Nonstationary Problem Research Articles

Development of algorithms for constructing two-dimensional optimal boundary-adaptive grids and their software implementation

Introduction. It is noted that the use of adaptive grids in calculations makes it possible to improve the accuracy and efficiency of computational algorithms without increasing the number of nodes. This approach is especially efficient when calculating nonstationary problems. The objective of this study is the development, construction and software implementation of methods for constructing computational two-dimensional optimal boundary-adaptive grids for complex configuration regions while maintaining the specified features of the shape and boundary of the region. The application of such methods contributes to improving the accuracy, efficiency, and cost-effectiveness of computational algorithms.Materials and Methods. The problem of automatic construction of an optimal boundary-adaptive grid in a simply connected region of arbitrary geometry, topologically equivalent to a rectangle, is considered. A solution is obtained for the minimum set of input information: the boundary of the region in the physical plane and the number of points on it are given. The creation of an algorithm and a mesh generation program is based on a model of particle dynamics. This provides determining the trajectories of individual particles and studying the dynamics of their pair interaction in the system under consideration. The interior and border nodes of the grid are separated through using the mask tool, and this makes it possible to determine the speed of movement of nodes, taking into account the specifics of the problem being solved.Results. The developed methods for constructing an optimal boundary-adaptive grid of a complex geometry region provides solving the problem on automatic grid construction in two-dimensional regions of any configuration. To evaluate the results of the algorithm research, a test problem was solved, and the solution stages were visualized. The computational domain of the test problem and the operation of the function for calculating the speed of movement of interior nodes are shown in the form of figures. Visualization confirms the advantage of this meshing method, which separates the border and interior nodes.Discussion and Conclusions. The theoretical and numerical studies results are important both for the investigation of the grids qualitative properties and for the computational grid methods that provide solving numerical modeling problems efficiently and with high accuracy.

Верифікація математичної моделі для розв’язання задачі Стефана в рамках методу “mushy layer”

The use of solar energy has limitations due to its periodic availability: solar plants do not operate at night and are ineffective in dull weather. The solution of this problem involves the introduction of energy storage and duplication systems into the conversion loop. Among the energy storage systems, solid–liquid phase transition modules have significant energy, ecologic, and cost advantages. Physical processes in modules of this type are described by a system of non-stationary nonlinear partial differential equations with specific boundary conditions at the phase interface. The verification of a method for solving the Stefan problem for a heat-storage material is presented in this paper. The use of the mushy layer method made it possible to simplify the classical mathematical model of the Stefan problem by reducing it to a nonstationary heat conduction problem with an implicit heat source that takes into account the latent heat of transition. The phase transition is considered to occur in an intermediate zone determined by the solidus and liquidus temperatures rather than in in infinite region. To develop a Python code, use was made of an implicit computational scheme in which the solidus and liquidus temperatures remain constant and are determined in the course of numerical experiments. The physical model chosen for computer simulation and algorithm verification is the process of ice layer formation on a water surface at a constant ambient temperature. The numerical results obtained allow one to determine the temperature fields in the solid and the liquid phase and the position of the phase interface and calculate its advance speed. The algorithm developed was verified by analyzing the classical analytical solution of the Stefan problem for the one-dimensional case at a constant advance speed of the phase interface. The value of the verification coefficient was determined from a numerical solution of a nonlinear equation with the use of special built-in Python functions. Substituting the data for the physical model under consideration into the analytical solution and comparing them with the numerical simulation data obtained with the use of the mushy layer method shows that the results are in close agreement, thus demonstrating the correctness of the computer algorithm developed. These studies will allow one to adapt the Python code developed on the basis of the mushy layer method to the calculation of heat storage systems with a solid-liquid phase transition with account for the features of their geometry, the temperature level, and actual boundary conditions.

