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A variation of the external load distribution function between nodes of a discrete grid in the static-geometric method allows discretely modeling curves of different shapes and solving problems of discrete interpolation on the area. The form of the continuous analogue of the discretely presented curve directly depends on the nature of the functions specified control load, which forms the discretely presented curve (DPC). There are known studies of the aspects of the relationship between the static geometric method of forming the DPC and the analytical description of a continuous curve through the synthesis of the static geometric method of forming discrete curves and the method of modeling them with numerical sequences. Separate issues of determining the correspondence of the equations of the continuous surface to the discrete function of the distribution of the external load are also investigated. This article examines the patterns of changes in the values of the superposition coefficients of two arbitrarily specified, both adjacent and non-adjacent nodal points, under the condition of a known distribution law of the magnitude of the finite difference, which in some cases will be a prototype of the external load between the nodes of the frame, which is a discrete model of a defined geometric image. If we change the uniformly distributed value of the finite difference or the value of the distribution function of the value of the finite difference, or the ordinates of one or two (marginal) nodal points at fixed values of the superposition coefficients, we can control the shape of the curve, discretely represented by the nodal points of its numerical sequence. The research data determine a general approach to obtaining similar patterns of changes in the values of the superposition coefficients of two arbitrarily specified, both adjacent and non-adjacent nodal points for determining the coordinates of points of modeling any one-dimensional functional dependencies and arbitrary one-dimensional sets of points. The results of the study of the regularities of changes in the values of the superposition coefficients given by two nodal points of different elementary functions, under the condition of a known distribution law of the magnitude of the finite difference, will allow solving the problems of continuous discrete interpolation and extrapolation by numerical sequences of any one-dimensional functional dependencies (determine the ordinates of the sought points of discrete curves) without time-consuming operations of assembling and solving large systems of linear equations, Варіювання функції розподілу зовнішнього навантаження між вузлами дискретної сітки у статико-геометричному методі дозволяє дискретно моделювати криві різної форми і розв’язувати задачі дискретної інтерполяції на площині. Форма континуального аналога дискретно представленої кривої безпосередньо залежить від характеру функціонально заданого управляючого навантаження, яке формує дискретно представлену криву (ДПК). Відомі дослідження аспектів зв’язку між статико геометричним методом формування ДПК і аналітичним описом неперервної кривої через синтез статико-геометричного методу формування дискретних кривих і способу моделювання їх числовими послідовностями. Також досліджені окремі питання визначення відповідності рівнянь неперервної поверхні дискретній функції розподілу зовнішнього навантаження. У даній статті досліджено закономірності змін величин коефіцієнтів суперпозиції двох довільно заданих, як суміжних, так і не суміжних вузлових точок, при умові відомого закону розподілення величини скінченої різниці, яка в окремих випадках буде прообразом зовнішнього навантаження між вузлами каркасу, що є дискретною моделлю визначеного геометричного образу. Якщо змінювати рівномірно розподілену величину скінченої різниці або значення функції розподілу величини скінченої різниці, або ординати однієї чи двох (крайових) вузлових точок при фіксованих величинах коефіцієнтів суперпозиції, зможемо управляти формою кривої, дискретно представленої вузловими точками її числової послідовності. Дані дослідження визначають загальний підхід до одержання подібних закономірностей зміни величин коефіцієнтів суперпозиції двох довільно заданих, як суміжних, так і не суміжних вузлових точок для визначення координат n точок модельованих будь-яких одновимірних функціональних залежностей та довільних одновимірних множин точок. Результати дослідження закономірностей зміни величин коефіцієнтів суперпозиції, заданих двох вузлових точок різних елементарних функцій, при умові відомого закону розподілення величини скінченої різниці, дозволять розв’язувати задачі суцільної дискретної інтерполяції та екстраполяції числовими послідовностями будь-яких одновимірних функціональних залежностей (визначати ординати шуканих точок дискретних кривих) без трудомістких операцій складання та розв’язання великих систем лінійних рівнянь.

