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The article proposes a model for assessing the potential strength of a composite material based on modern fibers with brittle fracture.

Materials consisting of parallel cylindrical fibers that are quasi-statically stretched in one direction are simulated. It is assumed that the sample is not less than 100 pieces, which corresponds to almost significant cases. It is known that the fibers have a distribution of ultimate deformation in the sample and are not destroyed at the same moment. Usually the distribution of their properties is described by the Weibull–Gnedenko statistical distribution. To simulate the strength of the composite, a model of fiber breaks accumulation is used. It is assumed that the fibers united by the polymer matrix are crushed to twice the inefficient length — the distance at which the stresses increase from the end of the broken fiber to the middle one. However, this model greatly overestimates the strength of composites with brittle fibers. For example, carbon and glass fibers are destroyed in this way.

In some cases, earlier attempts were made to take into account the stress concentration near the broken fiber (Hedgepest model, Ermolenko model, shear analysis), but such models either required a lot of initial data or did not coincide with the experiment. In addition, such models idealize the packing of fibers in the composite to the regular hexagonal packing.

The model combines the shear analysis approach to stress distribution near the destroyed fiber and the statistical approach of fiber strength based on the Weibull–Gnedenko distribution, while introducing a number of assumptions that simplify the calculation without loss of accuracy.

It is assumed that the stress concentration on the adjacent fiber increases the probability of its destruction in accordance with the Weibull distribution, and the number of such fibers with an increased probability of destruction is directly related to the number already destroyed before. All initial data can be obtained from simple experiments. It is shown that accounting for redistribution only for the nearest fibers gives an accurate forecast.

This allowed a complete calculation of the strength of the composite. The experimental data obtained by us on carbon fibers, glass fibers and model composites based on them (CFRP, GFRP), confirm some of the conclusions of the model.

Authors describe a two-stage traffic assignment model. It contains of two blocks. The first block consists of a model for calculating a correspondence (demand) matrix, whereas the second block is a traffic assignment model. The first model calculates a matrix of correspondences using a matrix of transport costs (it characterizes the required volumes of movement from one area to another, it is time in this case). To solve this problem, authors propose to use one of the most popular methods of calculating the correspondence matrix in urban studies — the entropy model. The second model describes exactly how the needs for displacement specified by the correspondence matrix are distributed along the possible paths. Knowing the ways of the flows distribution along the paths, it is possible to calculate the cost matrix. Equilibrium in a two-stage model is a fixed point in the sequence of these two models. In practice the problem of finding a fixed point can be solved by the fixed-point iteration method. Unfortunately, at the moment the issue of convergence and estimations of the convergence rate for this method has not been studied quite thoroughly. In addition, the numerical implementation of the algorithm results in many problems. In particular, if the starting point is incorrect, situations may arise where the algorithm requires extremely large numbers to be computed and exceeds the available memory even on the most modern computers. Therefore the article proposes a method for reducing the problem of finding the equilibrium to the problem of the convex non-smooth optimization. Also a numerical method for solving the obtained optimization problem is proposed. Numerical experiments were carried out for both methods of solving the problem. The authors used data for Vladivostok (for this city information from various sources was processed and collected in a new dataset) and two smaller cities in the USA. It was not possible to achieve convergence by the method of fixed-point iteration, whereas the second model for the same dataset demonstrated convergence rate $k^{-1.67}$.

A mathematical model of two-phase capillary-nonequilibrium isothermal flows of incompressible phases in a double porosity medium is constructed. A double porosity medium is considered, which is a composition of two porous media with contrasting capillary properties (absolute permeability, capillary pressure). One of the constituent media has high permeability and is conductive, the second is characterized by low permeability and forms an disconnected system of matrix blocks. A feature of the model is to take into account the influence of capillary nonequilibrium on mass transfer between subsystems of double porosity, while the nonequilibrium properties of two-phase flow in the constituent media are described in a linear approximation within the Hassanizadeh model. Homogenization by the method of formal asymptotic expansions leads to a system of partial differential equations, the coefficients of which depend on internal variables determined from the solution of cell problems. Numerical solution of cell problems for a system of partial differential equations is computationally expensive. Therefore, a thermodynamically consistent kinetic equation is formulated for the internal parameter characterizing the phase distribution between the subsystems of double porosity. Dynamic relative phase permeability and capillary pressure in the processes of drainage and impregnation are constructed. It is shown that the capillary nonequilibrium of flows in the constituent subsystems has a strong influence on them. Thus, the analysis and modeling of this factor is important in transfer problems in systems with double porosity.

