High-precision Calculations Research Articles

Спутниковая СВЧ-радиометрия для решения задач дистанционного зондирования Земли

The issues of development of the direction of satellite microwave radiometry in Russia in the interests of operational meteorology and oceanography are considered. The analysis of the current state of Russian and foreign radiometric ERS equipment in the microwave range is carried out. The technical characteristics of onboard multichannel microwave radiometers, combining the functions of a scanner and a sounder, are analyzed. The issues of metrological support of microwave measurements of equipment installed on Russian satellites of the Meteor-M series are considered. The original method of internal calibration of the MTVZA-GYA microwave scanner/sounding device is analyzed in detail in order to form the antenna temperature scale. The MTVZA-GYA calibration unit measures the radiation intensity of two matched loads with known brightness temperatures (“hot” and “cold”). An on-board calibrator is used as a “hot” load, it serves as an imitator of an absolutely black body, its brightness temperature of which is in the range of 240–300 K. Absolute (external) calibration is a transition from antenna to brightness temperatures and is performed using high-precision radiation calculations for specially selected natural testing sites. The issues of preliminary processing of MTVZA-GYA data are considered and examples of microwave images of the Earth in the scale of brightness temperatures are given.

Development of high-speed and high-precision distance calculation algorithm between two surfaces for Design Insights CAD system

Development of high-speed and high-precision distance calculation algorithm between two surfaces for Design Insights CAD system

ADAPTIVE SOLUTION OF THE NEUTRON DIFFUSION EQUATION WITH HETEROGENEOUS COEFFICIENTS USING THE MIXED FINITE ELEMENT METHOD ON STRUCTURED MESHES

The neutron transport equation can be used to model the physics of the nuclear reactor core. Its solution depends on several variables and requires a lot of high precision computations. One can simplify this model to obtain the SPN equation for a generalized eigenvalue problem. In order to solve this eigenvalue problem, we usually use the inverse power iteration by solving a source problem at each iteration. Classically, this problem can be recast in a mixed variational form, and then discretized by using the Raviart-Thomas-Nédélec Finite Element. In this article, we focus on the steady-state diffusion equation with heterogeneous coefficients discretized on Cartesian meshes. In this situation, it is expected that the solution has low regularity. Therefore, it is necessary to refine at the singular regions to get better accuracy. The Adaptive Mesh Refinement (AMR) is one of the most effective ways to do that and to improve the computational time. The main ingredient for the refinement techniques is the use of a posteriori error estimates, which gives a rigorous upper bound of the error between the exact and numerical solution. This indicator allows to refine the mesh in the regions where the error is large. In this work, some mesh refinement strategies are proposed on the Cartesian mesh for the source problem. Moreover, we numerically investigate an algorithm which combines the AMR process with the inverse power iteration to handle the generalized eigenvalue problem.

Энергия связи основного состояния бериллиеподобного молибдена: корреляционные и квантово-электродинамические эффекты

High-precision calculations of the ground-state binding energy of berylliumlike molybdenum ($Z=42$) are performed. The applied approach merges rigorous quantum-electrodynamics (QED) treatment up to the second order of perturbation theory in the framework of the Furry picture and the third- and higher-order correlation effects evaluated within the Breit approximation. The mixing of the close configurations of the same symmetry due to the electron--electron interaction is considered by employing the QED perturbation theory for quasi-degenerate levels. The most accurate up-to-date theoretical predictions for the binding energy are obtained.

Development the Measurement System of the Multi-Channel Frequency Standard Comparison

This paper introduces a measurement system and method of high-speed sampling and full digital for the comparison of multi-channel frequency standard. The hardware of the system consists of 4 parts: A: Impedance matching module (After matching with the impedance of the acquisition circuit, the two input signals are transformed into suitable amplitude, and then transformed into digital signals by the high-speed acquisition circuit). B: High speed data acquisition module. C: Data real-time processing module. D: Calculation module for phase difference and frequency stability. The system not only can achieve real-time processing under the minimum computing resources, but also provide the upper computer with complete information and equivalent frequency data with small amount of data. Moreover, the fast and high-precision calculation of phase difference and frequency stability can be achieved by improving the calculation process. The software of the system meets the requirements for the technical specifications of national metrology. Compared with domestic and foreign frequency standard comparators, the interface of software is more friendly, the operation is more convenient, and it has a strong ability to analyze and process data. The test results show that the uncertainty of the system is better than 1.5 ×10^-13 (sampling time is 1 second), which is consistent with Russian VCH-315 index.

