401 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
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
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
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
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
(%), (iv) the jet merging distance is the relevant scaling
length-scale of the turbulent flow, (v) high turbulence intensity (%) is achieved in the homogeneous and isotropic region, although the
Reynolds number based on the Taylor microscale remains moderate (). 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
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
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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
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