Cartesian Basis Research Articles

An augmented Lagrangian technique combined with a mortar algorithm for modelling mechanical contact problems

SUMMARYA finite element formulation for three dimensional (3D) contact mechanics using a mortar algorithm combined with a mixed penalty–duality formulation from an augmented Lagrangian approach is presented. In this method, no penalty parameter is introduced for the regularisation of the contact problem. The contact approach, based on the mortar method, gives a smooth representation of the contact forces across the bodies interface, and can be used in arbitrarily curved 3D configurations. The projection surface used for integrating the equations is built using a local Cartesian basis defined in each contact element. A unit normal to the contact surface is defined locally at each element, simplifying the implementation and linearisation of the equations. The displayed examples show that the algorithm verifies the contact patch tests exactly, and is applicable to large displacements problems with large sliding motions.Copyright © 2012 John Wiley & Sons, Ltd.

Radial orthogonality and Lebesgue constants on the disk

In polynomial interpolation, the choice of the polynomial basis and the location of the interpolation points play an important role numerically, even more so in the multivariate case. We explore the concept of spherical orthogonality for multivariate polynomials in more detail on the disk. We focus on two items: on the one hand the construction of a fully orthogonal cartesian basis for the space of multivariate polynomials starting from this sequence of spherical orthogonal polynomials, and on the other hand the connection between these orthogonal polynomials and the Lebesgue constant in multivariate polynomial interpolation on the disk. We point out the many links of the two topics under discussion with the existing literature. The new results are illustrated with an example of polynomial interpolation and approximation on the unit disk. The numerical example is also compared with the popular radial basis function interpolation.

Cartesian and polar Schmidt bases for down-converted photons

We derive an analytical form of the Schmidt modes of spontaneous parametric down-conversion (SPDC) biphotons in both Cartesian and polar coordinates. We show that these correspond to Hermite-Gauss (HG) or Laguerre-Gauss (LG) modes only for a specific value of their width, and we show how such value depends on the experimental parameters. The Schmidt modes that we explicitly derive allow one to set up an optimised projection basis that maximises the mutual information gained from a joint measurement. The possibility of doing so with LG modes makes it possible to take advantage of the properties of orbital angular momentum eigenmodes. We derive a general entropic entanglement measure using the Rényi entropy as a function of the Schmidt number, K, and then retrieve the von Neumann entropy, S. Using the relation between S and K we show that, for highly entangled states, a non-ideal measurement basis does not degrade the number of shared bits by a large extent. More specifically, given a non-ideal measurement which corresponds to the loss of a fraction of the total number of modes, we can quantify the experimental parameters needed to generate an entangled SPDC state with a sufficiently high dimensionality to retain any given fraction of shared bits.

GENERALIZED PARTON DISTRIBUTIONS IN THE CHIRAL ODD SECTOR AND THEIR ROLE IN NEUTRAL MESON ELECTROPRODUCTION

Our original suggestion to investigate exclusive π0 electroproduction as a method for extracting from data the tensor charge, transversity, and other quantities related to chiral odd generalized parton distributions is further examined. We now explain the details of the process: i) the connection between the helicity description and the cartesian basis; ii) the dependence on the momentum transfer squared, Q2, and iii) the angular momentum, parity, and charge conjugation constraints (JPC quantum numbers).

Eletroweak Form Factors in the Light-Front for Spin-1 Particles

The contribution of the light-front valence wave function to the electromagnetic current of spin-1 composite particles is not enough to warranty the proper transformation of the current under rotations. The naive derivation of the plus component of the current in the Drell-Yan-West frame within an analytical and covariant model of the vertex leads to the violation of the rotational symmetry. Computing the form-factors in a quasi Drell-Yan-West frame $q^+\rightarrow 0$, we were able to separate out in an analytical form the contributions from Z-diagrams or zero modes using the instant-form cartesian polarization basis.

