Self-consistent Formulation Research Articles

A comment on the Equation of State and the freezing point equation with respect to subglacial lake modelling

The empirical Equation of State ( EoS) allows the calculation of the density of water in dependence of salinity, temperature, and pressure. Water density is an important quantity to determine the internal structure and flow regime of ocean and lakes. Hence, its exact representation in numerical models is of utmost importance for the specific simulation results. The three parameters namely salinity, temperature, and pressure have a complex interdependency on the EoS. Whether warmer water parcels sink or rise, therefore depends on the surrounding salinity and pressure. The empirical Equation of Freezing Point ( EoFP) allows to calculate the pressure- and salinity-dependent freezing point of water. Both equations are necessary to model the basal mass balance below Antarctic ice shelves or at the ice–water interface of subglacial lakes. This article aims three tasks: first we comment on the most common formulations of the EoS and the EoFP applied in numerical ocean and lake models during the past decades. Then we describe the impact of the recent and self-consistent Gibbs thermodynamic potential formulation of the EoS and the EoFP on subglacial lake modelling. Finally, we show that the circulation regime of subglacial lakes covered by at least 3000 m of ice, in principle, is independent of the particular formulation, in contrast to lakes covered by a shallower ice sheet, like e.g., Subglacial Lake Ellsworth. However, as modelled values like the freezing and melting patterns or the distribution of accreted ice at the ice–lake interface are sensitive to different EoS and EoFP, we present updated values for Subglacial Lake Vostok and Subglacial Lake Concordia.

A Reformulation of von Neumann’s Postulates on Quantum Measurement by Using Two Theorems in Clifford Algebra

According to a procedure previously introduced from Y. Ilamed and N. Salingaros, we start giving proof of two existing Clifford algebras, the Si that has isomorphism with that one of Pauli matrices and the Ni,±1 where Ni stands for the dihedral Clifford algebra. The salient feature is that we show that the Ni,±1 may be obtained from the Si algebra when we attribute a numerical value (+1 or −1) to one of the basic elements (e1,e2,e3) of the Si. We utilize such result to advance a criterium under which the Si algebra has as counterpart the description of quantum systems that in standard quantum mechanics are considered in absence of observation and quantum measurement while the Ni,±1 attend when a quantum measurement is performed on such system with advent of wave function collapse. The physical content of the criterium is that the quantum measurement with wave function collapse induces the passage in the considered quantum system from the Si to Ni,+1 or to the Ni,−1 algebras, where each algebra has of course its proper rules of commutation. After a proper discussion on the difference between decoherence and wave function collapse, we re-examine the von Neumann postulate on quantum measurement, and we give a proper justification of such postulate by using the Si algebra. Soon after we study some applications of the above mentioned criterium to some cases of interest in standard quantum mechanics, analyzing in particular a two state quantum system, the case of time dependent interaction of such system with a measuring apparatus and finally the case of a quantum system plus measuring apparatus developed at the order n=4 of the considered Clifford algebras and of the corresponding density matrix in standard quantum mechanics. In each of such cases examined, we find that the passage from the algebra Si to Ni,±1, considered during the quantum measurement of the system, actually describes the collapse of the wave function. Therefore we conclude that the actual quantum measurement has as counterpart in the Clifford algebraic description, the passage from the Si to the Ni,±1 Clifford algebras, reaching in this manner the objective to reformulate von Neumann postulate on quantum measurement and proposing a self-consistent formulation of quantum theory.

Smoothed Particle Magnetohydrodynamics - IV. Using the vector potential

In this paper, we investigate the use of the vector potential as a means of maintaining the divergence constraint in the numerical solution to the equations of magnetohydrodynamics (MHD) using the Smoothed Particle Hydrodynamics (SPH) method. We derive a self-consistent formulation of the equations of motion using a variational principle that is constrained by the numerical formulation of both the induction equation and the curl operator used to obtain the magnetic field, which guarantees exact and simultaneous conservation of momentum, energy and entropy in the numerical scheme. This leads to a novel formulation of the MHD force term, unique to the vector potential, which differs from previous formulations. We also demonstrate how dissipative terms can be correctly formulated for the vector potential such that the contribution to the entropy is positive definite and the total energy is conserved. On a standard suite of numerical tests in one, two and three dimensions, we first find that the consistent formulation of the vector potential equations is unstable to the well-known SPH tensile instability, even more so than in the standard Smoothed Particle Magnetohydrodynamics (SPMHD) formulation where the magnetic field is evolved directly. Furthermore, we find that whilst a hybrid approach based on the vector potential evolution equation coupled with a standard force term gives good results for one- and two-dimensional problems (where dAz/dt= 0), such an approach suffers from numerical instability in three dimensions related to the unconstrained evolution of vector potential components. We conclude that use of the vector potential is not a viable approach for SPMHD.

