Nonorthogonal Terms Research Articles

A generic balanced-force algorithm for finite volume method on polyhedral unstructured grids with non-orthogonality

A generic balanced-force algorithm is proposed to solve incompressible multiphase flows with complex interfaces and large density ratios on the polyhedral unstructured grids of large non-orthogonality, skewness and non-uniformity. This algorithm combines the finite volume method based on the volume of fluid (VOF) approach with the fractional step method in a collocated framework, which retains a complete balance between the gravity, surface tension with exact curvature, and the resulting pressure gradients. In the original scheme, the gravity and surface tension terms are represented as gradients of the volume fraction and then discretized in the identical fashion as the pressure gradient, which results in a balanced formulation on the structured grids. However, this scheme cannot be readily applied to unstructured grids, despite that it has been misused for more than one decade. As demonstrated in the paper, it is insufficient to achieve force balance merely by the identical discretization of scalar gradients due to the non-orthogonality of unstructured grids, which will generate severe spurious currents that intensify with larger non-orthogonality angle and higher fluid density ratio. Therefore, we present a novel balanced-force algorithm to treat the non-orthogonal term in a different manner, which significantly suppresses, or even eliminates, the spurious velocity for two-phase flows with large density ratio on arbitrary mesh topology. Various numerical examples have demonstrated that the present algorithm and computational framework offer a promising platform to provide high-fidelity and robust predictions for practical multiphase flow simulations involving highly complex geometry.

P values and multivariate distributions: Non-orthogonal terms in regression models

The traditional use of p values both for univariate distributions and for Design of Experiments is described. In chemometrics they may additionally be calculated to provide an estimate of the significance of variables in non-orthogonal experimental matrices, for example as potential markers in a metabolomic analysis. The difficulties of interpreting these non-orthogonal variables is discussed. A set of simulations, each consisting of four variables, two variables which have varying levels of correlation, is described. As the correlation reduces between these two variables, so their calculated p values become smaller. An incorrect conclusion about the significance of individual variables may be obtained if there is a high correlation between them, although this depends on the intensity, relative to noise, of the variables of interest. The paper provides a numerical illustration of how serious this distortion is.

SCER Spoofing Attacks on the Galileo Open Service and Machine Learning Techniques for End-User Protection

Spoofing attacks pose a clear cybersecurity risk for all systems relying on Global Navigation Satellite Systems (GNSS) for time synchronization or positioning. Secure Code Estimation and Replay (SCER) spoofing attacks are the most challenging type of spoofing attacks, as these may be problematic even for future GNSS protection systems, like Navigation Message Authentication (NMA) or Spreading Code Authentication (SCA). This is one of the reasons that make the development of complementary protection techniques, like the one proposed in this work, necessary. In the first part of the paper, the spoofing SCER attacks are analyzed in detail for GPS and, particularly, for Galileo. The role of the Galileo Pseudorandom Noise (PRN) intra-satellite non-orthogonality distortion term in hindering the attacks is discussed and a detailed comparison between GPS and Galileo expected quality curves for the SCER attack is provided. A complementary detection method for end-user receivers (assuming NMA is used) against SCER attacks is proposed, based on the application of machine learning and a proposed set of features extracted from the receiver search space, assuming the attacker was not able to null the satellite signal.

Anomalous thermoelectric properties of a Floquet topological insulator with spin momentum non-orthogonality

The spin momentum non-orthogonality in 3D topological insulators leads to modification of the spin texture and brings in an out-of-plane spin polarization component. Apart from the spin texture, the anomalous thermoelectric properties of these materials are worth studying. In this paper, we have pointed out that the off resonant light used to irradiate the surface states, induces a gap, which becomes momentum dependent due to the presence of non-orthogonal terms in the Hamiltonian. Importantly, to maintain the off resonant condition of light, the momentum value should satisfy a bound. Furthermore, the momentum dependent gap causes a topological transition at a higher value of momentum, which is important to analyse the unusual double peak structure of the Nernst and electrical conductivities.

Buoyancy-induced flow, heat, and mass transfer in a porous annulus

ABSTRACTThis paper reports on double-diffusive natural convection heat transfer in a porous annulus between concentric horizontal circular and square cylinders. A pressure-based segregated finite volume method is used to solve the problem numerically. The diffusion fluxes are discretized using the MIND fully implicit scheme. Furthermore, a modified pressure correction equation is derived that implicitly accounts for the nonorthogonal diffusion terms, which are usually neglected in the standard SIMPLE algorithm. Results indicate that convection effects increase with an increase in Rayleigh number, Darcy number, porosity, and enclosure aspect ratio. Further, at low Darcy values, porosity has no effect on the flow, temperature, and concentration fields.

