Discrete Functions Research Articles

Efficiently and consistently energy-stable [formula omitted]-phase-field method for the incompressible ternary fluid problems

Ternary incompressible fluid flows extensively exist in atmospheric science, chemical engineering, and energy and power engineering, etc. The phase-field method is popular in multi-phase fluid modeling thanks to its efficient ability of interface capturing. This work aims to develop an energy dissipation law-preserving and temporally second-order accurate algorithm for a ternary phase-field fluid system. To establish a simple energy estimation, an adapted auxiliary variable approach is used to transform the original model into its equivalent form. Later, a second-order backward difference strategy is used to design the fully decoupled and linear time-marching scheme. To improve the consistency between original and modified discrete energy functionals, a practical energy correction technique is presented. We analytically prove the discrete energy dissipation property and show that the energy estimation can be easily established without considering the complex treatments of nonlinear and coupling terms. To facilitate the interested readers, we briefly describe the numerical implementation in each time step. The numerical tests indicate that the proposed method not only has desired accuracy, but also satisfies the energy stability even if a larger time step is used. Moreover, the proposed method can well simulate various ternary fluid phenomena, such as the liquid lens, phase separation, droplet dynamics, Kelvin–Helmholtz instability, and billowing cloud.

Discrete Pseudo-Quasi Overlap Functions and Their Applications in Fuzzy Multi-Attribute Group Decision-Making

The overlap function, a continuous aggregation function, is widely used in classification, decision-making, image processing, etc. Compared to applications, overlap functions have also achieved fruitful results in theory, such as studies on the fundamental properties of overlap functions, various generalizations of the concept, and the construction of additive and multiplicative generators based on overlap functions. However, most of the research studies on the overlap functions mentioned above assume commutativity and continuity, which can limit their practical applications. In this paper, we remove the symmetry and continuity from overlap functions and define discrete pseudo-quasi-overlap functions on finite chains. Meanwhile, we also discuss their related properties. Then, we introduce pseudo-quasi-overlap functions on sub-chains and construct discrete pseudo-quasi-overlap functions on finite chains using these sub-chain functions. Unlike quasi-overlap functions on finite chains generated by the ordinal sum, discrete pseudo-quasi-overlap functions on finite chains constructed through pseudo-quasi-overlap functions on different sub-chains are dissimilar. Eventually, we remove the continuity from pseudo-automorphisms and propose the concept of pseudo-quasi-automorphisms. Based on this, we utilize pseudo-overlap functions, pseudo-quasi-automorphisms, and integral functions to obtain discrete pseudo-quasi-overlap functions on finite chains; moreover, we apply them to fuzzy multi-attribute group decision-making. The results indicate that—compared to overlap functions and pseudo-overlap functions—discrete pseudo-quasi-overlap functions on finite chains have stronger flexibility and a wider range of practical applications.

Variational evolution of discrete one-dimensional second-order functionals

A variational scheme of evolution (minimizing movements) is applied to a sequence of discrete functionals converging, as the mesh size tends to zero, to the prototypical second-order functional with free-discontinuities. At fixed mesh size, a discrete evolution can be defined, depending on a (small) time parameter. We study the limit problem when both the mesh size and the time step tend to zero. The method provides a function which matches the expected evolution of the free-discontinuity limit functional. From a mechanical point of view, the model can be interpreted as the evolution from a non-equilibrium state, of a rod with possible crease discontinuities and fracture.

A new spectral measure of complexity and its capabilities for detecting signals in noise

This article is devoted to the improvement of signal recognition methods based on the information characteristics of the spectrum. A discrete function of the normalized ordered spectrum is established for a single window function included in the DFT. Lemmas on estimates of entropy, imbalance and statistical complexity in processing a time series of independent Gaussian quantities are proved. New concepts of one-dimensional and two-dimensional spectral complexities are proposed. The theoretical results obtained were verified by numerical experiments, which confirmed the effectiveness of the new information characteristic when detecting a signal mixed with white noise at low signal-to-noise ratios.

