Helical Coordinates Research Articles

On the Lundgren hierarchy of helically symmetric turbulence

Abstract This paper analyzes the reduction of the infinite Lundgren-Monin-Novikov (LMN) hierarchy of probability density functions (PDFs) in the statistical theory of helically symmetric turbulence. Lundgren's hierarchy is considered a complete model, i.e. fully describes the joint multi-point statistic of turbulence though at the expense of dealing with an infinite set of integro-differential equations. The LMN hierarchy and its respective side-conditions are transformed to helical coordinates and thus are dimesionally reduced. In the course of development, a number of key questions were solved, namely in particular the transformation of PDFs and sample space velocities into orthonormal coordinate systems. In a validity check it is shown, that the mean momentum equations derived from the helical LMN hierarchy via statistical moment integration are identical to the mean momentum equations derived by direct ensemble averaging the Navier-Stokes equation, in helically symmetric form. Finally, we derive the equation for the characteristic function equivalent to the PDF equation in a helically symmetric frame, which allows to generate arbitrary n^th-order statistical moments by simple differentiation.

Machine learning based prediction of the electronic structure of quasi-one-dimensional materials under strain

We present a machine learning based model that can predict the electronic structure of quasi-one-dimensional materials while they are subjected to deformation modes such as torsion and extension/compression. The technique described here applies to important classes of materials such as nanotubes, nanoribbons, nanowires, miscellaneous chiral structures and nano-assemblies, for all of which, tuning the interplay of mechanical deformations and electronic fields is an active area of investigation in the literature. Our model incorporates global structural symmetries and atomic relaxation effects, benefits from the use of helical coordinates to specify the electronic fields, and makes use of a specialized data generation process that solves the symmetry-adapted equations of Kohn-Sham Density Functional Theory in these coordinates. Using armchair single wall carbon nanotubes as a prototypical example, we demonstrate the use of the model to predict the fields associated with the ground state electron density and the nuclear pseudocharges, when three parameters - namely, the radius of the nanotube, its axial stretch, and the twist per unit length - are specified as inputs. Other electronic properties of interest, including the ground state electronic free energy, can then be evaluated with low-overhead post-processing, typically to chemical accuracy. We also show how the nuclear coordinates can be reliably determined from the pseudocharge field using a clustering based technique. Remarkably, only about 120 data points are found to be enough to predict the three dimensional electronic fields accurately, which we ascribe to the symmetry in the problem setup, the use of low-discrepancy sequences for sampling, and presence of intrinsic low-dimensional features in the electronic fields. We comment on the interpretability of our machine learning model and discuss its possible future applications.

Density functional theory method for twisted geometries with application to torsional deformations in group-IV nanotubes

We present a real-space formulation and implementation of Kohn-Sham Density Functional Theory suited to twisted geometries, and apply it to the study of torsional deformations of X (X = C, Si, Ge, Sn) nanotubes. Our formulation is based on higher order finite difference discretization in helical coordinates, uses ab initio pseudopotentials, and naturally incorporates rotational (cyclic) and screw operation (i.e., helical) symmetries. We discuss several aspects of the computational method, including the form of the governing equations, d ils of the numerical implementation, as well as its convergence, accuracy and efficiency properties.The technique presented here is particularly well suited to the first principles simulation of quasi-one-dimensional structures and their deformations, and many systems of interest can be investigated using small simulation cells containing just a few atoms. We apply the method to systematically study the properties of single-wall zigzag and armchair group-IV nanotubes in the range of (approximately) 1 to 3 nm radius, as they undergo twisting. For the range of deformations considered, the mechanical behavior of the tubes is found to be largely consistent with isotropic linear elasticity, with the torsional stiffness, ktwist, varying as the cube of the nanotube radius. Furthermore, for a given tube radius, ktwist is seen to be highest for carbon nanotubes and the lowest for those of tin, while nanotubes of silicon and germanium are found to have intermediate values of this quantity close to each other. We also describe different aspects of the variation in electronic properties of the nanotubes as they are twisted. In particular, we find that akin to the well known behavior of armchair carbon nanotubes, armchair nanotubes of silicon, germanium and tin also exhibit bandgaps that vary periodically with imposed rate of twist, and that the periodicity of the variation scales in an inverse quadratic manner with the tube radius. These examples highlight the utility of the proposed method in the accurate and efficient computational characterization of important nanomaterials from first principles.