The method of modelling of a compact bone tissue microstructure is presented. The modelling sample is considered as set of the structural elements containing reinforcing element – osteon and a matrix. The form of structural elements is defined by distances to next osteons and directions of next osteons arrangement. Calculation of the stress and strain state of the modelling sample is carried out at tension in program complex ANSYS. Results of calculation have shown, that haversian canals are stress concentrators.

This paper presents a method for obtaining fractal signals using fractal splines similar to signals generated by fractal functions. The hypothesis about the identity of discrete fractal functions and linear fractal splines is justified. There are considered the features of planning matrix calculation of cumulative fractal spline, examples of generated curves are shown.

A vector-matrix approach to the theoretical design of metabolic pathways converting chemical compounds, viz., preset substrates, into desirable products is described. It is a mathematical basis for computer–aided generation of alternative biochemical reaction sets executing the given substrate–product conversion. The pathways are retrieved from the used database of biochemical reactions and utilize the reaction stoichiometry and restrictions based on the irreversibility of a part of them. Particular attention is paid to the analysis of restriction interrelations. It is shown that the number of restrictions can be notably reduced due to the existence of families of parallel restricting planes in the space of reaction flows. Coinciding planes of contradirectional restrictions result in the existence of fixed reaction flow values. The problem of exclusion of so called futile cycles is also considered. Utilization of these factors allows essential lowering of the problem complexity and necessary computational resources. An example of alternative biochemical pathway computation for conversion of glucose and glycerol into succinic acid is given. It is found that for a preset “substrate–product” pair many pathways have the same high-energy bond balance.

In the framework of modern non-equilibrium thermodynamics (macroscopic approach of description and mathematical modeling of the dynamics of real physical and chemical processes), the authors developed a potential- flow method for describing and mathematical modeling of real physical and chemical processes applicable in the general case of real macroscopic physicochemical systems. In accordance with the potential-flow method, the description and mathematical modeling of these processes consists in determining through the interaction potentials of the thermodynamic forces driving these processes and the kinetic matrix determined by the kinetic properties of the system in question, which in turn determine the dynamics of the course of physicochemical processes in this system under the influence of the thermodynamic forces in it. Knowing the thermodynamic forces and the kinetic matrix of the system, the rates of the flow of physicochemical processes in the system are determined, and according to these conservation laws the rates of change of its state coordinates are determined. It turns out in this way a closed system of equations of physical and chemical processes in the system. Knowing the interaction potentials in the system, the kinetic matrices of its simple subsystems (individual processes that are conjugate to each other and not conjugate with other processes), the coefficients entering into the conservation laws, the initial state of the system under consideration, external flows into the system, one can obtain a complete dynamics of physicochemical processes in the system. However, in the case of a complex physico-chemical system in which a large number of physicochemical processes take place, the dimension of the system of equations for these processes becomes appropriate. Hence, the problem arises of automating the formation of the described system of equations of the dynamics of physical and chemical processes in the system under consideration. In this article, we develop a library of software data types that implement a user-defined physicochemical system at the level of its design scheme (coordinates of the state of the system, energy degrees of freedom, physico-chemical processes, flowing, external flows and the relationship between these listed components) and algorithms references in these types of data, as well as calculation of the described system parameters. This library includes both program types of the calculation scheme of the user-defined physicochemical system, and program data types of the components of this design scheme (coordinates of the system state, energy degrees of freedom, physicochemical processes, flowing, external flows). The relationship between these components is carried out by reference (index) addressing. This significantly speeds up the calculation of the system characteristics, because faster access to data.

