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 $H^1$ 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 $L^2$ or $H^1$ 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 $L^2$ 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 $H^1$ 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 $\omega$-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 ($\Omega$) above the trap frequency ($\omega_\perp$). Our numerical results are obtained by propagating the 3D Gross-Pitaevskii equation in imaginary time. For $\Omega \leq\omega_\perp$, we obtain an equilibrium vortex lattice similar (as size and number of vortices) to experimental observations. For $\Omega>\omega_\perp$ 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 $Omega$ 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 $Omega$, 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