Numerical and variational solutions of the dipolar Gross-Pitaevskii equation in reduced dimensions

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Freezing of 4He and its liquid-solid interface from density functional theory

We show that, at high densities, fully variational solutions of solidlike types can be obtained from a density functional formalism originally designed for liquid 4He . Motivated by this finding, we propose an extension of the method that accurately describes the solid phase and the freezing transition of liquid 4He at zero temperature. The density profile of the interface between liquid and the (0001) surface of the 4He crystal is also investigated, and its surface energy evaluated. The interfacial tension is found to be in semiquantitative agreement with experiments and with other microscopic calculations. This opens the possibility to use unbiased density functional (DF) methods to study highly nonhomogeneous systems, like 4He interacting with strongly attractive impurities and/or substrates, or the nucleation of the solid phase in the metastable liquid.

Vibrational-rotational analysis of the Hulthen potential using hydrogenic eigenfunction bases

Hulthen's potential admits analytical solutions for its energy eigenvalues and eigenfunctions corresponding to zero orbital angular momentum stales, but its non zero angular momentum states are not equally known. This work presents a vibrational-rotational analy sis of Hulthen's potential using hydrogenic eigenfunction bases, which may be of interest and useful to students of quantum mechanics at different stages.

Matter-wave localization in a random potential

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Dipolar droplet bound in a trapped Bose-Einstein condensate

We study the statics and dynamics of a dipolar Bose-Einstein condensate (BEC) droplet bound by interspecies contact interaction in a trapped nondipolar BEC. Our findings are demonstrated in terms of stability plots of a dipolar 164Dy droplet bound in a trapped nondipolar 87Rb BEC with a variable number of 164Dy atoms and interspecies scattering length. A trapped nondipolar BEC of a fixed number of atoms can bind only a dipolar droplet containing fewer atoms than a critical number for the interspecies scattering length between two critical values. The shape and size (statics) as well as the small breathing oscillation (dynamics) of the dipolar BEC droplet are studied using numerical and variational solutions of a mean-field model. We also suggest an experimental procedure for achieving such a 164Dy droplet by relaxing the trap on the 164Dy BEC in a trapped binary 87Rb-164Dy mixture. © 2013 American Physical Society.

Potential Analysis for Hypoelliptic Second Order PDEs with Nonnegative Characteristic Form

In this Thesis we consider a class of second order partial differential operators with non-negative characteristic form and with smooth coefficients. Main assumptions on the relevant operators are hypoellipticity and existence of a well-behaved global fundamental solution. We first make a deep analysis of the L-Green function for arbitrary open sets and of its applications to the Representation Theorems of Riesz-type for L-subharmonic and L-superharmonic functions. Then, we prove an Inverse Mean value Theorem characterizing the superlevel sets of the fundamental solution by means of L-harmonic functions. Furthermore, we establish a Lebesgue-type result showing the role of the mean-integal operator in solving the homogeneus Dirichlet problem related to L in the Perron-Wiener sense. Finally, we compare Perron-Wiener and weak variational solutions of the homogeneous Dirichlet problem, under specific hypothesis on the boundary datum.

Spectral and variational characterizations of solutions to semilinear eigenvalue problems

Die vorliegende Arbeit befaßt sich mit einer Klasse von nichtlinearen Eigenwertproblemen mit Variationsstrukturin einem reellen Hilbertraum. Die betrachteteEigenwertgleichung ergibt sich demnach als Euler-Lagrange-Gleichung eines stetig differenzierbarenFunktionals, zusätzlich sei der nichtlineare Anteil desProblems als ungerade und definit vorausgesetzt.Die wichtigsten Ergebnisse in diesem abstrakten Rahmen sindKriterien für die Existenz spektral charakterisierterLösungen, d.h. von Lösungen, deren Eigenwert gerade miteinem vorgegeben variationellen Eigenwert eines zugehörigen linearen Problems übereinstimmt. Die Herleitung dieserKriterien basiert auf einer Untersuchung kontinuierlicher Familien selbstadjungierterEigenwertprobleme und erfordert Verallgemeinerungenspektraltheoretischer Konzepte.Neben reinen Existenzsätzen werden auch Beziehungen zwischenspektralen Charakterisierungen und denLjusternik-Schnirelman-Niveaus des Funktionals erörtert.Wir betrachten Anwendungen auf semilineareDifferentialgleichungen (sowieIntegro-Differentialgleichungen) zweiter Ordnung. Diesliefert neue Informationen über die zugehörigenLösungsmengen im Hinblick auf Knoteneigenschaften. Diehergeleiteten Methoden eignen sich besonders für eindimensionale und radialsymmetrische Probleme, während einTeil der Resultate auch ohne Symmetrieforderungen gültigist.