Расчет напряженного состояния витой трубки некругового сечения на основе тензорной теории оболочек

The purpose of the study was to develop an algorithm for calculating helical-symmetric shells with a closed contour in oblique Gaussian coordinates. The twist and length of the shell were taken unchanged. The method is based on the representation of the generating contour of the helicoidal surface by a discrete set of points with the replacement of differentiation along the angular coordinate by finite differences. The unknown were the displacement vectors at the indicated points of the contour. Due to the helicoidal symmetry, the differentiation of vector quantities with respect to the helical coordinate was replaced by vector multiplication. The tensor of deformations and the tensor of the parameters of the change in curvature were calculated using the nabla operator, represented in oblique Gaussian coordinates. Integration over the contour coordinate was replaced by summation over discrete points. The tensors found, which characterize the deformed state, were used to calculate the strain energy of one period of the helicoidal shell, and then the total potential of the mechanical system was compiled. The unknown displacements were determined by minimizing the total potential, taking into account the constraints that prohibit the displacement of the shell as a rigid whole. The study gives a numerical example of the application of the developed approach.

Ab initio framework for systems with helical symmetry: Theory, numerical implementation and applications to torsional deformations in nanostructures

We formulate and implement Helical Density Functional Theory (Helical DFT) — a self-consistent first principles simulation method for nanostructures with helical symmetries. Such materials are well represented in all of nanotechnology, chemistry and biology, and prominent examples include nanotubes, nanosprings, nanowires, miscellaneous chiral structures and important proteins. The overwhelming preponderance of such helical structures in all of science and engineering and the likelihood of these systems being associated with exotic materials properties, provides the motivation to develop systematic and predictive tools for their study.Following this line of thought, we develop a mathematical and computational framework in this contribution, that allows helical structures to be studied ab initio, using Kohn–Sham theory. We first show that the electronic states in helical structures can be characterized by means of special solutions to the single electron problem called helical Bloch waves. We rigorously demonstrate the existence and completeness of such solutions, and then describe how they can be used to reduce the Kohn–Sham Density Functional Theory (KS-DFT) equations for helical structures to a suitable fundamental domain. Next, we develop a symmetry-adapted finite-difference strategy in helical coordinates to discretize the governing equations, and obtain a working realization of our proposed approach. We verify the accuracy and convergence properties of our numerical implementation through examples. Finally, we employ Helical DFT to study the properties of zigzag and chiral single wall black phosphorus (i.e., phosphorene) nanotubes. Specifically, we use our simulations to evaluate the torsional stiffness of a zigzag nanotube ab initio. Additionally, we observe an insulator-to-metal-like transition in the electronic properties of this nanotube as it is subjected to twisting. We also find that a similar transition can be effected in chiral phosphorene nanotubes by means of axial strains. The strong dependence of the band gap of these materials on various modes of strain suggests their possible use as nanomaterials with tunable electronic and transport properties. Notably, self-consistent ab initio simulations of this nature are unprecedented and well outside the scope of any other systematic first principles method in existence. We end with a discussion on various future avenues and applications.

Two-dimensional elastic Bloch waves in helical periodic structures

The goal of this paper is to investigate the propagation of two-dimensional elastic Bloch waves in helical periodic structures, obeying two discrete screw symmetries about the same axis. First, a three-dimensional coordinate system is built from the two helical directions of periodicity of the problem and the radial coordinate originating from the symmetry axis. The existence of Bloch waves in bi-helical structures can be justified owing to the independence of the metric tensor of the coordinate system on both helical coordinates. Considering the elastodynamic equilibrium equations, Bloch theorem is expressed in appropriate bases to project the vector wavefields, namely the covariant/contravariant bases of the bi-helical coordinate system (or, alternatively, the cylindrical basis). From a geometrical point of view, the three-dimensional unit cell is delimited by non-plane boundaries, which must be carefully parametrized. The so-called wave finite element method is then applied to numerically solve the Bloch wave eigenproblem and the implementation of the numerical method in a bi-helical system is detailed. Owing to the cylindrical nature of the geometry, the two-dimensional propagation constants are not independent to each other. The relationship between both constants is established. The calculation of wave mode properties (wavenumbers, group and energy velocities) is performed along the helical propagation directions, as well as the straight axial and circumferential directions. Numerical validations of the overall approach are carried out for cylindrical uniform tubes and a chiral nanotube. Finally, the method is applied to a complex multi-wire structure, often encountered in energy cables, consisting of two layers of helical wires twisted in opposite directions.