Investigation of the influence of the heterogeneous permeability distribution on the oil phase displacement processes

The object of research is the filtration processes of displacement of the oil phase under the influence of an injection well in a heterogeneous porous medium. It is possible to evaluate and take into account the effect of reservoir heterogeneity on the distribution of reservoir pressure (and, consequently, on the intensity of the filtration process) using numerical modeling of filtration processes based on the piezoelectric equation. To solve the non-stationary anisotropic problem of piezoconductivity, it is proposed to apply the combined finite-element-difference method of M. Lubkov, which makes it possible to take into account the inhomogeneous distribution of permeability inside the anisotropic oil-bearing formation and at its boundaries, and to adequately calculate the distribution of reservoir pressure. The use of the combined finite-element-difference method allows to combine the advantages of the finite-element method and the finite difference method: to model geometrically complex areas, to find the value at any point of the object under study. At the same time, the use of an implicit difference scheme when finding the nodal values of the grid provides high reliability and convergence of the results. The simulation results show that the distribution of the pressure field between the production and injection wells significantly depends on their location, both in the isotropic landslide and in the anisotropic oil-bearing reservoir. It is shown that the distance between the wells of more than 1 km levels out the effectiveness of the impact of the injection well on the oil filtration process. The influence of the permeability of the oil phase in the shear direction dominates the influence of the permeability in the axial directions (affects the pressure decrease by 4–9.5 %). In the case of a landslide-isotropic reservoir, the wells should be located in the shear (diagonal) direction, which will provide the lowest level of drop in the average reservoir pressure (by 4 %). Based on the information obtained, for the effective use of anisotropic low-permeability formations, it is necessary to place production and injection wells in areas with relatively low anisotropy of the formation permeability, especially to avoid places with the presence of landslide permeability of the formation. The location of the wells is important so that, on the one hand, there is no blockage of oil from the side of reduced permeability, and on the other hand, rapid depletion of the formation from the side of increased permeability does not occur. And also the mutual exchange between the production and injection wells did not stop. When placing a system of production and injection wells in anisotropic formations of an oil field, it is necessary to carry out a systematic analysis of the surrounding anisotropy of the formations in order to place them in such a way that would ensure effective dynamics of filtration processes around these wells. Using the method used, it is possible to predict the impact of an injection well on the distribution of reservoir pressure in the reservoir.

TWO-SIDED APPROXIMATIONS METHOD BASED ON THE GREEN’S FUNCTIONS USE FOR CONSTRUCTION OF A POSITIVE SOLUTION OF THE DIRICHLE PROBLEM FOR A SEMILINEAR ELLIPTIC EQUATION

Context. The question of constructing a method of two-sided approximations for finding a positive solution of the Dirichlet problem for a semilinear elliptic equation based on the use of the Green’s functions method is considered. The object of research is the first boundary value problem (the Dirichlet problem) for a second-order semilinear elliptic equation. 
 Objective. The purpose of the research is to develop a method of two-sided approximations for solving the Dirichlet problem for second-order semilinear elliptic equations based on the use of the Green’s functions method and to study its work in solving test problems.
 Method. Using the Green’s functions method, the initial first boundary value problem for a semilinear elliptic equation is replaced by the equivalent Hammerstein integral equation. The integral equation is represented in the form of a nonlinear operator equation with a heterotone operator and is considered in the space of continuous functions, which is semi-ordered using the cone of nonnegative functions. As a solution (generalized) of the boundary value problem, it was taken the solution of the equivalent integral equation. For a heterotone operator, a strongly invariant cone segment is found, the ends of which are the initial approximations for two iteration sequences. The first of these iterative sequences is monotonically increasing and approximates the desired solution to the boundary value problem from below, and the second is monotonically decreasing and approximates it from above. Conditions for the existence of a unique positive solution of the considered Dirichlet problem and two-sided convergence of successive approximations to it are given. General guidelines for constructing a strongly invariant cone segment are also given. The method developed has a simple computational implementation and a posteriori error estimate that is convenient for use in practice. 
 Results. The method developed was programmed and studied when solving test problems. The results of the computational experiment are illustrated with graphical and tabular informations.
 Conclusions. The experiments carried out have confirmed the efficiency and effectiveness of the developed method and make it possible to recommend it for practical use in solving problems of mathematical modeling of nonlinear processes. Prospects for further research may consist the development of two-sided methods for solving problems for systems of partial differential equations, partial differential equations of higher orders and nonstationary multidimensional problems, using semi-discrete methods (for example, the Rothe’s method of lines).