Codes of new generation for safety justification of power units with a closed nuclear fuel cycle developed for the “PRORYV” project

The article describes the status of development of codes of new generation for the “PRORYV” Project by the end of 2019: twenty-five commercial-grade software products to justify design solutions and safety of power units with fast neutron reactors and liquid metal coolant (sodium and lead) in a closed nuclear fuel cycle. The developed system of codes is multi-physical and multi-scale that allows performing both calculations of the whole installations and high precision calculations of their individual elements. The developed codes offer unique features. Twelve developed codes have already been certified by Rostechnadzor, and six more have been submitted for certification. In addition to creating the software products, a large-scale work is being carried out to conduct experimental studies for code validation that meet modern requirements imposed by the codes: unique measurement techniques have been created; experimental data on flow characteristics of heavy liquid metal coolant (HLMC) in a fuel assembly simulator have been obtained, as well as of “gas-HLMC” interphase interaction after inert gas injection in HLMC and characteristics of heat exchange between the inert gas and HLMC. The results are already used for validation of system and CFD codes used in the “PRORYV” Project.

Simulation of Microstructure Evolution in Metal Materials under Hot Plastic Deformation and Heat Treatment

The development of modern computational techniques and equipment enables one to perform high-precision calculations of complex processes for industry, including metallurgy. This review has classified the basic physical and mathematical models of structure formation during heat and deformation treatment. The Kocks–Mecking–Estrin model describing the dislocation structure at the initial stage of hot plastic deformation has been analyzed. The models of dynamic, metadynamic, and static recrystallization kinetics based on the Johnson–Mehl–Avrami–Kolmogorov equation have been considered. The models of the kinetics of phase transformation upon heating and cooling of steel have been reviewed. The Kampmann–Wagner model that describes the decomposition of supersaturated solid solution during the aging of aluminum alloys has also been considered. The main computational techniques to calculate microstructural evolution, such as the cellular automaton, Monte Carlo, and multiphase-field techniques have been considered. They exhibit high accuracy when calculating recrystallization processes and phase transformations. The systematization of the existing models that describe structural evolution has revealed the possibility to develop complex models for the comprehensive calculation of full cycles to process metal materials by heating and deformation. These models can be used for the optimization and development of new processing techniques.

Direct measurements for the fine-structure splitting of S viii and Cl ix

Using electron-beam ion traps (EBITs), we have made direct measurements for the fine-structure splitting $2{p}^{5}$ of S viii and Cl ix, which are important to the test of ab initio theory in F-like ions. Their wavelength are redetermined in the laboratory with an accuracy of 23 ppm, i.e., $10\phantom{\rule{4pt}{0ex}}085.40\ifmmode\pm\else\textpm\fi{}0.23\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ for S viii, where the early value observed in the solar spectrum is questionable, and $13\phantom{\rule{4pt}{0ex}}643.39\ifmmode\pm\else\textpm\fi{}0.32\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ for Cl ix, where only indirect measurements had been performed. High-precision calculations, including ab initio treatment of QED effects, are accomplished for $10\ensuremath{\le}Z\ensuremath{\le}18$ and compared with the experimental results. Good agreement is found between our experimental and theoretical values of the fine-structure energy among these ions except for Al v, which calls for further investigations. Besides, we have succeeded in pushing EBIT operation towards the infrared spectral region relevant to astrophysics.

Vacuum polarization and finite-nuclear-size effects in the two-photon decay of hydrogenlike ions

The total two-photon decay rate of hydrogen-like ions is studied using relativistic quantum electrodynamics. In particular, we analyse how finite nuclear size and QED vacuum polarization corrections affect the decay rate. To calculate these corrections, a finite basis set method based on $B$-splines is used for the generation of quasi-complete atomic spectra and, hence, of the relativistic Green's function. By making use of this $B$-spline approach, high precision calculations have been performed for the $2s_{1/2} \to 1s_{1/2} + 2\gamma$ and $2p_{1/2} \to 1s_{1/2} + 2\gamma$ decay of hydrogen-like ions along the entire isoelectronic sequence. The results of these calculations show that both, QED and finite nuclear size effects, are comparatively weak for the $2s_{1/2} \to 1s_{1/2} + 2\gamma$ transition. In contrast, they are much more pronounced for the $2p_{1/2}\to 1s_{1/2} + 2\gamma$ decay, where, for hydrogen-like Uranium, the decay rate is reduced by 0.484% due to the finite nuclear size and enhanced by 0.239% if the vacuum polarization is taken into account.