Application of the Gray Wolf Optimization Algorithm and Neural Networks for Solving Discrete Problems

Relevance. In recent decades, metaheuristic optimization methods have become popular for solving complex problems that require searching for global extrema. Algorithms such as genetic algorithm (GA), ant colony optimization (ACO), particle swarm optimization (PSO), as well as more modern approaches such as cat pack optimization (CSO) and gray wolf pack optimization (GWO) demonstrate high efficiency, but their application is often limited by the conditions of continuity and differentiability of the objective functions. This is a challenge when solving problems with discrete data, where such requirements are not met. In this context, the search for methods that allow adapting metaheuristic algorithms to work with discrete functions is of particular relevance.Aim. The study is aimed at testing the hypothesis about the possibility of using a neural network trained on a limited set of discrete data as an approximation of a function sufficient for the correct execution of the GWO algorithm when searching for a global minimum. The implementation of this hypothesis can significantly expand the scope of GWO, making it available for a wider range of problems where functions are defined on discrete sets.Methods. The study is based on the analysis of existing approaches and experimental verification of the hypothesis on two test functions: a linear function and a Booth function, which are widely used as standards for evaluating the performance of optimization algorithms. Numerical experiments were conducted using neural networks as an approximating model to obtain the results. Solution. During the experiments, an analysis of the applicability of neural networks for approximating discrete functions was carried out, which showed the success of this approach. It was found that neural networks can approximate discrete functions with high accuracy, creating conditions for a successful search for a global minimum using the GWO algorithm.Novelty. For the first time, a hypothesis was proposed and tested on the use of neural networks for approximating objective functions in metaheuristic optimization problems on discrete data. This direction has not previously received due coverage in the scientific literature, which adds significance to the obtained results and confirms the effectiveness of the proposed approach.Practical significance. The results of the study open up new prospects for the application of algorithms such as GWO in optimization problems based on discrete data, expanding the capabilities of metaheuristic methods and facilitating their implementation in a wider class of applied problems, including problems where the use of other methods is limited.

The Detection of Possible Quasiperiodic Oscillations in the BL Lac 4FGL J2139.4−4235

We present periodicity search analyses on the long-term γ-ray light curve of the BL Lacertae object 4FGL J2139.4−4235 observed by the Fermi Large Area Telescope, over a period of more than 15 yr, from 2008 August 4 to 2023 December 10. To determine the quasiperiodic oscillation (QPO) behavior of 4FGL J2139.4−4235 in the 0.3–300 GeV energy range, we used four methods, namely the Lomb–Scargle periodogram, the weighted wavelet z-transform, the phase dispersion minimization, and the autoregressive integrated moving average model. A Monte Carlo simulation technique is used to evaluate the significance level of the QPO signal. Significant levels above 3.5σ were detected in the γ-ray light curve at about 650 days QPO, which is presented throughout the observation period. Interestingly, there was some correlation between the three bands in the discrete correlation function method calculations, which may be an indication that the variability trends between the three bands are similar. We explore the possible physical models and show that a supermassive binary black hole system or a jet helical motion model seem to be reasonable explanations for the potential QPO behavior.

Gas-Kinetic Unified Algorithm for Aerodynamics Covering Various Flow Regimes by Computable Modeling of Boltzmann Equation

The gas-kinetic unified algorithm (GKUA), to solve the modeling of the Boltzmann equation, has been developed to study the aerothermodynamics problems with the effects of wall activation energy covering various flow regimes. The unified velocity distribution function equation could be accordingly presented on the basis of the Boltzmann-Shakhov model. To remove the dependence of the distribution function on velocity space, the conservational discrete velocity ordinate method has been developed for hypervelocity flows. The gas-kinetic finite difference scheme is constructed to directly solve the discrete velocity distribution functions by the operator splitting technique. The discrete velocity numerical integration method with the Gauss-type weight function has been developed to evaluate the macroscopic flow variables. Specially, to model the real physical process between gas molecules and the surface, the Maxwell-type gas-surface interaction model has been presented by the MD (molecular dynamic) simulation to obtain the energy adaptability coefficient. The multi-processing domain decomposition strategy and parallel implementation of high parallel efficiency and expansibility designed for the gas-kinetic numerical method is presented with good load balance and data communication efficiency. To validate the accuracy and feasibility of the present algorithm, the supersonic flows past two-dimensional circular cylinder are simulated covering various flow regimes. The results are in good agreement with the related theoretical, DSMC (Direct Simulation Monte Carlo), N-S (Navier-Stokes), and experimental data. The hypersonic reentry flows with the effects of wall activation energy around the Tianzhou-5 cargo spacecraft are simulated by the present GKUA and the massive parallel strategy. It has been confirmed that the present algorithm from the gas-kinetic point of view probably provides a promising approach to resolve the hypersonic aerothermodynamic problems with the complete spectrum of flow regimes during the re-entry and disintegration of the large-scale spacecraft.