Solution of the Skyrme–Hartree–Fock–Bogolyubov equations in the Cartesian deformed harmonic-oscillator basis.: (VII) hfodd (v2.49t): A new version of the program

We describe the new version (v2.49t) of the code HFODD which solves the nuclear Skyrme Hartree-Fock (HF) or Skyrme Hartree-Fock-Bogolyubov (HFB) problem by using the Cartesian deformed harmonic-oscillator basis. In the new version, we have implemented the following physics features: (i) the isospin mixing and projection, (ii) the finite temperature formalism for the HFB and HF+BCS methods, (iii) the Lipkin translational energy correction method, (iv) the calculation of the shell correction. A number of specific numerical methods have also been implemented in order to deal with large-scale multi-constraint calculations and hardware limitations: (i) the two-basis method for the HFB method, (ii) the Augmented Lagrangian Method (ALM) for multi-constraint calculations, (iii) the linear constraint method based on the approximation of the RPA matrix for multi-constraint calculations, (iv) an interface with the axial and parity-conserving Skyrme-HFB code HFBTHO, (v) the mixing of the HF or HFB matrix elements instead of the HF fields. Special care has been paid to using the code on massively parallel leadership class computers. For this purpose, the following features are now available with this version: (i) the Message Passing Interface (MPI) framework, (ii) scalable input data routines, (iii) multi-threading via OpenMP pragmas, (iv) parallel diagonalization of the HFB matrix in the simplex breaking case using the ScaLAPACK library. Finally, several little significant errors of the previous published version were corrected.

Polarization observables in the longitudinal basis for pseudo-scalar meson photoproduction using a density matrix approach

The complete expression for the intensity in pseudo-scalar meson photoproduction with a polarized beam, target, and recoil baryon is derived using a density matrix approach that offers great economy of notation. A Cartesian basis with spins for all particles quantized along a single direction, the longitudinal beam direction, is used for consistency and clarity in interpretation. A single spin-quantization axis for all particles enables the amplitudes to be written in a manifestly covariant fashion with simple relations to those of the well-known Chew-Goldberger-Low-Nambu formalism. Possible sign discrepancies between theoretical amplitude-level expressions and experimentally measurable intensity profiles are dealt with carefully. Our motivation is to provide a coherent framework for coupled-channel partial-wave analysis of several meson photoproduction reactions, incorporating recently published and forthcoming polarization data from Jefferson Lab.

The effect of scattering on single photon transmission of optical angular momentum

Schemes for the communication and registration of optical angular momentumdepend on the fidelity of transmission between optical system components. Itis known that electron spin can be faithfully relayed between exciton states inquantum dots; it has also been shown by several theoretical and experimental studiesthat the use of beams conveying orbital angular momentum can significantlyextend the density and efficiency of such information transfer. However, it remainsunclear to what extent the operation of such a concept at the single photon level ispracticable—especially where this involves optical propagation through a material system,in which forward scattering events can intervene. The possibility of transmitting anddecoding angular momentum over nanoscale distances itself raises other importantissues associated with near-field interrogation. This paper provides a framework toaddress these and related issues. A quantum electrodynamical representation isconstructed and used to pursue the consequences of individual photons, froma Laguerre–Gaussian beam, undergoing single and multiple scattering eventsin the course of propagation. In this context, issues concerning orbital angularmomentum conservation, and its possible compromise, are tackled by identifying therelevant components of the electromagnetic scattering and coupling tensors, using anirreducible Cartesian basis. The physical interpretation broadly supports the fidelity ofquantum information transmission, but it also identifies potential limitations ofprinciple.

Ab Initio Calculation of Resonance Raman Cross Sections Based on Excited State Geometry Optimization

A method for the calculation of resonance Raman cross sections is presented on the basis of calculation of structural differences between optimized ground and excited state geometries using density functional theory. A vibrational frequency calculation of the molecule is employed to obtain normal coordinate displacements for the modes of vibration. The excited state displacement relative to the ground state can be calculated in the normal coordinate basis by means of a linear transformation from a Cartesian basis to a normal coordinate one. The displacements in normal coordinates are then scaled by root-mean-square displacement of zero point motion to calculate dimensionless displacements for use in the two-time-correlator formalism for the calculation of resonance Raman spectra at an arbitrary temperature. The method is valid for Franck-Condon active modes within the harmonic approximation. The method was validated by calculation of resonance Raman cross sections and absorption spectra for chlorine dioxide, nitrate ion, trans-stilbene, 1,3,5-cycloheptatriene, and the aromatic amino acids. This method permits significant gains in the efficiency of calculating resonance Raman cross sections from first principles and, consequently, permits extension to large systems (>50 atoms).