Shifted Legendre polynomials algorithm used for the dynamic analysis of PMMA viscoelastic beam with an improved fractional model

• An improved fractional viscoelastic constitutive model is proposed. • The parameters in the proposed viscoelastic model are identified by DMA tests. • An algorithm based on shifted Legendre polynomials is proposed with high accuracy. • The fractional governing equation of the viscoelastic beam is solved directly in time domain. • Dynamic analysis of PMMA beam under different load conditions and temperatures. In this paper, a fractional viscoelastic model is proposed to describe the physical behaviour of polymeric material. The material parameters in the model are characterized by the experimental data obtained in the dynamical mechanical analysis. The proposed model is integrated into the fractional governing equation of polymethyl methacrylate (PMMA) above its glass transition temperature . The numerical algorithm based on the shifted Legendre polynomials is retained to solve the fractional governing equations in the time-domain. The accuracy and effectiveness of the algorithm are verified according to the mathematical examples. The advantage of this method is that Laplace transform and the inverse Laplace transform commonly used in fractional calculus are avoided. The dynamical response of the viscoelastic PMMA beam is determined with several loading conditions (uniformly distributed load and harmonic load). The effects of the loading condition and the temperature on the dynamic response of the beam are investigated in the results. The proposed approach shows great potentials for the high-precision calculation in solving the fractional equations in the science and engineering.

Separability of H2O molecular potential surfaces in hyperspherical coordinates via adiabatic approximation

An adiabatic scheme for separation of the three-dimensional (3D) nuclear dynamics on the ground electronic Born-Oppenheimer potential energy surface of an ${\mathrm{H}}_{2}\mathrm{O}$ molecule in hyperspherical coordinates is presented. It is found that the three vibrational modes are weakly coupled and the 3D vibrational wave function can be approximated as a product of three separable functions: one represented by the hyperradius and two by the two hyperangles individually. This framework is then used for investigation of the formation and the role of a saddlelike barrier arising in the two hyperspherical angles that is to moderate the OH $+$ H dissociation process. In order to test the validity of the framework, vibrational states with energies up to 19 500 ${\mathrm{cm}}^{\ensuremath{-}1}$ are constructed under the assumptions of adiabaticity and separability and compared to full three-dimensional high-precision numerical calculations yielding remarkable correspondence. As a result we present a simple construction scheme for separated molecular vibration states as the first step towards theoretical investigation of laser-driven molecular dynamics of triatomic molecules.

Model-Driven Architecture of Extreme Learning Machine to Extract Power Flow Features

Probabilistic power flow (PPF) calculation is an important power system analysis tool considering the increasing uncertainties. However, existing calculation methods cannot simultaneously achieve high precision and fast calculation, which limits the practical application of the PPF. This article designs a specific architecture of the extreme learning machine (ELM) in a model-driven pattern to extract the power flow features and therefore accelerate the calculation of PPF. ELM is selected because of the unique characteristics of fast training and less intervention. The key challenge is that the learning capability of the ELM for extracting complex features is limited compared with deep neural networks. In this article, we use the physical properties of the power flow model to assist the learning process. To reduce the learning complexity of the power flow features, the feature decomposition and nonlinearity reduction method is proposed to extract the features of the power flow model. An enhanced ELM network architecture is designed. An optimization model for the hidden node parameters is established to improve the learning performance. Based on the proposed model-driven ELM architecture, a fast and accurate PPF calculation method is proposed. The simulations on the IEEE 57-bus and Polish 2383-bus systems demonstrate the effectiveness of the proposed method.