Functional Principal Component Analysis for Continuous Non-Gaussian, Truncated, and Discrete Functional Data.

Mobile health studies often collect multiple within-day self-reported assessments of participants' behavior and well-being on different scales such as physical activity (continuous scale), pain levels (truncated scale), mood states (ordinal scale), and the occurrence of daily life events (binary scale). These assessments, when indexed by time of day, can be treated and analyzed as functional data corresponding to their respective types: continuous, truncated, ordinal, and binary. Motivated by these examples, we develop a functional principal component analysis that deals with all four types of functional data in a unified manner. It employs a semiparametric Gaussian copula model, assuming a generalized latent non-paranormal process as the underlying generating mechanism for these four types of functional data. We specify latent temporal dependence using a covariance estimated through Kendall's bridging method, incorporating smoothness in the bridging process. The approach is then extended with methods for handling both dense and sparse sampling designs, calculating subject-specific latent representations of observed data, latent principal components and principal component scores. Simulation studies demonstrate the method's competitive performance under both dense and sparse sampling designs. The method is applied to data from 497 participants in the National Institute of Mental Health Family Study of Mood Spectrum Disorders to characterize differences in within-day temporal patterns of mood in individuals with the major mood disorder subtypes, including Major Depressive Disorder and Type 1 and 2 Bipolar Disorder. Software implementation of the proposed method is provided in an R-package.

Approach for multi-valued integer programming in multi-material topology optimization: Random discrete steepest descent (RDSD) algorithm

The present study models the multi-material topology optimization problems as the multi-valued integer programming (MVIP) or named as combinatorial optimization. By extending classical convex analysis and convex programming to discrete point-set functions, the discrete convex analysis and discrete steepest descent (DSD) algorithm are introduced. To overcome combinatorial complexity of the DSD algorithm, we employ the sequential approximate integer programming (SAIP) to explicitly and linearly approximate the implicit objective and constraint functions. Considering the multiple potential changed directions for multi-valued design variables, the random discrete steepest descent (RDSD) algorithm is proposed, where a random strategy is implemented to select a definitive direction of change. To analytically calculate multi-material discrete variable sensitivities, topological derivatives with material contrast is applied. In all, the MVIP is finally transferred as the linear 0–1 programming that can be efficiently solved by the canonical relaxation algorithm (CRA). Explicit nonlinear examples demonstrate that the RDSD algorithm owns nearly three orders of magnitude improvement compared with the commercial software (GUROBI). The proposed approach, without using any continuous variable relaxation and interpolation penalization schemes, successfully solves the minimum compliance problem, strength-related problem, and frequency-related optimization problems. Given the algorithm efficiency, mathematical generality and merits over other algorithms, the proposed RDSD algorithm is meaningful for other structural and topology optimization problems involving multi-valued discrete design variables.

Matrix analysis of discrete functionals in compact difference method for nonlinear problems with higher derivatives and program code (I: 1D problem)

Matrix analysis of discrete functionals in compact difference method for nonlinear problems with higher derivatives and program code (I: 1D problem)

Optical and γ-ray variability analysis of BL Lacertae object TXS 1902+556

ABSTRACT The variability data for the BL Lacertae object TXS 1902+556 in the optical and $\gamma$-ray wavebands were obtained from the 0.76-m Katzman Automatic Imaging Telescope and the Fermi Large Area Telescope (Fermi-LAT), covering periods of 14.4 and 14.7 yr, respectively. The variability properties were systematically analysed, with particular emphasis on the first comprehensive investigation of radiation variation in the optical waveband. Four well-established techniques were employed for this purpose: the Lomb–Scargle periodogram, REDFIT program, Jurkevich method, and discrete correlation function (DCF) approach. The optical waveband exhibits quasi-periodic oscillations (QPO) with a time-scale of $P_{\rm O}=276.8\pm 6.1$ d at a significance level $3.87\sigma$, while the $\gamma$-ray waveband does not exhibit any significant periodicity. However, it should be noted that the QPO time-scale is consistent with the Sun-gaps in the optical light curve within 2$\sigma$ uncertainties. The optical QPO behaviour is most likely attributed to the helical motion of the jet driven by the orbital motion in a supermassive black hole binary system. Moreover, we have provided an explanation for the absence of QPO in the $\gamma$-ray light curves. Furthermore, utilizing the DCF method, a weak correlation between the variability in the optical and $\gamma$-ray wavebands was observed, suggesting that the emission of TXS 1902+556 may be generated through a combination of synchrotron self-Compton (SSC) and external Compton (EC) processes, or a leptonic–hadronic hybrid process.