Prediction of inter-laminar stresses in composite honeycomb sandwich panels under mechanical loading using Variational Asymptotic Method

This work focuses on the formulation of an asymptotically correct theory for symmetric composite honeycomb sandwich plate structures. In these panels, transverse stresses tremendously influence design. The conventional 2-D finite elements cannot predict the thickness-wise distributions of transverse shear or normal stresses and 3-D displacements. Unfortunately, the use of the more accurate three-dimensional finite elements is computationally prohibitive. The development of the present theory is based on the Variational Asymptotic Method (VAM). Its unique features are the identification and utilization of additional small parameters associated with the anisotropy and non-homogeneity of composite sandwich plate structures. These parameters are ratios of smallness of the thickness of both facial layers to that of the core and smallness of 3-D stiffness coefficients of the core to that of the face sheets. Finally, anisotropy in the core and face sheets is addressed by the small parameters within the 3-D stiffness matrices. Numerical results are illustrated for several sample problems. The 3-D responses recovered using VAM-based model are obtained in a much more computationally efficient manner than, and are in agreement with, those of available 3-D elasticity solutions and 3-D FE solutions of MSC NASTRAN. (c) 2012 Elsevier Ltd. All rights reserved.

Modeling of Piezocomposites using Variational Asymptotic Method

This work presents the development of piezocomposites made up of Macro Fiber Composites (MFCs) for aerospace applications and specifically involves, their computational analysis, material characterization and certain parametric studies. MFC was developed by NASA Langley Research Center in 1996 and currently is being distributed by Smart Material Co. 1] worldwide and finds applications both as an actuator as well as for sensor in various engineering applications. In this work, MFC is being modeled as an actuator and a theoretical formulation based on Variational Asymptotic Method (VAM) 2] is presented to analyse the laminates made up of MFCs. VAM minimizes the total electro-mechanical energy for the MFC laminate and approaches the exact solution asymptotically by making use of certain small parameters inherent to the problem through dimensional reduction. VAM provides closed form solutions for 1D constitutive law, recovery relations of warpings, 3D stress/strain fields and displacements and hence an ideal tool for carrying out parametric and design studies in such applications. VAM is geometrically exact and offers rigorous material characterization through cross-sectional analysis and dimensional reduction.

New variational principles for systems of partial differential equations

The question of finding variational principles for coupled systems of first order partial differential equations is considered. Using a potential representation for solutions of the first order system a higher order system is obtained. Existence of a variational principle follows if the original system can be transformed to a self-adjoint higher order system. Existence of variational principles for all linear wave equations with constant coefficients having real dispersion relations is established. The method of adjoining some of the equations of the original system to a suitable Lagrangian function by the method of Lagrange multipliers is used to construct new variational principles for a class of linear systems. The equations used as side conditions must satisfy highly-restrictive integrability conditions. In the more difficult nonlinear case the system of two equations in two independent variables can be analyzed completely. For systems determined by two conservation laws the side condition must be a conservation law in addition to satisfying the integrability conditions.

