Abel Transform Research Articles

Temperature Sensing in Space and Transparent Media: Advancements in Off-Axis Digital Holography and the Temperature Coefficient of Refractive Index

An off-axis digital holographic interferometry technique integrated with a Mach–Zehnder interferometer based setup is demonstrated for measuring the temperature and temperature profile of a transparent medium. This technique offers several advantages: it does not require precise optomechanical adjustments or accurate definition of the frequency carrier mask, making it simple and cost-effective. Additionally, high-quality optics are not necessary. The methodology relies on measuring the phase difference between two digitally reconstructed complex wave fields and utilizing the temperature coefficient of the refractive index. In this way, we presented an equation of the temperature as a function of phase changes and the temperature coefficient of refractive index. This approach simplifies the calculation process and avoids the burden of complicated mathematical inversions, such as the inverse Abel transformation. It also eliminates the need for additional work with the Lorentz–Lorentz equation and Gladstone–Dale relation and can be extend for 3D measurements.

A Weighted Generalization of Hardy–Hilbert-Type Inequality Involving Two Partial Sums

In this paper, we address Hardy–Hilbert-type inequality by virtue of constructing weight coefficients and introducing parameters. By using the Euler–Maclaurin summation formula, Abel’s partial summation formula, and differential mean value theorem, a new weighted Hardy–Hilbert-type inequality containing two partial sums can be proven, which is a further generalization of an existing result. Based on the obtained results, we provide the equivalent statements of the best possible constant factor related to several parameters. Also, we illustrate how the inequalities obtained in the main results can generate some new Hardy–Hilbert-type inequalities.

Boundary-optimized summation-by-parts operators for finite difference approximations of second derivatives with variable coefficients

Boundary-optimized summation-by-parts (SBP) finite difference operators for second derivatives with variable coefficients are presented. The operators achieve increased accuracy by utilizing non-equispaced grid points close to the boundaries of the grid. Using the optimized operators we formulate SBP schemes for the acoustic and elastic operators defined directly on curvilinear multiblock domains. Numerical studies of the acoustic and elastic wave equations demonstrate that, compared to traditional SBP difference operators, the new operators provide increased accuracy for surface waves as well as block interfaces in multiblock grids. For instance, simulations of Rayleigh waves demonstrate that the boundary-optimized operators more than halve the runtime required for a given error tolerance.

A discontinuous Galerkin discretization of elliptic problems with improved convergence properties using summation by parts operators

Nishikawa (2007) proposed to reformulate the classical Poisson equation as a steady state problem for a linear hyperbolic system. This results in optimal error estimates for both the solution of the elliptic equation and its gradient. However, it prevents the application of well-known solvers for elliptic problems. We show connections to a discontinuous Galerkin (DG) method analyzed by Cockburn, Guzmán, and Wang (2009) that is very difficult to implement in general. Next, we demonstrate how this method can be implemented efficiently using summation by parts (SBP) operators, in particular in the context of SBP DG methods such as the DG spectral element method (DGSEM). The resulting scheme combines nice properties of both the hyperbolic and the elliptic point of view, in particular a high order of convergence of the gradients, which is one order higher than what one would usually expect from DG methods for elliptic problems.

A Tree Pruning Technique for Decoding Complexity Reduction of Polar Codes and PAC Codes

Sorting operation is one of the main bottlenecks for the successive-cancellation list (SCL) decoding. This paper introduces an improvement to the SCL decoding for polar and pre-transformed polar codes that reduces the number of sorting operations without visible degradation in the code’s error-correction performance. In an SCL decoding with an optimum metric function we show that, on average, the correct branch’s bit-metric value must be equal to the bit-channel capacity, and on the other hand, the average bit-metric value of a wrong branch can be at most zero. This implies that a wrong path’s partial path metric value deviates from the bit-channel capacity’s partial summation. For relatively reliable bit-channels, the bit metric for a wrong branch becomes very large negative number, which enables us to detect and prune such paths. We prove that, for a threshold lower than the bit-channel cutoff rate, the probability of pruning the correct path decreases exponentially by the given threshold. Based on these findings, we presented a pruning technique, and the experimental results demonstrate a substantial decrease in the amount of sorting procedures required for SCL decoding. In the stack algorithm, a similar technique is used to significantly reduce the average number of paths in the stack.