An implicit adaptive unified gas-kinetic scheme for steady-state solutions of nonequilibrium flows

In recent years, nonequilibrium flows have been frequently encountered in various aerospace engineering and micro-electro-mechanical systems applications. To understand nonequilibrium physics, multiscale effects, and the dynamics in these applications, a reliable multiscale scheme for all flow regimes is required. Following the direct modeling methodology, the adaptive unified gas-kinetic scheme employs discrete velocity space to accurately capture the nonequilibrium physics, recovering the original unified gas-kinetic scheme (UGKS). By adaptively employing continuous distribution functions based on the Chapman–Enskog expansion, it efficiently handles near-equilibrium flow regions. The two regions are dynamically coupled at the cell interface through the fluxes from the discrete and continuous gas distribution functions, thereby avoiding any buffer zone between them. In this study, an implicit adaptive unified gas-kinetic scheme (IAUGKS) is constructed to further enhance the efficiency of steady-state solutions. The current scheme employs implicit macroscopic governing equations and couples them with implicit microscopic governing equations within the nonequilibrium region, resulting in high convergence efficiency in all flow regimes. To validate the efficiency and robustness of the IAUGKS, a series of numerical tests were conducted for high Mach number flows around diverse geometries. The current scheme can capture the nonequilibrium physics and provide accurate predictions of surface quantities. In comparison with the original UGKS, the velocity space adaptation, unstructured discrete velocity space, and implicit iteration significantly improve the efficiency by one or two orders of magnitude. Given its exceptional efficiency and accuracy, the IAUGKS serves as an effective tool for nonequilibrium flow simulations.

Paltanea Type Theorems on Estimation by Positive Discrete Functionals

Paltanea Type Theorems on Estimation by Positive Discrete Functionals

Discrete element method simulation of high-speed vehicle collisions with road barrier systems

AbstractThe behavior of road or perimeter protection barriers under vehicle impact are usually investigated based on crash tests and finite element (FE) numerical approaches, which are ether expensive or time-consuming. Several studies have proposed to reduce the computation time of the numerical analysis by substituting the complex FE models of vehicles using simplified mass–spring–damper system models. However, these models have drawbacks since consideration of different vehicle impact angles is difficult and they are unable to correctly simulate the risk of high-speed vehicle collision running over the barrier. In this paper, a new approach is proposed to simulate the collision of vehicles with barriers based on the discrete element method (DEM). Here, to save computation time only a handful of 3D non-spherical particles are used to represent the barrier and vehicle. These particles are generated based on the super-quadric function, which is capable of generating a variety of shapes needed for the model. The contact detection and evaluation are carried out based on discrete function representation of the particles with uniform sampling. The bond between two discrete elements is defined using a nonlinear cohesive beam model since the distance between the elements is relatively large. The simulation results obtained based on this approach are more accurate and complete than the simplified mass–spring models and computationally more efficient than the FE model.

Solar-Hydrogen system of additional power supply and digital control system for on-board power supply system

Even for short space flights, such as manned flights to the Moon back in the 60s, it became extremely obvious that electric batteries are completely unsuitable because of their huge mass needed to store the required supply of onboard energy. Therefore, already in the 60s of the last century, the USA and the USSR took the path of using a hydrogen storage system using hydrogen fuel cells.Solar-hydrogen energy systems for spacecraft reliably provide energy for carrying out various kinds of unscheduled work on board and for eliminating emergency situations. Surplus energy is constantly accumulated on board in the form of the chemical energy of cryogenic hydrogen. The article presents too the result of the development of a solar-hydrogen power supply system for spacecraft, the onboard network of which provides slow and fast processes. This paper delves into designing digital control systems for power conditioning devices on the example of charging-discharging device based on a boost DC-DC converter. The device was implemented to condition a storage battery by properly charging and discharging it. A technique for constructing a small signal model of the converter has been developed. This involves constructing a block diagram using the discontinuous switching functions and analog models of electrical circuit components. The transfer functions of the analog controllers for the two-loop subordinate control system were derived. Then the procedure of a bilinear z-transform to the given transfer functions was applied. As a result, the discrete transfer functions of the digital controllers were obtained. Realization both analog and digital controllers in Matlab Simulink confirmed their identical behavior and stability within the power converter. Experimental studies of the static and dynamic waveforms of the digital control system showed controllability and stability of the charging-discharging device for the storage battery in typical operating modes. Thus, the method of synthesizing digital control systems for energy converters is proved valid and applicable for designing new power conditioning devices within modern power supplies.