Establishing a theory for deuteron-induced surrogate reactions

Background: Deuteron-induced reactions serve as surrogates for neutron capture into compound states. Although these reactions are of great applicability, no theoretical efforts have been invested in this direction over the last decade.Purpose: The goal of this work is to establish on firm grounds a theory for deuteron-induced neutron-capture reactions. This includes formulating elastic and inelastic breakup in a consistent manner.Method: We describe this process both in post- and prior-form distorted wave Born approximation following previous works and discuss the differences in the formulation. While the convergence issues arising in the post formulation can be overcome in the prior formulation, in this case one still needs to take into account additional terms due to nonorthogonality.Results: We apply our method to the $^{93}\mathrm{Nb}(d,p)X$ at ${E}_{d}=15$ and 25 MeV and are able to obtain a good description of the data. We look at the various partial wave contributions, as well as elastic versus inelastic contributions. We also connect our formulation with transfer to neutron bound states.Conclusions: Our calculations demonstrate that the nonorthogonality term arising in the prior formulation is significant and is at the heart of the long-standing controversy between the post and the prior formulations of the theory. We also show that the cross sections for these reactions are angular-momentum dependent and therefore the commonly used Weisskopf limit is inadequate. Finally, we make important predictions for the relative contributions of elastic breakup and nonelastic breakup and call for elastic-breakup measurements to further constrain our model.

Evaluation of Assumptions in Soil Moisture Triple Collocation Analysis

Abstract Triple collocation analysis (TCA) enables estimation of error variances for three or more products that retrieve or estimate the same geophysical variable using mutually independent methods. Several statistical assumptions regarding the statistical nature of errors (e.g., mutual independence and orthogonality with respect to the truth) are required for TCA estimates to be unbiased. Even though soil moisture studies commonly acknowledge that these assumptions are required for an unbiased TCA, no study has specifically investigated the degree to which errors in existing soil moisture datasets conform to these assumptions. Here these assumptions are evaluated both analytically and numerically over four extensively instrumented watershed sites using soil moisture products derived from active microwave remote sensing, passive microwave remote sensing, and a land surface model. Results demonstrate that nonorthogonal and error cross-covariance terms represent a significant fraction of the total variance of these products. However, the overall impact of error cross correlation on TCA is found to be significantly larger than the impact of nonorthogonal errors. Because of the impact of cross-correlated errors, TCA error estimates generally underestimate the true random error of soil moisture products.

A New Eddy-Covariance Method Using Empirical Mode Decomposition

We introduce a new eddy-covariance method that uses a spectral decomposition algorithm called empirical mode decomposition. The technique is able to calculate contribu- tions to near-surface fluxes from different periodic components. Unlike traditional Fourier methods, this method allows for non-orthogonal contributions to the total flux, which are shown to be errors due to the undersampling of low-frequency processes. Inspection of the non-orthogonal terms with relation to sampling duration and periodicity reveals that a mea- sured periodic process requires approximately six cycles in order to be sufficiently sampled. This determines the maximum eddy size sufficiently measured given a particular sampling duration.

Productivity of Non-Orthogonal Term Rewrite Systems

Productivity is the property that finite prefixes of an infinite constructor term can be computed using a given term rewrite system. Hitherto, productivity has only been considered for orthogonal systems, where non-determinism is not allowed. This paper presents techniques to also prove productivity of non-orthogonal term rewrite systems. For such systems, it is desired that one does not have to guess the reduction steps to perform, instead any outermost-fair reduction should compute an infinite constructor term in the limit. As a main result, it is shown that for possibly non-orthogonal term rewrite systems this kind of productivity can be concluded from context-sensitive termination. This result can be applied to prove stabilization of digital circuits, as will be illustrated by means of an example.

