameters
in a three-dimensional numerical simulation of fast rotating Bose-Einstein
condensates. We simulate recent experiments [V. Bretin, S. Stock, Y. Seurin and
J. Dalibard, Phys. Rev. Lett. 92, 050403 (2004); S. Stock, V. Bretin, S. Stock,
F. Chevy and J. Dalibard, Europhys. Lett. 65, 594 (2004)] using an anharmonic
(quadratic-plus-quartic) confining potential to reach rotation frequencies
() above the trap frequency (). Our numerical results are
obtained by propagating the 3D Gross-Pitaevskii equation in imaginary time. For
, we obtain an equilibrium vortex lattice similar (as
size and number of vortices) to experimental observations. For
we observe the evolution of the vortex lattice into an
array of vortices with a central hole. Since this evolution was not visible in
experiments, we investigate the 3D structure of vortex configurations and
3D-effects on vortex contrast. Numerical data are also compared to recent
theory [D. E. Sheehy and L. Radzihovsky, Phys. Rev. A 70, 063620 (2004)]
describing vortex lattice inhomogeneities and a remarkably good agreement is
found.Comment: to appear in Phys Rev A 71 (2005
Optimal Reconstruction of Inviscid Vortices
We address the question of constructing simple inviscid vortex models which
optimally approximate realistic flows as solutions of an inverse problem.
Assuming the model to be incompressible, inviscid and stationary in the frame
of reference moving with the vortex, the "structure" of the vortex is uniquely
characterized by the functional relation between the streamfunction and
vorticity. It is demonstrated how the inverse problem of reconstructing this
functional relation from data can be framed as an optimization problem which
can be efficiently solved using variational techniques. In contrast to earlier
studies, the vorticity function defining the streamfunction-vorticity relation
is reconstructed in the continuous setting subject to a minimum number of
assumptions. To focus attention, we consider flows in 3D axisymmetric geometry
with vortex rings. To validate our approach, a test case involving Hill's
vortex is presented in which a very good reconstruction is obtained. In the
second example we construct an optimal inviscid vortex model for a realistic
flow in which a more accurate vorticity function is obtained than produced
through an empirical fit. When compared to available theoretical vortex-ring
models, our approach has the advantage of offering a good representation of
both the vortex structure and its integral characteristics.Comment: 33 pages, 10 figure
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
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
Giant vortices in combined harmonic and quartic traps
We consider a rotating Bose-Einstein condensate confined in combined harmonic
and quartic traps, following recent experiments [V. Bretin, S. Stock, Y. Seurin
and J. Dalibard, cond-mat/0307464]. We investigate numerically the behavior of
the wave function which solves the three-dimensional Gross Pitaevskii equation.
When the harmonic part of the potential is dominant, as the angular velocities
increases, the vortex lattice evolves into a giant vortex. We also
investigate a case not covered by the experiments or the previous numerical
works: for strong quartic potentials, the giant vortex is obtained for lower
, before the lattice is formed. We analyze in detail the three
dimensional structure of vortices
Brisure d'axisymétrie à l'instabilité primaire du jet rond
La plupart des études du jet rond libre considèrent des nombres de Reynolds (défini sur la vitesse de sortie de l'injecteur Vzo et le diamètre D de la buse) assez importants (Re > 300, cf., par ex.), de sorte que les effets non visqueux prévalent sur les effets visqueux. Pour ce type d'écoulements, le scénario qui caractérise l'évolution du jet dans la zone proche de la buse est bien connu. Récemment, nous avons réalisé des simulations de la zone proche de sortie d'un jet rond évoluant dans le temps et dans l'espace au nombre de Reynolds de 500. Bon nombre de résulats connus ont été retrouvés, prouvant la pertinence de ce type de simulations pour l'étude des détails du comportement dynamique du jet rond. La transition à l'instationnarité et ses mécanismes ont, en revanche, été très peu étudiés pour ce type d'écoulement. Nous avons alors considéré des variations fines dans une plage de nombres de Reynolds plus bas. Une instabilité primaire du jet accompagnée par la brisure de l'axisymétrie a ainsi été mise en évidence. Ses mécanismes principaux sont décrits dans cette étude
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
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