Stratospheric temperature measurements from nanosatellite stellar occultation observations of refractive bending

Abstract. Stellar occultation observations from space can probe the stratosphere and mesosphere at a fine vertical scale around the globe. Unlike other measurement techniques like radiosondes and aircraft, stellar occultation has the potential to observe the atmosphere above 30 km, and unlike radio occultation, stellar occultation probes fine-scale phenomena with potential to observe atmospheric turbulence. We imaged the refractive bending angle of a star centroid for a series of occultations by the atmosphere. Atmospheric refractivity, density, and then temperature are retrieved from the bending observations with the Abel transformation and Edlén's law, the hydrostatic equation, and the ideal gas law. The retrieval technique is applied to data collected by two nanosatellites operated by Terran Orbital. Measurements were primarily taken by the GEOStare SV2 mission, with a dedicated imaging telescope, supplemented with images captured by spacecraft bus sensors, namely the star trackers on other Terran Orbital missions. The bending angle noise floor is 10 and 30 arcsec for the star tracker and GEOStare SV2 data, respectively. The most significant sources of uncertainty are due to centroiding errors due to the fairly low-resolution stellar images and telescope pointing knowledge derived from noisy satellite attitude sensors. The former mainly affects the star tracker data, while the latter limits the GEOStare SV2 accuracy, with both providing low vertical resolution. This translates to a temperature profile retrieval up to roughly 20 km for both star tracker and GEOStare SV2 datasets. In preparation of an upcoming 2023 mission designed to correct these deficiencies, SOHIP, we simulated bending angle measurements with varying magnitudes of error. The expected maximum altitude of retrieved temperature is 41 km on average for these simulated measurements with a noise floor of 0.39 arcsec. Our work highlights the capabilities of stellar occultation observations from nanosatellites for atmospheric sounding. Future work will investigate high-frequency observations of atmospheric gravity waves and turbulence, mitigating the major uncertainties observed in these datasets.

Summation-by-Parts Operators for General Function Spaces

Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions.

Mechanical structure of a spin-1 particle

We investigate the mechanical structure of a spin-1 particle. Introducing three different frameworks, i.e., the three-dimensional (3D) Breit frame, the two-dimensional (2D) Breit frame, and the 2D infinite momentum frame (equivalently the two-dimensional Drell-Yan frame), we scrutinize the 2D and 3D energy-momentum tensor (EMT) distributions in these frames. We first derive the EMT distributions in the 2D Breit frame by performing the Abel transformation. The mass distribution in the 2D Breit frame contains an additional monopole contribution induced geometrically. The pressure distribution in the 2D Breit frame also gets an induced monopole structure. When the Lorentz boost is carried out, the mass distribution in the 2D infinite-momentum frame acquires the induced dipole term. Similarly, we also have the induced dipole contributions to the pressure and shear-force densities. We visualize the 2D mass distributions when the spin-1 particle is polarized along the $x$- and $z$-axes. We observe that the 2D mass distribution in the infinite momentum frame exhibit clearly the induced dipole structure when the spin-1 particle is polarized along the $x$-axis. We also discuss the strong force fields inside a polarized spin-1 particle.

Synchronization for a class of discrete complex dynamical networks in presence of Markov jump topology and actuator saturation

AbstractThis work addresses the synchronization problem for discrete complex dynamical networks (CDNs) that are associated with Markov jump outer coupling, parameter uncertainties, actuator saturation, and time delay. Especially, the uncertainties which occur in the inner coupling matrix are categorized with the use of the interval matrix approach. The main purpose of this work is to design an appropriate state feedback controller with a saturation effect that guarantees asymptotic synchronization and also satisfy a prescribed level of mixed and passivity performance for the addressed CDNs. Moreover, a new set of adequate criteria are expressed in the form of linear matrix inequalities to synchronize the considered network plant. The above synchronization criteria are retrieved with the support of an appropriate Lyapunov–Krasovskii functional, Kronecker product properties together with Jensen's inequality and Abel lemma. Eventually, the efficiency and superiority of our proposed synchronization criteria are demonstrated through two numerical examples.

A new variational discretization technique for initial value problems bypassing governing equations

Motivated by the fact that both the classical and quantum description of nature rest on causality and a variational principle, we develop a novel and highly versatile discretization prescription for classical initial value problems (IVPs). It is based on an optimization (action) functional with doubled degrees of freedom, which is discretized using a single regularized summation-by-parts (SBP) operator. Formulated as optimization task it allows us to obtain classical trajectories without the need to derive an equation of motion. The novel regularization we develop in this context is inspired by the weak imposition of initial data, often deployed in the modern treatment of IVPs and is implemented using affine coordinates. We demonstrate numerically the stability, accuracy and convergence properties of our approach in systems with classical equations of motion featuring both first and second order derivatives in time.