An Efficient Solver of Eigenmodes for a Class of Complex Optical Waveguides

In this paper, for a class of complex optical waveguide, the high-precision computation of the propagation constants β are studied. The corresponding Sturm-Liouville (S-L) problem is represented as in an open domain (open on one side), where x is a given value. Firstly, a perfectly matched layer is used to terminate the open domain. Secondly, both the equation and the complex coordinate stretching transformations are constructed. Thirdly, the S-L problem is turned to a simplified form such as in a bounded domain. Finally, the coefficient function is approximated by a piecewise polynomial of degree two. Since the simplified equation in each layer can be solved analytically by the Kummer functions, the approximate dispersion equation is established to the TE case. When the coefficient function is continuous, the approximate solutions converge fast to the exact ones, as the maximum value of the subinterval sizes tends to zero. Numerical simulations show that high-precision eigenmodes may be obtained by the Muller's method with suitable initial values.

Number Formats, Error Mitigation, and Scope for 16-Bit Arithmetics in Weather and Climate Modeling Analyzed With a Shallow Water Model.

The need for high‐precision calculations with 64‐bit or 32‐bit floating‐point arithmetic for weather and climate models is questioned. Lower‐precision numbers can accelerate simulations and are increasingly supported by modern computing hardware. This paper investigates the potential of 16‐bit arithmetic when applied within a shallow water model that serves as a medium complexity weather or climate application. There are several 16‐bit number formats that can potentially be used (IEEE half precision, BFloat16, posits, integer, and fixed‐point). It is evident that a simple change to 16‐bit arithmetic will not be possible for complex weather and climate applications as it will degrade model results by intolerable rounding errors that cause a stalling of model dynamics or model instabilities. However, if the posit number format is used as an alternative to the standard floating‐point numbers, the model degradation can be significantly reduced. Furthermore, mitigation methods, such as rescaling, reordering, and mixed precision, are available to make model simulations resilient against a precision reduction. If mitigation methods are applied, 16‐bit floating‐point arithmetic can be used successfully within the shallow water model. The results show the potential of 16‐bit formats for at least parts of complex weather and climate models where rounding errors would be entirely masked by initial condition, model, or discretization error.

Расчетный анализ особенностей аэротермодинамики высокоскоростного летательного аппарата X-51

The paper presents a numerical investigation of aero-dynamic coefficients for a model of an X-51-type high-velocity aircraft moving at Mach 6. The simulation made use of the original and modified versions of our custom hydrocodes (UST3D and UST3D-AUSMPW) designed for numerical simulation of aerodynamics and thermodynamics in high-velocity aircraft of arbitrary shapes. Such hydrocodes implement a model of viscous compressible thermally conductive gas described by a non-steady-state spatial system of Navier --- Stokes equations solved over unstructured three-dimensional tetrahedral meshes. The paper considers the theoretical aspects of simulating the aerodynamics and thermodynamics of high-velocity aircraft numerically. We describe the method for computing mass flow through mesh cell boundaries implemented in the modified custom hydrocode version. We performed cross-validation of the results obtained using our custom hydrocodes and compared our hydrocodes in terms of result convergence time. We show that these custom hydrocodes ensure adequately accurate distribution patterns concerning the fields of the values sought, and provide high-precision computation of aerodynamic characteristics as compared to each other

Exact Solution for Relativistic Trajectories Using Modal Transseries

In this article, we design a novel method for finding the exact solution of the geodesic equation in Schwarzschild spacetime, which represents the trajectories of the particles. This is a fundamental problem in astrophysics and astrodynamics if we want to incorporate relativistic effects in high precision calculations. Here, we show that exact analytical expressions can be given, in terms of modal transseries for the spiral orbits as they approach the limit cycles given by the two circular orbits that appear for each angular momentum value. The solution is expressed in terms of transseries generated by transmonomials of the form e−nθ, n=1, 2, …, where θ is the angle measured in the orbital plane. Examples are presented that verify the effect of the solutions.

Nuclear Magnetic Moments of Francium-207-213 from Precision Hyperfine Comparisons.

We report a fourfold improvement in the determination of nuclear magnetic moments for neutron-deficient francium isotopes 207-213, reducing the uncertainties from 2% for most isotopes to 0.5%. These are found by comparing our high-precision calculations of hyperfine structure constants for the ground states with experimental values. In particular, we show the importance of a careful modeling of the Bohr-Weisskopf effect, which arises due to the finite nuclear magnetization distribution. This effect is particularly large in Fr and until now has not been modeled with sufficiently high accuracy. An improved understanding of the nuclear magnetic moments and Bohr-Weisskopf effect are crucial for benchmarking the atomic theory required in precision tests of the standard model, in particular atomic parity violation studies, that are underway in francium.