New similarity reductions and exact solutions for helically symmetric viscous flows

In the present paper, we derive exact solutions for the helically invariant Navier–Stokes equations. The approach is based on an invariant solution ansatz emerging from the Galilean group in helical coordinates, which leads to linear functions in the helical coordinate ξ = az + bφ for the two helical velocity components uξ and uη. The variables z and φ are the usual cylinder coordinates. Starting from this approach, we derive a new equation for the radial velocity component ur in the helical frame, for which we found two special solutions. Moreover, we present an exact linearization of the Navier–Stokes equations by seeking exact solutions in the form of Beltrami flows. Using separation of variables, we found exponentially decaying time-dependent solutions, which consist of trigonometric functions in the helical coordinate ξ and of confluent Heun-type functions in the radial direction.

Investigation of wave propagation in piezoelectric helical waveguides with the spectral finite element method

The dispersion behaviors of wave propagation in waveguides of piezoelectric helical structures are investigated. By using the tensor analysis in the helical curve coordinate, the general strain-displacement relationship of piezoelectric helix is firstly considered. This paper's formulation is based on the spectral finite element which just requires the discretization of the cross-section with high-order spectral elements. The eigenvalue matrix of the dispersion relationship between wavenumbers and frequencies is obtained. Numerical examples on PZT5A and Ba2NaNb5O15 helical waveguides of a wide range of lay angles are presented. The effects of the piezoelectric on the dispersive properties and the variation tendency of dispersion curves on helix angles are shown. The mechanism of mode separation in piezoelectric helical waveguides is further analyzed through studying waves structures of the flexural modes.

Hydrodynamic characteristics of viscous fluid flow in screw channels formed by two ribs

The mathematical model of laminar viscous flows in screw channels, formed by two ribs, is developed using the helical coordinates. The numerical results of the flow with helical symmetry are presented.

Poly(U) RNA-templated synthesis of AppA.

Simple nucleotide templating activities are of interest as potential primordial reactions. Here we describe the acceleration of 5′-5′ AppA synthesis by 3′-5′ poly(U) under normal solution conditions. This reaction is apparently templated via complementary U:A base-pairing, despite the involvement of two different RNA backbones, because poly(U), unlike other polymers, significantly stimulates AppA synthesis. These interactions occur in moderate (K+) and (Mg2+) and are temperature sensitive, being more efficient at 10°C than at 4°C, but absent at 20°C. The reaction is only slightly pH sensitive, despite potentially relevant substrate pKa’s. Kinetic data explicitly support production of AppA by interaction of stacked 2MeImpA and pA nucleotides paired with a single molecule of U template. At a lower rate, AppA can also be produced by a chemical reaction between 2MeImpA and pA, without participation of poly(U). Molecular modeling suggests that 5′-5′ joining between stacked or concurrently paired A's can occur without major departures from normal U-A helical coordinates. So, coenzyme-like 5′-5′ purine dinucleotides might be readily synthesized from 3′-5′ RNAs with complementary sequences.

DNA exit ramps are revealed in the binding landscapes obtained from simulations in helical coordinates.

DNA molecules are highly charged semi-flexible polymers that are involved in a wide variety of dynamical processes such as transcription and replication. Characterizing the binding landscapes around DNA molecules is essential to understanding the energetics and kinetics of various biological processes. We present a curvilinear coordinate system that fully takes into account the helical symmetry of a DNA segment. The latter naturally allows to characterize the spatial organization and motions of ligands tracking the minor or major grooves, in a motion reminiscent of sliding. Using this approach, we performed umbrella sampling (US) molecular dynamics (MD) simulations to calculate the three-dimensional potentials of mean force (3D-PMFs) for a Na+ cation and for methyl guanidinium, an arginine analog. The computed PMFs show that, even for small ligands, the free energy landscapes are complex. In general, energy barriers of up to ~5 kcal/mol were measured for removing the ligands from the minor groove, and of ~1.5 kcal/mol for sliding along the minor groove. We shed light on the way the minor groove geometry, defined mainly by the DNA sequence, shapes the binding landscape around DNA, providing heterogeneous environments for recognition by various ligands. For example, we identified the presence of dissociation points or “exit ramps” that naturally would terminate sliding. We discuss how our findings have important implications for understanding how proteins and ligands associate and slide along DNA.