Research on probability statistics of train running safety indices based on pseudo-excitation method

In order to explore the random nature of high-speed railway train operation safety indices, the pseudo-excitation method, extreme value theory, and non-stationary harmonic superposition theory are used in this paper to study the statistics of train operation safety indices. The pseudo-excitation load formulation for track irregularity is obtained by the pseudo-excitation method, and the resulting non-stationary random vibration problem is transformed into a deterministic time history problem. The pseudo-excitation method is used to establish the dynamic equations of motion, and the separation iteration method is used to solve the equations, so as to obtain the power spectral density of the wheel-rail interaction forces. The wheel-rail interaction forces are obtained by using a modulation function and the harmonic superposition method. By fitting an extreme value distribution, the maximum values of the train running safety indices are explored. The proposed numerical approach is validated experimentally using the data from a 24.6 m long simply supported concrete bridge by studying the extreme value distributions of driving safety indices. Additional numerical simulation are conducted for varying train speeds and bridge spans. The results show that the Gumbel distribution can fit the extreme value of driving safety parameters for different speeds and different bridge span lengths. It is observed that the higher the speed, the sharper the extreme value distribution of train running safety indices, and the larger the train running safety index values corresponding to 99.87% confidence level. The corresponding extreme values at the 99.87% confidence level are greater than the maximum value of each time-domain sample.

Parallel technology for solving nonstationary heat conduction problems in axisymmetric domains

The paper develops a parallel algorithm and program for solving nonstationary heat conduction and diffusion problems in axisymmetric domains with axisymmetric boundary conditions. The numerical solution is based on the boundary element method. In order to optimize and enhance the effectiveness of the computer implementation of the algorithm, the computations are parallelized and the OpenMP application program interface is used. The program is tested by comparing the calculation results with the data of known exact solutions. The calculations confirm the correctness of the numerical solutions and the possibility of full scaling at different numbers of boundary elements according to the number of cores/processors available. The program is applicable to solving axisymmetric heat conduction and diffusion problems and, as a component of a software system, to solving nonlinear problems.

A novel deep interval type-2 fuzzy LSTM (DIT2FLSTM) model applied to COVID-19 pandemic time-series prediction

Currently, the novel COVID-19 coronavirus has been widely spread as a global pandemic. The COVID-19 pandemic has a major influence on human life, healthcare systems, and the economy. There are a large number of methods available for predicting the incidence of the virus. A complex and non-stationary problem such as the COVID-19 pandemic is characterized by high levels of uncertainty in its behavior during the pandemic time. The fuzzy logic, especially Type-2 Fuzzy Logic, is a robust and capable model to cope with high-order uncertainties associated with non-stationary time-dependent features. The main objective of the current study is to present a novel Deep Interval Type-2 Fuzzy LSTM (DIT2FLSTM) model for prediction of the COVID-19 incidence, including new cases, recovery cases, and mortality rate in both short and long time series. The proposed model was evaluated on real datasets produced by the world health organization (WHO) on top highly risked countries, including the USA, Brazil, Russia, India, Peru, Spain, Italy, Iran, Germany, and the U.K. The results confirm the superiority of the DIT2FLSTM model with an average area under the ROC curve (AUC) of 96% and a 95% confidence interval of [92–97] % in the short-term and long-term. The DIT2FLSTM was applied to a well-known standard benchmark, the Mackey-Glass time-series, to show the robustness and proficiency of the proposed model in uncertain and chaotic time series problems. The results were evaluated using a 10-fold cross-validation technique and statistically validated through the t-test method. The proposed DIT2FLSTM model is promising for the prediction of complex problems such as the COVID-19 pandemic and making strategic prevention decisions to save more lives.