Impurity-induced bound states in iron-based superconductors withs-wavecos kx⋅cos kypairing symmetry

Using both the self-consistent Bogoliubov-de Gennes formulation and non-self-consistent T-matrix approach, we perform a comprehensive investigation of the in-gap bound states induced by a localized single impurity in iron-based superconductors. We focus on studying signatures associated with the unconventional sign-changed s-wave pairing symmetry. For a non-magnetic impurity, we find that there are two in-gap bounds, symmetric with respect to zero energy, only in the sign changed s-wave pairing state, not in the sign-unchanged s-wave state, due to the existence of non-trivial Andreev bound states caused by the sign change. For a magnetic impurity, we find that due to the breakdown of the local time-reversal symmetry, there exist only bound state solutions (with orbital degeneracy) carrying one of the electron-spin polarizations around the impurity. As increasing the scattering strength, the system undergoes a quantum phase transition (level crossing) from a spin-unpolarized ground state to a spin-polarized one. While the results for the magnetic impurity are qualitatively similar in both the sign-changed and sign-unchanged s-wave superconducting states, the bound states in the first case are more robust and there is no $\pi$ phase shift of the SC gap near the impurity in the strong scattering regime.

Eulerian elastoplasticity: Basic issues and recent results

Traditional formulations of elastoplasticity in the presence of finite strain and large rotation are Eulerian type and widely used; they are based upon, among other things, the additive decomposition of the stretching or the Eulerian strain-rate into elastic and plastic parts. In such formulations, yield functions and objective rate constitutive equations are expressed in terms of objective Eulerian tensor quantities, including the stretching, the Kirchhoff stress, internal state variables, etc. Each of these quantities transforms in a corotational manner under a change of the observing frame. According to the principle of material frame-indifference or objectivity, each constitutive function should be invariant, whenever the observing frame is changed to another one by any given time-dependent rotation. In this work the general form of constitutive equations is discussed. Several frequently used objective rates are analyzed with respect to their serviceability to develop a self-consistent formulation, i.e. to be integrable to deliver an elastic in particular hyperelastic relation for vanishing plastic deformation. This would be of great importance, e.g., for so-called spring back calculations in metal forming.

Thermoelectrical regime of a vertical electrolyzer. Part 2: Numerical modeling of two-phase electrolyte convection

The results of numerical modeling of two-phase electrolyte thermo-and hydrodynamics in the working space of a separate electrolyzer cell with vertical electrode arrangement are presented. The problem is solved in a self-consistent formulation taking into account the electric current, Joule heat evolution, gas evolution, gas and liquid flows, and electric conductivity of the disperse phase. Within the framework of the proposed two-liquid approach, near-wall bubbly flow is calculated using a model of expanding gas plume, which is based on the introduction of an effective force acting upon the gas phase. The results of several variants of such calculations are compared. The numerical solution is also compared to approximate analytical solutions of the problem. The results of this study can be used to calculate estimations of the electrical and thermal regimes of vertical electrolyzers.

UNIVERSAL PRANDTL NUMBER IN TWO-DIMENSIONAL KRAICHNAN-BATCHELOR TURBULENCE

We evaluate the universal turbulent Prandtl numbers in the energy and enstrophy régimes of the Kraichnan-Batchelor spectra of two-dimensional turbulence using a self-consistent mode-coupling formulation coming from a renormalized perturbation expansion coupled with dynamic scaling ideas. The turbulent Prandtl number is found to be exactly unity in the (logarithmic) enstrophy régime, where the theory is infrared marginal. In the energy régime, the theory being finite, we extract singularities coming from both ultraviolet and infrared ends by means of Laurent expansions about these poles. This yields the turbulent Prandtl number σ ≈ 0.9 in the energy régime.

A numerical investigation of heat transfer of stratified melt with volumetric heat release in the bottom layer

The heat transfer of stratified melt in a massive steel vessel is investigated. Commercially available FLUENT 6.2 computer codes are employed. A steady-state problem in two-dimensional axisymmetric formulation is solved. The results of test calculations are given, including the calculation of heat transfer of steel melt heated from below by a preassigned heat flux. A numerical solution is obtained of the problem on stratified melt in self-consistent formulation. The value of steady heat flux to the vessel at typical values of input data is close to the critical value of heat flux under conditions of water boiling on a vertical wall.