Stokes flow through a twisted tube with square cross-section

Flow through a twisted tube with square cross-section and helical corrugations of arbitrary pitch is computed under conditions of Stokes flow. The governing equations are formulated in non-orthogonal helical coordinates in terms of a coupled system of linear differential equations describing the longitudinal and transverse flow over the tube cross-section. Numerical solutions are computed by a finite-difference method on a staggered grid incorporating pressure nodes and three sets of velocity nodes. The results illustrate the structure of the secondary flow developing over the tube cross-section and document the effect of the helical corrugations on the axial flow rate.

The torqued cylinder and Levi-Civita’s metric

When a static cylindrical system is subjected to equal and opposite torques top and bottom it transports angular momentum along its axis. The external metric of this static system can be transformed to Levi-Civita’s form by using helical coordinates. This gives the external metric of a static cylinder three dimensionless parameters corresponding to the mass per unit length, the total stress along the cylinder, and the total torque. The external vacuum metric of a spherical system is characterised by its mass alone. How many parameters characterise the external metric of a general stationary cylindrical system? Leaving aside the radius of the cylinder which defines the scale we find that there are five parameters, the three above mentioned, to which should be added the momentum along the cylinder per unit length and the angular momentum per unit length. We show how to transform Levi-Civita’s one parameter metric to include all five.

Numerical investigation of dispersion relations for helical waveguides using the Scaled Boundary Finite Element method

In this paper, the dispersion properties of elastic waves in helical waveguides are investigated. The formulation is based on the Scaled Boundary Finite Element method (SBEFM). With a set of orthogonal unit basis introduced as the contravariant basis, the helical coordinate is firstly considered, where components of tensor retain the dimension of original quantity. Based on the strain–displacement relation, the eigenvalue matrix is obtained about wavenumbers and frequencies. The cross section of the waveguides is discretized by using high-order spectral elements. Moreover, the formulated linear matrix is utilized to design efficient and accurate algorithms to compute the eigenvalues of helical waveguides. Compared with the Pochhammer–Chree curves, the convergence and accuracy of the SBFEM are discussed. Finally, we give some dispersion curves for a wide range of lay angles and analyze in detail properties of cut-off frequency, mode separation and mode transition for elastic wave propagation in the helical waveguides.

Transcription-factor binding and sliding on DNA studied using micro- and macroscopic models.

Transcription factors search for specific operator sequences by alternating rounds of 3D diffusion with rounds of 1D diffusion (sliding) along the DNA. The details of such sliding have largely been beyond direct experimental observation. For this purpose we devised an analytical formulation of umbrella sampling along a helical coordinate, and from extensive and fully atomistic simulations we quantified the free-energy landscapes that underlie the sliding dynamics and dissociation kinetics for the LacI dimer. The resulting potential of mean force distributions show a fine structure with an amplitude of 1 k(B)T for sliding and 12 k(B)T for dissociation. Based on the free-energy calculations the repressor slides in close contact with DNA for 8 bp on average before making a microscopic dissociation. By combining the microscopic molecular-dynamics calculations with Brownian simulation including rotational diffusion from the microscopically dissociated state we estimate a macroscopic residence time of 48 ms at the same DNA segment and an in vitro sliding distance of 240 bp. The sliding distance is in agreement with previous in vitro sliding-length estimates. The in vitro prediction for the macroscopic residence time also compares favorably to what we measure by single-molecule imaging of nonspecifically bound fluorescently labeled LacI in living cells. The investigation adds to our understanding of transcription-factor search kinetics and connects the macro-/mesoscopic rate constants to the microscopic dynamics.

Electron Pressure and Transport Analysis During QSH States in RFX-mod

In this paper, the pressure behavior is characterized for helical states in the Reversed Field eXperiment-mod plasmas; using the helical coordinates as a reference frame to match data coming from different diagnostics, pressure profile variations have been correlated to magnetic topology changes in plasma. The use of helical frame to integrate diagnostics is also exploited as a tool to identify changes in the electronic heat transport, within a single shot or a reduced shots ensemble, confirming via 1-D and 2-D power balance models the improvement in the plasma performances.