Combination of “generalized Trotter operator splitting” and “quadratic adaptive algorithm” method for tradeoff among speedup, stability, and accuracy in the Markov chain model of sodium ion channels in the ventricular cell model

The fast hybrid operator splitting (HOS) and stable uniformization (UNI) methods have been proposed to save computation cost and enhance stability for Markov chain model in cardiac cell simulations. Moreover, Chen-Chen-Luo's quadratic adaptive algorithm (CCL) combined with HOS or UNI was used to improve the tradeoff between speedup and stability, but without considering accuracy. To compromise among stability, acceleration, and accuracy, we propose a generalized Trotter operator splitting (GTOS) method combined with CCL independent of the asymptotic property of a particular ion-channel model. Due to the accuracy underestimation of the mixed root mean square error (MRMSE) method, threshold root mean square error (TRMSE) is proposed to evaluate computation accuracy. With the fixed time-step RK4 as a reference, the second-order GTOS combined with CCL (30.8-fold speedup) for the wild-type Markov chain model with nine states (WT-9 model) or (7.4-fold) for the wild-type Markov chain model with eight states (WT-8 model) is faster than UNI combined with CCL (15.6-fold) for WT-9 model or (1.2-fold) for WT-8 model, separately. Besides, the second-order GTOS combined with CCL has 3.81% TRMSE for WT-9 model or 4.32% TRMSE for WT-8 model more accurate than 72.43% TRMSE for WT-9 model or 136.17% TRMSE for WT-8 model of HOS combined with CCL. To compromise speedup and accuracy, low-order GTOS combined with CCL is suggested to have the advantages of high precision and low computation cost. For high-accuracy requirements, high-order GTOS combined with CCL is recommended. Graphical abstract.

General and consistent statistics for cosmological observations

This paper focuses on two aspects of the statistics of cosmological observables that are important for the next stages of precision cosmology. First, we note that the theory of reduced angular $N$-point spectra has only been developed in detail up to the trispectrum case and in a fashion that makes it difficult to go beyond. To fill this gap, here we present a constructive approach that provides a systematic description of reduced angular $N$-point spectra and their covariance matrices, for arbitrary $N$. Second, we focus on the common practice in the literature on cosmological observables, which consists in simply discarding a part of the expression, namely, the terms containing fields evaluated at the observer position. We point out that this is not justified beyond linear order in perturbation theory, as these terms contribute to all the multipoles of the corresponding spectra and with a magnitude that is of the same order as the rest of the non-linear corrections. We consider the possibility that the reason for neglecting these terms is a conceptual discomfort when using ensemble averages, which originates in an apparent tension between the ergodic hypothesis and the privileged position of the observer on the light-cone. We clarify this subtle issue by performing a careful derivation of the relation between the theoretical statistical predictions and the observational estimators for all $N$. We conclude that there is no inconsistency whatsoever in ensemble-averaging fields at and near the observer position, thus clearing the way for consistent and robust high-precision calculations.

Response surface method based on uniform design and weighted least squares for non‐probabilistic reliability analysis

AbstractThe non‐probabilistic reliability theory is a promising methodology for implementing structural reliability analysis in case of scarce statistical data. One of the main obstacles to implement non‐probabilistic reliability analysis is the implication of the limit state function (LSF) for complex structures. This paper aims to establish a surrogate model of the LSF with higher simulation precision, and whereby proposes a response surface method based on the combination of uniform design (UD) and weighted least squares (WLS). At first, the UD method is selected as the sampling method of interval variables to realize the uniform space‐filling of the initial samples, and the sample set is updated by gradually adding the approximate optimal points to increase the sampling density of critical domain. Then, the WLS method is applied to improve the precision of the response surface by adjusting the importance of samples to the function fitting. Finally, a method of constructing sample weights is developed. Two examples are applied to validate the feasibility and efficiency of the proposed method. Results show that the proposed method is effective for non‐probabilistic reliability analysis of complex structures owning to high computational precision and low computational cost in both numerical and case study.