A variational framework for spectral discretization of the density matrix in Kohn Sham density functional theory

Kohn-Sham density functional theory (KSDFT) is currently the main work-horse of quantum mechanical calculations in physics, chemistry, and materials science. From a mechanical engineering perspective, we are interested in studying the role of defects in the mechanical properties in materials. In real materials, defects are typically found at very small concentrations e.g., vacancies occur at parts per million, dislocation density in metals ranges from $10^{10} m^{-2}$ to $10^{15} m^{-2}$, and grain sizes vary from nanometers to micrometers in polycrystalline materials, etc. In order to model materials at realistic defect concentrations using DFT, we would need to work with system sizes beyond millions of atoms. Due to the cubic-scaling computational cost with respect to the number of atoms in conventional DFT implementations, such system sizes are unreachable. Since the early 1990s, there has been a huge interest in developing DFT implementations that have linear-scaling computational cost. A promising approach to achieving linear-scaling cost is to approximate the density matrix in KSDFT. The focus of this thesis is to provide a firm mathematical framework to study the convergence of these approximations. We reformulate the Kohn-Sham density functional theory as a nested variational problem in the density matrix, the electrostatic potential, and a field dual to the electron density. The corresponding functional is linear in the density matrix and thus amenable to spectral representation. Based on this reformulation, we introduce a new approximation scheme, called spectral binning, which does not require smoothing of the occupancy function and thus applies at arbitrarily low temperatures. We proof convergence of the approximate solutions with respect to spectral binning and with respect to an additional spatial discretization of the domain. For a standard one-dimensional benchmark problem, we present numerical experiments for which spectral binning exhibits excellent convergence characteristics and outperforms other linear-scaling methods.

Positive radial solutions of the Dirichlet problem for the Minkowski-Curvature equation in a ball

We study the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation { -div(del upsilon/root 1-vertical bar del upsilon vertical bar(2)) in B-R, upsilon=0 on partial derivative B-R,B- where B-R is a ball in R-N (N >= 2). According to the behaviour off = f (r, s) near s = 0, we prove the existence of either one, two or three positive solutions. All results are obtained by reduction to an equivalent non-singular one-dimensional problem, to which variational methods can be applied in a standard way.

The electronic structure of anionic halide complexes of element 105 in aqueous solutions and their extraction by aliphatic amines

To study the complex formation of group 5 elements (Nb, Ta, Ha, and pseudoanalog Pa) in aqueous HCI solutions of medium and high concentrations the electronic structures of anionic complexes of these elements [MCl_6]^-, [MOCl_4]^-, [M(OH)-2 Cl_4]^-, and [MOCl_5]^2- have been calculated using the relativistic Dirac-Slater Discrete-Variational Method. The charge density distribution analysis has shown that tantalum occupies a specific position in the group and has the highest tendency to form the pure halide complex, [TaCl_6-. This fact along with a high covalency of this complex explains its good extractability into aliphatic amines. Niobium has equal trends to form pure halide [NbCl_6]^- and oxyhalide [NbOCl_5]^2- species at medium and high acid concentrations. Protactinium has a slight preference for the [PaOCl_5]^2- form or for the pure halide complexes with coordination number higher than 6 under these conditions. Element 105 at high HCl concentrations will have a preference to form oxyhalide anionic complex [HaOCl_5]^2- rather than [HaCl_6]^-. For the same sort of anionic oxychloride complexes an estimate has been done of their partition between the organic and aqueous phases in the extraction by aliphatic amines, which shows the following succession of the partition coefficients: P_Nb < P_Ha < P_Pa.

Variational data assimilation for very large environmental problems

Variational data assimilation is commonly used in environmental forecasting to estimate the current state of the system from a model forecast and observational data. The assimilation problem can be written simply in the form of a nonlinear least squares optimization problem. However the practical solution of the problem in large systems requires many careful choices to be made in the implementation. In this article we present the theory of variational data assimilation and then discuss in detail how it is implemented in practice. Current solutions and open questions are discussed.

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF NONLINEAR SCHRODINGER EQUATIONS

This paper proves the multiplicity of positive solutions for the following class of quasilinear problems: {-epsilon(p)Delta(p)u+(lambda A(x) + 1)vertical bar u vertical bar(p-2)u = f(u), R(N) u(x)>0 in R(N), where Delta(p) is the p-Laplacian operator, N > p >= 2, lambda and epsilon are positive parameters, A is a nonnegative continuous function and f is a continuous function with subcritical growth. Here, we use variational methods to get multiplicity of positive solutions involving the Lusternick-Schnirelman category of intA(-1)(0) for all sufficiently large lambda and small epsilon.