A new discrete ecological-evolutionary mathematical model is presented, in which the search mechanisms for evolutionarily stable migration routes of fish populations are implemented. The proposed adaptive designs have a small dimension, and therefore have high speed. This allows carrying out calculations on long-term perspective for an acceptable machine time. Both geometric approaches of nonlinear analysis and computer “asymptotic” methods were used in the study of stability. The migration dynamics of the fish population is described by a certain Markov matrix, which can change during evolution. The “basis” matrices are selected in the family of Markov matrices (of fixed dimension), which are used to generate migration routes of mutant. A promising direction of the evolution of the spatial behavior of fish is revealed for a given fishery and food supply, as a result of competition of the initial population with mutants. This model was applied to solve the problem of optimal catch for the long term, provided that the reservoir is divided into two parts, each of which has its own owner. Dynamic programming is used, based on the construction of the Bellman function, when solving optimization problems. A paradoxical strategy of “luring” was discovered, when one of the participants in the fishery temporarily reduces the catch in its water area. In this case, the migrating fish spends more time in this area (on condition of equal food supply). This route is evolutionarily fixes and does not change even after the resumption of fishing in the area. The second participant in the fishery can restore the status quo by applying “luring” to its part of the water area. Endless sequence of “luring” arises as a kind of game “giveaway”. A new effective concept has been introduced — the internal price of the fish population, depending on the zone of the reservoir. In fact, these prices are Bellman's private derivatives, and can be used as a tax on caught fish. In this case, the problem of long-term fishing is reduced to solving the problem of one-year optimization.

This article discusses the mechanical properties of carbon nanotube (CNT)-reinforced platinum under uniaxial tensile loading using the molecular dynamics method. A review of current computational and experimental studies on the use of carbon nanotube-reinforced composites from a structural point of view. However, quantitative and qualitative studies of CNTs to improve the properties of composites are still rare. Composite selection is a promising application for platinum alloys in many cases where they may be subjected to mechanical stress, including in biocompatibility sources. Pt-reinforced with CNTs may have additional possibilities for implantation of the implant and at the same time obtain the required mechanical characteristics.

The structure of the composite is composed of a Pt crystal with a face-centered cubic lattice with a constant of 3.92 Å and a carbon nanotube. The Pt matrix has the shape of a cube with dimensions of $43.1541 Å \times 43.1541 Å \times 43.1541 Å$. The hole size in the average platinum dimension is the radius of the carbon nanotube of the «zigzag» type (8,0), which is 2.6 Å. A carbon nanotube is placed in a hole with a radius of 4.2 Å. At such parameters, the maximum energy level was mutually observed. The model under consideration is contained in 320 atomic bombs and 5181 atomic platinum. The volume fraction of deaths in the Pt-C composite is 5.8%. At the first stage of the study, the strain rate was analyzed for stress-strain and energy change during uniaxial action on the Pt-C composite.

Analysis of the strain rate study showed that the consumption yield strength increases with high strain rate, and the elasticity has increased density with decreasing strain rate. This work also increased by 40% for Pt-C, the elasticity of the composite decreased by 42.3%. In general, fracture processes are considered in detail, including plastic deformation on an atomistic scale.

This paper considers various approaches to averaging the generalized travel costs calculated for different modes of travel in the transportation network. The mode of transportation is understood to mean both the mode of transport, for example, a car or public transport, and movement without the use of transport, for example, on foot. The task of calculating the trip matrices includes the task of calculating the total matrices, in other words, estimating the total demand for movements by all modes, as well as the task of splitting the matrices according to the mode, also called modal splitting. To calculate trip matrices, gravitational, entropy and other models are used, in which the probability of movement between zones is estimated based on a certain measure of the distance of these zones from each other. Usually, the generalized cost of moving along the optimal path between zones is used as a distance measure. However, the generalized cost of movement differs for different modes of movement. When calculating the total trip matrices, it becomes necessary to average the generalized costs by modes of movement. The averaging procedure is subject to the natural requirement of monotonicity in all arguments. This requirement is not met by some commonly used averaging methods, for example, averaging with weights. The problem of modal splitting is solved by applying the methods of discrete choice theory. In particular, within the framework of the theory of discrete choice, correct methods have been developed for averaging the utility of alternatives that are monotonic in all arguments. The authors propose some adaptation of the methods of the theory of discrete choice for application to the calculation of the average cost of movements in the gravitational and entropy models. The transfer of averaging formulas from the context of the modal splitting model to the trip matrix calculation model requires the introduction of new parameters and the derivation of conditions for the possible value of these parameters, which was done in this article. The issues of recalibration of the gravitational function, which is necessary when switching to a new averaging method, if the existing function is calibrated taking into account the use of the weighted average cost, were also considered. The proposed methods were implemented on the example of a small fragment of the transport network. The results of calculations are presented, demonstrating the advantage of the proposed methods.