Prediction of Photovoltaic Panel Power Outputs using Time Series and Artificial Neural Network Methods

Solar energy is one of the renewable energy sources that has been in high demand in the last decades. With the increasing penetration of photovoltaic (PV) systems in around the world, accurate estimation of the power output of PV systems has become an important issue. Since PV systems directly convert sunlight into electrical energy, PV power output varies depending on environmental conditions. In order to deal with the periodic and non-stationary problems of PV output power, modelling methods are widely use for forecasting. The main purpose of this study is to lead an assessment of forecasting of the PV power outputs in short-time. For this purpose, data are obtained from experimental activities carried out on a real 250 kWp PV stystem, which is located in T.C Tekirdağ Namık Kemal University, Süleymanpaşa district of Tekirdağ province. All parametres are measured hourly with three times according to inclination of the panel setups (0˚, 30˚,60˚). In this sense, this study differs from the previously studies in literature, as it expands the forecasting model with considering of different panel angle. In the first stage, the significant variables for predicting PV power output are identified based on both correlation analysis and stepwise regression analysis. The findings are shown that solar radiation and angle of inclination of the panel are significant predictors of the generation of PV power. In the second stage, three different model are proposed based on Time Series Analysis (TSA) and Artificial Neural Network (ANN) approaches in order to predict power production of PV system. Furthermore, the accuracies of the models are analyzed in order to better understand the intrinsic errors caused and to evaluate its potential in energy forecasting applications. All models are compared in terms of the correlation coefficient (R), coefficient of determination (R2), mean absolute percentage error (MAPE). The results of analyses show that the ANN models have higher accuracy than the TSA model for forecasting PV power.

Development of a Mathematical Model for a Workpiece Heating and Cooling in a Protective Medium while Treatment under Pressure

The paper reviews and develops the mathematical model of plastic flow during the hot-forming processes. A flat non-stationary temperature problem for a cross-section of a long solid (rolled product) of arbitrary shape with different heat transfer conditions along the perimeter of the cross-section was considered. Equations for calculation of the thermal conductivity coefficient and heat capacity of tungsten billets were obtained in the temperature range of 700 - 1500°C, based on the literature data. Analytical dependences in form of regression equations were obtained, allowing for computer calculations of physical specifications of 11x11 mm VA grade tungsten billets in form of temperature functions with accuracy sufficient for practical calculations.

The inverse non-stationary problem of identification of defects in an elastic rod

Non-stationary inverse problems of deformed solid mechanics are among the most underexplored due to, inter alia, increasing dimension of non-stationary problems per unit as compared with stationary and static problems, as well as necessity to consider the initial conditions. In the context of the continuing progress of the aviation and aerospace industries, the question arises about technical condition monitoring of aircraft for the purposes of their safe operation. A large proportion of an aircraft structure consists of beam and rod elements exposed to various man-made and natural effects which cause defects inaccessible for visual inspection and required to be identified well in advance. It is well known that defects (such as cracks, cavities, rigid and elastic inclusions) are concentrators of stresses and largely cause processes, which lead to the destruction of elastic bodies. Therefore, the problem of identification of such defects and their parameters, i.e. the problem of identification, represents a great practical interest. Mathematically, the problem of identification represents a non-linear inverse problem. The development of methods of solving such problems is currently a live fundamental research issue.

Numerical modelling in problems of thermal control for three-layer structures with defects

This article describes solutions to the direct and inverse problems of the three-dimensional non-stationary heat conduction problem in a three-layer structure, using the finite element method for the direct problem and the gradient descent method for the inverse problem. A comparison of the FEM-solution and the analytical solution for a solid with a simple geometry is presented. Here are presented solutions of the direct and inverse three-dimensional non-stationary heat conductivity problem for a free three-stage turbine. The accuracy of the found and exact solutions is compared.

Asynchronous and Non-Stationary Interference Cancellation in Multiuser Interference Channels

A single antenna interference cancellation (SAIC) scheme is proposed. The scheme is motivated by the challenging asynchronous and non-stationary interference cancellation problem in wireless communications. Suppose the existence of the single-user region (SUR) of the desired signal, we propose to first recognize the SUR via a novel detection scheme. The SUR detection scheme proposed consists of the pseudo-observation matrix construction and the information theoretic criterion-based source number estimation. A basis set for the desired signal is then learnt over the SUR, via dictionary learning techniques. In the final signal recovery stage, a novel constrained sparse coding (CSC) process is proposed. The CSC reforms the conventional sparse coding (SC) via incorporating signal-specific constraints. Unlike existing SAIC algorithms, the proposed scheme is completely independent of a prior knowledge on the interferences. Numerical results are provided to demonstrate the effectiveness of the above proposed schemes under varied interference intensities and environmental noise levels. The proposed scheme outperformed competing SAIC schemes by about 5dB signal-to-interference-plus-noise ratio (SINR) improvement. The proposed CSC scheme outperformed the conventional SC by about 9dB SINR improvement, besides the superior recovery fidelity of signal property.