Tensor Transmission-Line Metamaterials

Transmission-line (TL) metamaterials possessing effective material parameters that are diagonal in the Cartesian basis have been previously studied. In this paper, TL metamaterials with arbitrary full tensors are introduced and analyzed. An approximate tensor analysis of the proposed metamaterials is first developed. Bloch analysis is then used to verify the approximate analysis and derive exact dispersion equations and impedance relations. Finally, simulation results are presented that validate the analysis and show the utility of this new class of metamaterials. The ability to design tensor metamaterials such as these is crucial to the practical implementation of novel devices derived through transformational optics/electromagnetics.

Origin of Hund’s multiplicity rule in quasi-two-dimensional two-electron quantum dots

The origin of Hund's multiplicity rules has been studied for a system of two electrons confined by a quasi-two-dimensional harmonic-oscillator potential by relying on a full configuration interaction wave function and Cartesian anisotropic Gaussian basis sets. In terms of appropriate normal-mode coordinates the wave function factors into a product of the center-of-mass and the internal components. The ${}^{1}{\ensuremath{\Pi}}_{u}$ singlet state and the ${}^{3}{\ensuremath{\Pi}}_{u}$ triplet state represent the energetically lowest pair of states to which Hund's multiplicity rule applies. They are shown to involve excitations into different degrees of freedom, namely, into the center-of-mass angular mode and the internal angular mode for the singlet and triplet states, respectively. The presence of an angular nodal line in the internal space allows then the triplet state to avoid the singularity in the electron-electron interaction potential, leading to the energy lowering of the triplet state relative to its counterpart singlet state.

Karhunen–Loève basis of Kolmogorov phase screens covering a rectangular stripe

Karhunen–Loève functions are statistically independent base functions to be multiplied with randomized coefficients to generate instances of the underlying statistics with specific covariances. In the context of numerical simulation of atmospheric turbulence, they are tools to generate phase distributions in the input pupil of telescope optics. Within the Taylor frozen screen approximation, the favorable geometry of phase screens is a stripe as wide as the telescope diameter, but much longer. A phase screen movie is generated by dragging the stripe across the telescope pupil. The aim is to avoid the wasteful alternative of cutting through a virtual circular phase screen with a diameter equal to the stripe's length. We calculate Karhunen–Loève functions over rectangular stripes by solving the well-known integral equation for isotropic variance of the phase distribution. They are expanded in products of Legendre functions along the two Cartesian coordinates, represented by spherical Bessel functions in Fourier space. The rectangular shape of the integral's domain breaks the symmetry of the structure function. Yet, the matrix elements of the eigenvalue equation can be calculated by separating the primitive Cartesian basis via a Neumann series of radial and azimuthal terms.

Solution of the Skyrme–Hartree–Fock–Bogolyubov equations in the Cartesian deformed harmonic-oscillator basis.: (VI) hfodd (v2.40h): A new version of the program

Solution of the Skyrme–Hartree–Fock–Bogolyubov equations in the Cartesian deformed harmonic-oscillator basis.: (VI) hfodd (v2.40h): A new version of the program

Development of basis sets for molecular calculations. Comparison of tetrahedral gaussian lobe functions and cartesian gaussian basis sets

In order to allow development of a range of basis sets for use of floating spherical Gaussian orbitals in molecular calculations, an analysis of numerical errors that result from using off-center functions was carried out. This analysis included both rotational variance and numerical differencing problems that can arise when lobe functions are employed. The analysis and numerical studies indicated that satisfactory accuracy in integral evaluation can be maintained if the ratio of distance from the origin to orbital radius is taken to be in the range 0.002 to 0.005 for computers having 72-bit words. 2 figures, 1 table.

Search for multiple chiral doublets in rhodium isotopes

The deformation in rhodium isotopes is investigated using adiabatic and configuration-fixed constrained triaxial relativistic mean field (RMF) approaches. The triaxial deformations are found in the ground states of (102,104,106,108,110)Rh, which is consistent with triaxial Skyrme Hartree-Fock calculations. Several minima with triaxial deformation in (104,106,108,110)Rh are obtained by the configuration-fixed constrained calculations. The corresponding configurations are characterized by the quantum numbers vertical bar nljm > obtained by transforming wave functions from a Cartesian basis to a spherical basis. The possible existence of multiple chiral doublets (M chi D) is demonstrated in (104,106,108,110)Rh isotopes, based on different particle-hole configurations and triaxial deformations.