Magnetic coordinate systems for helical SHAx states in reverse field pinch plasmas

Single helical axis (SHAx) states are the improved confinement RFP helical states, which have been observed at high plasma currents (beyond 1 MA) in the RFX-mod device. A reconstruction of these equilibrium configurations has been implemented in a code named SHEq, where SHAx states are modelled as pure single helicity (SH) states. We consider the helical flux in this work, which happens to be a good flux function for SH equilibrium, and its contours give the shape of the flux surfaces in the helical equilibrium. New curvilinear and non-orthogonal ‘helical’ coordinate systems are defined in SHEq, with the fundamental request that angles are defined with respect to the helical axis. We look for helical coordinates beginning from the canonical form of the magnetic field B and the Hamiltonian mechanics theory enables us to obtain straight-field-line helical coordinates computing action–angle coordinates.These coordinate systems are used to determine the helical safety factor q, using the simple formula valid in action–angle coordinates. Very good agreement with the computation of the safety factor using the ORBIT code has been found.

Helical flow of polymer melts in extruders, part 1: Model development

AbstractOperating and processing conditions as well as the selection of the screw design in injection molding industry are largely based on trial‐and‐error exercise, which is expensive and time consuming. A better approach is to develop mathematical models to help select the conditions and parameters and predict the process performance. However, most of the models developed and used so far contain unrealistic geometrical and mathematical simplifications. The objective of this work is to develop a steady‐state three‐dimensional mathematical model to describe the flow of an incompressible polymer melt inside a helical geometry, which represents the polymer's true motion in extrusion and injection molding processes. The mathematical model is first developed in a natural cylindrical system. Transformers are then derived to obtain the model in helical coordinates. A novel feature of this work is the consideration of tapered screws, i.e., screws tapered either upward or downward along the direction of the flow. © 2010 Wiley Periodicals, Inc. Adv Polym Techn 29:249–260, 2010; View this article online at wileyonlinelibrary.com. DOI 10.1002/adv.20195

Magnetic chaos healing in the helical reversed-field pinch: indications from the volume-preserving field line tracing code NEMATO

The emergence of a self-organized reversed-field pinch (RFP) helical regime, first shown by 3D MHD numerical simulations, has been highlighted in the RFX-mod experiment at high current operation (IP above 1 MA). In fact, a quasi-stationary helical configuration spontaneously appears, characterized by strong internal electron transport barriers. In such regime electron temperature and density become, to a very good approximation, functions of the helical flux coordinate related to the dominant helical magnetic component. In addition, this regime is diagnosed to be associated with the topological transition to a single-helical-axis (SHAx) state, achieved after the expulsion of the separatrix of the dominant mode's magnetic island. The SHAx state is theoretically predicted to be resilient to the magnetic chaos induced by secondary modes. In this paper, we present initial results of the volume-preserving field line tracing code NEMATO [Finn J M and Chacón L 2005 Phys. Plasmas 12 054503] applied to study the magnetic topology resulting from 3D MHD simulations of the RFP. First, a successful 2D verification test of the code is shown, then, initial application to a systematic study of chaos healing in the helical RFP is discussed. The separatrix disappearance is confirmed to play an essential role for chaos healing. The triggering effect of a reversed magnetic shear for the formation of ordered surfaces within magnetic chaos is also diagnosed.

Generation and conversion of optical vortices in long-period helical core optical fibers

We theoretically study generation and conversion of optical vortices (OVs) in long-period helical core fibers (HCFs) produced by drawing from a perform with an eccentric core. Solving scalar waveguide equation in the helical coordinates we demonstrate that in such fibers topologically induced corrections to scalar propagation constants lead to convergence of spectral curves. At their intersection points a resonance coupling takes place between the fundamental and the vortex modes, as well as the coupling between OVs, whose topological charges differ by unity. This coupling leads to energy exchange between the coupled modes, which is manifested either in conversion of the input fundamental mode into an OV or transformation of an OV into the OV with higher by 1 (or lower, depending on core's helicity) value of the topological charge. This effect can be used for creation of all-fiber generators of singular beams from regular input beams. We also study effect of ellipticity of the core's form on transformation properties of HCFs and show that this leads to splitting of resonance fiber's parameters, at which conversion of left and right circularly polarized input regular beams into OVs occurs. We prove that in long-period HCFs one can neglect this effect due to reduction of polarization mode dispersion in twisted fibers.