Influence of a Nonstationary Surface Load on an Elastic Porous Half-Space

In this paper, we develop a method for solving planar nonstationary problems on the influence of surface pressure on an elastic-porous half-space. Using surface influence functions, we obtain an integral relation, which expresses the desired displacements with the surface pressure. A numericalanalytical algorithm is constructed. For calculating singular integrals, we derive special quadrature formulas based on the canonical regularization of integrands.

Modeling of oil phase displacement processes in heterogeneous anisotropic reservoirs

Methods of computer modeling of slightly permeable anisotropic oil-bearing layers are needed nowadays because they give us a possibility to obtain a concept on filtration processes near producing and forcing wells in different practical situations and in this way to raise the level of exploitation of such layers significantly. On the other hand, they allow evaluation and taking into account some uncertainties which appear as a result of inefficient information on the structure and properties of the layer outside the wells. In order to investigate the practical aspects of supporting efficiency of oil production in anisotropic heterogeneous low permeable reservoirs on the base of combined finite-element-difference method for solving the non-stationary anisotropic piezoconductivity problem, modeling of distribution of layer pressure was carried out in the vicinity of the production and forcing wells taking into account the anisotropy of the permeability and conditions of the oil phase infiltration on the margins of the examined layer. It has been found that the intensity of filtration process between producing and forcing wells depends essentially on their spacing in both shear-isotropic and anisotropic oil-bearing layers. In addition the effect of oil phase permeability in shear direction dominates over the effect of permeability in axes directions. Starting from the obtained information for the effective exploitation of anisotropic slightly permeable layers we need to locate producing and forcing wells in the areas with relatively low permeability of the layer and especially to avoid the places with presence of shear permeability. It is important to locate the wells in such a way that blocking the oil in the direction of reduced permeability and fast depletion of the layer in the direction of increased permeability would not happen as well as mutual exchange between producing and forcing wells would not stop. While locating the system of specified wells within anisotropic layers of oil deposit it is necessary to conduct a systemic analysis of environmental anisotropy of layers aimed at such a location of these wells which would guarantee the effective dynamics of filtration processes around them. Application of quadratic isoparametric approximation of finite-elemental net of examined area of oil-bearing layer and implicit differential time approximation brings to increase of precision and stability of numerical solution of the problem.

Analytical Solution of the Non-Stationary Heat Conduction Problem in Thin-Walled Products during the Additive Manufacturing Process.

The work is devoted to the development of a model for calculating transient quasiperiodic temperature fields arising in the direct deposition process of thin walls with various configurations. The model allows calculating the temperature field, thermal cycles, temperature gradients, and the cooling rate in the wall during the direct deposition process at any time. The temperature field in the deposited wall is determined based on the analytical solution of the non-stationary heat conduction equation for a moving heat source, taking into account heat transfer to the environment. Heat accumulation and temperature change are calculated based on the superposition principle of transient temperature fields resulting from the heat source action at each pass. The proposed method for calculating temperature fields describes the heat-transfer process and heat accumulation in the wall with satisfactory accuracy. This was confirmed by comparisons with experimental thermocouple data. It takes into account the size of the wall and the substrate, the change in power from layer to layer, the pause time between passes, and the heat-source trajectory. In addition, this calculation method is easy to adapt to various additive manufacturing processes that use both laser and arc heat sources.

Thermoelastic State of Contacting Thermally Sensitive Half Space and Thermally Sensitive Layer

By using the numerical inverse Laplace transform based on the modified Prudnikov formula, we construct the solution of a nonstationary problem of heat conduction for a thermally sensitive half space operating in contact with a thermally sensitive layer under the conditions of convective heat exchange with a medium of constant temperature. The thermoelastic state of this thermally sensitive piecewise homogeneous structure is investigated for the case where a constant temperature is kept on the surface.