Making use of experimental data for bias extension, shearing, and point-load tests in large deformation regime for rectangular and square bi-pantographic specimens, we perform a numerical identification to fit the a priori parameters of a planar discrete spring model. The main objective of the work is to develop an automatized optimization process based on the Nelder–Mead simplex algorithm for identifying the constitutive parameters of discrete modeling of bi-pantographic structures, as well as assessing its descriptiveness and predictive capacity. The analysis allows to conclude that there exists a single set of parameters for the adopted discrete modeling such that it is descriptive and predictive for several different tests and for a wide range of deformations.

With the rapid development of the Internet, the amount of data increases sharply. In this network age, data have been generated, such as multimedia, social networks, and search records. Due to the generation of data, complex system discrete model technology is also widely used in all walks of life. Multimedia learning is a popular teaching mode in the world. Through the discrete model technology of the complex system, this paper deeply studies the impact of multimedia input on college students’ English ability. Because the modeling and sampling of complex systems are abstract and not fixed, approximation and control methods are adopted in discrete time, which greatly improves the reliability of the final research results. Compared with the original discrete modeling technology, this paper adds data sampling compression and assimilation technology on the basis of the original discrete modeling technology of the complex system. The results show that, by adding the technical application of controllable and observable systems, the discrete model of the whole complex system can reduce the error of the final output data. At the same time, this paper compresses and simplifies the data nodes at the microlevel. Macroscopically, it reduces the scale, especially the number of node data compression. The final result also realizes the data scale and greatly shortens the calculation time. This study has greatly improved the accuracy of the impact of multimedia on Chinese college students’ English ability in practical application.

Discrete modeling of continuous images by the static-geometric method in most cases is associated with certain errors. Therefore, it is relevant to study the formation of geometric images with a given accuracy, using a minimum amount of initial information. This will allow to create models with optimal discretization. Further development of this modeling method with rational decrease in initial information volume, is topical. The proposed line of research will open up new possibilities for its use in various industries, such as construction, engineering, economics etc. One of the objectives of this work is to continue research on the formation of discrete images of curved lines. The study is based on the classical finite difference method, static-geometric modeling method and the geometric apparatus of superpositions. The article proposes a method for determining the distribution functions of finite-difference values in the formation of discrete analogues of linear-fractional, exponential and hyperbolic functional dependencies using finite differences of different orders. These studies define a general method for obtaining similar patterns of distribution of the finite difference value in formation of discrete analogues of elementary functional dependencies using finite differences of different orders. Establishing a correspondence between the equation of the curve and the distribution function of a finite difference value can be the basis for assessing the accuracy of the formation of discrete frameworks of curves described by elementary functional dependencies. In the future, the results of this work will make it possible to determine coordinates of any point of a numerical sequence of the nth order and numerical sequences of elementary functional dependencies as a superposition of the coordinates of adjacent points. It can also be a superposition of arbitrarily given nodal points of these sequences, if the value of the final difference or its distribution function are already known., Дискретне моделювання неперервних образів статико-геометричним методом у більшості випадків пов’язано із певними похибками. Тому актуальним є дослідження щодо формування геометричних образів із заданою точністю дискретної моделі при умові мінімального обсягу вихідної інформації. Це дозволить створювати моделі із раціональною дискретизацією. Подальший розвиток даного методу моделювання при умові раціонального зменшення обсягу вихідної інформації є актуальним, а запропонований напрям досліджень дозволить розкрити нові можливості його використання у різних галузях: будівництві, машинобудуванні, економіці та інших. Однією із задач даної роботи є продовженні досліджень визначення дискретних образів кривих ліній на основі класичного методу скінчених різниць, статико-геометричного методу моделювання і геометричного апарату суперпозицій. У даній статті запропоновано спосіб визначення функцій розподілу величини скінченої різниці Pi у формуванні дискретних аналогів дробово-лінійної, показникової та гіперболічної функціональних залежностей скінченими різницями різних порядків. Дані дослідження визначають загальну методику до одержання подібних закономірностей розподілу величини скінченої різниці у формуванні дискретних аналогів елементарних функціональних залежностей скінченими різницями різних порядків. Визначення відповідності між аналітичним рівнянням кривої і функцією розподілу величини скінченної різниці може бути основою для оцінки точності формування дискретних каркасів кривих, що аналітично описуються елементарними функціональними залежностями. У подальшому результати даної роботи дозволять визначати координати будь-якої точки числової послідовності n-го порядку і числових послідовностей елементарних функціональних залежностей як суперпозицію координат суміжних, а також довільно заданих вузлових точок даних послідовностей за умови відомої величини чи функції розподілу величини скінченої різниці.