Re-entrant corner flows of PTT fluids in the Cartesian stress basis

We consider the planar flow of Phan-Thien–Tanner (PTT) fluids around a re-entrant corner of angle π / α where α ∈ [ 1 / 2 , 1 ) . The model is considered in the absence of a solvent viscosity and the flow situation assumes complete flow around the corner with the absence of a lip votex. The local asymptotic solution structure is similar to that for the upper convected Maxwell (UCM) model and is shown to comprise a core flow (outer) region in which the fluid behaves elastically, together with wall boundary layers (inner regions) of similar thickness as those in the UCM model. In the core flow, the stress singularity is that for UCM, namely O ( r − 2 ( 1 − α ) ) where r is the radial distance from the corner, although the stream function vanishes at the slower rate O ( r α ( 1 + α ) ) compared to O ( r α ( 3 − α ) ) for UCM—this latter feature being a consequence of the shear thinning property of the PTT model. The amplitudes of the velocity and stress fields are determined and are seen to be independent from this local analysis, any link between them appearing to require global flow information away from the corner. The analysis is performed here in the Cartesian stress formulation of the problem, allowing the description of a similarity solution for the core flow and upstream boundary layer. The analysis remains to be completed by a solution for the downstream boundary layer which requires the use of the natural stress basis.

Re-entrant corner flows of UCM fluids: The Cartesian stress basis

For a given re-entrant corner geometry, we describe a two parameter family of solutions for the local asymptotic behaviour of the flow and stress fields of UCM fluids. The two parameters used are the coefficients of the upstream wall shear rate and pressure gradient. In describing this parametric solution, the relationship between the Cartesian and natural stress basis is explained, reconciling these two equivalent formulations for the problem. The asymptotic solution structure investigated here comprises an outer (core) region together with inner regions (single wall boundary layers) located at the upstream and downstream walls. It is implicitly assumed that there are no regions of recirculation at the upstream wall, i.e. we consider flow in the absence of a lip vortex. The essential feature of the analysis is a full description of the matching between the outer and inner regions in both the Cartesian and natural stress bases, as well as the derivation of numerical estimates of important solution parameters such as the coefficients of the stream function and extra-stresses in the outer (core flow) region together with the downstream wall shear rate. This work is divided into two papers, the first one describing the solution structure in the Cartesian stress basis for the core and upstream boundary layer, and the second paper using the natural stress basis which allows the downstream boundary layer solution to be linked through the core to the upstream boundary layer solution. It is the latter formulation of this problem which allows a complete solution description.

The intramolecular tunnelling of hydrogen†

This paper reports an approach to the analysis of the local anharmonic states of motion and tunnelling of hydrogen illustrated with the molecule malonaldehyde. A potential energy function is used accurately to fit the local potential energy surface. The basis fitting functions in the potential resemble the Morse basis function, , but are intrinsically Gaussian: . After rearrangement to isolate the Gaussian operators, the new form of the potential allows the direct use of mathematically friendly (Gaussian) manipulations that are needed to carry out the quantum mechanical analysis of the anharmonic features, namely tunnelling. The quantal analysis uses Cartesian harmonic oscillator basis functions together with the Talmi–Moshinsky–Smirnov methods to evaluate matrix elements of the potential energy operator. The longer term objective of this kind of work is to develop practical methods of hybrid analysis with which to treat the local atom dynamics associated with a reactive species while the rest of the dynamics are modelled as simple harmonic oscillators. The goal is eventually to have a computational theory of proton/hydrogen or small molecular group transfer reactions in condensed-phase systems. This paper addresses only the detailed local dynamical analysis. †Dedicated to Pavel Rosmus on the occasion of the celebration of his scienti.c achievements, in Prague.