Pickup coupling effects in deuteron scattering: The case ofd+Ca40

The dynamic polarization potential (DPP) contribution to the effective deuteron-nucleus interaction is evaluated by means of coupled reaction channel (CRC) calculations followed by $S$-matrix-to-potential inversion. The full coupled channel $S$ matrix ${S}_{{l}^{'}l}^{J}$ is inverted using the iterative-perturbative algorithm to yield a potential that includes a complex ${T}_{R}$ tensor term as well as central and spin-orbit components. The differences between the various components of the inverted potential and the corresponding terms in the bare potential of the CRC calculation constitute a local equivalent representation of the complete DPP that is generated by the reaction channel coupling. The magnitude of the DPP, the real part in particular, is much less than that found in earlier calculations in which the nonorthogonality terms were omitted. The characteristic features of the tensor part of the DPP were traced to breakup and reorientation processes in the entrance deuteron channel that had been included with the pickup. The contribution of stripping to the deuteron-nucleus interaction is also discussed.

Numerical simulation of viscous flow around unrestrained cylinders

Abstract A 2-D analysis is made for the dynamic interactions between viscous flow and one or more circular cylinders. The cylinder is free to respond to the fluid excitation and its motions are part of the solution. The numerical procedure is based on the finite volume discretization of the Navier–Stokes equations on adaptive tri-tree grids which are unstructured and nonorthogonal. Both a fully implicit scheme and a semi-implicit scheme in the time domain have been used for the momentum equations, while the pressure correction method based on the SIMPLE technique is adopted to satisfy the continuity equation. A new upwind method is developed for the triangular and unstructured mesh, which requires information only from two neighbouring cells but is of order of accuracy higher than linear. A new procedure is also introduced to deal with the nonorthogonal term. The pressure on the body surface required in solving the momentum equation is obtained through the Poisson equation in the local cell. Results including flow field, pressure distribution and force are provided for fixed single and multiple cylinders and for an unrestrained cylinder in steady incoming flow with Reynolds numbers at 200 and 500 and in unsteady flow with Keulegan–Carpenter numbers at 5 and 10.

Efficient discretization of pressure‐correction equations on non‐orthogonal grids

AbstractAn alternative discretization of pressure‐correction equations within pressure‐correction schemes for the solution of the incompressible Navier–Stokes equations is introduced, which improves the convergence and robustness properties of such schemes for non‐orthogonal grids. As against standard approaches, where the non‐orthogonal terms usually are just neglected, the approach allows for a simplification of the pressure‐correction equation to correspond to 5‐point or 7‐point computational molecules in two or three dimensions, respectively, but still incorporates the effects of non‐orthogonality. As a result a wide range (including rather high values) of underrelaxation factors can be used, resulting in an increased overall performance of the underlying pressure‐correction schemes. Within this context, a second issue of the paper is the investigation of the accuracy to which the pressure‐correction equation should be solved in each pressure‐correction iteration. The scheme is investigated for standard test cases and, in order to show its applicability to practical flow problems, for a more complex configuration of a micro heat exchanger. Copyright © 2003 John Wiley & Sons, Ltd.

Nonorthogonal distortion function and appearance of gauge field in atomic scattering

In the scattering of electrons or positrons from atoms the target wave function is modified by a distortion term which contains the projectile coordinate as a parameter. It is shown that the close coupling approximation (CCA) with a nonorthogonal distortion term induces a gauge field in a reduced description of the projectile dynamics quite analogous to what is obtained in the Born–Oppenheimer approximation for molecules. As an illustration elastic scattering cross-sections are calculated for positron–hydrogen scattering and the results are shown to be in quite good agreement with those obtained from more involved studies.

An efficient solution method for incompressible N-S equations using non-orthogonal collocated grid

An efficient strategy for the solution of N-S Equations using collocated, non-orthogonal grids is presented. The governing equations have been discretized in the physical plane itself without co-ordinate transformation, thereby retaining the lucidity of the basic finite volume method. The non-orthogonal terms and QUICK type corrections for the convective terms in the momentum equations are treated explicitly, while the other terms are taken in implicit form. In the pressure correction equation, the non-orthogonal terms have been dropped altogether. The discretized equations have been solved by the preconditioned conjugate gradient square method. The specific combination of above steps has resulted in better convergence properties as compared to those of existing algorithms, even for highly skewed grids. The scheme has been validated against benchmark solutions such as lid-driven flow in square and skewed cavities and experi mental results of flow over a single cylinder. Its applicability has also been illustrated for flow through a bank of staggered cylinders, with anti-symmetric inlet and outlet boundary conditions. Copyright © 1999 John Wiley & Sons, Ltd.