An innovative method for modeling a complicated geotechnical problem combining the coarse and fine zones is to use the hybrid methods. The granular zone of a geotechnical system can be modeled by the discrete element method (DEM) and the fine zone by continuum-based methods. The advantage of this approach is the use of the capabilities of both methods. In the present research, by combining DEM and finite difference method (FDM), a 2D numerical laboratory framework is constructed. Simulation of a vertically loaded stone column installed in clay was chosen to examine the capabilities of the present approach. The contrast between the stone column aggregates, which usually consist of crushed stone with distinct behavior, and the host medium, which is clay with continuum behavior, makes the stone column suitable for this kind of coupled simulation. The coupled numerical model is validated by comparing the load–settlement response of the numerical model and the reported experimental results. Afterward, the failure mechanism of the stone column is evaluated. The column bulging was captured properly, and it was found that in addition to the bulging, the contraction occurs at the upper portion of the column. It was also found that at the early stages of loading, the compaction occurs in the column. However, at later stages of loading, the bulging will happen. Furthermore, the results showed that bulging decreases the internal stresses of the column. All of these achievements were obtained by micro and macro investigations. The obtained results indicate that the coupled DEM–FDM method is a robust approach to simulate the behavior of some geotechnical problems.

In this work, an including the switched evolution process discrete model of a DC–DC converter feeding a switched constant power load is proposed based on the Poincare mapping rule and matrix exponent method. Numerical simulations of the bifurcation scheme based on the proposed discrete model reveal the period-adding bifurcation structures of the system as multiple devils’ staircases. Further, the calculation of the stable regions of the multiple-period orbits of the nested period-adding bifurcation structures are illustrated. The experimental results verify the validity of the discrete model and analytical accuracy of the period-adding bifurcation characteristics of the system. The proposed model solves the difficulties associated with discrete modeling for power electronic nonlinear switched systems. The introduction of the switched evolution process in the proposed discrete model provides a specific mathematical model for studying the influence of the switched evolution process on the dynamics of power electronic nonlinear switched systems. This study is expected to be a foundation for the theory and further research on such power electronic nonlinear switched systems.

The discrete element method (DEM) is a general discrete modeling technique traditionally used to model granular material and discontinuous phenomena such as fracture and fragmentation of solids. We investigate the efficacy of DEM to model deformable elastic continua, cantilever beams in particular, and describe their mechanical behavior under loading. Of distinct interest is the ability of DEM to capture geometric nonlinearity in their large-displacement response. A case study is presented to demonstrate the utility of this approach in particle–structure problems that involve large structural displacements. We simulated an array of these cantilever beams in large elastic oscillations amidst a particulate environment, modeled using DEM. Mobile particles encounter the deflecting beams in their paths and can be redirected for capture based on net forces and moments that act on them. The dynamics of the ensuing particle–structure interactions is simulated, and factors regulating the efficiency of particle capture are examined.