Spectra and correlated wave functions of two electrons confined in a quasi-one-dimensional nanostructure

The energy spectra and wave functions of two electrons confined by a quasi-one-dimensional Gaussian potential have been calculated for different strengths of confinement, ${\ensuremath{\omega}}_{z}$, and anharmonicity by using the quantum chemical full configuration interaction method employing a Cartesian anisotropic Gaussian basis set. The energy spectra for a nearly harmonic Gaussian potential have been studied and analyzed in three regimes of ${\ensuremath{\omega}}_{z}$: namely, large $({\ensuremath{\omega}}_{z}=5.0)$, medium $({\ensuremath{\omega}}_{z}=1.0)$, and small $({\ensuremath{\omega}}_{z}=0.1)$. For large and medium ${\ensuremath{\omega}}_{z}$ the energy spectrum shows a band structure which is characterized by the polyad quantum number ${v}_{p}$ while for small ${\ensuremath{\omega}}_{z}$ it is characterized by the extended polyad quantum number ${v}_{p}^{*}$. The energy levels for small ${\ensuremath{\omega}}_{z}$ form doublet pairs each of which consists of a pair of singlet and triplet states. The nodal patterns of their wave functions are almost identical to each other except for their phases. The energy spectra for the strongly anharmonic Gaussian potential look quite similar to those of the nearly harmonic case except that an irregular level structure appears in the high-energy region for ${\ensuremath{\omega}}_{z}=0.1$. The wave functions of the states in this high-energy region have curved nodal lines which align along a pair of bent nodal coordinates. Two types of pairs of bent nodal coordinates have been identified: namely, those passing through the valley of the confining potential and the others passing on the hillside. It is shown that the wave functions with these nodal coordinates correspond to new types of classical local-mode motions of electrons.

Single particle calculations for a Woods–Saxon potential with triaxial deformations, and large Cartesian oscillator basis (new version code)

We present a new version of the computer program which solves the Schrödinger equation of the stationary states for an average nuclear potential of Woods–Saxon type. In this work, we take specifically into account triaxial (i.e. ellipsoidal) nuclear surfaces. The deformation is specified by the usual Bohr parameters. The calculations are carried out in two stages. In the first, one calculates the representative matrix of the Hamiltonian in the Cartesian oscillator basis. In the second stage one diagonalizes this matrix with the help of subroutines of the EISPACK library. This new version calculates all the eigenvalues up to a given cutoff energy, and gives the components of the corresponding eigenfunctions. For a more convenient handling, these results are stored simultaneously in the computer memory, and on a files. Program summary Title of program:Triaxial2007 Catalogue identifier:ADSK_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSK_v2_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Summary of revision:One input file instead two. Reduced number of input parameters. Storage of eigenvalues and eigenvectors in memory in a very simple way which makes the code very convenient to the user. Reasons for the new version: More convenient handling of the eigenvectors Catalogue number old version: ADSK Catalogue number new version:ADSK_v2_0 Journal: Computer Physics Commun. 156 (2004) 241–282 Licensing provisions: none Computer: PC Pentium 4, 2600 MHz Hard disk: 40 Gb RAM: 256 Mb Swap file: 4 Gb Operating system: WINDOWS XP Software used: Compaq Visual FORTRAN (with full optimizations in the settings project options) Programming language used:Fortran 77/90 (double precision) Number of bits in a word: 32 No. of lines in distributed program, including test data, etc.:4058 No. of bytes in distributed program, including test data, etc.:75 590 Distribution format:tar.gz Nature of the problem: The single particle energies and the single particle wave functions are calculated from one-body Hamiltonian including a central field of Woods–Saxon type, a spin–orbit interaction, and the Coulomb potential for the protons. We consider only ellipsoidal (triaxial) shapes. The deformation of the nuclear shape is fixed by the usual Bohr parameters ( β , γ ) . Method of solution: The representative matrix of the Hamiltonian is built by means of the Cartesian basis of the anisotropic harmonic oscillator, and then diagonalized by a set of subroutines of the EISPACK library. Two quadrature methods of Gauss are employed to calculate, respectively, the integrals of the matrix elements of the Hamiltonian, and the integral defining the Coulomb potential. Restrictions: There are two restrictions for the code: The number of the major shells of the basis does not have to exceed N max = 26 . For the largest values of N max (∼23–26), the diagonalization takes the major part of the running time, but the global run-time remains reasonable. Typical running time: (With full optimization in the project settings of the Compaq Visual Fortran on Windows XP) With N max = 23 , for the neutrons case, and for both parities, the running time is about 40 sec on the P4 computer at 2.6 GHz. In this case, the calculation of the matrix elements takes only about 17 sec. If all unbound states are required, the runtime becomes larger.