The inner valence photoionization of acetylene

We performed both single and coupled-channel scattering calculations of the photoionization of acetylene using an iterative Schwinger variational method. A comparison of the resulting single-channel and multichannel cross sections shows that the two results differ at photon energies of up to several hundred eV, which indicates that interchannel coupling and/or nonorthogonality terms may continue to be important even at these high photon energies. We compared the energy dependent satellite branching ratios with the corresponding ratios of spectroscopic intensity factors (SIFs) and found that the theoretically predicted high energy convergence of the former to the latter occurs by 1000 eV in only half of the cases studied. When the satellite provenance results from more than one main line, we do not observe this convergence behavior. We compare our theoretical results to experimental synchrotron radiation photoelectron spectroscopy (SRPES) results and illustrate how the superposition of one or more low intensity satellite lines may lead to the erroneous identification of convergence behavior. Based on our theoretical results for the photon energy dependence of the branching ratios of the four principle satellites of acetylene, results which are in excellent agreement with the experimental results, we conclude that all four of the satellites are dynamically correlated according to a phenomenological classification system. This conclusion opposes conclusions of a recent study that three of the satellites are intrinsically correlated, conclusions based upon the same experimental data that we use here, the same classification system, but differing in that the conclusion was formed in part on the basis of SIFs, i.e., static theoretical results. The results presented here underscore the difficulties inherent in analyzing the dynamics of photoionization using static theoretical results such as SIFs, and illustrate how a dynamic theory of photoionization can be used to interpret experimental data.

Event structures and non-orthogonal term graph rewriting

We show that for every term graph in a left-linear but non-orthogonal term graph rewrite system, one can construct an event structure that represents all the possible reductions that can occur in reduction sequences starting from that term graph. Every finite reduction sequence from that graph corresponds to a configuration of the event structure, and Lévy-equivalent sequences correspond to the same configuration.Garbage collection is modelled in the event structure by an ‘erases’ relation. The asymmetric conflicts that arise in non-orthogonal rewrite systems are modelled by introducing a ‘prevents’ relation. The configurations of the event structure then form the state space of an event automaton. Taking the directed completion of this space yields a prime algebraic domain.

Analytical expressions for power exchange in multiwaveguide systems

We present a complete analysis of power exchange in multiple coupled waveguides, using the modified scalar coupled-mode theory and give generalized expressions for power exchange between any two waveguides of a multiple-waveguide system. We thus obtain a reciprocity principle applicable to power exchange between any two waveguides. Completely generalized and accurate analytical expressions for all coupling coefficients, all perturbation correction terms, and all nonorthogonality terms involved in the analysis are presented.

Infinite normal forms for non-linear term rewriting systems

Nonterminating rewrite relations have recently been studied in order to set a framework within which infinite terms can be seen as limits of infinite converging derivations. Results about the existence of infinite normal forms have been given only for orthogonal term rewriting systems, namely left-linear and nonoverlapping systems. In this paper we show that some of those results can be extended to a particular class of nonorthogonal term rewriting systems. We deal with systems in which the nonterminating rules are unfolding rules that model the operational semantics of a recursive operator. The left-linearity requirement is replaced by a retraction property of the supporting term algebra, that allows the definition of a rewrite relation modulo a congruence relation induced on the set of terms by the unfolding rules.

Some Properties of Non-Orthogonal Term Graph Rewriting

This paper examines left-linear non-orthogonal term graph rewriting systems that allow asymmetric conflicts between redexes. Using a definition of compatibility of sequences based on Boudol's work on the semantics of term rewriting, it shows that two properties associated with functional languages are true of such graph rewriting systems. First, that a notion of standard computation can be defined and that an associated standardisation theorem can be proved. Second, that a set of events can be associated with a reduction sequence and hence event structures modelling all the possible reduction sequences from a given initial graph can be constructed.

NEW TREATMENT OF NONORTHOGONAL TERMS IN THE PRESSURE-CORRECTION EQUATION

Abstract This article proposes a new treatment method for nonorthogonal terms in the pressure-correction equation in order to enlarge the ranges of relaxation factors for convergence that are independent of the grid skewness and thus to ease the difficulty in determining the relaxation factors. The proposed method has been tested against four typical nonorthogonal two-dimensional cavity flows on a nonstaggered grid system. The results show that this method yields converged solution in wide ranges of relaxation factors, αu and α p, which is nearly independent of the grid skewness. In addition, since only seven nodes are required to define the coefficient matrix of the pressure-correction equation for three-dimensional flows, the proposed method makes computations of three-dimensional flows quite practical.