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

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

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

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

Multi-Scale Turbulence Injector: a new tool to generate intense homogeneous and isotropic turbulence for premixed combustion

Nearly homogeneous and isotropic, highly turbulent flow, generated by an original multi-scale injector is experimentally studied. This multi-scale injector is made of three perforated plates shifted in space such that the diameter of their holes and their blockage ratio increase with the downstream distance. The Multi-Scale Turbulence Injector (hereafter, MuSTI) is compared with a Mono-Scale Turbulence Injector (MoSTI), the latter being constituted by only the last plate of MuSTI. This comparison is done for both cold and reactive flows. For the cold flow, it is shown that, in comparison with the classical mono-scale injector, for the MuSTI injector: (i) the turbulent kinetic energy is roughly twice larger, and the kinetic energy supply is distributed over the whole range of scales. This is emphasized by second and third order structure functions. (ii) the transverse fluxes of momentum and energy are enhanced, (iii) the homogeneity and isotropy are reached earlier ($\approx 50$%), (iv) the jet merging distance is the relevant scaling length-scale of the turbulent flow, (v) high turbulence intensity ($\approx 15$%) is achieved in the homogeneous and isotropic region, although the Reynolds number based on the Taylor microscale remains moderate ($Re_\lambda \approx 80$). In a second part, the interaction between the multi-scale generated turbulence and the premixed flame front is investigated by laser tomography. A lean V-shaped methane/air flame is stabilised on a heated rod in the homogeneous and isotropic region of the turbulent flow. The main observation is that the flame wrinkling is hugely amplified with the multi-scale generated injector, as testified by the increase of the flame brush thickness.Comment: 29 pages, 21 figures, submitted to Journal of Turbulenc

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

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

Non-newtonian 3D ciliary fluid flow in a semi-infinite domain

This paper was presented at the 3rd Micro and Nano Flows Conference (MNF2011), which was held at the Makedonia Palace Hotel, Thessaloniki in Greece. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, Aristotle University of Thessaloniki, University of Thessaly, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute.Continuing our previous investigations in ciliary fluid transportation (Isvoranu et al., 2010) our present paper looks into the matter of non-newtonian fluid flow. Naturally, in constant properties fluids (newtonian fluids) ciliary transportation is based on a non-symmetric actuation mechanism meaning different geometrical configuration of the cilium during the active and passive stroke. Artificial cilia can mimick this behaviour through asymmetric magnetic actuation as discused in (Isvoranu et. al., 2008). What happens when fluid properties (eg. viscosity) are not constant throughout a beating cycle? Such situation is expected to be encountered when dealing with biological fluids like saliva, for example. In the case of a shear-thinning fluid, like the above mentioned one, the motion can also become asymetric due to deformation rate dependent viscosity that ultimately leads to different time scales of the forward and backward strokes. In the present paper we are investigating a 3D flow generated by an array of cilia embedded in a non-newtonian fluid whose viscosity is characterized by a power low shear rate dependency. The same magnetic actuation mechanism is considered