152 research outputs found

    A finite-element toolbox for the stationary Gross-Pitaevskii equation with rotation

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    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

    Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization

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    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 L2L^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 H1H^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

    Identification of vortices in quantum fluids: finite element algorithms and programs

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    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

    Acceleration of the imaginary time method for spectrally computing the stationary states of Gross-Pitaevskii equations

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    International audienceThe aim of this paper is to propose a simple accelerated spectral gradient flow formulation for solving the Gross-Pitaevskii Equation (GPE) when computing the stationary states of Bose-Einstein Condensates. The new algorithm, based on the recent iPiano minimization algorithm [35], converges three to four times faster than the standard implicit gradient scheme. To support the method, we provide a complete numerical study for 1d-2d-3d GPEs, including rotation and dipolar terms

    A Regularized Newton Method for Computing Ground States of Bose-Einstein condensates

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    In this paper, we propose a regularized Newton method for computing ground states of Bose-Einstein condensates (BECs), which can be formulated as an energy minimization problem with a spherical constraint. The energy functional and constraint are discretized by either the finite difference, or sine or Fourier pseudospectral discretization schemes and thus the original infinite dimensional nonconvex minimization problem is approximated by a finite dimensional constrained nonconvex minimization problem. Then an initial solution is first constructed by using a feasible gradient type method, which is an explicit scheme and maintains the spherical constraint automatically. To accelerate the convergence of the gradient type method, we approximate the energy functional by its second-order Taylor expansion with a regularized term at each Newton iteration and adopt a cascadic multigrid technique for selecting initial data. It leads to a standard trust-region subproblem and we solve it again by the feasible gradient type method. The convergence of the regularized Newton method is established by adjusting the regularization parameter as the standard trust-region strategy. Extensive numerical experiments on challenging examples, including a BEC in three dimensions with an optical lattice potential and rotating BECs in two dimensions with rapid rotation and strongly repulsive interaction, show that our method is efficient, accurate and robust.Comment: 25 pages, 6 figure

    BEC2HPC: a HPC spectral solver for nonlinear Schrödinger and Gross-Pitaevskii equations. Stationary states computation

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    International audienceWe present BEC2HPC which is a parallel HPC spectral solver for computing the ground states of the nonlinear Schrödinger equation and the Gross-Pitaevskii equation (GPE) modeling rotating Bose-Einstein condensates (BEC). Considering a standard pseudo-spectral discretization based on Fast Fourier Transforms (FFTs), the method consists in finding the numerical solution of the energy functional minimization problem under normalization constraint by using a preconditioned nonlinear conjugate gradient method. We present some numerical simulations and scalability results for the 2D and 3D problems to obtain the stationary states of BEC with fast rotation and large nonlinearities. The code takes advantage of existing HPC libraries and can itself be leveraged to implement other numerical methods like e.g. for the dynamics of BECs

    Super-localised wave function approximation of Bose-Einstein condensates

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    This paper presents a novel spatial discretisation method for the reliable and efficient simulation of Bose-Einstein condensates modelled by the Gross-Pitaevskii equation and the corresponding nonlinear eigenvector problem. The method combines the high-accuracy properties of numerical homogenisation methods with a novel super-localisation approach for the calculation of the basis functions. A rigorous numerical analysis demonstrates superconvergence of the approach compared to classical polynomial and multiscale finite element methods, even in low regularity regimes. Numerical tests reveal the method's competitiveness with spectral methods, particularly in capturing critical physical effects in extreme conditions, such as vortex lattice formation in fast-rotating potential traps. The method's potential is further highlighted through a dynamic simulation of a phase transition from Mott insulator to Bose-Einstein condensate, emphasising its capability for reliable exploration of physical phenomena

    Numerical model of the Gross-Pitaevskii equation for vortex lattice formation in rotating Bose-Einstein condensates using smoothed-particle hydrodynamics

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    This study proposed a new numerical scheme for vortex lattice formation in a rotating Bose-Einstein condensate (BEC) using smoothed particle hydrodynamics (SPH) with an explicit real-time integration scheme. Specifically, the Gross-Pitaevskii (GP) equation was described as a complex representation to obtain a pair of time-dependent equations, which were then solved simultaneously following discretization based on SPH particle approximation. We adopt the 4th-order Runge-Kutta method for time evolution. We performed simulations of a rotating Bose gas trapped in a harmonic potential, showing results that qualitatively agreed with previously reported experiments and simulations. The geometric patterns of formed lattices were successfully reproduced for several cases, for example, the pentagonal lattice observed in the experiments of rotating BECs. Consequently, it was confirmed that the simulation began with the periodic oscillation of the condensate, which attenuated and maintained a stable rotation with slanted elliptical shapes; however, the surface was excited to be unstable and generated ripples, which grew into vortices and then penetrated the inside the condensate, forming a lattice. We confirmed that each branch point of the phase of wavefunctions corresponds to each vortex. These results demonstrate our approach at a certain degree of accuracy. In conclusion, we successfully developed a new SPH scheme for the simulations of vortex lattice formation in rotating BECs

    Perfectly Matched Layer for computing the dynamics of nonlinear Schrödinger equations by pseudospectral methods. Application to rotating Bose-Einstein condensates

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    In this paper, we first propose a general strategy to implement the Perfectly Matched Layer (PML) approach in the most standard numerical schemes used for simulating the dynamics of nonlinear Schrödinger equations. The methods are based on the time-splitting [15] or relaxation [24] schemes in time, and finite element or FFT-based pseudospectral discretization methods in space. A thorough numerical study is developed for linear and nonlinear problems to understand how the PML approach behaves (absorbing function and tuning parameters) for a given scheme. The extension to the rotating Gross-Pitaevskii equation is then proposed by using the rotating Lagrangian coordinates transformation method [13, 16, 39], some numerical simulations illustrating the strength of the proposed approach

    Robust and efficient preconditioned Krylov spectral solvers for computing the ground states of fast rotating and strongly interacting Bose-Einstein condensates

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    International audienceWe consider the Backward Euler SPectral (BESP) scheme proposed in [10] for computing the stationary states of Bose-Einstein Condensates (BECs) through the Gross-Pitaevskii equation. We show that the fixed point approach introduced in [10] fails to converge for fast rotating BECs. A simple alternative approach based on Krylov subspace solvers with a Laplace or Thomas-Fermi preconditioner is given. Numerical simulations (obtained with the associated freely available Matlab toolbox GPELab) for complex configurations show that the method is accurate, fast and robust for 2D/3D problems and multi-components BECs
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