Mathematical models have shed light on the dynamics of HIV- 1 infection in vivo. In this paper, we generalize continuous mathematical models of drug therapy for HIV-1 by Perelson et al. (Science 271:1582-1586, 1996) and Perelson and Nelson (SIAM Rev 41:3-44, 1999) on time scales, i.e., a nonempty closed subset of real numbers in order to derive new discrete models that predict the total concentration of plasma virus as a function of time. One of our main goals is to compare discrete mathematical models with the continuous model in Perelson et al. (1996) where HIV infected patients were given protease inhibitors and sampled frequently thereafter. For the comparison, we use experimental data collected in Perelson et al. (1996) and estimate the parameters such as the virion clearance rate and the rate of loss of infected cells by fitting the total concentration of plasma virus to this data set. Our results show that discrete systems describe the best fit. In the previous models of this study, the efficacy of protease inhibitor is assumed to be perfect. Motivated by Perelson and Nelson (1999), we end the paper with a mathematical model of imperfect protease inhibitor and reverse transcriptase (RT) inhibitor combination therapy of HIV-1 infection on time scales with its stability analysis.

With the growing number of cars in the world, there are corresponding problems that society needs to solve. In metropolises, intersections are generators of traffic jams and traffic accidents. The introduction of unmanned autonomous vehicles is one of the solutions to increase safety on the road. To avoid traffic jams, it is necessary to optimize road traffic. This article proposes a model of the intersection management system. The system aims to organize a conflict-free, safe, and optimal traversal of AVs within the framework of the intelligent transport system of a smart city. The model assumes that at each intersection there are objects of transport infrastructure responsible for building routes at the intersection. To assess the feasibility of the presented model, the authors developed a custom software simulator of a network of urban intersections and conducted experiments. Based on simulation experiments, the use of the developed model can significantly reduce the time spent by unmanned vehicles to overcome the intersection compared to traffic lights.

Simulations of neural activity are commonly based on differential equations. We address the question what can be achieved with a simplified discrete model. The proposed model resembles artificial neural networks enriched with additional biologically inspired features. A neuron has several states, and the state transitions follow endogenous patterns which roughly correspond to firing behavior observed in biological neurons: oscillatory, tonic, plateauing, etc. Neural interactions consist of two components: synaptic connections and extrasynaptic emission of neurotransmitters. The dynamics is asynchronous and event-based; the events correspond to the changes in neurons activity. This model is innovative in introducing discrete framework for modeling neurotransmitter interactions which play the important role in neuromodulation. We simulate rhythmic activity of small neural ensembles like central pattern generators (CPG). The modeled examples include: the biphasic rhythm generated by the half-center mechanism with the post-inhibitory rebound (like the leech heartbeat CPG), the triphasic rhythm (like in pond snail feeding CPG) and the pattern switch in the system of several neurons (like the switch between ingestion and egestion in Aplysia feeding CPG). The asynchronous dynamics allows to obtain multi-phasic rhythms with phase durations close to their biological prototypes. The perspectives of discrete modeling in biological research are discussed in the conclusion.