A self-consistent boundary element, parametric dislocation dynamics formulation of plastic flow in finite volumes

We present a self-consistent formulation of 3-D parametric dislocation dynamics (PDD) with the boundary element method (BEM) to describe dislocation motion, and hence microscopic plastic flow in finite volumes. We develop quantitative measures of the accuracy and convergence of the method by considering a comparison with known analytical solutions. It is shown that the method displays absolute convergence with increasing the number of quadrature points on the dislocation loop and the surface mesh density. The error in the image force on a screw dislocation approaching a free surface is shown to increase as the dislocation approaches the surface, but is nevertheless controllable. For example, at a distance of one lattice parameter from the surface, the relative error is less than 5% for a surface mesh with an element size of 1000 × 2000 (in units of lattice parameter), and 64 quadrature points. The Eshelby twist angle in a finite-length cylinder containing a coaxial screw dislocation is also used to benchmark the method. Finally, large scale 3-D simulation results of single slip behavior in cylindrical microcrystals are presented. Plastic flow characteristics and the stress–strain behavior of cylindrical microcrystals under compression are shown to be in agreement with experimental observations. It is shown that the mean length of dislocations trapped at the surface is the dominant factor in determining the size effects on hardening of single crystals. The influence of surface image fields on the flow stress is finally explored. It is shown that the flow stress is reduced by as much as 20% for small single crystals of size less than 0.15 μ m .

Numerical simulations of plasma equilibrium in a one-dimensional current sheet with a nonzero normal magnetic field component

The force balance in a thin collisionless current sheet in the Earth’s magnetotail with a given constant magnetic field component Bz across the sheet is numerically studied for the first time in a self-consistent formulation of the problem. The current sheet is produced by oppositely directed plasma flows propagating from the periphery of the sheet toward the neutral plane. A substantially improved version of a macroparticle numerical model is used that makes it possible to simulate on the order of 107 macroparticles even with a personal computer and to calculate equilibrium configurations with a sufficiently low discrete noise level in the first-and second-order moments of the distribution function, which determine the stress tensor elements. Quasisteady configurations were calculated numerically for several sets of plasma parameters in some parts of the magnetotail. The force balance in the sheet was checked by calculating the longitudinal and transverse pressures as well as the elements of the full stress tensor. The stress tensor in the current sheet is found to be nondiagonal and to differ appreciably from the gyrotropic stress tensor in the Chew-Goldberger-Low model, although the Chew-Goldberger-Low theory and numerical calculations yield close results for large distances from the region of reversed magnetic field.

Dirac-Bogoliubov-deGennes quasiparticles in a vortex lattice

In the mixed state of an extreme type-II $d$-wave superconductor and within a broad regime of weak magnetic fields $({H}_{c1}⪡H⪡{H}_{c2})$, the low-energy Bogoliubov-deGennes quasiparticles can be effectively described as Dirac fermions moving in the field of singular scalar and vector potentials. Although the effective linearized Hamiltonian operator does not formally depend on the structure of vortex cores, a singular nature of the perturbation requires choosing a self-adjoint extension of the Hamiltonian by imposing additional boundary conditions at vortex locations. Each vortex is described by a single parameter $\ensuremath{\theta}$ that effectively represents all effects arising from the physics beyond linearization. With the value of $\ensuremath{\theta}$ properly fixed, the resulting density of states of Dirac Hamiltonian exhibits full invariance under arbitrary singular gauge transformations applied at vortex positions. We identify the self-adjoint extensions of the solutions found earlier, within the framework of the linearized Hamiltonian diagonalized by expansion in the plane wave basis, and analyze the relation between fully self-consistent formulation of the problem and the linearized model. In particular, we construct the low-field scaling form of the nodal quasiparticle spectra, which incorporates the self-adjoint extension parameter $\ensuremath{\theta}$ explicitly and generalizes the conventional Simon-Lee scaling. In a companion paper, we also present a detailed numerical study of the lattice $d$-wave superconductor model and examine its low-energy, low-magnetic-field behavior with an eye on determining the proper self-adjoint extension(s) of the linearized continuum limit. In general, we find that the density of quasiparticle states always vanishes at the chemical potential, either linearly or by virtue of a finite gap. The low-energy continuum limit is thus faithfully represented by Dirac-like fermions, which are either truly massless, massless at the linearized level (mass $\ensuremath{\sim}H$), or massive (mass $\ensuremath{\sim}{H}^{1∕2}$), depending on the mutual commensuration of magnetic length and lattice spacing.