Adaptive energy stable artificial dissipation for preserving scalar boundedness in turbulent flows

An adaptive dissipation scheme is developed that preserves scalar boundedness in a high-order finite difference framework satisfying the summation-by-parts (SBP) property. A sensor is introduced that switches from an SBP dissipation operator based on a high derivative order that is efficient at absorbing the highest energy modes to a second order derivative that targets energy across scales. Conditionally- and unconditionally-stable formulations are presented. Numerical experiments of a one-dimensional advection equation and a three-dimensional turbulent round jet demonstrate that scalar boundedness can be achieved within an acceptable threshold while retaining overall high-order accuracy. Although not demonstrated here, the same approach could be employed for shock capturing as well.

Discontinuous Galerkin spectral element method for shock capturing with summation by parts properties

This paper presents a computational methodology developed for a high-order approximation of compressible fluid dynamics equations with discontinuities. The methodology is based on a discontinuous Galerkin spectral-element method (DGSEM) built upon a split discretization framework with summation-by-parts (SBP) property, which mimics the integration-by-parts operation in a discrete sense. To extend the split DGSEM framework to discontinuous cases, we implement a shock capturing method based on the entropy viscosity formulation. The developed high-order split-form DGSEM with shock-capturing methodology is subject to a series of evaluation on both one-dimensional and two-dimensional, continuous and discontinuous cases. Convergence of the method is demonstrated both for smooth and shocked cases that have analytical solutions. The 2D Riemann problem tests illustrate an accurate representation of all the relevant flow phenomena, such as shocks, contact discontinuities, and rarefaction waves. All test cases are able to run with a polynomial order of 7 or higher. The values of the tunable parameters related to the entropy viscosity are robust for both 1D and 2D test problems. We also show that higher-order approximations yield smaller errors than lower-order approximations, for the same number of total degrees of freedom.

Griffith crack analysis in nonlocal magneto-elastic strip using Daubechies wavelets

The main focus of this work is to employ the Daubechies wavelet approximation to analyze the Griffith crack in nonlocal magneto-elastic horizontally shear (SH) wave propagation. It is assumed that the crack is located in a homogeneous isotropic nonlocal magneto-elastic strip of finite thickness and infinite length. The governing mixed boundary value problem is converted to a dual integral equation by employing Fourier transformation. This dual integral equation is further converted to a Fredholmn integral equation of the second kind by means of Abel transformation. A wavelet-based approximation scheme is employed to solve the integral equation as it is not solvable analytically. Griffith crack analysis using Daubechies wavelets has not been studied previously in the literature. The stress intensity factor (SIF) at the crack tip is determined numerically after solving the integral equation. In the end, the effect of the magnetic field, strip breath, incident angle of the shear wave and the nonlocal parameter on stress intensity factor (SIF) at the crack tip is explained through various graphical representations.

Electronic nonadiabatic effects in the state-to-state dynamics of the H + H2 → H2 + H exchange reaction with a vibrationally excited reagent.

Out of the many major breakthroughs that the hydrogen-exchange reaction has led to, electronic nonadiabatic effects that are mainly due to the geometric phase has intrigued many. In this work we investigate such effects in the state-to-state dynamics of the H + H2 (v = 3, 4, j = 0) → H2 (v', j') + H reaction with a vibrationally excited reagent at energies corresponding to thermal conditions. The dynamical calculations are performed by a time-dependent quantum mechanical method both on the lower adiabatic potential energy surface (PES) and also using a two-states coupled diabatic theoretical model to explicitly include all the nonadiabatic couplings present in the 1E' ground electronic manifold of the H3 system. The nonadiabatic couplings are considered here up to the quadratic term; however, the effect of the latter on the reaction dynamics is found to be very small. Adiabatic population analysis showed a minimal participation of the upper adiabatic surface even for the vibrationally excited reagent. A strong nonadiabatic effect appears in the state-to-state reaction probabilities and differential cross sections (DCSs). This effect is manifested as "out-of-phase" oscillations in the DCSs between the results of the uncoupled and coupled surface situations. The oscillations persist as a function of both scattering angle and collision energy in both the backward and forward scattering regions. The origins of these oscillations are examined in detail. The oscillations that appear in the forward direction are found to be different from those due to glory scattering, where the latter showed a negligibly small nonadiabatic effect. The nonadiabatic effects are reduced to a large extent when summed over all product quantum states, in addition to the cancellation due to integration over the scattering angle and partial wave summation.