Forecasting Underground Water Dynamics within the Technogenic Environment of a Mine Field. Case Study

The objective is to analyze the dynamics of the underground water of a mine field based on the study of the geofiltration process of the rock mass disturbed by mining to achieve safe extraction operations as well as subsurface territories at the stage of the mining enterprise closure. Numerical modeling, based on a finite difference method under the conditions of multifactority and definite uncertainty of processes of transformation of technogenic environment of a mine field, helps solve a problem concerning underground water dynamics forecasting. A hydrodynamic model of the M.I. Stashkov mine was developed while solving option series of epignosis problems in terms of the chronology of mine field stoping. The abovementioned made it possible to identify regularities of the history of filtration, the capacity parameters of rock mass and the expansion of areas of heightened hydraulic conductivity as well as to evaluate qualitatively the water balance components of a carbonic watered formation and an overlying one. The stage of mining closure helped obtain the forecasting hydrodynamic solutions. The efficiency of measures, concerning reduction of water ingress into mine workings and the mitigation of surface ecological effects of mine flooding was evaluated quantitatively. It was determined that implementation of the water control procedures makes it possible to perform a 10–38% decrease in water ingress. In this context, they may be applied both independently and simultaneously. In terms of mine closure and flooding, a period of complete underground water recovery takes three years; in the process, surface zones of potential waterlogging and swamping are developed within the floodplain of Samara River, located at the territory of Western Donbas (Ukraine). The scientific novelty is to define regularities of hydraulic conductivity transformation of the rock mass of a mine field starting from the mine working roof fall, up to its compaction during the mine operation period. To do that, nonstationary identification problems were solved, using numerical modeling. The abovementioned makes it possible to improve the reliability of hydrodynamic prognoses and develop technological schemes to control water at the state of the mine closure.

Approximations for non-stationary stochastic lot-sizing under ([formula omitted])-type policy

This paper addresses the single-item single-stocking location non-stationary stochastic lot-sizing problem under a reorder point – order quantity control strategy. The reorder points and order quantities are chosen at the beginning of the planning horizon. The reorder points are allowed to vary with time and we consider order quantities either to be a series of time-dependent constants or a fixed value; this leads to two variants of the policy: the (st,Qt) and the (st,Q) policies, respectively. For both policies, we present stochastic dynamic programs (SDP) to determine optimal policy parameters and introduce mixed integer non-linear programming (MINLP) heuristics that leverage piecewise-linear approximations of the cost function. Numerical experiments demonstrate that our solution method efficiently computes near-optimal parameters for a broad class of problem instances.

Method of Solving Nonstationary Problems of Electrodynamics in Material Media

Method of Solving Nonstationary Problems of Electrodynamics in Material Media

Multiagent Federated Reinforcement Learning for Secure Incentive Mechanism in Intelligent Cyber–Physical Systems

Federated learning (FL) is an emerging technology for empowering various applications that generate large amounts of data in intelligent cyber–physical systems (ICPS). Though FL can address users’ concerns about data privacy, its maintenance still depends on efficient incentive mechanisms. For long-term incentivization to participants in data federation under dynamic environments, deep reinforcement learning as a promising technology has been extensively studied. However, the nonstationary problem caused by the heterogeneity of ICPS devices results in a serious effect on the convergence rate of existing single-agent reinforcement learning. In this article, we propose a multiagent learning-based incentive mechanism to capture the stationarity approximation in FL with heterogeneous ICPS. First, we formulate the secure communication and data resource allocation problem as a Stackelberg game in FL with multiple participants. Then, to tackle the heterogeneous problem, we model this multiagent game as a partially observable Markov decision process. In particular, a multiagent federated reinforcement learning algorithm is proposed to learn the allocation policies efficiently by dwindling variances in policy evaluation caused by interaction among multiple devices without the requirement of sharing privacy information. Moreover, the proposed algorithm is proved to attain convergence at an expected rate. Finally, extensive experimental results demonstrate that our proposed algorithm significantly outperforms baseline approaches.