The discrete flow category: structure and computation

In this article, we use concepts and methods from the theory of simplicial sets to study discrete Morse theory. We focus on the discrete flow category introduced by Vidit Nanda, and investigate its properties in the case where it is defined from a discrete Morse function on a regular CW complex. We design an algorithm to efficiently compute the Hom posets of the discrete flow category in this case. Furthermore, we show that in the special case where the discrete Morse function is defined on a simplicial complex, then each Hom poset has the structure of a face poset of a regular CW complex. Finally, we prove that the spectral sequence associated to the double nerve of the discrete flow category collapses on page 2.

A Non-Standard Finite Difference Schemes for the Solution of Stiff Initial Value Problems

In this study, we introduce a novel non-standard finite difference (NSFD) scheme designed to address the challenges posed by stiff initial value problems. Stiffness in differential equations often leads to numerical instability and requires specialized methods for stable and accurate solutions. A novel set of numerical schemes for solving stiff ordinary differential equations caused by the decay of radioactive substances developed. This paper demonstrates the power of normalization in the discretization function. We employed non-local approximation and renormalization of the denominator function to create qualitatively stable schemes for a stiff ordinary differential equation. The schemes' stability properties were verified using numerical experiments. The schemes' performance is evaluated in comparison to other typical finite difference schemes

A mini immersed finite element method for two-phase Stokes problems on Cartesian meshes

Abstract This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.

Safe Spacecraft Inspection via Deep Reinforcement Learning and Discrete Control Barrier Functions

While reinforcement learning (RL) offers high-performance solutions to complex tasks, its trial-and-error paradigm presents safety concerns when exploring unsafe states may be unacceptable. Safety filtering or run-time assurance (RTA) approaches can be used to monitor the RL agent’s desired control and prevent the agent from taking unsafe actions by satisfying safety constraints during and after training. This study investigates the problem of on-orbit inspection of a chief spacecraft using a deputy spacecraft controlled by an RL agent with safety constraints. Several constraints on the state space are enforced using discrete control barrier functions and an optimization-based RTA system. This assures the safety of the deputy spacecraft and its sensors even with a neural-network-based primary controller. A detailed analysis of RL agent performance for eight separate experiments is presented, including each safety constraint individually, all constraints simultaneously, and no constraints. Results demonstrate that the RL agent can complete the inspection task while adhering to safety constraints in a simulated RL environment. These results also show that the RTA system can aid the RL agent in learning a successful policy over fewer time steps but may lead to higher fuel usage, depending on the constraint(s) enforced.

Axisymmetric phase-field-based lattice Boltzmann model for incompressible two-phase flow with phase change

In this work, a phase-field-based lattice Boltzmann equation (LBE) model for axisymmetric two-phase flow with phase change is proposed. Two sets of discrete particle distribution functions are employed to match the conserved Allen–Cahn equation and the hydrodynamic equations with phase change effect, respectively. Since phase change occurs at the interface, the divergence-free condition of the velocity field is no longer satisfied due to mass transfer, and the conserved Allen–Cahn equation needs to be equipped with a source term dependent on the phase change model. To deal with these, a novel source term in the hydrodynamic LBE is delicately designed to recover the correct target governing equations. Meanwhile, the LBE for the Allen–Cahn equation is modified with a discrete force term to model mass transfer. In particular, an additional correction term is added into the hydrodynamic LBE to reduce the spurious velocity and improve numerical stability. Several axisymmetric benchmark multiphase problems with phase change, including bubble growing in superheated liquid, D2 law, film boiling, bubble rising in superheated liquid under gravity, and droplet impact on a hot surface, have been conducted to test the performance of the proposed model. Numerical results agree well with analytical solutions and available published data in the literature.