In the article it was proposed the general approach to determine change patterns of superposition coefficients of two-dimensional point sets. This makes possible to solve problems of continuous discrete interpolation and extrapolation any two-dimensional functional dependencies by numerical sequences, using four arbitrary nodal points. Coordinates of any point of two-dimensional point set can be represented by a coordinates’ superposition of four arbitrary points from this set. This allows to get analytical dependences to determine superposition coefficients values, solving corresponding systems of equations. It was investigated the discrete analogues formation process of two-dimensional geometric images, using as an example surfaces, whose components are polynomial functional dependences. In the research we established change patterns of superposition coefficients values of three nodal basic contour points and an internal nodal point. These patterns are represented in the form of surfaces-graphs of two variables numerical sequences for a chosen calculation scheme. The received regularities allow to form surfaces, using the applicates of three basic contour points and an internal point. These surfaces are on a given calculation scheme and components of their frames are polynomial functional dependences. The studies determine the general approach of obtaining similar change patterns of four arbitrary points superposition coefficients. These points can be as adjacent as non-adjacent node points of a selected calculation scheme. The regularities are used to determine n points coordinates of any two-dimensional functional dependences and arbitrary two-dimensional point sets. In the future, the results of this work will allow to determine value change patterns of one of four superposition coefficients of given four node points (both adjacent and non-adjacent) of different two-dimensional numerical sequences. This will make possible to solve problems of continuous discrete interpolation and extrapolation of any two-dimensional functional dependences by numerical sequences (such as determination the applicates of required points of discrete frames of two-dimensional geometric image) without laborious operations of compilation and solving huge systems of linear and transcendental equations., У статті запропоновано загальний підхід до визначення закономірностей зміни величин коефіцієнтів суперпозицій двовимірних точкових множин що дозволяє розв’язувати задачі суцільної дискретної інтерполяції та екстраполяції числовими послідовностями будь-яких двовимірних функціональних залежностей за чотирма довільно заданими вузловими точками. Координати будь-якої точки двовимірної множини точок можна представити суперпозицією координат чотирьох довільних точок цієї множини і одержати аналітичні залежності для визначення величин коефіцієнтів суперпозиції із відповідної системи рівнянь. Досліджено процес формування дискретних аналогів двовимірних геометричних образів на прикладі поверхонь складовими каркаса яких є поліноміальні функціональні залежності. У процесі дослідження визначено закономірності зміни величин коефіцієнтів суперпозиції трьох вузлових точок опорного контуру і внутрішньої вузлової точки у вигляді поверхонь-графіків числових послідовностей двох змінних для обраної розрахункової схеми. Одержані закономірності дозволяють формувати поверхні на заданій у плані розрахунковій схемі, складовими каркаса яких будуть поліноміальні функціональні залежності, за даними аплікатами трьох точок опорного контуру і внутрішньої точки. Дані дослідження визначають загальний підхід до одержання подібних закономірностей зміни величин коефіцієнтів суперпозиції чотирьох довільно заданих, як суміжних, так і не суміжних вузлових точок обраної розрахункової схеми для визначення координат n точок модельованих будь-яких двовимірних функціональних залежностей та довільних двовимірних множин точок. У подальшому результати даної роботи дозволять визначати закономірності зміни величини одного із чотирьох коефіцієнтів суперпозиції, як для суміжних, так і для не суміжних заданих чотирьох вузлових точок різних двовимірних числових послідовностей, що дозволить розв’язувати задачі суцільної дискретної інтерполяції та екстраполяції числовими послідовностями будь-яких двовимірних функціональних залежностей (визначати аплікати шуканих точок дискретних каркасів двовимірних геометричних образів) без трудомістких операцій складання та розв’язання великих систем лінійних та трансцендентних рівнянь.

The mechanical responses of steel fiber–reinforced concrete (SFRC) thermally damaged at high temperature needs a deeper evaluation via a mesoscopic model that explicitly treats the fibers. ...

Trillions of chemical reactions occur in the human body every second, where the generated products are not only consumed locally but also transported to various locations in a systematic manner to sustain homeostasis. Current solutions to model these biological phenomena are restricted in computability and scalability due to the use of continuum approaches where it is practically impossible to encapsulate the complexity of the physiological processes occurring at diverse scales. Here we present a discrete modeling framework defined on an interacting graph that offers the flexibility to model multiscale systems by translating the physical space into a metamodel. We discretize the graph-based metamodel into functional units composed of well-mixed volumes with vascular and cellular subdomains; the operators defined over these volumes define the transport dynamics. We predict glucose drift governed by advective-dispersive transport in the vascular subdomains of an islet vasculature and cross-validate the flow and concentration fields with finite-element based COMSOL simulations. Vascular and cellular subdomains are coupled to model the nutrient exchange occurring in response to the gradient arising out of reaction and perfusion dynamics. The application of our framework for modeling biologically relevant test systems shows how our approach can assimilate both multi-omics data from in vitro - in vivo studies and vascular topology from imaging studies for examining the structure-function relationship of complex vasculatures. The framework can advance simulation of whole-body networks at user-defined levels and is expected to find major use in personalized medicine and drug discovery. Graphical abstract O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=199 SRC="FIGDIR/small/460337v1_ufig1.gif" ALT="Figure 1"> View larger version (41K): org.highwire.dtl.DTLVardef@16ff682org.highwire.dtl.DTLVardef@1a1101dorg.highwire.dtl.DTLVardef@1291b39org.highwire.dtl.DTLVardef@1ba65d6_HPS_FORMAT_FIGEXP M_FIG C_FIG SignificanceWithin a tissue, the short-range and long-range communications that sustain metabolism is mediated by the vessels perfusing through the cells. When blood flows through the vasculature, nutrient exchange occurs from cell-to-vessel in accordance with the concentration gradients arising out of cellular and vascular dynamics. Therefore, the topology of blood flow and the anatomical arrangement of cells act as primary determinants influencing the functional response of a tissue. We introduce a graph-based model for understanding the multi-scale connectivity observed across functional networks in a tissue. Our approach offers the flexibility to couple different scales by defining each component to the desired scale so as to balance the need for completeness without compromising on necessary details to answer the questions of interest.