The Derivation of the Riemann Analytic Continuation Formula from the Euler’s Quadratic Equation

The analysis of the derivation of the Riemann Analytic Continuation Formula from Euler’s Quadratic Equation is presented in this paper. The connections between the roots of Euler’s quadratic equation and the Analytic Continuation Formula of the Riemann Zeta equation are also considered. The method of partial summation is applied twice on the resulting series, thus leading to the Riemann Analytic Continuation Formula. A polynomial approach is anticipated to prove the Riemann hypothesis; thus, a general equation for the zeros of the Analytic Continuation Formula of the Riemann Zeta equation based on a polynomial function is also obtained. An expression in Terms of Prime numbers and their products is considered and obtained. A quadratic function, G(tn), that is required for Euler’s quadratic equation (EQE) to give the Analytic Continuation Formula of the Riemann Zeta equation (ACF) is presented. This function thus allows a new way of defining the Analytic Continuation Formula of the Riemann Zeta equation (ACF) via this equivalent equation. By and large, the Riemann Zeta function is shown to be a type of L function whose solutions are connected to some algebraic functions. These algebraic functions are shown and presented to be connected to some polynomials. These Polynomials are also shown to be some of the algebraic functions’ solutions. Conclusively, &(z) is redefined as the product of a new function which is called H(tn; z) and this new function is shown to be dependent on the polynomial function, G(tn).

On an Eigenvalue Property of Summation-By-Parts Operators

Summation-By-Parts (SBP) methods provide a systematic way of constructing provably stable numerical schemes. However, many proofs of convergence and accuracy rely on the assumption that the SBP operator possesses a particular eigenvalue property. In this note, three results pertaining to this property are proven. Firstly, the eigenvalue property does not hold for all nullspace consistent SBP operators. Secondly, this issue can be addressed without affecting the accuracy of the method by adding a specially designed, arbitrarily small perturbation term to the SBP operator. Thirdly, all pseudospectral methods satisfy the eigenvalue property.

Recent advancement of entropy split methods for compressible gas dynamics and MHD

The entropy splitting of the compressible Euler flux derivatives based on Harten’s entropy function Harten (1983), Gerritsen and Olsson (1996), Yee et al. (2000) in conjunction with classical spatial central and DRP (dispersion relation-preserving) finite discretizations with summation-by-parts (SBP) operators Strand (1994) for both periodic and non-periodic boundary conditions is proven to be entropy conservative and stable for a thermally-perfect gas by Sjögreen and Yee (2019), Sjögreen et al. (2020), Sjögreen and Yee (2021). The various high order methods resulting from applying classical spatial central, DRP and Padé (compact) methods to the split form of the Euler flux derivative are referred to as entropy split methods as a function of the splitting parameter β. These entropy split methods are entropy conserving and stable but they are usually not conservative numerical methods without additional reformulation; e.g., those proposed in Sjögreen and Yee (2021).This is an expanded version of Sjögreen and Yee (2022) and is intended to present a more in depth study for both the compressible gas dynamics and ideal MHD equations. Here the same Harten entropy function is used for the ideal magnetohydrodynamic (MHD) governing equation set. This approach was not examined in great depth by Sjögreen & Yee and Yee & Sjögreen in two conference proceedings papers Sjögreen and Yee (2022), Yee and Sjögreen (2022), which also included the high order Padé (compact) spatial discretizations Hirsh(1975). For the extension of the entropy split method to the ideal MHD, the Godunov symmetrizable non-conservative form of the formulation is used. Due to the non-conservative portion (commonly referred to as a nonlinear source term vector) of the symmetrizable system Godunov (1972), there are variants in formulating the entropy split methods. Two different numerical treatments for these particular source terms of the symmetrizable MHD governing equation set are presented.Systematic 3D formulation in curvilinear grids with the corresponding numerical fluxes among method comparisons for the Euler and ideal MHD equations is included. The curvilinear formulation includes high order classical central, DRP and compact spatial discretizations. For readers interested in the implementation of these methods into their computer code, it is hope that these formulations and the corresponding numerical fluxes in 3D curvilinear grids will be helpful.The comparative studies concentrate on the Tadmor-type of discrete entropy conserving Tadmor (2003), momentum conserving Ducros et al. (2000), kinetic energy preserving Kennedy and Gruber (2008), Pirozzoli (2010), Coppola et al. (2019), Yee et al., Sjögreen & Yee entropy split methods Yee et al.(2000), Sjögreen and Yee (2018), Sjögreen and Yee (2019), Sjögreen et al. (2020), Sjögreen and Yee (2021), as well as the combination of these physical-preserving methods Ranocha (2020), Yee and Sjögreen (2020). All of these methods are not only preserved certain physical properties of the chosen governing equations but are also known to either improve numerical stability, and/or minimize aliasing errors in long time integration of turbulent flow computations without the aid of added numerical dissipation for selected compressible flow types. Extensive error norm comparison with grid refinement were performed to show how well this methods conserve the entropy, momentum and mass, and preserve the kinetic energy for long time integration of the various flows. In general the Tadmor-type entropy conserving methods are at least twice the CPU per time step than the rest of the skew-symmetric splitting methods.