Self-consistent modelling of the mechanical behaviour of viscoplastic polycrystals incorporating intragranular field fluctuations

We present a detailed description of the numerical implementation, within the widely used viscoplastic self-consistent (VPSC) code, of a rigorous second-order (SO) homogenization procedure for non-linear polycrystals. The method is based on a linearization scheme, making explicit use of the covariance of the fluctuations of the local fields in a certain linear comparison material, whose properties are, in turn, determined by means of a suitably designed variational principle. We discuss the differences between this second-order approach and several first-order self-consistent (SC) formulations (secant, tangent and affine approximations) by comparing their predictions with exact full-field solutions. We do so for crystals with different symmetries, as a function of anisotropy, number of independent slip systems and degree of non-linearity. In this comparison, the second-order estimates show the best overall agreement with the full-field solutions. Finally, the different SC approaches are applied to simulate texture evolution in two strongly heterogeneous systems and, in both cases, the SO formulation yields results in better agreement with experimental evidence than the first-order approximations. In the case of cold-rolling of low-SFE fcc polycrystals, the SO formulation predicts the formation of a texture with most of the characteristic features of a brass-type texture. In the case of polycrystalline ice, deforming in uniaxial compression to large strain, the SO predicts a substantial and persistent accommodation of deformation by basal slip, even when the basal poles become strongly aligned with the compression direction.

Self-consistent molecular dynamics formulation for electric-field-mediated electrolyte transport through nanochannels

A self-consistent molecular dynamics (SCMD) formulation is presented for electric-field-mediated transport of water and ions through a nanochannel connected to reservoirs or baths. The SCMD formulation is compared with a uniform field MD approach, where the applied electric field is assumed to be uniform, for 2nm and 3.5nm wide nanochannels immersed in a 0.5M KCl solution. Reservoir ionic concentrations are maintained using the dual-control-volume grand canonical molecular dynamics technique. Simulation results with varying channel height indicate that the SCMD approach calculates the electrostatic potential in the simulation domain more accurately compared to the uniform field approach, with the deviation in results increasing with the channel height. The translocation times and ionic fluxes predicted by uniform field MD can be substantially different from those predicted by the SCMD approach. Our results also indicate that during a 2ns simulation time K+ ions can permeate through a 1nm channel when the applied electric field is computed self-consistently, while the permeation is not observed when the electric field is assumed to be uniform.

Simulation of Heme Using DFT + U: A Step toward Accurate Spin-State Energetics

We investigate the DFT + U approach as a viable solution to describe the low-lying states of ligated and unligated iron heme complexes. Besides their central role in organometallic chemistry, these compounds represent a paradigmatic case where LDA, GGA, and common hybrid functionals fail to reproduce the experimental magnetic splittings. In particular, the imidazole pentacoordinated heme is incorrectly described as a triplet by all usual DFT flavors. In this study, we show that a U parameter close to 4 eV leads to spin transitions and molecular geometries in quantitative agreement with experiments and that DFT + U represents an appealing tool in the description of iron porphyrin complexes, at a much reduced cost compared to correlated quantum-chemistry methods. The possibility of obtaining the U parameter from first principles is explored through a self-consistent linear-response formulation. We find that this approach, which proved to be successful in other iron systems, produces in this case some overestimation with respect to the optimal values of U.

Three-Dimensional Boundary Element – Dislocation Dynamics Modeling of Plastic Flow in Small Volumes

Format This is a copy of the slides presented at the meeting but not formally written up for the volume. Abstract We present a self-consistent formulation of 3-D Parametric Dislocation Dynamics (PDD) with the Boundary Element method (BEM) to describe dislocation motion, and hence microscopic plastic flow in finite volumes. We develop quantitative measures of the accuracy and convergence of the method by considering a comparison with known analytical solutions. It is shown that the method displays absolute convergence with increasing the number of quadrature points on the dislocation loop and the surface mesh density. For example, at a distance of one lattice parameter from the surface, the relative error is less than 5% for a surface mesh with an element size of 1000 x 2000 lattice parameters, and 64 quadrature points. The Eshelby twist in a finite length cylinder containing a coaxial screw dislocation is also used to benchmark the method. Simulation results of single slip behavior in cylindrical microcrystals is presented, and the general features are compared to single-crystal compression experiments. The method is utilized to study size scaling aspects of plastic flow in small volumes and assess the role of the dislocation starvation mechanism.