In this article, we study the issues of stabilization of the systems consisting of interacting objects – structures or processes. The results were obtained using the graph theory. To calculate the degree of influence of objects on each other, both direct influences between the vertices and indirect ones are taken into account. The problem of stability of systems is studied. Conditions for the stability of a system with nonlinear interactions are found. The questions of the system stabilization under the influence of the evolutionary mechanism, realizing the response of objects to the impact, are examined. An approach is proposed that sets the evolution of the original system by including response reactions that increase or decrease the weight of the arcs. Examples are given and conditions are specified under which the unbalanced original graph of mutual influences comes to a stable two-class structure.

Adaptation of metabolism is a response of many eukaryotic cells to nutrient heterogeneity in the cell microenvironment. One of these adaptations is the shift from respiratory to fermentative metabolism, also called the Warburg/Crabtree effect. It is a response to a very high nutrient increase in the cell microenvironment, even in the presence of oxygen. Understanding whether this metabolic transition can result from basic regulation signals between components of the central carbon metabolism are the the core question of this work. We use an extension of the René Thomas modeling framework for representing the regulations between the main catabolic and anabolic pathways of eukaryotic cells, and formal methods for confronting models with known biological properties in different microenvironments. The formal model of the regulation of eukaryote metabolism defined and validated here reveals the conditions under which this metabolic phenotype switch occurs. It clearly proves that currently known regulating signals within the main components of central carbon metabolism can be sufficient to bring out the Warburg/Crabtree effect. Moreover, this model offers a general perspective of the regulation of the central carbon metabolism that can be used to study other biological questions.

International audience; Whenever continuous dynamics and discrete control interact, hybrid systems arise. As hybrid systems become ubiquitous and more and more complex, analysis and synthesis techniques are in high demand to design safe hybrid systems. This is however challenging due to the nature of hybrid systems and their designs, and the question of how to formulate and reason their safety problems. Previous work has demonstrated how to extend the discrete modeling language Event-B with continuous support to integrate traditional refinement in hybrid system design. In the same spirit, we extend previous work by proposing a strategy that can coherently refine an abstract hybrid system design with safety constraints down to a concrete one, integrated with implementable discrete control, that can behave safely. We demonstrate our proposal on a smart heating system that regulates room temperature between two references, and we share our experience.

The dynamics of individual behavior are related to the dynamics of the social structures in which individuals are embedded. This implies that in order to study social mechanisms such as social selection or peer influence, we need to model the evolution of social networks and the attributes of network actors as interdependent processes. The stochastic actor-oriented model is a statistical approach to study network-attribute coevolution based on longitudinal data. In its standard specification, the coevolving actor attributes are assumed to be measured on an ordinal categorical scale. Continuous variables first need to be discretized to fit into such a modeling framework. This article presents an extension of the stochastic actor-oriented model that does away with this restriction by using a stochastic differential equation to model the evolution of a continuous attribute. We propose a measure for explained variance and give an interpretation of parameter sizes. The proposed method is illustrated by a study of the relationship between friendship, alcohol consumption, and self-esteem among adolescents.