Entropy-stable discontinuous Galerkin difference methods for hyperbolic conservation laws

The paper describes the construction of entropy-stable discontinuous Galerkin difference (DGD) discretizations for hyperbolic conservation laws on unstructured grids. The construction takes advantage of existing theory for entropy-stable (diagonal-norm) summation-by-parts (SBP) discretizations. In particular, the paper shows how DGD discretizations — both linear and nonlinear — can be constructed by defining the SBP trial and test functions in terms of interpolated DGD degrees of freedom. In the case of entropy-stable discretizations, the entropy variables rather than the conservative variables must be interpolated to the SBP nodes. A fully-discrete entropy-stable scheme is obtained by adopting a relaxation Runge–Kutta version of the midpoint method. In addition, DGD matrix operators for the first derivative are shown to be dense-norm SBP operators. Numerical results are presented to verify the entropy-stability of the DGD discretization in the context of the Euler equations. Accuracy studies reveal that the DGD method is efficient; indeed, like tensor-product DGD schemes, the unstructured DGD method exhibits superconvergent solution error for periodic problems. An investigation of the DGD spectra shows that the spectral radius is relatively insensitive to discretization order. Finally, the DGD scheme is applied to a one-dimensional Riemann problem, and global conservation and convergence in the L1 norm are observed.

Gravitational form factors and mechanical properties of the proton: Connections between distributions in 2D and 3D

The gravitational form factors which are obtained from the matrix elements of the energy-momentum tensor provide us information about internal distributions of mass, energy, pressure, and shear. The Druck term is the least understood among all the gravitational form factors. In a light front quark-diquark model of a proton, we investigate the Druck form factor. Using Abel transformation, we evaluate the 3D distribution in the Breit frame from the 2D light front distributions. The results are compared with other models and lattice predictions.

Analysis of the oscillations induced by a supersonic jet applied to produce nanofibers

High-performance fibers are key components for enhancing the mechanical properties of composite materials. The development of high strength nanofibers augurs the production of new nano-composites with outstanding features. However, the robust production of continuous glass nanofibers that can be feasible processed for efficiently manufacturing nanocomposites is still challenging. Recently, Cofiblas (Continuous Fiberizing by Laser melting and Supersonic dragging) was demonstrated as a technique capable of producing continuous glass nanofibers with unlimited length. Cofiblas process has some similarities with the widely known melt blowing technique for the production of polymeric fibers. In both techniques, the design of the gas nozzle is key to ensure the feasibility of the process since the turbulences of the gas jet may induce strong whipping of the filament.This paper gives novel experimental evidences on the correlation of the supersonic gas jet instabilities with the oscillation of the filament in the melt-blowing and Cofiblas processes, relating these oscillations with the presence of shock waves and unsteadiness in the flow, and gives valuable insight into the use of supersonic jets in the melt blowing process as an effective approach for the formation of nanofibers. A thin 3D-axisymmetric model in OpenFOAM® was put to test by comparing the performance of different solvers which were validated by flow visualization of the exit jet using digital holography (DH). In order to perform a realistic and thorough validation, we simulated the optical measurements of the flow from the CFD simulations of the mass density by Abel transform and numerical differentiation. The application of digital holography as the flow visualization technique makes possible both a precise validation of the density maps obtained from the Abel transformation of the 2D-alike results, and the analysis of the shockwave pattern in the air jet. Conversely, the numerical reconstruction of time-averaged holograms is employed to detect unsteadiness in the flow and to analyze the fiber oscillation, which is essential to assess the stability of the process. Lastly, the analysis and comparison of the vibration of the filament using the basic design and the optimized nozzle demonstrates a clear influence of the shock waves and flow unsteadiness in the stability of the filament.