Self-consistent multimode lasing theory for complex or random lasing media

A semiclassical theory of single and multimode lasing is derived for open complex or random media using a self-consistent linear response formulation. Unlike standard approaches which use closed cavity solutions to describe the lasing modes, we introduce an appropriate discrete basis of functions which describe also the intensity and angular emission pattern outside the cavity. This constant flux (CF) basis is dictated by the Green function which arises when formulating the steady state Maxwell-Bloch equations as a self-consistent linear response problem. This basis is similar to the quasibound state basis which is familiar in resonator theory and it obeys biorthogonality relations with a set of dual functions. Within a single-pole approximation for the Green function the lasing modes are proportional to these CF states and their intensities and lasing frequencies are determined by a set of nonlinear equations. When a near threshold approximation is made to these equations a generalized version of the Haken-Sauermann equations for multimode lasing is obtained, appropriate for open cavities. Illustrative results from these equations are given for single and few mode lasing states, for the case of dielectric cavity lasers. The standard near threshold approximation is found to be unreliable. Applications to wave-chaotic cavities and random lasers are discussed.

Rigorous carrier dynamic model of electroluminescent metal-oxide-semiconductor silicon tunneling diodes

The carrier dynamics of electroluminescent metal-oxide-semiconductor silicon tunneling diodes is rigorously modeled in this study. Various tunneling and recombination current densities are formulated without using the Maxwell-Boltzmann approximation for the carrier concentrations. This model satisfactorily explains the current-voltage characteristics in strong accumulation using a self-consistent formulation. It also relates light emission efficiency to the interface-state density and the bulk-trap density in a straightforward manner. The internal radiative recombination efficiency in the bulk Si substrate is estimated to be in tens of percents. The model also explains clearly the strongly influenced small-signal light-current response time with respect to the injection current density. It enables the small-signal method to be useful in extracting the parameters of the interface-state density and the bulk-trap density.

Vertex similarity in networks

We consider methods for quantifying the similarity of vertices in networks. We propose a measure of similarity based on the concept that two vertices are similar if their immediate neighbors in the network are themselves similar. This leads to a self-consistent matrix formulation of similarity that can be evaluated iteratively using only a knowledge of the adjacency matrix of the network. We test our similarity measure on computer-generated networks for which the expected results are known, and on a number of real-world networks.

A polycrystal hysteresis model for ferroelectric ceramics

Most key elements of ferroelectric properties are defined through the hysteresis loops. For a ferroelectric ceramic, its loop is contributed collectively by its constituent grains, each having its own hysteresis loop when the ceramic polycrystal is under a cyclic electric field. In this paper, we propose a polycrystal hysteresis model so that the hysteresis loop of a ceramic can be calculated from the loops of its constituent grains. In this model a micromechanics-based thermodynamic approach is developed to determine the hysteresis behaviour of the constituent grains, and a self-consistent scheme is introduced to translate these behaviours to the polycrystal level. This theory differs from the classical phenomenological ones in that it is a micromechanics-based thermodynamic approach and it can provide the evolution of new domain concentration among the constituent grains. It also differs from some recent micromechanics studies in its secant form of self-consistent formulation and in its application of irreversible thermodynamics to derive the kinetic equation of domain growth. To put this two-level micromechanics theory in perspective, it is applied to a ceramic PLZT 8/65/35, to calculate its hysteresis loop between the electric displacement and the electric field ( D versus E ), and the butterfly-shaped longitudinal strain versus the electric field relation ( ϵ versus E ). The calculated results are found to be in good quantitative agreement with the test data. The corresponding evolution of new domain concentration c 1 and the individual hysteresis loops of several selected grains—along with those of the overall polycrystal—are also illustrated.

Three-Dimensional Boundary Element-Dislocation Dynamics Modeling of Plastic Flow in Small Volumes [PowerPoint Submission

AbstractWe present a self-consistent formulation of 3-D Parametric Dislocation Dynamics (PDD) with the Boundary Element method (BEM) to describe dislocation motion, and hence microscopic plastic flow in finite volumes. We develop quantitative measures of the accuracy and convergence of the method by considering a comparison with known analytical solutions. It is shown that the method displays absolute convergence with increasing the number of quadrature points on the dislocation loop and the surface mesh density. For example, at a distance of one lattice parameter from the surface, the relative error is less than 5% for a surface mesh with an element size of 1000 × 2000 lattice parameters, and 64 quadrature points. The Eshelby twist in a finite length cylinder containing a coaxial screw dislocation is also used to benchmark the method. Simulation results of single slip behavior in cylindrical microcrystals is presented, and the general features are compared to single-crystal compression experiments. The method is utilized to study size scaling aspects of plastic flow in small volumes and assess the role of the dislocation starvation mechanism.